305.07/136.66	MAYBE
305.07/136.67	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
305.07/136.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
305.07/136.67	
305.07/136.67	
305.07/136.67	Outermost Termination of the given OTRS could not be shown:
305.07/136.67	
305.07/136.67	(0) OTRS
305.07/136.67	(1) Trivial-Transformation [SOUND, 0 ms]
305.07/136.67	(2) QTRS
305.07/136.67	    (3) DependencyPairsProof [EQUIVALENT, 0 ms]
305.07/136.67	    (4) QDP
305.07/136.67	    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
305.07/136.67	    (6) QDP
305.07/136.67	    (7) UsableRulesProof [EQUIVALENT, 0 ms]
305.07/136.67	    (8) QDP
305.07/136.67	    (9) NonTerminationLoopProof [COMPLETE, 0 ms]
305.07/136.67	    (10) NO
305.07/136.67	(11) Raffelsieper-Zantema-Transformation [SOUND, 0 ms]
305.07/136.67	(12) QTRS
305.07/136.67	    (13) AAECC Innermost [EQUIVALENT, 0 ms]
305.07/136.67	    (14) QTRS
305.07/136.67	    (15) DependencyPairsProof [EQUIVALENT, 0 ms]
305.07/136.67	    (16) QDP
305.07/136.67	    (17) DependencyGraphProof [EQUIVALENT, 0 ms]
305.07/136.67	    (18) AND
305.07/136.67	        (19) QDP
305.07/136.67	            (20) UsableRulesProof [EQUIVALENT, 0 ms]
305.07/136.67	            (21) QDP
305.07/136.67	            (22) QReductionProof [EQUIVALENT, 0 ms]
305.07/136.67	            (23) QDP
305.07/136.67	            (24) QDPSizeChangeProof [EQUIVALENT, 0 ms]
305.07/136.67	            (25) YES
305.07/136.67	        (26) QDP
305.07/136.67	            (27) UsableRulesProof [EQUIVALENT, 0 ms]
305.07/136.67	            (28) QDP
305.07/136.67	            (29) QReductionProof [EQUIVALENT, 0 ms]
305.07/136.67	            (30) QDP
305.07/136.67	            (31) TransformationProof [EQUIVALENT, 13 ms]
305.07/136.67	            (32) QDP
305.07/136.67	            (33) DependencyGraphProof [EQUIVALENT, 0 ms]
305.07/136.67	            (34) QDP
305.07/136.67	            (35) TransformationProof [EQUIVALENT, 0 ms]
305.07/136.67	            (36) QDP
305.07/136.67	            (37) TransformationProof [EQUIVALENT, 0 ms]
305.07/136.67	            (38) QDP
305.07/136.67	            (39) TransformationProof [EQUIVALENT, 0 ms]
305.07/136.67	            (40) QDP
305.07/136.67	            (41) TransformationProof [EQUIVALENT, 0 ms]
305.07/136.67	            (42) QDP
305.07/136.67	            (43) TransformationProof [EQUIVALENT, 0 ms]
305.07/136.67	            (44) QDP
305.07/136.67	            (45) TransformationProof [EQUIVALENT, 0 ms]
305.07/136.67	            (46) QDP
305.07/136.67	            (47) TransformationProof [EQUIVALENT, 0 ms]
305.07/136.67	            (48) QDP
305.07/136.67	            (49) TransformationProof [EQUIVALENT, 0 ms]
305.07/136.67	            (50) QDP
305.07/136.67	            (51) QDPOrderProof [EQUIVALENT, 24 ms]
305.07/136.67	            (52) QDP
305.07/136.67	            (53) DependencyGraphProof [EQUIVALENT, 0 ms]
305.07/136.67	            (54) QDP
305.07/136.67	            (55) QDPOrderProof [EQUIVALENT, 8 ms]
305.07/136.67	            (56) QDP
305.07/136.67	            (57) QDPOrderProof [EQUIVALENT, 747 ms]
305.07/136.67	            (58) QDP
305.07/136.67	            (59) QDPOrderProof [EQUIVALENT, 904 ms]
305.07/136.67	            (60) QDP
305.07/136.67	            (61) MNOCProof [EQUIVALENT, 0 ms]
305.07/136.67	            (62) QDP
305.07/136.67	            (63) SplitQDPProof [EQUIVALENT, 0 ms]
305.07/136.67	            (64) AND
305.07/136.67	                (65) QDP
305.07/136.67	                    (66) SemLabProof [SOUND, 0 ms]
305.07/136.67	                    (67) QDP
305.07/136.67	                    (68) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
305.07/136.67	                    (69) QDP
305.07/136.67	                    (70) MRRProof [EQUIVALENT, 0 ms]
305.07/136.67	                    (71) QDP
305.07/136.67	                    (72) DependencyGraphProof [EQUIVALENT, 0 ms]
305.07/136.67	                    (73) QDP
305.07/136.67	                    (74) MRRProof [EQUIVALENT, 6 ms]
305.07/136.67	                    (75) QDP
305.07/136.67	                    (76) MRRProof [EQUIVALENT, 0 ms]
305.07/136.67	                    (77) QDP
305.07/136.67	                    (78) PisEmptyProof [SOUND, 0 ms]
305.07/136.67	                    (79) TRUE
305.07/136.67	                (80) QDP
305.07/136.67	                    (81) QReductionProof [EQUIVALENT, 0 ms]
305.07/136.67	                    (82) QDP
305.07/136.67	                        (83) MNOCProof [EQUIVALENT, 0 ms]
305.07/136.67	                        (84) QDP
305.07/136.67	                    (85) SplitQDPProof [EQUIVALENT, 0 ms]
305.07/136.67	                    (86) AND
305.07/136.67	                        (87) QDP
305.07/136.67	                            (88) SemLabProof [SOUND, 0 ms]
305.07/136.67	                            (89) QDP
305.07/136.67	                            (90) MRRProof [EQUIVALENT, 0 ms]
305.07/136.67	                            (91) QDP
305.07/136.67	                            (92) DependencyGraphProof [EQUIVALENT, 0 ms]
305.07/136.67	                            (93) QDP
305.07/136.67	                            (94) PisEmptyProof [SOUND, 0 ms]
305.07/136.67	                            (95) TRUE
305.07/136.67	                        (96) QDP
305.07/136.67	                            (97) QReductionProof [EQUIVALENT, 0 ms]
305.07/136.67	                            (98) QDP
305.07/136.67	(99) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms]
305.07/136.67	(100) QTRS
305.07/136.67	    (101) DependencyPairsProof [EQUIVALENT, 0 ms]
305.07/136.67	    (102) QDP
305.07/136.67	    (103) DependencyGraphProof [EQUIVALENT, 0 ms]
305.07/136.68	    (104) AND
305.07/136.68	        (105) QDP
305.07/136.68	            (106) UsableRulesProof [EQUIVALENT, 0 ms]
305.07/136.68	            (107) QDP
305.07/136.68	            (108) QReductionProof [EQUIVALENT, 0 ms]
305.07/136.68	            (109) QDP
305.07/136.68	            (110) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
305.07/136.68	            (111) QDP
305.07/136.68	            (112) DependencyGraphProof [EQUIVALENT, 0 ms]
305.07/136.68	            (113) QDP
305.07/136.68	            (114) QReductionProof [EQUIVALENT, 1 ms]
305.07/136.68	            (115) QDP
305.07/136.68	            (116) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
305.07/136.68	            (117) QDP
305.07/136.68	            (118) DependencyGraphProof [EQUIVALENT, 0 ms]
305.07/136.68	            (119) TRUE
305.07/136.68	        (120) QDP
305.07/136.68	            (121) UsableRulesProof [EQUIVALENT, 0 ms]
305.07/136.68	            (122) QDP
305.07/136.68	                (123) QReductionProof [EQUIVALENT, 0 ms]
305.07/136.68	                (124) QDP
305.07/136.68	                (125) TransformationProof [EQUIVALENT, 0 ms]
305.07/136.68	                (126) QDP
305.07/136.68	                (127) QDPOrderProof [EQUIVALENT, 0 ms]
305.07/136.68	                (128) QDP
305.07/136.68	            (129) UsableRulesProof [EQUIVALENT, 0 ms]
305.07/136.68	            (130) QDP
305.07/136.68	                (131) QReductionProof [EQUIVALENT, 0 ms]
305.07/136.68	                (132) QDP
305.07/136.68	
305.07/136.68	
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(0)
305.07/136.68	Obligation:
305.07/136.68	Term rewrite system R:
305.07/136.68	The TRS R consists of the following rules:
305.07/136.68	
305.07/136.68	   a -> f(a)
305.07/136.68	   f(f(x)) -> g(x)
305.07/136.68	   g(g(x)) -> h(x)
305.07/136.68	   h(h(x)) -> c
305.07/136.68	
305.07/136.68	
305.07/136.68	
305.07/136.68	Outermost Strategy.
305.07/136.68	
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(1) Trivial-Transformation (SOUND)
305.07/136.68	We applied the Trivial transformation  to transform the outermost TRS to a standard TRS.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(2)
305.07/136.68	Obligation:
305.07/136.68	Q restricted rewrite system:
305.07/136.68	The TRS R consists of the following rules:
305.07/136.68	
305.07/136.68	   a -> f(a)
305.07/136.68	   f(f(x)) -> g(x)
305.07/136.68	   g(g(x)) -> h(x)
305.07/136.68	   h(h(x)) -> c
305.07/136.68	
305.07/136.68	Q is empty.
305.07/136.68	
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(3) DependencyPairsProof (EQUIVALENT)
305.07/136.68	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(4)
305.07/136.68	Obligation:
305.07/136.68	Q DP problem:
305.07/136.68	The TRS P consists of the following rules:
305.07/136.68	
305.07/136.68	   A -> F(a)
305.07/136.68	   A -> A
305.07/136.68	   F(f(x)) -> G(x)
305.07/136.68	   G(g(x)) -> H(x)
305.07/136.68	
305.07/136.68	The TRS R consists of the following rules:
305.07/136.68	
305.07/136.68	   a -> f(a)
305.07/136.68	   f(f(x)) -> g(x)
305.07/136.68	   g(g(x)) -> h(x)
305.07/136.68	   h(h(x)) -> c
305.07/136.68	
305.07/136.68	Q is empty.
305.07/136.68	We have to consider all minimal (P,Q,R)-chains.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(5) DependencyGraphProof (EQUIVALENT)
305.07/136.68	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(6)
305.07/136.68	Obligation:
305.07/136.68	Q DP problem:
305.07/136.68	The TRS P consists of the following rules:
305.07/136.68	
305.07/136.68	   A -> A
305.07/136.68	
305.07/136.68	The TRS R consists of the following rules:
305.07/136.68	
305.07/136.68	   a -> f(a)
305.07/136.68	   f(f(x)) -> g(x)
305.07/136.68	   g(g(x)) -> h(x)
305.07/136.68	   h(h(x)) -> c
305.07/136.68	
305.07/136.68	Q is empty.
305.07/136.68	We have to consider all minimal (P,Q,R)-chains.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(7) UsableRulesProof (EQUIVALENT)
305.07/136.68	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(8)
305.07/136.68	Obligation:
305.07/136.68	Q DP problem:
305.07/136.68	The TRS P consists of the following rules:
305.07/136.68	
305.07/136.68	   A -> A
305.07/136.68	
305.07/136.68	R is empty.
305.07/136.68	Q is empty.
305.07/136.68	We have to consider all minimal (P,Q,R)-chains.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(9) NonTerminationLoopProof (COMPLETE)
305.07/136.68	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
305.07/136.68	Found a loop by semiunifying a rule from P directly.
305.07/136.68	
305.07/136.68	s = A evaluates to  t =A
305.07/136.68	
305.07/136.68	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
305.07/136.68	* Matcher: [ ]
305.07/136.68	* Semiunifier: [ ]
305.07/136.68	
305.07/136.68	--------------------------------------------------------------------------------
305.07/136.68	Rewriting sequence
305.07/136.68	
305.07/136.68	The DP semiunifies directly so there is only one rewrite step from A to A.
305.07/136.68	
305.07/136.68	
305.07/136.68	
305.07/136.68	
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(10)
305.07/136.68	NO
305.07/136.68	
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(11) Raffelsieper-Zantema-Transformation (SOUND)
305.07/136.68	We applied the Raffelsieper-Zantema transformation  to transform the outermost TRS to a standard TRS.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(12)
305.07/136.68	Obligation:
305.07/136.68	Q restricted rewrite system:
305.07/136.68	The TRS R consists of the following rules:
305.07/136.68	
305.07/136.68	   down(a) -> up(f(a))
305.07/136.68	   down(f(f(x))) -> up(g(x))
305.07/136.68	   down(g(g(x))) -> up(h(x))
305.07/136.68	   down(h(h(x))) -> up(c)
305.07/136.68	   top(up(x)) -> top(down(x))
305.07/136.68	   down(f(a)) -> f_flat(down(a))
305.07/136.68	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.68	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.68	   down(f(c)) -> f_flat(down(c))
305.07/136.68	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.68	   down(g(a)) -> g_flat(down(a))
305.07/136.68	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.68	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.68	   down(g(c)) -> g_flat(down(c))
305.07/136.68	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.68	   down(h(a)) -> h_flat(down(a))
305.07/136.68	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.68	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.68	   down(h(c)) -> h_flat(down(c))
305.07/136.68	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.68	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.68	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.68	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.68	
305.07/136.68	Q is empty.
305.07/136.68	
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(13) AAECC Innermost (EQUIVALENT)
305.07/136.68	We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is 
305.07/136.68	   down(f(a)) -> f_flat(down(a))
305.07/136.68	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.68	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.68	   down(f(c)) -> f_flat(down(c))
305.07/136.68	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.68	   down(g(a)) -> g_flat(down(a))
305.07/136.68	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.68	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.68	   down(g(c)) -> g_flat(down(c))
305.07/136.68	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.68	   down(h(a)) -> h_flat(down(a))
305.07/136.68	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.68	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.68	   down(h(c)) -> h_flat(down(c))
305.07/136.68	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.68	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.68	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.68	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.68	   down(a) -> up(f(a))
305.07/136.68	   down(f(f(x))) -> up(g(x))
305.07/136.68	   down(g(g(x))) -> up(h(x))
305.07/136.68	   down(h(h(x))) -> up(c)
305.07/136.68	
305.07/136.68	The TRS R 2 is 
305.07/136.68	   top(up(x)) -> top(down(x))
305.07/136.68	
305.07/136.68	The signature Sigma is {top_1}
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(14)
305.07/136.68	Obligation:
305.07/136.68	Q restricted rewrite system:
305.07/136.68	The TRS R consists of the following rules:
305.07/136.68	
305.07/136.68	   down(a) -> up(f(a))
305.07/136.68	   down(f(f(x))) -> up(g(x))
305.07/136.68	   down(g(g(x))) -> up(h(x))
305.07/136.68	   down(h(h(x))) -> up(c)
305.07/136.68	   top(up(x)) -> top(down(x))
305.07/136.68	   down(f(a)) -> f_flat(down(a))
305.07/136.68	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.68	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.68	   down(f(c)) -> f_flat(down(c))
305.07/136.68	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.68	   down(g(a)) -> g_flat(down(a))
305.07/136.68	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.68	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.68	   down(g(c)) -> g_flat(down(c))
305.07/136.68	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.68	   down(h(a)) -> h_flat(down(a))
305.07/136.68	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.68	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.68	   down(h(c)) -> h_flat(down(c))
305.07/136.68	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.68	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.68	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.68	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.68	
305.07/136.68	The set Q consists of the following terms:
305.07/136.68	
305.07/136.68	   down(a)
305.07/136.68	   down(f(f(x0)))
305.07/136.68	   down(g(g(x0)))
305.07/136.68	   down(h(h(x0)))
305.07/136.68	   top(up(x0))
305.07/136.68	   down(f(a))
305.07/136.68	   down(f(g(x0)))
305.07/136.68	   down(f(h(x0)))
305.07/136.68	   down(f(c))
305.07/136.68	   down(f(fresh_constant))
305.07/136.68	   down(g(a))
305.07/136.68	   down(g(f(x0)))
305.07/136.68	   down(g(h(x0)))
305.07/136.68	   down(g(c))
305.07/136.68	   down(g(fresh_constant))
305.07/136.68	   down(h(a))
305.07/136.68	   down(h(f(x0)))
305.07/136.68	   down(h(g(x0)))
305.07/136.68	   down(h(c))
305.07/136.68	   down(h(fresh_constant))
305.07/136.68	   f_flat(up(x0))
305.07/136.68	   g_flat(up(x0))
305.07/136.68	   h_flat(up(x0))
305.07/136.68	
305.07/136.68	
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(15) DependencyPairsProof (EQUIVALENT)
305.07/136.68	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(16)
305.07/136.68	Obligation:
305.07/136.68	Q DP problem:
305.07/136.68	The TRS P consists of the following rules:
305.07/136.68	
305.07/136.68	   TOP(up(x)) -> TOP(down(x))
305.07/136.68	   TOP(up(x)) -> DOWN(x)
305.07/136.68	   DOWN(f(a)) -> F_FLAT(down(a))
305.07/136.68	   DOWN(f(a)) -> DOWN(a)
305.07/136.68	   DOWN(f(g(y5))) -> F_FLAT(down(g(y5)))
305.07/136.68	   DOWN(f(g(y5))) -> DOWN(g(y5))
305.07/136.68	   DOWN(f(h(y6))) -> F_FLAT(down(h(y6)))
305.07/136.68	   DOWN(f(h(y6))) -> DOWN(h(y6))
305.07/136.68	   DOWN(f(c)) -> F_FLAT(down(c))
305.07/136.68	   DOWN(f(c)) -> DOWN(c)
305.07/136.68	   DOWN(f(fresh_constant)) -> F_FLAT(down(fresh_constant))
305.07/136.68	   DOWN(f(fresh_constant)) -> DOWN(fresh_constant)
305.07/136.68	   DOWN(g(a)) -> G_FLAT(down(a))
305.07/136.68	   DOWN(g(a)) -> DOWN(a)
305.07/136.68	   DOWN(g(f(y8))) -> G_FLAT(down(f(y8)))
305.07/136.68	   DOWN(g(f(y8))) -> DOWN(f(y8))
305.07/136.68	   DOWN(g(h(y10))) -> G_FLAT(down(h(y10)))
305.07/136.68	   DOWN(g(h(y10))) -> DOWN(h(y10))
305.07/136.68	   DOWN(g(c)) -> G_FLAT(down(c))
305.07/136.68	   DOWN(g(c)) -> DOWN(c)
305.07/136.68	   DOWN(g(fresh_constant)) -> G_FLAT(down(fresh_constant))
305.07/136.68	   DOWN(g(fresh_constant)) -> DOWN(fresh_constant)
305.07/136.68	   DOWN(h(a)) -> H_FLAT(down(a))
305.07/136.68	   DOWN(h(a)) -> DOWN(a)
305.07/136.68	   DOWN(h(f(y12))) -> H_FLAT(down(f(y12)))
305.07/136.68	   DOWN(h(f(y12))) -> DOWN(f(y12))
305.07/136.68	   DOWN(h(g(y13))) -> H_FLAT(down(g(y13)))
305.07/136.68	   DOWN(h(g(y13))) -> DOWN(g(y13))
305.07/136.68	   DOWN(h(c)) -> H_FLAT(down(c))
305.07/136.68	   DOWN(h(c)) -> DOWN(c)
305.07/136.68	   DOWN(h(fresh_constant)) -> H_FLAT(down(fresh_constant))
305.07/136.68	   DOWN(h(fresh_constant)) -> DOWN(fresh_constant)
305.07/136.68	
305.07/136.68	The TRS R consists of the following rules:
305.07/136.68	
305.07/136.68	   down(a) -> up(f(a))
305.07/136.68	   down(f(f(x))) -> up(g(x))
305.07/136.68	   down(g(g(x))) -> up(h(x))
305.07/136.68	   down(h(h(x))) -> up(c)
305.07/136.68	   top(up(x)) -> top(down(x))
305.07/136.68	   down(f(a)) -> f_flat(down(a))
305.07/136.68	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.68	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.68	   down(f(c)) -> f_flat(down(c))
305.07/136.68	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.68	   down(g(a)) -> g_flat(down(a))
305.07/136.68	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.68	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.68	   down(g(c)) -> g_flat(down(c))
305.07/136.68	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.68	   down(h(a)) -> h_flat(down(a))
305.07/136.68	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.68	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.68	   down(h(c)) -> h_flat(down(c))
305.07/136.68	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.68	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.68	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.68	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.68	
305.07/136.68	The set Q consists of the following terms:
305.07/136.68	
305.07/136.68	   down(a)
305.07/136.68	   down(f(f(x0)))
305.07/136.68	   down(g(g(x0)))
305.07/136.68	   down(h(h(x0)))
305.07/136.68	   top(up(x0))
305.07/136.68	   down(f(a))
305.07/136.68	   down(f(g(x0)))
305.07/136.68	   down(f(h(x0)))
305.07/136.68	   down(f(c))
305.07/136.68	   down(f(fresh_constant))
305.07/136.68	   down(g(a))
305.07/136.68	   down(g(f(x0)))
305.07/136.68	   down(g(h(x0)))
305.07/136.68	   down(g(c))
305.07/136.68	   down(g(fresh_constant))
305.07/136.68	   down(h(a))
305.07/136.68	   down(h(f(x0)))
305.07/136.68	   down(h(g(x0)))
305.07/136.68	   down(h(c))
305.07/136.68	   down(h(fresh_constant))
305.07/136.68	   f_flat(up(x0))
305.07/136.68	   g_flat(up(x0))
305.07/136.68	   h_flat(up(x0))
305.07/136.68	
305.07/136.68	We have to consider all minimal (P,Q,R)-chains.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(17) DependencyGraphProof (EQUIVALENT)
305.07/136.68	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 25 less nodes.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(18)
305.07/136.68	Complex Obligation (AND)
305.07/136.68	
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(19)
305.07/136.68	Obligation:
305.07/136.68	Q DP problem:
305.07/136.68	The TRS P consists of the following rules:
305.07/136.68	
305.07/136.68	   DOWN(f(g(y5))) -> DOWN(g(y5))
305.07/136.68	   DOWN(g(f(y8))) -> DOWN(f(y8))
305.07/136.68	   DOWN(f(h(y6))) -> DOWN(h(y6))
305.07/136.68	   DOWN(h(f(y12))) -> DOWN(f(y12))
305.07/136.68	   DOWN(h(g(y13))) -> DOWN(g(y13))
305.07/136.68	   DOWN(g(h(y10))) -> DOWN(h(y10))
305.07/136.68	
305.07/136.68	The TRS R consists of the following rules:
305.07/136.68	
305.07/136.68	   down(a) -> up(f(a))
305.07/136.68	   down(f(f(x))) -> up(g(x))
305.07/136.68	   down(g(g(x))) -> up(h(x))
305.07/136.68	   down(h(h(x))) -> up(c)
305.07/136.68	   top(up(x)) -> top(down(x))
305.07/136.68	   down(f(a)) -> f_flat(down(a))
305.07/136.68	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.68	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.68	   down(f(c)) -> f_flat(down(c))
305.07/136.68	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.68	   down(g(a)) -> g_flat(down(a))
305.07/136.68	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.68	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.68	   down(g(c)) -> g_flat(down(c))
305.07/136.68	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.68	   down(h(a)) -> h_flat(down(a))
305.07/136.68	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.68	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.68	   down(h(c)) -> h_flat(down(c))
305.07/136.68	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.68	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.68	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.68	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.68	
305.07/136.68	The set Q consists of the following terms:
305.07/136.68	
305.07/136.68	   down(a)
305.07/136.68	   down(f(f(x0)))
305.07/136.68	   down(g(g(x0)))
305.07/136.68	   down(h(h(x0)))
305.07/136.68	   top(up(x0))
305.07/136.68	   down(f(a))
305.07/136.68	   down(f(g(x0)))
305.07/136.68	   down(f(h(x0)))
305.07/136.68	   down(f(c))
305.07/136.68	   down(f(fresh_constant))
305.07/136.68	   down(g(a))
305.07/136.68	   down(g(f(x0)))
305.07/136.68	   down(g(h(x0)))
305.07/136.68	   down(g(c))
305.07/136.68	   down(g(fresh_constant))
305.07/136.68	   down(h(a))
305.07/136.68	   down(h(f(x0)))
305.07/136.68	   down(h(g(x0)))
305.07/136.68	   down(h(c))
305.07/136.68	   down(h(fresh_constant))
305.07/136.68	   f_flat(up(x0))
305.07/136.68	   g_flat(up(x0))
305.07/136.68	   h_flat(up(x0))
305.07/136.68	
305.07/136.68	We have to consider all minimal (P,Q,R)-chains.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(20) UsableRulesProof (EQUIVALENT)
305.07/136.68	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.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(21)
305.07/136.68	Obligation:
305.07/136.68	Q DP problem:
305.07/136.68	The TRS P consists of the following rules:
305.07/136.68	
305.07/136.68	   DOWN(f(g(y5))) -> DOWN(g(y5))
305.07/136.68	   DOWN(g(f(y8))) -> DOWN(f(y8))
305.07/136.68	   DOWN(f(h(y6))) -> DOWN(h(y6))
305.07/136.68	   DOWN(h(f(y12))) -> DOWN(f(y12))
305.07/136.68	   DOWN(h(g(y13))) -> DOWN(g(y13))
305.07/136.68	   DOWN(g(h(y10))) -> DOWN(h(y10))
305.07/136.68	
305.07/136.68	R is empty.
305.07/136.68	The set Q consists of the following terms:
305.07/136.68	
305.07/136.68	   down(a)
305.07/136.68	   down(f(f(x0)))
305.07/136.68	   down(g(g(x0)))
305.07/136.68	   down(h(h(x0)))
305.07/136.68	   top(up(x0))
305.07/136.68	   down(f(a))
305.07/136.68	   down(f(g(x0)))
305.07/136.68	   down(f(h(x0)))
305.07/136.68	   down(f(c))
305.07/136.68	   down(f(fresh_constant))
305.07/136.68	   down(g(a))
305.07/136.68	   down(g(f(x0)))
305.07/136.68	   down(g(h(x0)))
305.07/136.68	   down(g(c))
305.07/136.68	   down(g(fresh_constant))
305.07/136.68	   down(h(a))
305.07/136.68	   down(h(f(x0)))
305.07/136.68	   down(h(g(x0)))
305.07/136.68	   down(h(c))
305.07/136.68	   down(h(fresh_constant))
305.07/136.68	   f_flat(up(x0))
305.07/136.68	   g_flat(up(x0))
305.07/136.68	   h_flat(up(x0))
305.07/136.68	
305.07/136.68	We have to consider all minimal (P,Q,R)-chains.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(22) QReductionProof (EQUIVALENT)
305.07/136.68	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
305.07/136.68	
305.07/136.68	   down(a)
305.07/136.68	   down(f(f(x0)))
305.07/136.68	   down(g(g(x0)))
305.07/136.68	   down(h(h(x0)))
305.07/136.68	   top(up(x0))
305.07/136.68	   down(f(a))
305.07/136.68	   down(f(g(x0)))
305.07/136.68	   down(f(h(x0)))
305.07/136.68	   down(f(c))
305.07/136.68	   down(f(fresh_constant))
305.07/136.68	   down(g(a))
305.07/136.68	   down(g(f(x0)))
305.07/136.68	   down(g(h(x0)))
305.07/136.68	   down(g(c))
305.07/136.68	   down(g(fresh_constant))
305.07/136.68	   down(h(a))
305.07/136.68	   down(h(f(x0)))
305.07/136.68	   down(h(g(x0)))
305.07/136.68	   down(h(c))
305.07/136.68	   down(h(fresh_constant))
305.07/136.68	   f_flat(up(x0))
305.07/136.68	   g_flat(up(x0))
305.07/136.68	   h_flat(up(x0))
305.07/136.68	
305.07/136.68	
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(23)
305.07/136.68	Obligation:
305.07/136.68	Q DP problem:
305.07/136.68	The TRS P consists of the following rules:
305.07/136.68	
305.07/136.68	   DOWN(f(g(y5))) -> DOWN(g(y5))
305.07/136.68	   DOWN(g(f(y8))) -> DOWN(f(y8))
305.07/136.68	   DOWN(f(h(y6))) -> DOWN(h(y6))
305.07/136.68	   DOWN(h(f(y12))) -> DOWN(f(y12))
305.07/136.68	   DOWN(h(g(y13))) -> DOWN(g(y13))
305.07/136.68	   DOWN(g(h(y10))) -> DOWN(h(y10))
305.07/136.68	
305.07/136.68	R is empty.
305.07/136.68	Q is empty.
305.07/136.68	We have to consider all minimal (P,Q,R)-chains.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(24) QDPSizeChangeProof (EQUIVALENT)
305.07/136.68	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. 
305.07/136.68	
305.07/136.68	From the DPs we obtained the following set of size-change graphs:
305.07/136.68	*DOWN(g(f(y8))) -> DOWN(f(y8))
305.07/136.68	The graph contains the following edges 1 > 1
305.07/136.68	
305.07/136.68	
305.07/136.68	*DOWN(g(h(y10))) -> DOWN(h(y10))
305.07/136.68	The graph contains the following edges 1 > 1
305.07/136.68	
305.07/136.68	
305.07/136.68	*DOWN(h(f(y12))) -> DOWN(f(y12))
305.07/136.68	The graph contains the following edges 1 > 1
305.07/136.68	
305.07/136.68	
305.07/136.68	*DOWN(f(g(y5))) -> DOWN(g(y5))
305.07/136.68	The graph contains the following edges 1 > 1
305.07/136.68	
305.07/136.68	
305.07/136.68	*DOWN(f(h(y6))) -> DOWN(h(y6))
305.07/136.68	The graph contains the following edges 1 > 1
305.07/136.68	
305.07/136.68	
305.07/136.68	*DOWN(h(g(y13))) -> DOWN(g(y13))
305.07/136.68	The graph contains the following edges 1 > 1
305.07/136.68	
305.07/136.68	
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(25)
305.07/136.68	YES
305.07/136.68	
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(26)
305.07/136.68	Obligation:
305.07/136.68	Q DP problem:
305.07/136.68	The TRS P consists of the following rules:
305.07/136.68	
305.07/136.68	   TOP(up(x)) -> TOP(down(x))
305.07/136.68	
305.07/136.68	The TRS R consists of the following rules:
305.07/136.68	
305.07/136.68	   down(a) -> up(f(a))
305.07/136.68	   down(f(f(x))) -> up(g(x))
305.07/136.68	   down(g(g(x))) -> up(h(x))
305.07/136.68	   down(h(h(x))) -> up(c)
305.07/136.68	   top(up(x)) -> top(down(x))
305.07/136.68	   down(f(a)) -> f_flat(down(a))
305.07/136.68	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.68	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.68	   down(f(c)) -> f_flat(down(c))
305.07/136.68	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.68	   down(g(a)) -> g_flat(down(a))
305.07/136.68	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.68	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.68	   down(g(c)) -> g_flat(down(c))
305.07/136.68	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.68	   down(h(a)) -> h_flat(down(a))
305.07/136.68	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.68	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.68	   down(h(c)) -> h_flat(down(c))
305.07/136.68	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.68	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.68	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.68	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.68	
305.07/136.68	The set Q consists of the following terms:
305.07/136.68	
305.07/136.68	   down(a)
305.07/136.68	   down(f(f(x0)))
305.07/136.68	   down(g(g(x0)))
305.07/136.68	   down(h(h(x0)))
305.07/136.68	   top(up(x0))
305.07/136.68	   down(f(a))
305.07/136.68	   down(f(g(x0)))
305.07/136.68	   down(f(h(x0)))
305.07/136.68	   down(f(c))
305.07/136.68	   down(f(fresh_constant))
305.07/136.68	   down(g(a))
305.07/136.68	   down(g(f(x0)))
305.07/136.68	   down(g(h(x0)))
305.07/136.68	   down(g(c))
305.07/136.68	   down(g(fresh_constant))
305.07/136.68	   down(h(a))
305.07/136.68	   down(h(f(x0)))
305.07/136.68	   down(h(g(x0)))
305.07/136.68	   down(h(c))
305.07/136.68	   down(h(fresh_constant))
305.07/136.68	   f_flat(up(x0))
305.07/136.68	   g_flat(up(x0))
305.07/136.68	   h_flat(up(x0))
305.07/136.68	
305.07/136.68	We have to consider all minimal (P,Q,R)-chains.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(27) UsableRulesProof (EQUIVALENT)
305.07/136.68	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.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(28)
305.07/136.68	Obligation:
305.07/136.68	Q DP problem:
305.07/136.68	The TRS P consists of the following rules:
305.07/136.68	
305.07/136.68	   TOP(up(x)) -> TOP(down(x))
305.07/136.68	
305.07/136.68	The TRS R consists of the following rules:
305.07/136.68	
305.07/136.68	   down(a) -> up(f(a))
305.07/136.68	   down(f(f(x))) -> up(g(x))
305.07/136.68	   down(g(g(x))) -> up(h(x))
305.07/136.68	   down(h(h(x))) -> up(c)
305.07/136.68	   down(f(a)) -> f_flat(down(a))
305.07/136.68	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.68	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.68	   down(f(c)) -> f_flat(down(c))
305.07/136.68	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.68	   down(g(a)) -> g_flat(down(a))
305.07/136.68	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.68	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.68	   down(g(c)) -> g_flat(down(c))
305.07/136.68	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.68	   down(h(a)) -> h_flat(down(a))
305.07/136.68	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.68	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.68	   down(h(c)) -> h_flat(down(c))
305.07/136.68	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.68	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.68	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.68	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.68	
305.07/136.68	The set Q consists of the following terms:
305.07/136.68	
305.07/136.68	   down(a)
305.07/136.68	   down(f(f(x0)))
305.07/136.68	   down(g(g(x0)))
305.07/136.68	   down(h(h(x0)))
305.07/136.68	   top(up(x0))
305.07/136.68	   down(f(a))
305.07/136.68	   down(f(g(x0)))
305.07/136.68	   down(f(h(x0)))
305.07/136.68	   down(f(c))
305.07/136.68	   down(f(fresh_constant))
305.07/136.68	   down(g(a))
305.07/136.68	   down(g(f(x0)))
305.07/136.68	   down(g(h(x0)))
305.07/136.68	   down(g(c))
305.07/136.68	   down(g(fresh_constant))
305.07/136.68	   down(h(a))
305.07/136.68	   down(h(f(x0)))
305.07/136.68	   down(h(g(x0)))
305.07/136.68	   down(h(c))
305.07/136.68	   down(h(fresh_constant))
305.07/136.68	   f_flat(up(x0))
305.07/136.68	   g_flat(up(x0))
305.07/136.68	   h_flat(up(x0))
305.07/136.68	
305.07/136.68	We have to consider all minimal (P,Q,R)-chains.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(29) QReductionProof (EQUIVALENT)
305.07/136.68	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
305.07/136.68	
305.07/136.68	   top(up(x0))
305.07/136.68	
305.07/136.68	
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(30)
305.07/136.68	Obligation:
305.07/136.68	Q DP problem:
305.07/136.68	The TRS P consists of the following rules:
305.07/136.68	
305.07/136.68	   TOP(up(x)) -> TOP(down(x))
305.07/136.68	
305.07/136.68	The TRS R consists of the following rules:
305.07/136.68	
305.07/136.68	   down(a) -> up(f(a))
305.07/136.68	   down(f(f(x))) -> up(g(x))
305.07/136.68	   down(g(g(x))) -> up(h(x))
305.07/136.68	   down(h(h(x))) -> up(c)
305.07/136.68	   down(f(a)) -> f_flat(down(a))
305.07/136.68	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.68	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.68	   down(f(c)) -> f_flat(down(c))
305.07/136.68	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.68	   down(g(a)) -> g_flat(down(a))
305.07/136.68	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.68	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.68	   down(g(c)) -> g_flat(down(c))
305.07/136.68	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.68	   down(h(a)) -> h_flat(down(a))
305.07/136.68	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.68	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.68	   down(h(c)) -> h_flat(down(c))
305.07/136.68	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.68	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.68	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.68	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.68	
305.07/136.68	The set Q consists of the following terms:
305.07/136.68	
305.07/136.68	   down(a)
305.07/136.68	   down(f(f(x0)))
305.07/136.68	   down(g(g(x0)))
305.07/136.68	   down(h(h(x0)))
305.07/136.68	   down(f(a))
305.07/136.68	   down(f(g(x0)))
305.07/136.68	   down(f(h(x0)))
305.07/136.68	   down(f(c))
305.07/136.68	   down(f(fresh_constant))
305.07/136.68	   down(g(a))
305.07/136.68	   down(g(f(x0)))
305.07/136.68	   down(g(h(x0)))
305.07/136.68	   down(g(c))
305.07/136.68	   down(g(fresh_constant))
305.07/136.68	   down(h(a))
305.07/136.68	   down(h(f(x0)))
305.07/136.68	   down(h(g(x0)))
305.07/136.68	   down(h(c))
305.07/136.68	   down(h(fresh_constant))
305.07/136.68	   f_flat(up(x0))
305.07/136.68	   g_flat(up(x0))
305.07/136.68	   h_flat(up(x0))
305.07/136.68	
305.07/136.68	We have to consider all minimal (P,Q,R)-chains.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(31) TransformationProof (EQUIVALENT)
305.07/136.68	By narrowing [LPAR04] the rule TOP(up(x)) -> TOP(down(x)) at position [0] we obtained the following new rules [LPAR04]:
305.07/136.68	
305.07/136.68	   (TOP(up(a)) -> TOP(up(f(a))),TOP(up(a)) -> TOP(up(f(a))))
305.07/136.68	   (TOP(up(f(f(x0)))) -> TOP(up(g(x0))),TOP(up(f(f(x0)))) -> TOP(up(g(x0))))
305.07/136.68	   (TOP(up(g(g(x0)))) -> TOP(up(h(x0))),TOP(up(g(g(x0)))) -> TOP(up(h(x0))))
305.07/136.68	   (TOP(up(h(h(x0)))) -> TOP(up(c)),TOP(up(h(h(x0)))) -> TOP(up(c)))
305.07/136.68	   (TOP(up(f(a))) -> TOP(f_flat(down(a))),TOP(up(f(a))) -> TOP(f_flat(down(a))))
305.07/136.68	   (TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))),TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))))
305.07/136.68	   (TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0)))),TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0)))))
305.07/136.68	   (TOP(up(f(c))) -> TOP(f_flat(down(c))),TOP(up(f(c))) -> TOP(f_flat(down(c))))
305.07/136.68	   (TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))),TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))))
305.07/136.68	   (TOP(up(g(a))) -> TOP(g_flat(down(a))),TOP(up(g(a))) -> TOP(g_flat(down(a))))
305.07/136.68	   (TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))),TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))))
305.07/136.68	   (TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0)))),TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0)))))
305.07/136.68	   (TOP(up(g(c))) -> TOP(g_flat(down(c))),TOP(up(g(c))) -> TOP(g_flat(down(c))))
305.07/136.68	   (TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))),TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))))
305.07/136.68	   (TOP(up(h(a))) -> TOP(h_flat(down(a))),TOP(up(h(a))) -> TOP(h_flat(down(a))))
305.07/136.68	   (TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0)))),TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0)))))
305.07/136.68	   (TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0)))),TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0)))))
305.07/136.68	   (TOP(up(h(c))) -> TOP(h_flat(down(c))),TOP(up(h(c))) -> TOP(h_flat(down(c))))
305.07/136.68	   (TOP(up(h(fresh_constant))) -> TOP(h_flat(down(fresh_constant))),TOP(up(h(fresh_constant))) -> TOP(h_flat(down(fresh_constant))))
305.07/136.68	
305.07/136.68	
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(32)
305.07/136.68	Obligation:
305.07/136.68	Q DP problem:
305.07/136.68	The TRS P consists of the following rules:
305.07/136.68	
305.07/136.68	   TOP(up(a)) -> TOP(up(f(a)))
305.07/136.68	   TOP(up(f(f(x0)))) -> TOP(up(g(x0)))
305.07/136.68	   TOP(up(g(g(x0)))) -> TOP(up(h(x0)))
305.07/136.68	   TOP(up(h(h(x0)))) -> TOP(up(c))
305.07/136.68	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
305.07/136.68	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.07/136.68	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.07/136.68	   TOP(up(f(c))) -> TOP(f_flat(down(c)))
305.07/136.68	   TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant)))
305.07/136.68	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
305.07/136.68	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.07/136.68	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.07/136.68	   TOP(up(g(c))) -> TOP(g_flat(down(c)))
305.07/136.68	   TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant)))
305.07/136.68	   TOP(up(h(a))) -> TOP(h_flat(down(a)))
305.07/136.68	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.07/136.68	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.07/136.68	   TOP(up(h(c))) -> TOP(h_flat(down(c)))
305.07/136.68	   TOP(up(h(fresh_constant))) -> TOP(h_flat(down(fresh_constant)))
305.07/136.68	
305.07/136.68	The TRS R consists of the following rules:
305.07/136.68	
305.07/136.68	   down(a) -> up(f(a))
305.07/136.68	   down(f(f(x))) -> up(g(x))
305.07/136.68	   down(g(g(x))) -> up(h(x))
305.07/136.68	   down(h(h(x))) -> up(c)
305.07/136.68	   down(f(a)) -> f_flat(down(a))
305.07/136.68	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.68	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.68	   down(f(c)) -> f_flat(down(c))
305.07/136.68	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.68	   down(g(a)) -> g_flat(down(a))
305.07/136.68	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.68	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.68	   down(g(c)) -> g_flat(down(c))
305.07/136.68	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.68	   down(h(a)) -> h_flat(down(a))
305.07/136.68	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.68	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.68	   down(h(c)) -> h_flat(down(c))
305.07/136.68	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.68	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.68	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.68	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.68	
305.07/136.68	The set Q consists of the following terms:
305.07/136.68	
305.07/136.68	   down(a)
305.07/136.68	   down(f(f(x0)))
305.07/136.68	   down(g(g(x0)))
305.07/136.68	   down(h(h(x0)))
305.07/136.68	   down(f(a))
305.07/136.68	   down(f(g(x0)))
305.07/136.68	   down(f(h(x0)))
305.07/136.68	   down(f(c))
305.07/136.68	   down(f(fresh_constant))
305.07/136.68	   down(g(a))
305.07/136.68	   down(g(f(x0)))
305.07/136.68	   down(g(h(x0)))
305.07/136.68	   down(g(c))
305.07/136.68	   down(g(fresh_constant))
305.07/136.68	   down(h(a))
305.07/136.68	   down(h(f(x0)))
305.07/136.68	   down(h(g(x0)))
305.07/136.68	   down(h(c))
305.07/136.68	   down(h(fresh_constant))
305.07/136.68	   f_flat(up(x0))
305.07/136.68	   g_flat(up(x0))
305.07/136.68	   h_flat(up(x0))
305.07/136.68	
305.07/136.68	We have to consider all minimal (P,Q,R)-chains.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(33) DependencyGraphProof (EQUIVALENT)
305.07/136.68	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 8 less nodes.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(34)
305.07/136.68	Obligation:
305.07/136.68	Q DP problem:
305.07/136.68	The TRS P consists of the following rules:
305.07/136.68	
305.07/136.68	   TOP(up(g(g(x0)))) -> TOP(up(h(x0)))
305.07/136.68	   TOP(up(h(a))) -> TOP(h_flat(down(a)))
305.07/136.68	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
305.07/136.68	   TOP(up(f(f(x0)))) -> TOP(up(g(x0)))
305.07/136.68	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
305.07/136.68	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.07/136.68	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.07/136.68	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.07/136.68	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.07/136.68	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.07/136.68	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.07/136.68	
305.07/136.68	The TRS R consists of the following rules:
305.07/136.68	
305.07/136.68	   down(a) -> up(f(a))
305.07/136.68	   down(f(f(x))) -> up(g(x))
305.07/136.68	   down(g(g(x))) -> up(h(x))
305.07/136.68	   down(h(h(x))) -> up(c)
305.07/136.68	   down(f(a)) -> f_flat(down(a))
305.07/136.68	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.68	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.68	   down(f(c)) -> f_flat(down(c))
305.07/136.68	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.68	   down(g(a)) -> g_flat(down(a))
305.07/136.68	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.68	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.68	   down(g(c)) -> g_flat(down(c))
305.07/136.68	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.68	   down(h(a)) -> h_flat(down(a))
305.07/136.68	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.68	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.68	   down(h(c)) -> h_flat(down(c))
305.07/136.68	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.68	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.68	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.68	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.68	
305.07/136.68	The set Q consists of the following terms:
305.07/136.68	
305.07/136.68	   down(a)
305.07/136.68	   down(f(f(x0)))
305.07/136.68	   down(g(g(x0)))
305.07/136.68	   down(h(h(x0)))
305.07/136.68	   down(f(a))
305.07/136.68	   down(f(g(x0)))
305.07/136.68	   down(f(h(x0)))
305.07/136.68	   down(f(c))
305.07/136.68	   down(f(fresh_constant))
305.07/136.68	   down(g(a))
305.07/136.68	   down(g(f(x0)))
305.07/136.68	   down(g(h(x0)))
305.07/136.68	   down(g(c))
305.07/136.68	   down(g(fresh_constant))
305.07/136.68	   down(h(a))
305.07/136.68	   down(h(f(x0)))
305.07/136.68	   down(h(g(x0)))
305.07/136.68	   down(h(c))
305.07/136.68	   down(h(fresh_constant))
305.07/136.68	   f_flat(up(x0))
305.07/136.68	   g_flat(up(x0))
305.07/136.68	   h_flat(up(x0))
305.07/136.68	
305.07/136.68	We have to consider all minimal (P,Q,R)-chains.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(35) TransformationProof (EQUIVALENT)
305.07/136.68	By rewriting [LPAR04] the rule TOP(up(h(a))) -> TOP(h_flat(down(a))) at position [0,0] we obtained the following new rules [LPAR04]:
305.07/136.68	
305.07/136.68	   (TOP(up(h(a))) -> TOP(h_flat(up(f(a)))),TOP(up(h(a))) -> TOP(h_flat(up(f(a)))))
305.07/136.68	
305.07/136.68	
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(36)
305.07/136.68	Obligation:
305.07/136.68	Q DP problem:
305.07/136.68	The TRS P consists of the following rules:
305.07/136.68	
305.07/136.68	   TOP(up(g(g(x0)))) -> TOP(up(h(x0)))
305.07/136.68	   TOP(up(f(a))) -> TOP(f_flat(down(a)))
305.07/136.68	   TOP(up(f(f(x0)))) -> TOP(up(g(x0)))
305.07/136.68	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
305.07/136.68	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.07/136.68	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.07/136.68	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.07/136.68	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.07/136.68	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.07/136.68	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.07/136.68	   TOP(up(h(a))) -> TOP(h_flat(up(f(a))))
305.07/136.68	
305.07/136.68	The TRS R consists of the following rules:
305.07/136.68	
305.07/136.68	   down(a) -> up(f(a))
305.07/136.68	   down(f(f(x))) -> up(g(x))
305.07/136.68	   down(g(g(x))) -> up(h(x))
305.07/136.68	   down(h(h(x))) -> up(c)
305.07/136.68	   down(f(a)) -> f_flat(down(a))
305.07/136.68	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.68	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.68	   down(f(c)) -> f_flat(down(c))
305.07/136.68	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.68	   down(g(a)) -> g_flat(down(a))
305.07/136.68	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.68	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.68	   down(g(c)) -> g_flat(down(c))
305.07/136.68	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.68	   down(h(a)) -> h_flat(down(a))
305.07/136.68	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.68	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.68	   down(h(c)) -> h_flat(down(c))
305.07/136.68	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.68	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.68	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.68	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.68	
305.07/136.68	The set Q consists of the following terms:
305.07/136.68	
305.07/136.68	   down(a)
305.07/136.68	   down(f(f(x0)))
305.07/136.68	   down(g(g(x0)))
305.07/136.68	   down(h(h(x0)))
305.07/136.68	   down(f(a))
305.07/136.68	   down(f(g(x0)))
305.07/136.68	   down(f(h(x0)))
305.07/136.68	   down(f(c))
305.07/136.68	   down(f(fresh_constant))
305.07/136.68	   down(g(a))
305.07/136.68	   down(g(f(x0)))
305.07/136.68	   down(g(h(x0)))
305.07/136.68	   down(g(c))
305.07/136.68	   down(g(fresh_constant))
305.07/136.68	   down(h(a))
305.07/136.68	   down(h(f(x0)))
305.07/136.68	   down(h(g(x0)))
305.07/136.68	   down(h(c))
305.07/136.68	   down(h(fresh_constant))
305.07/136.68	   f_flat(up(x0))
305.07/136.68	   g_flat(up(x0))
305.07/136.68	   h_flat(up(x0))
305.07/136.68	
305.07/136.68	We have to consider all minimal (P,Q,R)-chains.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(37) TransformationProof (EQUIVALENT)
305.07/136.68	By rewriting [LPAR04] the rule TOP(up(f(a))) -> TOP(f_flat(down(a))) at position [0,0] we obtained the following new rules [LPAR04]:
305.07/136.68	
305.07/136.68	   (TOP(up(f(a))) -> TOP(f_flat(up(f(a)))),TOP(up(f(a))) -> TOP(f_flat(up(f(a)))))
305.07/136.68	
305.07/136.68	
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(38)
305.07/136.68	Obligation:
305.07/136.68	Q DP problem:
305.07/136.68	The TRS P consists of the following rules:
305.07/136.68	
305.07/136.68	   TOP(up(g(g(x0)))) -> TOP(up(h(x0)))
305.07/136.68	   TOP(up(f(f(x0)))) -> TOP(up(g(x0)))
305.07/136.68	   TOP(up(g(a))) -> TOP(g_flat(down(a)))
305.07/136.68	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.07/136.68	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.07/136.68	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.07/136.68	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.07/136.68	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.07/136.68	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.07/136.68	   TOP(up(h(a))) -> TOP(h_flat(up(f(a))))
305.07/136.68	   TOP(up(f(a))) -> TOP(f_flat(up(f(a))))
305.07/136.68	
305.07/136.68	The TRS R consists of the following rules:
305.07/136.68	
305.07/136.68	   down(a) -> up(f(a))
305.07/136.68	   down(f(f(x))) -> up(g(x))
305.07/136.68	   down(g(g(x))) -> up(h(x))
305.07/136.68	   down(h(h(x))) -> up(c)
305.07/136.68	   down(f(a)) -> f_flat(down(a))
305.07/136.68	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.68	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.68	   down(f(c)) -> f_flat(down(c))
305.07/136.68	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.68	   down(g(a)) -> g_flat(down(a))
305.07/136.68	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.68	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.68	   down(g(c)) -> g_flat(down(c))
305.07/136.68	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.68	   down(h(a)) -> h_flat(down(a))
305.07/136.68	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.68	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.68	   down(h(c)) -> h_flat(down(c))
305.07/136.68	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.68	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.68	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.68	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.68	
305.07/136.68	The set Q consists of the following terms:
305.07/136.68	
305.07/136.68	   down(a)
305.07/136.68	   down(f(f(x0)))
305.07/136.68	   down(g(g(x0)))
305.07/136.68	   down(h(h(x0)))
305.07/136.68	   down(f(a))
305.07/136.68	   down(f(g(x0)))
305.07/136.68	   down(f(h(x0)))
305.07/136.68	   down(f(c))
305.07/136.68	   down(f(fresh_constant))
305.07/136.68	   down(g(a))
305.07/136.68	   down(g(f(x0)))
305.07/136.68	   down(g(h(x0)))
305.07/136.68	   down(g(c))
305.07/136.68	   down(g(fresh_constant))
305.07/136.68	   down(h(a))
305.07/136.68	   down(h(f(x0)))
305.07/136.68	   down(h(g(x0)))
305.07/136.68	   down(h(c))
305.07/136.68	   down(h(fresh_constant))
305.07/136.68	   f_flat(up(x0))
305.07/136.68	   g_flat(up(x0))
305.07/136.68	   h_flat(up(x0))
305.07/136.68	
305.07/136.68	We have to consider all minimal (P,Q,R)-chains.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(39) TransformationProof (EQUIVALENT)
305.07/136.68	By rewriting [LPAR04] the rule TOP(up(g(a))) -> TOP(g_flat(down(a))) at position [0,0] we obtained the following new rules [LPAR04]:
305.07/136.68	
305.07/136.68	   (TOP(up(g(a))) -> TOP(g_flat(up(f(a)))),TOP(up(g(a))) -> TOP(g_flat(up(f(a)))))
305.07/136.68	
305.07/136.68	
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(40)
305.07/136.68	Obligation:
305.07/136.68	Q DP problem:
305.07/136.68	The TRS P consists of the following rules:
305.07/136.68	
305.07/136.68	   TOP(up(g(g(x0)))) -> TOP(up(h(x0)))
305.07/136.68	   TOP(up(f(f(x0)))) -> TOP(up(g(x0)))
305.07/136.68	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.07/136.68	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.07/136.68	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.07/136.68	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.07/136.68	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.07/136.68	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.07/136.68	   TOP(up(h(a))) -> TOP(h_flat(up(f(a))))
305.07/136.68	   TOP(up(f(a))) -> TOP(f_flat(up(f(a))))
305.07/136.68	   TOP(up(g(a))) -> TOP(g_flat(up(f(a))))
305.07/136.68	
305.07/136.68	The TRS R consists of the following rules:
305.07/136.68	
305.07/136.68	   down(a) -> up(f(a))
305.07/136.68	   down(f(f(x))) -> up(g(x))
305.07/136.68	   down(g(g(x))) -> up(h(x))
305.07/136.68	   down(h(h(x))) -> up(c)
305.07/136.68	   down(f(a)) -> f_flat(down(a))
305.07/136.68	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.68	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.68	   down(f(c)) -> f_flat(down(c))
305.07/136.68	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.68	   down(g(a)) -> g_flat(down(a))
305.07/136.68	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.68	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.68	   down(g(c)) -> g_flat(down(c))
305.07/136.68	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.68	   down(h(a)) -> h_flat(down(a))
305.07/136.68	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.68	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.68	   down(h(c)) -> h_flat(down(c))
305.07/136.68	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.68	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.68	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.68	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.68	
305.07/136.68	The set Q consists of the following terms:
305.07/136.68	
305.07/136.68	   down(a)
305.07/136.68	   down(f(f(x0)))
305.07/136.68	   down(g(g(x0)))
305.07/136.68	   down(h(h(x0)))
305.07/136.68	   down(f(a))
305.07/136.68	   down(f(g(x0)))
305.07/136.68	   down(f(h(x0)))
305.07/136.68	   down(f(c))
305.07/136.68	   down(f(fresh_constant))
305.07/136.68	   down(g(a))
305.07/136.68	   down(g(f(x0)))
305.07/136.68	   down(g(h(x0)))
305.07/136.68	   down(g(c))
305.07/136.68	   down(g(fresh_constant))
305.07/136.68	   down(h(a))
305.07/136.68	   down(h(f(x0)))
305.07/136.68	   down(h(g(x0)))
305.07/136.68	   down(h(c))
305.07/136.68	   down(h(fresh_constant))
305.07/136.68	   f_flat(up(x0))
305.07/136.68	   g_flat(up(x0))
305.07/136.68	   h_flat(up(x0))
305.07/136.68	
305.07/136.68	We have to consider all minimal (P,Q,R)-chains.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(41) TransformationProof (EQUIVALENT)
305.07/136.68	By rewriting [LPAR04] the rule TOP(up(h(a))) -> TOP(h_flat(up(f(a)))) at position [0] we obtained the following new rules [LPAR04]:
305.07/136.68	
305.07/136.68	   (TOP(up(h(a))) -> TOP(up(h(f(a)))),TOP(up(h(a))) -> TOP(up(h(f(a)))))
305.07/136.68	
305.07/136.68	
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(42)
305.07/136.68	Obligation:
305.07/136.68	Q DP problem:
305.07/136.68	The TRS P consists of the following rules:
305.07/136.68	
305.07/136.68	   TOP(up(g(g(x0)))) -> TOP(up(h(x0)))
305.07/136.68	   TOP(up(f(f(x0)))) -> TOP(up(g(x0)))
305.07/136.68	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.07/136.68	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.07/136.68	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.07/136.68	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.07/136.68	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.07/136.68	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.07/136.68	   TOP(up(f(a))) -> TOP(f_flat(up(f(a))))
305.07/136.68	   TOP(up(g(a))) -> TOP(g_flat(up(f(a))))
305.07/136.68	   TOP(up(h(a))) -> TOP(up(h(f(a))))
305.07/136.68	
305.07/136.68	The TRS R consists of the following rules:
305.07/136.68	
305.07/136.68	   down(a) -> up(f(a))
305.07/136.68	   down(f(f(x))) -> up(g(x))
305.07/136.68	   down(g(g(x))) -> up(h(x))
305.07/136.68	   down(h(h(x))) -> up(c)
305.07/136.68	   down(f(a)) -> f_flat(down(a))
305.07/136.68	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.68	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.68	   down(f(c)) -> f_flat(down(c))
305.07/136.68	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.68	   down(g(a)) -> g_flat(down(a))
305.07/136.68	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.68	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.68	   down(g(c)) -> g_flat(down(c))
305.07/136.68	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.68	   down(h(a)) -> h_flat(down(a))
305.07/136.68	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.68	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.68	   down(h(c)) -> h_flat(down(c))
305.07/136.68	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.68	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.68	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.68	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.68	
305.07/136.68	The set Q consists of the following terms:
305.07/136.68	
305.07/136.68	   down(a)
305.07/136.68	   down(f(f(x0)))
305.07/136.68	   down(g(g(x0)))
305.07/136.68	   down(h(h(x0)))
305.07/136.68	   down(f(a))
305.07/136.68	   down(f(g(x0)))
305.07/136.68	   down(f(h(x0)))
305.07/136.68	   down(f(c))
305.07/136.68	   down(f(fresh_constant))
305.07/136.68	   down(g(a))
305.07/136.68	   down(g(f(x0)))
305.07/136.68	   down(g(h(x0)))
305.07/136.68	   down(g(c))
305.07/136.68	   down(g(fresh_constant))
305.07/136.68	   down(h(a))
305.07/136.68	   down(h(f(x0)))
305.07/136.68	   down(h(g(x0)))
305.07/136.68	   down(h(c))
305.07/136.68	   down(h(fresh_constant))
305.07/136.68	   f_flat(up(x0))
305.07/136.68	   g_flat(up(x0))
305.07/136.68	   h_flat(up(x0))
305.07/136.68	
305.07/136.68	We have to consider all minimal (P,Q,R)-chains.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(43) TransformationProof (EQUIVALENT)
305.07/136.68	By rewriting [LPAR04] the rule TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) at position [0] we obtained the following new rules [LPAR04]:
305.07/136.68	
305.07/136.68	   (TOP(up(f(a))) -> TOP(up(f(f(a)))),TOP(up(f(a))) -> TOP(up(f(f(a)))))
305.07/136.68	
305.07/136.68	
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(44)
305.07/136.68	Obligation:
305.07/136.68	Q DP problem:
305.07/136.68	The TRS P consists of the following rules:
305.07/136.68	
305.07/136.68	   TOP(up(g(g(x0)))) -> TOP(up(h(x0)))
305.07/136.68	   TOP(up(f(f(x0)))) -> TOP(up(g(x0)))
305.07/136.68	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.07/136.68	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.07/136.68	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.07/136.68	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.07/136.68	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.07/136.68	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.07/136.68	   TOP(up(g(a))) -> TOP(g_flat(up(f(a))))
305.07/136.68	   TOP(up(h(a))) -> TOP(up(h(f(a))))
305.07/136.68	   TOP(up(f(a))) -> TOP(up(f(f(a))))
305.07/136.68	
305.07/136.68	The TRS R consists of the following rules:
305.07/136.68	
305.07/136.68	   down(a) -> up(f(a))
305.07/136.68	   down(f(f(x))) -> up(g(x))
305.07/136.68	   down(g(g(x))) -> up(h(x))
305.07/136.68	   down(h(h(x))) -> up(c)
305.07/136.68	   down(f(a)) -> f_flat(down(a))
305.07/136.68	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.68	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.68	   down(f(c)) -> f_flat(down(c))
305.07/136.68	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.68	   down(g(a)) -> g_flat(down(a))
305.07/136.68	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.68	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.68	   down(g(c)) -> g_flat(down(c))
305.07/136.68	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.68	   down(h(a)) -> h_flat(down(a))
305.07/136.68	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.68	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.68	   down(h(c)) -> h_flat(down(c))
305.07/136.68	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.68	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.68	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.68	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.68	
305.07/136.68	The set Q consists of the following terms:
305.07/136.68	
305.07/136.68	   down(a)
305.07/136.68	   down(f(f(x0)))
305.07/136.68	   down(g(g(x0)))
305.07/136.68	   down(h(h(x0)))
305.07/136.68	   down(f(a))
305.07/136.68	   down(f(g(x0)))
305.07/136.68	   down(f(h(x0)))
305.07/136.68	   down(f(c))
305.07/136.68	   down(f(fresh_constant))
305.07/136.68	   down(g(a))
305.07/136.68	   down(g(f(x0)))
305.07/136.68	   down(g(h(x0)))
305.07/136.68	   down(g(c))
305.07/136.68	   down(g(fresh_constant))
305.07/136.68	   down(h(a))
305.07/136.68	   down(h(f(x0)))
305.07/136.68	   down(h(g(x0)))
305.07/136.68	   down(h(c))
305.07/136.68	   down(h(fresh_constant))
305.07/136.68	   f_flat(up(x0))
305.07/136.68	   g_flat(up(x0))
305.07/136.68	   h_flat(up(x0))
305.07/136.68	
305.07/136.68	We have to consider all minimal (P,Q,R)-chains.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(45) TransformationProof (EQUIVALENT)
305.07/136.68	By rewriting [LPAR04] the rule TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) at position [0] we obtained the following new rules [LPAR04]:
305.07/136.68	
305.07/136.68	   (TOP(up(g(a))) -> TOP(up(g(f(a)))),TOP(up(g(a))) -> TOP(up(g(f(a)))))
305.07/136.68	
305.07/136.68	
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(46)
305.07/136.68	Obligation:
305.07/136.68	Q DP problem:
305.07/136.68	The TRS P consists of the following rules:
305.07/136.68	
305.07/136.68	   TOP(up(g(g(x0)))) -> TOP(up(h(x0)))
305.07/136.68	   TOP(up(f(f(x0)))) -> TOP(up(g(x0)))
305.07/136.68	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.07/136.68	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.07/136.68	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.07/136.68	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.07/136.68	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.07/136.68	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.07/136.68	   TOP(up(h(a))) -> TOP(up(h(f(a))))
305.07/136.68	   TOP(up(f(a))) -> TOP(up(f(f(a))))
305.07/136.68	   TOP(up(g(a))) -> TOP(up(g(f(a))))
305.07/136.68	
305.07/136.68	The TRS R consists of the following rules:
305.07/136.68	
305.07/136.68	   down(a) -> up(f(a))
305.07/136.68	   down(f(f(x))) -> up(g(x))
305.07/136.68	   down(g(g(x))) -> up(h(x))
305.07/136.68	   down(h(h(x))) -> up(c)
305.07/136.68	   down(f(a)) -> f_flat(down(a))
305.07/136.68	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.68	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.68	   down(f(c)) -> f_flat(down(c))
305.07/136.68	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.68	   down(g(a)) -> g_flat(down(a))
305.07/136.68	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.68	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.68	   down(g(c)) -> g_flat(down(c))
305.07/136.68	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.68	   down(h(a)) -> h_flat(down(a))
305.07/136.68	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.68	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.68	   down(h(c)) -> h_flat(down(c))
305.07/136.68	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.68	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.68	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.68	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.68	
305.07/136.68	The set Q consists of the following terms:
305.07/136.68	
305.07/136.68	   down(a)
305.07/136.68	   down(f(f(x0)))
305.07/136.68	   down(g(g(x0)))
305.07/136.68	   down(h(h(x0)))
305.07/136.68	   down(f(a))
305.07/136.68	   down(f(g(x0)))
305.07/136.68	   down(f(h(x0)))
305.07/136.68	   down(f(c))
305.07/136.68	   down(f(fresh_constant))
305.07/136.68	   down(g(a))
305.07/136.68	   down(g(f(x0)))
305.07/136.68	   down(g(h(x0)))
305.07/136.68	   down(g(c))
305.07/136.68	   down(g(fresh_constant))
305.07/136.68	   down(h(a))
305.07/136.68	   down(h(f(x0)))
305.07/136.68	   down(h(g(x0)))
305.07/136.68	   down(h(c))
305.07/136.68	   down(h(fresh_constant))
305.07/136.68	   f_flat(up(x0))
305.07/136.68	   g_flat(up(x0))
305.07/136.68	   h_flat(up(x0))
305.07/136.68	
305.07/136.68	We have to consider all minimal (P,Q,R)-chains.
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(47) TransformationProof (EQUIVALENT)
305.07/136.68	By forward instantiating [JAR06] the rule TOP(up(g(g(x0)))) -> TOP(up(h(x0))) we obtained the following new rules [LPAR04]:
305.07/136.68	
305.07/136.68	   (TOP(up(g(g(f(y_0))))) -> TOP(up(h(f(y_0)))),TOP(up(g(g(f(y_0))))) -> TOP(up(h(f(y_0)))))
305.07/136.68	   (TOP(up(g(g(g(y_0))))) -> TOP(up(h(g(y_0)))),TOP(up(g(g(g(y_0))))) -> TOP(up(h(g(y_0)))))
305.07/136.68	   (TOP(up(g(g(a)))) -> TOP(up(h(a))),TOP(up(g(g(a)))) -> TOP(up(h(a))))
305.07/136.68	
305.07/136.68	
305.07/136.68	----------------------------------------
305.07/136.68	
305.07/136.68	(48)
305.07/136.68	Obligation:
305.07/136.68	Q DP problem:
305.07/136.68	The TRS P consists of the following rules:
305.07/136.68	
305.07/136.68	   TOP(up(f(f(x0)))) -> TOP(up(g(x0)))
305.07/136.68	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.07/136.68	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.07/136.68	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.07/136.68	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.07/136.68	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.07/136.68	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.07/136.68	   TOP(up(h(a))) -> TOP(up(h(f(a))))
305.07/136.68	   TOP(up(f(a))) -> TOP(up(f(f(a))))
305.07/136.68	   TOP(up(g(a))) -> TOP(up(g(f(a))))
305.07/136.68	   TOP(up(g(g(f(y_0))))) -> TOP(up(h(f(y_0))))
305.07/136.68	   TOP(up(g(g(g(y_0))))) -> TOP(up(h(g(y_0))))
305.07/136.68	   TOP(up(g(g(a)))) -> TOP(up(h(a)))
305.07/136.68	
305.07/136.68	The TRS R consists of the following rules:
305.07/136.68	
305.07/136.68	   down(a) -> up(f(a))
305.07/136.68	   down(f(f(x))) -> up(g(x))
305.07/136.68	   down(g(g(x))) -> up(h(x))
305.07/136.68	   down(h(h(x))) -> up(c)
305.07/136.68	   down(f(a)) -> f_flat(down(a))
305.07/136.68	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.68	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.68	   down(f(c)) -> f_flat(down(c))
305.07/136.68	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.69	   down(g(a)) -> g_flat(down(a))
305.07/136.69	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.69	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.69	   down(g(c)) -> g_flat(down(c))
305.07/136.69	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.69	   down(h(a)) -> h_flat(down(a))
305.07/136.69	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.69	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.69	   down(h(c)) -> h_flat(down(c))
305.07/136.69	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.69	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.69	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.69	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.69	
305.07/136.69	The set Q consists of the following terms:
305.07/136.69	
305.07/136.69	   down(a)
305.07/136.69	   down(f(f(x0)))
305.07/136.69	   down(g(g(x0)))
305.07/136.69	   down(h(h(x0)))
305.07/136.69	   down(f(a))
305.07/136.69	   down(f(g(x0)))
305.07/136.69	   down(f(h(x0)))
305.07/136.69	   down(f(c))
305.07/136.69	   down(f(fresh_constant))
305.07/136.69	   down(g(a))
305.07/136.69	   down(g(f(x0)))
305.07/136.69	   down(g(h(x0)))
305.07/136.69	   down(g(c))
305.07/136.69	   down(g(fresh_constant))
305.07/136.69	   down(h(a))
305.07/136.69	   down(h(f(x0)))
305.07/136.69	   down(h(g(x0)))
305.07/136.69	   down(h(c))
305.07/136.69	   down(h(fresh_constant))
305.07/136.69	   f_flat(up(x0))
305.07/136.69	   g_flat(up(x0))
305.07/136.69	   h_flat(up(x0))
305.07/136.69	
305.07/136.69	We have to consider all minimal (P,Q,R)-chains.
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(49) TransformationProof (EQUIVALENT)
305.07/136.69	By forward instantiating [JAR06] the rule TOP(up(f(f(x0)))) -> TOP(up(g(x0))) we obtained the following new rules [LPAR04]:
305.07/136.69	
305.07/136.69	   (TOP(up(f(f(f(y_0))))) -> TOP(up(g(f(y_0)))),TOP(up(f(f(f(y_0))))) -> TOP(up(g(f(y_0)))))
305.07/136.69	   (TOP(up(f(f(h(y_0))))) -> TOP(up(g(h(y_0)))),TOP(up(f(f(h(y_0))))) -> TOP(up(g(h(y_0)))))
305.07/136.69	   (TOP(up(f(f(a)))) -> TOP(up(g(a))),TOP(up(f(f(a)))) -> TOP(up(g(a))))
305.07/136.69	   (TOP(up(f(f(g(f(y_0)))))) -> TOP(up(g(g(f(y_0))))),TOP(up(f(f(g(f(y_0)))))) -> TOP(up(g(g(f(y_0))))))
305.07/136.69	   (TOP(up(f(f(g(g(y_0)))))) -> TOP(up(g(g(g(y_0))))),TOP(up(f(f(g(g(y_0)))))) -> TOP(up(g(g(g(y_0))))))
305.07/136.69	   (TOP(up(f(f(g(a))))) -> TOP(up(g(g(a)))),TOP(up(f(f(g(a))))) -> TOP(up(g(g(a)))))
305.07/136.69	
305.07/136.69	
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(50)
305.07/136.69	Obligation:
305.07/136.69	Q DP problem:
305.07/136.69	The TRS P consists of the following rules:
305.07/136.69	
305.07/136.69	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.07/136.69	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.07/136.69	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.07/136.69	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.07/136.69	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.07/136.69	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.07/136.69	   TOP(up(h(a))) -> TOP(up(h(f(a))))
305.07/136.69	   TOP(up(f(a))) -> TOP(up(f(f(a))))
305.07/136.69	   TOP(up(g(a))) -> TOP(up(g(f(a))))
305.07/136.69	   TOP(up(g(g(f(y_0))))) -> TOP(up(h(f(y_0))))
305.07/136.69	   TOP(up(g(g(g(y_0))))) -> TOP(up(h(g(y_0))))
305.07/136.69	   TOP(up(g(g(a)))) -> TOP(up(h(a)))
305.07/136.69	   TOP(up(f(f(f(y_0))))) -> TOP(up(g(f(y_0))))
305.07/136.69	   TOP(up(f(f(h(y_0))))) -> TOP(up(g(h(y_0))))
305.07/136.69	   TOP(up(f(f(a)))) -> TOP(up(g(a)))
305.07/136.69	   TOP(up(f(f(g(f(y_0)))))) -> TOP(up(g(g(f(y_0)))))
305.07/136.69	   TOP(up(f(f(g(g(y_0)))))) -> TOP(up(g(g(g(y_0)))))
305.07/136.69	   TOP(up(f(f(g(a))))) -> TOP(up(g(g(a))))
305.07/136.69	
305.07/136.69	The TRS R consists of the following rules:
305.07/136.69	
305.07/136.69	   down(a) -> up(f(a))
305.07/136.69	   down(f(f(x))) -> up(g(x))
305.07/136.69	   down(g(g(x))) -> up(h(x))
305.07/136.69	   down(h(h(x))) -> up(c)
305.07/136.69	   down(f(a)) -> f_flat(down(a))
305.07/136.69	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.69	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.69	   down(f(c)) -> f_flat(down(c))
305.07/136.69	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.69	   down(g(a)) -> g_flat(down(a))
305.07/136.69	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.69	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.69	   down(g(c)) -> g_flat(down(c))
305.07/136.69	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.69	   down(h(a)) -> h_flat(down(a))
305.07/136.69	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.69	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.69	   down(h(c)) -> h_flat(down(c))
305.07/136.69	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.69	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.69	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.69	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.69	
305.07/136.69	The set Q consists of the following terms:
305.07/136.69	
305.07/136.69	   down(a)
305.07/136.69	   down(f(f(x0)))
305.07/136.69	   down(g(g(x0)))
305.07/136.69	   down(h(h(x0)))
305.07/136.69	   down(f(a))
305.07/136.69	   down(f(g(x0)))
305.07/136.69	   down(f(h(x0)))
305.07/136.69	   down(f(c))
305.07/136.69	   down(f(fresh_constant))
305.07/136.69	   down(g(a))
305.07/136.69	   down(g(f(x0)))
305.07/136.69	   down(g(h(x0)))
305.07/136.69	   down(g(c))
305.07/136.69	   down(g(fresh_constant))
305.07/136.69	   down(h(a))
305.07/136.69	   down(h(f(x0)))
305.07/136.69	   down(h(g(x0)))
305.07/136.69	   down(h(c))
305.07/136.69	   down(h(fresh_constant))
305.07/136.69	   f_flat(up(x0))
305.07/136.69	   g_flat(up(x0))
305.07/136.69	   h_flat(up(x0))
305.07/136.69	
305.07/136.69	We have to consider all minimal (P,Q,R)-chains.
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(51) QDPOrderProof (EQUIVALENT)
305.07/136.69	We use the reduction pair processor [LPAR04,JAR06].
305.07/136.69	
305.07/136.69	
305.07/136.69	The following pairs can be oriented strictly and are deleted.
305.07/136.69	
305.07/136.69	   TOP(up(f(f(f(y_0))))) -> TOP(up(g(f(y_0))))
305.07/136.69	   TOP(up(f(f(h(y_0))))) -> TOP(up(g(h(y_0))))
305.07/136.69	   TOP(up(f(f(a)))) -> TOP(up(g(a)))
305.07/136.69	   TOP(up(f(f(g(f(y_0)))))) -> TOP(up(g(g(f(y_0)))))
305.07/136.69	   TOP(up(f(f(g(g(y_0)))))) -> TOP(up(g(g(g(y_0)))))
305.07/136.69	   TOP(up(f(f(g(a))))) -> TOP(up(g(g(a))))
305.07/136.69	The remaining pairs can at least be oriented weakly.
305.07/136.69	Used ordering:  Polynomial interpretation [POLO]:
305.07/136.69	
305.07/136.69	   POL(TOP(x_1)) = x_1
305.07/136.69	   POL(a) = 0
305.07/136.69	   POL(c) = 0
305.07/136.69	   POL(down(x_1)) = 0
305.07/136.69	   POL(f(x_1)) = 1
305.07/136.69	   POL(f_flat(x_1)) = 1
305.07/136.69	   POL(fresh_constant) = 0
305.07/136.69	   POL(g(x_1)) = 0
305.07/136.69	   POL(g_flat(x_1)) = 0
305.07/136.69	   POL(h(x_1)) = 0
305.07/136.69	   POL(h_flat(x_1)) = 0
305.07/136.69	   POL(up(x_1)) = x_1
305.07/136.69	
305.07/136.69	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
305.07/136.69	
305.07/136.69	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.69	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.69	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.69	
305.07/136.69	
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(52)
305.07/136.69	Obligation:
305.07/136.69	Q DP problem:
305.07/136.69	The TRS P consists of the following rules:
305.07/136.69	
305.07/136.69	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.07/136.69	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.07/136.69	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.07/136.69	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.07/136.69	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.07/136.69	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.07/136.69	   TOP(up(h(a))) -> TOP(up(h(f(a))))
305.07/136.69	   TOP(up(f(a))) -> TOP(up(f(f(a))))
305.07/136.69	   TOP(up(g(a))) -> TOP(up(g(f(a))))
305.07/136.69	   TOP(up(g(g(f(y_0))))) -> TOP(up(h(f(y_0))))
305.07/136.69	   TOP(up(g(g(g(y_0))))) -> TOP(up(h(g(y_0))))
305.07/136.69	   TOP(up(g(g(a)))) -> TOP(up(h(a)))
305.07/136.69	
305.07/136.69	The TRS R consists of the following rules:
305.07/136.69	
305.07/136.69	   down(a) -> up(f(a))
305.07/136.69	   down(f(f(x))) -> up(g(x))
305.07/136.69	   down(g(g(x))) -> up(h(x))
305.07/136.69	   down(h(h(x))) -> up(c)
305.07/136.69	   down(f(a)) -> f_flat(down(a))
305.07/136.69	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.69	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.69	   down(f(c)) -> f_flat(down(c))
305.07/136.69	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.69	   down(g(a)) -> g_flat(down(a))
305.07/136.69	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.69	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.69	   down(g(c)) -> g_flat(down(c))
305.07/136.69	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.69	   down(h(a)) -> h_flat(down(a))
305.07/136.69	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.69	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.69	   down(h(c)) -> h_flat(down(c))
305.07/136.69	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.69	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.69	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.69	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.69	
305.07/136.69	The set Q consists of the following terms:
305.07/136.69	
305.07/136.69	   down(a)
305.07/136.69	   down(f(f(x0)))
305.07/136.69	   down(g(g(x0)))
305.07/136.69	   down(h(h(x0)))
305.07/136.69	   down(f(a))
305.07/136.69	   down(f(g(x0)))
305.07/136.69	   down(f(h(x0)))
305.07/136.69	   down(f(c))
305.07/136.69	   down(f(fresh_constant))
305.07/136.69	   down(g(a))
305.07/136.69	   down(g(f(x0)))
305.07/136.69	   down(g(h(x0)))
305.07/136.69	   down(g(c))
305.07/136.69	   down(g(fresh_constant))
305.07/136.69	   down(h(a))
305.07/136.69	   down(h(f(x0)))
305.07/136.69	   down(h(g(x0)))
305.07/136.69	   down(h(c))
305.07/136.69	   down(h(fresh_constant))
305.07/136.69	   f_flat(up(x0))
305.07/136.69	   g_flat(up(x0))
305.07/136.69	   h_flat(up(x0))
305.07/136.69	
305.07/136.69	We have to consider all minimal (P,Q,R)-chains.
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(53) DependencyGraphProof (EQUIVALENT)
305.07/136.69	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(54)
305.07/136.69	Obligation:
305.07/136.69	Q DP problem:
305.07/136.69	The TRS P consists of the following rules:
305.07/136.69	
305.07/136.69	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.07/136.69	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.07/136.69	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.07/136.69	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.07/136.69	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.07/136.69	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.07/136.69	   TOP(up(h(a))) -> TOP(up(h(f(a))))
305.07/136.69	   TOP(up(g(a))) -> TOP(up(g(f(a))))
305.07/136.69	   TOP(up(g(g(f(y_0))))) -> TOP(up(h(f(y_0))))
305.07/136.69	   TOP(up(g(g(g(y_0))))) -> TOP(up(h(g(y_0))))
305.07/136.69	   TOP(up(g(g(a)))) -> TOP(up(h(a)))
305.07/136.69	
305.07/136.69	The TRS R consists of the following rules:
305.07/136.69	
305.07/136.69	   down(a) -> up(f(a))
305.07/136.69	   down(f(f(x))) -> up(g(x))
305.07/136.69	   down(g(g(x))) -> up(h(x))
305.07/136.69	   down(h(h(x))) -> up(c)
305.07/136.69	   down(f(a)) -> f_flat(down(a))
305.07/136.69	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.69	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.69	   down(f(c)) -> f_flat(down(c))
305.07/136.69	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.69	   down(g(a)) -> g_flat(down(a))
305.07/136.69	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.69	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.69	   down(g(c)) -> g_flat(down(c))
305.07/136.69	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.69	   down(h(a)) -> h_flat(down(a))
305.07/136.69	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.69	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.69	   down(h(c)) -> h_flat(down(c))
305.07/136.69	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.69	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.69	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.69	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.69	
305.07/136.69	The set Q consists of the following terms:
305.07/136.69	
305.07/136.69	   down(a)
305.07/136.69	   down(f(f(x0)))
305.07/136.69	   down(g(g(x0)))
305.07/136.69	   down(h(h(x0)))
305.07/136.69	   down(f(a))
305.07/136.69	   down(f(g(x0)))
305.07/136.69	   down(f(h(x0)))
305.07/136.69	   down(f(c))
305.07/136.69	   down(f(fresh_constant))
305.07/136.69	   down(g(a))
305.07/136.69	   down(g(f(x0)))
305.07/136.69	   down(g(h(x0)))
305.07/136.69	   down(g(c))
305.07/136.69	   down(g(fresh_constant))
305.07/136.69	   down(h(a))
305.07/136.69	   down(h(f(x0)))
305.07/136.69	   down(h(g(x0)))
305.07/136.69	   down(h(c))
305.07/136.69	   down(h(fresh_constant))
305.07/136.69	   f_flat(up(x0))
305.07/136.69	   g_flat(up(x0))
305.07/136.69	   h_flat(up(x0))
305.07/136.69	
305.07/136.69	We have to consider all minimal (P,Q,R)-chains.
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(55) QDPOrderProof (EQUIVALENT)
305.07/136.69	We use the reduction pair processor [LPAR04,JAR06].
305.07/136.69	
305.07/136.69	
305.07/136.69	The following pairs can be oriented strictly and are deleted.
305.07/136.69	
305.07/136.69	   TOP(up(g(g(f(y_0))))) -> TOP(up(h(f(y_0))))
305.07/136.69	   TOP(up(g(g(g(y_0))))) -> TOP(up(h(g(y_0))))
305.07/136.69	   TOP(up(g(g(a)))) -> TOP(up(h(a)))
305.07/136.69	The remaining pairs can at least be oriented weakly.
305.07/136.69	Used ordering:  Polynomial interpretation [POLO]:
305.07/136.69	
305.07/136.69	   POL(TOP(x_1)) = x_1
305.07/136.69	   POL(a) = 0
305.07/136.69	   POL(c) = 0
305.07/136.69	   POL(down(x_1)) = 0
305.07/136.69	   POL(f(x_1)) = 0
305.07/136.69	   POL(f_flat(x_1)) = 0
305.07/136.69	   POL(fresh_constant) = 0
305.07/136.69	   POL(g(x_1)) = 1
305.07/136.69	   POL(g_flat(x_1)) = 1
305.07/136.69	   POL(h(x_1)) = 0
305.07/136.69	   POL(h_flat(x_1)) = 0
305.07/136.69	   POL(up(x_1)) = x_1
305.07/136.69	
305.07/136.69	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
305.07/136.69	
305.07/136.69	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.69	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.69	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.69	
305.07/136.69	
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(56)
305.07/136.69	Obligation:
305.07/136.69	Q DP problem:
305.07/136.69	The TRS P consists of the following rules:
305.07/136.69	
305.07/136.69	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.07/136.69	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.07/136.69	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.07/136.69	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.07/136.69	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.07/136.69	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.07/136.69	   TOP(up(h(a))) -> TOP(up(h(f(a))))
305.07/136.69	   TOP(up(g(a))) -> TOP(up(g(f(a))))
305.07/136.69	
305.07/136.69	The TRS R consists of the following rules:
305.07/136.69	
305.07/136.69	   down(a) -> up(f(a))
305.07/136.69	   down(f(f(x))) -> up(g(x))
305.07/136.69	   down(g(g(x))) -> up(h(x))
305.07/136.69	   down(h(h(x))) -> up(c)
305.07/136.69	   down(f(a)) -> f_flat(down(a))
305.07/136.69	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.69	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.69	   down(f(c)) -> f_flat(down(c))
305.07/136.69	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.69	   down(g(a)) -> g_flat(down(a))
305.07/136.69	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.69	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.69	   down(g(c)) -> g_flat(down(c))
305.07/136.69	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.69	   down(h(a)) -> h_flat(down(a))
305.07/136.69	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.69	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.69	   down(h(c)) -> h_flat(down(c))
305.07/136.69	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.69	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.69	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.69	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.69	
305.07/136.69	The set Q consists of the following terms:
305.07/136.69	
305.07/136.69	   down(a)
305.07/136.69	   down(f(f(x0)))
305.07/136.69	   down(g(g(x0)))
305.07/136.69	   down(h(h(x0)))
305.07/136.69	   down(f(a))
305.07/136.69	   down(f(g(x0)))
305.07/136.69	   down(f(h(x0)))
305.07/136.69	   down(f(c))
305.07/136.69	   down(f(fresh_constant))
305.07/136.69	   down(g(a))
305.07/136.69	   down(g(f(x0)))
305.07/136.69	   down(g(h(x0)))
305.07/136.69	   down(g(c))
305.07/136.69	   down(g(fresh_constant))
305.07/136.69	   down(h(a))
305.07/136.69	   down(h(f(x0)))
305.07/136.69	   down(h(g(x0)))
305.07/136.69	   down(h(c))
305.07/136.69	   down(h(fresh_constant))
305.07/136.69	   f_flat(up(x0))
305.07/136.69	   g_flat(up(x0))
305.07/136.69	   h_flat(up(x0))
305.07/136.69	
305.07/136.69	We have to consider all minimal (P,Q,R)-chains.
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(57) QDPOrderProof (EQUIVALENT)
305.07/136.69	We use the reduction pair processor [LPAR04,JAR06].
305.07/136.69	
305.07/136.69	
305.07/136.69	The following pairs can be oriented strictly and are deleted.
305.07/136.69	
305.07/136.69	   TOP(up(g(a))) -> TOP(up(g(f(a))))
305.07/136.69	The remaining pairs can at least be oriented weakly.
305.07/136.69	Used ordering:  Matrix interpretation [MATRO]:
305.07/136.69	
305.07/136.69	Non-tuple symbols: 
305.07/136.69	<<<
305.07/136.69	 M( a ) =  	[[1], [0]]
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	<<<
305.07/136.69	 M( h_flat_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	<<<
305.07/136.69	 M( c ) =  	[[0], [0]]
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	<<<
305.07/136.69	 M( down_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [1, 1]] 	* 	x_1
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	<<<
305.07/136.69	 M( f_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [1, 1]] 	* 	x_1
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	<<<
305.07/136.69	 M( fresh_constant ) =  	[[0], [0]]
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	<<<
305.07/136.69	 M( up_1(x_1) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 1]] 	* 	x_1
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	<<<
305.07/136.69	 M( h_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	<<<
305.07/136.69	 M( f_flat_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [1, 1]] 	* 	x_1
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	<<<
305.07/136.69	 M( g_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [1, 0]] 	* 	x_1
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	<<<
305.07/136.69	 M( g_flat_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [1, 0]] 	* 	x_1
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	Tuple symbols: 
305.07/136.69	<<<
305.07/136.69	 M( TOP_1(x_1) ) =  	[[0]] 	+ 	[[0, 1]] 	* 	x_1
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	
305.07/136.69	
305.07/136.69	Matrix type: 
305.07/136.69	
305.07/136.69	We used a basic matrix type which is not further parametrizeable.
305.07/136.69	
305.07/136.69	
305.07/136.69	
305.07/136.69	
305.07/136.69	
305.07/136.69	As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
305.07/136.69	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
305.07/136.69	
305.07/136.69	   down(g(g(x))) -> up(h(x))
305.07/136.69	   down(g(a)) -> g_flat(down(a))
305.07/136.69	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.69	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.69	   down(g(c)) -> g_flat(down(c))
305.07/136.69	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.69	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.69	   down(h(h(x))) -> up(c)
305.07/136.69	   down(h(a)) -> h_flat(down(a))
305.07/136.69	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.69	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.69	   down(h(c)) -> h_flat(down(c))
305.07/136.69	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.69	   down(f(f(x))) -> up(g(x))
305.07/136.69	   down(f(a)) -> f_flat(down(a))
305.07/136.69	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.69	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.69	   down(f(c)) -> f_flat(down(c))
305.07/136.69	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.69	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.69	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.69	   down(a) -> up(f(a))
305.07/136.69	
305.07/136.69	
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(58)
305.07/136.69	Obligation:
305.07/136.69	Q DP problem:
305.07/136.69	The TRS P consists of the following rules:
305.07/136.69	
305.07/136.69	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.07/136.69	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.07/136.69	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.07/136.69	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.07/136.69	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.07/136.69	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.07/136.69	   TOP(up(h(a))) -> TOP(up(h(f(a))))
305.07/136.69	
305.07/136.69	The TRS R consists of the following rules:
305.07/136.69	
305.07/136.69	   down(a) -> up(f(a))
305.07/136.69	   down(f(f(x))) -> up(g(x))
305.07/136.69	   down(g(g(x))) -> up(h(x))
305.07/136.69	   down(h(h(x))) -> up(c)
305.07/136.69	   down(f(a)) -> f_flat(down(a))
305.07/136.69	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.69	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.69	   down(f(c)) -> f_flat(down(c))
305.07/136.69	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.69	   down(g(a)) -> g_flat(down(a))
305.07/136.69	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.69	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.69	   down(g(c)) -> g_flat(down(c))
305.07/136.69	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.69	   down(h(a)) -> h_flat(down(a))
305.07/136.69	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.69	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.69	   down(h(c)) -> h_flat(down(c))
305.07/136.69	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.69	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.69	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.69	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.69	
305.07/136.69	The set Q consists of the following terms:
305.07/136.69	
305.07/136.69	   down(a)
305.07/136.69	   down(f(f(x0)))
305.07/136.69	   down(g(g(x0)))
305.07/136.69	   down(h(h(x0)))
305.07/136.69	   down(f(a))
305.07/136.69	   down(f(g(x0)))
305.07/136.69	   down(f(h(x0)))
305.07/136.69	   down(f(c))
305.07/136.69	   down(f(fresh_constant))
305.07/136.69	   down(g(a))
305.07/136.69	   down(g(f(x0)))
305.07/136.69	   down(g(h(x0)))
305.07/136.69	   down(g(c))
305.07/136.69	   down(g(fresh_constant))
305.07/136.69	   down(h(a))
305.07/136.69	   down(h(f(x0)))
305.07/136.69	   down(h(g(x0)))
305.07/136.69	   down(h(c))
305.07/136.69	   down(h(fresh_constant))
305.07/136.69	   f_flat(up(x0))
305.07/136.69	   g_flat(up(x0))
305.07/136.69	   h_flat(up(x0))
305.07/136.69	
305.07/136.69	We have to consider all minimal (P,Q,R)-chains.
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(59) QDPOrderProof (EQUIVALENT)
305.07/136.69	We use the reduction pair processor [LPAR04,JAR06].
305.07/136.69	
305.07/136.69	
305.07/136.69	The following pairs can be oriented strictly and are deleted.
305.07/136.69	
305.07/136.69	   TOP(up(h(a))) -> TOP(up(h(f(a))))
305.07/136.69	The remaining pairs can at least be oriented weakly.
305.07/136.69	Used ordering:  Matrix interpretation [MATRO]:
305.07/136.69	
305.07/136.69	Non-tuple symbols: 
305.07/136.69	<<<
305.07/136.69	 M( a ) =  	[[0], [1]]
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	<<<
305.07/136.69	 M( h_flat_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [1, 0]] 	* 	x_1
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	<<<
305.07/136.69	 M( c ) =  	[[0], [0]]
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	<<<
305.07/136.69	 M( down_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [1, 1]] 	* 	x_1
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	<<<
305.07/136.69	 M( f_1(x_1) ) =  	[[0], [0]] 	+ 	[[1, 1], [0, 0]] 	* 	x_1
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	<<<
305.07/136.69	 M( fresh_constant ) =  	[[0], [0]]
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	<<<
305.07/136.69	 M( up_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 1], [1, 0]] 	* 	x_1
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	<<<
305.07/136.69	 M( h_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 1], [0, 0]] 	* 	x_1
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	<<<
305.07/136.69	 M( f_flat_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [1, 1]] 	* 	x_1
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	<<<
305.07/136.69	 M( g_1(x_1) ) =  	[[0], [0]] 	+ 	[[1, 1], [0, 0]] 	* 	x_1
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	<<<
305.07/136.69	 M( g_flat_1(x_1) ) =  	[[0], [0]] 	+ 	[[0, 0], [1, 1]] 	* 	x_1
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	Tuple symbols: 
305.07/136.69	<<<
305.07/136.69	 M( TOP_1(x_1) ) =  	[[0]] 	+ 	[[0, 1]] 	* 	x_1
305.07/136.69	>>>
305.07/136.69	
305.07/136.69	
305.07/136.69	
305.07/136.69	Matrix type: 
305.07/136.69	
305.07/136.69	We used a basic matrix type which is not further parametrizeable.
305.07/136.69	
305.07/136.69	
305.07/136.69	
305.07/136.69	
305.07/136.69	
305.07/136.69	As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
305.07/136.69	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
305.07/136.69	
305.07/136.69	   down(g(g(x))) -> up(h(x))
305.07/136.69	   down(g(a)) -> g_flat(down(a))
305.07/136.69	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.69	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.69	   down(g(c)) -> g_flat(down(c))
305.07/136.69	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.69	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.69	   down(h(h(x))) -> up(c)
305.07/136.69	   down(h(a)) -> h_flat(down(a))
305.07/136.69	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.69	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.69	   down(h(c)) -> h_flat(down(c))
305.07/136.69	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.69	   down(f(f(x))) -> up(g(x))
305.07/136.69	   down(f(a)) -> f_flat(down(a))
305.07/136.69	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.69	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.69	   down(f(c)) -> f_flat(down(c))
305.07/136.69	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.69	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.69	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.69	   down(a) -> up(f(a))
305.07/136.69	
305.07/136.69	
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(60)
305.07/136.69	Obligation:
305.07/136.69	Q DP problem:
305.07/136.69	The TRS P consists of the following rules:
305.07/136.69	
305.07/136.69	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.07/136.69	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.07/136.69	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.07/136.69	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.07/136.69	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.07/136.69	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.07/136.69	
305.07/136.69	The TRS R consists of the following rules:
305.07/136.69	
305.07/136.69	   down(a) -> up(f(a))
305.07/136.69	   down(f(f(x))) -> up(g(x))
305.07/136.69	   down(g(g(x))) -> up(h(x))
305.07/136.69	   down(h(h(x))) -> up(c)
305.07/136.69	   down(f(a)) -> f_flat(down(a))
305.07/136.69	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.69	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.69	   down(f(c)) -> f_flat(down(c))
305.07/136.69	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.69	   down(g(a)) -> g_flat(down(a))
305.07/136.69	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.69	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.69	   down(g(c)) -> g_flat(down(c))
305.07/136.69	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.69	   down(h(a)) -> h_flat(down(a))
305.07/136.69	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.69	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.69	   down(h(c)) -> h_flat(down(c))
305.07/136.69	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.69	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.69	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.69	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.69	
305.07/136.69	The set Q consists of the following terms:
305.07/136.69	
305.07/136.69	   down(a)
305.07/136.69	   down(f(f(x0)))
305.07/136.69	   down(g(g(x0)))
305.07/136.69	   down(h(h(x0)))
305.07/136.69	   down(f(a))
305.07/136.69	   down(f(g(x0)))
305.07/136.69	   down(f(h(x0)))
305.07/136.69	   down(f(c))
305.07/136.69	   down(f(fresh_constant))
305.07/136.69	   down(g(a))
305.07/136.69	   down(g(f(x0)))
305.07/136.69	   down(g(h(x0)))
305.07/136.69	   down(g(c))
305.07/136.69	   down(g(fresh_constant))
305.07/136.69	   down(h(a))
305.07/136.69	   down(h(f(x0)))
305.07/136.69	   down(h(g(x0)))
305.07/136.69	   down(h(c))
305.07/136.69	   down(h(fresh_constant))
305.07/136.69	   f_flat(up(x0))
305.07/136.69	   g_flat(up(x0))
305.07/136.69	   h_flat(up(x0))
305.07/136.69	
305.07/136.69	We have to consider all minimal (P,Q,R)-chains.
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(61) MNOCProof (EQUIVALENT)
305.07/136.69	We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(62)
305.07/136.69	Obligation:
305.07/136.69	Q DP problem:
305.07/136.69	The TRS P consists of the following rules:
305.07/136.69	
305.07/136.69	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.07/136.69	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.07/136.69	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.07/136.69	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.07/136.69	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.07/136.69	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.07/136.69	
305.07/136.69	The TRS R consists of the following rules:
305.07/136.69	
305.07/136.69	   down(a) -> up(f(a))
305.07/136.69	   down(f(f(x))) -> up(g(x))
305.07/136.69	   down(g(g(x))) -> up(h(x))
305.07/136.69	   down(h(h(x))) -> up(c)
305.07/136.69	   down(f(a)) -> f_flat(down(a))
305.07/136.69	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.69	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.69	   down(f(c)) -> f_flat(down(c))
305.07/136.69	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.69	   down(g(a)) -> g_flat(down(a))
305.07/136.69	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.69	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.69	   down(g(c)) -> g_flat(down(c))
305.07/136.69	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.69	   down(h(a)) -> h_flat(down(a))
305.07/136.69	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.69	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.69	   down(h(c)) -> h_flat(down(c))
305.07/136.69	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.69	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.69	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.69	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.69	
305.07/136.69	Q is empty.
305.07/136.69	We have to consider all (P,Q,R)-chains.
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(63) SplitQDPProof (EQUIVALENT)
305.07/136.69	We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem
305.07/136.69	
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(64)
305.07/136.69	Complex Obligation (AND)
305.07/136.69	
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(65)
305.07/136.69	Obligation:
305.07/136.69	Q DP problem:
305.07/136.69	The TRS P consists of the following rules:
305.07/136.69	
305.07/136.69	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.07/136.69	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.07/136.69	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.07/136.69	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.07/136.69	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.07/136.69	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.07/136.69	
305.07/136.69	The TRS R consists of the following rules:
305.07/136.69	
305.07/136.69	   down(a) -> up(f(a))
305.07/136.69	   down(f(f(x))) -> up(g(x))
305.07/136.69	   down(g(g(x))) -> up(h(x))
305.07/136.69	   down(h(h(x))) -> up(c)
305.07/136.69	   down(f(a)) -> f_flat(down(a))
305.07/136.69	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.07/136.69	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.07/136.69	   down(f(c)) -> f_flat(down(c))
305.07/136.69	   down(f(fresh_constant)) -> f_flat(down(fresh_constant))
305.07/136.69	   down(g(a)) -> g_flat(down(a))
305.07/136.69	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.07/136.69	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.07/136.69	   down(g(c)) -> g_flat(down(c))
305.07/136.69	   down(g(fresh_constant)) -> g_flat(down(fresh_constant))
305.07/136.69	   down(h(a)) -> h_flat(down(a))
305.07/136.69	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.07/136.69	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.07/136.69	   down(h(c)) -> h_flat(down(c))
305.07/136.69	   down(h(fresh_constant)) -> h_flat(down(fresh_constant))
305.07/136.69	   h_flat(up(x_1)) -> up(h(x_1))
305.07/136.69	   g_flat(up(x_1)) -> up(g(x_1))
305.07/136.69	   f_flat(up(x_1)) -> up(f(x_1))
305.07/136.69	
305.07/136.69	The set Q consists of the following terms:
305.07/136.69	
305.07/136.69	   down(a)
305.07/136.69	   down(f(f(x0)))
305.07/136.69	   down(g(g(x0)))
305.07/136.69	   down(h(h(x0)))
305.07/136.69	   down(f(a))
305.07/136.69	   down(f(g(x0)))
305.07/136.69	   down(f(h(x0)))
305.07/136.69	   down(f(c))
305.07/136.69	   down(f(fresh_constant))
305.07/136.69	   down(g(a))
305.07/136.69	   down(g(f(x0)))
305.07/136.69	   down(g(h(x0)))
305.07/136.69	   down(g(c))
305.07/136.69	   down(g(fresh_constant))
305.07/136.69	   down(h(a))
305.07/136.69	   down(h(f(x0)))
305.07/136.69	   down(h(g(x0)))
305.07/136.69	   down(h(c))
305.07/136.69	   down(h(fresh_constant))
305.07/136.69	   f_flat(up(x0))
305.07/136.69	   g_flat(up(x0))
305.07/136.69	   h_flat(up(x0))
305.07/136.69	
305.07/136.69	We have to consider all minimal (P,Q,R)-chains.
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(66) SemLabProof (SOUND)
305.07/136.69	We found the following model for the rules of the TRSs R and P.
305.07/136.69	Interpretation over the domain with elements from 0 to 1.
305.07/136.69	h_flat: 0
305.07/136.69	a: 0
305.07/136.69	c: 0
305.07/136.69	down: 0
305.07/136.69	f: 0
305.07/136.69	fresh_constant: 1
305.07/136.69	up: 0
305.07/136.69	h: 0
305.07/136.69	f_flat: 0
305.07/136.69	TOP: 0
305.07/136.69	g_flat: 0
305.07/136.69	g: 0
305.07/136.69	By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(67)
305.07/136.69	Obligation:
305.07/136.69	Q DP problem:
305.07/136.69	The TRS P consists of the following rules:
305.07/136.69	
305.07/136.69	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
305.07/136.69	   TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0))))
305.07/136.69	   TOP.0(up.0(f.0(h.0(x0)))) -> TOP.0(f_flat.0(down.0(h.0(x0))))
305.07/136.69	   TOP.0(up.0(f.0(h.1(x0)))) -> TOP.0(f_flat.0(down.0(h.1(x0))))
305.07/136.69	   TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0))))
305.07/136.69	   TOP.0(up.0(g.0(f.1(x0)))) -> TOP.0(g_flat.0(down.0(f.1(x0))))
305.07/136.69	   TOP.0(up.0(g.0(h.0(x0)))) -> TOP.0(g_flat.0(down.0(h.0(x0))))
305.07/136.69	   TOP.0(up.0(g.0(h.1(x0)))) -> TOP.0(g_flat.0(down.0(h.1(x0))))
305.07/136.69	   TOP.0(up.0(h.0(f.0(x0)))) -> TOP.0(h_flat.0(down.0(f.0(x0))))
305.07/136.69	   TOP.0(up.0(h.0(f.1(x0)))) -> TOP.0(h_flat.0(down.0(f.1(x0))))
305.07/136.69	   TOP.0(up.0(h.0(g.0(x0)))) -> TOP.0(h_flat.0(down.0(g.0(x0))))
305.07/136.69	   TOP.0(up.0(h.0(g.1(x0)))) -> TOP.0(h_flat.0(down.0(g.1(x0))))
305.07/136.69	
305.07/136.69	The TRS R consists of the following rules:
305.07/136.69	
305.07/136.69	   down.0(a.) -> up.0(f.0(a.))
305.07/136.69	   down.0(f.0(f.0(x))) -> up.0(g.0(x))
305.07/136.69	   down.0(f.0(f.1(x))) -> up.0(g.1(x))
305.07/136.69	   down.0(g.0(g.0(x))) -> up.0(h.0(x))
305.07/136.69	   down.0(g.0(g.1(x))) -> up.0(h.1(x))
305.07/136.69	   down.0(h.0(h.0(x))) -> up.0(c.)
305.07/136.69	   down.0(h.0(h.1(x))) -> up.0(c.)
305.07/136.69	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
305.07/136.69	   down.0(f.0(g.0(y5))) -> f_flat.0(down.0(g.0(y5)))
305.07/136.69	   down.0(f.0(g.1(y5))) -> f_flat.0(down.0(g.1(y5)))
305.07/136.69	   down.0(f.0(h.0(y6))) -> f_flat.0(down.0(h.0(y6)))
305.07/136.69	   down.0(f.0(h.1(y6))) -> f_flat.0(down.0(h.1(y6)))
305.07/136.69	   down.0(f.0(c.)) -> f_flat.0(down.0(c.))
305.07/136.69	   down.0(f.1(fresh_constant.)) -> f_flat.0(down.1(fresh_constant.))
305.07/136.69	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
305.07/136.69	   down.0(g.0(f.0(y8))) -> g_flat.0(down.0(f.0(y8)))
305.07/136.69	   down.0(g.0(f.1(y8))) -> g_flat.0(down.0(f.1(y8)))
305.07/136.69	   down.0(g.0(h.0(y10))) -> g_flat.0(down.0(h.0(y10)))
305.07/136.69	   down.0(g.0(h.1(y10))) -> g_flat.0(down.0(h.1(y10)))
305.07/136.69	   down.0(g.0(c.)) -> g_flat.0(down.0(c.))
305.07/136.69	   down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.))
305.07/136.69	   down.0(h.0(a.)) -> h_flat.0(down.0(a.))
305.07/136.69	   down.0(h.0(f.0(y12))) -> h_flat.0(down.0(f.0(y12)))
305.07/136.69	   down.0(h.0(f.1(y12))) -> h_flat.0(down.0(f.1(y12)))
305.07/136.69	   down.0(h.0(g.0(y13))) -> h_flat.0(down.0(g.0(y13)))
305.07/136.69	   down.0(h.0(g.1(y13))) -> h_flat.0(down.0(g.1(y13)))
305.07/136.69	   down.0(h.0(c.)) -> h_flat.0(down.0(c.))
305.07/136.69	   down.0(h.1(fresh_constant.)) -> h_flat.0(down.1(fresh_constant.))
305.07/136.69	   h_flat.0(up.0(x_1)) -> up.0(h.0(x_1))
305.07/136.69	   h_flat.0(up.1(x_1)) -> up.0(h.1(x_1))
305.07/136.69	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
305.07/136.69	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
305.07/136.69	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
305.07/136.69	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
305.07/136.69	
305.07/136.69	The set Q consists of the following terms:
305.07/136.69	
305.07/136.69	   down.0(a.)
305.07/136.69	   down.0(f.0(f.0(x0)))
305.07/136.69	   down.0(f.0(f.1(x0)))
305.07/136.69	   down.0(g.0(g.0(x0)))
305.07/136.69	   down.0(g.0(g.1(x0)))
305.07/136.69	   down.0(h.0(h.0(x0)))
305.07/136.69	   down.0(h.0(h.1(x0)))
305.07/136.69	   down.0(f.0(a.))
305.07/136.69	   down.0(f.0(g.0(x0)))
305.07/136.69	   down.0(f.0(g.1(x0)))
305.07/136.69	   down.0(f.0(h.0(x0)))
305.07/136.69	   down.0(f.0(h.1(x0)))
305.07/136.69	   down.0(f.0(c.))
305.07/136.69	   down.0(f.1(fresh_constant.))
305.07/136.69	   down.0(g.0(a.))
305.07/136.69	   down.0(g.0(f.0(x0)))
305.07/136.69	   down.0(g.0(f.1(x0)))
305.07/136.69	   down.0(g.0(h.0(x0)))
305.07/136.69	   down.0(g.0(h.1(x0)))
305.07/136.69	   down.0(g.0(c.))
305.07/136.69	   down.0(g.1(fresh_constant.))
305.07/136.69	   down.0(h.0(a.))
305.07/136.69	   down.0(h.0(f.0(x0)))
305.07/136.69	   down.0(h.0(f.1(x0)))
305.07/136.69	   down.0(h.0(g.0(x0)))
305.07/136.69	   down.0(h.0(g.1(x0)))
305.07/136.69	   down.0(h.0(c.))
305.07/136.69	   down.0(h.1(fresh_constant.))
305.07/136.69	   f_flat.0(up.0(x0))
305.07/136.69	   f_flat.0(up.1(x0))
305.07/136.69	   g_flat.0(up.0(x0))
305.07/136.69	   g_flat.0(up.1(x0))
305.07/136.69	   h_flat.0(up.0(x0))
305.07/136.69	   h_flat.0(up.1(x0))
305.07/136.69	
305.07/136.69	We have to consider all minimal (P,Q,R)-chains.
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(68) UsableRulesReductionPairsProof (EQUIVALENT)
305.07/136.69	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.
305.07/136.69	
305.07/136.69	No dependency pairs are removed.
305.07/136.69	
305.07/136.69	The following rules are removed from R:
305.07/136.69	
305.07/136.69	   h_flat.0(up.1(x_1)) -> up.0(h.1(x_1))
305.07/136.69	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
305.07/136.69	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
305.07/136.69	Used ordering: POLO with Polynomial interpretation [POLO]:
305.07/136.69	
305.07/136.69	   POL(TOP.0(x_1)) = x_1
305.07/136.69	   POL(a.) = 0
305.07/136.69	   POL(c.) = 0
305.07/136.69	   POL(down.0(x_1)) = 1 + x_1
305.07/136.69	   POL(down.1(x_1)) = 1 + x_1
305.07/136.69	   POL(f.0(x_1)) = x_1
305.07/136.69	   POL(f.1(x_1)) = x_1
305.07/136.69	   POL(f_flat.0(x_1)) = x_1
305.07/136.69	   POL(fresh_constant.) = 0
305.07/136.69	   POL(g.0(x_1)) = x_1
305.07/136.69	   POL(g.1(x_1)) = x_1
305.07/136.69	   POL(g_flat.0(x_1)) = x_1
305.07/136.69	   POL(h.0(x_1)) = x_1
305.07/136.69	   POL(h.1(x_1)) = x_1
305.07/136.69	   POL(h_flat.0(x_1)) = x_1
305.07/136.69	   POL(up.0(x_1)) = 1 + x_1
305.07/136.69	   POL(up.1(x_1)) = 1 + x_1
305.07/136.69	
305.07/136.69	
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(69)
305.07/136.69	Obligation:
305.07/136.69	Q DP problem:
305.07/136.69	The TRS P consists of the following rules:
305.07/136.69	
305.07/136.69	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
305.07/136.69	   TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0))))
305.07/136.69	   TOP.0(up.0(f.0(h.0(x0)))) -> TOP.0(f_flat.0(down.0(h.0(x0))))
305.07/136.69	   TOP.0(up.0(f.0(h.1(x0)))) -> TOP.0(f_flat.0(down.0(h.1(x0))))
305.07/136.69	   TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0))))
305.07/136.69	   TOP.0(up.0(g.0(f.1(x0)))) -> TOP.0(g_flat.0(down.0(f.1(x0))))
305.07/136.69	   TOP.0(up.0(g.0(h.0(x0)))) -> TOP.0(g_flat.0(down.0(h.0(x0))))
305.07/136.69	   TOP.0(up.0(g.0(h.1(x0)))) -> TOP.0(g_flat.0(down.0(h.1(x0))))
305.07/136.69	   TOP.0(up.0(h.0(f.0(x0)))) -> TOP.0(h_flat.0(down.0(f.0(x0))))
305.07/136.69	   TOP.0(up.0(h.0(f.1(x0)))) -> TOP.0(h_flat.0(down.0(f.1(x0))))
305.07/136.69	   TOP.0(up.0(h.0(g.0(x0)))) -> TOP.0(h_flat.0(down.0(g.0(x0))))
305.07/136.69	   TOP.0(up.0(h.0(g.1(x0)))) -> TOP.0(h_flat.0(down.0(g.1(x0))))
305.07/136.69	
305.07/136.69	The TRS R consists of the following rules:
305.07/136.69	
305.07/136.69	   down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.))
305.07/136.69	   h_flat.0(up.0(x_1)) -> up.0(h.0(x_1))
305.07/136.69	   down.0(g.0(g.0(x))) -> up.0(h.0(x))
305.07/136.69	   down.0(g.0(g.1(x))) -> up.0(h.1(x))
305.07/136.69	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
305.07/136.69	   down.0(g.0(f.0(y8))) -> g_flat.0(down.0(f.0(y8)))
305.07/136.69	   down.0(g.0(f.1(y8))) -> g_flat.0(down.0(f.1(y8)))
305.07/136.69	   down.0(g.0(h.0(y10))) -> g_flat.0(down.0(h.0(y10)))
305.07/136.69	   down.0(g.0(h.1(y10))) -> g_flat.0(down.0(h.1(y10)))
305.07/136.69	   down.0(g.0(c.)) -> g_flat.0(down.0(c.))
305.07/136.69	   down.0(h.1(fresh_constant.)) -> h_flat.0(down.1(fresh_constant.))
305.07/136.69	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
305.07/136.69	   down.0(h.0(h.0(x))) -> up.0(c.)
305.07/136.69	   down.0(h.0(h.1(x))) -> up.0(c.)
305.07/136.69	   down.0(h.0(a.)) -> h_flat.0(down.0(a.))
305.07/136.69	   down.0(h.0(f.0(y12))) -> h_flat.0(down.0(f.0(y12)))
305.07/136.69	   down.0(h.0(f.1(y12))) -> h_flat.0(down.0(f.1(y12)))
305.07/136.69	   down.0(h.0(g.0(y13))) -> h_flat.0(down.0(g.0(y13)))
305.07/136.69	   down.0(h.0(g.1(y13))) -> h_flat.0(down.0(g.1(y13)))
305.07/136.69	   down.0(h.0(c.)) -> h_flat.0(down.0(c.))
305.07/136.69	   down.0(f.1(fresh_constant.)) -> f_flat.0(down.1(fresh_constant.))
305.07/136.69	   down.0(f.0(f.0(x))) -> up.0(g.0(x))
305.07/136.69	   down.0(f.0(f.1(x))) -> up.0(g.1(x))
305.07/136.69	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
305.07/136.69	   down.0(f.0(g.0(y5))) -> f_flat.0(down.0(g.0(y5)))
305.07/136.69	   down.0(f.0(g.1(y5))) -> f_flat.0(down.0(g.1(y5)))
305.07/136.69	   down.0(f.0(h.0(y6))) -> f_flat.0(down.0(h.0(y6)))
305.07/136.69	   down.0(f.0(h.1(y6))) -> f_flat.0(down.0(h.1(y6)))
305.07/136.69	   down.0(f.0(c.)) -> f_flat.0(down.0(c.))
305.07/136.69	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
305.07/136.69	   down.0(a.) -> up.0(f.0(a.))
305.07/136.69	
305.07/136.69	The set Q consists of the following terms:
305.07/136.69	
305.07/136.69	   down.0(a.)
305.07/136.69	   down.0(f.0(f.0(x0)))
305.07/136.69	   down.0(f.0(f.1(x0)))
305.07/136.69	   down.0(g.0(g.0(x0)))
305.07/136.69	   down.0(g.0(g.1(x0)))
305.07/136.69	   down.0(h.0(h.0(x0)))
305.07/136.69	   down.0(h.0(h.1(x0)))
305.07/136.69	   down.0(f.0(a.))
305.07/136.69	   down.0(f.0(g.0(x0)))
305.07/136.69	   down.0(f.0(g.1(x0)))
305.07/136.69	   down.0(f.0(h.0(x0)))
305.07/136.69	   down.0(f.0(h.1(x0)))
305.07/136.69	   down.0(f.0(c.))
305.07/136.69	   down.0(f.1(fresh_constant.))
305.07/136.69	   down.0(g.0(a.))
305.07/136.69	   down.0(g.0(f.0(x0)))
305.07/136.69	   down.0(g.0(f.1(x0)))
305.07/136.69	   down.0(g.0(h.0(x0)))
305.07/136.69	   down.0(g.0(h.1(x0)))
305.07/136.69	   down.0(g.0(c.))
305.07/136.69	   down.0(g.1(fresh_constant.))
305.07/136.69	   down.0(h.0(a.))
305.07/136.69	   down.0(h.0(f.0(x0)))
305.07/136.69	   down.0(h.0(f.1(x0)))
305.07/136.69	   down.0(h.0(g.0(x0)))
305.07/136.69	   down.0(h.0(g.1(x0)))
305.07/136.69	   down.0(h.0(c.))
305.07/136.69	   down.0(h.1(fresh_constant.))
305.07/136.69	   f_flat.0(up.0(x0))
305.07/136.69	   f_flat.0(up.1(x0))
305.07/136.69	   g_flat.0(up.0(x0))
305.07/136.69	   g_flat.0(up.1(x0))
305.07/136.69	   h_flat.0(up.0(x0))
305.07/136.69	   h_flat.0(up.1(x0))
305.07/136.69	
305.07/136.69	We have to consider all minimal (P,Q,R)-chains.
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(70) MRRProof (EQUIVALENT)
305.07/136.69	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.
305.07/136.69	
305.07/136.69	
305.07/136.69	Strictly oriented rules of the TRS R:
305.07/136.69	
305.07/136.69	   down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.))
305.07/136.69	   down.0(h.1(fresh_constant.)) -> h_flat.0(down.1(fresh_constant.))
305.07/136.69	   down.0(h.0(h.1(x))) -> up.0(c.)
305.07/136.69	   down.0(f.1(fresh_constant.)) -> f_flat.0(down.1(fresh_constant.))
305.07/136.69	
305.07/136.69	Used ordering: Polynomial interpretation [POLO]:
305.07/136.69	
305.07/136.69	   POL(TOP.0(x_1)) = x_1
305.07/136.69	   POL(a.) = 0
305.07/136.69	   POL(c.) = 0
305.07/136.69	   POL(down.0(x_1)) = x_1
305.07/136.69	   POL(down.1(x_1)) = x_1
305.07/136.69	   POL(f.0(x_1)) = x_1
305.07/136.69	   POL(f.1(x_1)) = 1 + x_1
305.07/136.69	   POL(f_flat.0(x_1)) = x_1
305.07/136.69	   POL(fresh_constant.) = 0
305.07/136.69	   POL(g.0(x_1)) = x_1
305.07/136.69	   POL(g.1(x_1)) = 1 + x_1
305.07/136.69	   POL(g_flat.0(x_1)) = x_1
305.07/136.69	   POL(h.0(x_1)) = x_1
305.07/136.69	   POL(h.1(x_1)) = 1 + x_1
305.07/136.69	   POL(h_flat.0(x_1)) = x_1
305.07/136.69	   POL(up.0(x_1)) = x_1
305.07/136.69	
305.07/136.69	
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(71)
305.07/136.69	Obligation:
305.07/136.69	Q DP problem:
305.07/136.69	The TRS P consists of the following rules:
305.07/136.69	
305.07/136.69	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
305.07/136.69	   TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0))))
305.07/136.69	   TOP.0(up.0(f.0(h.0(x0)))) -> TOP.0(f_flat.0(down.0(h.0(x0))))
305.07/136.69	   TOP.0(up.0(f.0(h.1(x0)))) -> TOP.0(f_flat.0(down.0(h.1(x0))))
305.07/136.69	   TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0))))
305.07/136.69	   TOP.0(up.0(g.0(f.1(x0)))) -> TOP.0(g_flat.0(down.0(f.1(x0))))
305.07/136.69	   TOP.0(up.0(g.0(h.0(x0)))) -> TOP.0(g_flat.0(down.0(h.0(x0))))
305.07/136.69	   TOP.0(up.0(g.0(h.1(x0)))) -> TOP.0(g_flat.0(down.0(h.1(x0))))
305.07/136.69	   TOP.0(up.0(h.0(f.0(x0)))) -> TOP.0(h_flat.0(down.0(f.0(x0))))
305.07/136.69	   TOP.0(up.0(h.0(f.1(x0)))) -> TOP.0(h_flat.0(down.0(f.1(x0))))
305.07/136.69	   TOP.0(up.0(h.0(g.0(x0)))) -> TOP.0(h_flat.0(down.0(g.0(x0))))
305.07/136.69	   TOP.0(up.0(h.0(g.1(x0)))) -> TOP.0(h_flat.0(down.0(g.1(x0))))
305.07/136.69	
305.07/136.69	The TRS R consists of the following rules:
305.07/136.69	
305.07/136.69	   h_flat.0(up.0(x_1)) -> up.0(h.0(x_1))
305.07/136.69	   down.0(g.0(g.0(x))) -> up.0(h.0(x))
305.07/136.69	   down.0(g.0(g.1(x))) -> up.0(h.1(x))
305.07/136.69	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
305.07/136.69	   down.0(g.0(f.0(y8))) -> g_flat.0(down.0(f.0(y8)))
305.07/136.69	   down.0(g.0(f.1(y8))) -> g_flat.0(down.0(f.1(y8)))
305.07/136.69	   down.0(g.0(h.0(y10))) -> g_flat.0(down.0(h.0(y10)))
305.07/136.69	   down.0(g.0(h.1(y10))) -> g_flat.0(down.0(h.1(y10)))
305.07/136.69	   down.0(g.0(c.)) -> g_flat.0(down.0(c.))
305.07/136.69	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
305.07/136.69	   down.0(h.0(h.0(x))) -> up.0(c.)
305.07/136.69	   down.0(h.0(a.)) -> h_flat.0(down.0(a.))
305.07/136.69	   down.0(h.0(f.0(y12))) -> h_flat.0(down.0(f.0(y12)))
305.07/136.69	   down.0(h.0(f.1(y12))) -> h_flat.0(down.0(f.1(y12)))
305.07/136.69	   down.0(h.0(g.0(y13))) -> h_flat.0(down.0(g.0(y13)))
305.07/136.69	   down.0(h.0(g.1(y13))) -> h_flat.0(down.0(g.1(y13)))
305.07/136.69	   down.0(h.0(c.)) -> h_flat.0(down.0(c.))
305.07/136.69	   down.0(f.0(f.0(x))) -> up.0(g.0(x))
305.07/136.69	   down.0(f.0(f.1(x))) -> up.0(g.1(x))
305.07/136.69	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
305.07/136.69	   down.0(f.0(g.0(y5))) -> f_flat.0(down.0(g.0(y5)))
305.07/136.69	   down.0(f.0(g.1(y5))) -> f_flat.0(down.0(g.1(y5)))
305.07/136.69	   down.0(f.0(h.0(y6))) -> f_flat.0(down.0(h.0(y6)))
305.07/136.69	   down.0(f.0(h.1(y6))) -> f_flat.0(down.0(h.1(y6)))
305.07/136.69	   down.0(f.0(c.)) -> f_flat.0(down.0(c.))
305.07/136.69	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
305.07/136.69	   down.0(a.) -> up.0(f.0(a.))
305.07/136.69	
305.07/136.69	The set Q consists of the following terms:
305.07/136.69	
305.07/136.69	   down.0(a.)
305.07/136.69	   down.0(f.0(f.0(x0)))
305.07/136.69	   down.0(f.0(f.1(x0)))
305.07/136.69	   down.0(g.0(g.0(x0)))
305.07/136.69	   down.0(g.0(g.1(x0)))
305.07/136.69	   down.0(h.0(h.0(x0)))
305.07/136.69	   down.0(h.0(h.1(x0)))
305.07/136.69	   down.0(f.0(a.))
305.07/136.69	   down.0(f.0(g.0(x0)))
305.07/136.69	   down.0(f.0(g.1(x0)))
305.07/136.69	   down.0(f.0(h.0(x0)))
305.07/136.69	   down.0(f.0(h.1(x0)))
305.07/136.69	   down.0(f.0(c.))
305.07/136.69	   down.0(f.1(fresh_constant.))
305.07/136.69	   down.0(g.0(a.))
305.07/136.69	   down.0(g.0(f.0(x0)))
305.07/136.69	   down.0(g.0(f.1(x0)))
305.07/136.69	   down.0(g.0(h.0(x0)))
305.07/136.69	   down.0(g.0(h.1(x0)))
305.07/136.69	   down.0(g.0(c.))
305.07/136.69	   down.0(g.1(fresh_constant.))
305.07/136.69	   down.0(h.0(a.))
305.07/136.69	   down.0(h.0(f.0(x0)))
305.07/136.69	   down.0(h.0(f.1(x0)))
305.07/136.69	   down.0(h.0(g.0(x0)))
305.07/136.69	   down.0(h.0(g.1(x0)))
305.07/136.69	   down.0(h.0(c.))
305.07/136.69	   down.0(h.1(fresh_constant.))
305.07/136.69	   f_flat.0(up.0(x0))
305.07/136.69	   f_flat.0(up.1(x0))
305.07/136.69	   g_flat.0(up.0(x0))
305.07/136.69	   g_flat.0(up.1(x0))
305.07/136.69	   h_flat.0(up.0(x0))
305.07/136.69	   h_flat.0(up.1(x0))
305.07/136.69	
305.07/136.69	We have to consider all minimal (P,Q,R)-chains.
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(72) DependencyGraphProof (EQUIVALENT)
305.07/136.69	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(73)
305.07/136.69	Obligation:
305.07/136.69	Q DP problem:
305.07/136.69	The TRS P consists of the following rules:
305.07/136.69	
305.07/136.69	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
305.07/136.69	   TOP.0(up.0(f.0(h.0(x0)))) -> TOP.0(f_flat.0(down.0(h.0(x0))))
305.07/136.69	   TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0))))
305.07/136.69	   TOP.0(up.0(g.0(h.0(x0)))) -> TOP.0(g_flat.0(down.0(h.0(x0))))
305.07/136.69	   TOP.0(up.0(h.0(f.0(x0)))) -> TOP.0(h_flat.0(down.0(f.0(x0))))
305.07/136.69	   TOP.0(up.0(h.0(g.0(x0)))) -> TOP.0(h_flat.0(down.0(g.0(x0))))
305.07/136.69	
305.07/136.69	The TRS R consists of the following rules:
305.07/136.69	
305.07/136.69	   h_flat.0(up.0(x_1)) -> up.0(h.0(x_1))
305.07/136.69	   down.0(g.0(g.0(x))) -> up.0(h.0(x))
305.07/136.69	   down.0(g.0(g.1(x))) -> up.0(h.1(x))
305.07/136.69	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
305.07/136.69	   down.0(g.0(f.0(y8))) -> g_flat.0(down.0(f.0(y8)))
305.07/136.69	   down.0(g.0(f.1(y8))) -> g_flat.0(down.0(f.1(y8)))
305.07/136.69	   down.0(g.0(h.0(y10))) -> g_flat.0(down.0(h.0(y10)))
305.07/136.69	   down.0(g.0(h.1(y10))) -> g_flat.0(down.0(h.1(y10)))
305.07/136.69	   down.0(g.0(c.)) -> g_flat.0(down.0(c.))
305.07/136.69	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
305.07/136.69	   down.0(h.0(h.0(x))) -> up.0(c.)
305.07/136.69	   down.0(h.0(a.)) -> h_flat.0(down.0(a.))
305.07/136.69	   down.0(h.0(f.0(y12))) -> h_flat.0(down.0(f.0(y12)))
305.07/136.69	   down.0(h.0(f.1(y12))) -> h_flat.0(down.0(f.1(y12)))
305.07/136.69	   down.0(h.0(g.0(y13))) -> h_flat.0(down.0(g.0(y13)))
305.07/136.69	   down.0(h.0(g.1(y13))) -> h_flat.0(down.0(g.1(y13)))
305.07/136.69	   down.0(h.0(c.)) -> h_flat.0(down.0(c.))
305.07/136.69	   down.0(f.0(f.0(x))) -> up.0(g.0(x))
305.07/136.69	   down.0(f.0(f.1(x))) -> up.0(g.1(x))
305.07/136.69	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
305.07/136.69	   down.0(f.0(g.0(y5))) -> f_flat.0(down.0(g.0(y5)))
305.07/136.69	   down.0(f.0(g.1(y5))) -> f_flat.0(down.0(g.1(y5)))
305.07/136.69	   down.0(f.0(h.0(y6))) -> f_flat.0(down.0(h.0(y6)))
305.07/136.69	   down.0(f.0(h.1(y6))) -> f_flat.0(down.0(h.1(y6)))
305.07/136.69	   down.0(f.0(c.)) -> f_flat.0(down.0(c.))
305.07/136.69	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
305.07/136.69	   down.0(a.) -> up.0(f.0(a.))
305.07/136.69	
305.07/136.69	The set Q consists of the following terms:
305.07/136.69	
305.07/136.69	   down.0(a.)
305.07/136.69	   down.0(f.0(f.0(x0)))
305.07/136.69	   down.0(f.0(f.1(x0)))
305.07/136.69	   down.0(g.0(g.0(x0)))
305.07/136.69	   down.0(g.0(g.1(x0)))
305.07/136.69	   down.0(h.0(h.0(x0)))
305.07/136.69	   down.0(h.0(h.1(x0)))
305.07/136.69	   down.0(f.0(a.))
305.07/136.69	   down.0(f.0(g.0(x0)))
305.07/136.69	   down.0(f.0(g.1(x0)))
305.07/136.69	   down.0(f.0(h.0(x0)))
305.07/136.69	   down.0(f.0(h.1(x0)))
305.07/136.69	   down.0(f.0(c.))
305.07/136.69	   down.0(f.1(fresh_constant.))
305.07/136.69	   down.0(g.0(a.))
305.07/136.69	   down.0(g.0(f.0(x0)))
305.07/136.69	   down.0(g.0(f.1(x0)))
305.07/136.69	   down.0(g.0(h.0(x0)))
305.07/136.69	   down.0(g.0(h.1(x0)))
305.07/136.69	   down.0(g.0(c.))
305.07/136.69	   down.0(g.1(fresh_constant.))
305.07/136.69	   down.0(h.0(a.))
305.07/136.69	   down.0(h.0(f.0(x0)))
305.07/136.69	   down.0(h.0(f.1(x0)))
305.07/136.69	   down.0(h.0(g.0(x0)))
305.07/136.69	   down.0(h.0(g.1(x0)))
305.07/136.69	   down.0(h.0(c.))
305.07/136.69	   down.0(h.1(fresh_constant.))
305.07/136.69	   f_flat.0(up.0(x0))
305.07/136.69	   f_flat.0(up.1(x0))
305.07/136.69	   g_flat.0(up.0(x0))
305.07/136.69	   g_flat.0(up.1(x0))
305.07/136.69	   h_flat.0(up.0(x0))
305.07/136.69	   h_flat.0(up.1(x0))
305.07/136.69	
305.07/136.69	We have to consider all minimal (P,Q,R)-chains.
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(74) MRRProof (EQUIVALENT)
305.07/136.69	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.
305.07/136.69	
305.07/136.69	
305.07/136.69	Strictly oriented rules of the TRS R:
305.07/136.69	
305.07/136.69	   down.0(g.0(g.1(x))) -> up.0(h.1(x))
305.07/136.69	
305.07/136.69	Used ordering: Polynomial interpretation [POLO]:
305.07/136.69	
305.07/136.69	   POL(TOP.0(x_1)) = x_1
305.07/136.69	   POL(a.) = 0
305.07/136.69	   POL(c.) = 0
305.07/136.69	   POL(down.0(x_1)) = x_1
305.07/136.69	   POL(f.0(x_1)) = x_1
305.07/136.69	   POL(f.1(x_1)) = 1 + x_1
305.07/136.69	   POL(f_flat.0(x_1)) = x_1
305.07/136.69	   POL(g.0(x_1)) = x_1
305.07/136.69	   POL(g.1(x_1)) = 1 + x_1
305.07/136.69	   POL(g_flat.0(x_1)) = x_1
305.07/136.69	   POL(h.0(x_1)) = x_1
305.07/136.69	   POL(h.1(x_1)) = x_1
305.07/136.69	   POL(h_flat.0(x_1)) = x_1
305.07/136.69	   POL(up.0(x_1)) = x_1
305.07/136.69	
305.07/136.69	
305.07/136.69	----------------------------------------
305.07/136.69	
305.07/136.69	(75)
305.07/136.69	Obligation:
305.07/136.69	Q DP problem:
305.07/136.69	The TRS P consists of the following rules:
305.07/136.69	
305.07/136.69	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
305.07/136.69	   TOP.0(up.0(f.0(h.0(x0)))) -> TOP.0(f_flat.0(down.0(h.0(x0))))
305.07/136.69	   TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0))))
305.07/136.69	   TOP.0(up.0(g.0(h.0(x0)))) -> TOP.0(g_flat.0(down.0(h.0(x0))))
305.07/136.69	   TOP.0(up.0(h.0(f.0(x0)))) -> TOP.0(h_flat.0(down.0(f.0(x0))))
305.07/136.69	   TOP.0(up.0(h.0(g.0(x0)))) -> TOP.0(h_flat.0(down.0(g.0(x0))))
305.07/136.69	
305.07/136.69	The TRS R consists of the following rules:
305.07/136.69	
305.07/136.69	   h_flat.0(up.0(x_1)) -> up.0(h.0(x_1))
305.07/136.69	   down.0(g.0(g.0(x))) -> up.0(h.0(x))
305.07/136.69	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
305.07/136.69	   down.0(g.0(f.0(y8))) -> g_flat.0(down.0(f.0(y8)))
305.07/136.69	   down.0(g.0(f.1(y8))) -> g_flat.0(down.0(f.1(y8)))
305.07/136.69	   down.0(g.0(h.0(y10))) -> g_flat.0(down.0(h.0(y10)))
305.07/136.69	   down.0(g.0(h.1(y10))) -> g_flat.0(down.0(h.1(y10)))
305.07/136.69	   down.0(g.0(c.)) -> g_flat.0(down.0(c.))
305.07/136.69	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
305.07/136.69	   down.0(h.0(h.0(x))) -> up.0(c.)
305.07/136.69	   down.0(h.0(a.)) -> h_flat.0(down.0(a.))
305.07/136.69	   down.0(h.0(f.0(y12))) -> h_flat.0(down.0(f.0(y12)))
305.07/136.69	   down.0(h.0(f.1(y12))) -> h_flat.0(down.0(f.1(y12)))
305.07/136.69	   down.0(h.0(g.0(y13))) -> h_flat.0(down.0(g.0(y13)))
305.07/136.69	   down.0(h.0(g.1(y13))) -> h_flat.0(down.0(g.1(y13)))
305.07/136.69	   down.0(h.0(c.)) -> h_flat.0(down.0(c.))
305.07/136.69	   down.0(f.0(f.0(x))) -> up.0(g.0(x))
305.07/136.69	   down.0(f.0(f.1(x))) -> up.0(g.1(x))
305.07/136.69	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
305.07/136.69	   down.0(f.0(g.0(y5))) -> f_flat.0(down.0(g.0(y5)))
305.07/136.69	   down.0(f.0(g.1(y5))) -> f_flat.0(down.0(g.1(y5)))
305.07/136.69	   down.0(f.0(h.0(y6))) -> f_flat.0(down.0(h.0(y6)))
305.07/136.69	   down.0(f.0(h.1(y6))) -> f_flat.0(down.0(h.1(y6)))
305.07/136.69	   down.0(f.0(c.)) -> f_flat.0(down.0(c.))
305.07/136.69	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
305.07/136.69	   down.0(a.) -> up.0(f.0(a.))
305.07/136.69	
305.07/136.69	The set Q consists of the following terms:
305.07/136.69	
305.07/136.69	   down.0(a.)
305.07/136.69	   down.0(f.0(f.0(x0)))
305.07/136.69	   down.0(f.0(f.1(x0)))
305.07/136.69	   down.0(g.0(g.0(x0)))
305.07/136.69	   down.0(g.0(g.1(x0)))
305.07/136.69	   down.0(h.0(h.0(x0)))
305.07/136.69	   down.0(h.0(h.1(x0)))
305.07/136.69	   down.0(f.0(a.))
305.07/136.69	   down.0(f.0(g.0(x0)))
305.07/136.69	   down.0(f.0(g.1(x0)))
305.07/136.69	   down.0(f.0(h.0(x0)))
305.07/136.69	   down.0(f.0(h.1(x0)))
305.07/136.69	   down.0(f.0(c.))
305.07/136.69	   down.0(f.1(fresh_constant.))
305.07/136.69	   down.0(g.0(a.))
305.07/136.69	   down.0(g.0(f.0(x0)))
305.07/136.69	   down.0(g.0(f.1(x0)))
305.07/136.69	   down.0(g.0(h.0(x0)))
305.07/136.69	   down.0(g.0(h.1(x0)))
305.07/136.69	   down.0(g.0(c.))
305.07/136.69	   down.0(g.1(fresh_constant.))
305.07/136.69	   down.0(h.0(a.))
305.07/136.69	   down.0(h.0(f.0(x0)))
305.07/136.69	   down.0(h.0(f.1(x0)))
305.07/136.69	   down.0(h.0(g.0(x0)))
305.07/136.69	   down.0(h.0(g.1(x0)))
305.07/136.69	   down.0(h.0(c.))
305.07/136.69	   down.0(h.1(fresh_constant.))
305.20/136.69	   f_flat.0(up.0(x0))
305.20/136.69	   f_flat.0(up.1(x0))
305.20/136.69	   g_flat.0(up.0(x0))
305.20/136.69	   g_flat.0(up.1(x0))
305.20/136.69	   h_flat.0(up.0(x0))
305.20/136.69	   h_flat.0(up.1(x0))
305.20/136.69	
305.20/136.69	We have to consider all minimal (P,Q,R)-chains.
305.20/136.69	----------------------------------------
305.20/136.69	
305.20/136.69	(76) MRRProof (EQUIVALENT)
305.20/136.69	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.
305.20/136.69	
305.20/136.69	
305.20/136.69	Strictly oriented rules of the TRS R:
305.20/136.69	
305.20/136.69	   down.0(f.0(f.1(x))) -> up.0(g.1(x))
305.20/136.69	
305.20/136.69	Used ordering: Polynomial interpretation [POLO]:
305.20/136.69	
305.20/136.69	   POL(TOP.0(x_1)) = x_1
305.20/136.69	   POL(a.) = 0
305.20/136.69	   POL(c.) = 0
305.20/136.69	   POL(down.0(x_1)) = 1 + x_1
305.20/136.69	   POL(f.0(x_1)) = x_1
305.20/136.69	   POL(f.1(x_1)) = 1 + x_1
305.20/136.69	   POL(f_flat.0(x_1)) = x_1
305.20/136.69	   POL(g.0(x_1)) = x_1
305.20/136.69	   POL(g.1(x_1)) = x_1
305.20/136.69	   POL(g_flat.0(x_1)) = x_1
305.20/136.69	   POL(h.0(x_1)) = x_1
305.20/136.69	   POL(h.1(x_1)) = x_1
305.20/136.69	   POL(h_flat.0(x_1)) = x_1
305.20/136.69	   POL(up.0(x_1)) = 1 + x_1
305.20/136.69	
305.20/136.69	
305.20/136.69	----------------------------------------
305.20/136.69	
305.20/136.69	(77)
305.20/136.69	Obligation:
305.20/136.69	Q DP problem:
305.20/136.69	The TRS P consists of the following rules:
305.20/136.69	
305.20/136.69	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
305.20/136.69	   TOP.0(up.0(f.0(h.0(x0)))) -> TOP.0(f_flat.0(down.0(h.0(x0))))
305.20/136.69	   TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0))))
305.20/136.69	   TOP.0(up.0(g.0(h.0(x0)))) -> TOP.0(g_flat.0(down.0(h.0(x0))))
305.20/136.69	   TOP.0(up.0(h.0(f.0(x0)))) -> TOP.0(h_flat.0(down.0(f.0(x0))))
305.20/136.69	   TOP.0(up.0(h.0(g.0(x0)))) -> TOP.0(h_flat.0(down.0(g.0(x0))))
305.20/136.69	
305.20/136.69	The TRS R consists of the following rules:
305.20/136.69	
305.20/136.69	   h_flat.0(up.0(x_1)) -> up.0(h.0(x_1))
305.20/136.69	   down.0(g.0(g.0(x))) -> up.0(h.0(x))
305.20/136.69	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
305.20/136.69	   down.0(g.0(f.0(y8))) -> g_flat.0(down.0(f.0(y8)))
305.20/136.69	   down.0(g.0(f.1(y8))) -> g_flat.0(down.0(f.1(y8)))
305.20/136.69	   down.0(g.0(h.0(y10))) -> g_flat.0(down.0(h.0(y10)))
305.20/136.69	   down.0(g.0(h.1(y10))) -> g_flat.0(down.0(h.1(y10)))
305.20/136.69	   down.0(g.0(c.)) -> g_flat.0(down.0(c.))
305.20/136.69	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
305.20/136.69	   down.0(h.0(h.0(x))) -> up.0(c.)
305.20/136.69	   down.0(h.0(a.)) -> h_flat.0(down.0(a.))
305.20/136.69	   down.0(h.0(f.0(y12))) -> h_flat.0(down.0(f.0(y12)))
305.20/136.69	   down.0(h.0(f.1(y12))) -> h_flat.0(down.0(f.1(y12)))
305.20/136.69	   down.0(h.0(g.0(y13))) -> h_flat.0(down.0(g.0(y13)))
305.20/136.69	   down.0(h.0(g.1(y13))) -> h_flat.0(down.0(g.1(y13)))
305.20/136.69	   down.0(h.0(c.)) -> h_flat.0(down.0(c.))
305.20/136.69	   down.0(f.0(f.0(x))) -> up.0(g.0(x))
305.20/136.69	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
305.20/136.69	   down.0(f.0(g.0(y5))) -> f_flat.0(down.0(g.0(y5)))
305.20/136.69	   down.0(f.0(g.1(y5))) -> f_flat.0(down.0(g.1(y5)))
305.20/136.69	   down.0(f.0(h.0(y6))) -> f_flat.0(down.0(h.0(y6)))
305.20/136.69	   down.0(f.0(h.1(y6))) -> f_flat.0(down.0(h.1(y6)))
305.20/136.69	   down.0(f.0(c.)) -> f_flat.0(down.0(c.))
305.20/136.69	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
305.20/136.69	   down.0(a.) -> up.0(f.0(a.))
305.20/136.69	
305.20/136.69	The set Q consists of the following terms:
305.20/136.69	
305.20/136.69	   down.0(a.)
305.20/136.69	   down.0(f.0(f.0(x0)))
305.20/136.69	   down.0(f.0(f.1(x0)))
305.20/136.69	   down.0(g.0(g.0(x0)))
305.20/136.69	   down.0(g.0(g.1(x0)))
305.20/136.69	   down.0(h.0(h.0(x0)))
305.20/136.69	   down.0(h.0(h.1(x0)))
305.20/136.69	   down.0(f.0(a.))
305.20/136.69	   down.0(f.0(g.0(x0)))
305.20/136.69	   down.0(f.0(g.1(x0)))
305.20/136.69	   down.0(f.0(h.0(x0)))
305.20/136.69	   down.0(f.0(h.1(x0)))
305.20/136.69	   down.0(f.0(c.))
305.20/136.69	   down.0(f.1(fresh_constant.))
305.20/136.69	   down.0(g.0(a.))
305.20/136.69	   down.0(g.0(f.0(x0)))
305.20/136.69	   down.0(g.0(f.1(x0)))
305.20/136.69	   down.0(g.0(h.0(x0)))
305.20/136.69	   down.0(g.0(h.1(x0)))
305.20/136.69	   down.0(g.0(c.))
305.20/136.69	   down.0(g.1(fresh_constant.))
305.20/136.69	   down.0(h.0(a.))
305.20/136.69	   down.0(h.0(f.0(x0)))
305.20/136.69	   down.0(h.0(f.1(x0)))
305.20/136.69	   down.0(h.0(g.0(x0)))
305.20/136.69	   down.0(h.0(g.1(x0)))
305.20/136.69	   down.0(h.0(c.))
305.20/136.69	   down.0(h.1(fresh_constant.))
305.20/136.69	   f_flat.0(up.0(x0))
305.20/136.69	   f_flat.0(up.1(x0))
305.20/136.69	   g_flat.0(up.0(x0))
305.20/136.69	   g_flat.0(up.1(x0))
305.20/136.69	   h_flat.0(up.0(x0))
305.20/136.69	   h_flat.0(up.1(x0))
305.20/136.69	
305.20/136.69	We have to consider all minimal (P,Q,R)-chains.
305.20/136.69	----------------------------------------
305.20/136.69	
305.20/136.69	(78) PisEmptyProof (SOUND)
305.20/136.69	The TRS P is empty. Hence, there is no (P,Q,R) chain.
305.20/136.69	----------------------------------------
305.20/136.69	
305.20/136.69	(79)
305.20/136.69	TRUE
305.20/136.69	
305.20/136.69	----------------------------------------
305.20/136.69	
305.20/136.69	(80)
305.20/136.69	Obligation:
305.20/136.69	Q DP problem:
305.20/136.69	The TRS P consists of the following rules:
305.20/136.69	
305.20/136.69	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.20/136.69	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.20/136.69	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.20/136.69	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.20/136.69	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.20/136.69	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.20/136.69	
305.20/136.69	The TRS R consists of the following rules:
305.20/136.69	
305.20/136.69	   h_flat(up(x_1)) -> up(h(x_1))
305.20/136.69	   down(g(g(x))) -> up(h(x))
305.20/136.69	   down(g(a)) -> g_flat(down(a))
305.20/136.69	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.20/136.69	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.20/136.69	   down(g(c)) -> g_flat(down(c))
305.20/136.69	   g_flat(up(x_1)) -> up(g(x_1))
305.20/136.69	   down(h(h(x))) -> up(c)
305.20/136.69	   down(h(a)) -> h_flat(down(a))
305.20/136.69	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.20/136.69	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.20/136.69	   down(h(c)) -> h_flat(down(c))
305.20/136.69	   down(f(f(x))) -> up(g(x))
305.20/136.69	   down(f(a)) -> f_flat(down(a))
305.20/136.69	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.20/136.69	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.20/136.69	   down(f(c)) -> f_flat(down(c))
305.20/136.69	   f_flat(up(x_1)) -> up(f(x_1))
305.20/136.69	   down(a) -> up(f(a))
305.20/136.69	
305.20/136.69	The set Q consists of the following terms:
305.20/136.69	
305.20/136.69	   down(a)
305.20/136.69	   down(f(f(x0)))
305.20/136.69	   down(g(g(x0)))
305.20/136.69	   down(h(h(x0)))
305.20/136.69	   down(f(a))
305.20/136.69	   down(f(g(x0)))
305.20/136.69	   down(f(h(x0)))
305.20/136.69	   down(f(c))
305.20/136.69	   down(f(fresh_constant))
305.20/136.69	   down(g(a))
305.20/136.69	   down(g(f(x0)))
305.20/136.69	   down(g(h(x0)))
305.20/136.69	   down(g(c))
305.20/136.69	   down(g(fresh_constant))
305.20/136.69	   down(h(a))
305.20/136.69	   down(h(f(x0)))
305.20/136.69	   down(h(g(x0)))
305.20/136.69	   down(h(c))
305.20/136.69	   down(h(fresh_constant))
305.20/136.69	   f_flat(up(x0))
305.20/136.69	   g_flat(up(x0))
305.20/136.69	   h_flat(up(x0))
305.20/136.69	
305.20/136.69	We have to consider all minimal (P,Q,R)-chains.
305.20/136.69	----------------------------------------
305.20/136.69	
305.20/136.69	(81) QReductionProof (EQUIVALENT)
305.20/136.69	We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN].
305.20/136.69	
305.20/136.69	   down(f(fresh_constant))
305.20/136.69	   down(g(fresh_constant))
305.20/136.69	   down(h(fresh_constant))
305.20/136.69	
305.20/136.69	
305.20/136.69	----------------------------------------
305.20/136.69	
305.20/136.69	(82)
305.20/136.69	Obligation:
305.20/136.69	Q DP problem:
305.20/136.69	The TRS P consists of the following rules:
305.20/136.69	
305.20/136.69	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.20/136.69	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.20/136.69	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.20/136.69	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.20/136.69	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.20/136.69	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.20/136.69	
305.20/136.69	The TRS R consists of the following rules:
305.20/136.69	
305.20/136.69	   h_flat(up(x_1)) -> up(h(x_1))
305.20/136.69	   down(g(g(x))) -> up(h(x))
305.20/136.69	   down(g(a)) -> g_flat(down(a))
305.20/136.69	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.20/136.69	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.20/136.69	   down(g(c)) -> g_flat(down(c))
305.20/136.69	   g_flat(up(x_1)) -> up(g(x_1))
305.20/136.69	   down(h(h(x))) -> up(c)
305.20/136.69	   down(h(a)) -> h_flat(down(a))
305.20/136.69	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.20/136.69	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.20/136.69	   down(h(c)) -> h_flat(down(c))
305.20/136.69	   down(f(f(x))) -> up(g(x))
305.20/136.69	   down(f(a)) -> f_flat(down(a))
305.20/136.69	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.20/136.69	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.20/136.69	   down(f(c)) -> f_flat(down(c))
305.20/136.69	   f_flat(up(x_1)) -> up(f(x_1))
305.20/136.69	   down(a) -> up(f(a))
305.20/136.69	
305.20/136.69	The set Q consists of the following terms:
305.20/136.69	
305.20/136.69	   down(a)
305.20/136.69	   down(f(f(x0)))
305.20/136.69	   down(g(g(x0)))
305.20/136.69	   down(h(h(x0)))
305.20/136.69	   down(f(a))
305.20/136.69	   down(f(g(x0)))
305.20/136.69	   down(f(h(x0)))
305.20/136.69	   down(f(c))
305.20/136.69	   down(g(a))
305.20/136.69	   down(g(f(x0)))
305.20/136.69	   down(g(h(x0)))
305.20/136.69	   down(g(c))
305.20/136.69	   down(h(a))
305.20/136.69	   down(h(f(x0)))
305.20/136.69	   down(h(g(x0)))
305.20/136.69	   down(h(c))
305.20/136.69	   f_flat(up(x0))
305.20/136.69	   g_flat(up(x0))
305.20/136.69	   h_flat(up(x0))
305.20/136.69	
305.20/136.69	We have to consider all (P,Q,R)-chains.
305.20/136.69	----------------------------------------
305.20/136.69	
305.20/136.69	(83) MNOCProof (EQUIVALENT)
305.20/136.69	We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
305.20/136.69	----------------------------------------
305.20/136.69	
305.20/136.69	(84)
305.20/136.69	Obligation:
305.20/136.69	Q DP problem:
305.20/136.69	The TRS P consists of the following rules:
305.20/136.69	
305.20/136.69	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.20/136.69	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.20/136.69	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.20/136.69	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.20/136.69	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.20/136.69	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.20/136.69	
305.20/136.69	The TRS R consists of the following rules:
305.20/136.69	
305.20/136.69	   h_flat(up(x_1)) -> up(h(x_1))
305.20/136.69	   down(g(g(x))) -> up(h(x))
305.20/136.69	   down(g(a)) -> g_flat(down(a))
305.20/136.69	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.20/136.69	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.20/136.69	   down(g(c)) -> g_flat(down(c))
305.20/136.69	   g_flat(up(x_1)) -> up(g(x_1))
305.20/136.69	   down(h(h(x))) -> up(c)
305.20/136.69	   down(h(a)) -> h_flat(down(a))
305.20/136.69	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.20/136.69	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.20/136.69	   down(h(c)) -> h_flat(down(c))
305.20/136.69	   down(f(f(x))) -> up(g(x))
305.20/136.69	   down(f(a)) -> f_flat(down(a))
305.20/136.69	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.20/136.69	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.20/136.69	   down(f(c)) -> f_flat(down(c))
305.20/136.69	   f_flat(up(x_1)) -> up(f(x_1))
305.20/136.69	   down(a) -> up(f(a))
305.20/136.69	
305.20/136.69	Q is empty.
305.20/136.69	We have to consider all (P,Q,R)-chains.
305.20/136.69	----------------------------------------
305.20/136.69	
305.20/136.69	(85) SplitQDPProof (EQUIVALENT)
305.20/136.69	We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem
305.20/136.69	
305.20/136.69	----------------------------------------
305.20/136.69	
305.20/136.69	(86)
305.20/136.69	Complex Obligation (AND)
305.20/136.69	
305.20/136.69	----------------------------------------
305.20/136.69	
305.20/136.69	(87)
305.20/136.69	Obligation:
305.20/136.69	Q DP problem:
305.20/136.69	The TRS P consists of the following rules:
305.20/136.69	
305.20/136.69	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.20/136.69	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.20/136.69	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.20/136.69	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.20/136.69	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.20/136.69	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.20/136.69	
305.20/136.69	The TRS R consists of the following rules:
305.20/136.69	
305.20/136.69	   h_flat(up(x_1)) -> up(h(x_1))
305.20/136.69	   down(g(g(x))) -> up(h(x))
305.20/136.69	   down(g(a)) -> g_flat(down(a))
305.20/136.69	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.20/136.69	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.20/136.69	   down(g(c)) -> g_flat(down(c))
305.20/136.69	   g_flat(up(x_1)) -> up(g(x_1))
305.20/136.69	   down(h(h(x))) -> up(c)
305.20/136.69	   down(h(a)) -> h_flat(down(a))
305.20/136.69	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.20/136.69	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.20/136.69	   down(h(c)) -> h_flat(down(c))
305.20/136.69	   down(f(f(x))) -> up(g(x))
305.20/136.69	   down(f(a)) -> f_flat(down(a))
305.20/136.69	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.20/136.69	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.20/136.69	   down(f(c)) -> f_flat(down(c))
305.20/136.69	   f_flat(up(x_1)) -> up(f(x_1))
305.20/136.69	   down(a) -> up(f(a))
305.20/136.69	
305.20/136.69	The set Q consists of the following terms:
305.20/136.69	
305.20/136.69	   down(a)
305.20/136.69	   down(f(f(x0)))
305.20/136.69	   down(g(g(x0)))
305.20/136.69	   down(h(h(x0)))
305.20/136.69	   down(f(a))
305.20/136.69	   down(f(g(x0)))
305.20/136.69	   down(f(h(x0)))
305.20/136.69	   down(f(c))
305.20/136.69	   down(f(fresh_constant))
305.20/136.69	   down(g(a))
305.20/136.69	   down(g(f(x0)))
305.20/136.69	   down(g(h(x0)))
305.20/136.69	   down(g(c))
305.20/136.69	   down(g(fresh_constant))
305.20/136.69	   down(h(a))
305.20/136.69	   down(h(f(x0)))
305.20/136.69	   down(h(g(x0)))
305.20/136.69	   down(h(c))
305.20/136.69	   down(h(fresh_constant))
305.20/136.69	   f_flat(up(x0))
305.20/136.69	   g_flat(up(x0))
305.20/136.69	   h_flat(up(x0))
305.20/136.69	
305.20/136.69	We have to consider all minimal (P,Q,R)-chains.
305.20/136.69	----------------------------------------
305.20/136.69	
305.20/136.69	(88) SemLabProof (SOUND)
305.20/136.69	We found the following model for the rules of the TRSs R and P.
305.20/136.69	Interpretation over the domain with elements from 0 to 1.
305.20/136.69	h_flat: 0
305.20/136.69	a: 0
305.20/136.69	c: 1
305.20/136.69	down: 0
305.20/136.69	f: 0
305.20/136.69	fresh_constant: 0
305.20/136.69	up: 0
305.20/136.69	h: 0
305.20/136.69	f_flat: 0
305.20/136.69	TOP: 0
305.20/136.69	g_flat: 0
305.20/136.69	g: 0
305.20/136.69	By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.
305.20/136.69	----------------------------------------
305.20/136.69	
305.20/136.69	(89)
305.20/136.69	Obligation:
305.20/136.69	Q DP problem:
305.20/136.69	The TRS P consists of the following rules:
305.20/136.69	
305.20/136.69	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
305.20/136.69	   TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0))))
305.20/136.69	   TOP.0(up.0(f.0(h.0(x0)))) -> TOP.0(f_flat.0(down.0(h.0(x0))))
305.20/136.69	   TOP.0(up.0(f.0(h.1(x0)))) -> TOP.0(f_flat.0(down.0(h.1(x0))))
305.20/136.69	   TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0))))
305.20/136.69	   TOP.0(up.0(g.0(f.1(x0)))) -> TOP.0(g_flat.0(down.0(f.1(x0))))
305.20/136.69	   TOP.0(up.0(g.0(h.0(x0)))) -> TOP.0(g_flat.0(down.0(h.0(x0))))
305.20/136.69	   TOP.0(up.0(g.0(h.1(x0)))) -> TOP.0(g_flat.0(down.0(h.1(x0))))
305.20/136.69	   TOP.0(up.0(h.0(f.0(x0)))) -> TOP.0(h_flat.0(down.0(f.0(x0))))
305.20/136.69	   TOP.0(up.0(h.0(f.1(x0)))) -> TOP.0(h_flat.0(down.0(f.1(x0))))
305.20/136.69	   TOP.0(up.0(h.0(g.0(x0)))) -> TOP.0(h_flat.0(down.0(g.0(x0))))
305.20/136.69	   TOP.0(up.0(h.0(g.1(x0)))) -> TOP.0(h_flat.0(down.0(g.1(x0))))
305.20/136.69	
305.20/136.69	The TRS R consists of the following rules:
305.20/136.69	
305.20/136.69	   h_flat.0(up.0(x_1)) -> up.0(h.0(x_1))
305.20/136.69	   h_flat.0(up.1(x_1)) -> up.0(h.1(x_1))
305.20/136.69	   down.0(g.0(g.0(x))) -> up.0(h.0(x))
305.20/136.69	   down.0(g.0(g.1(x))) -> up.0(h.1(x))
305.20/136.69	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
305.20/136.69	   down.0(g.0(f.0(y8))) -> g_flat.0(down.0(f.0(y8)))
305.20/136.69	   down.0(g.0(f.1(y8))) -> g_flat.0(down.0(f.1(y8)))
305.20/136.69	   down.0(g.0(h.0(y10))) -> g_flat.0(down.0(h.0(y10)))
305.20/136.69	   down.0(g.0(h.1(y10))) -> g_flat.0(down.0(h.1(y10)))
305.20/136.69	   down.0(g.1(c.)) -> g_flat.0(down.1(c.))
305.20/136.69	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
305.20/136.69	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
305.20/136.69	   down.0(h.0(h.0(x))) -> up.1(c.)
305.20/136.69	   down.0(h.0(h.1(x))) -> up.1(c.)
305.20/136.69	   down.0(h.0(a.)) -> h_flat.0(down.0(a.))
305.20/136.69	   down.0(h.0(f.0(y12))) -> h_flat.0(down.0(f.0(y12)))
305.20/136.69	   down.0(h.0(f.1(y12))) -> h_flat.0(down.0(f.1(y12)))
305.20/136.69	   down.0(h.0(g.0(y13))) -> h_flat.0(down.0(g.0(y13)))
305.20/136.69	   down.0(h.0(g.1(y13))) -> h_flat.0(down.0(g.1(y13)))
305.20/136.69	   down.0(h.1(c.)) -> h_flat.0(down.1(c.))
305.20/136.69	   down.0(f.0(f.0(x))) -> up.0(g.0(x))
305.20/136.69	   down.0(f.0(f.1(x))) -> up.0(g.1(x))
305.20/136.69	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
305.20/136.69	   down.0(f.0(g.0(y5))) -> f_flat.0(down.0(g.0(y5)))
305.20/136.69	   down.0(f.0(g.1(y5))) -> f_flat.0(down.0(g.1(y5)))
305.20/136.69	   down.0(f.0(h.0(y6))) -> f_flat.0(down.0(h.0(y6)))
305.20/136.69	   down.0(f.0(h.1(y6))) -> f_flat.0(down.0(h.1(y6)))
305.20/136.69	   down.0(f.1(c.)) -> f_flat.0(down.1(c.))
305.20/136.69	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
305.20/136.69	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
305.20/136.69	   down.0(a.) -> up.0(f.0(a.))
305.20/136.69	
305.20/136.69	The set Q consists of the following terms:
305.20/136.69	
305.20/136.69	   down.0(a.)
305.20/136.69	   down.0(f.0(f.0(x0)))
305.20/136.69	   down.0(f.0(f.1(x0)))
305.20/136.69	   down.0(g.0(g.0(x0)))
305.20/136.69	   down.0(g.0(g.1(x0)))
305.20/136.69	   down.0(h.0(h.0(x0)))
305.20/136.69	   down.0(h.0(h.1(x0)))
305.20/136.69	   down.0(f.0(a.))
305.20/136.69	   down.0(f.0(g.0(x0)))
305.20/136.69	   down.0(f.0(g.1(x0)))
305.20/136.69	   down.0(f.0(h.0(x0)))
305.20/136.69	   down.0(f.0(h.1(x0)))
305.20/136.69	   down.0(f.1(c.))
305.20/136.69	   down.0(f.0(fresh_constant.))
305.20/136.69	   down.0(g.0(a.))
305.20/136.69	   down.0(g.0(f.0(x0)))
305.20/136.69	   down.0(g.0(f.1(x0)))
305.20/136.69	   down.0(g.0(h.0(x0)))
305.20/136.69	   down.0(g.0(h.1(x0)))
305.20/136.69	   down.0(g.1(c.))
305.20/136.69	   down.0(g.0(fresh_constant.))
305.20/136.69	   down.0(h.0(a.))
305.20/136.69	   down.0(h.0(f.0(x0)))
305.20/136.69	   down.0(h.0(f.1(x0)))
305.20/136.69	   down.0(h.0(g.0(x0)))
305.20/136.69	   down.0(h.0(g.1(x0)))
305.20/136.69	   down.0(h.1(c.))
305.20/136.69	   down.0(h.0(fresh_constant.))
305.20/136.69	   f_flat.0(up.0(x0))
305.20/136.69	   f_flat.0(up.1(x0))
305.20/136.69	   g_flat.0(up.0(x0))
305.20/136.69	   g_flat.0(up.1(x0))
305.20/136.69	   h_flat.0(up.0(x0))
305.20/136.69	   h_flat.0(up.1(x0))
305.20/136.69	
305.20/136.69	We have to consider all minimal (P,Q,R)-chains.
305.20/136.69	----------------------------------------
305.20/136.69	
305.20/136.69	(90) MRRProof (EQUIVALENT)
305.20/136.69	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.
305.20/136.69	
305.20/136.69	
305.20/136.69	Strictly oriented rules of the TRS R:
305.20/136.69	
305.20/136.69	   down.0(g.1(c.)) -> g_flat.0(down.1(c.))
305.20/136.69	   down.0(h.1(c.)) -> h_flat.0(down.1(c.))
305.20/136.69	   down.0(f.1(c.)) -> f_flat.0(down.1(c.))
305.20/136.69	
305.20/136.69	Used ordering: Polynomial interpretation [POLO]:
305.20/136.69	
305.20/136.69	   POL(TOP.0(x_1)) = x_1
305.20/136.69	   POL(a.) = 0
305.20/136.69	   POL(c.) = 0
305.20/136.69	   POL(down.0(x_1)) = 1 + x_1
305.20/136.69	   POL(down.1(x_1)) = x_1
305.20/136.69	   POL(f.0(x_1)) = x_1
305.20/136.69	   POL(f.1(x_1)) = x_1
305.20/136.69	   POL(f_flat.0(x_1)) = x_1
305.20/136.69	   POL(g.0(x_1)) = x_1
305.20/136.69	   POL(g.1(x_1)) = x_1
305.20/136.69	   POL(g_flat.0(x_1)) = x_1
305.20/136.69	   POL(h.0(x_1)) = x_1
305.20/136.69	   POL(h.1(x_1)) = x_1
305.20/136.69	   POL(h_flat.0(x_1)) = x_1
305.20/136.69	   POL(up.0(x_1)) = 1 + x_1
305.20/136.69	   POL(up.1(x_1)) = 1 + x_1
305.20/136.69	
305.20/136.69	
305.20/136.69	----------------------------------------
305.20/136.69	
305.20/136.69	(91)
305.20/136.69	Obligation:
305.20/136.69	Q DP problem:
305.20/136.69	The TRS P consists of the following rules:
305.20/136.69	
305.20/136.69	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
305.20/136.69	   TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0))))
305.20/136.69	   TOP.0(up.0(f.0(h.0(x0)))) -> TOP.0(f_flat.0(down.0(h.0(x0))))
305.20/136.69	   TOP.0(up.0(f.0(h.1(x0)))) -> TOP.0(f_flat.0(down.0(h.1(x0))))
305.20/136.69	   TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0))))
305.20/136.69	   TOP.0(up.0(g.0(f.1(x0)))) -> TOP.0(g_flat.0(down.0(f.1(x0))))
305.20/136.69	   TOP.0(up.0(g.0(h.0(x0)))) -> TOP.0(g_flat.0(down.0(h.0(x0))))
305.20/136.69	   TOP.0(up.0(g.0(h.1(x0)))) -> TOP.0(g_flat.0(down.0(h.1(x0))))
305.20/136.69	   TOP.0(up.0(h.0(f.0(x0)))) -> TOP.0(h_flat.0(down.0(f.0(x0))))
305.20/136.69	   TOP.0(up.0(h.0(f.1(x0)))) -> TOP.0(h_flat.0(down.0(f.1(x0))))
305.20/136.69	   TOP.0(up.0(h.0(g.0(x0)))) -> TOP.0(h_flat.0(down.0(g.0(x0))))
305.20/136.69	   TOP.0(up.0(h.0(g.1(x0)))) -> TOP.0(h_flat.0(down.0(g.1(x0))))
305.20/136.69	
305.20/136.69	The TRS R consists of the following rules:
305.20/136.69	
305.20/136.69	   h_flat.0(up.0(x_1)) -> up.0(h.0(x_1))
305.20/136.69	   h_flat.0(up.1(x_1)) -> up.0(h.1(x_1))
305.20/136.69	   down.0(g.0(g.0(x))) -> up.0(h.0(x))
305.20/136.69	   down.0(g.0(g.1(x))) -> up.0(h.1(x))
305.20/136.69	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
305.20/136.69	   down.0(g.0(f.0(y8))) -> g_flat.0(down.0(f.0(y8)))
305.20/136.69	   down.0(g.0(f.1(y8))) -> g_flat.0(down.0(f.1(y8)))
305.20/136.69	   down.0(g.0(h.0(y10))) -> g_flat.0(down.0(h.0(y10)))
305.20/136.69	   down.0(g.0(h.1(y10))) -> g_flat.0(down.0(h.1(y10)))
305.20/136.69	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
305.20/136.69	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
305.20/136.69	   down.0(h.0(h.0(x))) -> up.1(c.)
305.20/136.69	   down.0(h.0(h.1(x))) -> up.1(c.)
305.20/136.69	   down.0(h.0(a.)) -> h_flat.0(down.0(a.))
305.20/136.69	   down.0(h.0(f.0(y12))) -> h_flat.0(down.0(f.0(y12)))
305.20/136.69	   down.0(h.0(f.1(y12))) -> h_flat.0(down.0(f.1(y12)))
305.20/136.69	   down.0(h.0(g.0(y13))) -> h_flat.0(down.0(g.0(y13)))
305.20/136.69	   down.0(h.0(g.1(y13))) -> h_flat.0(down.0(g.1(y13)))
305.20/136.69	   down.0(f.0(f.0(x))) -> up.0(g.0(x))
305.20/136.69	   down.0(f.0(f.1(x))) -> up.0(g.1(x))
305.20/136.69	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
305.20/136.69	   down.0(f.0(g.0(y5))) -> f_flat.0(down.0(g.0(y5)))
305.20/136.69	   down.0(f.0(g.1(y5))) -> f_flat.0(down.0(g.1(y5)))
305.20/136.69	   down.0(f.0(h.0(y6))) -> f_flat.0(down.0(h.0(y6)))
305.20/136.69	   down.0(f.0(h.1(y6))) -> f_flat.0(down.0(h.1(y6)))
305.20/136.69	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
305.20/136.69	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
305.20/136.69	   down.0(a.) -> up.0(f.0(a.))
305.20/136.69	
305.20/136.69	The set Q consists of the following terms:
305.20/136.69	
305.20/136.69	   down.0(a.)
305.20/136.69	   down.0(f.0(f.0(x0)))
305.20/136.69	   down.0(f.0(f.1(x0)))
305.20/136.69	   down.0(g.0(g.0(x0)))
305.20/136.69	   down.0(g.0(g.1(x0)))
305.20/136.69	   down.0(h.0(h.0(x0)))
305.20/136.69	   down.0(h.0(h.1(x0)))
305.20/136.69	   down.0(f.0(a.))
305.20/136.69	   down.0(f.0(g.0(x0)))
305.20/136.69	   down.0(f.0(g.1(x0)))
305.20/136.69	   down.0(f.0(h.0(x0)))
305.20/136.69	   down.0(f.0(h.1(x0)))
305.20/136.69	   down.0(f.1(c.))
305.20/136.69	   down.0(f.0(fresh_constant.))
305.20/136.69	   down.0(g.0(a.))
305.20/136.69	   down.0(g.0(f.0(x0)))
305.20/136.69	   down.0(g.0(f.1(x0)))
305.20/136.69	   down.0(g.0(h.0(x0)))
305.20/136.69	   down.0(g.0(h.1(x0)))
305.20/136.69	   down.0(g.1(c.))
305.20/136.69	   down.0(g.0(fresh_constant.))
305.20/136.69	   down.0(h.0(a.))
305.20/136.69	   down.0(h.0(f.0(x0)))
305.20/136.69	   down.0(h.0(f.1(x0)))
305.20/136.69	   down.0(h.0(g.0(x0)))
305.20/136.69	   down.0(h.0(g.1(x0)))
305.20/136.69	   down.0(h.1(c.))
305.20/136.69	   down.0(h.0(fresh_constant.))
305.20/136.69	   f_flat.0(up.0(x0))
305.20/136.69	   f_flat.0(up.1(x0))
305.20/136.69	   g_flat.0(up.0(x0))
305.20/136.69	   g_flat.0(up.1(x0))
305.20/136.69	   h_flat.0(up.0(x0))
305.20/136.69	   h_flat.0(up.1(x0))
305.20/136.69	
305.20/136.69	We have to consider all minimal (P,Q,R)-chains.
305.20/136.69	----------------------------------------
305.20/136.69	
305.20/136.69	(92) DependencyGraphProof (EQUIVALENT)
305.20/136.69	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.
305.20/136.69	----------------------------------------
305.20/136.69	
305.20/136.69	(93)
305.20/136.69	Obligation:
305.20/136.69	Q DP problem:
305.20/136.69	The TRS P consists of the following rules:
305.20/136.69	
305.20/136.69	   TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0))))
305.20/136.69	   TOP.0(up.0(f.0(h.0(x0)))) -> TOP.0(f_flat.0(down.0(h.0(x0))))
305.20/136.69	   TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0))))
305.20/136.69	   TOP.0(up.0(g.0(h.0(x0)))) -> TOP.0(g_flat.0(down.0(h.0(x0))))
305.20/136.69	   TOP.0(up.0(h.0(f.0(x0)))) -> TOP.0(h_flat.0(down.0(f.0(x0))))
305.20/136.70	   TOP.0(up.0(h.0(g.0(x0)))) -> TOP.0(h_flat.0(down.0(g.0(x0))))
305.20/136.70	
305.20/136.70	The TRS R consists of the following rules:
305.20/136.70	
305.20/136.70	   h_flat.0(up.0(x_1)) -> up.0(h.0(x_1))
305.20/136.70	   h_flat.0(up.1(x_1)) -> up.0(h.1(x_1))
305.20/136.70	   down.0(g.0(g.0(x))) -> up.0(h.0(x))
305.20/136.70	   down.0(g.0(g.1(x))) -> up.0(h.1(x))
305.20/136.70	   down.0(g.0(a.)) -> g_flat.0(down.0(a.))
305.20/136.70	   down.0(g.0(f.0(y8))) -> g_flat.0(down.0(f.0(y8)))
305.20/136.70	   down.0(g.0(f.1(y8))) -> g_flat.0(down.0(f.1(y8)))
305.20/136.70	   down.0(g.0(h.0(y10))) -> g_flat.0(down.0(h.0(y10)))
305.20/136.70	   down.0(g.0(h.1(y10))) -> g_flat.0(down.0(h.1(y10)))
305.20/136.70	   g_flat.0(up.0(x_1)) -> up.0(g.0(x_1))
305.20/136.70	   g_flat.0(up.1(x_1)) -> up.0(g.1(x_1))
305.20/136.70	   down.0(h.0(h.0(x))) -> up.1(c.)
305.20/136.70	   down.0(h.0(h.1(x))) -> up.1(c.)
305.20/136.70	   down.0(h.0(a.)) -> h_flat.0(down.0(a.))
305.20/136.70	   down.0(h.0(f.0(y12))) -> h_flat.0(down.0(f.0(y12)))
305.20/136.70	   down.0(h.0(f.1(y12))) -> h_flat.0(down.0(f.1(y12)))
305.20/136.70	   down.0(h.0(g.0(y13))) -> h_flat.0(down.0(g.0(y13)))
305.20/136.70	   down.0(h.0(g.1(y13))) -> h_flat.0(down.0(g.1(y13)))
305.20/136.70	   down.0(f.0(f.0(x))) -> up.0(g.0(x))
305.20/136.70	   down.0(f.0(f.1(x))) -> up.0(g.1(x))
305.20/136.70	   down.0(f.0(a.)) -> f_flat.0(down.0(a.))
305.20/136.70	   down.0(f.0(g.0(y5))) -> f_flat.0(down.0(g.0(y5)))
305.20/136.70	   down.0(f.0(g.1(y5))) -> f_flat.0(down.0(g.1(y5)))
305.20/136.70	   down.0(f.0(h.0(y6))) -> f_flat.0(down.0(h.0(y6)))
305.20/136.70	   down.0(f.0(h.1(y6))) -> f_flat.0(down.0(h.1(y6)))
305.20/136.70	   f_flat.0(up.0(x_1)) -> up.0(f.0(x_1))
305.20/136.70	   f_flat.0(up.1(x_1)) -> up.0(f.1(x_1))
305.20/136.70	   down.0(a.) -> up.0(f.0(a.))
305.20/136.70	
305.20/136.70	The set Q consists of the following terms:
305.20/136.70	
305.20/136.70	   down.0(a.)
305.20/136.70	   down.0(f.0(f.0(x0)))
305.20/136.70	   down.0(f.0(f.1(x0)))
305.20/136.70	   down.0(g.0(g.0(x0)))
305.20/136.70	   down.0(g.0(g.1(x0)))
305.20/136.70	   down.0(h.0(h.0(x0)))
305.20/136.70	   down.0(h.0(h.1(x0)))
305.20/136.70	   down.0(f.0(a.))
305.20/136.70	   down.0(f.0(g.0(x0)))
305.20/136.70	   down.0(f.0(g.1(x0)))
305.20/136.70	   down.0(f.0(h.0(x0)))
305.20/136.70	   down.0(f.0(h.1(x0)))
305.20/136.70	   down.0(f.1(c.))
305.20/136.70	   down.0(f.0(fresh_constant.))
305.20/136.70	   down.0(g.0(a.))
305.20/136.70	   down.0(g.0(f.0(x0)))
305.20/136.70	   down.0(g.0(f.1(x0)))
305.20/136.70	   down.0(g.0(h.0(x0)))
305.20/136.70	   down.0(g.0(h.1(x0)))
305.20/136.70	   down.0(g.1(c.))
305.20/136.70	   down.0(g.0(fresh_constant.))
305.20/136.70	   down.0(h.0(a.))
305.20/136.70	   down.0(h.0(f.0(x0)))
305.20/136.70	   down.0(h.0(f.1(x0)))
305.20/136.70	   down.0(h.0(g.0(x0)))
305.20/136.70	   down.0(h.0(g.1(x0)))
305.20/136.70	   down.0(h.1(c.))
305.20/136.70	   down.0(h.0(fresh_constant.))
305.20/136.70	   f_flat.0(up.0(x0))
305.20/136.70	   f_flat.0(up.1(x0))
305.20/136.70	   g_flat.0(up.0(x0))
305.20/136.70	   g_flat.0(up.1(x0))
305.20/136.70	   h_flat.0(up.0(x0))
305.20/136.70	   h_flat.0(up.1(x0))
305.20/136.70	
305.20/136.70	We have to consider all minimal (P,Q,R)-chains.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(94) PisEmptyProof (SOUND)
305.20/136.70	The TRS P is empty. Hence, there is no (P,Q,R) chain.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(95)
305.20/136.70	TRUE
305.20/136.70	
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(96)
305.20/136.70	Obligation:
305.20/136.70	Q DP problem:
305.20/136.70	The TRS P consists of the following rules:
305.20/136.70	
305.20/136.70	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.20/136.70	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.20/136.70	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.20/136.70	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.20/136.70	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.20/136.70	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.20/136.70	
305.20/136.70	The TRS R consists of the following rules:
305.20/136.70	
305.20/136.70	   h_flat(up(x_1)) -> up(h(x_1))
305.20/136.70	   down(g(g(x))) -> up(h(x))
305.20/136.70	   down(g(a)) -> g_flat(down(a))
305.20/136.70	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.20/136.70	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.20/136.70	   g_flat(up(x_1)) -> up(g(x_1))
305.20/136.70	   down(h(h(x))) -> up(c)
305.20/136.70	   down(h(a)) -> h_flat(down(a))
305.20/136.70	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.20/136.70	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.20/136.70	   down(f(f(x))) -> up(g(x))
305.20/136.70	   down(f(a)) -> f_flat(down(a))
305.20/136.70	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.20/136.70	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.20/136.70	   f_flat(up(x_1)) -> up(f(x_1))
305.20/136.70	   down(a) -> up(f(a))
305.20/136.70	
305.20/136.70	The set Q consists of the following terms:
305.20/136.70	
305.20/136.70	   down(a)
305.20/136.70	   down(f(f(x0)))
305.20/136.70	   down(g(g(x0)))
305.20/136.70	   down(h(h(x0)))
305.20/136.70	   down(f(a))
305.20/136.70	   down(f(g(x0)))
305.20/136.70	   down(f(h(x0)))
305.20/136.70	   down(f(c))
305.20/136.70	   down(f(fresh_constant))
305.20/136.70	   down(g(a))
305.20/136.70	   down(g(f(x0)))
305.20/136.70	   down(g(h(x0)))
305.20/136.70	   down(g(c))
305.20/136.70	   down(g(fresh_constant))
305.20/136.70	   down(h(a))
305.20/136.70	   down(h(f(x0)))
305.20/136.70	   down(h(g(x0)))
305.20/136.70	   down(h(c))
305.20/136.70	   down(h(fresh_constant))
305.20/136.70	   f_flat(up(x0))
305.20/136.70	   g_flat(up(x0))
305.20/136.70	   h_flat(up(x0))
305.20/136.70	
305.20/136.70	We have to consider all minimal (P,Q,R)-chains.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(97) QReductionProof (EQUIVALENT)
305.20/136.70	We deleted the following terms from Q as they contain symbols which do neither occur in P nor in R.[THIEMANN].
305.20/136.70	
305.20/136.70	   down(f(fresh_constant))
305.20/136.70	   down(g(fresh_constant))
305.20/136.70	   down(h(fresh_constant))
305.20/136.70	
305.20/136.70	
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(98)
305.20/136.70	Obligation:
305.20/136.70	Q DP problem:
305.20/136.70	The TRS P consists of the following rules:
305.20/136.70	
305.20/136.70	   TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))
305.20/136.70	   TOP(up(f(h(x0)))) -> TOP(f_flat(down(h(x0))))
305.20/136.70	   TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))
305.20/136.70	   TOP(up(g(h(x0)))) -> TOP(g_flat(down(h(x0))))
305.20/136.70	   TOP(up(h(f(x0)))) -> TOP(h_flat(down(f(x0))))
305.20/136.70	   TOP(up(h(g(x0)))) -> TOP(h_flat(down(g(x0))))
305.20/136.70	
305.20/136.70	The TRS R consists of the following rules:
305.20/136.70	
305.20/136.70	   h_flat(up(x_1)) -> up(h(x_1))
305.20/136.70	   down(g(g(x))) -> up(h(x))
305.20/136.70	   down(g(a)) -> g_flat(down(a))
305.20/136.70	   down(g(f(y8))) -> g_flat(down(f(y8)))
305.20/136.70	   down(g(h(y10))) -> g_flat(down(h(y10)))
305.20/136.70	   g_flat(up(x_1)) -> up(g(x_1))
305.20/136.70	   down(h(h(x))) -> up(c)
305.20/136.70	   down(h(a)) -> h_flat(down(a))
305.20/136.70	   down(h(f(y12))) -> h_flat(down(f(y12)))
305.20/136.70	   down(h(g(y13))) -> h_flat(down(g(y13)))
305.20/136.70	   down(f(f(x))) -> up(g(x))
305.20/136.70	   down(f(a)) -> f_flat(down(a))
305.20/136.70	   down(f(g(y5))) -> f_flat(down(g(y5)))
305.20/136.70	   down(f(h(y6))) -> f_flat(down(h(y6)))
305.20/136.70	   f_flat(up(x_1)) -> up(f(x_1))
305.20/136.70	   down(a) -> up(f(a))
305.20/136.70	
305.20/136.70	The set Q consists of the following terms:
305.20/136.70	
305.20/136.70	   down(a)
305.20/136.70	   down(f(f(x0)))
305.20/136.70	   down(g(g(x0)))
305.20/136.70	   down(h(h(x0)))
305.20/136.70	   down(f(a))
305.20/136.70	   down(f(g(x0)))
305.20/136.70	   down(f(h(x0)))
305.20/136.70	   down(f(c))
305.20/136.70	   down(g(a))
305.20/136.70	   down(g(f(x0)))
305.20/136.70	   down(g(h(x0)))
305.20/136.70	   down(g(c))
305.20/136.70	   down(h(a))
305.20/136.70	   down(h(f(x0)))
305.20/136.70	   down(h(g(x0)))
305.20/136.70	   down(h(c))
305.20/136.70	   f_flat(up(x0))
305.20/136.70	   g_flat(up(x0))
305.20/136.70	   h_flat(up(x0))
305.20/136.70	
305.20/136.70	We have to consider all (P,Q,R)-chains.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(99) Thiemann-SpecialC-Transformation (EQUIVALENT)
305.20/136.70	We applied the Thiemann-SpecialC transformation  to transform the outermost TRS to an innermost TRS.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(100)
305.20/136.70	Obligation:
305.20/136.70	Q restricted rewrite system:
305.20/136.70	The TRS R consists of the following rules:
305.20/136.70	
305.20/136.70	   top(go_up(x)) -> top(reduce(x))
305.20/136.70	   reduce(f(x_1)) -> check_f(redex_f(x_1))
305.20/136.70	   reduce(g(x_1)) -> check_g(redex_g(x_1))
305.20/136.70	   reduce(h(x_1)) -> check_h(redex_h(x_1))
305.20/136.70	   reduce(a) -> go_up(f(a))
305.20/136.70	   redex_f(f(x)) -> result_f(g(x))
305.20/136.70	   redex_g(g(x)) -> result_g(h(x))
305.20/136.70	   redex_h(h(x)) -> result_h(c)
305.20/136.70	   check_f(result_f(x)) -> go_up(x)
305.20/136.70	   check_g(result_g(x)) -> go_up(x)
305.20/136.70	   check_h(result_h(x)) -> go_up(x)
305.20/136.70	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
305.20/136.70	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
305.20/136.70	   check_h(redex_h(x_1)) -> in_h_1(reduce(x_1))
305.20/136.70	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
305.20/136.70	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
305.20/136.70	   in_h_1(go_up(x_1)) -> go_up(h(x_1))
305.20/136.70	
305.20/136.70	The set Q consists of the following terms:
305.20/136.70	
305.20/136.70	   top(go_up(x0))
305.20/136.70	   reduce(f(x0))
305.20/136.70	   reduce(g(x0))
305.20/136.70	   reduce(h(x0))
305.20/136.70	   reduce(a)
305.20/136.70	   redex_f(f(x0))
305.20/136.70	   redex_g(g(x0))
305.20/136.70	   redex_h(h(x0))
305.20/136.70	   check_f(result_f(x0))
305.20/136.70	   check_g(result_g(x0))
305.20/136.70	   check_h(result_h(x0))
305.20/136.70	   check_f(redex_f(x0))
305.20/136.70	   check_g(redex_g(x0))
305.20/136.70	   check_h(redex_h(x0))
305.20/136.70	   in_f_1(go_up(x0))
305.20/136.70	   in_g_1(go_up(x0))
305.20/136.70	   in_h_1(go_up(x0))
305.20/136.70	
305.20/136.70	
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(101) DependencyPairsProof (EQUIVALENT)
305.20/136.70	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(102)
305.20/136.70	Obligation:
305.20/136.70	Q DP problem:
305.20/136.70	The TRS P consists of the following rules:
305.20/136.70	
305.20/136.70	   TOP(go_up(x)) -> TOP(reduce(x))
305.20/136.70	   TOP(go_up(x)) -> REDUCE(x)
305.20/136.70	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
305.20/136.70	   REDUCE(f(x_1)) -> REDEX_F(x_1)
305.20/136.70	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
305.20/136.70	   REDUCE(g(x_1)) -> REDEX_G(x_1)
305.20/136.70	   REDUCE(h(x_1)) -> CHECK_H(redex_h(x_1))
305.20/136.70	   REDUCE(h(x_1)) -> REDEX_H(x_1)
305.20/136.70	   CHECK_F(redex_f(x_1)) -> IN_F_1(reduce(x_1))
305.20/136.70	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
305.20/136.70	   CHECK_G(redex_g(x_1)) -> IN_G_1(reduce(x_1))
305.20/136.70	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
305.20/136.70	   CHECK_H(redex_h(x_1)) -> IN_H_1(reduce(x_1))
305.20/136.70	   CHECK_H(redex_h(x_1)) -> REDUCE(x_1)
305.20/136.70	
305.20/136.70	The TRS R consists of the following rules:
305.20/136.70	
305.20/136.70	   top(go_up(x)) -> top(reduce(x))
305.20/136.70	   reduce(f(x_1)) -> check_f(redex_f(x_1))
305.20/136.70	   reduce(g(x_1)) -> check_g(redex_g(x_1))
305.20/136.70	   reduce(h(x_1)) -> check_h(redex_h(x_1))
305.20/136.70	   reduce(a) -> go_up(f(a))
305.20/136.70	   redex_f(f(x)) -> result_f(g(x))
305.20/136.70	   redex_g(g(x)) -> result_g(h(x))
305.20/136.70	   redex_h(h(x)) -> result_h(c)
305.20/136.70	   check_f(result_f(x)) -> go_up(x)
305.20/136.70	   check_g(result_g(x)) -> go_up(x)
305.20/136.70	   check_h(result_h(x)) -> go_up(x)
305.20/136.70	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
305.20/136.70	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
305.20/136.70	   check_h(redex_h(x_1)) -> in_h_1(reduce(x_1))
305.20/136.70	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
305.20/136.70	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
305.20/136.70	   in_h_1(go_up(x_1)) -> go_up(h(x_1))
305.20/136.70	
305.20/136.70	The set Q consists of the following terms:
305.20/136.70	
305.20/136.70	   top(go_up(x0))
305.20/136.70	   reduce(f(x0))
305.20/136.70	   reduce(g(x0))
305.20/136.70	   reduce(h(x0))
305.20/136.70	   reduce(a)
305.20/136.70	   redex_f(f(x0))
305.20/136.70	   redex_g(g(x0))
305.20/136.70	   redex_h(h(x0))
305.20/136.70	   check_f(result_f(x0))
305.20/136.70	   check_g(result_g(x0))
305.20/136.70	   check_h(result_h(x0))
305.20/136.70	   check_f(redex_f(x0))
305.20/136.70	   check_g(redex_g(x0))
305.20/136.70	   check_h(redex_h(x0))
305.20/136.70	   in_f_1(go_up(x0))
305.20/136.70	   in_g_1(go_up(x0))
305.20/136.70	   in_h_1(go_up(x0))
305.20/136.70	
305.20/136.70	We have to consider all minimal (P,Q,R)-chains.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(103) DependencyGraphProof (EQUIVALENT)
305.20/136.70	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 7 less nodes.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(104)
305.20/136.70	Complex Obligation (AND)
305.20/136.70	
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(105)
305.20/136.70	Obligation:
305.20/136.70	Q DP problem:
305.20/136.70	The TRS P consists of the following rules:
305.20/136.70	
305.20/136.70	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
305.20/136.70	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
305.20/136.70	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
305.20/136.70	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
305.20/136.70	   REDUCE(h(x_1)) -> CHECK_H(redex_h(x_1))
305.20/136.70	   CHECK_H(redex_h(x_1)) -> REDUCE(x_1)
305.20/136.70	
305.20/136.70	The TRS R consists of the following rules:
305.20/136.70	
305.20/136.70	   top(go_up(x)) -> top(reduce(x))
305.20/136.70	   reduce(f(x_1)) -> check_f(redex_f(x_1))
305.20/136.70	   reduce(g(x_1)) -> check_g(redex_g(x_1))
305.20/136.70	   reduce(h(x_1)) -> check_h(redex_h(x_1))
305.20/136.70	   reduce(a) -> go_up(f(a))
305.20/136.70	   redex_f(f(x)) -> result_f(g(x))
305.20/136.70	   redex_g(g(x)) -> result_g(h(x))
305.20/136.70	   redex_h(h(x)) -> result_h(c)
305.20/136.70	   check_f(result_f(x)) -> go_up(x)
305.20/136.70	   check_g(result_g(x)) -> go_up(x)
305.20/136.70	   check_h(result_h(x)) -> go_up(x)
305.20/136.70	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
305.20/136.70	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
305.20/136.70	   check_h(redex_h(x_1)) -> in_h_1(reduce(x_1))
305.20/136.70	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
305.20/136.70	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
305.20/136.70	   in_h_1(go_up(x_1)) -> go_up(h(x_1))
305.20/136.70	
305.20/136.70	The set Q consists of the following terms:
305.20/136.70	
305.20/136.70	   top(go_up(x0))
305.20/136.70	   reduce(f(x0))
305.20/136.70	   reduce(g(x0))
305.20/136.70	   reduce(h(x0))
305.20/136.70	   reduce(a)
305.20/136.70	   redex_f(f(x0))
305.20/136.70	   redex_g(g(x0))
305.20/136.70	   redex_h(h(x0))
305.20/136.70	   check_f(result_f(x0))
305.20/136.70	   check_g(result_g(x0))
305.20/136.70	   check_h(result_h(x0))
305.20/136.70	   check_f(redex_f(x0))
305.20/136.70	   check_g(redex_g(x0))
305.20/136.70	   check_h(redex_h(x0))
305.20/136.70	   in_f_1(go_up(x0))
305.20/136.70	   in_g_1(go_up(x0))
305.20/136.70	   in_h_1(go_up(x0))
305.20/136.70	
305.20/136.70	We have to consider all minimal (P,Q,R)-chains.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(106) UsableRulesProof (EQUIVALENT)
305.20/136.70	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.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(107)
305.20/136.70	Obligation:
305.20/136.70	Q DP problem:
305.20/136.70	The TRS P consists of the following rules:
305.20/136.70	
305.20/136.70	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
305.20/136.70	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
305.20/136.70	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
305.20/136.70	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
305.20/136.70	   REDUCE(h(x_1)) -> CHECK_H(redex_h(x_1))
305.20/136.70	   CHECK_H(redex_h(x_1)) -> REDUCE(x_1)
305.20/136.70	
305.20/136.70	The TRS R consists of the following rules:
305.20/136.70	
305.20/136.70	   redex_h(h(x)) -> result_h(c)
305.20/136.70	   redex_f(f(x)) -> result_f(g(x))
305.20/136.70	   redex_g(g(x)) -> result_g(h(x))
305.20/136.70	
305.20/136.70	The set Q consists of the following terms:
305.20/136.70	
305.20/136.70	   top(go_up(x0))
305.20/136.70	   reduce(f(x0))
305.20/136.70	   reduce(g(x0))
305.20/136.70	   reduce(h(x0))
305.20/136.70	   reduce(a)
305.20/136.70	   redex_f(f(x0))
305.20/136.70	   redex_g(g(x0))
305.20/136.70	   redex_h(h(x0))
305.20/136.70	   check_f(result_f(x0))
305.20/136.70	   check_g(result_g(x0))
305.20/136.70	   check_h(result_h(x0))
305.20/136.70	   check_f(redex_f(x0))
305.20/136.70	   check_g(redex_g(x0))
305.20/136.70	   check_h(redex_h(x0))
305.20/136.70	   in_f_1(go_up(x0))
305.20/136.70	   in_g_1(go_up(x0))
305.20/136.70	   in_h_1(go_up(x0))
305.20/136.70	
305.20/136.70	We have to consider all minimal (P,Q,R)-chains.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(108) QReductionProof (EQUIVALENT)
305.20/136.70	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
305.20/136.70	
305.20/136.70	   top(go_up(x0))
305.20/136.70	   reduce(f(x0))
305.20/136.70	   reduce(g(x0))
305.20/136.70	   reduce(h(x0))
305.20/136.70	   reduce(a)
305.20/136.70	   check_f(result_f(x0))
305.20/136.70	   check_g(result_g(x0))
305.20/136.70	   check_h(result_h(x0))
305.20/136.70	   check_f(redex_f(x0))
305.20/136.70	   check_g(redex_g(x0))
305.20/136.70	   check_h(redex_h(x0))
305.20/136.70	   in_f_1(go_up(x0))
305.20/136.70	   in_g_1(go_up(x0))
305.20/136.70	   in_h_1(go_up(x0))
305.20/136.70	
305.20/136.70	
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(109)
305.20/136.70	Obligation:
305.20/136.70	Q DP problem:
305.20/136.70	The TRS P consists of the following rules:
305.20/136.70	
305.20/136.70	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
305.20/136.70	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
305.20/136.70	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
305.20/136.70	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
305.20/136.70	   REDUCE(h(x_1)) -> CHECK_H(redex_h(x_1))
305.20/136.70	   CHECK_H(redex_h(x_1)) -> REDUCE(x_1)
305.20/136.70	
305.20/136.70	The TRS R consists of the following rules:
305.20/136.70	
305.20/136.70	   redex_h(h(x)) -> result_h(c)
305.20/136.70	   redex_f(f(x)) -> result_f(g(x))
305.20/136.70	   redex_g(g(x)) -> result_g(h(x))
305.20/136.70	
305.20/136.70	The set Q consists of the following terms:
305.20/136.70	
305.20/136.70	   redex_f(f(x0))
305.20/136.70	   redex_g(g(x0))
305.20/136.70	   redex_h(h(x0))
305.20/136.70	
305.20/136.70	We have to consider all minimal (P,Q,R)-chains.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(110) UsableRulesReductionPairsProof (EQUIVALENT)
305.20/136.70	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.
305.20/136.70	
305.20/136.70	The following dependency pairs can be deleted:
305.20/136.70	
305.20/136.70	   REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1))
305.20/136.70	The following rules are removed from R:
305.20/136.70	
305.20/136.70	   redex_f(f(x)) -> result_f(g(x))
305.20/136.70	Used ordering: POLO with Polynomial interpretation [POLO]:
305.20/136.70	
305.20/136.70	   POL(CHECK_F(x_1)) = x_1
305.20/136.70	   POL(CHECK_G(x_1)) = x_1
305.20/136.70	   POL(CHECK_H(x_1)) = x_1
305.20/136.70	   POL(REDUCE(x_1)) = 2*x_1
305.20/136.70	   POL(c) = 0
305.20/136.70	   POL(f(x_1)) = 2*x_1
305.20/136.70	   POL(g(x_1)) = 2*x_1
305.20/136.70	   POL(h(x_1)) = 2*x_1
305.20/136.70	   POL(redex_f(x_1)) = 2*x_1
305.20/136.70	   POL(redex_g(x_1)) = 2*x_1
305.20/136.70	   POL(redex_h(x_1)) = 2*x_1
305.20/136.70	   POL(result_f(x_1)) = x_1
305.20/136.70	   POL(result_g(x_1)) = x_1
305.20/136.70	   POL(result_h(x_1)) = x_1
305.20/136.70	
305.20/136.70	
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(111)
305.20/136.70	Obligation:
305.20/136.70	Q DP problem:
305.20/136.70	The TRS P consists of the following rules:
305.20/136.70	
305.20/136.70	   CHECK_F(redex_f(x_1)) -> REDUCE(x_1)
305.20/136.70	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
305.20/136.70	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
305.20/136.70	   REDUCE(h(x_1)) -> CHECK_H(redex_h(x_1))
305.20/136.70	   CHECK_H(redex_h(x_1)) -> REDUCE(x_1)
305.20/136.70	
305.20/136.70	The TRS R consists of the following rules:
305.20/136.70	
305.20/136.70	   redex_h(h(x)) -> result_h(c)
305.20/136.70	   redex_g(g(x)) -> result_g(h(x))
305.20/136.70	
305.20/136.70	The set Q consists of the following terms:
305.20/136.70	
305.20/136.70	   redex_f(f(x0))
305.20/136.70	   redex_g(g(x0))
305.20/136.70	   redex_h(h(x0))
305.20/136.70	
305.20/136.70	We have to consider all minimal (P,Q,R)-chains.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(112) DependencyGraphProof (EQUIVALENT)
305.20/136.70	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(113)
305.20/136.70	Obligation:
305.20/136.70	Q DP problem:
305.20/136.70	The TRS P consists of the following rules:
305.20/136.70	
305.20/136.70	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
305.20/136.70	   REDUCE(h(x_1)) -> CHECK_H(redex_h(x_1))
305.20/136.70	   CHECK_H(redex_h(x_1)) -> REDUCE(x_1)
305.20/136.70	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
305.20/136.70	
305.20/136.70	The TRS R consists of the following rules:
305.20/136.70	
305.20/136.70	   redex_h(h(x)) -> result_h(c)
305.20/136.70	   redex_g(g(x)) -> result_g(h(x))
305.20/136.70	
305.20/136.70	The set Q consists of the following terms:
305.20/136.70	
305.20/136.70	   redex_f(f(x0))
305.20/136.70	   redex_g(g(x0))
305.20/136.70	   redex_h(h(x0))
305.20/136.70	
305.20/136.70	We have to consider all minimal (P,Q,R)-chains.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(114) QReductionProof (EQUIVALENT)
305.20/136.70	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
305.20/136.70	
305.20/136.70	   redex_f(f(x0))
305.20/136.70	
305.20/136.70	
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(115)
305.20/136.70	Obligation:
305.20/136.70	Q DP problem:
305.20/136.70	The TRS P consists of the following rules:
305.20/136.70	
305.20/136.70	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
305.20/136.70	   REDUCE(h(x_1)) -> CHECK_H(redex_h(x_1))
305.20/136.70	   CHECK_H(redex_h(x_1)) -> REDUCE(x_1)
305.20/136.70	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
305.20/136.70	
305.20/136.70	The TRS R consists of the following rules:
305.20/136.70	
305.20/136.70	   redex_h(h(x)) -> result_h(c)
305.20/136.70	   redex_g(g(x)) -> result_g(h(x))
305.20/136.70	
305.20/136.70	The set Q consists of the following terms:
305.20/136.70	
305.20/136.70	   redex_g(g(x0))
305.20/136.70	   redex_h(h(x0))
305.20/136.70	
305.20/136.70	We have to consider all minimal (P,Q,R)-chains.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(116) UsableRulesReductionPairsProof (EQUIVALENT)
305.20/136.70	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.
305.20/136.70	
305.20/136.70	The following dependency pairs can be deleted:
305.20/136.70	
305.20/136.70	   REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1))
305.20/136.70	The following rules are removed from R:
305.20/136.70	
305.20/136.70	   redex_g(g(x)) -> result_g(h(x))
305.20/136.70	Used ordering: POLO with Polynomial interpretation [POLO]:
305.20/136.70	
305.20/136.70	   POL(CHECK_G(x_1)) = x_1
305.20/136.70	   POL(CHECK_H(x_1)) = x_1
305.20/136.70	   POL(REDUCE(x_1)) = 2*x_1
305.20/136.70	   POL(c) = 0
305.20/136.70	   POL(g(x_1)) = 2*x_1
305.20/136.70	   POL(h(x_1)) = 2*x_1
305.20/136.70	   POL(redex_g(x_1)) = 2*x_1
305.20/136.70	   POL(redex_h(x_1)) = 2*x_1
305.20/136.70	   POL(result_g(x_1)) = x_1
305.20/136.70	   POL(result_h(x_1)) = x_1
305.20/136.70	
305.20/136.70	
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(117)
305.20/136.70	Obligation:
305.20/136.70	Q DP problem:
305.20/136.70	The TRS P consists of the following rules:
305.20/136.70	
305.20/136.70	   CHECK_G(redex_g(x_1)) -> REDUCE(x_1)
305.20/136.70	   REDUCE(h(x_1)) -> CHECK_H(redex_h(x_1))
305.20/136.70	   CHECK_H(redex_h(x_1)) -> REDUCE(x_1)
305.20/136.70	
305.20/136.70	The TRS R consists of the following rules:
305.20/136.70	
305.20/136.70	   redex_h(h(x)) -> result_h(c)
305.20/136.70	
305.20/136.70	The set Q consists of the following terms:
305.20/136.70	
305.20/136.70	   redex_g(g(x0))
305.20/136.70	   redex_h(h(x0))
305.20/136.70	
305.20/136.70	We have to consider all minimal (P,Q,R)-chains.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(118) DependencyGraphProof (EQUIVALENT)
305.20/136.70	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(119)
305.20/136.70	TRUE
305.20/136.70	
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(120)
305.20/136.70	Obligation:
305.20/136.70	Q DP problem:
305.20/136.70	The TRS P consists of the following rules:
305.20/136.70	
305.20/136.70	   TOP(go_up(x)) -> TOP(reduce(x))
305.20/136.70	
305.20/136.70	The TRS R consists of the following rules:
305.20/136.70	
305.20/136.70	   top(go_up(x)) -> top(reduce(x))
305.20/136.70	   reduce(f(x_1)) -> check_f(redex_f(x_1))
305.20/136.70	   reduce(g(x_1)) -> check_g(redex_g(x_1))
305.20/136.70	   reduce(h(x_1)) -> check_h(redex_h(x_1))
305.20/136.70	   reduce(a) -> go_up(f(a))
305.20/136.70	   redex_f(f(x)) -> result_f(g(x))
305.20/136.70	   redex_g(g(x)) -> result_g(h(x))
305.20/136.70	   redex_h(h(x)) -> result_h(c)
305.20/136.70	   check_f(result_f(x)) -> go_up(x)
305.20/136.70	   check_g(result_g(x)) -> go_up(x)
305.20/136.70	   check_h(result_h(x)) -> go_up(x)
305.20/136.70	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
305.20/136.70	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
305.20/136.70	   check_h(redex_h(x_1)) -> in_h_1(reduce(x_1))
305.20/136.70	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
305.20/136.70	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
305.20/136.70	   in_h_1(go_up(x_1)) -> go_up(h(x_1))
305.20/136.70	
305.20/136.70	The set Q consists of the following terms:
305.20/136.70	
305.20/136.70	   top(go_up(x0))
305.20/136.70	   reduce(f(x0))
305.20/136.70	   reduce(g(x0))
305.20/136.70	   reduce(h(x0))
305.20/136.70	   reduce(a)
305.20/136.70	   redex_f(f(x0))
305.20/136.70	   redex_g(g(x0))
305.20/136.70	   redex_h(h(x0))
305.20/136.70	   check_f(result_f(x0))
305.20/136.70	   check_g(result_g(x0))
305.20/136.70	   check_h(result_h(x0))
305.20/136.70	   check_f(redex_f(x0))
305.20/136.70	   check_g(redex_g(x0))
305.20/136.70	   check_h(redex_h(x0))
305.20/136.70	   in_f_1(go_up(x0))
305.20/136.70	   in_g_1(go_up(x0))
305.20/136.70	   in_h_1(go_up(x0))
305.20/136.70	
305.20/136.70	We have to consider all minimal (P,Q,R)-chains.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(121) UsableRulesProof (EQUIVALENT)
305.20/136.70	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.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(122)
305.20/136.70	Obligation:
305.20/136.70	Q DP problem:
305.20/136.70	The TRS P consists of the following rules:
305.20/136.70	
305.20/136.70	   TOP(go_up(x)) -> TOP(reduce(x))
305.20/136.70	
305.20/136.70	The TRS R consists of the following rules:
305.20/136.70	
305.20/136.70	   reduce(f(x_1)) -> check_f(redex_f(x_1))
305.20/136.70	   reduce(g(x_1)) -> check_g(redex_g(x_1))
305.20/136.70	   reduce(h(x_1)) -> check_h(redex_h(x_1))
305.20/136.70	   reduce(a) -> go_up(f(a))
305.20/136.70	   redex_h(h(x)) -> result_h(c)
305.20/136.70	   check_h(result_h(x)) -> go_up(x)
305.20/136.70	   check_h(redex_h(x_1)) -> in_h_1(reduce(x_1))
305.20/136.70	   in_h_1(go_up(x_1)) -> go_up(h(x_1))
305.20/136.70	   redex_g(g(x)) -> result_g(h(x))
305.20/136.70	   check_g(result_g(x)) -> go_up(x)
305.20/136.70	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
305.20/136.70	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
305.20/136.70	   redex_f(f(x)) -> result_f(g(x))
305.20/136.70	   check_f(result_f(x)) -> go_up(x)
305.20/136.70	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
305.20/136.70	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
305.20/136.70	
305.20/136.70	The set Q consists of the following terms:
305.20/136.70	
305.20/136.70	   top(go_up(x0))
305.20/136.70	   reduce(f(x0))
305.20/136.70	   reduce(g(x0))
305.20/136.70	   reduce(h(x0))
305.20/136.70	   reduce(a)
305.20/136.70	   redex_f(f(x0))
305.20/136.70	   redex_g(g(x0))
305.20/136.70	   redex_h(h(x0))
305.20/136.70	   check_f(result_f(x0))
305.20/136.70	   check_g(result_g(x0))
305.20/136.70	   check_h(result_h(x0))
305.20/136.70	   check_f(redex_f(x0))
305.20/136.70	   check_g(redex_g(x0))
305.20/136.70	   check_h(redex_h(x0))
305.20/136.70	   in_f_1(go_up(x0))
305.20/136.70	   in_g_1(go_up(x0))
305.20/136.70	   in_h_1(go_up(x0))
305.20/136.70	
305.20/136.70	We have to consider all minimal (P,Q,R)-chains.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(123) QReductionProof (EQUIVALENT)
305.20/136.70	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
305.20/136.70	
305.20/136.70	   top(go_up(x0))
305.20/136.70	
305.20/136.70	
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(124)
305.20/136.70	Obligation:
305.20/136.70	Q DP problem:
305.20/136.70	The TRS P consists of the following rules:
305.20/136.70	
305.20/136.70	   TOP(go_up(x)) -> TOP(reduce(x))
305.20/136.70	
305.20/136.70	The TRS R consists of the following rules:
305.20/136.70	
305.20/136.70	   reduce(f(x_1)) -> check_f(redex_f(x_1))
305.20/136.70	   reduce(g(x_1)) -> check_g(redex_g(x_1))
305.20/136.70	   reduce(h(x_1)) -> check_h(redex_h(x_1))
305.20/136.70	   reduce(a) -> go_up(f(a))
305.20/136.70	   redex_h(h(x)) -> result_h(c)
305.20/136.70	   check_h(result_h(x)) -> go_up(x)
305.20/136.70	   check_h(redex_h(x_1)) -> in_h_1(reduce(x_1))
305.20/136.70	   in_h_1(go_up(x_1)) -> go_up(h(x_1))
305.20/136.70	   redex_g(g(x)) -> result_g(h(x))
305.20/136.70	   check_g(result_g(x)) -> go_up(x)
305.20/136.70	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
305.20/136.70	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
305.20/136.70	   redex_f(f(x)) -> result_f(g(x))
305.20/136.70	   check_f(result_f(x)) -> go_up(x)
305.20/136.70	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
305.20/136.70	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
305.20/136.70	
305.20/136.70	The set Q consists of the following terms:
305.20/136.70	
305.20/136.70	   reduce(f(x0))
305.20/136.70	   reduce(g(x0))
305.20/136.70	   reduce(h(x0))
305.20/136.70	   reduce(a)
305.20/136.70	   redex_f(f(x0))
305.20/136.70	   redex_g(g(x0))
305.20/136.70	   redex_h(h(x0))
305.20/136.70	   check_f(result_f(x0))
305.20/136.70	   check_g(result_g(x0))
305.20/136.70	   check_h(result_h(x0))
305.20/136.70	   check_f(redex_f(x0))
305.20/136.70	   check_g(redex_g(x0))
305.20/136.70	   check_h(redex_h(x0))
305.20/136.70	   in_f_1(go_up(x0))
305.20/136.70	   in_g_1(go_up(x0))
305.20/136.70	   in_h_1(go_up(x0))
305.20/136.70	
305.20/136.70	We have to consider all minimal (P,Q,R)-chains.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(125) TransformationProof (EQUIVALENT)
305.20/136.70	By narrowing [LPAR04] the rule TOP(go_up(x)) -> TOP(reduce(x)) at position [0] we obtained the following new rules [LPAR04]:
305.20/136.70	
305.20/136.70	   (TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))),TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))))
305.20/136.70	   (TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))),TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))))
305.20/136.70	   (TOP(go_up(h(x0))) -> TOP(check_h(redex_h(x0))),TOP(go_up(h(x0))) -> TOP(check_h(redex_h(x0))))
305.20/136.70	   (TOP(go_up(a)) -> TOP(go_up(f(a))),TOP(go_up(a)) -> TOP(go_up(f(a))))
305.20/136.70	
305.20/136.70	
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(126)
305.20/136.70	Obligation:
305.20/136.70	Q DP problem:
305.20/136.70	The TRS P consists of the following rules:
305.20/136.70	
305.20/136.70	   TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0)))
305.20/136.70	   TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0)))
305.20/136.70	   TOP(go_up(h(x0))) -> TOP(check_h(redex_h(x0)))
305.20/136.70	   TOP(go_up(a)) -> TOP(go_up(f(a)))
305.20/136.70	
305.20/136.70	The TRS R consists of the following rules:
305.20/136.70	
305.20/136.70	   reduce(f(x_1)) -> check_f(redex_f(x_1))
305.20/136.70	   reduce(g(x_1)) -> check_g(redex_g(x_1))
305.20/136.70	   reduce(h(x_1)) -> check_h(redex_h(x_1))
305.20/136.70	   reduce(a) -> go_up(f(a))
305.20/136.70	   redex_h(h(x)) -> result_h(c)
305.20/136.70	   check_h(result_h(x)) -> go_up(x)
305.20/136.70	   check_h(redex_h(x_1)) -> in_h_1(reduce(x_1))
305.20/136.70	   in_h_1(go_up(x_1)) -> go_up(h(x_1))
305.20/136.70	   redex_g(g(x)) -> result_g(h(x))
305.20/136.70	   check_g(result_g(x)) -> go_up(x)
305.20/136.70	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
305.20/136.70	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
305.20/136.70	   redex_f(f(x)) -> result_f(g(x))
305.20/136.70	   check_f(result_f(x)) -> go_up(x)
305.20/136.70	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
305.20/136.70	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
305.20/136.70	
305.20/136.70	The set Q consists of the following terms:
305.20/136.70	
305.20/136.70	   reduce(f(x0))
305.20/136.70	   reduce(g(x0))
305.20/136.70	   reduce(h(x0))
305.20/136.70	   reduce(a)
305.20/136.70	   redex_f(f(x0))
305.20/136.70	   redex_g(g(x0))
305.20/136.70	   redex_h(h(x0))
305.20/136.70	   check_f(result_f(x0))
305.20/136.70	   check_g(result_g(x0))
305.20/136.70	   check_h(result_h(x0))
305.20/136.70	   check_f(redex_f(x0))
305.20/136.70	   check_g(redex_g(x0))
305.20/136.70	   check_h(redex_h(x0))
305.20/136.70	   in_f_1(go_up(x0))
305.20/136.70	   in_g_1(go_up(x0))
305.20/136.70	   in_h_1(go_up(x0))
305.20/136.70	
305.20/136.70	We have to consider all minimal (P,Q,R)-chains.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(127) QDPOrderProof (EQUIVALENT)
305.20/136.70	We use the reduction pair processor [LPAR04,JAR06].
305.20/136.70	
305.20/136.70	
305.20/136.70	The following pairs can be oriented strictly and are deleted.
305.20/136.70	
305.20/136.70	   TOP(go_up(a)) -> TOP(go_up(f(a)))
305.20/136.70	The remaining pairs can at least be oriented weakly.
305.20/136.70	Used ordering:  Polynomial interpretation [POLO]:
305.20/136.70	
305.20/136.70	   POL(TOP(x_1)) = x_1
305.20/136.70	   POL(a) = 1
305.20/136.70	   POL(c) = 0
305.20/136.70	   POL(check_f(x_1)) = x_1
305.20/136.70	   POL(check_g(x_1)) = x_1
305.20/136.70	   POL(check_h(x_1)) = x_1
305.20/136.70	   POL(f(x_1)) = 0
305.20/136.70	   POL(g(x_1)) = 0
305.20/136.70	   POL(go_up(x_1)) = x_1
305.20/136.70	   POL(h(x_1)) = 0
305.20/136.70	   POL(in_f_1(x_1)) = 0
305.20/136.70	   POL(in_g_1(x_1)) = 0
305.20/136.70	   POL(in_h_1(x_1)) = 0
305.20/136.70	   POL(redex_f(x_1)) = 0
305.20/136.70	   POL(redex_g(x_1)) = 0
305.20/136.70	   POL(redex_h(x_1)) = 0
305.20/136.70	   POL(reduce(x_1)) = 0
305.20/136.70	   POL(result_f(x_1)) = x_1
305.20/136.70	   POL(result_g(x_1)) = x_1
305.20/136.70	   POL(result_h(x_1)) = x_1
305.20/136.70	
305.20/136.70	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
305.20/136.70	
305.20/136.70	   redex_f(f(x)) -> result_f(g(x))
305.20/136.70	   check_f(result_f(x)) -> go_up(x)
305.20/136.70	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
305.20/136.70	   redex_g(g(x)) -> result_g(h(x))
305.20/136.70	   check_g(result_g(x)) -> go_up(x)
305.20/136.70	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
305.20/136.70	   redex_h(h(x)) -> result_h(c)
305.20/136.70	   check_h(result_h(x)) -> go_up(x)
305.20/136.70	   check_h(redex_h(x_1)) -> in_h_1(reduce(x_1))
305.20/136.70	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
305.20/136.70	   reduce(g(x_1)) -> check_g(redex_g(x_1))
305.20/136.70	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
305.20/136.70	   reduce(f(x_1)) -> check_f(redex_f(x_1))
305.20/136.70	   reduce(h(x_1)) -> check_h(redex_h(x_1))
305.20/136.70	   in_h_1(go_up(x_1)) -> go_up(h(x_1))
305.20/136.70	
305.20/136.70	
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(128)
305.20/136.70	Obligation:
305.20/136.70	Q DP problem:
305.20/136.70	The TRS P consists of the following rules:
305.20/136.70	
305.20/136.70	   TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0)))
305.20/136.70	   TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0)))
305.20/136.70	   TOP(go_up(h(x0))) -> TOP(check_h(redex_h(x0)))
305.20/136.70	
305.20/136.70	The TRS R consists of the following rules:
305.20/136.70	
305.20/136.70	   reduce(f(x_1)) -> check_f(redex_f(x_1))
305.20/136.70	   reduce(g(x_1)) -> check_g(redex_g(x_1))
305.20/136.70	   reduce(h(x_1)) -> check_h(redex_h(x_1))
305.20/136.70	   reduce(a) -> go_up(f(a))
305.20/136.70	   redex_h(h(x)) -> result_h(c)
305.20/136.70	   check_h(result_h(x)) -> go_up(x)
305.20/136.70	   check_h(redex_h(x_1)) -> in_h_1(reduce(x_1))
305.20/136.70	   in_h_1(go_up(x_1)) -> go_up(h(x_1))
305.20/136.70	   redex_g(g(x)) -> result_g(h(x))
305.20/136.70	   check_g(result_g(x)) -> go_up(x)
305.20/136.70	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
305.20/136.70	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
305.20/136.70	   redex_f(f(x)) -> result_f(g(x))
305.20/136.70	   check_f(result_f(x)) -> go_up(x)
305.20/136.70	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
305.20/136.70	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
305.20/136.70	
305.20/136.70	The set Q consists of the following terms:
305.20/136.70	
305.20/136.70	   reduce(f(x0))
305.20/136.70	   reduce(g(x0))
305.20/136.70	   reduce(h(x0))
305.20/136.70	   reduce(a)
305.20/136.70	   redex_f(f(x0))
305.20/136.70	   redex_g(g(x0))
305.20/136.70	   redex_h(h(x0))
305.20/136.70	   check_f(result_f(x0))
305.20/136.70	   check_g(result_g(x0))
305.20/136.70	   check_h(result_h(x0))
305.20/136.70	   check_f(redex_f(x0))
305.20/136.70	   check_g(redex_g(x0))
305.20/136.70	   check_h(redex_h(x0))
305.20/136.70	   in_f_1(go_up(x0))
305.20/136.70	   in_g_1(go_up(x0))
305.20/136.70	   in_h_1(go_up(x0))
305.20/136.70	
305.20/136.70	We have to consider all minimal (P,Q,R)-chains.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(129) UsableRulesProof (EQUIVALENT)
305.20/136.70	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.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(130)
305.20/136.70	Obligation:
305.20/136.70	Q DP problem:
305.20/136.70	The TRS P consists of the following rules:
305.20/136.70	
305.20/136.70	   TOP(go_up(x)) -> TOP(reduce(x))
305.20/136.70	
305.20/136.70	The TRS R consists of the following rules:
305.20/136.70	
305.20/136.70	   reduce(f(x_1)) -> check_f(redex_f(x_1))
305.20/136.70	   reduce(g(x_1)) -> check_g(redex_g(x_1))
305.20/136.70	   reduce(h(x_1)) -> check_h(redex_h(x_1))
305.20/136.70	   reduce(a) -> go_up(f(a))
305.20/136.70	   redex_h(h(x)) -> result_h(c)
305.20/136.70	   check_h(result_h(x)) -> go_up(x)
305.20/136.70	   check_h(redex_h(x_1)) -> in_h_1(reduce(x_1))
305.20/136.70	   in_h_1(go_up(x_1)) -> go_up(h(x_1))
305.20/136.70	   redex_g(g(x)) -> result_g(h(x))
305.20/136.70	   check_g(result_g(x)) -> go_up(x)
305.20/136.70	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
305.20/136.70	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
305.20/136.70	   redex_f(f(x)) -> result_f(g(x))
305.20/136.70	   check_f(result_f(x)) -> go_up(x)
305.20/136.70	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
305.20/136.70	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
305.20/136.70	
305.20/136.70	The set Q consists of the following terms:
305.20/136.70	
305.20/136.70	   top(go_up(x0))
305.20/136.70	   reduce(f(x0))
305.20/136.70	   reduce(g(x0))
305.20/136.70	   reduce(h(x0))
305.20/136.70	   reduce(a)
305.20/136.70	   redex_f(f(x0))
305.20/136.70	   redex_g(g(x0))
305.20/136.70	   redex_h(h(x0))
305.20/136.70	   check_f(result_f(x0))
305.20/136.70	   check_g(result_g(x0))
305.20/136.70	   check_h(result_h(x0))
305.20/136.70	   check_f(redex_f(x0))
305.20/136.70	   check_g(redex_g(x0))
305.20/136.70	   check_h(redex_h(x0))
305.20/136.70	   in_f_1(go_up(x0))
305.20/136.70	   in_g_1(go_up(x0))
305.20/136.70	   in_h_1(go_up(x0))
305.20/136.70	
305.20/136.70	We have to consider all minimal (P,Q,R)-chains.
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(131) QReductionProof (EQUIVALENT)
305.20/136.70	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
305.20/136.70	
305.20/136.70	   top(go_up(x0))
305.20/136.70	
305.20/136.70	
305.20/136.70	----------------------------------------
305.20/136.70	
305.20/136.70	(132)
305.20/136.70	Obligation:
305.20/136.70	Q DP problem:
305.20/136.70	The TRS P consists of the following rules:
305.20/136.70	
305.20/136.70	   TOP(go_up(x)) -> TOP(reduce(x))
305.20/136.70	
305.20/136.70	The TRS R consists of the following rules:
305.20/136.70	
305.20/136.70	   reduce(f(x_1)) -> check_f(redex_f(x_1))
305.20/136.70	   reduce(g(x_1)) -> check_g(redex_g(x_1))
305.20/136.70	   reduce(h(x_1)) -> check_h(redex_h(x_1))
305.20/136.70	   reduce(a) -> go_up(f(a))
305.20/136.70	   redex_h(h(x)) -> result_h(c)
305.20/136.70	   check_h(result_h(x)) -> go_up(x)
305.20/136.70	   check_h(redex_h(x_1)) -> in_h_1(reduce(x_1))
305.20/136.70	   in_h_1(go_up(x_1)) -> go_up(h(x_1))
305.20/136.70	   redex_g(g(x)) -> result_g(h(x))
305.20/136.70	   check_g(result_g(x)) -> go_up(x)
305.20/136.70	   check_g(redex_g(x_1)) -> in_g_1(reduce(x_1))
305.20/136.70	   in_g_1(go_up(x_1)) -> go_up(g(x_1))
305.20/136.70	   redex_f(f(x)) -> result_f(g(x))
305.20/136.70	   check_f(result_f(x)) -> go_up(x)
305.20/136.70	   check_f(redex_f(x_1)) -> in_f_1(reduce(x_1))
305.20/136.70	   in_f_1(go_up(x_1)) -> go_up(f(x_1))
305.20/136.70	
305.20/136.70	The set Q consists of the following terms:
305.20/136.70	
305.20/136.70	   reduce(f(x0))
305.20/136.70	   reduce(g(x0))
305.20/136.70	   reduce(h(x0))
305.20/136.70	   reduce(a)
305.20/136.70	   redex_f(f(x0))
305.20/136.70	   redex_g(g(x0))
305.20/136.70	   redex_h(h(x0))
305.20/136.70	   check_f(result_f(x0))
305.20/136.70	   check_g(result_g(x0))
305.20/136.70	   check_h(result_h(x0))
305.20/136.70	   check_f(redex_f(x0))
305.20/136.70	   check_g(redex_g(x0))
305.20/136.70	   check_h(redex_h(x0))
305.20/136.70	   in_f_1(go_up(x0))
305.20/136.70	   in_g_1(go_up(x0))
305.20/136.70	   in_h_1(go_up(x0))
305.20/136.70	
305.20/136.70	We have to consider all minimal (P,Q,R)-chains.
305.20/136.77	EOF