15.10/5.66	MAYBE
16.14/6.82	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
16.14/6.82	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
16.14/6.82	
16.14/6.82	
16.14/6.82	Left Termination of the query pattern
16.14/6.82	
16.14/6.82	reach(g,g,g)
16.14/6.82	
16.14/6.82	w.r.t. the given Prolog program could not be shown:
16.14/6.82	
16.14/6.82	(0) Prolog
16.14/6.82	(1) PrologToPiTRSProof [SOUND, 0 ms]
16.14/6.82	(2) PiTRS
16.14/6.82	    (3) DependencyPairsProof [EQUIVALENT, 12 ms]
16.14/6.82	    (4) PiDP
16.14/6.82	    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
16.14/6.82	    (6) AND
16.14/6.82	        (7) PiDP
16.14/6.82	            (8) UsableRulesProof [EQUIVALENT, 0 ms]
16.14/6.82	            (9) PiDP
16.14/6.82	            (10) PiDPToQDPProof [SOUND, 0 ms]
16.14/6.82	            (11) QDP
16.14/6.82	            (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
16.14/6.82	            (13) YES
16.14/6.82	        (14) PiDP
16.14/6.82	            (15) UsableRulesProof [EQUIVALENT, 0 ms]
16.14/6.82	            (16) PiDP
16.14/6.82	            (17) PiDPToQDPProof [EQUIVALENT, 0 ms]
16.14/6.82	            (18) QDP
16.14/6.82	            (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
16.14/6.82	            (20) YES
16.14/6.82	        (21) PiDP
16.14/6.82	            (22) UsableRulesProof [EQUIVALENT, 0 ms]
16.14/6.82	            (23) PiDP
16.14/6.82	            (24) PiDPToQDPProof [SOUND, 0 ms]
16.14/6.82	            (25) QDP
16.14/6.82	            (26) TransformationProof [SOUND, 0 ms]
16.14/6.82	            (27) QDP
16.14/6.82	            (28) TransformationProof [EQUIVALENT, 0 ms]
16.14/6.82	            (29) QDP
16.14/6.82	            (30) TransformationProof [EQUIVALENT, 0 ms]
16.14/6.82	            (31) QDP
16.14/6.82	(32) PrologToPiTRSProof [SOUND, 0 ms]
16.14/6.82	(33) PiTRS
16.14/6.82	    (34) DependencyPairsProof [EQUIVALENT, 8 ms]
16.14/6.82	    (35) PiDP
16.14/6.82	    (36) DependencyGraphProof [EQUIVALENT, 0 ms]
16.14/6.82	    (37) AND
16.14/6.82	        (38) PiDP
16.14/6.82	            (39) UsableRulesProof [EQUIVALENT, 0 ms]
16.14/6.82	            (40) PiDP
16.14/6.82	            (41) PiDPToQDPProof [SOUND, 10 ms]
16.14/6.82	            (42) QDP
16.14/6.82	            (43) QDPSizeChangeProof [EQUIVALENT, 0 ms]
16.14/6.82	            (44) YES
16.14/6.82	        (45) PiDP
16.14/6.82	            (46) UsableRulesProof [EQUIVALENT, 0 ms]
16.14/6.82	            (47) PiDP
16.14/6.82	            (48) PiDPToQDPProof [EQUIVALENT, 0 ms]
16.14/6.82	            (49) QDP
16.14/6.82	            (50) QDPSizeChangeProof [EQUIVALENT, 0 ms]
16.14/6.82	            (51) YES
16.14/6.82	        (52) PiDP
16.14/6.82	            (53) UsableRulesProof [EQUIVALENT, 0 ms]
16.14/6.82	            (54) PiDP
16.14/6.82	            (55) PiDPToQDPProof [SOUND, 1 ms]
16.14/6.82	            (56) QDP
16.14/6.82	            (57) TransformationProof [SOUND, 0 ms]
16.14/6.82	            (58) QDP
16.14/6.82	            (59) TransformationProof [EQUIVALENT, 0 ms]
16.14/6.82	            (60) QDP
16.14/6.82	            (61) TransformationProof [EQUIVALENT, 0 ms]
16.14/6.82	            (62) QDP
16.14/6.82	(63) PrologToTRSTransformerProof [SOUND, 20 ms]
16.14/6.82	(64) QTRS
16.14/6.82	    (65) DependencyPairsProof [EQUIVALENT, 0 ms]
16.14/6.82	    (66) QDP
16.14/6.82	    (67) DependencyGraphProof [EQUIVALENT, 0 ms]
16.14/6.82	    (68) AND
16.14/6.82	        (69) QDP
16.14/6.82	            (70) UsableRulesProof [EQUIVALENT, 0 ms]
16.14/6.82	            (71) QDP
16.14/6.82	            (72) QDPSizeChangeProof [EQUIVALENT, 0 ms]
16.14/6.82	            (73) YES
16.14/6.82	        (74) QDP
16.14/6.82	            (75) UsableRulesProof [EQUIVALENT, 0 ms]
16.14/6.82	            (76) QDP
16.14/6.82	            (77) QDPSizeChangeProof [EQUIVALENT, 0 ms]
16.14/6.82	            (78) YES
16.14/6.82	        (79) QDP
16.14/6.82	            (80) NonTerminationLoopProof [COMPLETE, 562 ms]
16.14/6.82	            (81) NO
16.14/6.82	(82) PrologToDTProblemTransformerProof [SOUND, 41 ms]
16.14/6.82	(83) TRIPLES
16.14/6.82	    (84) TriplesToPiDPProof [SOUND, 0 ms]
16.14/6.82	    (85) PiDP
16.14/6.82	    (86) DependencyGraphProof [EQUIVALENT, 0 ms]
16.14/6.82	    (87) AND
16.14/6.82	        (88) PiDP
16.14/6.82	            (89) UsableRulesProof [EQUIVALENT, 0 ms]
16.14/6.82	            (90) PiDP
16.14/6.82	            (91) PiDPToQDPProof [SOUND, 0 ms]
16.14/6.82	            (92) QDP
16.14/6.82	            (93) QDPSizeChangeProof [EQUIVALENT, 0 ms]
16.14/6.82	            (94) YES
16.14/6.82	        (95) PiDP
16.14/6.82	            (96) UsableRulesProof [EQUIVALENT, 0 ms]
16.14/6.82	            (97) PiDP
16.14/6.82	            (98) PiDPToQDPProof [EQUIVALENT, 0 ms]
16.14/6.82	            (99) QDP
16.14/6.82	            (100) QDPSizeChangeProof [EQUIVALENT, 0 ms]
16.14/6.82	            (101) YES
16.14/6.82	        (102) PiDP
16.14/6.82	            (103) PiDPToQDPProof [SOUND, 0 ms]
16.14/6.82	            (104) QDP
16.14/6.82	            (105) TransformationProof [SOUND, 0 ms]
16.14/6.82	            (106) QDP
16.14/6.82	(107) PrologToIRSwTTransformerProof [SOUND, 34 ms]
16.14/6.82	(108) AND
16.14/6.82	    (109) IRSwT
16.14/6.82	        (110) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
16.14/6.82	        (111) IRSwT
16.14/6.82	        (112) IntTRSCompressionProof [EQUIVALENT, 3 ms]
16.14/6.82	        (113) IRSwT
16.14/6.82	        (114) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
16.14/6.82	        (115) IRSwT
16.14/6.82	        (116) IRSwTTerminationDigraphProof [EQUIVALENT, 2 ms]
16.14/6.82	        (117) IRSwT
16.14/6.82	        (118) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 33 ms]
16.14/6.82	        (119) IRSwT
16.14/6.82	        (120) TempFilterProof [SOUND, 3 ms]
16.14/6.82	        (121) IRSwT
16.14/6.82	        (122) IRSwTToQDPProof [SOUND, 0 ms]
16.14/6.82	        (123) QDP
16.14/6.82	        (124) QDPSizeChangeProof [EQUIVALENT, 0 ms]
16.14/6.82	        (125) YES
16.14/6.82	    (126) IRSwT
16.14/6.82	        (127) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
16.14/6.82	        (128) IRSwT
16.14/6.82	        (129) IntTRSCompressionProof [EQUIVALENT, 0 ms]
16.14/6.82	        (130) IRSwT
16.14/6.82	        (131) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
16.14/6.82	        (132) IRSwT
16.14/6.82	        (133) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
16.14/6.82	        (134) IRSwT
16.14/6.82	        (135) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
16.14/6.82	        (136) IRSwT
16.14/6.82	        (137) TempFilterProof [SOUND, 1 ms]
16.14/6.82	        (138) IRSwT
16.14/6.82	        (139) IRSwTToQDPProof [SOUND, 0 ms]
16.14/6.82	        (140) QDP
16.14/6.82	        (141) QDPSizeChangeProof [EQUIVALENT, 0 ms]
16.14/6.82	        (142) YES
16.14/6.82	    (143) IRSwT
16.14/6.82	        (144) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
16.14/6.82	        (145) IRSwT
16.14/6.82	        (146) IntTRSCompressionProof [EQUIVALENT, 10 ms]
16.14/6.82	        (147) IRSwT
16.14/6.82	        (148) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
16.14/6.82	        (149) IRSwT
16.14/6.82	
16.14/6.82	
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(0)
16.14/6.82	Obligation:
16.14/6.82	Clauses:
16.14/6.82	
16.14/6.82	reach(X, Y, Edges) :- member(.(X, .(Y, [])), Edges).
16.14/6.82	reach(X, Z, Edges) :- ','(member1(.(X, .(Y, [])), Edges), reach(Y, Z, Edges)).
16.14/6.82	member(H, .(H, L)).
16.14/6.82	member(X, .(H, L)) :- member(X, L).
16.14/6.82	member1(H, .(H, L)).
16.14/6.82	member1(X, .(H, L)) :- member1(X, L).
16.14/6.82	
16.14/6.82	
16.14/6.82	Query: reach(g,g,g)
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(1) PrologToPiTRSProof (SOUND)
16.14/6.82	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
16.14/6.82	
16.14/6.82	reach_in_3: (b,b,b)
16.14/6.82	
16.14/6.82	member_in_2: (b,b)
16.14/6.82	
16.14/6.82	member1_in_2: (f,b)
16.14/6.82	
16.14/6.82	Transforming Prolog into the following Term Rewriting System:
16.14/6.82	
16.14/6.82	Pi-finite rewrite system:
16.14/6.82	The TRS R consists of the following rules:
16.14/6.82	
16.14/6.82	   reach_in_ggg(X, Y, Edges) -> U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
16.14/6.82	   member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L))
16.14/6.82	   member_in_gg(X, .(H, L)) -> U4_gg(X, H, L, member_in_gg(X, L))
16.14/6.82	   U4_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L))
16.14/6.82	   U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_ggg(X, Y, Edges)
16.14/6.82	   reach_in_ggg(X, Z, Edges) -> U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.82	   member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L))
16.14/6.82	   member1_in_ag(X, .(H, L)) -> U5_ag(X, H, L, member1_in_ag(X, L))
16.14/6.82	   U5_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L))
16.14/6.82	   U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
16.14/6.82	   U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) -> reach_out_ggg(X, Z, Edges)
16.14/6.82	
16.14/6.82	The argument filtering Pi contains the following mapping:
16.14/6.82	reach_in_ggg(x1, x2, x3)  =  reach_in_ggg(x1, x2, x3)
16.14/6.82	
16.14/6.82	U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	member_in_gg(x1, x2)  =  member_in_gg(x1, x2)
16.14/6.82	
16.14/6.82	.(x1, x2)  =  .(x1, x2)
16.14/6.82	
16.14/6.82	member_out_gg(x1, x2)  =  member_out_gg(x1, x2)
16.14/6.82	
16.14/6.82	U4_gg(x1, x2, x3, x4)  =  U4_gg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	[]  =  []
16.14/6.82	
16.14/6.82	reach_out_ggg(x1, x2, x3)  =  reach_out_ggg(x1, x2, x3)
16.14/6.82	
16.14/6.82	U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	member1_in_ag(x1, x2)  =  member1_in_ag(x2)
16.14/6.82	
16.14/6.82	member1_out_ag(x1, x2)  =  member1_out_ag(x1, x2)
16.14/6.82	
16.14/6.82	U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
16.14/6.82	
16.14/6.82	U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	
16.14/6.82	
16.14/6.82	
16.14/6.82	
16.14/6.82	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
16.14/6.82	
16.14/6.82	
16.14/6.82	
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(2)
16.14/6.82	Obligation:
16.14/6.82	Pi-finite rewrite system:
16.14/6.82	The TRS R consists of the following rules:
16.14/6.82	
16.14/6.82	   reach_in_ggg(X, Y, Edges) -> U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
16.14/6.82	   member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L))
16.14/6.82	   member_in_gg(X, .(H, L)) -> U4_gg(X, H, L, member_in_gg(X, L))
16.14/6.82	   U4_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L))
16.14/6.82	   U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_ggg(X, Y, Edges)
16.14/6.82	   reach_in_ggg(X, Z, Edges) -> U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.82	   member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L))
16.14/6.82	   member1_in_ag(X, .(H, L)) -> U5_ag(X, H, L, member1_in_ag(X, L))
16.14/6.82	   U5_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L))
16.14/6.82	   U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
16.14/6.82	   U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) -> reach_out_ggg(X, Z, Edges)
16.14/6.82	
16.14/6.82	The argument filtering Pi contains the following mapping:
16.14/6.82	reach_in_ggg(x1, x2, x3)  =  reach_in_ggg(x1, x2, x3)
16.14/6.82	
16.14/6.82	U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	member_in_gg(x1, x2)  =  member_in_gg(x1, x2)
16.14/6.82	
16.14/6.82	.(x1, x2)  =  .(x1, x2)
16.14/6.82	
16.14/6.82	member_out_gg(x1, x2)  =  member_out_gg(x1, x2)
16.14/6.82	
16.14/6.82	U4_gg(x1, x2, x3, x4)  =  U4_gg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	[]  =  []
16.14/6.82	
16.14/6.82	reach_out_ggg(x1, x2, x3)  =  reach_out_ggg(x1, x2, x3)
16.14/6.82	
16.14/6.82	U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	member1_in_ag(x1, x2)  =  member1_in_ag(x2)
16.14/6.82	
16.14/6.82	member1_out_ag(x1, x2)  =  member1_out_ag(x1, x2)
16.14/6.82	
16.14/6.82	U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
16.14/6.82	
16.14/6.82	U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	
16.14/6.82	
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(3) DependencyPairsProof (EQUIVALENT)
16.14/6.82	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
16.14/6.82	Pi DP problem:
16.14/6.82	The TRS P consists of the following rules:
16.14/6.82	
16.14/6.82	   REACH_IN_GGG(X, Y, Edges) -> U1_GGG(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
16.14/6.82	   REACH_IN_GGG(X, Y, Edges) -> MEMBER_IN_GG(.(X, .(Y, [])), Edges)
16.14/6.82	   MEMBER_IN_GG(X, .(H, L)) -> U4_GG(X, H, L, member_in_gg(X, L))
16.14/6.82	   MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L)
16.14/6.82	   REACH_IN_GGG(X, Z, Edges) -> U2_GGG(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.82	   REACH_IN_GGG(X, Z, Edges) -> MEMBER1_IN_AG(.(X, .(Y, [])), Edges)
16.14/6.82	   MEMBER1_IN_AG(X, .(H, L)) -> U5_AG(X, H, L, member1_in_ag(X, L))
16.14/6.82	   MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L)
16.14/6.82	   U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGG(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
16.14/6.82	   U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> REACH_IN_GGG(Y, Z, Edges)
16.14/6.82	
16.14/6.82	The TRS R consists of the following rules:
16.14/6.82	
16.14/6.82	   reach_in_ggg(X, Y, Edges) -> U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
16.14/6.82	   member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L))
16.14/6.82	   member_in_gg(X, .(H, L)) -> U4_gg(X, H, L, member_in_gg(X, L))
16.14/6.82	   U4_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L))
16.14/6.82	   U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_ggg(X, Y, Edges)
16.14/6.82	   reach_in_ggg(X, Z, Edges) -> U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.82	   member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L))
16.14/6.82	   member1_in_ag(X, .(H, L)) -> U5_ag(X, H, L, member1_in_ag(X, L))
16.14/6.82	   U5_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L))
16.14/6.82	   U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
16.14/6.82	   U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) -> reach_out_ggg(X, Z, Edges)
16.14/6.82	
16.14/6.82	The argument filtering Pi contains the following mapping:
16.14/6.82	reach_in_ggg(x1, x2, x3)  =  reach_in_ggg(x1, x2, x3)
16.14/6.82	
16.14/6.82	U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	member_in_gg(x1, x2)  =  member_in_gg(x1, x2)
16.14/6.82	
16.14/6.82	.(x1, x2)  =  .(x1, x2)
16.14/6.82	
16.14/6.82	member_out_gg(x1, x2)  =  member_out_gg(x1, x2)
16.14/6.82	
16.14/6.82	U4_gg(x1, x2, x3, x4)  =  U4_gg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	[]  =  []
16.14/6.82	
16.14/6.82	reach_out_ggg(x1, x2, x3)  =  reach_out_ggg(x1, x2, x3)
16.14/6.82	
16.14/6.82	U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	member1_in_ag(x1, x2)  =  member1_in_ag(x2)
16.14/6.82	
16.14/6.82	member1_out_ag(x1, x2)  =  member1_out_ag(x1, x2)
16.14/6.82	
16.14/6.82	U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
16.14/6.82	
16.14/6.82	U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	REACH_IN_GGG(x1, x2, x3)  =  REACH_IN_GGG(x1, x2, x3)
16.14/6.82	
16.14/6.82	U1_GGG(x1, x2, x3, x4)  =  U1_GGG(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	MEMBER_IN_GG(x1, x2)  =  MEMBER_IN_GG(x1, x2)
16.14/6.82	
16.14/6.82	U4_GG(x1, x2, x3, x4)  =  U4_GG(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	MEMBER1_IN_AG(x1, x2)  =  MEMBER1_IN_AG(x2)
16.14/6.82	
16.14/6.82	U5_AG(x1, x2, x3, x4)  =  U5_AG(x2, x3, x4)
16.14/6.82	
16.14/6.82	U3_GGG(x1, x2, x3, x4)  =  U3_GGG(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	
16.14/6.82	We have to consider all (P,R,Pi)-chains
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(4)
16.14/6.82	Obligation:
16.14/6.82	Pi DP problem:
16.14/6.82	The TRS P consists of the following rules:
16.14/6.82	
16.14/6.82	   REACH_IN_GGG(X, Y, Edges) -> U1_GGG(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
16.14/6.82	   REACH_IN_GGG(X, Y, Edges) -> MEMBER_IN_GG(.(X, .(Y, [])), Edges)
16.14/6.82	   MEMBER_IN_GG(X, .(H, L)) -> U4_GG(X, H, L, member_in_gg(X, L))
16.14/6.82	   MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L)
16.14/6.82	   REACH_IN_GGG(X, Z, Edges) -> U2_GGG(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.82	   REACH_IN_GGG(X, Z, Edges) -> MEMBER1_IN_AG(.(X, .(Y, [])), Edges)
16.14/6.82	   MEMBER1_IN_AG(X, .(H, L)) -> U5_AG(X, H, L, member1_in_ag(X, L))
16.14/6.82	   MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L)
16.14/6.82	   U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGG(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
16.14/6.82	   U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> REACH_IN_GGG(Y, Z, Edges)
16.14/6.82	
16.14/6.82	The TRS R consists of the following rules:
16.14/6.82	
16.14/6.82	   reach_in_ggg(X, Y, Edges) -> U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
16.14/6.82	   member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L))
16.14/6.82	   member_in_gg(X, .(H, L)) -> U4_gg(X, H, L, member_in_gg(X, L))
16.14/6.82	   U4_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L))
16.14/6.82	   U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_ggg(X, Y, Edges)
16.14/6.82	   reach_in_ggg(X, Z, Edges) -> U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.82	   member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L))
16.14/6.82	   member1_in_ag(X, .(H, L)) -> U5_ag(X, H, L, member1_in_ag(X, L))
16.14/6.82	   U5_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L))
16.14/6.82	   U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
16.14/6.82	   U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) -> reach_out_ggg(X, Z, Edges)
16.14/6.82	
16.14/6.82	The argument filtering Pi contains the following mapping:
16.14/6.82	reach_in_ggg(x1, x2, x3)  =  reach_in_ggg(x1, x2, x3)
16.14/6.82	
16.14/6.82	U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	member_in_gg(x1, x2)  =  member_in_gg(x1, x2)
16.14/6.82	
16.14/6.82	.(x1, x2)  =  .(x1, x2)
16.14/6.82	
16.14/6.82	member_out_gg(x1, x2)  =  member_out_gg(x1, x2)
16.14/6.82	
16.14/6.82	U4_gg(x1, x2, x3, x4)  =  U4_gg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	[]  =  []
16.14/6.82	
16.14/6.82	reach_out_ggg(x1, x2, x3)  =  reach_out_ggg(x1, x2, x3)
16.14/6.82	
16.14/6.82	U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	member1_in_ag(x1, x2)  =  member1_in_ag(x2)
16.14/6.82	
16.14/6.82	member1_out_ag(x1, x2)  =  member1_out_ag(x1, x2)
16.14/6.82	
16.14/6.82	U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
16.14/6.82	
16.14/6.82	U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	REACH_IN_GGG(x1, x2, x3)  =  REACH_IN_GGG(x1, x2, x3)
16.14/6.82	
16.14/6.82	U1_GGG(x1, x2, x3, x4)  =  U1_GGG(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	MEMBER_IN_GG(x1, x2)  =  MEMBER_IN_GG(x1, x2)
16.14/6.82	
16.14/6.82	U4_GG(x1, x2, x3, x4)  =  U4_GG(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	MEMBER1_IN_AG(x1, x2)  =  MEMBER1_IN_AG(x2)
16.14/6.82	
16.14/6.82	U5_AG(x1, x2, x3, x4)  =  U5_AG(x2, x3, x4)
16.14/6.82	
16.14/6.82	U3_GGG(x1, x2, x3, x4)  =  U3_GGG(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	
16.14/6.82	We have to consider all (P,R,Pi)-chains
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(5) DependencyGraphProof (EQUIVALENT)
16.14/6.82	The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 6 less nodes.
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(6)
16.14/6.82	Complex Obligation (AND)
16.14/6.82	
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(7)
16.14/6.82	Obligation:
16.14/6.82	Pi DP problem:
16.14/6.82	The TRS P consists of the following rules:
16.14/6.82	
16.14/6.82	   MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L)
16.14/6.82	
16.14/6.82	The TRS R consists of the following rules:
16.14/6.82	
16.14/6.82	   reach_in_ggg(X, Y, Edges) -> U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
16.14/6.82	   member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L))
16.14/6.82	   member_in_gg(X, .(H, L)) -> U4_gg(X, H, L, member_in_gg(X, L))
16.14/6.82	   U4_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L))
16.14/6.82	   U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_ggg(X, Y, Edges)
16.14/6.82	   reach_in_ggg(X, Z, Edges) -> U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.82	   member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L))
16.14/6.82	   member1_in_ag(X, .(H, L)) -> U5_ag(X, H, L, member1_in_ag(X, L))
16.14/6.82	   U5_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L))
16.14/6.82	   U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
16.14/6.82	   U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) -> reach_out_ggg(X, Z, Edges)
16.14/6.82	
16.14/6.82	The argument filtering Pi contains the following mapping:
16.14/6.82	reach_in_ggg(x1, x2, x3)  =  reach_in_ggg(x1, x2, x3)
16.14/6.82	
16.14/6.82	U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	member_in_gg(x1, x2)  =  member_in_gg(x1, x2)
16.14/6.82	
16.14/6.82	.(x1, x2)  =  .(x1, x2)
16.14/6.82	
16.14/6.82	member_out_gg(x1, x2)  =  member_out_gg(x1, x2)
16.14/6.82	
16.14/6.82	U4_gg(x1, x2, x3, x4)  =  U4_gg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	[]  =  []
16.14/6.82	
16.14/6.82	reach_out_ggg(x1, x2, x3)  =  reach_out_ggg(x1, x2, x3)
16.14/6.82	
16.14/6.82	U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	member1_in_ag(x1, x2)  =  member1_in_ag(x2)
16.14/6.82	
16.14/6.82	member1_out_ag(x1, x2)  =  member1_out_ag(x1, x2)
16.14/6.82	
16.14/6.82	U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
16.14/6.82	
16.14/6.82	U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	MEMBER1_IN_AG(x1, x2)  =  MEMBER1_IN_AG(x2)
16.14/6.82	
16.14/6.82	
16.14/6.82	We have to consider all (P,R,Pi)-chains
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(8) UsableRulesProof (EQUIVALENT)
16.14/6.82	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(9)
16.14/6.82	Obligation:
16.14/6.82	Pi DP problem:
16.14/6.82	The TRS P consists of the following rules:
16.14/6.82	
16.14/6.82	   MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L)
16.14/6.82	
16.14/6.82	R is empty.
16.14/6.82	The argument filtering Pi contains the following mapping:
16.14/6.82	.(x1, x2)  =  .(x1, x2)
16.14/6.82	
16.14/6.82	MEMBER1_IN_AG(x1, x2)  =  MEMBER1_IN_AG(x2)
16.14/6.82	
16.14/6.82	
16.14/6.82	We have to consider all (P,R,Pi)-chains
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(10) PiDPToQDPProof (SOUND)
16.14/6.82	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(11)
16.14/6.82	Obligation:
16.14/6.82	Q DP problem:
16.14/6.82	The TRS P consists of the following rules:
16.14/6.82	
16.14/6.82	   MEMBER1_IN_AG(.(H, L)) -> MEMBER1_IN_AG(L)
16.14/6.82	
16.14/6.82	R is empty.
16.14/6.82	Q is empty.
16.14/6.82	We have to consider all (P,Q,R)-chains.
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(12) QDPSizeChangeProof (EQUIVALENT)
16.14/6.82	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. 
16.14/6.82	
16.14/6.82	From the DPs we obtained the following set of size-change graphs:
16.14/6.82	*MEMBER1_IN_AG(.(H, L)) -> MEMBER1_IN_AG(L)
16.14/6.82	The graph contains the following edges 1 > 1
16.14/6.82	
16.14/6.82	
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(13)
16.14/6.82	YES
16.14/6.82	
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(14)
16.14/6.82	Obligation:
16.14/6.82	Pi DP problem:
16.14/6.82	The TRS P consists of the following rules:
16.14/6.82	
16.14/6.82	   MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L)
16.14/6.82	
16.14/6.82	The TRS R consists of the following rules:
16.14/6.82	
16.14/6.82	   reach_in_ggg(X, Y, Edges) -> U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
16.14/6.82	   member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L))
16.14/6.82	   member_in_gg(X, .(H, L)) -> U4_gg(X, H, L, member_in_gg(X, L))
16.14/6.82	   U4_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L))
16.14/6.82	   U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_ggg(X, Y, Edges)
16.14/6.82	   reach_in_ggg(X, Z, Edges) -> U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.82	   member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L))
16.14/6.82	   member1_in_ag(X, .(H, L)) -> U5_ag(X, H, L, member1_in_ag(X, L))
16.14/6.82	   U5_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L))
16.14/6.82	   U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
16.14/6.82	   U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) -> reach_out_ggg(X, Z, Edges)
16.14/6.82	
16.14/6.82	The argument filtering Pi contains the following mapping:
16.14/6.82	reach_in_ggg(x1, x2, x3)  =  reach_in_ggg(x1, x2, x3)
16.14/6.82	
16.14/6.82	U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	member_in_gg(x1, x2)  =  member_in_gg(x1, x2)
16.14/6.82	
16.14/6.82	.(x1, x2)  =  .(x1, x2)
16.14/6.82	
16.14/6.82	member_out_gg(x1, x2)  =  member_out_gg(x1, x2)
16.14/6.82	
16.14/6.82	U4_gg(x1, x2, x3, x4)  =  U4_gg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	[]  =  []
16.14/6.82	
16.14/6.82	reach_out_ggg(x1, x2, x3)  =  reach_out_ggg(x1, x2, x3)
16.14/6.82	
16.14/6.82	U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	member1_in_ag(x1, x2)  =  member1_in_ag(x2)
16.14/6.82	
16.14/6.82	member1_out_ag(x1, x2)  =  member1_out_ag(x1, x2)
16.14/6.82	
16.14/6.82	U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
16.14/6.82	
16.14/6.82	U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	MEMBER_IN_GG(x1, x2)  =  MEMBER_IN_GG(x1, x2)
16.14/6.82	
16.14/6.82	
16.14/6.82	We have to consider all (P,R,Pi)-chains
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(15) UsableRulesProof (EQUIVALENT)
16.14/6.82	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(16)
16.14/6.82	Obligation:
16.14/6.82	Pi DP problem:
16.14/6.82	The TRS P consists of the following rules:
16.14/6.82	
16.14/6.82	   MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L)
16.14/6.82	
16.14/6.82	R is empty.
16.14/6.82	Pi is empty.
16.14/6.82	We have to consider all (P,R,Pi)-chains
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(17) PiDPToQDPProof (EQUIVALENT)
16.14/6.82	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(18)
16.14/6.82	Obligation:
16.14/6.82	Q DP problem:
16.14/6.82	The TRS P consists of the following rules:
16.14/6.82	
16.14/6.82	   MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L)
16.14/6.82	
16.14/6.82	R is empty.
16.14/6.82	Q is empty.
16.14/6.82	We have to consider all (P,Q,R)-chains.
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(19) QDPSizeChangeProof (EQUIVALENT)
16.14/6.82	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. 
16.14/6.82	
16.14/6.82	From the DPs we obtained the following set of size-change graphs:
16.14/6.82	*MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L)
16.14/6.82	The graph contains the following edges 1 >= 1, 2 > 2
16.14/6.82	
16.14/6.82	
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(20)
16.14/6.82	YES
16.14/6.82	
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(21)
16.14/6.82	Obligation:
16.14/6.82	Pi DP problem:
16.14/6.82	The TRS P consists of the following rules:
16.14/6.82	
16.14/6.82	   REACH_IN_GGG(X, Z, Edges) -> U2_GGG(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.82	   U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> REACH_IN_GGG(Y, Z, Edges)
16.14/6.82	
16.14/6.82	The TRS R consists of the following rules:
16.14/6.82	
16.14/6.82	   reach_in_ggg(X, Y, Edges) -> U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
16.14/6.82	   member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L))
16.14/6.82	   member_in_gg(X, .(H, L)) -> U4_gg(X, H, L, member_in_gg(X, L))
16.14/6.82	   U4_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L))
16.14/6.82	   U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_ggg(X, Y, Edges)
16.14/6.82	   reach_in_ggg(X, Z, Edges) -> U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.82	   member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L))
16.14/6.82	   member1_in_ag(X, .(H, L)) -> U5_ag(X, H, L, member1_in_ag(X, L))
16.14/6.82	   U5_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L))
16.14/6.82	   U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
16.14/6.82	   U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) -> reach_out_ggg(X, Z, Edges)
16.14/6.82	
16.14/6.82	The argument filtering Pi contains the following mapping:
16.14/6.82	reach_in_ggg(x1, x2, x3)  =  reach_in_ggg(x1, x2, x3)
16.14/6.82	
16.14/6.82	U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	member_in_gg(x1, x2)  =  member_in_gg(x1, x2)
16.14/6.82	
16.14/6.82	.(x1, x2)  =  .(x1, x2)
16.14/6.82	
16.14/6.82	member_out_gg(x1, x2)  =  member_out_gg(x1, x2)
16.14/6.82	
16.14/6.82	U4_gg(x1, x2, x3, x4)  =  U4_gg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	[]  =  []
16.14/6.82	
16.14/6.82	reach_out_ggg(x1, x2, x3)  =  reach_out_ggg(x1, x2, x3)
16.14/6.82	
16.14/6.82	U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	member1_in_ag(x1, x2)  =  member1_in_ag(x2)
16.14/6.82	
16.14/6.82	member1_out_ag(x1, x2)  =  member1_out_ag(x1, x2)
16.14/6.82	
16.14/6.82	U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
16.14/6.82	
16.14/6.82	U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	REACH_IN_GGG(x1, x2, x3)  =  REACH_IN_GGG(x1, x2, x3)
16.14/6.82	
16.14/6.82	U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	
16.14/6.82	We have to consider all (P,R,Pi)-chains
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(22) UsableRulesProof (EQUIVALENT)
16.14/6.82	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(23)
16.14/6.82	Obligation:
16.14/6.82	Pi DP problem:
16.14/6.82	The TRS P consists of the following rules:
16.14/6.82	
16.14/6.82	   REACH_IN_GGG(X, Z, Edges) -> U2_GGG(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.82	   U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> REACH_IN_GGG(Y, Z, Edges)
16.14/6.82	
16.14/6.82	The TRS R consists of the following rules:
16.14/6.82	
16.14/6.82	   member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L))
16.14/6.82	   member1_in_ag(X, .(H, L)) -> U5_ag(X, H, L, member1_in_ag(X, L))
16.14/6.82	   U5_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L))
16.14/6.82	
16.14/6.82	The argument filtering Pi contains the following mapping:
16.14/6.82	.(x1, x2)  =  .(x1, x2)
16.14/6.82	
16.14/6.82	[]  =  []
16.14/6.82	
16.14/6.82	member1_in_ag(x1, x2)  =  member1_in_ag(x2)
16.14/6.82	
16.14/6.82	member1_out_ag(x1, x2)  =  member1_out_ag(x1, x2)
16.14/6.82	
16.14/6.82	U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
16.14/6.82	
16.14/6.82	REACH_IN_GGG(x1, x2, x3)  =  REACH_IN_GGG(x1, x2, x3)
16.14/6.82	
16.14/6.82	U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x1, x2, x3, x4)
16.14/6.82	
16.14/6.82	
16.14/6.82	We have to consider all (P,R,Pi)-chains
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(24) PiDPToQDPProof (SOUND)
16.14/6.82	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(25)
16.14/6.82	Obligation:
16.14/6.82	Q DP problem:
16.14/6.82	The TRS P consists of the following rules:
16.14/6.82	
16.14/6.82	   REACH_IN_GGG(X, Z, Edges) -> U2_GGG(X, Z, Edges, member1_in_ag(Edges))
16.14/6.82	   U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> REACH_IN_GGG(Y, Z, Edges)
16.14/6.82	
16.14/6.82	The TRS R consists of the following rules:
16.14/6.82	
16.14/6.82	   member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L))
16.14/6.82	   member1_in_ag(.(H, L)) -> U5_ag(H, L, member1_in_ag(L))
16.14/6.82	   U5_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L))
16.14/6.82	
16.14/6.82	The set Q consists of the following terms:
16.14/6.82	
16.14/6.82	   member1_in_ag(x0)
16.14/6.82	   U5_ag(x0, x1, x2)
16.14/6.82	
16.14/6.82	We have to consider all (P,Q,R)-chains.
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(26) TransformationProof (SOUND)
16.14/6.82	By narrowing [LPAR04] the rule REACH_IN_GGG(X, Z, Edges) -> U2_GGG(X, Z, Edges, member1_in_ag(Edges)) at position [3] we obtained the following new rules [LPAR04]:
16.14/6.82	
16.14/6.82	   (REACH_IN_GGG(y0, y1, .(x0, x1)) -> U2_GGG(y0, y1, .(x0, x1), member1_out_ag(x0, .(x0, x1))),REACH_IN_GGG(y0, y1, .(x0, x1)) -> U2_GGG(y0, y1, .(x0, x1), member1_out_ag(x0, .(x0, x1))))
16.14/6.82	   (REACH_IN_GGG(y0, y1, .(x0, x1)) -> U2_GGG(y0, y1, .(x0, x1), U5_ag(x0, x1, member1_in_ag(x1))),REACH_IN_GGG(y0, y1, .(x0, x1)) -> U2_GGG(y0, y1, .(x0, x1), U5_ag(x0, x1, member1_in_ag(x1))))
16.14/6.82	
16.14/6.82	
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(27)
16.14/6.82	Obligation:
16.14/6.82	Q DP problem:
16.14/6.82	The TRS P consists of the following rules:
16.14/6.82	
16.14/6.82	   U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> REACH_IN_GGG(Y, Z, Edges)
16.14/6.82	   REACH_IN_GGG(y0, y1, .(x0, x1)) -> U2_GGG(y0, y1, .(x0, x1), member1_out_ag(x0, .(x0, x1)))
16.14/6.82	   REACH_IN_GGG(y0, y1, .(x0, x1)) -> U2_GGG(y0, y1, .(x0, x1), U5_ag(x0, x1, member1_in_ag(x1)))
16.14/6.82	
16.14/6.82	The TRS R consists of the following rules:
16.14/6.82	
16.14/6.82	   member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L))
16.14/6.82	   member1_in_ag(.(H, L)) -> U5_ag(H, L, member1_in_ag(L))
16.14/6.82	   U5_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L))
16.14/6.82	
16.14/6.82	The set Q consists of the following terms:
16.14/6.82	
16.14/6.82	   member1_in_ag(x0)
16.14/6.82	   U5_ag(x0, x1, x2)
16.14/6.82	
16.14/6.82	We have to consider all (P,Q,R)-chains.
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(28) TransformationProof (EQUIVALENT)
16.14/6.82	By instantiating [LPAR04] the rule U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> REACH_IN_GGG(Y, Z, Edges) we obtained the following new rules [LPAR04]:
16.14/6.82	
16.14/6.82	   (U2_GGG(z0, z1, .(.(z0, .(x3, [])), z3), member1_out_ag(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) -> REACH_IN_GGG(x3, z1, .(.(z0, .(x3, [])), z3)),U2_GGG(z0, z1, .(.(z0, .(x3, [])), z3), member1_out_ag(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) -> REACH_IN_GGG(x3, z1, .(.(z0, .(x3, [])), z3)))
16.14/6.82	   (U2_GGG(z0, z1, .(z2, z3), member1_out_ag(.(z0, .(x3, [])), .(z2, z3))) -> REACH_IN_GGG(x3, z1, .(z2, z3)),U2_GGG(z0, z1, .(z2, z3), member1_out_ag(.(z0, .(x3, [])), .(z2, z3))) -> REACH_IN_GGG(x3, z1, .(z2, z3)))
16.14/6.82	
16.14/6.82	
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(29)
16.14/6.82	Obligation:
16.14/6.82	Q DP problem:
16.14/6.82	The TRS P consists of the following rules:
16.14/6.82	
16.14/6.82	   REACH_IN_GGG(y0, y1, .(x0, x1)) -> U2_GGG(y0, y1, .(x0, x1), member1_out_ag(x0, .(x0, x1)))
16.14/6.82	   REACH_IN_GGG(y0, y1, .(x0, x1)) -> U2_GGG(y0, y1, .(x0, x1), U5_ag(x0, x1, member1_in_ag(x1)))
16.14/6.82	   U2_GGG(z0, z1, .(.(z0, .(x3, [])), z3), member1_out_ag(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) -> REACH_IN_GGG(x3, z1, .(.(z0, .(x3, [])), z3))
16.14/6.82	   U2_GGG(z0, z1, .(z2, z3), member1_out_ag(.(z0, .(x3, [])), .(z2, z3))) -> REACH_IN_GGG(x3, z1, .(z2, z3))
16.14/6.82	
16.14/6.82	The TRS R consists of the following rules:
16.14/6.82	
16.14/6.82	   member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L))
16.14/6.82	   member1_in_ag(.(H, L)) -> U5_ag(H, L, member1_in_ag(L))
16.14/6.82	   U5_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L))
16.14/6.82	
16.14/6.82	The set Q consists of the following terms:
16.14/6.82	
16.14/6.82	   member1_in_ag(x0)
16.14/6.82	   U5_ag(x0, x1, x2)
16.14/6.82	
16.14/6.82	We have to consider all (P,Q,R)-chains.
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(30) TransformationProof (EQUIVALENT)
16.14/6.82	By forward instantiating [JAR06] the rule REACH_IN_GGG(y0, y1, .(x0, x1)) -> U2_GGG(y0, y1, .(x0, x1), member1_out_ag(x0, .(x0, x1))) we obtained the following new rules [LPAR04]:
16.14/6.82	
16.14/6.82	   (REACH_IN_GGG(x0, x1, .(.(y_2, .(y_3, [])), x3)) -> U2_GGG(x0, x1, .(.(y_2, .(y_3, [])), x3), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3))),REACH_IN_GGG(x0, x1, .(.(y_2, .(y_3, [])), x3)) -> U2_GGG(x0, x1, .(.(y_2, .(y_3, [])), x3), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3))))
16.14/6.82	
16.14/6.82	
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(31)
16.14/6.82	Obligation:
16.14/6.82	Q DP problem:
16.14/6.82	The TRS P consists of the following rules:
16.14/6.82	
16.14/6.82	   REACH_IN_GGG(y0, y1, .(x0, x1)) -> U2_GGG(y0, y1, .(x0, x1), U5_ag(x0, x1, member1_in_ag(x1)))
16.14/6.82	   U2_GGG(z0, z1, .(.(z0, .(x3, [])), z3), member1_out_ag(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) -> REACH_IN_GGG(x3, z1, .(.(z0, .(x3, [])), z3))
16.14/6.82	   U2_GGG(z0, z1, .(z2, z3), member1_out_ag(.(z0, .(x3, [])), .(z2, z3))) -> REACH_IN_GGG(x3, z1, .(z2, z3))
16.14/6.82	   REACH_IN_GGG(x0, x1, .(.(y_2, .(y_3, [])), x3)) -> U2_GGG(x0, x1, .(.(y_2, .(y_3, [])), x3), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3)))
16.14/6.82	
16.14/6.82	The TRS R consists of the following rules:
16.14/6.82	
16.14/6.82	   member1_in_ag(.(H, L)) -> member1_out_ag(H, .(H, L))
16.14/6.82	   member1_in_ag(.(H, L)) -> U5_ag(H, L, member1_in_ag(L))
16.14/6.82	   U5_ag(H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L))
16.14/6.82	
16.14/6.82	The set Q consists of the following terms:
16.14/6.82	
16.14/6.82	   member1_in_ag(x0)
16.14/6.82	   U5_ag(x0, x1, x2)
16.14/6.82	
16.14/6.82	We have to consider all (P,Q,R)-chains.
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(32) PrologToPiTRSProof (SOUND)
16.14/6.82	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
16.14/6.82	
16.14/6.82	reach_in_3: (b,b,b)
16.14/6.82	
16.14/6.82	member_in_2: (b,b)
16.14/6.82	
16.14/6.82	member1_in_2: (f,b)
16.14/6.82	
16.14/6.82	Transforming Prolog into the following Term Rewriting System:
16.14/6.82	
16.14/6.82	Pi-finite rewrite system:
16.14/6.82	The TRS R consists of the following rules:
16.14/6.82	
16.14/6.82	   reach_in_ggg(X, Y, Edges) -> U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
16.14/6.82	   member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L))
16.14/6.82	   member_in_gg(X, .(H, L)) -> U4_gg(X, H, L, member_in_gg(X, L))
16.14/6.82	   U4_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L))
16.14/6.82	   U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_ggg(X, Y, Edges)
16.14/6.82	   reach_in_ggg(X, Z, Edges) -> U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.82	   member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L))
16.14/6.82	   member1_in_ag(X, .(H, L)) -> U5_ag(X, H, L, member1_in_ag(X, L))
16.14/6.82	   U5_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L))
16.14/6.82	   U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
16.14/6.82	   U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) -> reach_out_ggg(X, Z, Edges)
16.14/6.82	
16.14/6.82	The argument filtering Pi contains the following mapping:
16.14/6.82	reach_in_ggg(x1, x2, x3)  =  reach_in_ggg(x1, x2, x3)
16.14/6.82	
16.14/6.82	U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x4)
16.14/6.82	
16.14/6.82	member_in_gg(x1, x2)  =  member_in_gg(x1, x2)
16.14/6.82	
16.14/6.82	.(x1, x2)  =  .(x1, x2)
16.14/6.82	
16.14/6.82	member_out_gg(x1, x2)  =  member_out_gg
16.14/6.82	
16.14/6.82	U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
16.14/6.82	
16.14/6.82	[]  =  []
16.14/6.82	
16.14/6.82	reach_out_ggg(x1, x2, x3)  =  reach_out_ggg
16.14/6.82	
16.14/6.82	U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
16.14/6.82	
16.14/6.82	member1_in_ag(x1, x2)  =  member1_in_ag(x2)
16.14/6.82	
16.14/6.82	member1_out_ag(x1, x2)  =  member1_out_ag(x1)
16.14/6.82	
16.14/6.82	U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
16.14/6.82	
16.14/6.82	U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
16.14/6.82	
16.14/6.82	
16.14/6.82	
16.14/6.82	
16.14/6.82	
16.14/6.82	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
16.14/6.82	
16.14/6.82	
16.14/6.82	
16.14/6.82	----------------------------------------
16.14/6.82	
16.14/6.82	(33)
16.14/6.82	Obligation:
16.14/6.82	Pi-finite rewrite system:
16.14/6.82	The TRS R consists of the following rules:
16.14/6.82	
16.14/6.82	   reach_in_ggg(X, Y, Edges) -> U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
16.14/6.82	   member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L))
16.14/6.82	   member_in_gg(X, .(H, L)) -> U4_gg(X, H, L, member_in_gg(X, L))
16.14/6.82	   U4_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L))
16.14/6.82	   U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_ggg(X, Y, Edges)
16.14/6.82	   reach_in_ggg(X, Z, Edges) -> U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.82	   member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L))
16.14/6.83	   member1_in_ag(X, .(H, L)) -> U5_ag(X, H, L, member1_in_ag(X, L))
16.14/6.83	   U5_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L))
16.14/6.83	   U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
16.14/6.83	   U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) -> reach_out_ggg(X, Z, Edges)
16.14/6.83	
16.14/6.83	The argument filtering Pi contains the following mapping:
16.14/6.83	reach_in_ggg(x1, x2, x3)  =  reach_in_ggg(x1, x2, x3)
16.14/6.83	
16.14/6.83	U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x4)
16.14/6.83	
16.14/6.83	member_in_gg(x1, x2)  =  member_in_gg(x1, x2)
16.14/6.83	
16.14/6.83	.(x1, x2)  =  .(x1, x2)
16.14/6.83	
16.14/6.83	member_out_gg(x1, x2)  =  member_out_gg
16.14/6.83	
16.14/6.83	U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
16.14/6.83	
16.14/6.83	[]  =  []
16.14/6.83	
16.14/6.83	reach_out_ggg(x1, x2, x3)  =  reach_out_ggg
16.14/6.83	
16.14/6.83	U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
16.14/6.83	
16.14/6.83	member1_in_ag(x1, x2)  =  member1_in_ag(x2)
16.14/6.83	
16.14/6.83	member1_out_ag(x1, x2)  =  member1_out_ag(x1)
16.14/6.83	
16.14/6.83	U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
16.14/6.83	
16.14/6.83	U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
16.14/6.83	
16.14/6.83	
16.14/6.83	
16.14/6.83	----------------------------------------
16.14/6.83	
16.14/6.83	(34) DependencyPairsProof (EQUIVALENT)
16.14/6.83	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
16.14/6.83	Pi DP problem:
16.14/6.83	The TRS P consists of the following rules:
16.14/6.83	
16.14/6.83	   REACH_IN_GGG(X, Y, Edges) -> U1_GGG(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
16.14/6.83	   REACH_IN_GGG(X, Y, Edges) -> MEMBER_IN_GG(.(X, .(Y, [])), Edges)
16.14/6.83	   MEMBER_IN_GG(X, .(H, L)) -> U4_GG(X, H, L, member_in_gg(X, L))
16.14/6.83	   MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L)
16.14/6.83	   REACH_IN_GGG(X, Z, Edges) -> U2_GGG(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.83	   REACH_IN_GGG(X, Z, Edges) -> MEMBER1_IN_AG(.(X, .(Y, [])), Edges)
16.14/6.83	   MEMBER1_IN_AG(X, .(H, L)) -> U5_AG(X, H, L, member1_in_ag(X, L))
16.14/6.83	   MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L)
16.14/6.83	   U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGG(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
16.14/6.83	   U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> REACH_IN_GGG(Y, Z, Edges)
16.14/6.83	
16.14/6.83	The TRS R consists of the following rules:
16.14/6.83	
16.14/6.83	   reach_in_ggg(X, Y, Edges) -> U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
16.14/6.83	   member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L))
16.14/6.83	   member_in_gg(X, .(H, L)) -> U4_gg(X, H, L, member_in_gg(X, L))
16.14/6.83	   U4_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L))
16.14/6.83	   U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_ggg(X, Y, Edges)
16.14/6.83	   reach_in_ggg(X, Z, Edges) -> U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.83	   member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L))
16.14/6.83	   member1_in_ag(X, .(H, L)) -> U5_ag(X, H, L, member1_in_ag(X, L))
16.14/6.83	   U5_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L))
16.14/6.83	   U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
16.14/6.83	   U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) -> reach_out_ggg(X, Z, Edges)
16.14/6.83	
16.14/6.83	The argument filtering Pi contains the following mapping:
16.14/6.83	reach_in_ggg(x1, x2, x3)  =  reach_in_ggg(x1, x2, x3)
16.14/6.83	
16.14/6.83	U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x4)
16.14/6.83	
16.14/6.83	member_in_gg(x1, x2)  =  member_in_gg(x1, x2)
16.14/6.83	
16.14/6.83	.(x1, x2)  =  .(x1, x2)
16.14/6.83	
16.14/6.83	member_out_gg(x1, x2)  =  member_out_gg
16.14/6.83	
16.14/6.83	U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
16.14/6.83	
16.14/6.83	[]  =  []
16.14/6.83	
16.14/6.83	reach_out_ggg(x1, x2, x3)  =  reach_out_ggg
16.14/6.83	
16.14/6.83	U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
16.14/6.83	
16.14/6.83	member1_in_ag(x1, x2)  =  member1_in_ag(x2)
16.14/6.83	
16.14/6.83	member1_out_ag(x1, x2)  =  member1_out_ag(x1)
16.14/6.83	
16.14/6.83	U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
16.14/6.83	
16.14/6.83	U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
16.14/6.83	
16.14/6.83	REACH_IN_GGG(x1, x2, x3)  =  REACH_IN_GGG(x1, x2, x3)
16.14/6.83	
16.14/6.83	U1_GGG(x1, x2, x3, x4)  =  U1_GGG(x4)
16.14/6.83	
16.14/6.83	MEMBER_IN_GG(x1, x2)  =  MEMBER_IN_GG(x1, x2)
16.14/6.83	
16.14/6.83	U4_GG(x1, x2, x3, x4)  =  U4_GG(x4)
16.14/6.83	
16.14/6.83	U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x2, x3, x4)
16.14/6.83	
16.14/6.83	MEMBER1_IN_AG(x1, x2)  =  MEMBER1_IN_AG(x2)
16.14/6.83	
16.14/6.83	U5_AG(x1, x2, x3, x4)  =  U5_AG(x4)
16.14/6.83	
16.14/6.83	U3_GGG(x1, x2, x3, x4)  =  U3_GGG(x4)
16.14/6.83	
16.14/6.83	
16.14/6.83	We have to consider all (P,R,Pi)-chains
16.14/6.83	----------------------------------------
16.14/6.83	
16.14/6.83	(35)
16.14/6.83	Obligation:
16.14/6.83	Pi DP problem:
16.14/6.83	The TRS P consists of the following rules:
16.14/6.83	
16.14/6.83	   REACH_IN_GGG(X, Y, Edges) -> U1_GGG(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
16.14/6.83	   REACH_IN_GGG(X, Y, Edges) -> MEMBER_IN_GG(.(X, .(Y, [])), Edges)
16.14/6.83	   MEMBER_IN_GG(X, .(H, L)) -> U4_GG(X, H, L, member_in_gg(X, L))
16.14/6.83	   MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L)
16.14/6.83	   REACH_IN_GGG(X, Z, Edges) -> U2_GGG(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.83	   REACH_IN_GGG(X, Z, Edges) -> MEMBER1_IN_AG(.(X, .(Y, [])), Edges)
16.14/6.83	   MEMBER1_IN_AG(X, .(H, L)) -> U5_AG(X, H, L, member1_in_ag(X, L))
16.14/6.83	   MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L)
16.14/6.83	   U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_GGG(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
16.14/6.83	   U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> REACH_IN_GGG(Y, Z, Edges)
16.14/6.83	
16.14/6.83	The TRS R consists of the following rules:
16.14/6.83	
16.14/6.83	   reach_in_ggg(X, Y, Edges) -> U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
16.14/6.83	   member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L))
16.14/6.83	   member_in_gg(X, .(H, L)) -> U4_gg(X, H, L, member_in_gg(X, L))
16.14/6.83	   U4_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L))
16.14/6.83	   U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_ggg(X, Y, Edges)
16.14/6.83	   reach_in_ggg(X, Z, Edges) -> U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.83	   member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L))
16.14/6.83	   member1_in_ag(X, .(H, L)) -> U5_ag(X, H, L, member1_in_ag(X, L))
16.14/6.83	   U5_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L))
16.14/6.83	   U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
16.14/6.83	   U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) -> reach_out_ggg(X, Z, Edges)
16.14/6.83	
16.14/6.83	The argument filtering Pi contains the following mapping:
16.14/6.83	reach_in_ggg(x1, x2, x3)  =  reach_in_ggg(x1, x2, x3)
16.14/6.83	
16.14/6.83	U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x4)
16.14/6.83	
16.14/6.83	member_in_gg(x1, x2)  =  member_in_gg(x1, x2)
16.14/6.83	
16.14/6.83	.(x1, x2)  =  .(x1, x2)
16.14/6.83	
16.14/6.83	member_out_gg(x1, x2)  =  member_out_gg
16.14/6.83	
16.14/6.83	U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
16.14/6.83	
16.14/6.83	[]  =  []
16.14/6.83	
16.14/6.83	reach_out_ggg(x1, x2, x3)  =  reach_out_ggg
16.14/6.83	
16.14/6.83	U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
16.14/6.83	
16.14/6.83	member1_in_ag(x1, x2)  =  member1_in_ag(x2)
16.14/6.83	
16.14/6.83	member1_out_ag(x1, x2)  =  member1_out_ag(x1)
16.14/6.83	
16.14/6.83	U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
16.14/6.83	
16.14/6.83	U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
16.14/6.83	
16.14/6.83	REACH_IN_GGG(x1, x2, x3)  =  REACH_IN_GGG(x1, x2, x3)
16.14/6.83	
16.14/6.83	U1_GGG(x1, x2, x3, x4)  =  U1_GGG(x4)
16.14/6.83	
16.14/6.83	MEMBER_IN_GG(x1, x2)  =  MEMBER_IN_GG(x1, x2)
16.14/6.83	
16.14/6.83	U4_GG(x1, x2, x3, x4)  =  U4_GG(x4)
16.14/6.83	
16.14/6.83	U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x2, x3, x4)
16.14/6.83	
16.14/6.83	MEMBER1_IN_AG(x1, x2)  =  MEMBER1_IN_AG(x2)
16.14/6.83	
16.14/6.83	U5_AG(x1, x2, x3, x4)  =  U5_AG(x4)
16.14/6.83	
16.14/6.83	U3_GGG(x1, x2, x3, x4)  =  U3_GGG(x4)
16.14/6.83	
16.14/6.83	
16.14/6.83	We have to consider all (P,R,Pi)-chains
16.14/6.83	----------------------------------------
16.14/6.83	
16.14/6.83	(36) DependencyGraphProof (EQUIVALENT)
16.14/6.83	The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 6 less nodes.
16.14/6.83	----------------------------------------
16.14/6.83	
16.14/6.83	(37)
16.14/6.83	Complex Obligation (AND)
16.14/6.83	
16.14/6.83	----------------------------------------
16.14/6.83	
16.14/6.83	(38)
16.14/6.83	Obligation:
16.14/6.83	Pi DP problem:
16.14/6.83	The TRS P consists of the following rules:
16.14/6.83	
16.14/6.83	   MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L)
16.14/6.83	
16.14/6.83	The TRS R consists of the following rules:
16.14/6.83	
16.14/6.83	   reach_in_ggg(X, Y, Edges) -> U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
16.14/6.83	   member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L))
16.14/6.83	   member_in_gg(X, .(H, L)) -> U4_gg(X, H, L, member_in_gg(X, L))
16.14/6.83	   U4_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L))
16.14/6.83	   U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_ggg(X, Y, Edges)
16.14/6.83	   reach_in_ggg(X, Z, Edges) -> U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.83	   member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L))
16.14/6.83	   member1_in_ag(X, .(H, L)) -> U5_ag(X, H, L, member1_in_ag(X, L))
16.14/6.83	   U5_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L))
16.14/6.83	   U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
16.14/6.83	   U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) -> reach_out_ggg(X, Z, Edges)
16.14/6.83	
16.14/6.83	The argument filtering Pi contains the following mapping:
16.14/6.83	reach_in_ggg(x1, x2, x3)  =  reach_in_ggg(x1, x2, x3)
16.14/6.83	
16.14/6.83	U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x4)
16.14/6.83	
16.14/6.83	member_in_gg(x1, x2)  =  member_in_gg(x1, x2)
16.14/6.83	
16.14/6.83	.(x1, x2)  =  .(x1, x2)
16.14/6.83	
16.14/6.83	member_out_gg(x1, x2)  =  member_out_gg
16.14/6.83	
16.14/6.83	U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
16.14/6.83	
16.14/6.83	[]  =  []
16.14/6.83	
16.14/6.83	reach_out_ggg(x1, x2, x3)  =  reach_out_ggg
16.14/6.83	
16.14/6.83	U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
16.14/6.83	
16.14/6.83	member1_in_ag(x1, x2)  =  member1_in_ag(x2)
16.14/6.84	
16.14/6.84	member1_out_ag(x1, x2)  =  member1_out_ag(x1)
16.14/6.84	
16.14/6.84	U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
16.14/6.84	
16.14/6.84	U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
16.14/6.84	
16.14/6.84	MEMBER1_IN_AG(x1, x2)  =  MEMBER1_IN_AG(x2)
16.14/6.84	
16.14/6.84	
16.14/6.84	We have to consider all (P,R,Pi)-chains
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(39) UsableRulesProof (EQUIVALENT)
16.14/6.84	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(40)
16.14/6.84	Obligation:
16.14/6.84	Pi DP problem:
16.14/6.84	The TRS P consists of the following rules:
16.14/6.84	
16.14/6.84	   MEMBER1_IN_AG(X, .(H, L)) -> MEMBER1_IN_AG(X, L)
16.14/6.84	
16.14/6.84	R is empty.
16.14/6.84	The argument filtering Pi contains the following mapping:
16.14/6.84	.(x1, x2)  =  .(x1, x2)
16.14/6.84	
16.14/6.84	MEMBER1_IN_AG(x1, x2)  =  MEMBER1_IN_AG(x2)
16.14/6.84	
16.14/6.84	
16.14/6.84	We have to consider all (P,R,Pi)-chains
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(41) PiDPToQDPProof (SOUND)
16.14/6.84	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(42)
16.14/6.84	Obligation:
16.14/6.84	Q DP problem:
16.14/6.84	The TRS P consists of the following rules:
16.14/6.84	
16.14/6.84	   MEMBER1_IN_AG(.(H, L)) -> MEMBER1_IN_AG(L)
16.14/6.84	
16.14/6.84	R is empty.
16.14/6.84	Q is empty.
16.14/6.84	We have to consider all (P,Q,R)-chains.
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(43) QDPSizeChangeProof (EQUIVALENT)
16.14/6.84	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. 
16.14/6.84	
16.14/6.84	From the DPs we obtained the following set of size-change graphs:
16.14/6.84	*MEMBER1_IN_AG(.(H, L)) -> MEMBER1_IN_AG(L)
16.14/6.84	The graph contains the following edges 1 > 1
16.14/6.84	
16.14/6.84	
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(44)
16.14/6.84	YES
16.14/6.84	
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(45)
16.14/6.84	Obligation:
16.14/6.84	Pi DP problem:
16.14/6.84	The TRS P consists of the following rules:
16.14/6.84	
16.14/6.84	   MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L)
16.14/6.84	
16.14/6.84	The TRS R consists of the following rules:
16.14/6.84	
16.14/6.84	   reach_in_ggg(X, Y, Edges) -> U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
16.14/6.84	   member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L))
16.14/6.84	   member_in_gg(X, .(H, L)) -> U4_gg(X, H, L, member_in_gg(X, L))
16.14/6.84	   U4_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L))
16.14/6.84	   U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_ggg(X, Y, Edges)
16.14/6.84	   reach_in_ggg(X, Z, Edges) -> U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.84	   member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L))
16.14/6.84	   member1_in_ag(X, .(H, L)) -> U5_ag(X, H, L, member1_in_ag(X, L))
16.14/6.84	   U5_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L))
16.14/6.84	   U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
16.14/6.84	   U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) -> reach_out_ggg(X, Z, Edges)
16.14/6.84	
16.14/6.84	The argument filtering Pi contains the following mapping:
16.14/6.84	reach_in_ggg(x1, x2, x3)  =  reach_in_ggg(x1, x2, x3)
16.14/6.84	
16.14/6.84	U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x4)
16.14/6.84	
16.14/6.84	member_in_gg(x1, x2)  =  member_in_gg(x1, x2)
16.14/6.84	
16.14/6.84	.(x1, x2)  =  .(x1, x2)
16.14/6.84	
16.14/6.84	member_out_gg(x1, x2)  =  member_out_gg
16.14/6.84	
16.14/6.84	U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
16.14/6.84	
16.14/6.84	[]  =  []
16.14/6.84	
16.14/6.84	reach_out_ggg(x1, x2, x3)  =  reach_out_ggg
16.14/6.84	
16.14/6.84	U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
16.14/6.84	
16.14/6.84	member1_in_ag(x1, x2)  =  member1_in_ag(x2)
16.14/6.84	
16.14/6.84	member1_out_ag(x1, x2)  =  member1_out_ag(x1)
16.14/6.84	
16.14/6.84	U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
16.14/6.84	
16.14/6.84	U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
16.14/6.84	
16.14/6.84	MEMBER_IN_GG(x1, x2)  =  MEMBER_IN_GG(x1, x2)
16.14/6.84	
16.14/6.84	
16.14/6.84	We have to consider all (P,R,Pi)-chains
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(46) UsableRulesProof (EQUIVALENT)
16.14/6.84	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(47)
16.14/6.84	Obligation:
16.14/6.84	Pi DP problem:
16.14/6.84	The TRS P consists of the following rules:
16.14/6.84	
16.14/6.84	   MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L)
16.14/6.84	
16.14/6.84	R is empty.
16.14/6.84	Pi is empty.
16.14/6.84	We have to consider all (P,R,Pi)-chains
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(48) PiDPToQDPProof (EQUIVALENT)
16.14/6.84	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(49)
16.14/6.84	Obligation:
16.14/6.84	Q DP problem:
16.14/6.84	The TRS P consists of the following rules:
16.14/6.84	
16.14/6.84	   MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L)
16.14/6.84	
16.14/6.84	R is empty.
16.14/6.84	Q is empty.
16.14/6.84	We have to consider all (P,Q,R)-chains.
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(50) QDPSizeChangeProof (EQUIVALENT)
16.14/6.84	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. 
16.14/6.84	
16.14/6.84	From the DPs we obtained the following set of size-change graphs:
16.14/6.84	*MEMBER_IN_GG(X, .(H, L)) -> MEMBER_IN_GG(X, L)
16.14/6.84	The graph contains the following edges 1 >= 1, 2 > 2
16.14/6.84	
16.14/6.84	
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(51)
16.14/6.84	YES
16.14/6.84	
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(52)
16.14/6.84	Obligation:
16.14/6.84	Pi DP problem:
16.14/6.84	The TRS P consists of the following rules:
16.14/6.84	
16.14/6.84	   REACH_IN_GGG(X, Z, Edges) -> U2_GGG(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.84	   U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> REACH_IN_GGG(Y, Z, Edges)
16.14/6.84	
16.14/6.84	The TRS R consists of the following rules:
16.14/6.84	
16.14/6.84	   reach_in_ggg(X, Y, Edges) -> U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
16.14/6.84	   member_in_gg(H, .(H, L)) -> member_out_gg(H, .(H, L))
16.14/6.84	   member_in_gg(X, .(H, L)) -> U4_gg(X, H, L, member_in_gg(X, L))
16.14/6.84	   U4_gg(X, H, L, member_out_gg(X, L)) -> member_out_gg(X, .(H, L))
16.14/6.84	   U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) -> reach_out_ggg(X, Y, Edges)
16.14/6.84	   reach_in_ggg(X, Z, Edges) -> U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.84	   member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L))
16.14/6.84	   member1_in_ag(X, .(H, L)) -> U5_ag(X, H, L, member1_in_ag(X, L))
16.14/6.84	   U5_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L))
16.14/6.84	   U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
16.14/6.84	   U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) -> reach_out_ggg(X, Z, Edges)
16.14/6.84	
16.14/6.84	The argument filtering Pi contains the following mapping:
16.14/6.84	reach_in_ggg(x1, x2, x3)  =  reach_in_ggg(x1, x2, x3)
16.14/6.84	
16.14/6.84	U1_ggg(x1, x2, x3, x4)  =  U1_ggg(x4)
16.14/6.84	
16.14/6.84	member_in_gg(x1, x2)  =  member_in_gg(x1, x2)
16.14/6.84	
16.14/6.84	.(x1, x2)  =  .(x1, x2)
16.14/6.84	
16.14/6.84	member_out_gg(x1, x2)  =  member_out_gg
16.14/6.84	
16.14/6.84	U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
16.14/6.84	
16.14/6.84	[]  =  []
16.14/6.84	
16.14/6.84	reach_out_ggg(x1, x2, x3)  =  reach_out_ggg
16.14/6.84	
16.14/6.84	U2_ggg(x1, x2, x3, x4)  =  U2_ggg(x2, x3, x4)
16.14/6.84	
16.14/6.84	member1_in_ag(x1, x2)  =  member1_in_ag(x2)
16.14/6.84	
16.14/6.84	member1_out_ag(x1, x2)  =  member1_out_ag(x1)
16.14/6.84	
16.14/6.84	U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
16.14/6.84	
16.14/6.84	U3_ggg(x1, x2, x3, x4)  =  U3_ggg(x4)
16.14/6.84	
16.14/6.84	REACH_IN_GGG(x1, x2, x3)  =  REACH_IN_GGG(x1, x2, x3)
16.14/6.84	
16.14/6.84	U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x2, x3, x4)
16.14/6.84	
16.14/6.84	
16.14/6.84	We have to consider all (P,R,Pi)-chains
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(53) UsableRulesProof (EQUIVALENT)
16.14/6.84	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(54)
16.14/6.84	Obligation:
16.14/6.84	Pi DP problem:
16.14/6.84	The TRS P consists of the following rules:
16.14/6.84	
16.14/6.84	   REACH_IN_GGG(X, Z, Edges) -> U2_GGG(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
16.14/6.84	   U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) -> REACH_IN_GGG(Y, Z, Edges)
16.14/6.84	
16.14/6.84	The TRS R consists of the following rules:
16.14/6.84	
16.14/6.84	   member1_in_ag(H, .(H, L)) -> member1_out_ag(H, .(H, L))
16.14/6.84	   member1_in_ag(X, .(H, L)) -> U5_ag(X, H, L, member1_in_ag(X, L))
16.14/6.84	   U5_ag(X, H, L, member1_out_ag(X, L)) -> member1_out_ag(X, .(H, L))
16.14/6.84	
16.14/6.84	The argument filtering Pi contains the following mapping:
16.14/6.84	.(x1, x2)  =  .(x1, x2)
16.14/6.84	
16.14/6.84	[]  =  []
16.14/6.84	
16.14/6.84	member1_in_ag(x1, x2)  =  member1_in_ag(x2)
16.14/6.84	
16.14/6.84	member1_out_ag(x1, x2)  =  member1_out_ag(x1)
16.14/6.84	
16.14/6.84	U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
16.14/6.84	
16.14/6.84	REACH_IN_GGG(x1, x2, x3)  =  REACH_IN_GGG(x1, x2, x3)
16.14/6.84	
16.14/6.84	U2_GGG(x1, x2, x3, x4)  =  U2_GGG(x2, x3, x4)
16.14/6.84	
16.14/6.84	
16.14/6.84	We have to consider all (P,R,Pi)-chains
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(55) PiDPToQDPProof (SOUND)
16.14/6.84	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(56)
16.14/6.84	Obligation:
16.14/6.84	Q DP problem:
16.14/6.84	The TRS P consists of the following rules:
16.14/6.84	
16.14/6.84	   REACH_IN_GGG(X, Z, Edges) -> U2_GGG(Z, Edges, member1_in_ag(Edges))
16.14/6.84	   U2_GGG(Z, Edges, member1_out_ag(.(X, .(Y, [])))) -> REACH_IN_GGG(Y, Z, Edges)
16.14/6.84	
16.14/6.84	The TRS R consists of the following rules:
16.14/6.84	
16.14/6.84	   member1_in_ag(.(H, L)) -> member1_out_ag(H)
16.14/6.84	   member1_in_ag(.(H, L)) -> U5_ag(member1_in_ag(L))
16.14/6.84	   U5_ag(member1_out_ag(X)) -> member1_out_ag(X)
16.14/6.84	
16.14/6.84	The set Q consists of the following terms:
16.14/6.84	
16.14/6.84	   member1_in_ag(x0)
16.14/6.84	   U5_ag(x0)
16.14/6.84	
16.14/6.84	We have to consider all (P,Q,R)-chains.
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(57) TransformationProof (SOUND)
16.14/6.84	By narrowing [LPAR04] the rule REACH_IN_GGG(X, Z, Edges) -> U2_GGG(Z, Edges, member1_in_ag(Edges)) at position [2] we obtained the following new rules [LPAR04]:
16.14/6.84	
16.14/6.84	   (REACH_IN_GGG(y0, y1, .(x0, x1)) -> U2_GGG(y1, .(x0, x1), member1_out_ag(x0)),REACH_IN_GGG(y0, y1, .(x0, x1)) -> U2_GGG(y1, .(x0, x1), member1_out_ag(x0)))
16.14/6.84	   (REACH_IN_GGG(y0, y1, .(x0, x1)) -> U2_GGG(y1, .(x0, x1), U5_ag(member1_in_ag(x1))),REACH_IN_GGG(y0, y1, .(x0, x1)) -> U2_GGG(y1, .(x0, x1), U5_ag(member1_in_ag(x1))))
16.14/6.84	
16.14/6.84	
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(58)
16.14/6.84	Obligation:
16.14/6.84	Q DP problem:
16.14/6.84	The TRS P consists of the following rules:
16.14/6.84	
16.14/6.84	   U2_GGG(Z, Edges, member1_out_ag(.(X, .(Y, [])))) -> REACH_IN_GGG(Y, Z, Edges)
16.14/6.84	   REACH_IN_GGG(y0, y1, .(x0, x1)) -> U2_GGG(y1, .(x0, x1), member1_out_ag(x0))
16.14/6.84	   REACH_IN_GGG(y0, y1, .(x0, x1)) -> U2_GGG(y1, .(x0, x1), U5_ag(member1_in_ag(x1)))
16.14/6.84	
16.14/6.84	The TRS R consists of the following rules:
16.14/6.84	
16.14/6.84	   member1_in_ag(.(H, L)) -> member1_out_ag(H)
16.14/6.84	   member1_in_ag(.(H, L)) -> U5_ag(member1_in_ag(L))
16.14/6.84	   U5_ag(member1_out_ag(X)) -> member1_out_ag(X)
16.14/6.84	
16.14/6.84	The set Q consists of the following terms:
16.14/6.84	
16.14/6.84	   member1_in_ag(x0)
16.14/6.84	   U5_ag(x0)
16.14/6.84	
16.14/6.84	We have to consider all (P,Q,R)-chains.
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(59) TransformationProof (EQUIVALENT)
16.14/6.84	By instantiating [LPAR04] the rule U2_GGG(Z, Edges, member1_out_ag(.(X, .(Y, [])))) -> REACH_IN_GGG(Y, Z, Edges) we obtained the following new rules [LPAR04]:
16.14/6.84	
16.14/6.84	   (U2_GGG(z1, .(.(x2, .(x3, [])), z3), member1_out_ag(.(x2, .(x3, [])))) -> REACH_IN_GGG(x3, z1, .(.(x2, .(x3, [])), z3)),U2_GGG(z1, .(.(x2, .(x3, [])), z3), member1_out_ag(.(x2, .(x3, [])))) -> REACH_IN_GGG(x3, z1, .(.(x2, .(x3, [])), z3)))
16.14/6.84	   (U2_GGG(z1, .(z2, z3), member1_out_ag(.(x2, .(x3, [])))) -> REACH_IN_GGG(x3, z1, .(z2, z3)),U2_GGG(z1, .(z2, z3), member1_out_ag(.(x2, .(x3, [])))) -> REACH_IN_GGG(x3, z1, .(z2, z3)))
16.14/6.84	
16.14/6.84	
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(60)
16.14/6.84	Obligation:
16.14/6.84	Q DP problem:
16.14/6.84	The TRS P consists of the following rules:
16.14/6.84	
16.14/6.84	   REACH_IN_GGG(y0, y1, .(x0, x1)) -> U2_GGG(y1, .(x0, x1), member1_out_ag(x0))
16.14/6.84	   REACH_IN_GGG(y0, y1, .(x0, x1)) -> U2_GGG(y1, .(x0, x1), U5_ag(member1_in_ag(x1)))
16.14/6.84	   U2_GGG(z1, .(.(x2, .(x3, [])), z3), member1_out_ag(.(x2, .(x3, [])))) -> REACH_IN_GGG(x3, z1, .(.(x2, .(x3, [])), z3))
16.14/6.84	   U2_GGG(z1, .(z2, z3), member1_out_ag(.(x2, .(x3, [])))) -> REACH_IN_GGG(x3, z1, .(z2, z3))
16.14/6.84	
16.14/6.84	The TRS R consists of the following rules:
16.14/6.84	
16.14/6.84	   member1_in_ag(.(H, L)) -> member1_out_ag(H)
16.14/6.84	   member1_in_ag(.(H, L)) -> U5_ag(member1_in_ag(L))
16.14/6.84	   U5_ag(member1_out_ag(X)) -> member1_out_ag(X)
16.14/6.84	
16.14/6.84	The set Q consists of the following terms:
16.14/6.84	
16.14/6.84	   member1_in_ag(x0)
16.14/6.84	   U5_ag(x0)
16.14/6.84	
16.14/6.84	We have to consider all (P,Q,R)-chains.
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(61) TransformationProof (EQUIVALENT)
16.14/6.84	By forward instantiating [JAR06] the rule REACH_IN_GGG(y0, y1, .(x0, x1)) -> U2_GGG(y1, .(x0, x1), member1_out_ag(x0)) we obtained the following new rules [LPAR04]:
16.14/6.84	
16.14/6.84	   (REACH_IN_GGG(x0, x1, .(.(y_1, .(y_2, [])), x3)) -> U2_GGG(x1, .(.(y_1, .(y_2, [])), x3), member1_out_ag(.(y_1, .(y_2, [])))),REACH_IN_GGG(x0, x1, .(.(y_1, .(y_2, [])), x3)) -> U2_GGG(x1, .(.(y_1, .(y_2, [])), x3), member1_out_ag(.(y_1, .(y_2, [])))))
16.14/6.84	
16.14/6.84	
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(62)
16.14/6.84	Obligation:
16.14/6.84	Q DP problem:
16.14/6.84	The TRS P consists of the following rules:
16.14/6.84	
16.14/6.84	   REACH_IN_GGG(y0, y1, .(x0, x1)) -> U2_GGG(y1, .(x0, x1), U5_ag(member1_in_ag(x1)))
16.14/6.84	   U2_GGG(z1, .(.(x2, .(x3, [])), z3), member1_out_ag(.(x2, .(x3, [])))) -> REACH_IN_GGG(x3, z1, .(.(x2, .(x3, [])), z3))
16.14/6.84	   U2_GGG(z1, .(z2, z3), member1_out_ag(.(x2, .(x3, [])))) -> REACH_IN_GGG(x3, z1, .(z2, z3))
16.14/6.84	   REACH_IN_GGG(x0, x1, .(.(y_1, .(y_2, [])), x3)) -> U2_GGG(x1, .(.(y_1, .(y_2, [])), x3), member1_out_ag(.(y_1, .(y_2, []))))
16.14/6.84	
16.14/6.84	The TRS R consists of the following rules:
16.14/6.84	
16.14/6.84	   member1_in_ag(.(H, L)) -> member1_out_ag(H)
16.14/6.84	   member1_in_ag(.(H, L)) -> U5_ag(member1_in_ag(L))
16.14/6.84	   U5_ag(member1_out_ag(X)) -> member1_out_ag(X)
16.14/6.84	
16.14/6.84	The set Q consists of the following terms:
16.14/6.84	
16.14/6.84	   member1_in_ag(x0)
16.14/6.84	   U5_ag(x0)
16.14/6.84	
16.14/6.84	We have to consider all (P,Q,R)-chains.
16.14/6.84	----------------------------------------
16.14/6.84	
16.14/6.84	(63) PrologToTRSTransformerProof (SOUND)
16.14/6.84	Transformed Prolog program to TRS.
16.14/6.84	
16.14/6.84	{
16.14/6.84	    "root": 1,
16.14/6.84	    "program": {
16.14/6.84	        "directives": [],
16.14/6.84	        "clauses": [
16.14/6.84	            [
16.14/6.84	                "(reach X Y Edges)",
16.14/6.84	                "(member (. X (. Y ([]))) Edges)"
16.14/6.84	            ],
16.14/6.84	            [
16.14/6.84	                "(reach X Z Edges)",
16.14/6.84	                "(',' (member1 (. X (. Y ([]))) Edges) (reach Y Z Edges))"
16.14/6.84	            ],
16.14/6.84	            [
16.14/6.84	                "(member H (. H L))",
16.14/6.84	                null
16.14/6.84	            ],
16.14/6.84	            [
16.14/6.84	                "(member X (. H L))",
16.14/6.84	                "(member X L)"
16.14/6.84	            ],
16.14/6.84	            [
16.14/6.84	                "(member1 H (. H L))",
16.14/6.84	                null
16.14/6.84	            ],
16.14/6.84	            [
16.14/6.84	                "(member1 X (. H L))",
16.14/6.84	                "(member1 X L)"
16.14/6.84	            ]
16.14/6.84	        ]
16.14/6.84	    },
16.14/6.84	    "graph": {
16.14/6.84	        "nodes": {
16.14/6.84	            "270": {
16.14/6.84	                "goal": [],
16.14/6.84	                "kb": {
16.14/6.84	                    "nonunifying": [],
16.14/6.84	                    "intvars": {},
16.14/6.84	                    "arithmetic": {
16.14/6.84	                        "type": "PlainIntegerRelationState",
16.14/6.84	                        "relations": []
16.14/6.84	                    },
16.14/6.84	                    "ground": [],
16.14/6.84	                    "free": [],
16.14/6.84	                    "exprvars": []
16.14/6.84	                }
16.14/6.84	            },
16.14/6.84	            "type": "Nodes",
16.14/6.84	            "298": {
16.14/6.84	                "goal": [{
16.14/6.84	                    "clause": -1,
16.14/6.84	                    "scope": -1,
16.14/6.84	                    "term": "(member1 (. T119 (. X106 ([]))) T121)"
16.14/6.84	                }],
16.14/6.84	                "kb": {
16.14/6.84	                    "nonunifying": [],
16.14/6.84	                    "intvars": {},
16.14/6.84	                    "arithmetic": {
16.14/6.84	                        "type": "PlainIntegerRelationState",
16.14/6.84	                        "relations": []
16.14/6.84	                    },
16.14/6.84	                    "ground": [
16.14/6.84	                        "T119",
16.14/6.84	                        "T121"
16.14/6.84	                    ],
16.14/6.84	                    "free": ["X106"],
16.14/6.84	                    "exprvars": []
16.14/6.84	                }
16.14/6.84	            },
16.14/6.84	            "299": {
16.14/6.84	                "goal": [],
16.14/6.84	                "kb": {
16.14/6.84	                    "nonunifying": [],
16.14/6.84	                    "intvars": {},
16.14/6.84	                    "arithmetic": {
16.14/6.84	                        "type": "PlainIntegerRelationState",
16.14/6.84	                        "relations": []
16.14/6.84	                    },
16.14/6.84	                    "ground": [],
16.14/6.84	                    "free": [],
16.14/6.84	                    "exprvars": []
16.14/6.84	                }
16.14/6.84	            },
16.14/6.84	            "279": {
16.14/6.84	                "goal": [{
16.14/6.84	                    "clause": -1,
16.14/6.84	                    "scope": -1,
16.14/6.84	                    "term": "(',' (member1 (. T82 (. X67 ([]))) T84) (reach X67 T83 T84))"
16.14/6.84	                }],
16.14/6.84	                "kb": {
16.14/6.84	                    "nonunifying": [],
16.14/6.84	                    "intvars": {},
16.14/6.84	                    "arithmetic": {
16.14/6.84	                        "type": "PlainIntegerRelationState",
16.14/6.84	                        "relations": []
16.14/6.84	                    },
16.14/6.84	                    "ground": [
16.14/6.84	                        "T82",
16.14/6.84	                        "T83",
16.14/6.84	                        "T84"
16.14/6.84	                    ],
16.14/6.84	                    "free": ["X67"],
16.14/6.84	                    "exprvars": []
16.14/6.84	                }
16.14/6.84	            },
16.14/6.84	            "259": {
16.14/6.84	                "goal": [{
16.14/6.84	                    "clause": -1,
16.14/6.84	                    "scope": -1,
16.14/6.84	                    "term": "(member (. T28 (. T29 ([]))) T30)"
16.14/6.84	                }],
16.14/6.84	                "kb": {
16.14/6.84	                    "nonunifying": [],
16.14/6.84	                    "intvars": {},
16.14/6.84	                    "arithmetic": {
16.14/6.84	                        "type": "PlainIntegerRelationState",
16.14/6.84	                        "relations": []
16.14/6.84	                    },
16.14/6.84	                    "ground": [
16.14/6.84	                        "T28",
16.14/6.84	                        "T29",
16.14/6.84	                        "T30"
16.14/6.84	                    ],
16.14/6.84	                    "free": [],
16.14/6.84	                    "exprvars": []
16.14/6.84	                }
16.14/6.84	            },
16.14/6.84	            "32": {
16.14/6.84	                "goal": [{
16.14/6.84	                    "clause": 0,
16.14/6.84	                    "scope": 1,
16.14/6.84	                    "term": "(reach T1 T2 T3)"
16.14/6.84	                }],
16.14/6.84	                "kb": {
16.14/6.84	                    "nonunifying": [],
16.14/6.84	                    "intvars": {},
16.14/6.84	                    "arithmetic": {
16.14/6.84	                        "type": "PlainIntegerRelationState",
16.14/6.84	                        "relations": []
16.14/6.84	                    },
16.14/6.84	                    "ground": [
16.14/6.84	                        "T1",
16.14/6.84	                        "T2",
16.14/6.84	                        "T3"
16.14/6.84	                    ],
16.14/6.84	                    "free": [],
16.14/6.84	                    "exprvars": []
16.14/6.84	                }
16.14/6.84	            },
16.14/6.84	            "33": {
16.14/6.84	                "goal": [{
16.14/6.84	                    "clause": 1,
16.14/6.84	                    "scope": 1,
16.14/6.84	                    "term": "(reach T1 T2 T3)"
16.14/6.84	                }],
16.14/6.84	                "kb": {
16.14/6.84	                    "nonunifying": [],
16.14/6.84	                    "intvars": {},
16.14/6.84	                    "arithmetic": {
16.14/6.84	                        "type": "PlainIntegerRelationState",
16.14/6.84	                        "relations": []
16.14/6.84	                    },
16.14/6.84	                    "ground": [
16.14/6.84	                        "T1",
16.14/6.84	                        "T2",
16.14/6.84	                        "T3"
16.14/6.84	                    ],
16.14/6.84	                    "free": [],
16.14/6.84	                    "exprvars": []
16.14/6.84	                }
16.14/6.84	            },
16.14/6.84	            "19": {
16.14/6.84	                "goal": [
16.14/6.84	                    {
16.14/6.84	                        "clause": 0,
16.14/6.84	                        "scope": 1,
16.14/6.84	                        "term": "(reach T1 T2 T3)"
16.14/6.84	                    },
16.14/6.84	                    {
16.14/6.84	                        "clause": 1,
16.14/6.84	                        "scope": 1,
16.14/6.84	                        "term": "(reach T1 T2 T3)"
16.14/6.84	                    }
16.14/6.84	                ],
16.14/6.84	                "kb": {
16.14/6.84	                    "nonunifying": [],
16.14/6.84	                    "intvars": {},
16.14/6.84	                    "arithmetic": {
16.14/6.84	                        "type": "PlainIntegerRelationState",
16.14/6.84	                        "relations": []
16.14/6.84	                    },
16.14/6.84	                    "ground": [
16.14/6.84	                        "T1",
16.14/6.84	                        "T2",
16.14/6.84	                        "T3"
16.14/6.84	                    ],
16.14/6.84	                    "free": [],
16.14/6.84	                    "exprvars": []
16.14/6.84	                }
16.14/6.84	            },
16.14/6.84	            "280": {
16.14/6.84	                "goal": [{
16.14/6.84	                    "clause": -1,
16.14/6.84	                    "scope": -1,
16.14/6.84	                    "term": "(member1 (. T82 (. X67 ([]))) T84)"
16.14/6.84	                }],
16.14/6.84	                "kb": {
16.14/6.84	                    "nonunifying": [],
16.14/6.84	                    "intvars": {},
16.14/6.84	                    "arithmetic": {
16.14/6.84	                        "type": "PlainIntegerRelationState",
16.14/6.84	                        "relations": []
16.14/6.84	                    },
16.14/6.84	                    "ground": [
16.14/6.84	                        "T82",
16.14/6.84	                        "T84"
16.14/6.84	                    ],
16.14/6.84	                    "free": ["X67"],
16.14/6.84	                    "exprvars": []
16.14/6.84	                }
16.14/6.84	            },
16.14/6.84	            "281": {
16.14/6.84	                "goal": [{
16.14/6.84	                    "clause": -1,
16.14/6.84	                    "scope": -1,
16.14/6.84	                    "term": "(reach T91 T83 T84)"
16.14/6.84	                }],
16.14/6.84	                "kb": {
16.14/6.84	                    "nonunifying": [],
16.14/6.84	                    "intvars": {},
16.14/6.84	                    "arithmetic": {
16.14/6.84	                        "type": "PlainIntegerRelationState",
16.14/6.84	                        "relations": []
16.14/6.84	                    },
16.14/6.84	                    "ground": [
16.14/6.84	                        "T83",
16.14/6.84	                        "T84",
16.14/6.84	                        "T91"
16.14/6.84	                    ],
16.14/6.84	                    "free": [],
16.14/6.84	                    "exprvars": []
16.14/6.84	                }
16.14/6.84	            },
16.14/6.84	            "282": {
16.14/6.84	                "goal": [
16.14/6.84	                    {
16.14/6.84	                        "clause": 4,
16.14/6.84	                        "scope": 3,
16.14/6.84	                        "term": "(member1 (. T82 (. X67 ([]))) T84)"
16.14/6.84	                    },
16.14/6.84	                    {
16.14/6.84	                        "clause": 5,
16.14/6.84	                        "scope": 3,
16.14/6.84	                        "term": "(member1 (. T82 (. X67 ([]))) T84)"
16.14/6.84	                    }
16.14/6.84	                ],
16.14/6.84	                "kb": {
16.14/6.84	                    "nonunifying": [],
16.14/6.84	                    "intvars": {},
16.14/6.84	                    "arithmetic": {
16.14/6.84	                        "type": "PlainIntegerRelationState",
16.14/6.84	                        "relations": []
16.14/6.84	                    },
16.14/6.84	                    "ground": [
16.14/6.84	                        "T82",
16.14/6.84	                        "T84"
16.14/6.84	                    ],
16.14/6.84	                    "free": ["X67"],
16.14/6.84	                    "exprvars": []
16.14/6.84	                }
16.14/6.84	            },
16.14/6.84	            "283": {
16.14/6.84	                "goal": [{
16.14/6.84	                    "clause": 4,
16.14/6.84	                    "scope": 3,
16.14/6.84	                    "term": "(member1 (. T82 (. X67 ([]))) T84)"
16.14/6.84	                }],
16.14/6.84	                "kb": {
16.14/6.84	                    "nonunifying": [],
16.14/6.84	                    "intvars": {},
16.14/6.84	                    "arithmetic": {
16.14/6.84	                        "type": "PlainIntegerRelationState",
16.14/6.84	                        "relations": []
16.14/6.84	                    },
16.14/6.84	                    "ground": [
16.14/6.84	                        "T82",
16.14/6.84	                        "T84"
16.14/6.84	                    ],
16.14/6.84	                    "free": ["X67"],
16.14/6.84	                    "exprvars": []
16.14/6.84	                }
16.14/6.84	            },
16.14/6.84	            "262": {
16.14/6.84	                "goal": [
16.14/6.84	                    {
16.14/6.84	                        "clause": 2,
16.14/6.84	                        "scope": 2,
16.14/6.84	                        "term": "(member (. T28 (. T29 ([]))) T30)"
16.14/6.84	                    },
16.14/6.84	                    {
16.14/6.84	                        "clause": 3,
16.14/6.84	                        "scope": 2,
16.14/6.84	                        "term": "(member (. T28 (. T29 ([]))) T30)"
16.14/6.84	                    }
16.14/6.84	                ],
16.14/6.84	                "kb": {
16.14/6.84	                    "nonunifying": [],
16.14/6.84	                    "intvars": {},
16.14/6.84	                    "arithmetic": {
16.14/6.84	                        "type": "PlainIntegerRelationState",
16.14/6.84	                        "relations": []
16.14/6.84	                    },
16.14/6.84	                    "ground": [
16.14/6.84	                        "T28",
16.14/6.84	                        "T29",
16.14/6.84	                        "T30"
16.14/6.84	                    ],
16.14/6.84	                    "free": [],
16.14/6.84	                    "exprvars": []
16.14/6.84	                }
16.14/6.84	            },
16.14/6.84	            "284": {
16.14/6.84	                "goal": [{
16.14/6.84	                    "clause": 5,
16.14/6.84	                    "scope": 3,
16.14/6.84	                    "term": "(member1 (. T82 (. X67 ([]))) T84)"
16.14/6.84	                }],
16.14/6.84	                "kb": {
16.14/6.84	                    "nonunifying": [],
16.14/6.84	                    "intvars": {},
16.14/6.84	                    "arithmetic": {
16.14/6.84	                        "type": "PlainIntegerRelationState",
16.14/6.84	                        "relations": []
16.14/6.84	                    },
16.14/6.84	                    "ground": [
16.14/6.84	                        "T82",
16.14/6.84	                        "T84"
16.14/6.84	                    ],
16.14/6.84	                    "free": ["X67"],
16.14/6.84	                    "exprvars": []
16.14/6.84	                }
16.14/6.84	            },
16.14/6.84	            "263": {
16.14/6.84	                "goal": [{
16.14/6.84	                    "clause": 2,
16.14/6.84	                    "scope": 2,
16.14/6.84	                    "term": "(member (. T28 (. T29 ([]))) T30)"
16.14/6.84	                }],
16.14/6.84	                "kb": {
16.14/6.84	                    "nonunifying": [],
16.14/6.84	                    "intvars": {},
16.14/6.84	                    "arithmetic": {
16.14/6.84	                        "type": "PlainIntegerRelationState",
16.14/6.84	                        "relations": []
16.14/6.84	                    },
16.14/6.84	                    "ground": [
16.14/6.84	                        "T28",
16.14/6.84	                        "T29",
16.14/6.84	                        "T30"
16.14/6.84	                    ],
16.14/6.84	                    "free": [],
16.14/6.84	                    "exprvars": []
16.14/6.84	                }
16.14/6.84	            },
16.14/6.84	            "285": {
16.14/6.84	                "goal": [{
16.14/6.84	                    "clause": -1,
16.14/6.84	                    "scope": -1,
16.14/6.84	                    "term": "(true)"
16.14/6.84	                }],
16.14/6.84	                "kb": {
16.14/6.84	                    "nonunifying": [],
16.14/6.84	                    "intvars": {},
16.14/6.84	                    "arithmetic": {
16.14/6.84	                        "type": "PlainIntegerRelationState",
16.14/6.84	                        "relations": []
16.14/6.84	                    },
16.14/6.84	                    "ground": [],
16.14/6.84	                    "free": [],
16.14/6.84	                    "exprvars": []
16.14/6.84	                }
16.14/6.84	            },
16.14/6.85	            "264": {
16.14/6.85	                "goal": [{
16.14/6.85	                    "clause": 3,
16.14/6.85	                    "scope": 2,
16.14/6.85	                    "term": "(member (. T28 (. T29 ([]))) T30)"
16.14/6.85	                }],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [
16.14/6.85	                        "T28",
16.14/6.85	                        "T29",
16.14/6.85	                        "T30"
16.14/6.85	                    ],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "286": {
16.14/6.85	                "goal": [],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "1": {
16.14/6.85	                "goal": [{
16.14/6.85	                    "clause": -1,
16.14/6.85	                    "scope": -1,
16.14/6.85	                    "term": "(reach T1 T2 T3)"
16.14/6.85	                }],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [
16.14/6.85	                        "T1",
16.14/6.85	                        "T2",
16.14/6.85	                        "T3"
16.14/6.85	                    ],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "287": {
16.14/6.85	                "goal": [],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "266": {
16.14/6.85	                "goal": [{
16.14/6.85	                    "clause": -1,
16.14/6.85	                    "scope": -1,
16.14/6.85	                    "term": "(true)"
16.14/6.85	                }],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "267": {
16.14/6.85	                "goal": [],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "268": {
16.14/6.85	                "goal": [],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "269": {
16.14/6.85	                "goal": [{
16.14/6.85	                    "clause": -1,
16.14/6.85	                    "scope": -1,
16.14/6.85	                    "term": "(member (. T60 (. T61 ([]))) T63)"
16.14/6.85	                }],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [
16.14/6.85	                        "T60",
16.14/6.85	                        "T61",
16.14/6.85	                        "T63"
16.14/6.85	                    ],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            }
16.14/6.85	        },
16.14/6.85	        "edges": [
16.14/6.85	            {
16.14/6.85	                "from": 1,
16.14/6.85	                "to": 19,
16.14/6.85	                "label": "CASE"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 19,
16.14/6.85	                "to": 32,
16.14/6.85	                "label": "PARALLEL"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 19,
16.14/6.85	                "to": 33,
16.14/6.85	                "label": "PARALLEL"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 32,
16.14/6.85	                "to": 259,
16.14/6.85	                "label": "ONLY EVAL with clause\nreach(X21, X22, X23) :- member(.(X21, .(X22, [])), X23).\nand substitutionT1 -> T28,\nX21 -> T28,\nT2 -> T29,\nX22 -> T29,\nT3 -> T30,\nX23 -> T30"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 33,
16.14/6.85	                "to": 279,
16.14/6.85	                "label": "ONLY EVAL with clause\nreach(X64, X65, X66) :- ','(member1(.(X64, .(X67, [])), X66), reach(X67, X65, X66)).\nand substitutionT1 -> T82,\nX64 -> T82,\nT2 -> T83,\nX65 -> T83,\nT3 -> T84,\nX66 -> T84"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 259,
16.14/6.85	                "to": 262,
16.14/6.85	                "label": "CASE"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 262,
16.14/6.85	                "to": 263,
16.14/6.85	                "label": "PARALLEL"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 262,
16.14/6.85	                "to": 264,
16.14/6.85	                "label": "PARALLEL"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 263,
16.14/6.85	                "to": 266,
16.14/6.85	                "label": "EVAL with clause\nmember(X36, .(X36, X37)).\nand substitutionT28 -> T49,\nT29 -> T50,\nX36 -> .(T49, .(T50, [])),\nX37 -> T51,\nT30 -> .(.(T49, .(T50, [])), T51)"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 263,
16.14/6.85	                "to": 267,
16.14/6.85	                "label": "EVAL-BACKTRACK"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 264,
16.14/6.85	                "to": 269,
16.14/6.85	                "label": "EVAL with clause\nmember(X44, .(X45, X46)) :- member(X44, X46).\nand substitutionT28 -> T60,\nT29 -> T61,\nX44 -> .(T60, .(T61, [])),\nX45 -> T62,\nX46 -> T63,\nT30 -> .(T62, T63)"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 264,
16.14/6.85	                "to": 270,
16.14/6.85	                "label": "EVAL-BACKTRACK"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 266,
16.14/6.85	                "to": 268,
16.14/6.85	                "label": "SUCCESS"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 269,
16.14/6.85	                "to": 259,
16.14/6.85	                "label": "INSTANCE with matching:\nT28 -> T60\nT29 -> T61\nT30 -> T63"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 279,
16.14/6.85	                "to": 280,
16.14/6.85	                "label": "SPLIT 1"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 279,
16.14/6.85	                "to": 281,
16.14/6.85	                "label": "SPLIT 2\nnew knowledge:\nT82 is ground\nT91 is ground\nT84 is ground\nreplacements:X67 -> T91"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 280,
16.14/6.85	                "to": 282,
16.14/6.85	                "label": "CASE"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 281,
16.14/6.85	                "to": 1,
16.14/6.85	                "label": "INSTANCE with matching:\nT1 -> T91\nT2 -> T83\nT3 -> T84"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 282,
16.14/6.85	                "to": 283,
16.14/6.85	                "label": "PARALLEL"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 282,
16.14/6.85	                "to": 284,
16.14/6.85	                "label": "PARALLEL"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 283,
16.14/6.85	                "to": 285,
16.14/6.85	                "label": "EVAL with clause\nmember1(X92, .(X92, X93)).\nand substitutionT82 -> T110,\nX67 -> T111,\nX92 -> .(T110, .(T111, [])),\nX94 -> T111,\nX93 -> T112,\nT84 -> .(.(T110, .(T111, [])), T112)"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 283,
16.14/6.85	                "to": 286,
16.14/6.85	                "label": "EVAL-BACKTRACK"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 284,
16.14/6.85	                "to": 298,
16.14/6.85	                "label": "EVAL with clause\nmember1(X103, .(X104, X105)) :- member1(X103, X105).\nand substitutionT82 -> T119,\nX67 -> X106,\nX103 -> .(T119, .(X106, [])),\nX104 -> T120,\nX105 -> T121,\nT84 -> .(T120, T121)"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 284,
16.14/6.85	                "to": 299,
16.14/6.85	                "label": "EVAL-BACKTRACK"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 285,
16.14/6.85	                "to": 287,
16.14/6.85	                "label": "SUCCESS"
16.14/6.85	            },
16.14/6.85	            {
16.14/6.85	                "from": 298,
16.14/6.85	                "to": 280,
16.14/6.85	                "label": "INSTANCE with matching:\nT82 -> T119\nX67 -> X106\nT84 -> T121"
16.14/6.85	            }
16.14/6.85	        ],
16.14/6.85	        "type": "Graph"
16.14/6.85	    }
16.14/6.85	}
16.14/6.85	
16.14/6.85	----------------------------------------
16.14/6.85	
16.14/6.85	(64)
16.14/6.85	Obligation:
16.14/6.85	Q restricted rewrite system:
16.14/6.85	The TRS R consists of the following rules:
16.14/6.85	
16.14/6.85	   f1_in(T28, T29, T30) -> U1(f259_in(T28, T29, T30), T28, T29, T30)
16.14/6.85	   U1(f259_out1, T28, T29, T30) -> f1_out1
16.14/6.85	   f1_in(T82, T83, T84) -> U2(f279_in(T82, T84, T83), T82, T83, T84)
16.14/6.85	   U2(f279_out1(X67), T82, T83, T84) -> f1_out1
16.14/6.85	   f259_in(T49, T50, .(.(T49, .(T50, [])), T51)) -> f259_out1
16.14/6.85	   f259_in(T60, T61, .(T62, T63)) -> U3(f259_in(T60, T61, T63), T60, T61, .(T62, T63))
16.14/6.85	   U3(f259_out1, T60, T61, .(T62, T63)) -> f259_out1
16.14/6.85	   f280_in(T110, .(.(T110, .(T111, [])), T112)) -> f280_out1(T111)
16.14/6.85	   f280_in(T119, .(T120, T121)) -> U4(f280_in(T119, T121), T119, .(T120, T121))
16.14/6.85	   U4(f280_out1(X106), T119, .(T120, T121)) -> f280_out1(X106)
16.14/6.85	   f279_in(T82, T84, T83) -> U5(f280_in(T82, T84), T82, T84, T83)
16.14/6.85	   U5(f280_out1(T91), T82, T84, T83) -> U6(f1_in(T91, T83, T84), T82, T84, T83, T91)
16.14/6.85	   U6(f1_out1, T82, T84, T83, T91) -> f279_out1(T91)
16.14/6.85	
16.14/6.85	Q is empty.
16.14/6.85	
16.14/6.85	----------------------------------------
16.14/6.85	
16.14/6.85	(65) DependencyPairsProof (EQUIVALENT)
16.14/6.85	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
16.14/6.85	----------------------------------------
16.14/6.85	
16.14/6.85	(66)
16.14/6.85	Obligation:
16.14/6.85	Q DP problem:
16.14/6.85	The TRS P consists of the following rules:
16.14/6.85	
16.14/6.85	   F1_IN(T28, T29, T30) -> U1^1(f259_in(T28, T29, T30), T28, T29, T30)
16.14/6.85	   F1_IN(T28, T29, T30) -> F259_IN(T28, T29, T30)
16.14/6.85	   F1_IN(T82, T83, T84) -> U2^1(f279_in(T82, T84, T83), T82, T83, T84)
16.14/6.85	   F1_IN(T82, T83, T84) -> F279_IN(T82, T84, T83)
16.14/6.85	   F259_IN(T60, T61, .(T62, T63)) -> U3^1(f259_in(T60, T61, T63), T60, T61, .(T62, T63))
16.14/6.85	   F259_IN(T60, T61, .(T62, T63)) -> F259_IN(T60, T61, T63)
16.14/6.85	   F280_IN(T119, .(T120, T121)) -> U4^1(f280_in(T119, T121), T119, .(T120, T121))
16.14/6.85	   F280_IN(T119, .(T120, T121)) -> F280_IN(T119, T121)
16.14/6.85	   F279_IN(T82, T84, T83) -> U5^1(f280_in(T82, T84), T82, T84, T83)
16.14/6.85	   F279_IN(T82, T84, T83) -> F280_IN(T82, T84)
16.14/6.85	   U5^1(f280_out1(T91), T82, T84, T83) -> U6^1(f1_in(T91, T83, T84), T82, T84, T83, T91)
16.14/6.85	   U5^1(f280_out1(T91), T82, T84, T83) -> F1_IN(T91, T83, T84)
16.14/6.85	
16.14/6.85	The TRS R consists of the following rules:
16.14/6.85	
16.14/6.85	   f1_in(T28, T29, T30) -> U1(f259_in(T28, T29, T30), T28, T29, T30)
16.14/6.85	   U1(f259_out1, T28, T29, T30) -> f1_out1
16.14/6.85	   f1_in(T82, T83, T84) -> U2(f279_in(T82, T84, T83), T82, T83, T84)
16.14/6.85	   U2(f279_out1(X67), T82, T83, T84) -> f1_out1
16.14/6.85	   f259_in(T49, T50, .(.(T49, .(T50, [])), T51)) -> f259_out1
16.14/6.85	   f259_in(T60, T61, .(T62, T63)) -> U3(f259_in(T60, T61, T63), T60, T61, .(T62, T63))
16.14/6.85	   U3(f259_out1, T60, T61, .(T62, T63)) -> f259_out1
16.14/6.85	   f280_in(T110, .(.(T110, .(T111, [])), T112)) -> f280_out1(T111)
16.14/6.85	   f280_in(T119, .(T120, T121)) -> U4(f280_in(T119, T121), T119, .(T120, T121))
16.14/6.85	   U4(f280_out1(X106), T119, .(T120, T121)) -> f280_out1(X106)
16.14/6.85	   f279_in(T82, T84, T83) -> U5(f280_in(T82, T84), T82, T84, T83)
16.14/6.85	   U5(f280_out1(T91), T82, T84, T83) -> U6(f1_in(T91, T83, T84), T82, T84, T83, T91)
16.14/6.85	   U6(f1_out1, T82, T84, T83, T91) -> f279_out1(T91)
16.14/6.85	
16.14/6.85	Q is empty.
16.14/6.85	We have to consider all minimal (P,Q,R)-chains.
16.14/6.85	----------------------------------------
16.14/6.85	
16.14/6.85	(67) DependencyGraphProof (EQUIVALENT)
16.14/6.85	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 7 less nodes.
16.14/6.85	----------------------------------------
16.14/6.85	
16.14/6.85	(68)
16.14/6.85	Complex Obligation (AND)
16.14/6.85	
16.14/6.85	----------------------------------------
16.14/6.85	
16.14/6.85	(69)
16.14/6.85	Obligation:
16.14/6.85	Q DP problem:
16.14/6.85	The TRS P consists of the following rules:
16.14/6.85	
16.14/6.85	   F280_IN(T119, .(T120, T121)) -> F280_IN(T119, T121)
16.14/6.85	
16.14/6.85	The TRS R consists of the following rules:
16.14/6.85	
16.14/6.85	   f1_in(T28, T29, T30) -> U1(f259_in(T28, T29, T30), T28, T29, T30)
16.14/6.85	   U1(f259_out1, T28, T29, T30) -> f1_out1
16.14/6.85	   f1_in(T82, T83, T84) -> U2(f279_in(T82, T84, T83), T82, T83, T84)
16.14/6.85	   U2(f279_out1(X67), T82, T83, T84) -> f1_out1
16.14/6.85	   f259_in(T49, T50, .(.(T49, .(T50, [])), T51)) -> f259_out1
16.14/6.85	   f259_in(T60, T61, .(T62, T63)) -> U3(f259_in(T60, T61, T63), T60, T61, .(T62, T63))
16.14/6.85	   U3(f259_out1, T60, T61, .(T62, T63)) -> f259_out1
16.14/6.85	   f280_in(T110, .(.(T110, .(T111, [])), T112)) -> f280_out1(T111)
16.14/6.85	   f280_in(T119, .(T120, T121)) -> U4(f280_in(T119, T121), T119, .(T120, T121))
16.14/6.85	   U4(f280_out1(X106), T119, .(T120, T121)) -> f280_out1(X106)
16.14/6.85	   f279_in(T82, T84, T83) -> U5(f280_in(T82, T84), T82, T84, T83)
16.14/6.85	   U5(f280_out1(T91), T82, T84, T83) -> U6(f1_in(T91, T83, T84), T82, T84, T83, T91)
16.14/6.85	   U6(f1_out1, T82, T84, T83, T91) -> f279_out1(T91)
16.14/6.85	
16.14/6.85	Q is empty.
16.14/6.85	We have to consider all minimal (P,Q,R)-chains.
16.14/6.85	----------------------------------------
16.14/6.85	
16.14/6.85	(70) UsableRulesProof (EQUIVALENT)
16.14/6.85	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.
16.14/6.85	----------------------------------------
16.14/6.85	
16.14/6.85	(71)
16.14/6.85	Obligation:
16.14/6.85	Q DP problem:
16.14/6.85	The TRS P consists of the following rules:
16.14/6.85	
16.14/6.85	   F280_IN(T119, .(T120, T121)) -> F280_IN(T119, T121)
16.14/6.85	
16.14/6.85	R is empty.
16.14/6.85	Q is empty.
16.14/6.85	We have to consider all minimal (P,Q,R)-chains.
16.14/6.85	----------------------------------------
16.14/6.85	
16.14/6.85	(72) QDPSizeChangeProof (EQUIVALENT)
16.14/6.85	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. 
16.14/6.85	
16.14/6.85	From the DPs we obtained the following set of size-change graphs:
16.14/6.85	*F280_IN(T119, .(T120, T121)) -> F280_IN(T119, T121)
16.14/6.85	The graph contains the following edges 1 >= 1, 2 > 2
16.14/6.85	
16.14/6.85	
16.14/6.85	----------------------------------------
16.14/6.85	
16.14/6.85	(73)
16.14/6.85	YES
16.14/6.85	
16.14/6.85	----------------------------------------
16.14/6.85	
16.14/6.85	(74)
16.14/6.85	Obligation:
16.14/6.85	Q DP problem:
16.14/6.85	The TRS P consists of the following rules:
16.14/6.85	
16.14/6.85	   F259_IN(T60, T61, .(T62, T63)) -> F259_IN(T60, T61, T63)
16.14/6.85	
16.14/6.85	The TRS R consists of the following rules:
16.14/6.85	
16.14/6.85	   f1_in(T28, T29, T30) -> U1(f259_in(T28, T29, T30), T28, T29, T30)
16.14/6.85	   U1(f259_out1, T28, T29, T30) -> f1_out1
16.14/6.85	   f1_in(T82, T83, T84) -> U2(f279_in(T82, T84, T83), T82, T83, T84)
16.14/6.85	   U2(f279_out1(X67), T82, T83, T84) -> f1_out1
16.14/6.85	   f259_in(T49, T50, .(.(T49, .(T50, [])), T51)) -> f259_out1
16.14/6.85	   f259_in(T60, T61, .(T62, T63)) -> U3(f259_in(T60, T61, T63), T60, T61, .(T62, T63))
16.14/6.85	   U3(f259_out1, T60, T61, .(T62, T63)) -> f259_out1
16.14/6.85	   f280_in(T110, .(.(T110, .(T111, [])), T112)) -> f280_out1(T111)
16.14/6.85	   f280_in(T119, .(T120, T121)) -> U4(f280_in(T119, T121), T119, .(T120, T121))
16.14/6.85	   U4(f280_out1(X106), T119, .(T120, T121)) -> f280_out1(X106)
16.14/6.85	   f279_in(T82, T84, T83) -> U5(f280_in(T82, T84), T82, T84, T83)
16.14/6.85	   U5(f280_out1(T91), T82, T84, T83) -> U6(f1_in(T91, T83, T84), T82, T84, T83, T91)
16.14/6.85	   U6(f1_out1, T82, T84, T83, T91) -> f279_out1(T91)
16.14/6.85	
16.14/6.85	Q is empty.
16.14/6.85	We have to consider all minimal (P,Q,R)-chains.
16.14/6.85	----------------------------------------
16.14/6.85	
16.14/6.85	(75) UsableRulesProof (EQUIVALENT)
16.14/6.85	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.
16.14/6.85	----------------------------------------
16.14/6.85	
16.14/6.85	(76)
16.14/6.85	Obligation:
16.14/6.85	Q DP problem:
16.14/6.85	The TRS P consists of the following rules:
16.14/6.85	
16.14/6.85	   F259_IN(T60, T61, .(T62, T63)) -> F259_IN(T60, T61, T63)
16.14/6.85	
16.14/6.85	R is empty.
16.14/6.85	Q is empty.
16.14/6.85	We have to consider all minimal (P,Q,R)-chains.
16.14/6.85	----------------------------------------
16.14/6.85	
16.14/6.85	(77) QDPSizeChangeProof (EQUIVALENT)
16.14/6.85	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. 
16.14/6.85	
16.14/6.85	From the DPs we obtained the following set of size-change graphs:
16.14/6.85	*F259_IN(T60, T61, .(T62, T63)) -> F259_IN(T60, T61, T63)
16.14/6.85	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
16.14/6.85	
16.14/6.85	
16.14/6.85	----------------------------------------
16.14/6.85	
16.14/6.85	(78)
16.14/6.85	YES
16.14/6.85	
16.14/6.85	----------------------------------------
16.14/6.85	
16.14/6.85	(79)
16.14/6.85	Obligation:
16.14/6.85	Q DP problem:
16.14/6.85	The TRS P consists of the following rules:
16.14/6.85	
16.14/6.85	   F1_IN(T82, T83, T84) -> F279_IN(T82, T84, T83)
16.14/6.85	   F279_IN(T82, T84, T83) -> U5^1(f280_in(T82, T84), T82, T84, T83)
16.14/6.85	   U5^1(f280_out1(T91), T82, T84, T83) -> F1_IN(T91, T83, T84)
16.14/6.85	
16.14/6.85	The TRS R consists of the following rules:
16.14/6.85	
16.14/6.85	   f1_in(T28, T29, T30) -> U1(f259_in(T28, T29, T30), T28, T29, T30)
16.14/6.85	   U1(f259_out1, T28, T29, T30) -> f1_out1
16.14/6.85	   f1_in(T82, T83, T84) -> U2(f279_in(T82, T84, T83), T82, T83, T84)
16.14/6.85	   U2(f279_out1(X67), T82, T83, T84) -> f1_out1
16.14/6.85	   f259_in(T49, T50, .(.(T49, .(T50, [])), T51)) -> f259_out1
16.14/6.85	   f259_in(T60, T61, .(T62, T63)) -> U3(f259_in(T60, T61, T63), T60, T61, .(T62, T63))
16.14/6.85	   U3(f259_out1, T60, T61, .(T62, T63)) -> f259_out1
16.14/6.85	   f280_in(T110, .(.(T110, .(T111, [])), T112)) -> f280_out1(T111)
16.14/6.85	   f280_in(T119, .(T120, T121)) -> U4(f280_in(T119, T121), T119, .(T120, T121))
16.14/6.85	   U4(f280_out1(X106), T119, .(T120, T121)) -> f280_out1(X106)
16.14/6.85	   f279_in(T82, T84, T83) -> U5(f280_in(T82, T84), T82, T84, T83)
16.14/6.85	   U5(f280_out1(T91), T82, T84, T83) -> U6(f1_in(T91, T83, T84), T82, T84, T83, T91)
16.14/6.85	   U6(f1_out1, T82, T84, T83, T91) -> f279_out1(T91)
16.14/6.85	
16.14/6.85	Q is empty.
16.14/6.85	We have to consider all minimal (P,Q,R)-chains.
16.14/6.85	----------------------------------------
16.14/6.85	
16.14/6.85	(80) NonTerminationLoopProof (COMPLETE)
16.14/6.85	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
16.14/6.85	Found a loop by narrowing to the left:
16.14/6.85	
16.14/6.85	s = F279_IN(T82, .(.(T82, .(T111, [])), T112), T83) evaluates to  t =F279_IN(T111, .(.(T82, .(T111, [])), T112), T83)
16.14/6.85	
16.14/6.85	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
16.14/6.85	* Matcher: [ ]
16.14/6.85	* Semiunifier: [T111 / T82]
16.14/6.85	
16.14/6.85	--------------------------------------------------------------------------------
16.14/6.85	Rewriting sequence
16.14/6.85	
16.14/6.85	F279_IN(T82, .(.(T82, .(T82, [])), T112), T83) -> U5^1(f280_in(T82, .(.(T82, .(T82, [])), T112)), T82, .(.(T82, .(T82, [])), T112), T83)
16.14/6.85	with rule F279_IN(T82', T84, T83') -> U5^1(f280_in(T82', T84), T82', T84, T83') at position [] and matcher [T82' / T82, T84 / .(.(T82, .(T82, [])), T112), T83' / T83]
16.14/6.85	
16.14/6.85	U5^1(f280_in(T82, .(.(T82, .(T82, [])), T112)), T82, .(.(T82, .(T82, [])), T112), T83) -> U5^1(f280_out1(T82), T82, .(.(T82, .(T82, [])), T112), T83)
16.14/6.85	with rule f280_in(T110, .(.(T110, .(T111, [])), T112')) -> f280_out1(T111) at position [0] and matcher [T110 / T82, T111 / T82, T112' / T112]
16.14/6.85	
16.14/6.85	U5^1(f280_out1(T82), T82, .(.(T82, .(T82, [])), T112), T83) -> F1_IN(T82, T83, .(.(T82, .(T82, [])), T112))
16.14/6.85	with rule U5^1(f280_out1(T91), T82', T84', T83') -> F1_IN(T91, T83', T84') at position [] and matcher [T91 / T82, T82' / T82, T84' / .(.(T82, .(T82, [])), T112), T83' / T83]
16.14/6.85	
16.14/6.85	F1_IN(T82, T83, .(.(T82, .(T82, [])), T112)) -> F279_IN(T82, .(.(T82, .(T82, [])), T112), T83)
16.14/6.85	with rule F1_IN(T82, T83, T84) -> F279_IN(T82, T84, T83)
16.14/6.85	
16.14/6.85	Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
16.14/6.85	
16.14/6.85	
16.14/6.85	All these steps are and every following step will be a correct step w.r.t to Q.
16.14/6.85	
16.14/6.85	
16.14/6.85	
16.14/6.85	
16.14/6.85	----------------------------------------
16.14/6.85	
16.14/6.85	(81)
16.14/6.85	NO
16.14/6.85	
16.14/6.85	----------------------------------------
16.14/6.85	
16.14/6.85	(82) PrologToDTProblemTransformerProof (SOUND)
16.14/6.85	Built DT problem from termination graph DT10.
16.14/6.85	
16.14/6.85	{
16.14/6.85	    "root": 3,
16.14/6.85	    "program": {
16.14/6.85	        "directives": [],
16.14/6.85	        "clauses": [
16.14/6.85	            [
16.14/6.85	                "(reach X Y Edges)",
16.14/6.85	                "(member (. X (. Y ([]))) Edges)"
16.14/6.85	            ],
16.14/6.85	            [
16.14/6.85	                "(reach X Z Edges)",
16.14/6.85	                "(',' (member1 (. X (. Y ([]))) Edges) (reach Y Z Edges))"
16.14/6.85	            ],
16.14/6.85	            [
16.14/6.85	                "(member H (. H L))",
16.14/6.85	                null
16.14/6.85	            ],
16.14/6.85	            [
16.14/6.85	                "(member X (. H L))",
16.14/6.85	                "(member X L)"
16.14/6.85	            ],
16.14/6.85	            [
16.14/6.85	                "(member1 H (. H L))",
16.14/6.85	                null
16.14/6.85	            ],
16.14/6.85	            [
16.14/6.85	                "(member1 X (. H L))",
16.14/6.85	                "(member1 X L)"
16.14/6.85	            ]
16.14/6.85	        ]
16.14/6.85	    },
16.14/6.85	    "graph": {
16.14/6.85	        "nodes": {
16.14/6.85	            "type": "Nodes",
16.14/6.85	            "271": {
16.14/6.85	                "goal": [{
16.14/6.85	                    "clause": -1,
16.14/6.85	                    "scope": -1,
16.14/6.85	                    "term": "(',' (member1 (. T96 (. X75 ([]))) T98) (reach X75 T97 T98))"
16.14/6.85	                }],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [
16.14/6.85	                        "T96",
16.14/6.85	                        "T97",
16.14/6.85	                        "T98"
16.14/6.85	                    ],
16.14/6.85	                    "free": ["X75"],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "272": {
16.14/6.85	                "goal": [{
16.14/6.85	                    "clause": -1,
16.14/6.85	                    "scope": -1,
16.14/6.85	                    "term": "(member1 (. T96 (. X75 ([]))) T98)"
16.14/6.85	                }],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [
16.14/6.85	                        "T96",
16.14/6.85	                        "T98"
16.14/6.85	                    ],
16.14/6.85	                    "free": ["X75"],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "273": {
16.14/6.85	                "goal": [{
16.14/6.85	                    "clause": -1,
16.14/6.85	                    "scope": -1,
16.14/6.85	                    "term": "(reach T105 T97 T98)"
16.14/6.85	                }],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [
16.14/6.85	                        "T97",
16.14/6.85	                        "T98",
16.14/6.85	                        "T105"
16.14/6.85	                    ],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "274": {
16.14/6.85	                "goal": [
16.14/6.85	                    {
16.14/6.85	                        "clause": 4,
16.14/6.85	                        "scope": 4,
16.14/6.85	                        "term": "(member1 (. T96 (. X75 ([]))) T98)"
16.14/6.85	                    },
16.14/6.85	                    {
16.14/6.85	                        "clause": 5,
16.14/6.85	                        "scope": 4,
16.14/6.85	                        "term": "(member1 (. T96 (. X75 ([]))) T98)"
16.14/6.85	                    }
16.14/6.85	                ],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [
16.14/6.85	                        "T96",
16.14/6.85	                        "T98"
16.14/6.85	                    ],
16.14/6.85	                    "free": ["X75"],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "254": {
16.14/6.85	                "goal": [{
16.14/6.85	                    "clause": 2,
16.14/6.85	                    "scope": 3,
16.14/6.85	                    "term": "(member (. T41 (. T42 ([]))) T44)"
16.14/6.85	                }],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [
16.14/6.85	                        "T41",
16.14/6.85	                        "T42",
16.14/6.85	                        "T44"
16.14/6.85	                    ],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "233": {
16.14/6.85	                "goal": [{
16.14/6.85	                    "clause": 3,
16.14/6.85	                    "scope": 2,
16.14/6.85	                    "term": "(member (. T7 (. T8 ([]))) T9)"
16.14/6.85	                }],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [
16.14/6.85	                        "T7",
16.14/6.85	                        "T8",
16.14/6.85	                        "T9"
16.14/6.85	                    ],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "255": {
16.14/6.85	                "goal": [{
16.14/6.85	                    "clause": 3,
16.14/6.85	                    "scope": 3,
16.14/6.85	                    "term": "(member (. T41 (. T42 ([]))) T44)"
16.14/6.85	                }],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [
16.14/6.85	                        "T41",
16.14/6.85	                        "T42",
16.14/6.85	                        "T44"
16.14/6.85	                    ],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "277": {
16.14/6.85	                "goal": [{
16.14/6.85	                    "clause": 4,
16.14/6.85	                    "scope": 4,
16.14/6.85	                    "term": "(member1 (. T96 (. X75 ([]))) T98)"
16.14/6.85	                }],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [
16.14/6.85	                        "T96",
16.14/6.85	                        "T98"
16.14/6.85	                    ],
16.14/6.85	                    "free": ["X75"],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "234": {
16.14/6.85	                "goal": [
16.14/6.85	                    {
16.14/6.85	                        "clause": -1,
16.14/6.85	                        "scope": 2,
16.14/6.85	                        "term": null
16.14/6.85	                    },
16.14/6.85	                    {
16.14/6.85	                        "clause": 1,
16.14/6.85	                        "scope": 1,
16.14/6.85	                        "term": "(reach T7 T8 T9)"
16.14/6.85	                    }
16.14/6.85	                ],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [
16.14/6.85	                        "T7",
16.14/6.85	                        "T8",
16.14/6.85	                        "T9"
16.14/6.85	                    ],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "256": {
16.14/6.85	                "goal": [{
16.14/6.85	                    "clause": -1,
16.14/6.85	                    "scope": -1,
16.14/6.85	                    "term": "(true)"
16.14/6.85	                }],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "278": {
16.14/6.85	                "goal": [{
16.14/6.85	                    "clause": 5,
16.14/6.85	                    "scope": 4,
16.14/6.85	                    "term": "(member1 (. T96 (. X75 ([]))) T98)"
16.14/6.85	                }],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [
16.14/6.85	                        "T96",
16.14/6.85	                        "T98"
16.14/6.85	                    ],
16.14/6.85	                    "free": ["X75"],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "257": {
16.14/6.85	                "goal": [],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "258": {
16.14/6.85	                "goal": [],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "238": {
16.14/6.85	                "goal": [{
16.14/6.85	                    "clause": -1,
16.14/6.85	                    "scope": -1,
16.14/6.85	                    "term": "(member (. T41 (. T42 ([]))) T44)"
16.14/6.85	                }],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [
16.14/6.85	                        "T41",
16.14/6.85	                        "T42",
16.14/6.85	                        "T44"
16.14/6.85	                    ],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "219": {
16.14/6.85	                "goal": [
16.14/6.85	                    {
16.14/6.85	                        "clause": 0,
16.14/6.85	                        "scope": 1,
16.14/6.85	                        "term": "(reach T1 T2 T3)"
16.14/6.85	                    },
16.14/6.85	                    {
16.14/6.85	                        "clause": 1,
16.14/6.85	                        "scope": 1,
16.14/6.85	                        "term": "(reach T1 T2 T3)"
16.14/6.85	                    }
16.14/6.85	                ],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [
16.14/6.85	                        "T1",
16.14/6.85	                        "T2",
16.14/6.85	                        "T3"
16.14/6.85	                    ],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "260": {
16.14/6.85	                "goal": [{
16.14/6.85	                    "clause": -1,
16.14/6.85	                    "scope": -1,
16.14/6.85	                    "term": "(member (. T74 (. T75 ([]))) T77)"
16.14/6.85	                }],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [
16.14/6.85	                        "T74",
16.14/6.85	                        "T75",
16.14/6.85	                        "T77"
16.14/6.85	                    ],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "261": {
16.14/6.85	                "goal": [],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "240": {
16.14/6.85	                "goal": [],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "220": {
16.14/6.85	                "goal": [
16.14/6.85	                    {
16.14/6.85	                        "clause": -1,
16.14/6.85	                        "scope": -1,
16.14/6.85	                        "term": "(member (. T7 (. T8 ([]))) T9)"
16.14/6.85	                    },
16.14/6.85	                    {
16.14/6.85	                        "clause": 1,
16.14/6.85	                        "scope": 1,
16.14/6.85	                        "term": "(reach T7 T8 T9)"
16.14/6.85	                    }
16.14/6.85	                ],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [
16.14/6.85	                        "T7",
16.14/6.85	                        "T8",
16.14/6.85	                        "T9"
16.14/6.85	                    ],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "221": {
16.14/6.85	                "goal": [
16.14/6.85	                    {
16.14/6.85	                        "clause": 2,
16.14/6.85	                        "scope": 2,
16.14/6.85	                        "term": "(member (. T7 (. T8 ([]))) T9)"
16.14/6.85	                    },
16.14/6.85	                    {
16.14/6.85	                        "clause": 3,
16.14/6.85	                        "scope": 2,
16.14/6.85	                        "term": "(member (. T7 (. T8 ([]))) T9)"
16.14/6.85	                    },
16.14/6.85	                    {
16.14/6.85	                        "clause": -1,
16.14/6.85	                        "scope": 2,
16.14/6.85	                        "term": null
16.14/6.85	                    },
16.14/6.85	                    {
16.14/6.85	                        "clause": 1,
16.14/6.85	                        "scope": 1,
16.14/6.85	                        "term": "(reach T7 T8 T9)"
16.14/6.85	                    }
16.14/6.85	                ],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [
16.14/6.85	                        "T7",
16.14/6.85	                        "T8",
16.14/6.85	                        "T9"
16.14/6.85	                    ],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "265": {
16.14/6.85	                "goal": [{
16.14/6.85	                    "clause": 1,
16.14/6.85	                    "scope": 1,
16.14/6.85	                    "term": "(reach T7 T8 T9)"
16.14/6.85	                }],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.85	                        "relations": []
16.14/6.85	                    },
16.14/6.85	                    "ground": [
16.14/6.85	                        "T7",
16.14/6.85	                        "T8",
16.14/6.85	                        "T9"
16.14/6.85	                    ],
16.14/6.85	                    "free": [],
16.14/6.85	                    "exprvars": []
16.14/6.85	                }
16.14/6.85	            },
16.14/6.85	            "222": {
16.14/6.85	                "goal": [{
16.14/6.85	                    "clause": 2,
16.14/6.85	                    "scope": 2,
16.14/6.85	                    "term": "(member (. T7 (. T8 ([]))) T9)"
16.14/6.85	                }],
16.14/6.85	                "kb": {
16.14/6.85	                    "nonunifying": [],
16.14/6.85	                    "intvars": {},
16.14/6.85	                    "arithmetic": {
16.14/6.85	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [
16.14/6.86	                        "T7",
16.14/6.86	                        "T8",
16.14/6.86	                        "T9"
16.14/6.86	                    ],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "3": {
16.14/6.86	                "goal": [{
16.14/6.86	                    "clause": -1,
16.14/6.86	                    "scope": -1,
16.14/6.86	                    "term": "(reach T1 T2 T3)"
16.14/6.86	                }],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [
16.14/6.86	                        "T1",
16.14/6.86	                        "T2",
16.14/6.86	                        "T3"
16.14/6.86	                    ],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "223": {
16.14/6.86	                "goal": [
16.14/6.86	                    {
16.14/6.86	                        "clause": 3,
16.14/6.86	                        "scope": 2,
16.14/6.86	                        "term": "(member (. T7 (. T8 ([]))) T9)"
16.14/6.86	                    },
16.14/6.86	                    {
16.14/6.86	                        "clause": -1,
16.14/6.86	                        "scope": 2,
16.14/6.86	                        "term": null
16.14/6.86	                    },
16.14/6.86	                    {
16.14/6.86	                        "clause": 1,
16.14/6.86	                        "scope": 1,
16.14/6.86	                        "term": "(reach T7 T8 T9)"
16.14/6.86	                    }
16.14/6.86	                ],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [
16.14/6.86	                        "T7",
16.14/6.86	                        "T8",
16.14/6.86	                        "T9"
16.14/6.86	                    ],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "300": {
16.14/6.86	                "goal": [{
16.14/6.86	                    "clause": -1,
16.14/6.86	                    "scope": -1,
16.14/6.86	                    "term": "(true)"
16.14/6.86	                }],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "224": {
16.14/6.86	                "goal": [{
16.14/6.86	                    "clause": -1,
16.14/6.86	                    "scope": -1,
16.14/6.86	                    "term": "(true)"
16.14/6.86	                }],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "301": {
16.14/6.86	                "goal": [],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "302": {
16.14/6.86	                "goal": [],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "226": {
16.14/6.86	                "goal": [],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "303": {
16.14/6.86	                "goal": [{
16.14/6.86	                    "clause": -1,
16.14/6.86	                    "scope": -1,
16.14/6.86	                    "term": "(member1 (. T133 (. X114 ([]))) T135)"
16.14/6.86	                }],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [
16.14/6.86	                        "T133",
16.14/6.86	                        "T135"
16.14/6.86	                    ],
16.14/6.86	                    "free": ["X114"],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "227": {
16.14/6.86	                "goal": [],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "249": {
16.14/6.86	                "goal": [
16.14/6.86	                    {
16.14/6.86	                        "clause": 2,
16.14/6.86	                        "scope": 3,
16.14/6.86	                        "term": "(member (. T41 (. T42 ([]))) T44)"
16.14/6.86	                    },
16.14/6.86	                    {
16.14/6.86	                        "clause": 3,
16.14/6.86	                        "scope": 3,
16.14/6.86	                        "term": "(member (. T41 (. T42 ([]))) T44)"
16.14/6.86	                    }
16.14/6.86	                ],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [
16.14/6.86	                        "T41",
16.14/6.86	                        "T42",
16.14/6.86	                        "T44"
16.14/6.86	                    ],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "304": {
16.14/6.86	                "goal": [],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            }
16.14/6.86	        },
16.14/6.86	        "edges": [
16.14/6.86	            {
16.14/6.86	                "from": 3,
16.14/6.86	                "to": 219,
16.14/6.86	                "label": "CASE"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 219,
16.14/6.86	                "to": 220,
16.14/6.86	                "label": "ONLY EVAL with clause\nreach(X4, X5, X6) :- member(.(X4, .(X5, [])), X6).\nand substitutionT1 -> T7,\nX4 -> T7,\nT2 -> T8,\nX5 -> T8,\nT3 -> T9,\nX6 -> T9"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 220,
16.14/6.86	                "to": 221,
16.14/6.86	                "label": "CASE"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 221,
16.14/6.86	                "to": 222,
16.14/6.86	                "label": "PARALLEL"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 221,
16.14/6.86	                "to": 223,
16.14/6.86	                "label": "PARALLEL"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 222,
16.14/6.86	                "to": 224,
16.14/6.86	                "label": "EVAL with clause\nmember(X15, .(X15, X16)).\nand substitutionT7 -> T22,\nT8 -> T23,\nX15 -> .(T22, .(T23, [])),\nX16 -> T24,\nT9 -> .(.(T22, .(T23, [])), T24)"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 222,
16.14/6.86	                "to": 226,
16.14/6.86	                "label": "EVAL-BACKTRACK"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 223,
16.14/6.86	                "to": 233,
16.14/6.86	                "label": "PARALLEL"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 223,
16.14/6.86	                "to": 234,
16.14/6.86	                "label": "PARALLEL"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 224,
16.14/6.86	                "to": 227,
16.14/6.86	                "label": "SUCCESS"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 233,
16.14/6.86	                "to": 238,
16.14/6.86	                "label": "EVAL with clause\nmember(X29, .(X30, X31)) :- member(X29, X31).\nand substitutionT7 -> T41,\nT8 -> T42,\nX29 -> .(T41, .(T42, [])),\nX30 -> T43,\nX31 -> T44,\nT9 -> .(T43, T44)"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 233,
16.14/6.86	                "to": 240,
16.14/6.86	                "label": "EVAL-BACKTRACK"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 234,
16.14/6.86	                "to": 265,
16.14/6.86	                "label": "FAILURE"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 238,
16.14/6.86	                "to": 249,
16.14/6.86	                "label": "CASE"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 249,
16.14/6.86	                "to": 254,
16.14/6.86	                "label": "PARALLEL"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 249,
16.14/6.86	                "to": 255,
16.14/6.86	                "label": "PARALLEL"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 254,
16.14/6.86	                "to": 256,
16.14/6.86	                "label": "EVAL with clause\nmember(X44, .(X44, X45)).\nand substitutionT41 -> T63,\nT42 -> T64,\nX44 -> .(T63, .(T64, [])),\nX45 -> T65,\nT44 -> .(.(T63, .(T64, [])), T65)"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 254,
16.14/6.86	                "to": 257,
16.14/6.86	                "label": "EVAL-BACKTRACK"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 255,
16.14/6.86	                "to": 260,
16.14/6.86	                "label": "EVAL with clause\nmember(X52, .(X53, X54)) :- member(X52, X54).\nand substitutionT41 -> T74,\nT42 -> T75,\nX52 -> .(T74, .(T75, [])),\nX53 -> T76,\nX54 -> T77,\nT44 -> .(T76, T77)"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 255,
16.14/6.86	                "to": 261,
16.14/6.86	                "label": "EVAL-BACKTRACK"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 256,
16.14/6.86	                "to": 258,
16.14/6.86	                "label": "SUCCESS"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 260,
16.14/6.86	                "to": 238,
16.14/6.86	                "label": "INSTANCE with matching:\nT41 -> T74\nT42 -> T75\nT44 -> T77"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 265,
16.14/6.86	                "to": 271,
16.14/6.86	                "label": "ONLY EVAL with clause\nreach(X72, X73, X74) :- ','(member1(.(X72, .(X75, [])), X74), reach(X75, X73, X74)).\nand substitutionT7 -> T96,\nX72 -> T96,\nT8 -> T97,\nX73 -> T97,\nT9 -> T98,\nX74 -> T98"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 271,
16.14/6.86	                "to": 272,
16.14/6.86	                "label": "SPLIT 1"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 271,
16.14/6.86	                "to": 273,
16.14/6.86	                "label": "SPLIT 2\nnew knowledge:\nT96 is ground\nT105 is ground\nT98 is ground\nreplacements:X75 -> T105"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 272,
16.14/6.86	                "to": 274,
16.14/6.86	                "label": "CASE"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 273,
16.14/6.86	                "to": 3,
16.14/6.86	                "label": "INSTANCE with matching:\nT1 -> T105\nT2 -> T97\nT3 -> T98"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 274,
16.14/6.86	                "to": 277,
16.14/6.86	                "label": "PARALLEL"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 274,
16.14/6.86	                "to": 278,
16.14/6.86	                "label": "PARALLEL"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 277,
16.14/6.86	                "to": 300,
16.14/6.86	                "label": "EVAL with clause\nmember1(X100, .(X100, X101)).\nand substitutionT96 -> T124,\nX75 -> T125,\nX100 -> .(T124, .(T125, [])),\nX102 -> T125,\nX101 -> T126,\nT98 -> .(.(T124, .(T125, [])), T126)"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 277,
16.14/6.86	                "to": 301,
16.14/6.86	                "label": "EVAL-BACKTRACK"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 278,
16.14/6.86	                "to": 303,
16.14/6.86	                "label": "EVAL with clause\nmember1(X111, .(X112, X113)) :- member1(X111, X113).\nand substitutionT96 -> T133,\nX75 -> X114,\nX111 -> .(T133, .(X114, [])),\nX112 -> T134,\nX113 -> T135,\nT98 -> .(T134, T135)"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 278,
16.14/6.86	                "to": 304,
16.14/6.86	                "label": "EVAL-BACKTRACK"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 300,
16.14/6.86	                "to": 302,
16.14/6.86	                "label": "SUCCESS"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 303,
16.14/6.86	                "to": 272,
16.14/6.86	                "label": "INSTANCE with matching:\nT96 -> T133\nX75 -> X114\nT98 -> T135"
16.14/6.86	            }
16.14/6.86	        ],
16.14/6.86	        "type": "Graph"
16.14/6.86	    }
16.14/6.86	}
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(83)
16.14/6.86	Obligation:
16.14/6.86	Triples:
16.14/6.86	
16.14/6.86	memberA(X1, X2, .(X3, X4)) :- memberA(X1, X2, X4).
16.14/6.86	member1C(X1, X2, .(X3, X4)) :- member1C(X1, X2, X4).
16.14/6.86	reachB(X1, X2, .(X3, X4)) :- memberA(X1, X2, X4).
16.14/6.86	reachB(X1, X2, X3) :- member1C(X1, X4, X3).
16.14/6.86	reachB(X1, X2, X3) :- ','(member1cC(X1, X4, X3), reachB(X4, X2, X3)).
16.14/6.86	
16.14/6.86	Clauses:
16.14/6.86	
16.14/6.86	membercA(X1, X2, .(.(X1, .(X2, [])), X3)).
16.14/6.86	membercA(X1, X2, .(X3, X4)) :- membercA(X1, X2, X4).
16.14/6.86	reachcB(X1, X2, .(.(X1, .(X2, [])), X3)).
16.14/6.86	reachcB(X1, X2, .(X3, X4)) :- membercA(X1, X2, X4).
16.14/6.86	reachcB(X1, X2, X3) :- ','(member1cC(X1, X4, X3), reachcB(X4, X2, X3)).
16.14/6.86	member1cC(X1, X2, .(.(X1, .(X2, [])), X3)).
16.14/6.86	member1cC(X1, X2, .(X3, X4)) :- member1cC(X1, X2, X4).
16.14/6.86	
16.14/6.86	Afs:
16.14/6.86	
16.14/6.86	reachB(x1, x2, x3)  =  reachB(x1, x2, x3)
16.14/6.86	
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(84) TriplesToPiDPProof (SOUND)
16.14/6.86	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
16.14/6.86	
16.14/6.86	reachB_in_3: (b,b,b)
16.14/6.86	
16.14/6.86	memberA_in_3: (b,b,b)
16.14/6.86	
16.14/6.86	member1C_in_3: (b,f,b)
16.14/6.86	
16.14/6.86	member1cC_in_3: (b,f,b)
16.14/6.86	
16.14/6.86	Transforming TRIPLES into the following Term Rewriting System:
16.14/6.86	
16.14/6.86	Pi DP problem:
16.14/6.86	The TRS P consists of the following rules:
16.14/6.86	
16.14/6.86	   REACHB_IN_GGG(X1, X2, .(X3, X4)) -> U3_GGG(X1, X2, X3, X4, memberA_in_ggg(X1, X2, X4))
16.14/6.86	   REACHB_IN_GGG(X1, X2, .(X3, X4)) -> MEMBERA_IN_GGG(X1, X2, X4)
16.14/6.86	   MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> U1_GGG(X1, X2, X3, X4, memberA_in_ggg(X1, X2, X4))
16.14/6.86	   MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> MEMBERA_IN_GGG(X1, X2, X4)
16.14/6.86	   REACHB_IN_GGG(X1, X2, X3) -> U4_GGG(X1, X2, X3, member1C_in_gag(X1, X4, X3))
16.14/6.86	   REACHB_IN_GGG(X1, X2, X3) -> MEMBER1C_IN_GAG(X1, X4, X3)
16.14/6.86	   MEMBER1C_IN_GAG(X1, X2, .(X3, X4)) -> U2_GAG(X1, X2, X3, X4, member1C_in_gag(X1, X2, X4))
16.14/6.86	   MEMBER1C_IN_GAG(X1, X2, .(X3, X4)) -> MEMBER1C_IN_GAG(X1, X2, X4)
16.14/6.86	   REACHB_IN_GGG(X1, X2, X3) -> U5_GGG(X1, X2, X3, member1cC_in_gag(X1, X4, X3))
16.14/6.86	   U5_GGG(X1, X2, X3, member1cC_out_gag(X1, X4, X3)) -> U6_GGG(X1, X2, X3, reachB_in_ggg(X4, X2, X3))
16.14/6.86	   U5_GGG(X1, X2, X3, member1cC_out_gag(X1, X4, X3)) -> REACHB_IN_GGG(X4, X2, X3)
16.14/6.86	
16.14/6.86	The TRS R consists of the following rules:
16.14/6.86	
16.14/6.86	   member1cC_in_gag(X1, X2, .(.(X1, .(X2, [])), X3)) -> member1cC_out_gag(X1, X2, .(.(X1, .(X2, [])), X3))
16.14/6.86	   member1cC_in_gag(X1, X2, .(X3, X4)) -> U12_gag(X1, X2, X3, X4, member1cC_in_gag(X1, X2, X4))
16.14/6.86	   U12_gag(X1, X2, X3, X4, member1cC_out_gag(X1, X2, X4)) -> member1cC_out_gag(X1, X2, .(X3, X4))
16.14/6.86	
16.14/6.86	The argument filtering Pi contains the following mapping:
16.14/6.86	reachB_in_ggg(x1, x2, x3)  =  reachB_in_ggg(x1, x2, x3)
16.14/6.86	
16.14/6.86	.(x1, x2)  =  .(x1, x2)
16.14/6.86	
16.14/6.86	memberA_in_ggg(x1, x2, x3)  =  memberA_in_ggg(x1, x2, x3)
16.14/6.86	
16.14/6.86	member1C_in_gag(x1, x2, x3)  =  member1C_in_gag(x1, x3)
16.14/6.86	
16.14/6.86	member1cC_in_gag(x1, x2, x3)  =  member1cC_in_gag(x1, x3)
16.14/6.86	
16.14/6.86	[]  =  []
16.14/6.86	
16.14/6.86	member1cC_out_gag(x1, x2, x3)  =  member1cC_out_gag(x1, x2, x3)
16.14/6.86	
16.14/6.86	U12_gag(x1, x2, x3, x4, x5)  =  U12_gag(x1, x3, x4, x5)
16.14/6.86	
16.14/6.86	REACHB_IN_GGG(x1, x2, x3)  =  REACHB_IN_GGG(x1, x2, x3)
16.14/6.86	
16.14/6.86	U3_GGG(x1, x2, x3, x4, x5)  =  U3_GGG(x1, x2, x3, x4, x5)
16.14/6.86	
16.14/6.86	MEMBERA_IN_GGG(x1, x2, x3)  =  MEMBERA_IN_GGG(x1, x2, x3)
16.14/6.86	
16.14/6.86	U1_GGG(x1, x2, x3, x4, x5)  =  U1_GGG(x1, x2, x3, x4, x5)
16.14/6.86	
16.14/6.86	U4_GGG(x1, x2, x3, x4)  =  U4_GGG(x1, x2, x3, x4)
16.14/6.86	
16.14/6.86	MEMBER1C_IN_GAG(x1, x2, x3)  =  MEMBER1C_IN_GAG(x1, x3)
16.14/6.86	
16.14/6.86	U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x1, x3, x4, x5)
16.14/6.86	
16.14/6.86	U5_GGG(x1, x2, x3, x4)  =  U5_GGG(x1, x2, x3, x4)
16.14/6.86	
16.14/6.86	U6_GGG(x1, x2, x3, x4)  =  U6_GGG(x1, x2, x3, x4)
16.14/6.86	
16.14/6.86	
16.14/6.86	We have to consider all (P,R,Pi)-chains
16.14/6.86	
16.14/6.86	
16.14/6.86	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
16.14/6.86	
16.14/6.86	
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(85)
16.14/6.86	Obligation:
16.14/6.86	Pi DP problem:
16.14/6.86	The TRS P consists of the following rules:
16.14/6.86	
16.14/6.86	   REACHB_IN_GGG(X1, X2, .(X3, X4)) -> U3_GGG(X1, X2, X3, X4, memberA_in_ggg(X1, X2, X4))
16.14/6.86	   REACHB_IN_GGG(X1, X2, .(X3, X4)) -> MEMBERA_IN_GGG(X1, X2, X4)
16.14/6.86	   MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> U1_GGG(X1, X2, X3, X4, memberA_in_ggg(X1, X2, X4))
16.14/6.86	   MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> MEMBERA_IN_GGG(X1, X2, X4)
16.14/6.86	   REACHB_IN_GGG(X1, X2, X3) -> U4_GGG(X1, X2, X3, member1C_in_gag(X1, X4, X3))
16.14/6.86	   REACHB_IN_GGG(X1, X2, X3) -> MEMBER1C_IN_GAG(X1, X4, X3)
16.14/6.86	   MEMBER1C_IN_GAG(X1, X2, .(X3, X4)) -> U2_GAG(X1, X2, X3, X4, member1C_in_gag(X1, X2, X4))
16.14/6.86	   MEMBER1C_IN_GAG(X1, X2, .(X3, X4)) -> MEMBER1C_IN_GAG(X1, X2, X4)
16.14/6.86	   REACHB_IN_GGG(X1, X2, X3) -> U5_GGG(X1, X2, X3, member1cC_in_gag(X1, X4, X3))
16.14/6.86	   U5_GGG(X1, X2, X3, member1cC_out_gag(X1, X4, X3)) -> U6_GGG(X1, X2, X3, reachB_in_ggg(X4, X2, X3))
16.14/6.86	   U5_GGG(X1, X2, X3, member1cC_out_gag(X1, X4, X3)) -> REACHB_IN_GGG(X4, X2, X3)
16.14/6.86	
16.14/6.86	The TRS R consists of the following rules:
16.14/6.86	
16.14/6.86	   member1cC_in_gag(X1, X2, .(.(X1, .(X2, [])), X3)) -> member1cC_out_gag(X1, X2, .(.(X1, .(X2, [])), X3))
16.14/6.86	   member1cC_in_gag(X1, X2, .(X3, X4)) -> U12_gag(X1, X2, X3, X4, member1cC_in_gag(X1, X2, X4))
16.14/6.86	   U12_gag(X1, X2, X3, X4, member1cC_out_gag(X1, X2, X4)) -> member1cC_out_gag(X1, X2, .(X3, X4))
16.14/6.86	
16.14/6.86	The argument filtering Pi contains the following mapping:
16.14/6.86	reachB_in_ggg(x1, x2, x3)  =  reachB_in_ggg(x1, x2, x3)
16.14/6.86	
16.14/6.86	.(x1, x2)  =  .(x1, x2)
16.14/6.86	
16.14/6.86	memberA_in_ggg(x1, x2, x3)  =  memberA_in_ggg(x1, x2, x3)
16.14/6.86	
16.14/6.86	member1C_in_gag(x1, x2, x3)  =  member1C_in_gag(x1, x3)
16.14/6.86	
16.14/6.86	member1cC_in_gag(x1, x2, x3)  =  member1cC_in_gag(x1, x3)
16.14/6.86	
16.14/6.86	[]  =  []
16.14/6.86	
16.14/6.86	member1cC_out_gag(x1, x2, x3)  =  member1cC_out_gag(x1, x2, x3)
16.14/6.86	
16.14/6.86	U12_gag(x1, x2, x3, x4, x5)  =  U12_gag(x1, x3, x4, x5)
16.14/6.86	
16.14/6.86	REACHB_IN_GGG(x1, x2, x3)  =  REACHB_IN_GGG(x1, x2, x3)
16.14/6.86	
16.14/6.86	U3_GGG(x1, x2, x3, x4, x5)  =  U3_GGG(x1, x2, x3, x4, x5)
16.14/6.86	
16.14/6.86	MEMBERA_IN_GGG(x1, x2, x3)  =  MEMBERA_IN_GGG(x1, x2, x3)
16.14/6.86	
16.14/6.86	U1_GGG(x1, x2, x3, x4, x5)  =  U1_GGG(x1, x2, x3, x4, x5)
16.14/6.86	
16.14/6.86	U4_GGG(x1, x2, x3, x4)  =  U4_GGG(x1, x2, x3, x4)
16.14/6.86	
16.14/6.86	MEMBER1C_IN_GAG(x1, x2, x3)  =  MEMBER1C_IN_GAG(x1, x3)
16.14/6.86	
16.14/6.86	U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x1, x3, x4, x5)
16.14/6.86	
16.14/6.86	U5_GGG(x1, x2, x3, x4)  =  U5_GGG(x1, x2, x3, x4)
16.14/6.86	
16.14/6.86	U6_GGG(x1, x2, x3, x4)  =  U6_GGG(x1, x2, x3, x4)
16.14/6.86	
16.14/6.86	
16.14/6.86	We have to consider all (P,R,Pi)-chains
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(86) DependencyGraphProof (EQUIVALENT)
16.14/6.86	The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 7 less nodes.
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(87)
16.14/6.86	Complex Obligation (AND)
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(88)
16.14/6.86	Obligation:
16.14/6.86	Pi DP problem:
16.14/6.86	The TRS P consists of the following rules:
16.14/6.86	
16.14/6.86	   MEMBER1C_IN_GAG(X1, X2, .(X3, X4)) -> MEMBER1C_IN_GAG(X1, X2, X4)
16.14/6.86	
16.14/6.86	The TRS R consists of the following rules:
16.14/6.86	
16.14/6.86	   member1cC_in_gag(X1, X2, .(.(X1, .(X2, [])), X3)) -> member1cC_out_gag(X1, X2, .(.(X1, .(X2, [])), X3))
16.14/6.86	   member1cC_in_gag(X1, X2, .(X3, X4)) -> U12_gag(X1, X2, X3, X4, member1cC_in_gag(X1, X2, X4))
16.14/6.86	   U12_gag(X1, X2, X3, X4, member1cC_out_gag(X1, X2, X4)) -> member1cC_out_gag(X1, X2, .(X3, X4))
16.14/6.86	
16.14/6.86	The argument filtering Pi contains the following mapping:
16.14/6.86	.(x1, x2)  =  .(x1, x2)
16.14/6.86	
16.14/6.86	member1cC_in_gag(x1, x2, x3)  =  member1cC_in_gag(x1, x3)
16.14/6.86	
16.14/6.86	[]  =  []
16.14/6.86	
16.14/6.86	member1cC_out_gag(x1, x2, x3)  =  member1cC_out_gag(x1, x2, x3)
16.14/6.86	
16.14/6.86	U12_gag(x1, x2, x3, x4, x5)  =  U12_gag(x1, x3, x4, x5)
16.14/6.86	
16.14/6.86	MEMBER1C_IN_GAG(x1, x2, x3)  =  MEMBER1C_IN_GAG(x1, x3)
16.14/6.86	
16.14/6.86	
16.14/6.86	We have to consider all (P,R,Pi)-chains
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(89) UsableRulesProof (EQUIVALENT)
16.14/6.86	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(90)
16.14/6.86	Obligation:
16.14/6.86	Pi DP problem:
16.14/6.86	The TRS P consists of the following rules:
16.14/6.86	
16.14/6.86	   MEMBER1C_IN_GAG(X1, X2, .(X3, X4)) -> MEMBER1C_IN_GAG(X1, X2, X4)
16.14/6.86	
16.14/6.86	R is empty.
16.14/6.86	The argument filtering Pi contains the following mapping:
16.14/6.86	.(x1, x2)  =  .(x1, x2)
16.14/6.86	
16.14/6.86	MEMBER1C_IN_GAG(x1, x2, x3)  =  MEMBER1C_IN_GAG(x1, x3)
16.14/6.86	
16.14/6.86	
16.14/6.86	We have to consider all (P,R,Pi)-chains
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(91) PiDPToQDPProof (SOUND)
16.14/6.86	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(92)
16.14/6.86	Obligation:
16.14/6.86	Q DP problem:
16.14/6.86	The TRS P consists of the following rules:
16.14/6.86	
16.14/6.86	   MEMBER1C_IN_GAG(X1, .(X3, X4)) -> MEMBER1C_IN_GAG(X1, X4)
16.14/6.86	
16.14/6.86	R is empty.
16.14/6.86	Q is empty.
16.14/6.86	We have to consider all (P,Q,R)-chains.
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(93) QDPSizeChangeProof (EQUIVALENT)
16.14/6.86	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. 
16.14/6.86	
16.14/6.86	From the DPs we obtained the following set of size-change graphs:
16.14/6.86	*MEMBER1C_IN_GAG(X1, .(X3, X4)) -> MEMBER1C_IN_GAG(X1, X4)
16.14/6.86	The graph contains the following edges 1 >= 1, 2 > 2
16.14/6.86	
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(94)
16.14/6.86	YES
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(95)
16.14/6.86	Obligation:
16.14/6.86	Pi DP problem:
16.14/6.86	The TRS P consists of the following rules:
16.14/6.86	
16.14/6.86	   MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> MEMBERA_IN_GGG(X1, X2, X4)
16.14/6.86	
16.14/6.86	The TRS R consists of the following rules:
16.14/6.86	
16.14/6.86	   member1cC_in_gag(X1, X2, .(.(X1, .(X2, [])), X3)) -> member1cC_out_gag(X1, X2, .(.(X1, .(X2, [])), X3))
16.14/6.86	   member1cC_in_gag(X1, X2, .(X3, X4)) -> U12_gag(X1, X2, X3, X4, member1cC_in_gag(X1, X2, X4))
16.14/6.86	   U12_gag(X1, X2, X3, X4, member1cC_out_gag(X1, X2, X4)) -> member1cC_out_gag(X1, X2, .(X3, X4))
16.14/6.86	
16.14/6.86	The argument filtering Pi contains the following mapping:
16.14/6.86	.(x1, x2)  =  .(x1, x2)
16.14/6.86	
16.14/6.86	member1cC_in_gag(x1, x2, x3)  =  member1cC_in_gag(x1, x3)
16.14/6.86	
16.14/6.86	[]  =  []
16.14/6.86	
16.14/6.86	member1cC_out_gag(x1, x2, x3)  =  member1cC_out_gag(x1, x2, x3)
16.14/6.86	
16.14/6.86	U12_gag(x1, x2, x3, x4, x5)  =  U12_gag(x1, x3, x4, x5)
16.14/6.86	
16.14/6.86	MEMBERA_IN_GGG(x1, x2, x3)  =  MEMBERA_IN_GGG(x1, x2, x3)
16.14/6.86	
16.14/6.86	
16.14/6.86	We have to consider all (P,R,Pi)-chains
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(96) UsableRulesProof (EQUIVALENT)
16.14/6.86	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(97)
16.14/6.86	Obligation:
16.14/6.86	Pi DP problem:
16.14/6.86	The TRS P consists of the following rules:
16.14/6.86	
16.14/6.86	   MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> MEMBERA_IN_GGG(X1, X2, X4)
16.14/6.86	
16.14/6.86	R is empty.
16.14/6.86	Pi is empty.
16.14/6.86	We have to consider all (P,R,Pi)-chains
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(98) PiDPToQDPProof (EQUIVALENT)
16.14/6.86	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(99)
16.14/6.86	Obligation:
16.14/6.86	Q DP problem:
16.14/6.86	The TRS P consists of the following rules:
16.14/6.86	
16.14/6.86	   MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> MEMBERA_IN_GGG(X1, X2, X4)
16.14/6.86	
16.14/6.86	R is empty.
16.14/6.86	Q is empty.
16.14/6.86	We have to consider all (P,Q,R)-chains.
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(100) QDPSizeChangeProof (EQUIVALENT)
16.14/6.86	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. 
16.14/6.86	
16.14/6.86	From the DPs we obtained the following set of size-change graphs:
16.14/6.86	*MEMBERA_IN_GGG(X1, X2, .(X3, X4)) -> MEMBERA_IN_GGG(X1, X2, X4)
16.14/6.86	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
16.14/6.86	
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(101)
16.14/6.86	YES
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(102)
16.14/6.86	Obligation:
16.14/6.86	Pi DP problem:
16.14/6.86	The TRS P consists of the following rules:
16.14/6.86	
16.14/6.86	   REACHB_IN_GGG(X1, X2, X3) -> U5_GGG(X1, X2, X3, member1cC_in_gag(X1, X4, X3))
16.14/6.86	   U5_GGG(X1, X2, X3, member1cC_out_gag(X1, X4, X3)) -> REACHB_IN_GGG(X4, X2, X3)
16.14/6.86	
16.14/6.86	The TRS R consists of the following rules:
16.14/6.86	
16.14/6.86	   member1cC_in_gag(X1, X2, .(.(X1, .(X2, [])), X3)) -> member1cC_out_gag(X1, X2, .(.(X1, .(X2, [])), X3))
16.14/6.86	   member1cC_in_gag(X1, X2, .(X3, X4)) -> U12_gag(X1, X2, X3, X4, member1cC_in_gag(X1, X2, X4))
16.14/6.86	   U12_gag(X1, X2, X3, X4, member1cC_out_gag(X1, X2, X4)) -> member1cC_out_gag(X1, X2, .(X3, X4))
16.14/6.86	
16.14/6.86	The argument filtering Pi contains the following mapping:
16.14/6.86	.(x1, x2)  =  .(x1, x2)
16.14/6.86	
16.14/6.86	member1cC_in_gag(x1, x2, x3)  =  member1cC_in_gag(x1, x3)
16.14/6.86	
16.14/6.86	[]  =  []
16.14/6.86	
16.14/6.86	member1cC_out_gag(x1, x2, x3)  =  member1cC_out_gag(x1, x2, x3)
16.14/6.86	
16.14/6.86	U12_gag(x1, x2, x3, x4, x5)  =  U12_gag(x1, x3, x4, x5)
16.14/6.86	
16.14/6.86	REACHB_IN_GGG(x1, x2, x3)  =  REACHB_IN_GGG(x1, x2, x3)
16.14/6.86	
16.14/6.86	U5_GGG(x1, x2, x3, x4)  =  U5_GGG(x1, x2, x3, x4)
16.14/6.86	
16.14/6.86	
16.14/6.86	We have to consider all (P,R,Pi)-chains
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(103) PiDPToQDPProof (SOUND)
16.14/6.86	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(104)
16.14/6.86	Obligation:
16.14/6.86	Q DP problem:
16.14/6.86	The TRS P consists of the following rules:
16.14/6.86	
16.14/6.86	   REACHB_IN_GGG(X1, X2, X3) -> U5_GGG(X1, X2, X3, member1cC_in_gag(X1, X3))
16.14/6.86	   U5_GGG(X1, X2, X3, member1cC_out_gag(X1, X4, X3)) -> REACHB_IN_GGG(X4, X2, X3)
16.14/6.86	
16.14/6.86	The TRS R consists of the following rules:
16.14/6.86	
16.14/6.86	   member1cC_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cC_out_gag(X1, X2, .(.(X1, .(X2, [])), X3))
16.14/6.86	   member1cC_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cC_in_gag(X1, X4))
16.14/6.86	   U12_gag(X1, X3, X4, member1cC_out_gag(X1, X2, X4)) -> member1cC_out_gag(X1, X2, .(X3, X4))
16.14/6.86	
16.14/6.86	The set Q consists of the following terms:
16.14/6.86	
16.14/6.86	   member1cC_in_gag(x0, x1)
16.14/6.86	   U12_gag(x0, x1, x2, x3)
16.14/6.86	
16.14/6.86	We have to consider all (P,Q,R)-chains.
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(105) TransformationProof (SOUND)
16.14/6.86	By narrowing [LPAR04] the rule REACHB_IN_GGG(X1, X2, X3) -> U5_GGG(X1, X2, X3, member1cC_in_gag(X1, X3)) at position [3] we obtained the following new rules [LPAR04]:
16.14/6.86	
16.14/6.86	   (REACHB_IN_GGG(x0, y1, .(.(x0, .(x1, [])), x2)) -> U5_GGG(x0, y1, .(.(x0, .(x1, [])), x2), member1cC_out_gag(x0, x1, .(.(x0, .(x1, [])), x2))),REACHB_IN_GGG(x0, y1, .(.(x0, .(x1, [])), x2)) -> U5_GGG(x0, y1, .(.(x0, .(x1, [])), x2), member1cC_out_gag(x0, x1, .(.(x0, .(x1, [])), x2))))
16.14/6.86	   (REACHB_IN_GGG(x0, y1, .(x1, x2)) -> U5_GGG(x0, y1, .(x1, x2), U12_gag(x0, x1, x2, member1cC_in_gag(x0, x2))),REACHB_IN_GGG(x0, y1, .(x1, x2)) -> U5_GGG(x0, y1, .(x1, x2), U12_gag(x0, x1, x2, member1cC_in_gag(x0, x2))))
16.14/6.86	
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(106)
16.14/6.86	Obligation:
16.14/6.86	Q DP problem:
16.14/6.86	The TRS P consists of the following rules:
16.14/6.86	
16.14/6.86	   U5_GGG(X1, X2, X3, member1cC_out_gag(X1, X4, X3)) -> REACHB_IN_GGG(X4, X2, X3)
16.14/6.86	   REACHB_IN_GGG(x0, y1, .(.(x0, .(x1, [])), x2)) -> U5_GGG(x0, y1, .(.(x0, .(x1, [])), x2), member1cC_out_gag(x0, x1, .(.(x0, .(x1, [])), x2)))
16.14/6.86	   REACHB_IN_GGG(x0, y1, .(x1, x2)) -> U5_GGG(x0, y1, .(x1, x2), U12_gag(x0, x1, x2, member1cC_in_gag(x0, x2)))
16.14/6.86	
16.14/6.86	The TRS R consists of the following rules:
16.14/6.86	
16.14/6.86	   member1cC_in_gag(X1, .(.(X1, .(X2, [])), X3)) -> member1cC_out_gag(X1, X2, .(.(X1, .(X2, [])), X3))
16.14/6.86	   member1cC_in_gag(X1, .(X3, X4)) -> U12_gag(X1, X3, X4, member1cC_in_gag(X1, X4))
16.14/6.86	   U12_gag(X1, X3, X4, member1cC_out_gag(X1, X2, X4)) -> member1cC_out_gag(X1, X2, .(X3, X4))
16.14/6.86	
16.14/6.86	The set Q consists of the following terms:
16.14/6.86	
16.14/6.86	   member1cC_in_gag(x0, x1)
16.14/6.86	   U12_gag(x0, x1, x2, x3)
16.14/6.86	
16.14/6.86	We have to consider all (P,Q,R)-chains.
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(107) PrologToIRSwTTransformerProof (SOUND)
16.14/6.86	Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert
16.14/6.86	
16.14/6.86	{
16.14/6.86	    "root": 2,
16.14/6.86	    "program": {
16.14/6.86	        "directives": [],
16.14/6.86	        "clauses": [
16.14/6.86	            [
16.14/6.86	                "(reach X Y Edges)",
16.14/6.86	                "(member (. X (. Y ([]))) Edges)"
16.14/6.86	            ],
16.14/6.86	            [
16.14/6.86	                "(reach X Z Edges)",
16.14/6.86	                "(',' (member1 (. X (. Y ([]))) Edges) (reach Y Z Edges))"
16.14/6.86	            ],
16.14/6.86	            [
16.14/6.86	                "(member H (. H L))",
16.14/6.86	                null
16.14/6.86	            ],
16.14/6.86	            [
16.14/6.86	                "(member X (. H L))",
16.14/6.86	                "(member X L)"
16.14/6.86	            ],
16.14/6.86	            [
16.14/6.86	                "(member1 H (. H L))",
16.14/6.86	                null
16.14/6.86	            ],
16.14/6.86	            [
16.14/6.86	                "(member1 X (. H L))",
16.14/6.86	                "(member1 X L)"
16.14/6.86	            ]
16.14/6.86	        ]
16.14/6.86	    },
16.14/6.86	    "graph": {
16.14/6.86	        "nodes": {
16.14/6.86	            "type": "Nodes",
16.14/6.86	            "275": {
16.14/6.86	                "goal": [{
16.14/6.86	                    "clause": -1,
16.14/6.86	                    "scope": -1,
16.14/6.86	                    "term": "(member1 (. T119 (. X106 ([]))) T121)"
16.14/6.86	                }],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [
16.14/6.86	                        "T119",
16.14/6.86	                        "T121"
16.14/6.86	                    ],
16.14/6.86	                    "free": ["X106"],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "111": {
16.14/6.86	                "goal": [{
16.14/6.86	                    "clause": -1,
16.14/6.86	                    "scope": -1,
16.14/6.86	                    "term": "(member1 (. T82 (. X67 ([]))) T84)"
16.14/6.86	                }],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [
16.14/6.86	                        "T82",
16.14/6.86	                        "T84"
16.14/6.86	                    ],
16.14/6.86	                    "free": ["X67"],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "276": {
16.14/6.86	                "goal": [],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "113": {
16.14/6.86	                "goal": [{
16.14/6.86	                    "clause": -1,
16.14/6.86	                    "scope": -1,
16.14/6.86	                    "term": "(reach T91 T83 T84)"
16.14/6.86	                }],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [
16.14/6.86	                        "T83",
16.14/6.86	                        "T84",
16.14/6.86	                        "T91"
16.14/6.86	                    ],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "212": {
16.14/6.86	                "goal": [{
16.14/6.86	                    "clause": 4,
16.14/6.86	                    "scope": 3,
16.14/6.86	                    "term": "(member1 (. T82 (. X67 ([]))) T84)"
16.14/6.86	                }],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [
16.14/6.86	                        "T82",
16.14/6.86	                        "T84"
16.14/6.86	                    ],
16.14/6.86	                    "free": ["X67"],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "214": {
16.14/6.86	                "goal": [{
16.14/6.86	                    "clause": 5,
16.14/6.86	                    "scope": 3,
16.14/6.86	                    "term": "(member1 (. T82 (. X67 ([]))) T84)"
16.14/6.86	                }],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [
16.14/6.86	                        "T82",
16.14/6.86	                        "T84"
16.14/6.86	                    ],
16.14/6.86	                    "free": ["X67"],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "216": {
16.14/6.86	                "goal": [{
16.14/6.86	                    "clause": -1,
16.14/6.86	                    "scope": -1,
16.14/6.86	                    "term": "(true)"
16.14/6.86	                }],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "217": {
16.14/6.86	                "goal": [],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "218": {
16.14/6.86	                "goal": [],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "51": {
16.14/6.86	                "goal": [{
16.14/6.86	                    "clause": -1,
16.14/6.86	                    "scope": -1,
16.14/6.86	                    "term": "(',' (member1 (. T82 (. X67 ([]))) T84) (reach X67 T83 T84))"
16.14/6.86	                }],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [
16.14/6.86	                        "T82",
16.14/6.86	                        "T83",
16.14/6.86	                        "T84"
16.14/6.86	                    ],
16.14/6.86	                    "free": ["X67"],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "34": {
16.14/6.86	                "goal": [{
16.14/6.86	                    "clause": -1,
16.14/6.86	                    "scope": -1,
16.14/6.86	                    "term": "(member (. T28 (. T29 ([]))) T30)"
16.14/6.86	                }],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [
16.14/6.86	                        "T28",
16.14/6.86	                        "T29",
16.14/6.86	                        "T30"
16.14/6.86	                    ],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "35": {
16.14/6.86	                "goal": [
16.14/6.86	                    {
16.14/6.86	                        "clause": 2,
16.14/6.86	                        "scope": 2,
16.14/6.86	                        "term": "(member (. T28 (. T29 ([]))) T30)"
16.14/6.86	                    },
16.14/6.86	                    {
16.14/6.86	                        "clause": 3,
16.14/6.86	                        "scope": 2,
16.14/6.86	                        "term": "(member (. T28 (. T29 ([]))) T30)"
16.14/6.86	                    }
16.14/6.86	                ],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [
16.14/6.86	                        "T28",
16.14/6.86	                        "T29",
16.14/6.86	                        "T30"
16.14/6.86	                    ],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "36": {
16.14/6.86	                "goal": [{
16.14/6.86	                    "clause": 2,
16.14/6.86	                    "scope": 2,
16.14/6.86	                    "term": "(member (. T28 (. T29 ([]))) T30)"
16.14/6.86	                }],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [
16.14/6.86	                        "T28",
16.14/6.86	                        "T29",
16.14/6.86	                        "T30"
16.14/6.86	                    ],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "37": {
16.14/6.86	                "goal": [{
16.14/6.86	                    "clause": 3,
16.14/6.86	                    "scope": 2,
16.14/6.86	                    "term": "(member (. T28 (. T29 ([]))) T30)"
16.14/6.86	                }],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [
16.14/6.86	                        "T28",
16.14/6.86	                        "T29",
16.14/6.86	                        "T30"
16.14/6.86	                    ],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "38": {
16.14/6.86	                "goal": [{
16.14/6.86	                    "clause": -1,
16.14/6.86	                    "scope": -1,
16.14/6.86	                    "term": "(true)"
16.14/6.86	                }],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "17": {
16.14/6.86	                "goal": [{
16.14/6.86	                    "clause": 0,
16.14/6.86	                    "scope": 1,
16.14/6.86	                    "term": "(reach T1 T2 T3)"
16.14/6.86	                }],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [
16.14/6.86	                        "T1",
16.14/6.86	                        "T2",
16.14/6.86	                        "T3"
16.14/6.86	                    ],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "39": {
16.14/6.86	                "goal": [],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "18": {
16.14/6.86	                "goal": [{
16.14/6.86	                    "clause": 1,
16.14/6.86	                    "scope": 1,
16.14/6.86	                    "term": "(reach T1 T2 T3)"
16.14/6.86	                }],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [
16.14/6.86	                        "T1",
16.14/6.86	                        "T2",
16.14/6.86	                        "T3"
16.14/6.86	                    ],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "2": {
16.14/6.86	                "goal": [{
16.14/6.86	                    "clause": -1,
16.14/6.86	                    "scope": -1,
16.14/6.86	                    "term": "(reach T1 T2 T3)"
16.14/6.86	                }],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [
16.14/6.86	                        "T1",
16.14/6.86	                        "T2",
16.14/6.86	                        "T3"
16.14/6.86	                    ],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "4": {
16.14/6.86	                "goal": [
16.14/6.86	                    {
16.14/6.86	                        "clause": 0,
16.14/6.86	                        "scope": 1,
16.14/6.86	                        "term": "(reach T1 T2 T3)"
16.14/6.86	                    },
16.14/6.86	                    {
16.14/6.86	                        "clause": 1,
16.14/6.86	                        "scope": 1,
16.14/6.86	                        "term": "(reach T1 T2 T3)"
16.14/6.86	                    }
16.14/6.86	                ],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [
16.14/6.86	                        "T1",
16.14/6.86	                        "T2",
16.14/6.86	                        "T3"
16.14/6.86	                    ],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "208": {
16.14/6.86	                "goal": [
16.14/6.86	                    {
16.14/6.86	                        "clause": 4,
16.14/6.86	                        "scope": 3,
16.14/6.86	                        "term": "(member1 (. T82 (. X67 ([]))) T84)"
16.14/6.86	                    },
16.14/6.86	                    {
16.14/6.86	                        "clause": 5,
16.14/6.86	                        "scope": 3,
16.14/6.86	                        "term": "(member1 (. T82 (. X67 ([]))) T84)"
16.14/6.86	                    }
16.14/6.86	                ],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [
16.14/6.86	                        "T82",
16.14/6.86	                        "T84"
16.14/6.86	                    ],
16.14/6.86	                    "free": ["X67"],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "40": {
16.14/6.86	                "goal": [],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "41": {
16.14/6.86	                "goal": [{
16.14/6.86	                    "clause": -1,
16.14/6.86	                    "scope": -1,
16.14/6.86	                    "term": "(member (. T60 (. T61 ([]))) T63)"
16.14/6.86	                }],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [
16.14/6.86	                        "T60",
16.14/6.86	                        "T61",
16.14/6.86	                        "T63"
16.14/6.86	                    ],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            },
16.14/6.86	            "42": {
16.14/6.86	                "goal": [],
16.14/6.86	                "kb": {
16.14/6.86	                    "nonunifying": [],
16.14/6.86	                    "intvars": {},
16.14/6.86	                    "arithmetic": {
16.14/6.86	                        "type": "PlainIntegerRelationState",
16.14/6.86	                        "relations": []
16.14/6.86	                    },
16.14/6.86	                    "ground": [],
16.14/6.86	                    "free": [],
16.14/6.86	                    "exprvars": []
16.14/6.86	                }
16.14/6.86	            }
16.14/6.86	        },
16.14/6.86	        "edges": [
16.14/6.86	            {
16.14/6.86	                "from": 2,
16.14/6.86	                "to": 4,
16.14/6.86	                "label": "CASE"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 4,
16.14/6.86	                "to": 17,
16.14/6.86	                "label": "PARALLEL"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 4,
16.14/6.86	                "to": 18,
16.14/6.86	                "label": "PARALLEL"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 17,
16.14/6.86	                "to": 34,
16.14/6.86	                "label": "ONLY EVAL with clause\nreach(X21, X22, X23) :- member(.(X21, .(X22, [])), X23).\nand substitutionT1 -> T28,\nX21 -> T28,\nT2 -> T29,\nX22 -> T29,\nT3 -> T30,\nX23 -> T30"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 18,
16.14/6.86	                "to": 51,
16.14/6.86	                "label": "ONLY EVAL with clause\nreach(X64, X65, X66) :- ','(member1(.(X64, .(X67, [])), X66), reach(X67, X65, X66)).\nand substitutionT1 -> T82,\nX64 -> T82,\nT2 -> T83,\nX65 -> T83,\nT3 -> T84,\nX66 -> T84"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 34,
16.14/6.86	                "to": 35,
16.14/6.86	                "label": "CASE"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 35,
16.14/6.86	                "to": 36,
16.14/6.86	                "label": "PARALLEL"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 35,
16.14/6.86	                "to": 37,
16.14/6.86	                "label": "PARALLEL"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 36,
16.14/6.86	                "to": 38,
16.14/6.86	                "label": "EVAL with clause\nmember(X36, .(X36, X37)).\nand substitutionT28 -> T49,\nT29 -> T50,\nX36 -> .(T49, .(T50, [])),\nX37 -> T51,\nT30 -> .(.(T49, .(T50, [])), T51)"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 36,
16.14/6.86	                "to": 39,
16.14/6.86	                "label": "EVAL-BACKTRACK"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 37,
16.14/6.86	                "to": 41,
16.14/6.86	                "label": "EVAL with clause\nmember(X44, .(X45, X46)) :- member(X44, X46).\nand substitutionT28 -> T60,\nT29 -> T61,\nX44 -> .(T60, .(T61, [])),\nX45 -> T62,\nX46 -> T63,\nT30 -> .(T62, T63)"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 37,
16.14/6.86	                "to": 42,
16.14/6.86	                "label": "EVAL-BACKTRACK"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 38,
16.14/6.86	                "to": 40,
16.14/6.86	                "label": "SUCCESS"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 41,
16.14/6.86	                "to": 34,
16.14/6.86	                "label": "INSTANCE with matching:\nT28 -> T60\nT29 -> T61\nT30 -> T63"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 51,
16.14/6.86	                "to": 111,
16.14/6.86	                "label": "SPLIT 1"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 51,
16.14/6.86	                "to": 113,
16.14/6.86	                "label": "SPLIT 2\nnew knowledge:\nT82 is ground\nT91 is ground\nT84 is ground\nreplacements:X67 -> T91"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 111,
16.14/6.86	                "to": 208,
16.14/6.86	                "label": "CASE"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 113,
16.14/6.86	                "to": 2,
16.14/6.86	                "label": "INSTANCE with matching:\nT1 -> T91\nT2 -> T83\nT3 -> T84"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 208,
16.14/6.86	                "to": 212,
16.14/6.86	                "label": "PARALLEL"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 208,
16.14/6.86	                "to": 214,
16.14/6.86	                "label": "PARALLEL"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 212,
16.14/6.86	                "to": 216,
16.14/6.86	                "label": "EVAL with clause\nmember1(X92, .(X92, X93)).\nand substitutionT82 -> T110,\nX67 -> T111,\nX92 -> .(T110, .(T111, [])),\nX94 -> T111,\nX93 -> T112,\nT84 -> .(.(T110, .(T111, [])), T112)"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 212,
16.14/6.86	                "to": 217,
16.14/6.86	                "label": "EVAL-BACKTRACK"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 214,
16.14/6.86	                "to": 275,
16.14/6.86	                "label": "EVAL with clause\nmember1(X103, .(X104, X105)) :- member1(X103, X105).\nand substitutionT82 -> T119,\nX67 -> X106,\nX103 -> .(T119, .(X106, [])),\nX104 -> T120,\nX105 -> T121,\nT84 -> .(T120, T121)"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 214,
16.14/6.86	                "to": 276,
16.14/6.86	                "label": "EVAL-BACKTRACK"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 216,
16.14/6.86	                "to": 218,
16.14/6.86	                "label": "SUCCESS"
16.14/6.86	            },
16.14/6.86	            {
16.14/6.86	                "from": 275,
16.14/6.86	                "to": 111,
16.14/6.86	                "label": "INSTANCE with matching:\nT82 -> T119\nX67 -> X106\nT84 -> T121"
16.14/6.86	            }
16.14/6.86	        ],
16.14/6.86	        "type": "Graph"
16.14/6.86	    }
16.14/6.86	}
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(108)
16.14/6.86	Complex Obligation (AND)
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(109)
16.14/6.86	Obligation:
16.14/6.86	Rules:
16.14/6.86	f111_in(T82, T84) -> f208_in(T82, T84) :|: TRUE
16.14/6.86	f208_out(x, x1) -> f111_out(x, x1) :|: TRUE
16.14/6.86	f214_in(T119, .(T120, T121)) -> f275_in(T119, T121) :|: TRUE
16.14/6.86	f276_out -> f214_out(x2, x3) :|: TRUE
16.14/6.86	f214_in(x4, x5) -> f276_in :|: TRUE
16.14/6.86	f275_out(x6, x7) -> f214_out(x6, .(x8, x7)) :|: TRUE
16.14/6.86	f208_in(x9, x10) -> f214_in(x9, x10) :|: TRUE
16.14/6.86	f208_in(x11, x12) -> f212_in(x11, x12) :|: TRUE
16.14/6.86	f214_out(x13, x14) -> f208_out(x13, x14) :|: TRUE
16.14/6.86	f212_out(x15, x16) -> f208_out(x15, x16) :|: TRUE
16.14/6.86	f111_out(x17, x18) -> f275_out(x17, x18) :|: TRUE
16.14/6.86	f275_in(x19, x20) -> f111_in(x19, x20) :|: TRUE
16.14/6.86	f2_in(T1, T2, T3) -> f4_in(T1, T2, T3) :|: TRUE
16.14/6.86	f4_out(x21, x22, x23) -> f2_out(x21, x22, x23) :|: TRUE
16.14/6.86	f18_out(x24, x25, x26) -> f4_out(x24, x25, x26) :|: TRUE
16.14/6.86	f4_in(x27, x28, x29) -> f17_in(x27, x28, x29) :|: TRUE
16.14/6.86	f17_out(x30, x31, x32) -> f4_out(x30, x31, x32) :|: TRUE
16.14/6.86	f4_in(x33, x34, x35) -> f18_in(x33, x34, x35) :|: TRUE
16.14/6.86	f18_in(x36, x37, x38) -> f51_in(x36, x38, x37) :|: TRUE
16.14/6.86	f51_out(x39, x40, x41) -> f18_out(x39, x41, x40) :|: TRUE
16.14/6.86	f51_in(x42, x43, x44) -> f111_in(x42, x43) :|: TRUE
16.14/6.86	f111_out(x45, x46) -> f113_in(x47, x48, x46) :|: TRUE
16.14/6.86	f113_out(x49, x50, x51) -> f51_out(x52, x51, x50) :|: TRUE
16.14/6.86	Start term: f2_in(T1, T2, T3)
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(110) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
16.14/6.86	Constructed simple dependency graph.
16.14/6.86	
16.14/6.86	Simplified to the following IRSwTs:
16.14/6.86	
16.14/6.86	intTRSProblem:
16.14/6.86	f111_in(T82, T84) -> f208_in(T82, T84) :|: TRUE
16.14/6.86	f214_in(T119, .(T120, T121)) -> f275_in(T119, T121) :|: TRUE
16.14/6.86	f208_in(x9, x10) -> f214_in(x9, x10) :|: TRUE
16.14/6.86	f275_in(x19, x20) -> f111_in(x19, x20) :|: TRUE
16.14/6.86	
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(111)
16.14/6.86	Obligation:
16.14/6.86	Rules:
16.14/6.86	f111_in(T82, T84) -> f208_in(T82, T84) :|: TRUE
16.14/6.86	f214_in(T119, .(T120, T121)) -> f275_in(T119, T121) :|: TRUE
16.14/6.86	f208_in(x9, x10) -> f214_in(x9, x10) :|: TRUE
16.14/6.86	f275_in(x19, x20) -> f111_in(x19, x20) :|: TRUE
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(112) IntTRSCompressionProof (EQUIVALENT)
16.14/6.86	Compressed rules.
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(113)
16.14/6.86	Obligation:
16.14/6.86	Rules:
16.14/6.86	f111_in(T82:0, .(T120:0, T121:0)) -> f111_in(T82:0, T121:0) :|: TRUE
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(114) IRSFormatTransformerProof (EQUIVALENT)
16.14/6.86	Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(115)
16.14/6.86	Obligation:
16.14/6.86	Rules:
16.14/6.86	f111_in(T82:0, .(T120:0, T121:0)) -> f111_in(T82:0, T121:0) :|: TRUE
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(116) IRSwTTerminationDigraphProof (EQUIVALENT)
16.14/6.86	Constructed termination digraph!
16.14/6.86	Nodes:
16.14/6.86	(1) f111_in(T82:0, .(T120:0, T121:0)) -> f111_in(T82:0, T121:0) :|: TRUE
16.14/6.86	
16.14/6.86	Arcs:
16.14/6.86	(1) -> (1)
16.14/6.86	
16.14/6.86	This digraph is fully evaluated!
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(117)
16.14/6.86	Obligation:
16.14/6.86	
16.14/6.86	Termination digraph:
16.14/6.86	Nodes:
16.14/6.86	(1) f111_in(T82:0, .(T120:0, T121:0)) -> f111_in(T82:0, T121:0) :|: TRUE
16.14/6.86	
16.14/6.86	Arcs:
16.14/6.86	(1) -> (1)
16.14/6.86	
16.14/6.86	This digraph is fully evaluated!
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(118) IntTRSUnneededArgumentFilterProof (EQUIVALENT)
16.14/6.86	Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements:
16.14/6.86	
16.14/6.86	   f111_in(x1, x2) -> f111_in(x2)
16.14/6.86	   .(x1, x2) -> .(x2)
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(119)
16.14/6.86	Obligation:
16.14/6.86	Rules:
16.14/6.86	f111_in(.(T121:0)) -> f111_in(T121:0) :|: TRUE
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(120) TempFilterProof (SOUND)
16.14/6.86	Used the following sort dictionary for filtering: 
16.14/6.86	f111_in(VARIABLE)
16.14/6.86	.(VARIABLE)
16.14/6.86	Removed predefined arithmetic.
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(121)
16.14/6.86	Obligation:
16.14/6.86	Rules:
16.14/6.86	f111_in(.(T121:0)) -> f111_in(T121:0)
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(122) IRSwTToQDPProof (SOUND)
16.14/6.86	Removed the integers and created a QDP-Problem.
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(123)
16.14/6.86	Obligation:
16.14/6.86	Q DP problem:
16.14/6.86	The TRS P consists of the following rules:
16.14/6.86	
16.14/6.86	   f111_in(.(T121:0)) -> f111_in(T121:0)
16.14/6.86	
16.14/6.86	R is empty.
16.14/6.86	Q is empty.
16.14/6.86	We have to consider all (P,Q,R)-chains.
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(124) QDPSizeChangeProof (EQUIVALENT)
16.14/6.86	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. 
16.14/6.86	
16.14/6.86	From the DPs we obtained the following set of size-change graphs:
16.14/6.86	*f111_in(.(T121:0)) -> f111_in(T121:0)
16.14/6.86	The graph contains the following edges 1 > 1
16.14/6.86	
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(125)
16.14/6.86	YES
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(126)
16.14/6.86	Obligation:
16.14/6.86	Rules:
16.14/6.86	f37_out(T28, T29, T30) -> f35_out(T28, T29, T30) :|: TRUE
16.14/6.86	f36_out(x, x1, x2) -> f35_out(x, x1, x2) :|: TRUE
16.14/6.86	f35_in(x3, x4, x5) -> f37_in(x3, x4, x5) :|: TRUE
16.14/6.86	f35_in(x6, x7, x8) -> f36_in(x6, x7, x8) :|: TRUE
16.14/6.86	f42_out -> f37_out(x9, x10, x11) :|: TRUE
16.14/6.86	f41_out(T60, T61, T63) -> f37_out(T60, T61, .(T62, T63)) :|: TRUE
16.14/6.86	f37_in(x12, x13, x14) -> f42_in :|: TRUE
16.14/6.86	f37_in(x15, x16, .(x17, x18)) -> f41_in(x15, x16, x18) :|: TRUE
16.14/6.86	f41_in(x19, x20, x21) -> f34_in(x19, x20, x21) :|: TRUE
16.14/6.86	f34_out(x22, x23, x24) -> f41_out(x22, x23, x24) :|: TRUE
16.14/6.86	f35_out(x25, x26, x27) -> f34_out(x25, x26, x27) :|: TRUE
16.14/6.86	f34_in(x28, x29, x30) -> f35_in(x28, x29, x30) :|: TRUE
16.14/6.86	f2_in(T1, T2, T3) -> f4_in(T1, T2, T3) :|: TRUE
16.14/6.86	f4_out(x31, x32, x33) -> f2_out(x31, x32, x33) :|: TRUE
16.14/6.86	f18_out(x34, x35, x36) -> f4_out(x34, x35, x36) :|: TRUE
16.14/6.86	f4_in(x37, x38, x39) -> f17_in(x37, x38, x39) :|: TRUE
16.14/6.86	f17_out(x40, x41, x42) -> f4_out(x40, x41, x42) :|: TRUE
16.14/6.86	f4_in(x43, x44, x45) -> f18_in(x43, x44, x45) :|: TRUE
16.14/6.86	f34_out(x46, x47, x48) -> f17_out(x46, x47, x48) :|: TRUE
16.14/6.86	f17_in(x49, x50, x51) -> f34_in(x49, x50, x51) :|: TRUE
16.14/6.86	Start term: f2_in(T1, T2, T3)
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(127) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
16.14/6.86	Constructed simple dependency graph.
16.14/6.86	
16.14/6.86	Simplified to the following IRSwTs:
16.14/6.86	
16.14/6.86	intTRSProblem:
16.14/6.86	f35_in(x3, x4, x5) -> f37_in(x3, x4, x5) :|: TRUE
16.14/6.86	f37_in(x15, x16, .(x17, x18)) -> f41_in(x15, x16, x18) :|: TRUE
16.14/6.86	f41_in(x19, x20, x21) -> f34_in(x19, x20, x21) :|: TRUE
16.14/6.86	f34_in(x28, x29, x30) -> f35_in(x28, x29, x30) :|: TRUE
16.14/6.86	
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(128)
16.14/6.86	Obligation:
16.14/6.86	Rules:
16.14/6.86	f35_in(x3, x4, x5) -> f37_in(x3, x4, x5) :|: TRUE
16.14/6.86	f37_in(x15, x16, .(x17, x18)) -> f41_in(x15, x16, x18) :|: TRUE
16.14/6.86	f41_in(x19, x20, x21) -> f34_in(x19, x20, x21) :|: TRUE
16.14/6.86	f34_in(x28, x29, x30) -> f35_in(x28, x29, x30) :|: TRUE
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(129) IntTRSCompressionProof (EQUIVALENT)
16.14/6.86	Compressed rules.
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(130)
16.14/6.86	Obligation:
16.14/6.86	Rules:
16.14/6.86	f35_in(x3:0, x4:0, .(x17:0, x18:0)) -> f35_in(x3:0, x4:0, x18:0) :|: TRUE
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(131) IRSFormatTransformerProof (EQUIVALENT)
16.14/6.86	Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(132)
16.14/6.86	Obligation:
16.14/6.86	Rules:
16.14/6.86	f35_in(x3:0, x4:0, .(x17:0, x18:0)) -> f35_in(x3:0, x4:0, x18:0) :|: TRUE
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(133) IRSwTTerminationDigraphProof (EQUIVALENT)
16.14/6.86	Constructed termination digraph!
16.14/6.86	Nodes:
16.14/6.86	(1) f35_in(x3:0, x4:0, .(x17:0, x18:0)) -> f35_in(x3:0, x4:0, x18:0) :|: TRUE
16.14/6.86	
16.14/6.86	Arcs:
16.14/6.86	(1) -> (1)
16.14/6.86	
16.14/6.86	This digraph is fully evaluated!
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(134)
16.14/6.86	Obligation:
16.14/6.86	
16.14/6.86	Termination digraph:
16.14/6.86	Nodes:
16.14/6.86	(1) f35_in(x3:0, x4:0, .(x17:0, x18:0)) -> f35_in(x3:0, x4:0, x18:0) :|: TRUE
16.14/6.86	
16.14/6.86	Arcs:
16.14/6.86	(1) -> (1)
16.14/6.86	
16.14/6.86	This digraph is fully evaluated!
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(135) IntTRSUnneededArgumentFilterProof (EQUIVALENT)
16.14/6.86	Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements:
16.14/6.86	
16.14/6.86	   f35_in(x1, x2, x3) -> f35_in(x3)
16.14/6.86	   .(x1, x2) -> .(x2)
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(136)
16.14/6.86	Obligation:
16.14/6.86	Rules:
16.14/6.86	f35_in(.(x18:0)) -> f35_in(x18:0) :|: TRUE
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(137) TempFilterProof (SOUND)
16.14/6.86	Used the following sort dictionary for filtering: 
16.14/6.86	f35_in(VARIABLE)
16.14/6.86	.(VARIABLE)
16.14/6.86	Removed predefined arithmetic.
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(138)
16.14/6.86	Obligation:
16.14/6.86	Rules:
16.14/6.86	f35_in(.(x18:0)) -> f35_in(x18:0)
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(139) IRSwTToQDPProof (SOUND)
16.14/6.86	Removed the integers and created a QDP-Problem.
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(140)
16.14/6.86	Obligation:
16.14/6.86	Q DP problem:
16.14/6.86	The TRS P consists of the following rules:
16.14/6.86	
16.14/6.86	   f35_in(.(x18:0)) -> f35_in(x18:0)
16.14/6.86	
16.14/6.86	R is empty.
16.14/6.86	Q is empty.
16.14/6.86	We have to consider all (P,Q,R)-chains.
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(141) QDPSizeChangeProof (EQUIVALENT)
16.14/6.86	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. 
16.14/6.86	
16.14/6.86	From the DPs we obtained the following set of size-change graphs:
16.14/6.86	*f35_in(.(x18:0)) -> f35_in(x18:0)
16.14/6.86	The graph contains the following edges 1 > 1
16.14/6.86	
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(142)
16.14/6.86	YES
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(143)
16.14/6.86	Obligation:
16.14/6.86	Rules:
16.14/6.86	f217_out -> f212_out(T82, T84) :|: TRUE
16.14/6.86	f216_out -> f212_out(T110, .(.(T110, .(T111, [])), T112)) :|: TRUE
16.14/6.86	f212_in(x, .(.(x, .(x1, [])), x2)) -> f216_in :|: TRUE
16.14/6.86	f212_in(x3, x4) -> f217_in :|: TRUE
16.14/6.86	f51_in(x5, x6, x7) -> f111_in(x5, x6) :|: TRUE
16.14/6.86	f111_out(x8, x9) -> f113_in(x10, x11, x9) :|: TRUE
16.14/6.86	f113_out(x12, x13, x14) -> f51_out(x15, x14, x13) :|: TRUE
16.14/6.86	f111_in(x16, x17) -> f208_in(x16, x17) :|: TRUE
16.14/6.86	f208_out(x18, x19) -> f111_out(x18, x19) :|: TRUE
16.14/6.86	f214_in(T119, .(T120, T121)) -> f275_in(T119, T121) :|: TRUE
16.14/6.86	f276_out -> f214_out(x20, x21) :|: TRUE
16.14/6.86	f214_in(x22, x23) -> f276_in :|: TRUE
16.14/6.86	f275_out(x24, x25) -> f214_out(x24, .(x26, x25)) :|: TRUE
16.14/6.86	f18_in(x27, x28, x29) -> f51_in(x27, x29, x28) :|: TRUE
16.14/6.86	f51_out(x30, x31, x32) -> f18_out(x30, x32, x31) :|: TRUE
16.14/6.86	f111_out(x33, x34) -> f275_out(x33, x34) :|: TRUE
16.14/6.86	f275_in(x35, x36) -> f111_in(x35, x36) :|: TRUE
16.14/6.86	f18_out(T1, T2, T3) -> f4_out(T1, T2, T3) :|: TRUE
16.14/6.86	f4_in(x37, x38, x39) -> f17_in(x37, x38, x39) :|: TRUE
16.14/6.86	f17_out(x40, x41, x42) -> f4_out(x40, x41, x42) :|: TRUE
16.14/6.86	f4_in(x43, x44, x45) -> f18_in(x43, x44, x45) :|: TRUE
16.14/6.86	f2_out(x46, x47, x48) -> f113_out(x46, x47, x48) :|: TRUE
16.14/6.86	f113_in(x49, x50, x51) -> f2_in(x49, x50, x51) :|: TRUE
16.14/6.86	f2_in(x52, x53, x54) -> f4_in(x52, x53, x54) :|: TRUE
16.14/6.86	f4_out(x55, x56, x57) -> f2_out(x55, x56, x57) :|: TRUE
16.14/6.86	f216_in -> f216_out :|: TRUE
16.14/6.86	f208_in(x58, x59) -> f214_in(x58, x59) :|: TRUE
16.14/6.86	f208_in(x60, x61) -> f212_in(x60, x61) :|: TRUE
16.14/6.86	f214_out(x62, x63) -> f208_out(x62, x63) :|: TRUE
16.14/6.86	f212_out(x64, x65) -> f208_out(x64, x65) :|: TRUE
16.14/6.86	Start term: f2_in(T1, T2, T3)
16.14/6.86	
16.14/6.86	----------------------------------------
16.14/6.86	
16.14/6.86	(144) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
16.14/6.86	Constructed simple dependency graph.
16.14/6.86	
16.14/6.86	Simplified to the following IRSwTs:
16.14/6.86	
16.14/6.86	intTRSProblem:
16.14/6.86	f216_out -> f212_out(T110, .(.(T110, .(T111, [])), T112)) :|: TRUE
16.14/6.86	f212_in(x, .(.(x, .(x1, [])), x2)) -> f216_in :|: TRUE
16.14/6.86	f51_in(x5, x6, x7) -> f111_in(x5, x6) :|: TRUE
16.14/6.86	f111_out(x8, x9) -> f113_in(x10, x11, x9) :|: TRUE
16.14/6.86	f111_in(x16, x17) -> f208_in(x16, x17) :|: TRUE
16.14/6.86	f208_out(x18, x19) -> f111_out(x18, x19) :|: TRUE
16.14/6.86	f214_in(T119, .(T120, T121)) -> f275_in(T119, T121) :|: TRUE
16.14/6.87	f275_out(x24, x25) -> f214_out(x24, .(x26, x25)) :|: TRUE
16.14/6.87	f18_in(x27, x28, x29) -> f51_in(x27, x29, x28) :|: TRUE
16.14/6.87	f111_out(x33, x34) -> f275_out(x33, x34) :|: TRUE
16.14/6.87	f275_in(x35, x36) -> f111_in(x35, x36) :|: TRUE
16.14/6.87	f4_in(x43, x44, x45) -> f18_in(x43, x44, x45) :|: TRUE
16.14/6.87	f113_in(x49, x50, x51) -> f2_in(x49, x50, x51) :|: TRUE
16.14/6.87	f2_in(x52, x53, x54) -> f4_in(x52, x53, x54) :|: TRUE
16.14/6.87	f216_in -> f216_out :|: TRUE
16.14/6.87	f208_in(x58, x59) -> f214_in(x58, x59) :|: TRUE
16.14/6.87	f208_in(x60, x61) -> f212_in(x60, x61) :|: TRUE
16.14/6.87	f214_out(x62, x63) -> f208_out(x62, x63) :|: TRUE
16.14/6.87	f212_out(x64, x65) -> f208_out(x64, x65) :|: TRUE
16.14/6.87	
16.14/6.87	
16.14/6.87	----------------------------------------
16.14/6.87	
16.14/6.87	(145)
16.14/6.87	Obligation:
16.14/6.87	Rules:
16.14/6.87	f216_out -> f212_out(T110, .(.(T110, .(T111, [])), T112)) :|: TRUE
16.14/6.87	f212_in(x, .(.(x, .(x1, [])), x2)) -> f216_in :|: TRUE
16.14/6.87	f51_in(x5, x6, x7) -> f111_in(x5, x6) :|: TRUE
16.14/6.87	f111_out(x8, x9) -> f113_in(x10, x11, x9) :|: TRUE
16.14/6.87	f111_in(x16, x17) -> f208_in(x16, x17) :|: TRUE
16.14/6.87	f208_out(x18, x19) -> f111_out(x18, x19) :|: TRUE
16.14/6.87	f214_in(T119, .(T120, T121)) -> f275_in(T119, T121) :|: TRUE
16.14/6.87	f275_out(x24, x25) -> f214_out(x24, .(x26, x25)) :|: TRUE
16.14/6.87	f18_in(x27, x28, x29) -> f51_in(x27, x29, x28) :|: TRUE
16.14/6.87	f111_out(x33, x34) -> f275_out(x33, x34) :|: TRUE
16.14/6.87	f275_in(x35, x36) -> f111_in(x35, x36) :|: TRUE
16.14/6.87	f4_in(x43, x44, x45) -> f18_in(x43, x44, x45) :|: TRUE
16.14/6.87	f113_in(x49, x50, x51) -> f2_in(x49, x50, x51) :|: TRUE
16.14/6.87	f2_in(x52, x53, x54) -> f4_in(x52, x53, x54) :|: TRUE
16.14/6.87	f216_in -> f216_out :|: TRUE
16.14/6.87	f208_in(x58, x59) -> f214_in(x58, x59) :|: TRUE
16.14/6.87	f208_in(x60, x61) -> f212_in(x60, x61) :|: TRUE
16.14/6.87	f214_out(x62, x63) -> f208_out(x62, x63) :|: TRUE
16.14/6.87	f212_out(x64, x65) -> f208_out(x64, x65) :|: TRUE
16.14/6.87	
16.14/6.87	----------------------------------------
16.14/6.87	
16.14/6.87	(146) IntTRSCompressionProof (EQUIVALENT)
16.14/6.87	Compressed rules.
16.14/6.87	----------------------------------------
16.14/6.87	
16.14/6.87	(147)
16.14/6.87	Obligation:
16.14/6.87	Rules:
16.14/6.87	f208_out(x18:0, x19:0) -> f208_out(x18:0, .(x26:0, x19:0)) :|: TRUE
16.14/6.87	f208_in(x60:0, .(.(x60:0, .(x1:0, [])), x2:0)) -> f208_out(T110:0, .(.(T110:0, .(T111:0, [])), T112:0)) :|: TRUE
16.14/6.87	f208_out(x, x1) -> f208_in(x2, x1) :|: TRUE
16.14/6.87	f208_in(x58:0, .(T120:0, T121:0)) -> f208_in(x58:0, T121:0) :|: TRUE
16.14/6.87	
16.14/6.87	----------------------------------------
16.14/6.87	
16.14/6.87	(148) IRSFormatTransformerProof (EQUIVALENT)
16.14/6.87	Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
16.14/6.87	----------------------------------------
16.14/6.87	
16.14/6.87	(149)
16.14/6.87	Obligation:
16.14/6.87	Rules:
16.14/6.87	f208_out(x18:0, x19:0) -> f208_out(x18:0, .(x26:0, x19:0)) :|: TRUE
16.14/6.87	f208_in(x60:0, .(.(x60:0, .(x1:0, [])), x2:0)) -> f208_out(T110:0, .(.(T110:0, .(T111:0, [])), T112:0)) :|: TRUE
16.14/6.87	f208_out(x, x1) -> f208_in(x2, x1) :|: TRUE
16.14/6.87	f208_in(x58:0, .(T120:0, T121:0)) -> f208_in(x58:0, T121:0) :|: TRUE
16.33/7.04	EOF