276.40/148.20	MAYBE
276.40/148.21	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
276.40/148.21	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
276.40/148.21	
276.40/148.21	
276.40/148.21	Outermost Termination of the given OTRS could not be shown:
276.40/148.21	
276.40/148.21	(0) OTRS
276.40/148.21	(1) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms]
276.40/148.21	(2) QTRS
276.40/148.21	    (3) QTRSRRRProof [EQUIVALENT, 76 ms]
276.40/148.21	    (4) QTRS
276.40/148.21	    (5) DependencyPairsProof [EQUIVALENT, 0 ms]
276.40/148.21	    (6) QDP
276.40/148.21	    (7) DependencyGraphProof [EQUIVALENT, 0 ms]
276.40/148.21	    (8) AND
276.40/148.21	        (9) QDP
276.40/148.21	            (10) UsableRulesProof [EQUIVALENT, 0 ms]
276.40/148.21	            (11) QDP
276.40/148.21	            (12) QReductionProof [EQUIVALENT, 0 ms]
276.40/148.21	            (13) QDP
276.40/148.21	            (14) MRRProof [EQUIVALENT, 12 ms]
276.40/148.21	            (15) QDP
276.40/148.21	            (16) MRRProof [EQUIVALENT, 0 ms]
276.40/148.21	            (17) QDP
276.40/148.21	            (18) DependencyGraphProof [EQUIVALENT, 0 ms]
276.40/148.21	            (19) QDP
276.40/148.21	            (20) UsableRulesProof [EQUIVALENT, 0 ms]
276.40/148.21	            (21) QDP
276.40/148.21	            (22) QReductionProof [EQUIVALENT, 0 ms]
276.40/148.21	            (23) QDP
276.40/148.21	            (24) UsableRulesReductionPairsProof [EQUIVALENT, 2 ms]
276.40/148.21	            (25) QDP
276.40/148.21	            (26) DependencyGraphProof [EQUIVALENT, 0 ms]
276.40/148.21	            (27) TRUE
276.40/148.21	        (28) QDP
276.40/148.21	            (29) UsableRulesProof [EQUIVALENT, 0 ms]
276.40/148.21	            (30) QDP
276.40/148.21	                (31) QReductionProof [EQUIVALENT, 0 ms]
276.40/148.21	                (32) QDP
276.40/148.21	                (33) TransformationProof [SOUND, 0 ms]
276.40/148.21	                (34) QDP
276.40/148.21	            (35) UsableRulesProof [EQUIVALENT, 0 ms]
276.40/148.21	            (36) QDP
276.40/148.21	                (37) QReductionProof [EQUIVALENT, 0 ms]
276.40/148.21	                (38) QDP
276.40/148.21	(39) Trivial-Transformation [SOUND, 0 ms]
276.40/148.21	(40) QTRS
276.40/148.21	    (41) QTRSRRRProof [EQUIVALENT, 69 ms]
276.40/148.21	    (42) QTRS
276.40/148.21	    (43) AAECC Innermost [EQUIVALENT, 0 ms]
276.40/148.21	    (44) QTRS
276.40/148.21	    (45) DependencyPairsProof [EQUIVALENT, 0 ms]
276.40/148.21	    (46) QDP
276.40/148.21	    (47) DependencyGraphProof [EQUIVALENT, 0 ms]
276.40/148.21	    (48) QDP
276.40/148.21	    (49) UsableRulesProof [EQUIVALENT, 0 ms]
276.40/148.21	    (50) QDP
276.40/148.21	    (51) QReductionProof [EQUIVALENT, 0 ms]
276.40/148.21	    (52) QDP
276.40/148.21	    (53) TransformationProof [EQUIVALENT, 0 ms]
276.40/148.21	    (54) QDP
276.40/148.21	    (55) TransformationProof [EQUIVALENT, 0 ms]
276.40/148.21	    (56) QDP
276.40/148.21	    (57) TransformationProof [EQUIVALENT, 0 ms]
276.40/148.21	    (58) QDP
276.40/148.21	    (59) TransformationProof [EQUIVALENT, 0 ms]
276.40/148.21	    (60) QDP
276.40/148.21	    (61) TransformationProof [EQUIVALENT, 0 ms]
276.40/148.21	    (62) QDP
276.40/148.21	    (63) UsableRulesProof [EQUIVALENT, 0 ms]
276.40/148.21	    (64) QDP
276.40/148.21	    (65) QReductionProof [EQUIVALENT, 0 ms]
276.40/148.21	    (66) QDP
276.40/148.21	    (67) NonTerminationLoopProof [COMPLETE, 0 ms]
276.40/148.21	    (68) NO
276.40/148.21	(69) Raffelsieper-Zantema-Transformation [SOUND, 0 ms]
276.40/148.21	(70) QTRS
276.40/148.21	    (71) QTRSRRRProof [EQUIVALENT, 87 ms]
276.40/148.21	    (72) QTRS
276.40/148.21	    (73) AAECC Innermost [EQUIVALENT, 0 ms]
276.40/148.21	    (74) QTRS
276.40/148.21	    (75) DependencyPairsProof [EQUIVALENT, 2 ms]
276.40/148.21	    (76) QDP
276.40/148.21	    (77) DependencyGraphProof [EQUIVALENT, 0 ms]
276.40/148.21	    (78) AND
276.40/148.21	        (79) QDP
276.40/148.21	            (80) UsableRulesProof [EQUIVALENT, 0 ms]
276.40/148.21	            (81) QDP
276.40/148.21	            (82) QReductionProof [EQUIVALENT, 0 ms]
276.40/148.21	            (83) QDP
276.40/148.21	            (84) QDPSizeChangeProof [EQUIVALENT, 0 ms]
276.40/148.21	            (85) YES
276.40/148.21	        (86) QDP
276.40/148.21	            (87) UsableRulesProof [EQUIVALENT, 0 ms]
276.40/148.21	            (88) QDP
276.40/148.21	            (89) QReductionProof [EQUIVALENT, 0 ms]
276.40/148.21	            (90) QDP
276.40/148.21	            (91) TransformationProof [EQUIVALENT, 0 ms]
276.40/148.21	            (92) QDP
276.40/148.21	            (93) DependencyGraphProof [EQUIVALENT, 0 ms]
276.40/148.21	            (94) QDP
276.40/148.21	            (95) TransformationProof [EQUIVALENT, 0 ms]
276.40/148.21	            (96) QDP
276.40/148.21	            (97) TransformationProof [EQUIVALENT, 0 ms]
276.40/148.21	            (98) QDP
276.40/148.21	            (99) TransformationProof [EQUIVALENT, 0 ms]
276.40/148.21	            (100) QDP
276.40/148.21	            (101) DependencyGraphProof [EQUIVALENT, 0 ms]
276.40/148.21	            (102) QDP
276.40/148.21	            (103) TransformationProof [EQUIVALENT, 0 ms]
276.40/148.21	            (104) QDP
276.40/148.21	            (105) DependencyGraphProof [EQUIVALENT, 0 ms]
276.40/148.21	            (106) QDP
276.40/148.21	            (107) QDPOrderProof [EQUIVALENT, 319 ms]
276.40/148.21	            (108) QDP
276.40/148.21	            (109) MNOCProof [EQUIVALENT, 0 ms]
276.40/148.21	            (110) QDP
276.40/148.21	            (111) SplitQDPProof [EQUIVALENT, 0 ms]
276.40/148.21	            (112) AND
276.40/148.21	                (113) QDP
276.40/148.21	                    (114) SemLabProof [SOUND, 0 ms]
276.40/148.21	                    (115) QDP
276.40/148.21	                    (116) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
276.40/148.21	                    (117) QDP
276.40/148.21	                    (118) DependencyGraphProof [EQUIVALENT, 0 ms]
276.40/148.21	                    (119) QDP
276.40/148.21	                    (120) PisEmptyProof [SOUND, 0 ms]
276.40/148.21	                    (121) TRUE
276.40/148.21	                (122) QDP
276.40/148.21	                    (123) SplitQDPProof [EQUIVALENT, 0 ms]
276.40/148.21	                    (124) AND
276.40/148.21	                        (125) QDP
276.40/148.21	                            (126) SemLabProof [SOUND, 0 ms]
276.40/148.21	                            (127) QDP
276.40/148.21	                            (128) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
276.40/148.21	                            (129) QDP
276.40/148.21	                            (130) DependencyGraphProof [EQUIVALENT, 0 ms]
276.40/148.21	                            (131) QDP
276.40/148.21	                            (132) PisEmptyProof [SOUND, 0 ms]
276.40/148.21	                            (133) TRUE
276.40/148.21	                        (134) QDP
276.40/148.21	                            (135) SplitQDPProof [EQUIVALENT, 0 ms]
276.40/148.21	                            (136) AND
276.40/148.21	                                (137) QDP
276.40/148.21	                                    (138) SemLabProof [SOUND, 0 ms]
276.40/148.21	                                    (139) QDP
276.40/148.21	                                    (140) MRRProof [EQUIVALENT, 4 ms]
276.40/148.21	                                    (141) QDP
276.40/148.21	                                    (142) DependencyGraphProof [EQUIVALENT, 0 ms]
276.40/148.21	                                    (143) QDP
276.40/148.21	                                    (144) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
276.40/148.21	                                    (145) QDP
276.40/148.21	                                    (146) MRRProof [EQUIVALENT, 0 ms]
276.40/148.21	                                    (147) QDP
276.40/148.21	                                    (148) PisEmptyProof [SOUND, 0 ms]
276.40/148.21	                                    (149) TRUE
276.40/148.21	                                (150) QDP
276.40/148.21	                                    (151) QReductionProof [EQUIVALENT, 0 ms]
276.40/148.21	                                    (152) QDP
276.40/148.21	                                    (153) SplitQDPProof [EQUIVALENT, 0 ms]
276.40/148.21	                                    (154) AND
276.40/148.21	                                        (155) QDP
276.40/148.21	                                            (156) SemLabProof [SOUND, 0 ms]
276.40/148.21	                                            (157) QDP
276.40/148.21	                                            (158) MRRProof [EQUIVALENT, 0 ms]
276.40/148.21	                                            (159) QDP
276.40/148.21	                                            (160) DependencyGraphProof [EQUIVALENT, 0 ms]
276.40/148.21	                                            (161) QDP
276.40/148.21	                                            (162) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
276.40/148.21	                                            (163) QDP
276.40/148.21	                                            (164) MRRProof [EQUIVALENT, 0 ms]
276.40/148.21	                                            (165) QDP
276.40/148.21	                                            (166) PisEmptyProof [SOUND, 0 ms]
276.40/148.21	                                            (167) TRUE
276.40/148.21	                                        (168) QDP
276.40/148.21	                                            (169) QReductionProof [EQUIVALENT, 0 ms]
276.40/148.21	                                            (170) QDP
276.40/148.21	                                            (171) MNOCProof [EQUIVALENT, 0 ms]
276.40/148.21	                                            (172) QDP
276.40/148.21	
276.40/148.21	
276.40/148.21	----------------------------------------
276.40/148.21	
276.40/148.21	(0)
276.40/148.21	Obligation:
276.40/148.21	Term rewrite system R:
276.40/148.21	The TRS R consists of the following rules:
276.40/148.21	
276.40/148.21	   g(b) -> g(f(f(f(f(f(b))))))
276.40/148.21	   f(b) -> b
276.40/148.21	   g(f(b)) -> c
276.40/148.21	
276.40/148.21	
276.40/148.21	
276.40/148.21	Outermost Strategy.
276.40/148.21	
276.40/148.21	----------------------------------------
276.40/148.21	
276.40/148.21	(1) Thiemann-SpecialC-Transformation (EQUIVALENT)
276.40/148.21	We applied the Thiemann-SpecialC transformation  to transform the outermost TRS to an innermost TRS.
276.40/148.21	----------------------------------------
276.40/148.21	
276.40/148.21	(2)
276.40/148.21	Obligation:
276.40/148.21	Q restricted rewrite system:
276.40/148.21	The TRS R consists of the following rules:
276.40/148.21	
276.40/148.21	   top(go_up(x)) -> top(reduce(x))
276.40/148.21	   reduce(g(x_1)) -> check_g(redex_g(x_1))
276.40/148.21	   reduce(f(x_1)) -> check_f(redex_f(x_1))
276.40/148.21	   redex_g(b) -> result_g(g(f(f(f(f(f(b)))))))
276.40/148.21	   redex_f(b) -> result_f(b)
276.40/148.21	   redex_g(f(b)) -> result_g(c)
276.40/148.21	   check_g(result_g(x)) -> go_up(x)
276.40/148.21	   check_f(result_f(x)) -> go_up(x)
276.40/148.21	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
276.40/148.21	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
276.40/148.21	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
276.40/148.21	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
276.40/148.21	
276.40/148.21	The set Q consists of the following terms:
276.40/148.21	
276.40/148.21	   top(go_up(x0))
276.40/148.21	   reduce(g(x0))
276.40/148.21	   reduce(f(x0))
276.40/148.21	   redex_g(b)
276.40/148.21	   redex_f(b)
276.40/148.21	   redex_g(f(b))
276.40/148.21	   check_g(result_g(x0))
276.40/148.21	   check_f(result_f(x0))
276.40/148.21	   check_g(redex_g(x0))
276.40/148.21	   check_f(redex_f(x0))
276.40/148.21	   in_g_1(go_up(x0))
276.40/148.21	   in_f_1(go_up(x0))
276.40/148.21	
276.40/148.21	
276.40/148.21	----------------------------------------
276.40/148.21	
276.40/148.21	(3) QTRSRRRProof (EQUIVALENT)
276.40/148.21	Used ordering:
276.40/148.21	Polynomial interpretation [POLO]:
276.40/148.21	
276.40/148.21	   POL(b) = 0
276.40/148.21	   POL(c) = 0
276.40/148.21	   POL(check_f(x_1)) = 2*x_1
276.40/148.21	   POL(check_g(x_1)) = 2*x_1
276.40/148.21	   POL(f(x_1)) = 2*x_1
276.40/148.21	   POL(g(x_1)) = 1 + 2*x_1
276.40/148.21	   POL(go_up(x_1)) = 2*x_1
276.40/148.21	   POL(in_f_1(x_1)) = 2*x_1
276.40/148.21	   POL(in_g_1(x_1)) = 2 + 2*x_1
276.40/148.21	   POL(redex_f(x_1)) = 2*x_1
276.40/148.21	   POL(redex_g(x_1)) = 1 + 2*x_1
276.40/148.21	   POL(reduce(x_1)) = 2*x_1
276.40/148.21	   POL(result_f(x_1)) = 2*x_1
276.40/148.21	   POL(result_g(x_1)) = x_1
276.40/148.21	   POL(top(x_1)) = 2*x_1
276.40/148.21	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
276.40/148.21	
276.40/148.21	   redex_g(f(b)) -> result_g(c)
276.40/148.21	
276.40/148.21	
276.40/148.21	
276.40/148.21	
276.40/148.21	----------------------------------------
276.40/148.21	
276.40/148.21	(4)
276.40/148.21	Obligation:
276.40/148.21	Q restricted rewrite system:
276.40/148.21	The TRS R consists of the following rules:
276.40/148.21	
276.40/148.21	   top(go_up(x)) -> top(reduce(x))
276.40/148.21	   reduce(g(x_1)) -> check_g(redex_g(x_1))
276.40/148.21	   reduce(f(x_1)) -> check_f(redex_f(x_1))
276.40/148.21	   redex_g(b) -> result_g(g(f(f(f(f(f(b)))))))
276.40/148.21	   redex_f(b) -> result_f(b)
276.40/148.21	   check_g(result_g(x)) -> go_up(x)
276.40/148.21	   check_f(result_f(x)) -> go_up(x)
276.40/148.21	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
276.40/148.21	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
276.40/148.21	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
276.40/148.21	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
276.40/148.21	
276.40/148.21	The set Q consists of the following terms:
276.40/148.21	
276.40/148.21	   top(go_up(x0))
276.40/148.21	   reduce(g(x0))
276.40/148.21	   reduce(f(x0))
276.40/148.21	   redex_g(b)
276.40/148.21	   redex_f(b)
276.40/148.21	   redex_g(f(b))
276.40/148.21	   check_g(result_g(x0))
276.40/148.21	   check_f(result_f(x0))
276.40/148.21	   check_g(redex_g(x0))
276.40/148.21	   check_f(redex_f(x0))
276.40/148.21	   in_g_1(go_up(x0))
276.40/148.21	   in_f_1(go_up(x0))
276.40/148.21	
276.40/148.21	
276.40/148.21	----------------------------------------
276.40/148.21	
276.40/148.21	(5) DependencyPairsProof (EQUIVALENT)
276.40/148.21	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
276.40/148.21	----------------------------------------
276.40/148.21	
276.40/148.21	(6)
276.40/148.21	Obligation:
276.40/148.21	Q DP problem:
276.40/148.21	The TRS P consists of the following rules:
276.40/148.21	
276.40/148.21	   TOP(go_up(x)) -> TOP(reduce(x))
276.40/148.21	   TOP(go_up(x)) -> REDUCE(x)
276.40/148.21	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
276.40/148.21	   REDUCE(g(x_1)) -> REDEX_G(x_1)
276.40/148.21	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
276.40/148.21	   REDUCE(f(x_1)) -> REDEX_F(x_1)
276.40/148.21	   CHECK_G(redex_g(x_1)) -> IN_G_1(reduce(x_1))
276.40/148.21	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
276.40/148.21	   CHECK_F(redex_f(x_1)) -> IN_F_1(reduce(x_1))
276.40/148.21	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
276.40/148.21	
276.40/148.21	The TRS R consists of the following rules:
276.40/148.21	
276.40/148.21	   top(go_up(x)) -> top(reduce(x))
276.40/148.21	   reduce(g(x_1)) -> check_g(redex_g(x_1))
276.40/148.21	   reduce(f(x_1)) -> check_f(redex_f(x_1))
276.40/148.21	   redex_g(b) -> result_g(g(f(f(f(f(f(b)))))))
276.40/148.21	   redex_f(b) -> result_f(b)
276.40/148.21	   check_g(result_g(x)) -> go_up(x)
276.40/148.21	   check_f(result_f(x)) -> go_up(x)
276.40/148.21	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
276.40/148.21	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
276.40/148.21	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
276.40/148.21	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
276.40/148.21	
276.40/148.21	The set Q consists of the following terms:
276.40/148.21	
276.40/148.21	   top(go_up(x0))
276.40/148.21	   reduce(g(x0))
276.40/148.21	   reduce(f(x0))
276.40/148.21	   redex_g(b)
276.40/148.21	   redex_f(b)
276.40/148.21	   redex_g(f(b))
276.40/148.21	   check_g(result_g(x0))
276.40/148.21	   check_f(result_f(x0))
276.40/148.21	   check_g(redex_g(x0))
276.40/148.21	   check_f(redex_f(x0))
276.40/148.21	   in_g_1(go_up(x0))
276.40/148.21	   in_f_1(go_up(x0))
276.40/148.21	
276.40/148.21	We have to consider all minimal (P,Q,R)-chains.
276.40/148.21	----------------------------------------
276.40/148.21	
276.40/148.21	(7) DependencyGraphProof (EQUIVALENT)
276.40/148.21	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 5 less nodes.
276.40/148.21	----------------------------------------
276.40/148.21	
276.40/148.21	(8)
276.40/148.21	Complex Obligation (AND)
276.40/148.21	
276.40/148.21	----------------------------------------
276.40/148.21	
276.40/148.21	(9)
276.40/148.21	Obligation:
276.40/148.21	Q DP problem:
276.40/148.21	The TRS P consists of the following rules:
276.40/148.21	
276.40/148.21	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
276.40/148.21	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
276.40/148.21	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
276.40/148.21	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
276.40/148.21	
276.40/148.21	The TRS R consists of the following rules:
276.40/148.21	
276.40/148.21	   top(go_up(x)) -> top(reduce(x))
276.40/148.21	   reduce(g(x_1)) -> check_g(redex_g(x_1))
276.40/148.21	   reduce(f(x_1)) -> check_f(redex_f(x_1))
276.40/148.21	   redex_g(b) -> result_g(g(f(f(f(f(f(b)))))))
276.40/148.21	   redex_f(b) -> result_f(b)
276.40/148.21	   check_g(result_g(x)) -> go_up(x)
276.40/148.21	   check_f(result_f(x)) -> go_up(x)
276.40/148.21	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
276.40/148.21	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
276.40/148.21	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
276.40/148.21	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
276.40/148.21	
276.40/148.21	The set Q consists of the following terms:
276.40/148.21	
276.40/148.21	   top(go_up(x0))
276.40/148.21	   reduce(g(x0))
276.40/148.21	   reduce(f(x0))
276.40/148.21	   redex_g(b)
276.40/148.21	   redex_f(b)
276.40/148.21	   redex_g(f(b))
276.40/148.21	   check_g(result_g(x0))
276.40/148.21	   check_f(result_f(x0))
276.40/148.21	   check_g(redex_g(x0))
276.40/148.21	   check_f(redex_f(x0))
276.40/148.21	   in_g_1(go_up(x0))
276.40/148.21	   in_f_1(go_up(x0))
276.40/148.21	
276.40/148.21	We have to consider all minimal (P,Q,R)-chains.
276.40/148.21	----------------------------------------
276.40/148.21	
276.40/148.21	(10) UsableRulesProof (EQUIVALENT)
276.40/148.21	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.
276.40/148.21	----------------------------------------
276.40/148.21	
276.40/148.21	(11)
276.40/148.21	Obligation:
276.40/148.21	Q DP problem:
276.40/148.21	The TRS P consists of the following rules:
276.40/148.21	
276.40/148.21	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
276.40/148.21	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
276.40/148.21	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
276.40/148.21	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
276.40/148.21	
276.40/148.21	The TRS R consists of the following rules:
276.40/148.21	
276.40/148.21	   redex_f(b) -> result_f(b)
276.40/148.21	   redex_g(b) -> result_g(g(f(f(f(f(f(b)))))))
276.40/148.21	
276.40/148.21	The set Q consists of the following terms:
276.40/148.21	
276.40/148.21	   top(go_up(x0))
276.40/148.21	   reduce(g(x0))
276.40/148.21	   reduce(f(x0))
276.40/148.21	   redex_g(b)
276.40/148.21	   redex_f(b)
276.40/148.21	   redex_g(f(b))
276.40/148.21	   check_g(result_g(x0))
276.40/148.21	   check_f(result_f(x0))
276.40/148.21	   check_g(redex_g(x0))
276.40/148.21	   check_f(redex_f(x0))
276.40/148.21	   in_g_1(go_up(x0))
276.40/148.21	   in_f_1(go_up(x0))
276.40/148.21	
276.40/148.21	We have to consider all minimal (P,Q,R)-chains.
276.40/148.21	----------------------------------------
276.40/148.21	
276.40/148.21	(12) QReductionProof (EQUIVALENT)
276.40/148.21	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
276.40/148.21	
276.40/148.21	   top(go_up(x0))
276.40/148.21	   reduce(g(x0))
276.40/148.21	   reduce(f(x0))
276.40/148.21	   check_g(result_g(x0))
276.40/148.21	   check_f(result_f(x0))
276.40/148.21	   check_g(redex_g(x0))
276.40/148.21	   check_f(redex_f(x0))
276.40/148.21	   in_g_1(go_up(x0))
276.40/148.21	   in_f_1(go_up(x0))
276.40/148.21	
276.40/148.21	
276.40/148.21	----------------------------------------
276.40/148.21	
276.40/148.21	(13)
276.40/148.21	Obligation:
276.40/148.21	Q DP problem:
276.40/148.21	The TRS P consists of the following rules:
276.40/148.21	
276.40/148.21	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
276.40/148.21	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
276.40/148.21	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
276.40/148.21	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
276.40/148.21	
276.40/148.21	The TRS R consists of the following rules:
276.40/148.21	
276.40/148.21	   redex_f(b) -> result_f(b)
276.40/148.21	   redex_g(b) -> result_g(g(f(f(f(f(f(b)))))))
276.40/148.21	
276.40/148.21	The set Q consists of the following terms:
276.40/148.21	
276.40/148.21	   redex_g(b)
276.40/148.21	   redex_f(b)
276.40/148.21	   redex_g(f(b))
276.40/148.21	
276.40/148.21	We have to consider all minimal (P,Q,R)-chains.
276.40/148.21	----------------------------------------
276.40/148.21	
276.40/148.21	(14) MRRProof (EQUIVALENT)
276.40/148.21	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.
276.40/148.21	
276.40/148.21	
276.40/148.21	Strictly oriented rules of the TRS R:
276.40/148.21	
276.40/148.21	   redex_f(b) -> result_f(b)
276.40/148.21	
276.40/148.21	Used ordering: Polynomial interpretation [POLO]:
276.40/148.21	
276.40/148.21	   POL(CHECK_F(x_1)) = x_1
276.40/148.21	   POL(CHECK_G(x_1)) = 1 + 2*x_1
276.40/148.21	   POL(REDUCE(x_1)) = 1 + 2*x_1
276.40/148.21	   POL(b) = 0
276.40/148.21	   POL(f(x_1)) = 2*x_1
276.40/148.21	   POL(g(x_1)) = x_1
276.40/148.21	   POL(redex_f(x_1)) = 1 + 2*x_1
276.40/148.21	   POL(redex_g(x_1)) = x_1
276.40/148.21	   POL(result_f(x_1)) = x_1
276.40/148.21	   POL(result_g(x_1)) = 2*x_1
276.40/148.21	
276.40/148.21	
276.40/148.21	----------------------------------------
276.40/148.21	
276.40/148.21	(15)
276.40/148.21	Obligation:
276.40/148.21	Q DP problem:
276.40/148.21	The TRS P consists of the following rules:
276.40/148.21	
276.40/148.21	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
276.40/148.21	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
276.40/148.21	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
276.40/148.21	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
276.40/148.21	
276.40/148.21	The TRS R consists of the following rules:
276.40/148.21	
276.40/148.21	   redex_g(b) -> result_g(g(f(f(f(f(f(b)))))))
276.40/148.21	
276.40/148.21	The set Q consists of the following terms:
276.40/148.21	
276.40/148.21	   redex_g(b)
276.40/148.21	   redex_f(b)
276.40/148.21	   redex_g(f(b))
276.40/148.21	
276.40/148.21	We have to consider all minimal (P,Q,R)-chains.
276.40/148.21	----------------------------------------
276.40/148.21	
276.40/148.21	(16) MRRProof (EQUIVALENT)
276.40/148.21	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.
276.40/148.21	
276.40/148.21	Strictly oriented dependency pairs:
276.40/148.22	
276.40/148.22	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
276.40/148.22	
276.40/148.22	
276.40/148.22	Used ordering: Polynomial interpretation [POLO]:
276.40/148.22	
276.40/148.22	   POL(CHECK_F(x_1)) = 2*x_1
276.40/148.22	   POL(CHECK_G(x_1)) = 2*x_1
276.40/148.22	   POL(REDUCE(x_1)) = 2*x_1
276.40/148.22	   POL(b) = 0
276.40/148.22	   POL(f(x_1)) = 2*x_1
276.40/148.22	   POL(g(x_1)) = 1 + 2*x_1
276.40/148.22	   POL(redex_f(x_1)) = 2*x_1
276.40/148.22	   POL(redex_g(x_1)) = 1 + x_1
276.40/148.22	   POL(result_g(x_1)) = x_1
276.40/148.22	
276.40/148.22	
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(17)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
276.40/148.22	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
276.40/148.22	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
276.40/148.22	
276.40/148.22	The TRS R consists of the following rules:
276.40/148.22	
276.40/148.22	   redex_g(b) -> result_g(g(f(f(f(f(f(b)))))))
276.40/148.22	
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   redex_g(b)
276.40/148.22	   redex_f(b)
276.40/148.22	   redex_g(f(b))
276.40/148.22	
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(18) DependencyGraphProof (EQUIVALENT)
276.40/148.22	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(19)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
276.40/148.22	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
276.40/148.22	
276.40/148.22	The TRS R consists of the following rules:
276.40/148.22	
276.40/148.22	   redex_g(b) -> result_g(g(f(f(f(f(f(b)))))))
276.40/148.22	
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   redex_g(b)
276.40/148.22	   redex_f(b)
276.40/148.22	   redex_g(f(b))
276.40/148.22	
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(20) UsableRulesProof (EQUIVALENT)
276.40/148.22	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.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(21)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
276.40/148.22	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
276.40/148.22	
276.40/148.22	R is empty.
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   redex_g(b)
276.40/148.22	   redex_f(b)
276.40/148.22	   redex_g(f(b))
276.40/148.22	
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(22) QReductionProof (EQUIVALENT)
276.40/148.22	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
276.40/148.22	
276.40/148.22	   redex_g(b)
276.40/148.22	   redex_g(f(b))
276.40/148.22	
276.40/148.22	
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(23)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
276.40/148.22	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
276.40/148.22	
276.40/148.22	R is empty.
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   redex_f(b)
276.40/148.22	
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(24) UsableRulesReductionPairsProof (EQUIVALENT)
276.40/148.22	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.
276.40/148.22	
276.40/148.22	The following dependency pairs can be deleted:
276.40/148.22	
276.40/148.22	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
276.40/148.22	No rules are removed from R.
276.40/148.22	
276.40/148.22	Used ordering: POLO with Polynomial interpretation [POLO]:
276.40/148.22	
276.40/148.22	   POL(CHECK_F(x_1)) = 2*x_1
276.40/148.22	   POL(REDUCE(x_1)) = 2*x_1
276.40/148.22	   POL(f(x_1)) = 2*x_1
276.40/148.22	   POL(redex_f(x_1)) = x_1
276.40/148.22	
276.40/148.22	
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(25)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
276.40/148.22	
276.40/148.22	R is empty.
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   redex_f(b)
276.40/148.22	
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(26) DependencyGraphProof (EQUIVALENT)
276.40/148.22	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(27)
276.40/148.22	TRUE
276.40/148.22	
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(28)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   TOP(go_up(x)) -> TOP(reduce(x))
276.40/148.22	
276.40/148.22	The TRS R consists of the following rules:
276.40/148.22	
276.40/148.22	   top(go_up(x)) -> top(reduce(x))
276.40/148.22	   reduce(g(x_1)) -> check_g(redex_g(x_1))
276.40/148.22	   reduce(f(x_1)) -> check_f(redex_f(x_1))
276.40/148.22	   redex_g(b) -> result_g(g(f(f(f(f(f(b)))))))
276.40/148.22	   redex_f(b) -> result_f(b)
276.40/148.22	   check_g(result_g(x)) -> go_up(x)
276.40/148.22	   check_f(result_f(x)) -> go_up(x)
276.40/148.22	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
276.40/148.22	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
276.40/148.22	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
276.40/148.22	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
276.40/148.22	
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   top(go_up(x0))
276.40/148.22	   reduce(g(x0))
276.40/148.22	   reduce(f(x0))
276.40/148.22	   redex_g(b)
276.40/148.22	   redex_f(b)
276.40/148.22	   redex_g(f(b))
276.40/148.22	   check_g(result_g(x0))
276.40/148.22	   check_f(result_f(x0))
276.40/148.22	   check_g(redex_g(x0))
276.40/148.22	   check_f(redex_f(x0))
276.40/148.22	   in_g_1(go_up(x0))
276.40/148.22	   in_f_1(go_up(x0))
276.40/148.22	
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(29) UsableRulesProof (EQUIVALENT)
276.40/148.22	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.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(30)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   TOP(go_up(x)) -> TOP(reduce(x))
276.40/148.22	
276.40/148.22	The TRS R consists of the following rules:
276.40/148.22	
276.40/148.22	   reduce(g(x_1)) -> check_g(redex_g(x_1))
276.40/148.22	   reduce(f(x_1)) -> check_f(redex_f(x_1))
276.40/148.22	   redex_f(b) -> result_f(b)
276.40/148.22	   check_f(result_f(x)) -> go_up(x)
276.40/148.22	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
276.40/148.22	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
276.40/148.22	   redex_g(b) -> result_g(g(f(f(f(f(f(b)))))))
276.40/148.22	   check_g(result_g(x)) -> go_up(x)
276.40/148.22	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
276.40/148.22	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
276.40/148.22	
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   top(go_up(x0))
276.40/148.22	   reduce(g(x0))
276.40/148.22	   reduce(f(x0))
276.40/148.22	   redex_g(b)
276.40/148.22	   redex_f(b)
276.40/148.22	   redex_g(f(b))
276.40/148.22	   check_g(result_g(x0))
276.40/148.22	   check_f(result_f(x0))
276.40/148.22	   check_g(redex_g(x0))
276.40/148.22	   check_f(redex_f(x0))
276.40/148.22	   in_g_1(go_up(x0))
276.40/148.22	   in_f_1(go_up(x0))
276.40/148.22	
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(31) QReductionProof (EQUIVALENT)
276.40/148.22	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
276.40/148.22	
276.40/148.22	   top(go_up(x0))
276.40/148.22	
276.40/148.22	
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(32)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   TOP(go_up(x)) -> TOP(reduce(x))
276.40/148.22	
276.40/148.22	The TRS R consists of the following rules:
276.40/148.22	
276.40/148.22	   reduce(g(x_1)) -> check_g(redex_g(x_1))
276.40/148.22	   reduce(f(x_1)) -> check_f(redex_f(x_1))
276.40/148.22	   redex_f(b) -> result_f(b)
276.40/148.22	   check_f(result_f(x)) -> go_up(x)
276.40/148.22	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
276.40/148.22	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
276.40/148.22	   redex_g(b) -> result_g(g(f(f(f(f(f(b)))))))
276.40/148.22	   check_g(result_g(x)) -> go_up(x)
276.40/148.22	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
276.40/148.22	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
276.40/148.22	
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   reduce(g(x0))
276.40/148.22	   reduce(f(x0))
276.40/148.22	   redex_g(b)
276.40/148.22	   redex_f(b)
276.40/148.22	   redex_g(f(b))
276.40/148.22	   check_g(result_g(x0))
276.40/148.22	   check_f(result_f(x0))
276.40/148.22	   check_g(redex_g(x0))
276.40/148.22	   check_f(redex_f(x0))
276.40/148.22	   in_g_1(go_up(x0))
276.40/148.22	   in_f_1(go_up(x0))
276.40/148.22	
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(33) TransformationProof (SOUND)
276.40/148.22	By narrowing [LPAR04] the rule TOP(go_up(x)) -> TOP(reduce(x)) at position [0] we obtained the following new rules [LPAR04]:
276.40/148.22	
276.40/148.22	   (TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))),TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))))
276.40/148.22	   (TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))),TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))))
276.40/148.22	
276.40/148.22	
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(34)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0)))
276.40/148.22	   TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0)))
276.40/148.22	
276.40/148.22	The TRS R consists of the following rules:
276.40/148.22	
276.40/148.22	   reduce(g(x_1)) -> check_g(redex_g(x_1))
276.40/148.22	   reduce(f(x_1)) -> check_f(redex_f(x_1))
276.40/148.22	   redex_f(b) -> result_f(b)
276.40/148.22	   check_f(result_f(x)) -> go_up(x)
276.40/148.22	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
276.40/148.22	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
276.40/148.22	   redex_g(b) -> result_g(g(f(f(f(f(f(b)))))))
276.40/148.22	   check_g(result_g(x)) -> go_up(x)
276.40/148.22	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
276.40/148.22	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
276.40/148.22	
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   reduce(g(x0))
276.40/148.22	   reduce(f(x0))
276.40/148.22	   redex_g(b)
276.40/148.22	   redex_f(b)
276.40/148.22	   redex_g(f(b))
276.40/148.22	   check_g(result_g(x0))
276.40/148.22	   check_f(result_f(x0))
276.40/148.22	   check_g(redex_g(x0))
276.40/148.22	   check_f(redex_f(x0))
276.40/148.22	   in_g_1(go_up(x0))
276.40/148.22	   in_f_1(go_up(x0))
276.40/148.22	
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(35) UsableRulesProof (EQUIVALENT)
276.40/148.22	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.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(36)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   TOP(go_up(x)) -> TOP(reduce(x))
276.40/148.22	
276.40/148.22	The TRS R consists of the following rules:
276.40/148.22	
276.40/148.22	   reduce(g(x_1)) -> check_g(redex_g(x_1))
276.40/148.22	   reduce(f(x_1)) -> check_f(redex_f(x_1))
276.40/148.22	   redex_f(b) -> result_f(b)
276.40/148.22	   check_f(result_f(x)) -> go_up(x)
276.40/148.22	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
276.40/148.22	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
276.40/148.22	   redex_g(b) -> result_g(g(f(f(f(f(f(b)))))))
276.40/148.22	   check_g(result_g(x)) -> go_up(x)
276.40/148.22	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
276.40/148.22	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
276.40/148.22	
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   top(go_up(x0))
276.40/148.22	   reduce(g(x0))
276.40/148.22	   reduce(f(x0))
276.40/148.22	   redex_g(b)
276.40/148.22	   redex_f(b)
276.40/148.22	   redex_g(f(b))
276.40/148.22	   check_g(result_g(x0))
276.40/148.22	   check_f(result_f(x0))
276.40/148.22	   check_g(redex_g(x0))
276.40/148.22	   check_f(redex_f(x0))
276.40/148.22	   in_g_1(go_up(x0))
276.40/148.22	   in_f_1(go_up(x0))
276.40/148.22	
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(37) QReductionProof (EQUIVALENT)
276.40/148.22	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
276.40/148.22	
276.40/148.22	   top(go_up(x0))
276.40/148.22	
276.40/148.22	
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(38)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   TOP(go_up(x)) -> TOP(reduce(x))
276.40/148.22	
276.40/148.22	The TRS R consists of the following rules:
276.40/148.22	
276.40/148.22	   reduce(g(x_1)) -> check_g(redex_g(x_1))
276.40/148.22	   reduce(f(x_1)) -> check_f(redex_f(x_1))
276.40/148.22	   redex_f(b) -> result_f(b)
276.40/148.22	   check_f(result_f(x)) -> go_up(x)
276.40/148.22	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
276.40/148.22	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
276.40/148.22	   redex_g(b) -> result_g(g(f(f(f(f(f(b)))))))
276.40/148.22	   check_g(result_g(x)) -> go_up(x)
276.40/148.22	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
276.40/148.22	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
276.40/148.22	
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   reduce(g(x0))
276.40/148.22	   reduce(f(x0))
276.40/148.22	   redex_g(b)
276.40/148.22	   redex_f(b)
276.40/148.22	   redex_g(f(b))
276.40/148.22	   check_g(result_g(x0))
276.40/148.22	   check_f(result_f(x0))
276.40/148.22	   check_g(redex_g(x0))
276.40/148.22	   check_f(redex_f(x0))
276.40/148.22	   in_g_1(go_up(x0))
276.40/148.22	   in_f_1(go_up(x0))
276.40/148.22	
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(39) Trivial-Transformation (SOUND)
276.40/148.22	We applied the Trivial transformation  to transform the outermost TRS to a standard TRS.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(40)
276.40/148.22	Obligation:
276.40/148.22	Q restricted rewrite system:
276.40/148.22	The TRS R consists of the following rules:
276.40/148.22	
276.40/148.22	   g(b) -> g(f(f(f(f(f(b))))))
276.40/148.22	   f(b) -> b
276.40/148.22	   g(f(b)) -> c
276.40/148.22	
276.40/148.22	Q is empty.
276.40/148.22	
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(41) QTRSRRRProof (EQUIVALENT)
276.40/148.22	Used ordering:
276.40/148.22	Polynomial interpretation [POLO]:
276.40/148.22	
276.40/148.22	   POL(b) = 0
276.40/148.22	   POL(c) = 0
276.40/148.22	   POL(f(x_1)) = x_1
276.40/148.22	   POL(g(x_1)) = 1 + 2*x_1
276.40/148.22	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
276.40/148.22	
276.40/148.22	   g(f(b)) -> c
276.40/148.22	
276.40/148.22	
276.40/148.22	
276.40/148.22	
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(42)
276.40/148.22	Obligation:
276.40/148.22	Q restricted rewrite system:
276.40/148.22	The TRS R consists of the following rules:
276.40/148.22	
276.40/148.22	   g(b) -> g(f(f(f(f(f(b))))))
276.40/148.22	   f(b) -> b
276.40/148.22	
276.40/148.22	Q is empty.
276.40/148.22	
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(43) AAECC Innermost (EQUIVALENT)
276.40/148.22	We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is 
276.40/148.22	   f(b) -> b
276.40/148.22	
276.40/148.22	The TRS R 2 is 
276.40/148.22	   g(b) -> g(f(f(f(f(f(b))))))
276.40/148.22	
276.40/148.22	The signature Sigma is {g_1}
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(44)
276.40/148.22	Obligation:
276.40/148.22	Q restricted rewrite system:
276.40/148.22	The TRS R consists of the following rules:
276.40/148.22	
276.40/148.22	   g(b) -> g(f(f(f(f(f(b))))))
276.40/148.22	   f(b) -> b
276.40/148.22	
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   g(b)
276.40/148.22	   f(b)
276.40/148.22	
276.40/148.22	
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(45) DependencyPairsProof (EQUIVALENT)
276.40/148.22	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(46)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   G(b) -> G(f(f(f(f(f(b))))))
276.40/148.22	   G(b) -> F(f(f(f(f(b)))))
276.40/148.22	   G(b) -> F(f(f(f(b))))
276.40/148.22	   G(b) -> F(f(f(b)))
276.40/148.22	   G(b) -> F(f(b))
276.40/148.22	   G(b) -> F(b)
276.40/148.22	
276.40/148.22	The TRS R consists of the following rules:
276.40/148.22	
276.40/148.22	   g(b) -> g(f(f(f(f(f(b))))))
276.40/148.22	   f(b) -> b
276.40/148.22	
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   g(b)
276.40/148.22	   f(b)
276.40/148.22	
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(47) DependencyGraphProof (EQUIVALENT)
276.40/148.22	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(48)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   G(b) -> G(f(f(f(f(f(b))))))
276.40/148.22	
276.40/148.22	The TRS R consists of the following rules:
276.40/148.22	
276.40/148.22	   g(b) -> g(f(f(f(f(f(b))))))
276.40/148.22	   f(b) -> b
276.40/148.22	
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   g(b)
276.40/148.22	   f(b)
276.40/148.22	
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(49) UsableRulesProof (EQUIVALENT)
276.40/148.22	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.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(50)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   G(b) -> G(f(f(f(f(f(b))))))
276.40/148.22	
276.40/148.22	The TRS R consists of the following rules:
276.40/148.22	
276.40/148.22	   f(b) -> b
276.40/148.22	
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   g(b)
276.40/148.22	   f(b)
276.40/148.22	
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(51) QReductionProof (EQUIVALENT)
276.40/148.22	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
276.40/148.22	
276.40/148.22	   g(b)
276.40/148.22	
276.40/148.22	
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(52)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   G(b) -> G(f(f(f(f(f(b))))))
276.40/148.22	
276.40/148.22	The TRS R consists of the following rules:
276.40/148.22	
276.40/148.22	   f(b) -> b
276.40/148.22	
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   f(b)
276.40/148.22	
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(53) TransformationProof (EQUIVALENT)
276.40/148.22	By rewriting [LPAR04] the rule G(b) -> G(f(f(f(f(f(b)))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]:
276.40/148.22	
276.40/148.22	   (G(b) -> G(f(f(f(f(b))))),G(b) -> G(f(f(f(f(b))))))
276.40/148.22	
276.40/148.22	
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(54)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   G(b) -> G(f(f(f(f(b)))))
276.40/148.22	
276.40/148.22	The TRS R consists of the following rules:
276.40/148.22	
276.40/148.22	   f(b) -> b
276.40/148.22	
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   f(b)
276.40/148.22	
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(55) TransformationProof (EQUIVALENT)
276.40/148.22	By rewriting [LPAR04] the rule G(b) -> G(f(f(f(f(b))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]:
276.40/148.22	
276.40/148.22	   (G(b) -> G(f(f(f(b)))),G(b) -> G(f(f(f(b)))))
276.40/148.22	
276.40/148.22	
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(56)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   G(b) -> G(f(f(f(b))))
276.40/148.22	
276.40/148.22	The TRS R consists of the following rules:
276.40/148.22	
276.40/148.22	   f(b) -> b
276.40/148.22	
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   f(b)
276.40/148.22	
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(57) TransformationProof (EQUIVALENT)
276.40/148.22	By rewriting [LPAR04] the rule G(b) -> G(f(f(f(b)))) at position [0,0,0] we obtained the following new rules [LPAR04]:
276.40/148.22	
276.40/148.22	   (G(b) -> G(f(f(b))),G(b) -> G(f(f(b))))
276.40/148.22	
276.40/148.22	
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(58)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   G(b) -> G(f(f(b)))
276.40/148.22	
276.40/148.22	The TRS R consists of the following rules:
276.40/148.22	
276.40/148.22	   f(b) -> b
276.40/148.22	
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   f(b)
276.40/148.22	
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(59) TransformationProof (EQUIVALENT)
276.40/148.22	By rewriting [LPAR04] the rule G(b) -> G(f(f(b))) at position [0,0] we obtained the following new rules [LPAR04]:
276.40/148.22	
276.40/148.22	   (G(b) -> G(f(b)),G(b) -> G(f(b)))
276.40/148.22	
276.40/148.22	
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(60)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   G(b) -> G(f(b))
276.40/148.22	
276.40/148.22	The TRS R consists of the following rules:
276.40/148.22	
276.40/148.22	   f(b) -> b
276.40/148.22	
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   f(b)
276.40/148.22	
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(61) TransformationProof (EQUIVALENT)
276.40/148.22	By rewriting [LPAR04] the rule G(b) -> G(f(b)) at position [0] we obtained the following new rules [LPAR04]:
276.40/148.22	
276.40/148.22	   (G(b) -> G(b),G(b) -> G(b))
276.40/148.22	
276.40/148.22	
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(62)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   G(b) -> G(b)
276.40/148.22	
276.40/148.22	The TRS R consists of the following rules:
276.40/148.22	
276.40/148.22	   f(b) -> b
276.40/148.22	
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   f(b)
276.40/148.22	
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(63) UsableRulesProof (EQUIVALENT)
276.40/148.22	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.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(64)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   G(b) -> G(b)
276.40/148.22	
276.40/148.22	R is empty.
276.40/148.22	The set Q consists of the following terms:
276.40/148.22	
276.40/148.22	   f(b)
276.40/148.22	
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(65) QReductionProof (EQUIVALENT)
276.40/148.22	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
276.40/148.22	
276.40/148.22	   f(b)
276.40/148.22	
276.40/148.22	
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(66)
276.40/148.22	Obligation:
276.40/148.22	Q DP problem:
276.40/148.22	The TRS P consists of the following rules:
276.40/148.22	
276.40/148.22	   G(b) -> G(b)
276.40/148.22	
276.40/148.22	R is empty.
276.40/148.22	Q is empty.
276.40/148.22	We have to consider all minimal (P,Q,R)-chains.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(67) NonTerminationLoopProof (COMPLETE)
276.40/148.22	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
276.40/148.22	Found a loop by semiunifying a rule from P directly.
276.40/148.22	
276.40/148.22	s = G(b) evaluates to  t =G(b)
276.40/148.22	
276.40/148.22	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
276.40/148.22	* Matcher: [ ]
276.40/148.22	* Semiunifier: [ ]
276.40/148.22	
276.40/148.22	--------------------------------------------------------------------------------
276.40/148.22	Rewriting sequence
276.40/148.22	
276.40/148.22	The DP semiunifies directly so there is only one rewrite step from G(b) to G(b).
276.40/148.22	
276.40/148.22	
276.40/148.22	
276.40/148.22	
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(68)
276.40/148.22	NO
276.40/148.22	
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(69) Raffelsieper-Zantema-Transformation (SOUND)
276.40/148.22	We applied the Raffelsieper-Zantema transformation  to transform the outermost TRS to a standard TRS.
276.40/148.22	----------------------------------------
276.40/148.22	
276.40/148.22	(70)
276.40/148.22	Obligation:
276.40/148.22	Q restricted rewrite system:
276.40/148.22	The TRS R consists of the following rules:
276.40/148.22	
276.40/148.22	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.40/148.22	   down(f(b)) -> up(b)
276.40/148.22	   down(g(f(b))) -> up(c)
276.40/148.22	   top(up(x)) -> top(down(x))
276.40/148.22	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.40/148.22	   down(g(c)) -> g_flat(down(c))
276.40/148.22	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
276.40/148.22	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.40/148.22	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.40/148.22	   down(f(c)) -> f_flat(down(c))
276.40/148.22	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	
276.49/148.23	Q is empty.
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(71) QTRSRRRProof (EQUIVALENT)
276.49/148.23	Used ordering:
276.49/148.23	Polynomial interpretation [POLO]:
276.49/148.23	
276.49/148.23	   POL(b) = 0
276.49/148.23	   POL(c) = 0
276.49/148.23	   POL(down(x_1)) = x_1
276.49/148.23	   POL(f(x_1)) = x_1
276.49/148.23	   POL(f_flat(x_1)) = x_1
276.49/148.23	   POL(fresh_constant) = 0
276.49/148.23	   POL(g(x_1)) = 1 + 2*x_1
276.49/148.23	   POL(g_flat(x_1)) = 1 + 2*x_1
276.49/148.23	   POL(top(x_1)) = 2*x_1
276.49/148.23	   POL(up(x_1)) = x_1
276.49/148.23	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
276.49/148.23	
276.49/148.23	   down(g(f(b))) -> up(c)
276.49/148.23	
276.49/148.23	
276.49/148.23	
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(72)
276.49/148.23	Obligation:
276.49/148.23	Q restricted rewrite system:
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   top(up(x)) -> top(down(x))
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	
276.49/148.23	Q is empty.
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(73) AAECC Innermost (EQUIVALENT)
276.49/148.23	We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is 
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	
276.49/148.23	The TRS R 2 is 
276.49/148.23	   top(up(x)) -> top(down(x))
276.49/148.23	
276.49/148.23	The signature Sigma is {top_1}
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(74)
276.49/148.23	Obligation:
276.49/148.23	Q restricted rewrite system:
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   top(up(x)) -> top(down(x))
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   top(up(x0))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(75) DependencyPairsProof (EQUIVALENT)
276.49/148.23	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(76)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP(up(x)) -> TOP(down(x))
276.49/148.23	   TOP(up(x)) -> DOWN(x)
276.49/148.23	   DOWN(g(g(y3))) -> G_FLAT(down(g(y3)))
276.49/148.23	   DOWN(g(g(y3))) -> DOWN(g(y3))
276.49/148.23	   DOWN(g(c)) -> G_FLAT(down(c))
276.49/148.23	   DOWN(g(c)) -> DOWN(c)
276.49/148.23	   DOWN(g(fresh_constant)) -> G_FLAT(down(fresh_constant))
276.49/148.23	   DOWN(g(fresh_constant)) -> DOWN(fresh_constant)
276.49/148.23	   DOWN(f(g(y6))) -> F_FLAT(down(g(y6)))
276.49/148.23	   DOWN(f(g(y6))) -> DOWN(g(y6))
276.49/148.23	   DOWN(f(f(y7))) -> F_FLAT(down(f(y7)))
276.49/148.23	   DOWN(f(f(y7))) -> DOWN(f(y7))
276.49/148.23	   DOWN(f(c)) -> F_FLAT(down(c))
276.49/148.23	   DOWN(f(c)) -> DOWN(c)
276.49/148.23	   DOWN(f(fresh_constant)) -> F_FLAT(down(fresh_constant))
276.49/148.23	   DOWN(f(fresh_constant)) -> DOWN(fresh_constant)
276.49/148.23	   DOWN(g(f(g(y9)))) -> G_FLAT(down(f(g(y9))))
276.49/148.23	   DOWN(g(f(g(y9)))) -> DOWN(f(g(y9)))
276.49/148.23	   DOWN(g(f(f(y10)))) -> G_FLAT(down(f(f(y10))))
276.49/148.23	   DOWN(g(f(f(y10)))) -> DOWN(f(f(y10)))
276.49/148.23	   DOWN(g(f(c))) -> G_FLAT(down(f(c)))
276.49/148.23	   DOWN(g(f(c))) -> DOWN(f(c))
276.49/148.23	   DOWN(g(f(fresh_constant))) -> G_FLAT(down(f(fresh_constant)))
276.49/148.23	   DOWN(g(f(fresh_constant))) -> DOWN(f(fresh_constant))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   top(up(x)) -> top(down(x))
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   top(up(x0))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(77) DependencyGraphProof (EQUIVALENT)
276.49/148.23	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 18 less nodes.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(78)
276.49/148.23	Complex Obligation (AND)
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(79)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   DOWN(g(f(g(y9)))) -> DOWN(f(g(y9)))
276.49/148.23	   DOWN(f(g(y6))) -> DOWN(g(y6))
276.49/148.23	   DOWN(g(g(y3))) -> DOWN(g(y3))
276.49/148.23	   DOWN(g(f(f(y10)))) -> DOWN(f(f(y10)))
276.49/148.23	   DOWN(f(f(y7))) -> DOWN(f(y7))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   top(up(x)) -> top(down(x))
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   top(up(x0))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(80) UsableRulesProof (EQUIVALENT)
276.49/148.23	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.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(81)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   DOWN(g(f(g(y9)))) -> DOWN(f(g(y9)))
276.49/148.23	   DOWN(f(g(y6))) -> DOWN(g(y6))
276.49/148.23	   DOWN(g(g(y3))) -> DOWN(g(y3))
276.49/148.23	   DOWN(g(f(f(y10)))) -> DOWN(f(f(y10)))
276.49/148.23	   DOWN(f(f(y7))) -> DOWN(f(y7))
276.49/148.23	
276.49/148.23	R is empty.
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   top(up(x0))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(82) QReductionProof (EQUIVALENT)
276.49/148.23	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   top(up(x0))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(83)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   DOWN(g(f(g(y9)))) -> DOWN(f(g(y9)))
276.49/148.23	   DOWN(f(g(y6))) -> DOWN(g(y6))
276.49/148.23	   DOWN(g(g(y3))) -> DOWN(g(y3))
276.49/148.23	   DOWN(g(f(f(y10)))) -> DOWN(f(f(y10)))
276.49/148.23	   DOWN(f(f(y7))) -> DOWN(f(y7))
276.49/148.23	
276.49/148.23	R is empty.
276.49/148.23	Q is empty.
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(84) QDPSizeChangeProof (EQUIVALENT)
276.49/148.23	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. 
276.49/148.23	
276.49/148.23	From the DPs we obtained the following set of size-change graphs:
276.49/148.23	*DOWN(f(g(y6))) -> DOWN(g(y6))
276.49/148.23	The graph contains the following edges 1 > 1
276.49/148.23	
276.49/148.23	
276.49/148.23	*DOWN(g(g(y3))) -> DOWN(g(y3))
276.49/148.23	The graph contains the following edges 1 > 1
276.49/148.23	
276.49/148.23	
276.49/148.23	*DOWN(g(f(g(y9)))) -> DOWN(f(g(y9)))
276.49/148.23	The graph contains the following edges 1 > 1
276.49/148.23	
276.49/148.23	
276.49/148.23	*DOWN(g(f(f(y10)))) -> DOWN(f(f(y10)))
276.49/148.23	The graph contains the following edges 1 > 1
276.49/148.23	
276.49/148.23	
276.49/148.23	*DOWN(f(f(y7))) -> DOWN(f(y7))
276.49/148.23	The graph contains the following edges 1 > 1
276.49/148.23	
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(85)
276.49/148.23	YES
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(86)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP(up(x)) -> TOP(down(x))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   top(up(x)) -> top(down(x))
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   top(up(x0))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(87) UsableRulesProof (EQUIVALENT)
276.49/148.23	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.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(88)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP(up(x)) -> TOP(down(x))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   top(up(x0))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(89) QReductionProof (EQUIVALENT)
276.49/148.23	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
276.49/148.23	
276.49/148.23	   top(up(x0))
276.49/148.23	
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(90)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP(up(x)) -> TOP(down(x))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(91) TransformationProof (EQUIVALENT)
276.49/148.23	By narrowing [LPAR04] the rule TOP(up(x)) -> TOP(down(x)) at position [0] we obtained the following new rules [LPAR04]:
276.49/148.23	
276.49/148.23	   (TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b)))))))),TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b)))))))))
276.49/148.23	   (TOP(up(f(b))) -> TOP(up(b)),TOP(up(f(b))) -> TOP(up(b)))
276.49/148.23	   (TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))),TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0)))))
276.49/148.23	   (TOP(up(g(c))) -> TOP(g_flat(down(c))),TOP(up(g(c))) -> TOP(g_flat(down(c))))
276.49/148.23	   (TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))),TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))))
276.49/148.23	   (TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))),TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))))
276.49/148.23	   (TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0)))),TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0)))))
276.49/148.23	   (TOP(up(f(c))) -> TOP(f_flat(down(c))),TOP(up(f(c))) -> TOP(f_flat(down(c))))
276.49/148.23	   (TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))),TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))))
276.49/148.23	   (TOP(up(g(f(g(x0))))) -> TOP(g_flat(down(f(g(x0))))),TOP(up(g(f(g(x0))))) -> TOP(g_flat(down(f(g(x0))))))
276.49/148.23	   (TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))),TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))))
276.49/148.23	   (TOP(up(g(f(c)))) -> TOP(g_flat(down(f(c)))),TOP(up(g(f(c)))) -> TOP(g_flat(down(f(c)))))
276.49/148.23	   (TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant)))),TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant)))))
276.49/148.23	
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(92)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b))))))))
276.49/148.23	   TOP(up(f(b))) -> TOP(up(b))
276.49/148.23	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
276.49/148.23	   TOP(up(g(c))) -> TOP(g_flat(down(c)))
276.49/148.23	   TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant)))
276.49/148.23	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))
276.49/148.23	   TOP(up(f(c))) -> TOP(f_flat(down(c)))
276.49/148.23	   TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant)))
276.49/148.23	   TOP(up(g(f(g(x0))))) -> TOP(g_flat(down(f(g(x0)))))
276.49/148.23	   TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0)))))
276.49/148.23	   TOP(up(g(f(c)))) -> TOP(g_flat(down(f(c))))
276.49/148.23	   TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(93) DependencyGraphProof (EQUIVALENT)
276.49/148.23	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(94)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0)))))
276.49/148.23	   TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b))))))))
276.49/148.23	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))
276.49/148.23	   TOP(up(g(f(g(x0))))) -> TOP(g_flat(down(f(g(x0)))))
276.49/148.23	   TOP(up(g(f(c)))) -> TOP(g_flat(down(f(c))))
276.49/148.23	   TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(95) TransformationProof (EQUIVALENT)
276.49/148.23	By rewriting [LPAR04] the rule TOP(up(g(f(f(x0))))) -> TOP(g_flat(down(f(f(x0))))) at position [0,0] we obtained the following new rules [LPAR04]:
276.49/148.23	
276.49/148.23	   (TOP(up(g(f(f(x0))))) -> TOP(g_flat(f_flat(down(f(x0))))),TOP(up(g(f(f(x0))))) -> TOP(g_flat(f_flat(down(f(x0))))))
276.49/148.23	
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(96)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b))))))))
276.49/148.23	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))
276.49/148.23	   TOP(up(g(f(g(x0))))) -> TOP(g_flat(down(f(g(x0)))))
276.49/148.23	   TOP(up(g(f(c)))) -> TOP(g_flat(down(f(c))))
276.49/148.23	   TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant))))
276.49/148.23	   TOP(up(g(f(f(x0))))) -> TOP(g_flat(f_flat(down(f(x0)))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(97) TransformationProof (EQUIVALENT)
276.49/148.23	By rewriting [LPAR04] the rule TOP(up(g(f(g(x0))))) -> TOP(g_flat(down(f(g(x0))))) at position [0,0] we obtained the following new rules [LPAR04]:
276.49/148.23	
276.49/148.23	   (TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))),TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0))))))
276.49/148.23	
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(98)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b))))))))
276.49/148.23	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))
276.49/148.23	   TOP(up(g(f(c)))) -> TOP(g_flat(down(f(c))))
276.49/148.23	   TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant))))
276.49/148.23	   TOP(up(g(f(f(x0))))) -> TOP(g_flat(f_flat(down(f(x0)))))
276.49/148.23	   TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0)))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(99) TransformationProof (EQUIVALENT)
276.49/148.23	By rewriting [LPAR04] the rule TOP(up(g(f(c)))) -> TOP(g_flat(down(f(c)))) at position [0,0] we obtained the following new rules [LPAR04]:
276.49/148.23	
276.49/148.23	   (TOP(up(g(f(c)))) -> TOP(g_flat(f_flat(down(c)))),TOP(up(g(f(c)))) -> TOP(g_flat(f_flat(down(c)))))
276.49/148.23	
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(100)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b))))))))
276.49/148.23	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))
276.49/148.23	   TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant))))
276.49/148.23	   TOP(up(g(f(f(x0))))) -> TOP(g_flat(f_flat(down(f(x0)))))
276.49/148.23	   TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0)))))
276.49/148.23	   TOP(up(g(f(c)))) -> TOP(g_flat(f_flat(down(c))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(101) DependencyGraphProof (EQUIVALENT)
276.49/148.23	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(102)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP(up(g(f(f(x0))))) -> TOP(g_flat(f_flat(down(f(x0)))))
276.49/148.23	   TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b))))))))
276.49/148.23	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))
276.49/148.23	   TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant))))
276.49/148.23	   TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0)))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(103) TransformationProof (EQUIVALENT)
276.49/148.23	By rewriting [LPAR04] the rule TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(down(f(fresh_constant)))) at position [0,0] we obtained the following new rules [LPAR04]:
276.49/148.23	
276.49/148.23	   (TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(f_flat(down(fresh_constant)))),TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(f_flat(down(fresh_constant)))))
276.49/148.23	
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(104)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP(up(g(f(f(x0))))) -> TOP(g_flat(f_flat(down(f(x0)))))
276.49/148.23	   TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b))))))))
276.49/148.23	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))
276.49/148.23	   TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0)))))
276.49/148.23	   TOP(up(g(f(fresh_constant)))) -> TOP(g_flat(f_flat(down(fresh_constant))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(105) DependencyGraphProof (EQUIVALENT)
276.49/148.23	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(106)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b))))))))
276.49/148.23	   TOP(up(g(f(f(x0))))) -> TOP(g_flat(f_flat(down(f(x0)))))
276.49/148.23	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))
276.49/148.23	   TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0)))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(107) QDPOrderProof (EQUIVALENT)
276.49/148.23	We use the reduction pair processor [LPAR04,JAR06].
276.49/148.23	
276.49/148.23	
276.49/148.23	The following pairs can be oriented strictly and are deleted.
276.49/148.23	
276.49/148.23	   TOP(up(g(b))) -> TOP(up(g(f(f(f(f(f(b))))))))
276.49/148.23	The remaining pairs can at least be oriented weakly.
276.49/148.23	Used ordering:  Matrix interpretation [MATRO]:
276.49/148.23	
276.49/148.23	Non-tuple symbols: 
276.49/148.23	<<<
276.49/148.23	 M( b ) =  	[[1], [1]]
276.49/148.23	>>>
276.49/148.23	
276.49/148.23	<<<
276.49/148.23	 M( c ) =  	[[0], [0]]
276.49/148.23	>>>
276.49/148.23	
276.49/148.23	<<<
276.49/148.23	 M( down_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
276.49/148.23	>>>
276.49/148.23	
276.49/148.23	<<<
276.49/148.23	 M( f_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
276.49/148.23	>>>
276.49/148.23	
276.49/148.23	<<<
276.49/148.23	 M( fresh_constant ) =  	[[0], [0]]
276.49/148.23	>>>
276.49/148.23	
276.49/148.23	<<<
276.49/148.23	 M( up_1(x_1) ) =  	[[0], [0]] 	+ 	[[1, 1], [1, 0]] 	* 	x_1
276.49/148.23	>>>
276.49/148.23	
276.49/148.23	<<<
276.49/148.23	 M( f_flat_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
276.49/148.23	>>>
276.49/148.23	
276.49/148.23	<<<
276.49/148.23	 M( g_1(x_1) ) =  	[[0], [0]] 	+ 	[[1, 1], [0, 0]] 	* 	x_1
276.49/148.23	>>>
276.49/148.23	
276.49/148.23	<<<
276.49/148.23	 M( g_flat_1(x_1) ) =  	[[0], [0]] 	+ 	[[1, 0], [1, 0]] 	* 	x_1
276.49/148.23	>>>
276.49/148.23	
276.49/148.23	Tuple symbols: 
276.49/148.23	<<<
276.49/148.23	 M( TOP_1(x_1) ) =  	[[0]] 	+ 	[[0, 1]] 	* 	x_1
276.49/148.23	>>>
276.49/148.23	
276.49/148.23	
276.49/148.23	
276.49/148.23	Matrix type: 
276.49/148.23	
276.49/148.23	We used a basic matrix type which is not further parametrizeable.
276.49/148.23	
276.49/148.23	
276.49/148.23	
276.49/148.23	
276.49/148.23	
276.49/148.23	As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
276.49/148.23	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
276.49/148.23	
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(108)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP(up(g(f(f(x0))))) -> TOP(g_flat(f_flat(down(f(x0)))))
276.49/148.23	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))
276.49/148.23	   TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0)))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(109) MNOCProof (EQUIVALENT)
276.49/148.23	We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(110)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP(up(g(f(f(x0))))) -> TOP(g_flat(f_flat(down(f(x0)))))
276.49/148.23	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))
276.49/148.23	   TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0)))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	
276.49/148.23	Q is empty.
276.49/148.23	We have to consider all (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(111) SplitQDPProof (EQUIVALENT)
276.49/148.23	We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(112)
276.49/148.23	Complex Obligation (AND)
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(113)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP(up(g(f(f(x0))))) -> TOP(g_flat(f_flat(down(f(x0)))))
276.49/148.23	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))
276.49/148.23	   TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0)))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(114) SemLabProof (SOUND)
276.49/148.23	We found the following model for the rules of the TRSs R and P.
276.49/148.23	Interpretation over the domain with elements from 0 to 1.
276.49/148.23	b: 0
276.49/148.23	c: 0
276.49/148.23	down: 0
276.49/148.23	f: 0
276.49/148.23	fresh_constant: 1
276.49/148.23	up: 0
276.49/148.23	f_flat: 0
276.49/148.23	TOP: 0
276.49/148.23	g_flat: 0
276.49/148.23	g: 0
276.49/148.23	By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(115)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(f.0(x0)))))
276.49/148.23	   TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0))))
276.49/148.23	   TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0))))
276.49/148.23	   TOP.0(up.0(g.0(f.0(f.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(f.1(x0)))))
276.49/148.23	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
276.49/148.23	   TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0))))
276.49/148.23	   TOP.0(up.0(f.0(f.0(x0)))) -> TOP.0(f_flat.0(down.0(f.0(x0))))
276.49/148.23	   TOP.0(up.0(f.0(f.1(x0)))) -> TOP.0(f_flat.0(down.0(f.1(x0))))
276.49/148.23	   TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0)))))
276.49/148.23	   TOP.0(up.0(g.0(f.0(g.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.1(x0)))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.)))))))
276.49/148.23	   down.0(f.0(b.)) -> up.0(b.)
276.49/148.23	   down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3)))
276.49/148.23	   down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3)))
276.49/148.23	   down.0(g.0(c.)) -> g_flat.0(down.0(c.))
276.49/148.23	   down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.))
276.49/148.23	   down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6)))
276.49/148.23	   down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6)))
276.49/148.23	   down.0(f.0(f.0(y7))) -> f_flat.0(down.0(f.0(y7)))
276.49/148.23	   down.0(f.0(f.1(y7))) -> f_flat.0(down.0(f.1(y7)))
276.49/148.23	   down.0(f.0(c.)) -> f_flat.0(down.0(c.))
276.49/148.23	   down.0(f.1(fresh_constant.)) -> f_flat.0(down.1(fresh_constant.))
276.49/148.23	   down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9))))
276.49/148.23	   down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9))))
276.49/148.23	   down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10))))
276.49/148.23	   down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10))))
276.49/148.23	   down.0(g.0(f.0(c.))) -> g_flat.0(down.0(f.0(c.)))
276.49/148.23	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
276.49/148.23	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
276.49/148.23	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
276.49/148.23	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
276.49/148.23	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down.0(g.0(b.))
276.49/148.23	   down.0(f.0(b.))
276.49/148.23	   down.0(g.0(g.0(x0)))
276.49/148.23	   down.0(g.0(g.1(x0)))
276.49/148.23	   down.0(g.0(c.))
276.49/148.23	   down.0(g.1(fresh_constant.))
276.49/148.23	   down.0(f.0(g.0(x0)))
276.49/148.23	   down.0(f.0(g.1(x0)))
276.49/148.23	   down.0(f.0(f.0(x0)))
276.49/148.23	   down.0(f.0(f.1(x0)))
276.49/148.23	   down.0(f.0(c.))
276.49/148.23	   down.0(f.1(fresh_constant.))
276.49/148.23	   down.0(g.0(f.0(g.0(x0))))
276.49/148.23	   down.0(g.0(f.0(g.1(x0))))
276.49/148.23	   down.0(g.0(f.0(f.0(x0))))
276.49/148.23	   down.0(g.0(f.0(f.1(x0))))
276.49/148.23	   down.0(g.0(f.0(c.)))
276.49/148.23	   down.0(g.0(f.1(fresh_constant.)))
276.49/148.23	   g_flat.0(up.0(x0))
276.49/148.23	   g_flat.0(up.1(x0))
276.49/148.23	   f_flat.0(up.0(x0))
276.49/148.23	   f_flat.0(up.1(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(116) UsableRulesReductionPairsProof (EQUIVALENT)
276.49/148.23	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.
276.49/148.23	
276.49/148.23	No dependency pairs are removed.
276.49/148.23	
276.49/148.23	The following rules are removed from R:
276.49/148.23	
276.49/148.23	   down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.))
276.49/148.23	   down.0(f.1(fresh_constant.)) -> f_flat.0(down.1(fresh_constant.))
276.49/148.23	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
276.49/148.23	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
276.49/148.23	Used ordering: POLO with Polynomial interpretation [POLO]:
276.49/148.23	
276.49/148.23	   POL(TOP.0(x_1)) = x_1
276.49/148.23	   POL(b.) = 0
276.49/148.23	   POL(c.) = 0
276.49/148.23	   POL(down.0(x_1)) = 1 + x_1
276.49/148.23	   POL(down.1(x_1)) = x_1
276.49/148.23	   POL(f.0(x_1)) = x_1
276.49/148.23	   POL(f.1(x_1)) = x_1
276.49/148.23	   POL(f_flat.0(x_1)) = x_1
276.49/148.23	   POL(fresh_constant.) = 0
276.49/148.23	   POL(g.0(x_1)) = 1 + x_1
276.49/148.23	   POL(g.1(x_1)) = 1 + x_1
276.49/148.23	   POL(g_flat.0(x_1)) = 1 + x_1
276.49/148.23	   POL(up.0(x_1)) = 1 + x_1
276.49/148.23	   POL(up.1(x_1)) = 1 + x_1
276.49/148.23	
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(117)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(f.0(x0)))))
276.49/148.23	   TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0))))
276.49/148.23	   TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0))))
276.49/148.23	   TOP.0(up.0(g.0(f.0(f.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(f.1(x0)))))
276.49/148.23	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
276.49/148.23	   TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0))))
276.49/148.23	   TOP.0(up.0(f.0(f.0(x0)))) -> TOP.0(f_flat.0(down.0(f.0(x0))))
276.49/148.23	   TOP.0(up.0(f.0(f.1(x0)))) -> TOP.0(f_flat.0(down.0(f.1(x0))))
276.49/148.23	   TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0)))))
276.49/148.23	   TOP.0(up.0(g.0(f.0(g.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.1(x0)))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
276.49/148.23	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
276.49/148.23	   down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.)))))))
276.49/148.23	   down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3)))
276.49/148.23	   down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3)))
276.49/148.23	   down.0(g.0(c.)) -> g_flat.0(down.0(c.))
276.49/148.23	   down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9))))
276.49/148.23	   down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9))))
276.49/148.23	   down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10))))
276.49/148.23	   down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10))))
276.49/148.23	   down.0(g.0(f.0(c.))) -> g_flat.0(down.0(f.0(c.)))
276.49/148.23	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
276.49/148.23	   down.0(f.0(c.)) -> f_flat.0(down.0(c.))
276.49/148.23	   down.0(f.0(f.1(y7))) -> f_flat.0(down.0(f.1(y7)))
276.49/148.23	   down.0(f.0(f.0(y7))) -> f_flat.0(down.0(f.0(y7)))
276.49/148.23	   down.0(f.0(b.)) -> up.0(b.)
276.49/148.23	   down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6)))
276.49/148.23	   down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6)))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down.0(g.0(b.))
276.49/148.23	   down.0(f.0(b.))
276.49/148.23	   down.0(g.0(g.0(x0)))
276.49/148.23	   down.0(g.0(g.1(x0)))
276.49/148.23	   down.0(g.0(c.))
276.49/148.23	   down.0(g.1(fresh_constant.))
276.49/148.23	   down.0(f.0(g.0(x0)))
276.49/148.23	   down.0(f.0(g.1(x0)))
276.49/148.23	   down.0(f.0(f.0(x0)))
276.49/148.23	   down.0(f.0(f.1(x0)))
276.49/148.23	   down.0(f.0(c.))
276.49/148.23	   down.0(f.1(fresh_constant.))
276.49/148.23	   down.0(g.0(f.0(g.0(x0))))
276.49/148.23	   down.0(g.0(f.0(g.1(x0))))
276.49/148.23	   down.0(g.0(f.0(f.0(x0))))
276.49/148.23	   down.0(g.0(f.0(f.1(x0))))
276.49/148.23	   down.0(g.0(f.0(c.)))
276.49/148.23	   down.0(g.0(f.1(fresh_constant.)))
276.49/148.23	   g_flat.0(up.0(x0))
276.49/148.23	   g_flat.0(up.1(x0))
276.49/148.23	   f_flat.0(up.0(x0))
276.49/148.23	   f_flat.0(up.1(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(118) DependencyGraphProof (EQUIVALENT)
276.49/148.23	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(119)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0))))
276.49/148.23	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
276.49/148.23	   TOP.0(up.0(f.0(f.0(x0)))) -> TOP.0(f_flat.0(down.0(f.0(x0))))
276.49/148.23	   TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(f.0(x0)))))
276.49/148.23	   TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0)))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
276.49/148.23	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
276.49/148.23	   down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.)))))))
276.49/148.23	   down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3)))
276.49/148.23	   down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3)))
276.49/148.23	   down.0(g.0(c.)) -> g_flat.0(down.0(c.))
276.49/148.23	   down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9))))
276.49/148.23	   down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9))))
276.49/148.23	   down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10))))
276.49/148.23	   down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10))))
276.49/148.23	   down.0(g.0(f.0(c.))) -> g_flat.0(down.0(f.0(c.)))
276.49/148.23	   down.0(g.0(f.1(fresh_constant.))) -> g_flat.0(down.0(f.1(fresh_constant.)))
276.49/148.23	   down.0(f.0(c.)) -> f_flat.0(down.0(c.))
276.49/148.23	   down.0(f.0(f.1(y7))) -> f_flat.0(down.0(f.1(y7)))
276.49/148.23	   down.0(f.0(f.0(y7))) -> f_flat.0(down.0(f.0(y7)))
276.49/148.23	   down.0(f.0(b.)) -> up.0(b.)
276.49/148.23	   down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6)))
276.49/148.23	   down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6)))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down.0(g.0(b.))
276.49/148.23	   down.0(f.0(b.))
276.49/148.23	   down.0(g.0(g.0(x0)))
276.49/148.23	   down.0(g.0(g.1(x0)))
276.49/148.23	   down.0(g.0(c.))
276.49/148.23	   down.0(g.1(fresh_constant.))
276.49/148.23	   down.0(f.0(g.0(x0)))
276.49/148.23	   down.0(f.0(g.1(x0)))
276.49/148.23	   down.0(f.0(f.0(x0)))
276.49/148.23	   down.0(f.0(f.1(x0)))
276.49/148.23	   down.0(f.0(c.))
276.49/148.23	   down.0(f.1(fresh_constant.))
276.49/148.23	   down.0(g.0(f.0(g.0(x0))))
276.49/148.23	   down.0(g.0(f.0(g.1(x0))))
276.49/148.23	   down.0(g.0(f.0(f.0(x0))))
276.49/148.23	   down.0(g.0(f.0(f.1(x0))))
276.49/148.23	   down.0(g.0(f.0(c.)))
276.49/148.23	   down.0(g.0(f.1(fresh_constant.)))
276.49/148.23	   g_flat.0(up.0(x0))
276.49/148.23	   g_flat.0(up.1(x0))
276.49/148.23	   f_flat.0(up.0(x0))
276.49/148.23	   f_flat.0(up.1(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(120) PisEmptyProof (SOUND)
276.49/148.23	The TRS P is empty. Hence, there is no (P,Q,R) chain.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(121)
276.49/148.23	TRUE
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(122)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))
276.49/148.23	   TOP(up(g(f(f(x0))))) -> TOP(g_flat(f_flat(down(f(x0)))))
276.49/148.23	   TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0)))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(123) SplitQDPProof (EQUIVALENT)
276.49/148.23	We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(124)
276.49/148.23	Complex Obligation (AND)
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(125)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))
276.49/148.23	   TOP(up(g(f(f(x0))))) -> TOP(g_flat(f_flat(down(f(x0)))))
276.49/148.23	   TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0)))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(c)) -> g_flat(down(c))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   down(f(c)) -> f_flat(down(c))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(126) SemLabProof (SOUND)
276.49/148.23	We found the following model for the rules of the TRSs R and P.
276.49/148.23	Interpretation over the domain with elements from 0 to 1.
276.49/148.23	b: 0
276.49/148.23	c: 1
276.49/148.23	down: 0
276.49/148.23	f: 0
276.49/148.23	fresh_constant: 0
276.49/148.23	up: 0
276.49/148.23	f_flat: 0
276.49/148.23	TOP: 0
276.49/148.23	g_flat: 0
276.49/148.23	g: 0
276.49/148.23	By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(127)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0))))
276.49/148.23	   TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0))))
276.49/148.23	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
276.49/148.23	   TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0))))
276.49/148.23	   TOP.0(up.0(f.0(f.0(x0)))) -> TOP.0(f_flat.0(down.0(f.0(x0))))
276.49/148.23	   TOP.0(up.0(f.0(f.1(x0)))) -> TOP.0(f_flat.0(down.0(f.1(x0))))
276.49/148.23	   TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(f.0(x0)))))
276.49/148.23	   TOP.0(up.0(g.0(f.0(f.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(f.1(x0)))))
276.49/148.23	   TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0)))))
276.49/148.23	   TOP.0(up.0(g.0(f.0(g.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.1(x0)))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
276.49/148.23	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
276.49/148.23	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
276.49/148.23	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
276.49/148.23	   down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.)))))))
276.49/148.23	   down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3)))
276.49/148.23	   down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3)))
276.49/148.23	   down.0(g.1(c.)) -> g_flat.0(down.1(c.))
276.49/148.23	   down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9))))
276.49/148.23	   down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9))))
276.49/148.23	   down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10))))
276.49/148.23	   down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10))))
276.49/148.23	   down.0(g.0(f.1(c.))) -> g_flat.0(down.0(f.1(c.)))
276.49/148.23	   down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.)))
276.49/148.23	   down.0(f.1(c.)) -> f_flat.0(down.1(c.))
276.49/148.23	   down.0(f.0(f.0(y7))) -> f_flat.0(down.0(f.0(y7)))
276.49/148.23	   down.0(f.0(f.1(y7))) -> f_flat.0(down.0(f.1(y7)))
276.49/148.23	   down.0(f.0(b.)) -> up.0(b.)
276.49/148.23	   down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6)))
276.49/148.23	   down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6)))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down.0(g.0(b.))
276.49/148.23	   down.0(f.0(b.))
276.49/148.23	   down.0(g.0(g.0(x0)))
276.49/148.23	   down.0(g.0(g.1(x0)))
276.49/148.23	   down.0(g.1(c.))
276.49/148.23	   down.0(g.0(fresh_constant.))
276.49/148.23	   down.0(f.0(g.0(x0)))
276.49/148.23	   down.0(f.0(g.1(x0)))
276.49/148.23	   down.0(f.0(f.0(x0)))
276.49/148.23	   down.0(f.0(f.1(x0)))
276.49/148.23	   down.0(f.1(c.))
276.49/148.23	   down.0(f.0(fresh_constant.))
276.49/148.23	   down.0(g.0(f.0(g.0(x0))))
276.49/148.23	   down.0(g.0(f.0(g.1(x0))))
276.49/148.23	   down.0(g.0(f.0(f.0(x0))))
276.49/148.23	   down.0(g.0(f.0(f.1(x0))))
276.49/148.23	   down.0(g.0(f.1(c.)))
276.49/148.23	   down.0(g.0(f.0(fresh_constant.)))
276.49/148.23	   g_flat.0(up.0(x0))
276.49/148.23	   g_flat.0(up.1(x0))
276.49/148.23	   f_flat.0(up.0(x0))
276.49/148.23	   f_flat.0(up.1(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(128) UsableRulesReductionPairsProof (EQUIVALENT)
276.49/148.23	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.
276.49/148.23	
276.49/148.23	No dependency pairs are removed.
276.49/148.23	
276.49/148.23	The following rules are removed from R:
276.49/148.23	
276.49/148.23	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
276.49/148.23	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
276.49/148.23	   down.0(g.1(c.)) -> g_flat.0(down.1(c.))
276.49/148.23	   down.0(f.1(c.)) -> f_flat.0(down.1(c.))
276.49/148.23	Used ordering: POLO with Polynomial interpretation [POLO]:
276.49/148.23	
276.49/148.23	   POL(TOP.0(x_1)) = x_1
276.49/148.23	   POL(b.) = 0
276.49/148.23	   POL(c.) = 0
276.49/148.23	   POL(down.0(x_1)) = 1 + x_1
276.49/148.23	   POL(down.1(x_1)) = x_1
276.49/148.23	   POL(f.0(x_1)) = x_1
276.49/148.23	   POL(f.1(x_1)) = x_1
276.49/148.23	   POL(f_flat.0(x_1)) = x_1
276.49/148.23	   POL(fresh_constant.) = 0
276.49/148.23	   POL(g.0(x_1)) = 1 + x_1
276.49/148.23	   POL(g.1(x_1)) = 1 + x_1
276.49/148.23	   POL(g_flat.0(x_1)) = 1 + x_1
276.49/148.23	   POL(up.0(x_1)) = 1 + x_1
276.49/148.23	   POL(up.1(x_1)) = 1 + x_1
276.49/148.23	
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(129)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0))))
276.49/148.23	   TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0))))
276.49/148.23	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
276.49/148.23	   TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0))))
276.49/148.23	   TOP.0(up.0(f.0(f.0(x0)))) -> TOP.0(f_flat.0(down.0(f.0(x0))))
276.49/148.23	   TOP.0(up.0(f.0(f.1(x0)))) -> TOP.0(f_flat.0(down.0(f.1(x0))))
276.49/148.23	   TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(f.0(x0)))))
276.49/148.23	   TOP.0(up.0(g.0(f.0(f.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(f.1(x0)))))
276.49/148.23	   TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0)))))
276.49/148.23	   TOP.0(up.0(g.0(f.0(g.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.1(x0)))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
276.49/148.23	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
276.49/148.23	   down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.)))))))
276.49/148.23	   down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3)))
276.49/148.23	   down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3)))
276.49/148.23	   down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9))))
276.49/148.23	   down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9))))
276.49/148.23	   down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10))))
276.49/148.23	   down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10))))
276.49/148.23	   down.0(g.0(f.1(c.))) -> g_flat.0(down.0(f.1(c.)))
276.49/148.23	   down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.)))
276.49/148.23	   down.0(f.0(f.1(y7))) -> f_flat.0(down.0(f.1(y7)))
276.49/148.23	   down.0(f.0(f.0(y7))) -> f_flat.0(down.0(f.0(y7)))
276.49/148.23	   down.0(f.0(b.)) -> up.0(b.)
276.49/148.23	   down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6)))
276.49/148.23	   down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6)))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down.0(g.0(b.))
276.49/148.23	   down.0(f.0(b.))
276.49/148.23	   down.0(g.0(g.0(x0)))
276.49/148.23	   down.0(g.0(g.1(x0)))
276.49/148.23	   down.0(g.1(c.))
276.49/148.23	   down.0(g.0(fresh_constant.))
276.49/148.23	   down.0(f.0(g.0(x0)))
276.49/148.23	   down.0(f.0(g.1(x0)))
276.49/148.23	   down.0(f.0(f.0(x0)))
276.49/148.23	   down.0(f.0(f.1(x0)))
276.49/148.23	   down.0(f.1(c.))
276.49/148.23	   down.0(f.0(fresh_constant.))
276.49/148.23	   down.0(g.0(f.0(g.0(x0))))
276.49/148.23	   down.0(g.0(f.0(g.1(x0))))
276.49/148.23	   down.0(g.0(f.0(f.0(x0))))
276.49/148.23	   down.0(g.0(f.0(f.1(x0))))
276.49/148.23	   down.0(g.0(f.1(c.)))
276.49/148.23	   down.0(g.0(f.0(fresh_constant.)))
276.49/148.23	   g_flat.0(up.0(x0))
276.49/148.23	   g_flat.0(up.1(x0))
276.49/148.23	   f_flat.0(up.0(x0))
276.49/148.23	   f_flat.0(up.1(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(130) DependencyGraphProof (EQUIVALENT)
276.49/148.23	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(131)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0))))
276.49/148.23	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
276.49/148.23	   TOP.0(up.0(f.0(f.0(x0)))) -> TOP.0(f_flat.0(down.0(f.0(x0))))
276.49/148.23	   TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(f.0(x0)))))
276.49/148.23	   TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0)))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
276.49/148.23	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
276.49/148.23	   down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.)))))))
276.49/148.23	   down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3)))
276.49/148.23	   down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3)))
276.49/148.23	   down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9))))
276.49/148.23	   down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9))))
276.49/148.23	   down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10))))
276.49/148.23	   down.0(g.0(f.0(f.1(y10)))) -> g_flat.0(down.0(f.0(f.1(y10))))
276.49/148.23	   down.0(g.0(f.1(c.))) -> g_flat.0(down.0(f.1(c.)))
276.49/148.23	   down.0(g.0(f.0(fresh_constant.))) -> g_flat.0(down.0(f.0(fresh_constant.)))
276.49/148.23	   down.0(f.0(f.1(y7))) -> f_flat.0(down.0(f.1(y7)))
276.49/148.23	   down.0(f.0(f.0(y7))) -> f_flat.0(down.0(f.0(y7)))
276.49/148.23	   down.0(f.0(b.)) -> up.0(b.)
276.49/148.23	   down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6)))
276.49/148.23	   down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6)))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down.0(g.0(b.))
276.49/148.23	   down.0(f.0(b.))
276.49/148.23	   down.0(g.0(g.0(x0)))
276.49/148.23	   down.0(g.0(g.1(x0)))
276.49/148.23	   down.0(g.1(c.))
276.49/148.23	   down.0(g.0(fresh_constant.))
276.49/148.23	   down.0(f.0(g.0(x0)))
276.49/148.23	   down.0(f.0(g.1(x0)))
276.49/148.23	   down.0(f.0(f.0(x0)))
276.49/148.23	   down.0(f.0(f.1(x0)))
276.49/148.23	   down.0(f.1(c.))
276.49/148.23	   down.0(f.0(fresh_constant.))
276.49/148.23	   down.0(g.0(f.0(g.0(x0))))
276.49/148.23	   down.0(g.0(f.0(g.1(x0))))
276.49/148.23	   down.0(g.0(f.0(f.0(x0))))
276.49/148.23	   down.0(g.0(f.0(f.1(x0))))
276.49/148.23	   down.0(g.0(f.1(c.)))
276.49/148.23	   down.0(g.0(f.0(fresh_constant.)))
276.49/148.23	   g_flat.0(up.0(x0))
276.49/148.23	   g_flat.0(up.1(x0))
276.49/148.23	   f_flat.0(up.0(x0))
276.49/148.23	   f_flat.0(up.1(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(132) PisEmptyProof (SOUND)
276.49/148.23	The TRS P is empty. Hence, there is no (P,Q,R) chain.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(133)
276.49/148.23	TRUE
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(134)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))
276.49/148.23	   TOP(up(g(f(f(x0))))) -> TOP(g_flat(f_flat(down(f(x0)))))
276.49/148.23	   TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0)))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(135) SplitQDPProof (EQUIVALENT)
276.49/148.23	We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(136)
276.49/148.23	Complex Obligation (AND)
276.49/148.23	
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(137)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
276.49/148.23	   TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))
276.49/148.23	   TOP(up(g(f(f(x0))))) -> TOP(g_flat(f_flat(down(f(x0)))))
276.49/148.23	   TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0)))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.23	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.23	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.23	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.23	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.23	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.23	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.23	   down(g(f(fresh_constant))) -> g_flat(down(f(fresh_constant)))
276.49/148.23	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.23	   down(f(b)) -> up(b)
276.49/148.23	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down(g(b))
276.49/148.23	   down(f(b))
276.49/148.23	   down(g(g(x0)))
276.49/148.23	   down(g(c))
276.49/148.23	   down(g(fresh_constant))
276.49/148.23	   down(f(g(x0)))
276.49/148.23	   down(f(f(x0)))
276.49/148.23	   down(f(c))
276.49/148.23	   down(f(fresh_constant))
276.49/148.23	   down(g(f(g(x0))))
276.49/148.23	   down(g(f(f(x0))))
276.49/148.23	   down(g(f(c)))
276.49/148.23	   down(g(f(fresh_constant)))
276.49/148.23	   g_flat(up(x0))
276.49/148.23	   f_flat(up(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(138) SemLabProof (SOUND)
276.49/148.23	We found the following model for the rules of the TRSs R and P.
276.49/148.23	Interpretation over the domain with elements from 0 to 1.
276.49/148.23	b: 0
276.49/148.23	c: 0
276.49/148.23	down: 0
276.49/148.23	f: x<sub>0</sub>
276.49/148.23	fresh_constant: 1
276.49/148.23	up: 0
276.49/148.23	f_flat: 0
276.49/148.23	TOP: 0
276.49/148.23	g_flat: 0
276.49/148.23	g: 0
276.49/148.23	By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(139)
276.49/148.23	Obligation:
276.49/148.23	Q DP problem:
276.49/148.23	The TRS P consists of the following rules:
276.49/148.23	
276.49/148.23	   TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0))))
276.49/148.23	   TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0))))
276.49/148.23	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
276.49/148.23	   TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0))))
276.49/148.23	   TOP.0(up.0(f.0(f.0(x0)))) -> TOP.0(f_flat.0(down.0(f.0(x0))))
276.49/148.23	   TOP.0(up.1(f.1(f.1(x0)))) -> TOP.0(f_flat.0(down.1(f.1(x0))))
276.49/148.23	   TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(f.0(x0)))))
276.49/148.23	   TOP.0(up.0(g.1(f.1(f.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.1(f.1(x0)))))
276.49/148.23	   TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0)))))
276.49/148.23	   TOP.0(up.0(g.0(f.0(g.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.1(x0)))))
276.49/148.23	
276.49/148.23	The TRS R consists of the following rules:
276.49/148.23	
276.49/148.23	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
276.49/148.23	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
276.49/148.23	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
276.49/148.23	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
276.49/148.23	   down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.)))))))
276.49/148.23	   down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3)))
276.49/148.23	   down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3)))
276.49/148.23	   down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9))))
276.49/148.23	   down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9))))
276.49/148.23	   down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10))))
276.49/148.23	   down.0(g.1(f.1(f.1(y10)))) -> g_flat.0(down.1(f.1(f.1(y10))))
276.49/148.23	   down.0(g.0(f.0(c.))) -> g_flat.0(down.0(f.0(c.)))
276.49/148.23	   down.0(g.1(f.1(fresh_constant.))) -> g_flat.0(down.1(f.1(fresh_constant.)))
276.49/148.23	   down.0(f.0(f.0(y7))) -> f_flat.0(down.0(f.0(y7)))
276.49/148.23	   down.1(f.1(f.1(y7))) -> f_flat.0(down.1(f.1(y7)))
276.49/148.23	   down.0(f.0(b.)) -> up.0(b.)
276.49/148.23	   down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6)))
276.49/148.23	   down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6)))
276.49/148.23	
276.49/148.23	The set Q consists of the following terms:
276.49/148.23	
276.49/148.23	   down.0(g.0(b.))
276.49/148.23	   down.0(f.0(b.))
276.49/148.23	   down.0(g.0(g.0(x0)))
276.49/148.23	   down.0(g.0(g.1(x0)))
276.49/148.23	   down.0(g.0(c.))
276.49/148.23	   down.0(g.1(fresh_constant.))
276.49/148.23	   down.0(f.0(g.0(x0)))
276.49/148.23	   down.0(f.0(g.1(x0)))
276.49/148.23	   down.0(f.0(f.0(x0)))
276.49/148.23	   down.1(f.1(f.1(x0)))
276.49/148.23	   down.0(f.0(c.))
276.49/148.23	   down.1(f.1(fresh_constant.))
276.49/148.23	   down.0(g.0(f.0(g.0(x0))))
276.49/148.23	   down.0(g.0(f.0(g.1(x0))))
276.49/148.23	   down.0(g.0(f.0(f.0(x0))))
276.49/148.23	   down.0(g.1(f.1(f.1(x0))))
276.49/148.23	   down.0(g.0(f.0(c.)))
276.49/148.23	   down.0(g.1(f.1(fresh_constant.)))
276.49/148.23	   g_flat.0(up.0(x0))
276.49/148.23	   g_flat.0(up.1(x0))
276.49/148.23	   f_flat.0(up.0(x0))
276.49/148.23	   f_flat.0(up.1(x0))
276.49/148.23	
276.49/148.23	We have to consider all minimal (P,Q,R)-chains.
276.49/148.23	----------------------------------------
276.49/148.23	
276.49/148.23	(140) MRRProof (EQUIVALENT)
276.49/148.23	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.
276.49/148.23	
276.49/148.23	Strictly oriented dependency pairs:
276.49/148.23	
276.49/148.23	   TOP.0(up.1(f.1(f.1(x0)))) -> TOP.0(f_flat.0(down.1(f.1(x0))))
276.49/148.23	   TOP.0(up.0(g.1(f.1(f.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.1(f.1(x0)))))
276.49/148.23	
276.49/148.23	Strictly oriented rules of the TRS R:
276.49/148.23	
276.49/148.23	   down.0(g.1(f.1(f.1(y10)))) -> g_flat.0(down.1(f.1(f.1(y10))))
276.49/148.23	   down.0(g.1(f.1(fresh_constant.))) -> g_flat.0(down.1(f.1(fresh_constant.)))
276.49/148.23	
276.49/148.23	Used ordering: Polynomial interpretation [POLO]:
276.49/148.24	
276.49/148.24	   POL(TOP.0(x_1)) = x_1
276.49/148.24	   POL(b.) = 0
276.49/148.24	   POL(c.) = 0
276.49/148.24	   POL(down.0(x_1)) = x_1
276.49/148.24	   POL(down.1(x_1)) = x_1
276.49/148.24	   POL(f.0(x_1)) = x_1
276.49/148.24	   POL(f.1(x_1)) = x_1
276.49/148.24	   POL(f_flat.0(x_1)) = x_1
276.49/148.24	   POL(fresh_constant.) = 0
276.49/148.24	   POL(g.0(x_1)) = x_1
276.49/148.24	   POL(g.1(x_1)) = 1 + x_1
276.49/148.24	   POL(g_flat.0(x_1)) = x_1
276.49/148.24	   POL(up.0(x_1)) = x_1
276.49/148.24	   POL(up.1(x_1)) = 1 + x_1
276.49/148.24	
276.49/148.24	
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(141)
276.49/148.24	Obligation:
276.49/148.24	Q DP problem:
276.49/148.24	The TRS P consists of the following rules:
276.49/148.24	
276.49/148.24	   TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0))))
276.49/148.24	   TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0))))
276.49/148.24	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
276.49/148.24	   TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0))))
276.49/148.24	   TOP.0(up.0(f.0(f.0(x0)))) -> TOP.0(f_flat.0(down.0(f.0(x0))))
276.49/148.24	   TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(f.0(x0)))))
276.49/148.24	   TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0)))))
276.49/148.24	   TOP.0(up.0(g.0(f.0(g.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.1(x0)))))
276.49/148.24	
276.49/148.24	The TRS R consists of the following rules:
276.49/148.24	
276.49/148.24	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
276.49/148.24	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
276.49/148.24	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
276.49/148.24	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
276.49/148.24	   down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.)))))))
276.49/148.24	   down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3)))
276.49/148.24	   down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3)))
276.49/148.24	   down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9))))
276.49/148.24	   down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9))))
276.49/148.24	   down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10))))
276.49/148.24	   down.0(g.0(f.0(c.))) -> g_flat.0(down.0(f.0(c.)))
276.49/148.24	   down.0(f.0(f.0(y7))) -> f_flat.0(down.0(f.0(y7)))
276.49/148.24	   down.1(f.1(f.1(y7))) -> f_flat.0(down.1(f.1(y7)))
276.49/148.24	   down.0(f.0(b.)) -> up.0(b.)
276.49/148.24	   down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6)))
276.49/148.24	   down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6)))
276.49/148.24	
276.49/148.24	The set Q consists of the following terms:
276.49/148.24	
276.49/148.24	   down.0(g.0(b.))
276.49/148.24	   down.0(f.0(b.))
276.49/148.24	   down.0(g.0(g.0(x0)))
276.49/148.24	   down.0(g.0(g.1(x0)))
276.49/148.24	   down.0(g.0(c.))
276.49/148.24	   down.0(g.1(fresh_constant.))
276.49/148.24	   down.0(f.0(g.0(x0)))
276.49/148.24	   down.0(f.0(g.1(x0)))
276.49/148.24	   down.0(f.0(f.0(x0)))
276.49/148.24	   down.1(f.1(f.1(x0)))
276.49/148.24	   down.0(f.0(c.))
276.49/148.24	   down.1(f.1(fresh_constant.))
276.49/148.24	   down.0(g.0(f.0(g.0(x0))))
276.49/148.24	   down.0(g.0(f.0(g.1(x0))))
276.49/148.24	   down.0(g.0(f.0(f.0(x0))))
276.49/148.24	   down.0(g.1(f.1(f.1(x0))))
276.49/148.24	   down.0(g.0(f.0(c.)))
276.49/148.24	   down.0(g.1(f.1(fresh_constant.)))
276.49/148.24	   g_flat.0(up.0(x0))
276.49/148.24	   g_flat.0(up.1(x0))
276.49/148.24	   f_flat.0(up.0(x0))
276.49/148.24	   f_flat.0(up.1(x0))
276.49/148.24	
276.49/148.24	We have to consider all minimal (P,Q,R)-chains.
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(142) DependencyGraphProof (EQUIVALENT)
276.49/148.24	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(143)
276.49/148.24	Obligation:
276.49/148.24	Q DP problem:
276.49/148.24	The TRS P consists of the following rules:
276.49/148.24	
276.49/148.24	   TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0))))
276.49/148.24	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
276.49/148.24	   TOP.0(up.0(f.0(f.0(x0)))) -> TOP.0(f_flat.0(down.0(f.0(x0))))
276.49/148.24	   TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(f.0(x0)))))
276.49/148.24	   TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0)))))
276.49/148.24	
276.49/148.24	The TRS R consists of the following rules:
276.49/148.24	
276.49/148.24	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
276.49/148.24	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
276.49/148.24	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
276.49/148.24	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
276.49/148.24	   down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.)))))))
276.49/148.24	   down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3)))
276.49/148.24	   down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3)))
276.49/148.24	   down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9))))
276.49/148.24	   down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9))))
276.49/148.24	   down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10))))
276.49/148.24	   down.0(g.0(f.0(c.))) -> g_flat.0(down.0(f.0(c.)))
276.49/148.24	   down.0(f.0(f.0(y7))) -> f_flat.0(down.0(f.0(y7)))
276.49/148.24	   down.1(f.1(f.1(y7))) -> f_flat.0(down.1(f.1(y7)))
276.49/148.24	   down.0(f.0(b.)) -> up.0(b.)
276.49/148.24	   down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6)))
276.49/148.24	   down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6)))
276.49/148.24	
276.49/148.24	The set Q consists of the following terms:
276.49/148.24	
276.49/148.24	   down.0(g.0(b.))
276.49/148.24	   down.0(f.0(b.))
276.49/148.24	   down.0(g.0(g.0(x0)))
276.49/148.24	   down.0(g.0(g.1(x0)))
276.49/148.24	   down.0(g.0(c.))
276.49/148.24	   down.0(g.1(fresh_constant.))
276.49/148.24	   down.0(f.0(g.0(x0)))
276.49/148.24	   down.0(f.0(g.1(x0)))
276.49/148.24	   down.0(f.0(f.0(x0)))
276.49/148.24	   down.1(f.1(f.1(x0)))
276.49/148.24	   down.0(f.0(c.))
276.49/148.24	   down.1(f.1(fresh_constant.))
276.49/148.24	   down.0(g.0(f.0(g.0(x0))))
276.49/148.24	   down.0(g.0(f.0(g.1(x0))))
276.49/148.24	   down.0(g.0(f.0(f.0(x0))))
276.49/148.24	   down.0(g.1(f.1(f.1(x0))))
276.49/148.24	   down.0(g.0(f.0(c.)))
276.49/148.24	   down.0(g.1(f.1(fresh_constant.)))
276.49/148.24	   g_flat.0(up.0(x0))
276.49/148.24	   g_flat.0(up.1(x0))
276.49/148.24	   f_flat.0(up.0(x0))
276.49/148.24	   f_flat.0(up.1(x0))
276.49/148.24	
276.49/148.24	We have to consider all minimal (P,Q,R)-chains.
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(144) UsableRulesReductionPairsProof (EQUIVALENT)
276.49/148.24	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.
276.49/148.24	
276.49/148.24	No dependency pairs are removed.
276.49/148.24	
276.49/148.24	The following rules are removed from R:
276.49/148.24	
276.49/148.24	   down.1(f.1(f.1(y7))) -> f_flat.0(down.1(f.1(y7)))
276.49/148.24	Used ordering: POLO with Polynomial interpretation [POLO]:
276.49/148.24	
276.49/148.24	   POL(TOP.0(x_1)) = x_1
276.49/148.24	   POL(b.) = 0
276.49/148.24	   POL(c.) = 0
276.49/148.24	   POL(down.0(x_1)) = 1 + x_1
276.49/148.24	   POL(f.0(x_1)) = x_1
276.49/148.24	   POL(f.1(x_1)) = x_1
276.49/148.24	   POL(f_flat.0(x_1)) = x_1
276.49/148.24	   POL(g.0(x_1)) = 1 + x_1
276.49/148.24	   POL(g.1(x_1)) = 1 + x_1
276.49/148.24	   POL(g_flat.0(x_1)) = 1 + x_1
276.49/148.24	   POL(up.0(x_1)) = 1 + x_1
276.49/148.24	   POL(up.1(x_1)) = 1 + x_1
276.49/148.24	
276.49/148.24	
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(145)
276.49/148.24	Obligation:
276.49/148.24	Q DP problem:
276.49/148.24	The TRS P consists of the following rules:
276.49/148.24	
276.49/148.24	   TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0))))
276.49/148.24	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
276.49/148.24	   TOP.0(up.0(f.0(f.0(x0)))) -> TOP.0(f_flat.0(down.0(f.0(x0))))
276.49/148.24	   TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(f.0(x0)))))
276.49/148.24	   TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0)))))
276.49/148.24	
276.49/148.24	The TRS R consists of the following rules:
276.49/148.24	
276.49/148.24	   down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.)))))))
276.49/148.24	   down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3)))
276.49/148.24	   down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3)))
276.49/148.24	   down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9))))
276.49/148.24	   down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9))))
276.49/148.24	   down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10))))
276.49/148.24	   down.0(g.0(f.0(c.))) -> g_flat.0(down.0(f.0(c.)))
276.49/148.24	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
276.49/148.24	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
276.49/148.24	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
276.49/148.24	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
276.49/148.24	   down.0(f.0(f.0(y7))) -> f_flat.0(down.0(f.0(y7)))
276.49/148.24	   down.0(f.0(b.)) -> up.0(b.)
276.49/148.24	   down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6)))
276.49/148.24	   down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6)))
276.49/148.24	
276.49/148.24	The set Q consists of the following terms:
276.49/148.24	
276.49/148.24	   down.0(g.0(b.))
276.49/148.24	   down.0(f.0(b.))
276.49/148.24	   down.0(g.0(g.0(x0)))
276.49/148.24	   down.0(g.0(g.1(x0)))
276.49/148.24	   down.0(g.0(c.))
276.49/148.24	   down.0(g.1(fresh_constant.))
276.49/148.24	   down.0(f.0(g.0(x0)))
276.49/148.24	   down.0(f.0(g.1(x0)))
276.49/148.24	   down.0(f.0(f.0(x0)))
276.49/148.24	   down.1(f.1(f.1(x0)))
276.49/148.24	   down.0(f.0(c.))
276.49/148.24	   down.1(f.1(fresh_constant.))
276.49/148.24	   down.0(g.0(f.0(g.0(x0))))
276.49/148.24	   down.0(g.0(f.0(g.1(x0))))
276.49/148.24	   down.0(g.0(f.0(f.0(x0))))
276.49/148.24	   down.0(g.1(f.1(f.1(x0))))
276.49/148.24	   down.0(g.0(f.0(c.)))
276.49/148.24	   down.0(g.1(f.1(fresh_constant.)))
276.49/148.24	   g_flat.0(up.0(x0))
276.49/148.24	   g_flat.0(up.1(x0))
276.49/148.24	   f_flat.0(up.0(x0))
276.49/148.24	   f_flat.0(up.1(x0))
276.49/148.24	
276.49/148.24	We have to consider all minimal (P,Q,R)-chains.
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(146) MRRProof (EQUIVALENT)
276.49/148.24	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.
276.49/148.24	
276.49/148.24	
276.49/148.24	Strictly oriented rules of the TRS R:
276.49/148.24	
276.49/148.24	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
276.49/148.24	
276.49/148.24	Used ordering: Polynomial interpretation [POLO]:
276.49/148.24	
276.49/148.24	   POL(TOP.0(x_1)) = x_1
276.49/148.24	   POL(b.) = 0
276.49/148.24	   POL(c.) = 0
276.49/148.24	   POL(down.0(x_1)) = x_1
276.49/148.24	   POL(f.0(x_1)) = x_1
276.49/148.24	   POL(f.1(x_1)) = x_1
276.49/148.24	   POL(f_flat.0(x_1)) = x_1
276.49/148.24	   POL(g.0(x_1)) = 1 + x_1
276.49/148.24	   POL(g.1(x_1)) = x_1
276.49/148.24	   POL(g_flat.0(x_1)) = 1 + x_1
276.49/148.24	   POL(up.0(x_1)) = x_1
276.49/148.24	   POL(up.1(x_1)) = 1 + x_1
276.49/148.24	
276.49/148.24	
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(147)
276.49/148.24	Obligation:
276.49/148.24	Q DP problem:
276.49/148.24	The TRS P consists of the following rules:
276.49/148.24	
276.49/148.24	   TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0))))
276.49/148.24	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
276.49/148.24	   TOP.0(up.0(f.0(f.0(x0)))) -> TOP.0(f_flat.0(down.0(f.0(x0))))
276.49/148.24	   TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(f.0(x0)))))
276.49/148.24	   TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0)))))
276.49/148.24	
276.49/148.24	The TRS R consists of the following rules:
276.49/148.24	
276.49/148.24	   down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.)))))))
276.49/148.24	   down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3)))
276.49/148.24	   down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3)))
276.49/148.24	   down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9))))
276.49/148.24	   down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9))))
276.49/148.24	   down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10))))
276.49/148.24	   down.0(g.0(f.0(c.))) -> g_flat.0(down.0(f.0(c.)))
276.49/148.24	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
276.49/148.24	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
276.49/148.24	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
276.49/148.24	   down.0(f.0(f.0(y7))) -> f_flat.0(down.0(f.0(y7)))
276.49/148.24	   down.0(f.0(b.)) -> up.0(b.)
276.49/148.24	   down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6)))
276.49/148.24	   down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6)))
276.49/148.24	
276.49/148.24	The set Q consists of the following terms:
276.49/148.24	
276.49/148.24	   down.0(g.0(b.))
276.49/148.24	   down.0(f.0(b.))
276.49/148.24	   down.0(g.0(g.0(x0)))
276.49/148.24	   down.0(g.0(g.1(x0)))
276.49/148.24	   down.0(g.0(c.))
276.49/148.24	   down.0(g.1(fresh_constant.))
276.49/148.24	   down.0(f.0(g.0(x0)))
276.49/148.24	   down.0(f.0(g.1(x0)))
276.49/148.24	   down.0(f.0(f.0(x0)))
276.49/148.24	   down.1(f.1(f.1(x0)))
276.49/148.24	   down.0(f.0(c.))
276.49/148.24	   down.1(f.1(fresh_constant.))
276.49/148.24	   down.0(g.0(f.0(g.0(x0))))
276.49/148.24	   down.0(g.0(f.0(g.1(x0))))
276.49/148.24	   down.0(g.0(f.0(f.0(x0))))
276.49/148.24	   down.0(g.1(f.1(f.1(x0))))
276.49/148.24	   down.0(g.0(f.0(c.)))
276.49/148.24	   down.0(g.1(f.1(fresh_constant.)))
276.49/148.24	   g_flat.0(up.0(x0))
276.49/148.24	   g_flat.0(up.1(x0))
276.49/148.24	   f_flat.0(up.0(x0))
276.49/148.24	   f_flat.0(up.1(x0))
276.49/148.24	
276.49/148.24	We have to consider all minimal (P,Q,R)-chains.
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(148) PisEmptyProof (SOUND)
276.49/148.24	The TRS P is empty. Hence, there is no (P,Q,R) chain.
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(149)
276.49/148.24	TRUE
276.49/148.24	
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(150)
276.49/148.24	Obligation:
276.49/148.24	Q DP problem:
276.49/148.24	The TRS P consists of the following rules:
276.49/148.24	
276.49/148.24	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
276.49/148.24	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
276.49/148.24	   TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))
276.49/148.24	   TOP(up(g(f(f(x0))))) -> TOP(g_flat(f_flat(down(f(x0)))))
276.49/148.24	   TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0)))))
276.49/148.24	
276.49/148.24	The TRS R consists of the following rules:
276.49/148.24	
276.49/148.24	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.24	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.24	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.24	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.24	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.24	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.24	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.24	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.24	   down(f(b)) -> up(b)
276.49/148.24	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.24	
276.49/148.24	The set Q consists of the following terms:
276.49/148.24	
276.49/148.24	   down(g(b))
276.49/148.24	   down(f(b))
276.49/148.24	   down(g(g(x0)))
276.49/148.24	   down(g(c))
276.49/148.24	   down(g(fresh_constant))
276.49/148.24	   down(f(g(x0)))
276.49/148.24	   down(f(f(x0)))
276.49/148.24	   down(f(c))
276.49/148.24	   down(f(fresh_constant))
276.49/148.24	   down(g(f(g(x0))))
276.49/148.24	   down(g(f(f(x0))))
276.49/148.24	   down(g(f(c)))
276.49/148.24	   down(g(f(fresh_constant)))
276.49/148.24	   g_flat(up(x0))
276.49/148.24	   f_flat(up(x0))
276.49/148.24	
276.49/148.24	We have to consider all minimal (P,Q,R)-chains.
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(151) QReductionProof (EQUIVALENT)
276.49/148.24	We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN].
276.49/148.24	
276.49/148.24	   down(g(fresh_constant))
276.49/148.24	   down(f(fresh_constant))
276.49/148.24	   down(g(f(fresh_constant)))
276.49/148.24	
276.49/148.24	
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(152)
276.49/148.24	Obligation:
276.49/148.24	Q DP problem:
276.49/148.24	The TRS P consists of the following rules:
276.49/148.24	
276.49/148.24	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
276.49/148.24	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
276.49/148.24	   TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))
276.49/148.24	   TOP(up(g(f(f(x0))))) -> TOP(g_flat(f_flat(down(f(x0)))))
276.49/148.24	   TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0)))))
276.49/148.24	
276.49/148.24	The TRS R consists of the following rules:
276.49/148.24	
276.49/148.24	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.24	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.24	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.24	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.24	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.24	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.24	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.24	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.24	   down(f(b)) -> up(b)
276.49/148.24	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.24	
276.49/148.24	The set Q consists of the following terms:
276.49/148.24	
276.49/148.24	   down(g(b))
276.49/148.24	   down(f(b))
276.49/148.24	   down(g(g(x0)))
276.49/148.24	   down(g(c))
276.49/148.24	   down(f(g(x0)))
276.49/148.24	   down(f(f(x0)))
276.49/148.24	   down(f(c))
276.49/148.24	   down(g(f(g(x0))))
276.49/148.24	   down(g(f(f(x0))))
276.49/148.24	   down(g(f(c)))
276.49/148.24	   g_flat(up(x0))
276.49/148.24	   f_flat(up(x0))
276.49/148.24	
276.49/148.24	We have to consider all (P,Q,R)-chains.
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(153) SplitQDPProof (EQUIVALENT)
276.49/148.24	We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem
276.49/148.24	
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(154)
276.49/148.24	Complex Obligation (AND)
276.49/148.24	
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(155)
276.49/148.24	Obligation:
276.49/148.24	Q DP problem:
276.49/148.24	The TRS P consists of the following rules:
276.49/148.24	
276.49/148.24	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
276.49/148.24	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
276.49/148.24	   TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))
276.49/148.24	   TOP(up(g(f(f(x0))))) -> TOP(g_flat(f_flat(down(f(x0)))))
276.49/148.24	   TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0)))))
276.49/148.24	
276.49/148.24	The TRS R consists of the following rules:
276.49/148.24	
276.49/148.24	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.24	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.24	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.24	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.24	   down(g(f(c))) -> g_flat(down(f(c)))
276.49/148.24	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.24	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.24	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.24	   down(f(b)) -> up(b)
276.49/148.24	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.24	
276.49/148.24	The set Q consists of the following terms:
276.49/148.24	
276.49/148.24	   down(g(b))
276.49/148.24	   down(f(b))
276.49/148.24	   down(g(g(x0)))
276.49/148.24	   down(g(c))
276.49/148.24	   down(g(fresh_constant))
276.49/148.24	   down(f(g(x0)))
276.49/148.24	   down(f(f(x0)))
276.49/148.24	   down(f(c))
276.49/148.24	   down(f(fresh_constant))
276.49/148.24	   down(g(f(g(x0))))
276.49/148.24	   down(g(f(f(x0))))
276.49/148.24	   down(g(f(c)))
276.49/148.24	   down(g(f(fresh_constant)))
276.49/148.24	   g_flat(up(x0))
276.49/148.24	   f_flat(up(x0))
276.49/148.24	
276.49/148.24	We have to consider all minimal (P,Q,R)-chains.
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(156) SemLabProof (SOUND)
276.49/148.24	We found the following model for the rules of the TRSs R and P.
276.49/148.24	Interpretation over the domain with elements from 0 to 1.
276.49/148.24	b: 0
276.49/148.24	c: 1
276.49/148.24	down: 0
276.49/148.24	f: x<sub>0</sub>
276.49/148.24	fresh_constant: 0
276.49/148.24	up: 0
276.49/148.24	f_flat: 0
276.49/148.24	TOP: 0
276.49/148.24	g_flat: 0
276.49/148.24	g: 0
276.49/148.24	By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(157)
276.49/148.24	Obligation:
276.49/148.24	Q DP problem:
276.49/148.24	The TRS P consists of the following rules:
276.49/148.24	
276.49/148.24	   TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0))))
276.49/148.24	   TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0))))
276.49/148.24	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
276.49/148.24	   TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0))))
276.49/148.24	   TOP.0(up.0(f.0(f.0(x0)))) -> TOP.0(f_flat.0(down.0(f.0(x0))))
276.49/148.24	   TOP.0(up.1(f.1(f.1(x0)))) -> TOP.0(f_flat.0(down.1(f.1(x0))))
276.49/148.24	   TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(f.0(x0)))))
276.49/148.24	   TOP.0(up.0(g.1(f.1(f.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.1(f.1(x0)))))
276.49/148.24	   TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0)))))
276.49/148.24	   TOP.0(up.0(g.0(f.0(g.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.1(x0)))))
276.49/148.24	
276.49/148.24	The TRS R consists of the following rules:
276.49/148.24	
276.49/148.24	   down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.)))))))
276.49/148.24	   down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3)))
276.49/148.24	   down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3)))
276.49/148.24	   down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9))))
276.49/148.24	   down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9))))
276.49/148.24	   down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10))))
276.49/148.24	   down.0(g.1(f.1(f.1(y10)))) -> g_flat.0(down.1(f.1(f.1(y10))))
276.49/148.24	   down.0(g.1(f.1(c.))) -> g_flat.0(down.1(f.1(c.)))
276.49/148.24	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
276.49/148.24	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
276.49/148.24	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
276.49/148.24	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
276.49/148.24	   down.0(f.0(f.0(y7))) -> f_flat.0(down.0(f.0(y7)))
276.49/148.24	   down.1(f.1(f.1(y7))) -> f_flat.0(down.1(f.1(y7)))
276.49/148.24	   down.0(f.0(b.)) -> up.0(b.)
276.49/148.24	   down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6)))
276.49/148.24	   down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6)))
276.49/148.24	
276.49/148.24	The set Q consists of the following terms:
276.49/148.24	
276.49/148.24	   down.0(g.0(b.))
276.49/148.24	   down.0(f.0(b.))
276.49/148.24	   down.0(g.0(g.0(x0)))
276.49/148.24	   down.0(g.0(g.1(x0)))
276.49/148.24	   down.0(g.1(c.))
276.49/148.24	   down.0(g.0(fresh_constant.))
276.49/148.24	   down.0(f.0(g.0(x0)))
276.49/148.24	   down.0(f.0(g.1(x0)))
276.49/148.24	   down.0(f.0(f.0(x0)))
276.49/148.24	   down.1(f.1(f.1(x0)))
276.49/148.24	   down.1(f.1(c.))
276.49/148.24	   down.0(f.0(fresh_constant.))
276.49/148.24	   down.0(g.0(f.0(g.0(x0))))
276.49/148.24	   down.0(g.0(f.0(g.1(x0))))
276.49/148.24	   down.0(g.0(f.0(f.0(x0))))
276.49/148.24	   down.0(g.1(f.1(f.1(x0))))
276.49/148.24	   down.0(g.1(f.1(c.)))
276.49/148.24	   down.0(g.0(f.0(fresh_constant.)))
276.49/148.24	   g_flat.0(up.0(x0))
276.49/148.24	   g_flat.0(up.1(x0))
276.49/148.24	   f_flat.0(up.0(x0))
276.49/148.24	   f_flat.0(up.1(x0))
276.49/148.24	
276.49/148.24	We have to consider all minimal (P,Q,R)-chains.
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(158) MRRProof (EQUIVALENT)
276.49/148.24	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.
276.49/148.24	
276.49/148.24	Strictly oriented dependency pairs:
276.49/148.24	
276.49/148.24	   TOP.0(up.1(f.1(f.1(x0)))) -> TOP.0(f_flat.0(down.1(f.1(x0))))
276.49/148.24	   TOP.0(up.0(g.1(f.1(f.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.1(f.1(x0)))))
276.49/148.24	
276.49/148.24	Strictly oriented rules of the TRS R:
276.49/148.24	
276.49/148.24	   down.0(g.1(f.1(f.1(y10)))) -> g_flat.0(down.1(f.1(f.1(y10))))
276.49/148.24	   down.0(g.1(f.1(c.))) -> g_flat.0(down.1(f.1(c.)))
276.49/148.24	
276.49/148.24	Used ordering: Polynomial interpretation [POLO]:
276.49/148.24	
276.49/148.24	   POL(TOP.0(x_1)) = x_1
276.49/148.24	   POL(b.) = 0
276.49/148.24	   POL(c.) = 0
276.49/148.24	   POL(down.0(x_1)) = 1 + x_1
276.49/148.24	   POL(down.1(x_1)) = x_1
276.49/148.24	   POL(f.0(x_1)) = x_1
276.49/148.24	   POL(f.1(x_1)) = x_1
276.49/148.24	   POL(f_flat.0(x_1)) = x_1
276.49/148.24	   POL(g.0(x_1)) = x_1
276.49/148.24	   POL(g.1(x_1)) = x_1
276.49/148.24	   POL(g_flat.0(x_1)) = x_1
276.49/148.24	   POL(up.0(x_1)) = 1 + x_1
276.49/148.24	   POL(up.1(x_1)) = 1 + x_1
276.49/148.24	
276.49/148.24	
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(159)
276.49/148.24	Obligation:
276.49/148.24	Q DP problem:
276.49/148.24	The TRS P consists of the following rules:
276.49/148.24	
276.49/148.24	   TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0))))
276.49/148.24	   TOP.0(up.0(g.0(g.1(x0)))) -> TOP.0(g_flat.0(down.0(g.1(x0))))
276.49/148.24	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
276.49/148.24	   TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0))))
276.49/148.24	   TOP.0(up.0(f.0(f.0(x0)))) -> TOP.0(f_flat.0(down.0(f.0(x0))))
276.49/148.24	   TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(f.0(x0)))))
276.49/148.24	   TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0)))))
276.49/148.24	   TOP.0(up.0(g.0(f.0(g.1(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.1(x0)))))
276.49/148.24	
276.49/148.24	The TRS R consists of the following rules:
276.49/148.24	
276.49/148.24	   down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.)))))))
276.49/148.24	   down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3)))
276.49/148.24	   down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3)))
276.49/148.24	   down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9))))
276.49/148.24	   down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9))))
276.49/148.24	   down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10))))
276.49/148.24	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
276.49/148.24	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
276.49/148.24	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
276.49/148.24	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
276.49/148.24	   down.0(f.0(f.0(y7))) -> f_flat.0(down.0(f.0(y7)))
276.49/148.24	   down.1(f.1(f.1(y7))) -> f_flat.0(down.1(f.1(y7)))
276.49/148.24	   down.0(f.0(b.)) -> up.0(b.)
276.49/148.24	   down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6)))
276.49/148.24	   down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6)))
276.49/148.24	
276.49/148.24	The set Q consists of the following terms:
276.49/148.24	
276.49/148.24	   down.0(g.0(b.))
276.49/148.24	   down.0(f.0(b.))
276.49/148.24	   down.0(g.0(g.0(x0)))
276.49/148.24	   down.0(g.0(g.1(x0)))
276.49/148.24	   down.0(g.1(c.))
276.49/148.24	   down.0(g.0(fresh_constant.))
276.49/148.24	   down.0(f.0(g.0(x0)))
276.49/148.24	   down.0(f.0(g.1(x0)))
276.49/148.24	   down.0(f.0(f.0(x0)))
276.49/148.24	   down.1(f.1(f.1(x0)))
276.49/148.24	   down.1(f.1(c.))
276.49/148.24	   down.0(f.0(fresh_constant.))
276.49/148.24	   down.0(g.0(f.0(g.0(x0))))
276.49/148.24	   down.0(g.0(f.0(g.1(x0))))
276.49/148.24	   down.0(g.0(f.0(f.0(x0))))
276.49/148.24	   down.0(g.1(f.1(f.1(x0))))
276.49/148.24	   down.0(g.1(f.1(c.)))
276.49/148.24	   down.0(g.0(f.0(fresh_constant.)))
276.49/148.24	   g_flat.0(up.0(x0))
276.49/148.24	   g_flat.0(up.1(x0))
276.49/148.24	   f_flat.0(up.0(x0))
276.49/148.24	   f_flat.0(up.1(x0))
276.49/148.24	
276.49/148.24	We have to consider all minimal (P,Q,R)-chains.
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(160) DependencyGraphProof (EQUIVALENT)
276.49/148.24	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(161)
276.49/148.24	Obligation:
276.49/148.24	Q DP problem:
276.49/148.24	The TRS P consists of the following rules:
276.49/148.24	
276.49/148.24	   TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0))))
276.49/148.24	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
276.49/148.24	   TOP.0(up.0(f.0(f.0(x0)))) -> TOP.0(f_flat.0(down.0(f.0(x0))))
276.49/148.24	   TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(f.0(x0)))))
276.49/148.24	   TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0)))))
276.49/148.24	
276.49/148.24	The TRS R consists of the following rules:
276.49/148.24	
276.49/148.24	   down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.)))))))
276.49/148.24	   down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3)))
276.49/148.24	   down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3)))
276.49/148.24	   down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9))))
276.49/148.24	   down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9))))
276.49/148.24	   down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10))))
276.49/148.24	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
276.49/148.24	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
276.49/148.24	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
276.49/148.24	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
276.49/148.24	   down.0(f.0(f.0(y7))) -> f_flat.0(down.0(f.0(y7)))
276.49/148.24	   down.1(f.1(f.1(y7))) -> f_flat.0(down.1(f.1(y7)))
276.49/148.24	   down.0(f.0(b.)) -> up.0(b.)
276.49/148.24	   down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6)))
276.49/148.24	   down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6)))
276.49/148.24	
276.49/148.24	The set Q consists of the following terms:
276.49/148.24	
276.49/148.24	   down.0(g.0(b.))
276.49/148.24	   down.0(f.0(b.))
276.49/148.24	   down.0(g.0(g.0(x0)))
276.49/148.24	   down.0(g.0(g.1(x0)))
276.49/148.24	   down.0(g.1(c.))
276.49/148.24	   down.0(g.0(fresh_constant.))
276.49/148.24	   down.0(f.0(g.0(x0)))
276.49/148.24	   down.0(f.0(g.1(x0)))
276.49/148.24	   down.0(f.0(f.0(x0)))
276.49/148.24	   down.1(f.1(f.1(x0)))
276.49/148.24	   down.1(f.1(c.))
276.49/148.24	   down.0(f.0(fresh_constant.))
276.49/148.24	   down.0(g.0(f.0(g.0(x0))))
276.49/148.24	   down.0(g.0(f.0(g.1(x0))))
276.49/148.24	   down.0(g.0(f.0(f.0(x0))))
276.49/148.24	   down.0(g.1(f.1(f.1(x0))))
276.49/148.24	   down.0(g.1(f.1(c.)))
276.49/148.24	   down.0(g.0(f.0(fresh_constant.)))
276.49/148.24	   g_flat.0(up.0(x0))
276.49/148.24	   g_flat.0(up.1(x0))
276.49/148.24	   f_flat.0(up.0(x0))
276.49/148.24	   f_flat.0(up.1(x0))
276.49/148.24	
276.49/148.24	We have to consider all minimal (P,Q,R)-chains.
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(162) UsableRulesReductionPairsProof (EQUIVALENT)
276.49/148.24	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.
276.49/148.24	
276.49/148.24	No dependency pairs are removed.
276.49/148.24	
276.49/148.24	The following rules are removed from R:
276.49/148.24	
276.49/148.24	   down.1(f.1(f.1(y7))) -> f_flat.0(down.1(f.1(y7)))
276.49/148.24	Used ordering: POLO with Polynomial interpretation [POLO]:
276.49/148.24	
276.49/148.24	   POL(TOP.0(x_1)) = x_1
276.49/148.24	   POL(b.) = 0
276.49/148.24	   POL(down.0(x_1)) = 1 + x_1
276.49/148.24	   POL(f.0(x_1)) = x_1
276.49/148.24	   POL(f.1(x_1)) = x_1
276.49/148.24	   POL(f_flat.0(x_1)) = x_1
276.49/148.24	   POL(g.0(x_1)) = 1 + x_1
276.49/148.24	   POL(g.1(x_1)) = 1 + x_1
276.49/148.24	   POL(g_flat.0(x_1)) = 1 + x_1
276.49/148.24	   POL(up.0(x_1)) = 1 + x_1
276.49/148.24	   POL(up.1(x_1)) = 1 + x_1
276.49/148.24	
276.49/148.24	
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(163)
276.49/148.24	Obligation:
276.49/148.24	Q DP problem:
276.49/148.24	The TRS P consists of the following rules:
276.49/148.24	
276.49/148.24	   TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0))))
276.49/148.24	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
276.49/148.24	   TOP.0(up.0(f.0(f.0(x0)))) -> TOP.0(f_flat.0(down.0(f.0(x0))))
276.49/148.24	   TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(f.0(x0)))))
276.49/148.24	   TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0)))))
276.49/148.24	
276.49/148.24	The TRS R consists of the following rules:
276.49/148.24	
276.49/148.24	   down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.)))))))
276.49/148.24	   down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3)))
276.49/148.24	   down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3)))
276.49/148.24	   down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9))))
276.49/148.24	   down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9))))
276.49/148.24	   down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10))))
276.49/148.24	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
276.49/148.24	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
276.49/148.24	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
276.49/148.24	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
276.49/148.24	   down.0(f.0(f.0(y7))) -> f_flat.0(down.0(f.0(y7)))
276.49/148.24	   down.0(f.0(b.)) -> up.0(b.)
276.49/148.24	   down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6)))
276.49/148.24	   down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6)))
276.49/148.24	
276.49/148.24	The set Q consists of the following terms:
276.49/148.24	
276.49/148.24	   down.0(g.0(b.))
276.49/148.24	   down.0(f.0(b.))
276.49/148.24	   down.0(g.0(g.0(x0)))
276.49/148.24	   down.0(g.0(g.1(x0)))
276.49/148.24	   down.0(g.1(c.))
276.49/148.24	   down.0(g.0(fresh_constant.))
276.49/148.24	   down.0(f.0(g.0(x0)))
276.49/148.24	   down.0(f.0(g.1(x0)))
276.49/148.24	   down.0(f.0(f.0(x0)))
276.49/148.24	   down.1(f.1(f.1(x0)))
276.49/148.24	   down.1(f.1(c.))
276.49/148.24	   down.0(f.0(fresh_constant.))
276.49/148.24	   down.0(g.0(f.0(g.0(x0))))
276.49/148.24	   down.0(g.0(f.0(g.1(x0))))
276.49/148.24	   down.0(g.0(f.0(f.0(x0))))
276.49/148.24	   down.0(g.1(f.1(f.1(x0))))
276.49/148.24	   down.0(g.1(f.1(c.)))
276.49/148.24	   down.0(g.0(f.0(fresh_constant.)))
276.49/148.24	   g_flat.0(up.0(x0))
276.49/148.24	   g_flat.0(up.1(x0))
276.49/148.24	   f_flat.0(up.0(x0))
276.49/148.24	   f_flat.0(up.1(x0))
276.49/148.24	
276.49/148.24	We have to consider all minimal (P,Q,R)-chains.
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(164) MRRProof (EQUIVALENT)
276.49/148.24	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.
276.49/148.24	
276.49/148.24	
276.49/148.24	Strictly oriented rules of the TRS R:
276.49/148.24	
276.49/148.24	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
276.49/148.24	
276.49/148.24	Used ordering: Polynomial interpretation [POLO]:
276.49/148.24	
276.49/148.24	   POL(TOP.0(x_1)) = x_1
276.49/148.24	   POL(b.) = 0
276.49/148.24	   POL(down.0(x_1)) = x_1
276.49/148.24	   POL(f.0(x_1)) = x_1
276.49/148.24	   POL(f.1(x_1)) = x_1
276.49/148.24	   POL(f_flat.0(x_1)) = x_1
276.49/148.24	   POL(g.0(x_1)) = 1 + x_1
276.49/148.24	   POL(g.1(x_1)) = x_1
276.49/148.24	   POL(g_flat.0(x_1)) = 1 + x_1
276.49/148.24	   POL(up.0(x_1)) = x_1
276.49/148.24	   POL(up.1(x_1)) = 1 + x_1
276.49/148.24	
276.49/148.24	
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(165)
276.49/148.24	Obligation:
276.49/148.24	Q DP problem:
276.49/148.24	The TRS P consists of the following rules:
276.49/148.24	
276.49/148.24	   TOP.0(up.0(g.0(g.0(x0)))) -> TOP.0(g_flat.0(down.0(g.0(x0))))
276.49/148.24	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
276.49/148.24	   TOP.0(up.0(f.0(f.0(x0)))) -> TOP.0(f_flat.0(down.0(f.0(x0))))
276.49/148.24	   TOP.0(up.0(g.0(f.0(f.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(f.0(x0)))))
276.49/148.24	   TOP.0(up.0(g.0(f.0(g.0(x0))))) -> TOP.0(g_flat.0(f_flat.0(down.0(g.0(x0)))))
276.49/148.24	
276.49/148.24	The TRS R consists of the following rules:
276.49/148.24	
276.49/148.24	   down.0(g.0(b.)) -> up.0(g.0(f.0(f.0(f.0(f.0(f.0(b.)))))))
276.49/148.24	   down.0(g.0(g.0(y3))) -> g_flat.0(down.0(g.0(y3)))
276.49/148.24	   down.0(g.0(g.1(y3))) -> g_flat.0(down.0(g.1(y3)))
276.49/148.24	   down.0(g.0(f.0(g.0(y9)))) -> g_flat.0(down.0(f.0(g.0(y9))))
276.49/148.24	   down.0(g.0(f.0(g.1(y9)))) -> g_flat.0(down.0(f.0(g.1(y9))))
276.49/148.24	   down.0(g.0(f.0(f.0(y10)))) -> g_flat.0(down.0(f.0(f.0(y10))))
276.49/148.24	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
276.49/148.24	   f_flat.0(up.1(x_1)) -> up.1(f.1(x_1))
276.49/148.24	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
276.49/148.24	   down.0(f.0(f.0(y7))) -> f_flat.0(down.0(f.0(y7)))
276.49/148.24	   down.0(f.0(b.)) -> up.0(b.)
276.49/148.24	   down.0(f.0(g.0(y6))) -> f_flat.0(down.0(g.0(y6)))
276.49/148.24	   down.0(f.0(g.1(y6))) -> f_flat.0(down.0(g.1(y6)))
276.49/148.24	
276.49/148.24	The set Q consists of the following terms:
276.49/148.24	
276.49/148.24	   down.0(g.0(b.))
276.49/148.24	   down.0(f.0(b.))
276.49/148.24	   down.0(g.0(g.0(x0)))
276.49/148.24	   down.0(g.0(g.1(x0)))
276.49/148.24	   down.0(g.1(c.))
276.49/148.24	   down.0(g.0(fresh_constant.))
276.49/148.24	   down.0(f.0(g.0(x0)))
276.49/148.24	   down.0(f.0(g.1(x0)))
276.49/148.24	   down.0(f.0(f.0(x0)))
276.49/148.24	   down.1(f.1(f.1(x0)))
276.49/148.24	   down.1(f.1(c.))
276.49/148.24	   down.0(f.0(fresh_constant.))
276.49/148.24	   down.0(g.0(f.0(g.0(x0))))
276.49/148.24	   down.0(g.0(f.0(g.1(x0))))
276.49/148.24	   down.0(g.0(f.0(f.0(x0))))
276.49/148.24	   down.0(g.1(f.1(f.1(x0))))
276.49/148.24	   down.0(g.1(f.1(c.)))
276.49/148.24	   down.0(g.0(f.0(fresh_constant.)))
276.49/148.24	   g_flat.0(up.0(x0))
276.49/148.24	   g_flat.0(up.1(x0))
276.49/148.24	   f_flat.0(up.0(x0))
276.49/148.24	   f_flat.0(up.1(x0))
276.49/148.24	
276.49/148.24	We have to consider all minimal (P,Q,R)-chains.
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(166) PisEmptyProof (SOUND)
276.49/148.24	The TRS P is empty. Hence, there is no (P,Q,R) chain.
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(167)
276.49/148.24	TRUE
276.49/148.24	
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(168)
276.49/148.24	Obligation:
276.49/148.24	Q DP problem:
276.49/148.24	The TRS P consists of the following rules:
276.49/148.24	
276.49/148.24	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
276.49/148.24	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
276.49/148.24	   TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))
276.49/148.24	   TOP(up(g(f(f(x0))))) -> TOP(g_flat(f_flat(down(f(x0)))))
276.49/148.24	   TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0)))))
276.49/148.24	
276.49/148.24	The TRS R consists of the following rules:
276.49/148.24	
276.49/148.24	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.24	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.24	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.24	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.24	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.24	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.24	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.24	   down(f(b)) -> up(b)
276.49/148.24	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.24	
276.49/148.24	The set Q consists of the following terms:
276.49/148.24	
276.49/148.24	   down(g(b))
276.49/148.24	   down(f(b))
276.49/148.24	   down(g(g(x0)))
276.49/148.24	   down(g(c))
276.49/148.24	   down(g(fresh_constant))
276.49/148.24	   down(f(g(x0)))
276.49/148.24	   down(f(f(x0)))
276.49/148.24	   down(f(c))
276.49/148.24	   down(f(fresh_constant))
276.49/148.24	   down(g(f(g(x0))))
276.49/148.24	   down(g(f(f(x0))))
276.49/148.24	   down(g(f(c)))
276.49/148.24	   down(g(f(fresh_constant)))
276.49/148.24	   g_flat(up(x0))
276.49/148.24	   f_flat(up(x0))
276.49/148.24	
276.49/148.24	We have to consider all minimal (P,Q,R)-chains.
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(169) QReductionProof (EQUIVALENT)
276.49/148.24	We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN].
276.49/148.24	
276.49/148.24	   down(g(c))
276.49/148.24	   down(g(fresh_constant))
276.49/148.24	   down(f(c))
276.49/148.24	   down(f(fresh_constant))
276.49/148.24	   down(g(f(c)))
276.49/148.24	   down(g(f(fresh_constant)))
276.49/148.24	
276.49/148.24	
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(170)
276.49/148.24	Obligation:
276.49/148.24	Q DP problem:
276.49/148.24	The TRS P consists of the following rules:
276.49/148.24	
276.49/148.24	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
276.49/148.24	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
276.49/148.24	   TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))
276.49/148.24	   TOP(up(g(f(f(x0))))) -> TOP(g_flat(f_flat(down(f(x0)))))
276.49/148.24	   TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0)))))
276.49/148.24	
276.49/148.24	The TRS R consists of the following rules:
276.49/148.24	
276.49/148.24	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.24	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.24	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.24	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.24	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.24	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.24	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.24	   down(f(b)) -> up(b)
276.49/148.24	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.24	
276.49/148.24	The set Q consists of the following terms:
276.49/148.24	
276.49/148.24	   down(g(b))
276.49/148.24	   down(f(b))
276.49/148.24	   down(g(g(x0)))
276.49/148.24	   down(f(g(x0)))
276.49/148.24	   down(f(f(x0)))
276.49/148.24	   down(g(f(g(x0))))
276.49/148.24	   down(g(f(f(x0))))
276.49/148.24	   g_flat(up(x0))
276.49/148.24	   f_flat(up(x0))
276.49/148.24	
276.49/148.24	We have to consider all (P,Q,R)-chains.
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(171) MNOCProof (EQUIVALENT)
276.49/148.24	We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
276.49/148.24	----------------------------------------
276.49/148.24	
276.49/148.24	(172)
276.49/148.24	Obligation:
276.49/148.24	Q DP problem:
276.49/148.24	The TRS P consists of the following rules:
276.49/148.24	
276.49/148.24	   TOP(up(g(g(x0)))) -> TOP(g_flat(down(g(x0))))
276.49/148.24	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
276.49/148.24	   TOP(up(f(f(x0)))) -> TOP(f_flat(down(f(x0))))
276.49/148.24	   TOP(up(g(f(f(x0))))) -> TOP(g_flat(f_flat(down(f(x0)))))
276.49/148.24	   TOP(up(g(f(g(x0))))) -> TOP(g_flat(f_flat(down(g(x0)))))
276.49/148.24	
276.49/148.24	The TRS R consists of the following rules:
276.49/148.24	
276.49/148.24	   down(g(b)) -> up(g(f(f(f(f(f(b)))))))
276.49/148.24	   down(g(g(y3))) -> g_flat(down(g(y3)))
276.49/148.24	   down(g(f(g(y9)))) -> g_flat(down(f(g(y9))))
276.49/148.24	   down(g(f(f(y10)))) -> g_flat(down(f(f(y10))))
276.49/148.24	   f_flat(up(x_1)) -> up(f(x_1))
276.49/148.24	   g_flat(up(x_1)) -> up(g(x_1))
276.49/148.24	   down(f(f(y7))) -> f_flat(down(f(y7)))
276.49/148.24	   down(f(b)) -> up(b)
276.49/148.24	   down(f(g(y6))) -> f_flat(down(g(y6)))
276.49/148.24	
276.49/148.24	Q is empty.
276.49/148.24	We have to consider all (P,Q,R)-chains.
277.79/149.49	EOF