10.98/3.69	MAYBE
11.13/3.72	proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl
11.13/3.72	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.13/3.72	
11.13/3.72	
11.13/3.72	Left Termination of the query pattern
11.13/3.72	
11.13/3.72	ss(a,g)
11.13/3.72	
11.13/3.72	w.r.t. the given Prolog program could not be shown:
11.13/3.72	
11.13/3.72	(0) Prolog
11.13/3.72	(1) PrologToPiTRSProof [SOUND, 0 ms]
11.13/3.72	(2) PiTRS
11.13/3.72	    (3) DependencyPairsProof [EQUIVALENT, 0 ms]
11.13/3.72	    (4) PiDP
11.13/3.72	    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
11.13/3.72	    (6) AND
11.13/3.72	        (7) PiDP
11.13/3.72	            (8) UsableRulesProof [EQUIVALENT, 0 ms]
11.13/3.72	            (9) PiDP
11.13/3.72	            (10) PiDPToQDPProof [EQUIVALENT, 0 ms]
11.13/3.72	            (11) QDP
11.13/3.72	            (12) QDPSizeChangeProof [EQUIVALENT, 1 ms]
11.13/3.72	            (13) YES
11.13/3.72	        (14) PiDP
11.13/3.72	            (15) UsableRulesProof [EQUIVALENT, 0 ms]
11.13/3.72	            (16) PiDP
11.13/3.72	            (17) PiDPToQDPProof [SOUND, 0 ms]
11.13/3.72	            (18) QDP
11.13/3.72	            (19) UsableRulesReductionPairsProof [EQUIVALENT, 17 ms]
11.13/3.72	            (20) QDP
11.13/3.72	            (21) MRRProof [EQUIVALENT, 0 ms]
11.13/3.72	            (22) QDP
11.13/3.72	            (23) PisEmptyProof [EQUIVALENT, 0 ms]
11.13/3.72	            (24) YES
11.13/3.72	        (25) PiDP
11.13/3.72	            (26) UsableRulesProof [EQUIVALENT, 0 ms]
11.13/3.72	            (27) PiDP
11.13/3.72	            (28) PiDPToQDPProof [SOUND, 0 ms]
11.13/3.72	            (29) QDP
11.13/3.72	            (30) NonTerminationLoopProof [COMPLETE, 2 ms]
11.13/3.72	            (31) NO
11.13/3.72	        (32) PiDP
11.13/3.72	            (33) UsableRulesProof [EQUIVALENT, 0 ms]
11.13/3.72	            (34) PiDP
11.13/3.72	            (35) PiDPToQDPProof [SOUND, 0 ms]
11.13/3.72	            (36) QDP
11.13/3.72	            (37) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.13/3.72	            (38) YES
11.13/3.72	(39) PrologToPiTRSProof [SOUND, 0 ms]
11.13/3.72	(40) PiTRS
11.13/3.72	    (41) DependencyPairsProof [EQUIVALENT, 0 ms]
11.13/3.72	    (42) PiDP
11.13/3.72	    (43) DependencyGraphProof [EQUIVALENT, 0 ms]
11.13/3.72	    (44) AND
11.13/3.72	        (45) PiDP
11.13/3.72	            (46) UsableRulesProof [EQUIVALENT, 0 ms]
11.13/3.72	            (47) PiDP
11.13/3.72	            (48) PiDPToQDPProof [EQUIVALENT, 0 ms]
11.13/3.72	            (49) QDP
11.13/3.72	            (50) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.13/3.72	            (51) YES
11.13/3.72	        (52) PiDP
11.13/3.72	            (53) UsableRulesProof [EQUIVALENT, 0 ms]
11.13/3.72	            (54) PiDP
11.13/3.72	            (55) PiDPToQDPProof [EQUIVALENT, 1 ms]
11.13/3.72	            (56) QDP
11.13/3.72	            (57) MRRProof [EQUIVALENT, 31 ms]
11.13/3.72	            (58) QDP
11.13/3.72	            (59) PisEmptyProof [EQUIVALENT, 0 ms]
11.13/3.72	            (60) YES
11.13/3.72	        (61) PiDP
11.13/3.72	            (62) UsableRulesProof [EQUIVALENT, 0 ms]
11.13/3.72	            (63) PiDP
11.13/3.72	            (64) PiDPToQDPProof [SOUND, 0 ms]
11.13/3.72	            (65) QDP
11.13/3.72	            (66) NonTerminationLoopProof [COMPLETE, 1 ms]
11.13/3.72	            (67) NO
11.13/3.72	        (68) PiDP
11.13/3.72	            (69) UsableRulesProof [EQUIVALENT, 0 ms]
11.13/3.72	            (70) PiDP
11.13/3.72	            (71) PiDPToQDPProof [SOUND, 0 ms]
11.13/3.72	            (72) QDP
11.13/3.72	            (73) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.13/3.72	            (74) YES
11.13/3.72	(75) PrologToDTProblemTransformerProof [SOUND, 55 ms]
11.13/3.72	(76) TRIPLES
11.13/3.72	    (77) TriplesToPiDPProof [SOUND, 0 ms]
11.13/3.72	    (78) PiDP
11.13/3.72	    (79) DependencyGraphProof [EQUIVALENT, 0 ms]
11.13/3.72	    (80) AND
11.13/3.72	        (81) PiDP
11.13/3.72	            (82) UsableRulesProof [EQUIVALENT, 0 ms]
11.13/3.72	            (83) PiDP
11.13/3.72	            (84) PiDPToQDPProof [EQUIVALENT, 0 ms]
11.13/3.72	            (85) QDP
11.13/3.72	            (86) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.13/3.72	            (87) YES
11.13/3.72	        (88) PiDP
11.13/3.72	            (89) UsableRulesProof [EQUIVALENT, 0 ms]
11.13/3.72	            (90) PiDP
11.13/3.72	            (91) PiDPToQDPProof [EQUIVALENT, 0 ms]
11.13/3.72	            (92) QDP
11.13/3.72	            (93) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.13/3.72	            (94) YES
11.13/3.72	        (95) PiDP
11.13/3.72	            (96) UsableRulesProof [EQUIVALENT, 0 ms]
11.13/3.72	            (97) PiDP
11.13/3.72	            (98) PiDPToQDPProof [SOUND, 0 ms]
11.13/3.72	            (99) QDP
11.13/3.72	            (100) NonTerminationLoopProof [COMPLETE, 7 ms]
11.13/3.72	            (101) NO
11.13/3.72	        (102) PiDP
11.13/3.72	            (103) UsableRulesProof [EQUIVALENT, 0 ms]
11.13/3.72	            (104) PiDP
11.13/3.72	            (105) PiDPToQDPProof [SOUND, 0 ms]
11.13/3.72	            (106) QDP
11.13/3.72	        (107) PiDP
11.13/3.72	            (108) UsableRulesProof [EQUIVALENT, 0 ms]
11.13/3.72	            (109) PiDP
11.13/3.72	(110) PrologToTRSTransformerProof [SOUND, 46 ms]
11.13/3.72	(111) QTRS
11.13/3.72	    (112) DependencyPairsProof [EQUIVALENT, 4 ms]
11.13/3.72	    (113) QDP
11.13/3.72	    (114) DependencyGraphProof [EQUIVALENT, 0 ms]
11.13/3.72	    (115) AND
11.13/3.72	        (116) QDP
11.13/3.72	            (117) MNOCProof [EQUIVALENT, 0 ms]
11.13/3.72	            (118) QDP
11.13/3.72	            (119) UsableRulesProof [EQUIVALENT, 0 ms]
11.13/3.72	            (120) QDP
11.13/3.72	            (121) QReductionProof [EQUIVALENT, 0 ms]
11.13/3.72	            (122) QDP
11.13/3.72	            (123) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.13/3.72	            (124) YES
11.13/3.72	        (125) QDP
11.13/3.72	            (126) MNOCProof [EQUIVALENT, 0 ms]
11.13/3.72	            (127) QDP
11.13/3.72	            (128) UsableRulesProof [EQUIVALENT, 0 ms]
11.13/3.72	            (129) QDP
11.13/3.72	            (130) QReductionProof [EQUIVALENT, 0 ms]
11.13/3.72	            (131) QDP
11.13/3.72	            (132) QDPOrderProof [EQUIVALENT, 32 ms]
11.13/3.72	            (133) QDP
11.13/3.72	            (134) DependencyGraphProof [EQUIVALENT, 0 ms]
11.13/3.72	            (135) TRUE
11.13/3.72	        (136) QDP
11.13/3.72	            (137) MNOCProof [EQUIVALENT, 0 ms]
11.13/3.72	            (138) QDP
11.13/3.72	            (139) UsableRulesProof [EQUIVALENT, 0 ms]
11.13/3.72	            (140) QDP
11.13/3.72	            (141) QReductionProof [EQUIVALENT, 0 ms]
11.13/3.72	            (142) QDP
11.13/3.72	            (143) NonTerminationLoopProof [COMPLETE, 0 ms]
11.13/3.72	            (144) NO
11.13/3.72	        (145) QDP
11.13/3.72	            (146) MNOCProof [EQUIVALENT, 0 ms]
11.13/3.72	            (147) QDP
11.13/3.72	            (148) UsableRulesProof [EQUIVALENT, 0 ms]
11.13/3.72	            (149) QDP
11.13/3.72	            (150) QReductionProof [EQUIVALENT, 0 ms]
11.13/3.72	            (151) QDP
11.13/3.72	            (152) NonTerminationLoopProof [COMPLETE, 0 ms]
11.13/3.72	            (153) NO
11.13/3.72	        (154) QDP
11.13/3.72	            (155) MNOCProof [EQUIVALENT, 0 ms]
11.13/3.72	            (156) QDP
11.13/3.72	            (157) UsableRulesProof [EQUIVALENT, 0 ms]
11.13/3.72	            (158) QDP
11.13/3.72	            (159) QReductionProof [EQUIVALENT, 0 ms]
11.13/3.72	            (160) QDP
11.13/3.72	            (161) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.13/3.72	            (162) YES
11.13/3.72	(163) PrologToIRSwTTransformerProof [SOUND, 39 ms]
11.13/3.72	(164) AND
11.13/3.72	    (165) IRSwT
11.13/3.72	        (166) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
11.13/3.72	        (167) TRUE
11.13/3.72	    (168) IRSwT
11.13/3.72	        (169) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
11.13/3.72	        (170) TRUE
11.13/3.72	    (171) IRSwT
11.13/3.72	        (172) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
11.13/3.72	        (173) IRSwT
11.13/3.72	        (174) IntTRSCompressionProof [EQUIVALENT, 20 ms]
11.13/3.72	        (175) IRSwT
11.13/3.72	        (176) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
11.13/3.72	        (177) IRSwT
11.13/3.72	        (178) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
11.13/3.72	        (179) IRSwT
11.13/3.72	        (180) FilterProof [EQUIVALENT, 0 ms]
11.13/3.72	        (181) IntTRS
11.13/3.72	        (182) IntTRSPeriodicNontermProof [COMPLETE, 1 ms]
11.13/3.72	        (183) NO
11.13/3.72	    (184) IRSwT
11.13/3.72	        (185) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
11.13/3.72	        (186) TRUE
11.13/3.72	    (187) IRSwT
11.13/3.72	        (188) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
11.13/3.72	        (189) IRSwT
11.13/3.72	        (190) IntTRSCompressionProof [EQUIVALENT, 15 ms]
11.13/3.72	        (191) IRSwT
11.13/3.72	        (192) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
11.13/3.72	        (193) IRSwT
11.13/3.72	
11.13/3.72	
11.13/3.72	----------------------------------------
11.13/3.72	
11.13/3.72	(0)
11.13/3.72	Obligation:
11.13/3.72	Clauses:
11.13/3.72	
11.13/3.72	ss(Xs, Ys) :- ','(perm(Xs, Ys), ordered(Ys)).
11.13/3.72	perm([], []).
11.13/3.72	perm(Xs, .(X, Ys)) :- ','(app(X1s, .(X, X2s), Xs), ','(app(X1s, X2s, Zs), perm(Zs, Ys))).
11.13/3.72	app([], X, X).
11.13/3.72	app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
11.13/3.72	ordered([]).
11.13/3.72	ordered(.(X1, [])).
11.13/3.72	ordered(.(X, .(Y, Xs))) :- ','(less(X, s(Y)), ordered(.(Y, Xs))).
11.13/3.72	less(0, s(X2)).
11.13/3.72	less(s(X), s(Y)) :- less(X, Y).
11.13/3.72	
11.13/3.72	
11.13/3.72	Query: ss(a,g)
11.13/3.72	----------------------------------------
11.13/3.72	
11.13/3.72	(1) PrologToPiTRSProof (SOUND)
11.13/3.72	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
11.13/3.72	
11.13/3.72	ss_in_2: (f,b)
11.13/3.72	
11.13/3.72	perm_in_2: (f,b)
11.13/3.72	
11.13/3.72	app_in_3: (f,f,f)
11.13/3.72	
11.13/3.72	ordered_in_1: (b)
11.13/3.72	
11.13/3.72	less_in_2: (b,b)
11.13/3.72	
11.13/3.72	Transforming Prolog into the following Term Rewriting System:
11.13/3.72	
11.13/3.72	Pi-finite rewrite system:
11.13/3.72	The TRS R consists of the following rules:
11.13/3.72	
11.13/3.72	   ss_in_ag(Xs, Ys) -> U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
11.13/3.72	   perm_in_ag([], []) -> perm_out_ag([], [])
11.13/3.72	   perm_in_ag(Xs, .(X, Ys)) -> U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.13/3.72	   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
11.13/3.72	   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.13/3.72	   U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
11.13/3.72	   U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.13/3.73	   U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
11.13/3.73	   U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) -> perm_out_ag(Xs, .(X, Ys))
11.13/3.73	   U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) -> U2_ag(Xs, Ys, ordered_in_g(Ys))
11.13/3.73	   ordered_in_g([]) -> ordered_out_g([])
11.13/3.73	   ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, []))
11.13/3.73	   ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	   less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2))
11.13/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y))
11.13/3.73	   U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
11.13/3.73	   U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
11.13/3.73	   U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs)))
11.13/3.73	   U2_ag(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ag(Xs, Ys)
11.13/3.73	
11.13/3.73	The argument filtering Pi contains the following mapping:
11.13/3.73	ss_in_ag(x1, x2)  =  ss_in_ag(x2)
11.13/3.73	
11.13/3.73	U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
11.13/3.73	
11.13/3.73	perm_in_ag(x1, x2)  =  perm_in_ag(x2)
11.13/3.73	
11.13/3.73	[]  =  []
11.13/3.73	
11.13/3.73	perm_out_ag(x1, x2)  =  perm_out_ag
11.13/3.73	
11.13/3.73	.(x1, x2)  =  .(x1, x2)
11.13/3.73	
11.13/3.73	U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
11.13/3.73	
11.13/3.73	app_in_aaa(x1, x2, x3)  =  app_in_aaa
11.13/3.73	
11.13/3.73	app_out_aaa(x1, x2, x3)  =  app_out_aaa
11.13/3.73	
11.13/3.73	U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
11.13/3.73	
11.13/3.73	U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
11.13/3.73	
11.13/3.73	U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
11.13/3.73	
11.13/3.73	U2_ag(x1, x2, x3)  =  U2_ag(x3)
11.13/3.73	
11.13/3.73	ordered_in_g(x1)  =  ordered_in_g(x1)
11.13/3.73	
11.13/3.73	ordered_out_g(x1)  =  ordered_out_g
11.13/3.73	
11.13/3.73	U7_g(x1, x2, x3, x4)  =  U7_g(x2, x3, x4)
11.13/3.73	
11.13/3.73	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
11.13/3.73	
11.13/3.73	0  =  0
11.13/3.73	
11.13/3.73	s(x1)  =  s(x1)
11.13/3.73	
11.13/3.73	less_out_gg(x1, x2)  =  less_out_gg
11.13/3.73	
11.13/3.73	U9_gg(x1, x2, x3)  =  U9_gg(x3)
11.13/3.73	
11.13/3.73	U8_g(x1, x2, x3, x4)  =  U8_g(x4)
11.13/3.73	
11.13/3.73	ss_out_ag(x1, x2)  =  ss_out_ag
11.13/3.73	
11.13/3.73	
11.13/3.73	
11.13/3.73	
11.13/3.73	
11.13/3.73	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
11.13/3.73	
11.13/3.73	
11.13/3.73	
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(2)
11.13/3.73	Obligation:
11.13/3.73	Pi-finite rewrite system:
11.13/3.73	The TRS R consists of the following rules:
11.13/3.73	
11.13/3.73	   ss_in_ag(Xs, Ys) -> U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
11.13/3.73	   perm_in_ag([], []) -> perm_out_ag([], [])
11.13/3.73	   perm_in_ag(Xs, .(X, Ys)) -> U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.13/3.73	   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
11.13/3.73	   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.13/3.73	   U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
11.13/3.73	   U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.13/3.73	   U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
11.13/3.73	   U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) -> perm_out_ag(Xs, .(X, Ys))
11.13/3.73	   U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) -> U2_ag(Xs, Ys, ordered_in_g(Ys))
11.13/3.73	   ordered_in_g([]) -> ordered_out_g([])
11.13/3.73	   ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, []))
11.13/3.73	   ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	   less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2))
11.13/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y))
11.13/3.73	   U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
11.13/3.73	   U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
11.13/3.73	   U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs)))
11.13/3.73	   U2_ag(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ag(Xs, Ys)
11.13/3.73	
11.13/3.73	The argument filtering Pi contains the following mapping:
11.13/3.73	ss_in_ag(x1, x2)  =  ss_in_ag(x2)
11.13/3.73	
11.13/3.73	U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
11.13/3.73	
11.13/3.73	perm_in_ag(x1, x2)  =  perm_in_ag(x2)
11.13/3.73	
11.13/3.73	[]  =  []
11.13/3.73	
11.13/3.73	perm_out_ag(x1, x2)  =  perm_out_ag
11.13/3.73	
11.13/3.73	.(x1, x2)  =  .(x1, x2)
11.13/3.73	
11.13/3.73	U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
11.13/3.73	
11.13/3.73	app_in_aaa(x1, x2, x3)  =  app_in_aaa
11.13/3.73	
11.13/3.73	app_out_aaa(x1, x2, x3)  =  app_out_aaa
11.13/3.73	
11.13/3.73	U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
11.13/3.73	
11.13/3.73	U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
11.13/3.73	
11.13/3.73	U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
11.13/3.73	
11.13/3.73	U2_ag(x1, x2, x3)  =  U2_ag(x3)
11.13/3.73	
11.13/3.73	ordered_in_g(x1)  =  ordered_in_g(x1)
11.13/3.73	
11.13/3.73	ordered_out_g(x1)  =  ordered_out_g
11.13/3.73	
11.13/3.73	U7_g(x1, x2, x3, x4)  =  U7_g(x2, x3, x4)
11.13/3.73	
11.13/3.73	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
11.13/3.73	
11.13/3.73	0  =  0
11.13/3.73	
11.13/3.73	s(x1)  =  s(x1)
11.13/3.73	
11.13/3.73	less_out_gg(x1, x2)  =  less_out_gg
11.13/3.73	
11.13/3.73	U9_gg(x1, x2, x3)  =  U9_gg(x3)
11.13/3.73	
11.13/3.73	U8_g(x1, x2, x3, x4)  =  U8_g(x4)
11.13/3.73	
11.13/3.73	ss_out_ag(x1, x2)  =  ss_out_ag
11.13/3.73	
11.13/3.73	
11.13/3.73	
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(3) DependencyPairsProof (EQUIVALENT)
11.13/3.73	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
11.13/3.73	Pi DP problem:
11.13/3.73	The TRS P consists of the following rules:
11.13/3.73	
11.13/3.73	   SS_IN_AG(Xs, Ys) -> U1_AG(Xs, Ys, perm_in_ag(Xs, Ys))
11.13/3.73	   SS_IN_AG(Xs, Ys) -> PERM_IN_AG(Xs, Ys)
11.13/3.73	   PERM_IN_AG(Xs, .(X, Ys)) -> U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.13/3.73	   PERM_IN_AG(Xs, .(X, Ys)) -> APP_IN_AAA(X1s, .(X, X2s), Xs)
11.13/3.73	   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U6_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.13/3.73	   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs)
11.13/3.73	   U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.13/3.73	   U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> APP_IN_AAA(X1s, X2s, Zs)
11.13/3.73	   U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> U5_AG(Xs, X, Ys, perm_in_ag(Zs, Ys))
11.13/3.73	   U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> PERM_IN_AG(Zs, Ys)
11.13/3.73	   U1_AG(Xs, Ys, perm_out_ag(Xs, Ys)) -> U2_AG(Xs, Ys, ordered_in_g(Ys))
11.13/3.73	   U1_AG(Xs, Ys, perm_out_ag(Xs, Ys)) -> ORDERED_IN_G(Ys)
11.13/3.73	   ORDERED_IN_G(.(X, .(Y, Xs))) -> U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	   ORDERED_IN_G(.(X, .(Y, Xs))) -> LESS_IN_GG(X, s(Y))
11.13/3.73	   LESS_IN_GG(s(X), s(Y)) -> U9_GG(X, Y, less_in_gg(X, Y))
11.13/3.73	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
11.13/3.73	   U7_G(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_G(X, Y, Xs, ordered_in_g(.(Y, Xs)))
11.13/3.73	   U7_G(X, Y, Xs, less_out_gg(X, s(Y))) -> ORDERED_IN_G(.(Y, Xs))
11.13/3.73	
11.13/3.73	The TRS R consists of the following rules:
11.13/3.73	
11.13/3.73	   ss_in_ag(Xs, Ys) -> U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
11.13/3.73	   perm_in_ag([], []) -> perm_out_ag([], [])
11.13/3.73	   perm_in_ag(Xs, .(X, Ys)) -> U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.13/3.73	   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
11.13/3.73	   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.13/3.73	   U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
11.13/3.73	   U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.13/3.73	   U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
11.13/3.73	   U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) -> perm_out_ag(Xs, .(X, Ys))
11.13/3.73	   U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) -> U2_ag(Xs, Ys, ordered_in_g(Ys))
11.13/3.73	   ordered_in_g([]) -> ordered_out_g([])
11.13/3.73	   ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, []))
11.13/3.73	   ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	   less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2))
11.13/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y))
11.13/3.73	   U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
11.13/3.73	   U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
11.13/3.73	   U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs)))
11.13/3.73	   U2_ag(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ag(Xs, Ys)
11.13/3.73	
11.13/3.73	The argument filtering Pi contains the following mapping:
11.13/3.73	ss_in_ag(x1, x2)  =  ss_in_ag(x2)
11.13/3.73	
11.13/3.73	U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
11.13/3.73	
11.13/3.73	perm_in_ag(x1, x2)  =  perm_in_ag(x2)
11.13/3.73	
11.13/3.73	[]  =  []
11.13/3.73	
11.13/3.73	perm_out_ag(x1, x2)  =  perm_out_ag
11.13/3.73	
11.13/3.73	.(x1, x2)  =  .(x1, x2)
11.13/3.73	
11.13/3.73	U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
11.13/3.73	
11.13/3.73	app_in_aaa(x1, x2, x3)  =  app_in_aaa
11.13/3.73	
11.13/3.73	app_out_aaa(x1, x2, x3)  =  app_out_aaa
11.13/3.73	
11.13/3.73	U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
11.13/3.73	
11.13/3.73	U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
11.13/3.73	
11.13/3.73	U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
11.13/3.73	
11.13/3.73	U2_ag(x1, x2, x3)  =  U2_ag(x3)
11.13/3.73	
11.13/3.73	ordered_in_g(x1)  =  ordered_in_g(x1)
11.13/3.73	
11.13/3.73	ordered_out_g(x1)  =  ordered_out_g
11.13/3.73	
11.13/3.73	U7_g(x1, x2, x3, x4)  =  U7_g(x2, x3, x4)
11.13/3.73	
11.13/3.73	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
11.13/3.73	
11.13/3.73	0  =  0
11.13/3.73	
11.13/3.73	s(x1)  =  s(x1)
11.13/3.73	
11.13/3.73	less_out_gg(x1, x2)  =  less_out_gg
11.13/3.73	
11.13/3.73	U9_gg(x1, x2, x3)  =  U9_gg(x3)
11.13/3.73	
11.13/3.73	U8_g(x1, x2, x3, x4)  =  U8_g(x4)
11.13/3.73	
11.13/3.73	ss_out_ag(x1, x2)  =  ss_out_ag
11.13/3.73	
11.13/3.73	SS_IN_AG(x1, x2)  =  SS_IN_AG(x2)
11.13/3.73	
11.13/3.73	U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
11.13/3.73	
11.13/3.73	PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
11.13/3.73	
11.13/3.73	U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
11.13/3.73	
11.13/3.73	APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
11.13/3.73	
11.13/3.73	U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)
11.13/3.73	
11.13/3.73	U4_AG(x1, x2, x3, x4)  =  U4_AG(x3, x4)
11.13/3.73	
11.13/3.73	U5_AG(x1, x2, x3, x4)  =  U5_AG(x4)
11.13/3.73	
11.13/3.73	U2_AG(x1, x2, x3)  =  U2_AG(x3)
11.13/3.73	
11.13/3.73	ORDERED_IN_G(x1)  =  ORDERED_IN_G(x1)
11.13/3.73	
11.13/3.73	U7_G(x1, x2, x3, x4)  =  U7_G(x2, x3, x4)
11.13/3.73	
11.13/3.73	LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
11.13/3.73	
11.13/3.73	U9_GG(x1, x2, x3)  =  U9_GG(x3)
11.13/3.73	
11.13/3.73	U8_G(x1, x2, x3, x4)  =  U8_G(x4)
11.13/3.73	
11.13/3.73	
11.13/3.73	We have to consider all (P,R,Pi)-chains
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(4)
11.13/3.73	Obligation:
11.13/3.73	Pi DP problem:
11.13/3.73	The TRS P consists of the following rules:
11.13/3.73	
11.13/3.73	   SS_IN_AG(Xs, Ys) -> U1_AG(Xs, Ys, perm_in_ag(Xs, Ys))
11.13/3.73	   SS_IN_AG(Xs, Ys) -> PERM_IN_AG(Xs, Ys)
11.13/3.73	   PERM_IN_AG(Xs, .(X, Ys)) -> U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.13/3.73	   PERM_IN_AG(Xs, .(X, Ys)) -> APP_IN_AAA(X1s, .(X, X2s), Xs)
11.13/3.73	   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U6_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.13/3.73	   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs)
11.13/3.73	   U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.13/3.73	   U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> APP_IN_AAA(X1s, X2s, Zs)
11.13/3.73	   U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> U5_AG(Xs, X, Ys, perm_in_ag(Zs, Ys))
11.13/3.73	   U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> PERM_IN_AG(Zs, Ys)
11.13/3.73	   U1_AG(Xs, Ys, perm_out_ag(Xs, Ys)) -> U2_AG(Xs, Ys, ordered_in_g(Ys))
11.13/3.73	   U1_AG(Xs, Ys, perm_out_ag(Xs, Ys)) -> ORDERED_IN_G(Ys)
11.13/3.73	   ORDERED_IN_G(.(X, .(Y, Xs))) -> U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	   ORDERED_IN_G(.(X, .(Y, Xs))) -> LESS_IN_GG(X, s(Y))
11.13/3.73	   LESS_IN_GG(s(X), s(Y)) -> U9_GG(X, Y, less_in_gg(X, Y))
11.13/3.73	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
11.13/3.73	   U7_G(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_G(X, Y, Xs, ordered_in_g(.(Y, Xs)))
11.13/3.73	   U7_G(X, Y, Xs, less_out_gg(X, s(Y))) -> ORDERED_IN_G(.(Y, Xs))
11.13/3.73	
11.13/3.73	The TRS R consists of the following rules:
11.13/3.73	
11.13/3.73	   ss_in_ag(Xs, Ys) -> U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
11.13/3.73	   perm_in_ag([], []) -> perm_out_ag([], [])
11.13/3.73	   perm_in_ag(Xs, .(X, Ys)) -> U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.13/3.73	   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
11.13/3.73	   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.13/3.73	   U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
11.13/3.73	   U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.13/3.73	   U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
11.13/3.73	   U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) -> perm_out_ag(Xs, .(X, Ys))
11.13/3.73	   U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) -> U2_ag(Xs, Ys, ordered_in_g(Ys))
11.13/3.73	   ordered_in_g([]) -> ordered_out_g([])
11.13/3.73	   ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, []))
11.13/3.73	   ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	   less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2))
11.13/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y))
11.13/3.73	   U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
11.13/3.73	   U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
11.13/3.73	   U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs)))
11.13/3.73	   U2_ag(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ag(Xs, Ys)
11.13/3.73	
11.13/3.73	The argument filtering Pi contains the following mapping:
11.13/3.73	ss_in_ag(x1, x2)  =  ss_in_ag(x2)
11.13/3.73	
11.13/3.73	U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
11.13/3.73	
11.13/3.73	perm_in_ag(x1, x2)  =  perm_in_ag(x2)
11.13/3.73	
11.13/3.73	[]  =  []
11.13/3.73	
11.13/3.73	perm_out_ag(x1, x2)  =  perm_out_ag
11.13/3.73	
11.13/3.73	.(x1, x2)  =  .(x1, x2)
11.13/3.73	
11.13/3.73	U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
11.13/3.73	
11.13/3.73	app_in_aaa(x1, x2, x3)  =  app_in_aaa
11.13/3.73	
11.13/3.73	app_out_aaa(x1, x2, x3)  =  app_out_aaa
11.13/3.73	
11.13/3.73	U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
11.13/3.73	
11.13/3.73	U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
11.13/3.73	
11.13/3.73	U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
11.13/3.73	
11.13/3.73	U2_ag(x1, x2, x3)  =  U2_ag(x3)
11.13/3.73	
11.13/3.73	ordered_in_g(x1)  =  ordered_in_g(x1)
11.13/3.73	
11.13/3.73	ordered_out_g(x1)  =  ordered_out_g
11.13/3.73	
11.13/3.73	U7_g(x1, x2, x3, x4)  =  U7_g(x2, x3, x4)
11.13/3.73	
11.13/3.73	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
11.13/3.73	
11.13/3.73	0  =  0
11.13/3.73	
11.13/3.73	s(x1)  =  s(x1)
11.13/3.73	
11.13/3.73	less_out_gg(x1, x2)  =  less_out_gg
11.13/3.73	
11.13/3.73	U9_gg(x1, x2, x3)  =  U9_gg(x3)
11.13/3.73	
11.13/3.73	U8_g(x1, x2, x3, x4)  =  U8_g(x4)
11.13/3.73	
11.13/3.73	ss_out_ag(x1, x2)  =  ss_out_ag
11.13/3.73	
11.13/3.73	SS_IN_AG(x1, x2)  =  SS_IN_AG(x2)
11.13/3.73	
11.13/3.73	U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
11.13/3.73	
11.13/3.73	PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
11.13/3.73	
11.13/3.73	U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
11.13/3.73	
11.13/3.73	APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
11.13/3.73	
11.13/3.73	U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)
11.13/3.73	
11.13/3.73	U4_AG(x1, x2, x3, x4)  =  U4_AG(x3, x4)
11.13/3.73	
11.13/3.73	U5_AG(x1, x2, x3, x4)  =  U5_AG(x4)
11.13/3.73	
11.13/3.73	U2_AG(x1, x2, x3)  =  U2_AG(x3)
11.13/3.73	
11.13/3.73	ORDERED_IN_G(x1)  =  ORDERED_IN_G(x1)
11.13/3.73	
11.13/3.73	U7_G(x1, x2, x3, x4)  =  U7_G(x2, x3, x4)
11.13/3.73	
11.13/3.73	LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
11.13/3.73	
11.13/3.73	U9_GG(x1, x2, x3)  =  U9_GG(x3)
11.13/3.73	
11.13/3.73	U8_G(x1, x2, x3, x4)  =  U8_G(x4)
11.13/3.73	
11.13/3.73	
11.13/3.73	We have to consider all (P,R,Pi)-chains
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(5) DependencyGraphProof (EQUIVALENT)
11.13/3.73	The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 11 less nodes.
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(6)
11.13/3.73	Complex Obligation (AND)
11.13/3.73	
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(7)
11.13/3.73	Obligation:
11.13/3.73	Pi DP problem:
11.13/3.73	The TRS P consists of the following rules:
11.13/3.73	
11.13/3.73	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
11.13/3.73	
11.13/3.73	The TRS R consists of the following rules:
11.13/3.73	
11.13/3.73	   ss_in_ag(Xs, Ys) -> U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
11.13/3.73	   perm_in_ag([], []) -> perm_out_ag([], [])
11.13/3.73	   perm_in_ag(Xs, .(X, Ys)) -> U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.13/3.73	   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
11.13/3.73	   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.13/3.73	   U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
11.13/3.73	   U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.13/3.73	   U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
11.13/3.73	   U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) -> perm_out_ag(Xs, .(X, Ys))
11.13/3.73	   U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) -> U2_ag(Xs, Ys, ordered_in_g(Ys))
11.13/3.73	   ordered_in_g([]) -> ordered_out_g([])
11.13/3.73	   ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, []))
11.13/3.73	   ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	   less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2))
11.13/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y))
11.13/3.73	   U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
11.13/3.73	   U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
11.13/3.73	   U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs)))
11.13/3.73	   U2_ag(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ag(Xs, Ys)
11.13/3.73	
11.13/3.73	The argument filtering Pi contains the following mapping:
11.13/3.73	ss_in_ag(x1, x2)  =  ss_in_ag(x2)
11.13/3.73	
11.13/3.73	U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
11.13/3.73	
11.13/3.73	perm_in_ag(x1, x2)  =  perm_in_ag(x2)
11.13/3.73	
11.13/3.73	[]  =  []
11.13/3.73	
11.13/3.73	perm_out_ag(x1, x2)  =  perm_out_ag
11.13/3.73	
11.13/3.73	.(x1, x2)  =  .(x1, x2)
11.13/3.73	
11.13/3.73	U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
11.13/3.73	
11.13/3.73	app_in_aaa(x1, x2, x3)  =  app_in_aaa
11.13/3.73	
11.13/3.73	app_out_aaa(x1, x2, x3)  =  app_out_aaa
11.13/3.73	
11.13/3.73	U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
11.13/3.73	
11.13/3.73	U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
11.13/3.73	
11.13/3.73	U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
11.13/3.73	
11.13/3.73	U2_ag(x1, x2, x3)  =  U2_ag(x3)
11.13/3.73	
11.13/3.73	ordered_in_g(x1)  =  ordered_in_g(x1)
11.13/3.73	
11.13/3.73	ordered_out_g(x1)  =  ordered_out_g
11.13/3.73	
11.13/3.73	U7_g(x1, x2, x3, x4)  =  U7_g(x2, x3, x4)
11.13/3.73	
11.13/3.73	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
11.13/3.73	
11.13/3.73	0  =  0
11.13/3.73	
11.13/3.73	s(x1)  =  s(x1)
11.13/3.73	
11.13/3.73	less_out_gg(x1, x2)  =  less_out_gg
11.13/3.73	
11.13/3.73	U9_gg(x1, x2, x3)  =  U9_gg(x3)
11.13/3.73	
11.13/3.73	U8_g(x1, x2, x3, x4)  =  U8_g(x4)
11.13/3.73	
11.13/3.73	ss_out_ag(x1, x2)  =  ss_out_ag
11.13/3.73	
11.13/3.73	LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
11.13/3.73	
11.13/3.73	
11.13/3.73	We have to consider all (P,R,Pi)-chains
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(8) UsableRulesProof (EQUIVALENT)
11.13/3.73	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(9)
11.13/3.73	Obligation:
11.13/3.73	Pi DP problem:
11.13/3.73	The TRS P consists of the following rules:
11.13/3.73	
11.13/3.73	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
11.13/3.73	
11.13/3.73	R is empty.
11.13/3.73	Pi is empty.
11.13/3.73	We have to consider all (P,R,Pi)-chains
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(10) PiDPToQDPProof (EQUIVALENT)
11.13/3.73	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(11)
11.13/3.73	Obligation:
11.13/3.73	Q DP problem:
11.13/3.73	The TRS P consists of the following rules:
11.13/3.73	
11.13/3.73	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
11.13/3.73	
11.13/3.73	R is empty.
11.13/3.73	Q is empty.
11.13/3.73	We have to consider all (P,Q,R)-chains.
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(12) QDPSizeChangeProof (EQUIVALENT)
11.13/3.73	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. 
11.13/3.73	
11.13/3.73	From the DPs we obtained the following set of size-change graphs:
11.13/3.73	*LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
11.13/3.73	The graph contains the following edges 1 > 1, 2 > 2
11.13/3.73	
11.13/3.73	
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(13)
11.13/3.73	YES
11.13/3.73	
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(14)
11.13/3.73	Obligation:
11.13/3.73	Pi DP problem:
11.13/3.73	The TRS P consists of the following rules:
11.13/3.73	
11.13/3.73	   U7_G(X, Y, Xs, less_out_gg(X, s(Y))) -> ORDERED_IN_G(.(Y, Xs))
11.13/3.73	   ORDERED_IN_G(.(X, .(Y, Xs))) -> U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	
11.13/3.73	The TRS R consists of the following rules:
11.13/3.73	
11.13/3.73	   ss_in_ag(Xs, Ys) -> U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
11.13/3.73	   perm_in_ag([], []) -> perm_out_ag([], [])
11.13/3.73	   perm_in_ag(Xs, .(X, Ys)) -> U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.13/3.73	   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
11.13/3.73	   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.13/3.73	   U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
11.13/3.73	   U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.13/3.73	   U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
11.13/3.73	   U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) -> perm_out_ag(Xs, .(X, Ys))
11.13/3.73	   U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) -> U2_ag(Xs, Ys, ordered_in_g(Ys))
11.13/3.73	   ordered_in_g([]) -> ordered_out_g([])
11.13/3.73	   ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, []))
11.13/3.73	   ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	   less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2))
11.13/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y))
11.13/3.73	   U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
11.13/3.73	   U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
11.13/3.73	   U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs)))
11.13/3.73	   U2_ag(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ag(Xs, Ys)
11.13/3.73	
11.13/3.73	The argument filtering Pi contains the following mapping:
11.13/3.73	ss_in_ag(x1, x2)  =  ss_in_ag(x2)
11.13/3.73	
11.13/3.73	U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
11.13/3.73	
11.13/3.73	perm_in_ag(x1, x2)  =  perm_in_ag(x2)
11.13/3.73	
11.13/3.73	[]  =  []
11.13/3.73	
11.13/3.73	perm_out_ag(x1, x2)  =  perm_out_ag
11.13/3.73	
11.13/3.73	.(x1, x2)  =  .(x1, x2)
11.13/3.73	
11.13/3.73	U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
11.13/3.73	
11.13/3.73	app_in_aaa(x1, x2, x3)  =  app_in_aaa
11.13/3.73	
11.13/3.73	app_out_aaa(x1, x2, x3)  =  app_out_aaa
11.13/3.73	
11.13/3.73	U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
11.13/3.73	
11.13/3.73	U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
11.13/3.73	
11.13/3.73	U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
11.13/3.73	
11.13/3.73	U2_ag(x1, x2, x3)  =  U2_ag(x3)
11.13/3.73	
11.13/3.73	ordered_in_g(x1)  =  ordered_in_g(x1)
11.13/3.73	
11.13/3.73	ordered_out_g(x1)  =  ordered_out_g
11.13/3.73	
11.13/3.73	U7_g(x1, x2, x3, x4)  =  U7_g(x2, x3, x4)
11.13/3.73	
11.13/3.73	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
11.13/3.73	
11.13/3.73	0  =  0
11.13/3.73	
11.13/3.73	s(x1)  =  s(x1)
11.13/3.73	
11.13/3.73	less_out_gg(x1, x2)  =  less_out_gg
11.13/3.73	
11.13/3.73	U9_gg(x1, x2, x3)  =  U9_gg(x3)
11.13/3.73	
11.13/3.73	U8_g(x1, x2, x3, x4)  =  U8_g(x4)
11.13/3.73	
11.13/3.73	ss_out_ag(x1, x2)  =  ss_out_ag
11.13/3.73	
11.13/3.73	ORDERED_IN_G(x1)  =  ORDERED_IN_G(x1)
11.13/3.73	
11.13/3.73	U7_G(x1, x2, x3, x4)  =  U7_G(x2, x3, x4)
11.13/3.73	
11.13/3.73	
11.13/3.73	We have to consider all (P,R,Pi)-chains
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(15) UsableRulesProof (EQUIVALENT)
11.13/3.73	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(16)
11.13/3.73	Obligation:
11.13/3.73	Pi DP problem:
11.13/3.73	The TRS P consists of the following rules:
11.13/3.73	
11.13/3.73	   U7_G(X, Y, Xs, less_out_gg(X, s(Y))) -> ORDERED_IN_G(.(Y, Xs))
11.13/3.73	   ORDERED_IN_G(.(X, .(Y, Xs))) -> U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	
11.13/3.73	The TRS R consists of the following rules:
11.13/3.73	
11.13/3.73	   less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2))
11.13/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y))
11.13/3.73	   U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
11.13/3.73	
11.13/3.73	The argument filtering Pi contains the following mapping:
11.13/3.73	.(x1, x2)  =  .(x1, x2)
11.13/3.73	
11.13/3.73	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
11.13/3.73	
11.13/3.73	0  =  0
11.13/3.73	
11.13/3.73	s(x1)  =  s(x1)
11.13/3.73	
11.13/3.73	less_out_gg(x1, x2)  =  less_out_gg
11.13/3.73	
11.13/3.73	U9_gg(x1, x2, x3)  =  U9_gg(x3)
11.13/3.73	
11.13/3.73	ORDERED_IN_G(x1)  =  ORDERED_IN_G(x1)
11.13/3.73	
11.13/3.73	U7_G(x1, x2, x3, x4)  =  U7_G(x2, x3, x4)
11.13/3.73	
11.13/3.73	
11.13/3.73	We have to consider all (P,R,Pi)-chains
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(17) PiDPToQDPProof (SOUND)
11.13/3.73	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(18)
11.13/3.73	Obligation:
11.13/3.73	Q DP problem:
11.13/3.73	The TRS P consists of the following rules:
11.13/3.73	
11.13/3.73	   U7_G(Y, Xs, less_out_gg) -> ORDERED_IN_G(.(Y, Xs))
11.13/3.73	   ORDERED_IN_G(.(X, .(Y, Xs))) -> U7_G(Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	
11.13/3.73	The TRS R consists of the following rules:
11.13/3.73	
11.13/3.73	   less_in_gg(0, s(X2)) -> less_out_gg
11.13/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(less_in_gg(X, Y))
11.13/3.73	   U9_gg(less_out_gg) -> less_out_gg
11.13/3.73	
11.13/3.73	The set Q consists of the following terms:
11.13/3.73	
11.13/3.73	   less_in_gg(x0, x1)
11.13/3.73	   U9_gg(x0)
11.13/3.73	
11.13/3.73	We have to consider all (P,Q,R)-chains.
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(19) UsableRulesReductionPairsProof (EQUIVALENT)
11.13/3.73	By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
11.13/3.73	
11.13/3.73	No dependency pairs are removed.
11.13/3.73	
11.13/3.73	The following rules are removed from R:
11.13/3.73	
11.13/3.73	   less_in_gg(0, s(X2)) -> less_out_gg
11.13/3.73	Used ordering: POLO with Polynomial interpretation [POLO]:
11.13/3.73	
11.13/3.73	   POL(.(x_1, x_2)) = 2*x_1 + 2*x_2
11.13/3.73	   POL(0) = 0
11.13/3.73	   POL(ORDERED_IN_G(x_1)) = x_1
11.13/3.73	   POL(U7_G(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3
11.13/3.73	   POL(U9_gg(x_1)) = x_1
11.13/3.73	   POL(less_in_gg(x_1, x_2)) = x_1 + x_2
11.13/3.73	   POL(less_out_gg) = 0
11.13/3.73	   POL(s(x_1)) = x_1
11.13/3.73	
11.13/3.73	
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(20)
11.13/3.73	Obligation:
11.13/3.73	Q DP problem:
11.13/3.73	The TRS P consists of the following rules:
11.13/3.73	
11.13/3.73	   U7_G(Y, Xs, less_out_gg) -> ORDERED_IN_G(.(Y, Xs))
11.13/3.73	   ORDERED_IN_G(.(X, .(Y, Xs))) -> U7_G(Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	
11.13/3.73	The TRS R consists of the following rules:
11.13/3.73	
11.13/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(less_in_gg(X, Y))
11.13/3.73	   U9_gg(less_out_gg) -> less_out_gg
11.13/3.73	
11.13/3.73	The set Q consists of the following terms:
11.13/3.73	
11.13/3.73	   less_in_gg(x0, x1)
11.13/3.73	   U9_gg(x0)
11.13/3.73	
11.13/3.73	We have to consider all (P,Q,R)-chains.
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(21) MRRProof (EQUIVALENT)
11.13/3.73	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
11.13/3.73	
11.13/3.73	Strictly oriented dependency pairs:
11.13/3.73	
11.13/3.73	   U7_G(Y, Xs, less_out_gg) -> ORDERED_IN_G(.(Y, Xs))
11.13/3.73	   ORDERED_IN_G(.(X, .(Y, Xs))) -> U7_G(Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	
11.13/3.73	
11.13/3.73	Used ordering: Polynomial interpretation [POLO]:
11.13/3.73	
11.13/3.73	   POL(.(x_1, x_2)) = 2*x_1 + 2*x_2
11.13/3.73	   POL(ORDERED_IN_G(x_1)) = 2 + x_1
11.13/3.73	   POL(U7_G(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3
11.13/3.73	   POL(U9_gg(x_1)) = x_1
11.13/3.73	   POL(less_in_gg(x_1, x_2)) = x_1 + x_2
11.13/3.73	   POL(less_out_gg) = 2
11.13/3.73	   POL(s(x_1)) = x_1
11.13/3.73	
11.13/3.73	
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(22)
11.13/3.73	Obligation:
11.13/3.73	Q DP problem:
11.13/3.73	P is empty.
11.13/3.73	The TRS R consists of the following rules:
11.13/3.73	
11.13/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(less_in_gg(X, Y))
11.13/3.73	   U9_gg(less_out_gg) -> less_out_gg
11.13/3.73	
11.13/3.73	The set Q consists of the following terms:
11.13/3.73	
11.13/3.73	   less_in_gg(x0, x1)
11.13/3.73	   U9_gg(x0)
11.13/3.73	
11.13/3.73	We have to consider all (P,Q,R)-chains.
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(23) PisEmptyProof (EQUIVALENT)
11.13/3.73	The TRS P is empty. Hence, there is no (P,Q,R) chain.
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(24)
11.13/3.73	YES
11.13/3.73	
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(25)
11.13/3.73	Obligation:
11.13/3.73	Pi DP problem:
11.13/3.73	The TRS P consists of the following rules:
11.13/3.73	
11.13/3.73	   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs)
11.13/3.73	
11.13/3.73	The TRS R consists of the following rules:
11.13/3.73	
11.13/3.73	   ss_in_ag(Xs, Ys) -> U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
11.13/3.73	   perm_in_ag([], []) -> perm_out_ag([], [])
11.13/3.73	   perm_in_ag(Xs, .(X, Ys)) -> U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.13/3.73	   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
11.13/3.73	   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.13/3.73	   U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
11.13/3.73	   U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.13/3.73	   U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
11.13/3.73	   U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) -> perm_out_ag(Xs, .(X, Ys))
11.13/3.73	   U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) -> U2_ag(Xs, Ys, ordered_in_g(Ys))
11.13/3.73	   ordered_in_g([]) -> ordered_out_g([])
11.13/3.73	   ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, []))
11.13/3.73	   ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	   less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2))
11.13/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y))
11.13/3.73	   U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
11.13/3.73	   U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
11.13/3.73	   U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs)))
11.13/3.73	   U2_ag(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ag(Xs, Ys)
11.13/3.73	
11.13/3.73	The argument filtering Pi contains the following mapping:
11.13/3.73	ss_in_ag(x1, x2)  =  ss_in_ag(x2)
11.13/3.73	
11.13/3.73	U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
11.13/3.73	
11.13/3.73	perm_in_ag(x1, x2)  =  perm_in_ag(x2)
11.13/3.73	
11.13/3.73	[]  =  []
11.13/3.73	
11.13/3.73	perm_out_ag(x1, x2)  =  perm_out_ag
11.13/3.73	
11.13/3.73	.(x1, x2)  =  .(x1, x2)
11.13/3.73	
11.13/3.73	U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
11.13/3.73	
11.13/3.73	app_in_aaa(x1, x2, x3)  =  app_in_aaa
11.13/3.73	
11.13/3.73	app_out_aaa(x1, x2, x3)  =  app_out_aaa
11.13/3.73	
11.13/3.73	U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
11.13/3.73	
11.13/3.73	U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
11.13/3.73	
11.13/3.73	U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
11.13/3.73	
11.13/3.73	U2_ag(x1, x2, x3)  =  U2_ag(x3)
11.13/3.73	
11.13/3.73	ordered_in_g(x1)  =  ordered_in_g(x1)
11.13/3.73	
11.13/3.73	ordered_out_g(x1)  =  ordered_out_g
11.13/3.73	
11.13/3.73	U7_g(x1, x2, x3, x4)  =  U7_g(x2, x3, x4)
11.13/3.73	
11.13/3.73	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
11.13/3.73	
11.13/3.73	0  =  0
11.13/3.73	
11.13/3.73	s(x1)  =  s(x1)
11.13/3.73	
11.13/3.73	less_out_gg(x1, x2)  =  less_out_gg
11.13/3.73	
11.13/3.73	U9_gg(x1, x2, x3)  =  U9_gg(x3)
11.13/3.73	
11.13/3.73	U8_g(x1, x2, x3, x4)  =  U8_g(x4)
11.13/3.73	
11.13/3.73	ss_out_ag(x1, x2)  =  ss_out_ag
11.13/3.73	
11.13/3.73	APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
11.13/3.73	
11.13/3.73	
11.13/3.73	We have to consider all (P,R,Pi)-chains
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(26) UsableRulesProof (EQUIVALENT)
11.13/3.73	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(27)
11.13/3.73	Obligation:
11.13/3.73	Pi DP problem:
11.13/3.73	The TRS P consists of the following rules:
11.13/3.73	
11.13/3.73	   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs)
11.13/3.73	
11.13/3.73	R is empty.
11.13/3.73	The argument filtering Pi contains the following mapping:
11.13/3.73	.(x1, x2)  =  .(x1, x2)
11.13/3.73	
11.13/3.73	APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
11.13/3.73	
11.13/3.73	
11.13/3.73	We have to consider all (P,R,Pi)-chains
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(28) PiDPToQDPProof (SOUND)
11.13/3.73	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(29)
11.13/3.73	Obligation:
11.13/3.73	Q DP problem:
11.13/3.73	The TRS P consists of the following rules:
11.13/3.73	
11.13/3.73	   APP_IN_AAA -> APP_IN_AAA
11.13/3.73	
11.13/3.73	R is empty.
11.13/3.73	Q is empty.
11.13/3.73	We have to consider all (P,Q,R)-chains.
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(30) NonTerminationLoopProof (COMPLETE)
11.13/3.73	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
11.13/3.73	Found a loop by semiunifying a rule from P directly.
11.13/3.73	
11.13/3.73	s = APP_IN_AAA evaluates to  t =APP_IN_AAA
11.13/3.73	
11.13/3.73	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
11.13/3.73	* Matcher: [ ]
11.13/3.73	* Semiunifier: [ ]
11.13/3.73	
11.13/3.73	--------------------------------------------------------------------------------
11.13/3.73	Rewriting sequence
11.13/3.73	
11.13/3.73	The DP semiunifies directly so there is only one rewrite step from APP_IN_AAA to APP_IN_AAA.
11.13/3.73	
11.13/3.73	
11.13/3.73	
11.13/3.73	
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(31)
11.13/3.73	NO
11.13/3.73	
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(32)
11.13/3.73	Obligation:
11.13/3.73	Pi DP problem:
11.13/3.73	The TRS P consists of the following rules:
11.13/3.73	
11.13/3.73	   U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.13/3.73	   U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> PERM_IN_AG(Zs, Ys)
11.13/3.73	   PERM_IN_AG(Xs, .(X, Ys)) -> U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.13/3.73	
11.13/3.73	The TRS R consists of the following rules:
11.13/3.73	
11.13/3.73	   ss_in_ag(Xs, Ys) -> U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
11.13/3.73	   perm_in_ag([], []) -> perm_out_ag([], [])
11.13/3.73	   perm_in_ag(Xs, .(X, Ys)) -> U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.13/3.73	   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
11.13/3.73	   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.13/3.73	   U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
11.13/3.73	   U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.13/3.73	   U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
11.13/3.73	   U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) -> perm_out_ag(Xs, .(X, Ys))
11.13/3.73	   U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) -> U2_ag(Xs, Ys, ordered_in_g(Ys))
11.13/3.73	   ordered_in_g([]) -> ordered_out_g([])
11.13/3.73	   ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, []))
11.13/3.73	   ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	   less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2))
11.13/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y))
11.13/3.73	   U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
11.13/3.73	   U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
11.13/3.73	   U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs)))
11.13/3.73	   U2_ag(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ag(Xs, Ys)
11.13/3.73	
11.13/3.73	The argument filtering Pi contains the following mapping:
11.13/3.73	ss_in_ag(x1, x2)  =  ss_in_ag(x2)
11.13/3.73	
11.13/3.73	U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
11.13/3.73	
11.13/3.73	perm_in_ag(x1, x2)  =  perm_in_ag(x2)
11.13/3.73	
11.13/3.73	[]  =  []
11.13/3.73	
11.13/3.73	perm_out_ag(x1, x2)  =  perm_out_ag
11.13/3.73	
11.13/3.73	.(x1, x2)  =  .(x1, x2)
11.13/3.73	
11.13/3.73	U3_ag(x1, x2, x3, x4)  =  U3_ag(x3, x4)
11.13/3.73	
11.13/3.73	app_in_aaa(x1, x2, x3)  =  app_in_aaa
11.13/3.73	
11.13/3.73	app_out_aaa(x1, x2, x3)  =  app_out_aaa
11.13/3.73	
11.13/3.73	U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
11.13/3.73	
11.13/3.73	U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)
11.13/3.73	
11.13/3.73	U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)
11.13/3.73	
11.13/3.73	U2_ag(x1, x2, x3)  =  U2_ag(x3)
11.13/3.73	
11.13/3.73	ordered_in_g(x1)  =  ordered_in_g(x1)
11.13/3.73	
11.13/3.73	ordered_out_g(x1)  =  ordered_out_g
11.13/3.73	
11.13/3.73	U7_g(x1, x2, x3, x4)  =  U7_g(x2, x3, x4)
11.13/3.73	
11.13/3.73	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
11.13/3.73	
11.13/3.73	0  =  0
11.13/3.73	
11.13/3.73	s(x1)  =  s(x1)
11.13/3.73	
11.13/3.73	less_out_gg(x1, x2)  =  less_out_gg
11.13/3.73	
11.13/3.73	U9_gg(x1, x2, x3)  =  U9_gg(x3)
11.13/3.73	
11.13/3.73	U8_g(x1, x2, x3, x4)  =  U8_g(x4)
11.13/3.73	
11.13/3.73	ss_out_ag(x1, x2)  =  ss_out_ag
11.13/3.73	
11.13/3.73	PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
11.13/3.73	
11.13/3.73	U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
11.13/3.73	
11.13/3.73	U4_AG(x1, x2, x3, x4)  =  U4_AG(x3, x4)
11.13/3.73	
11.13/3.73	
11.13/3.73	We have to consider all (P,R,Pi)-chains
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(33) UsableRulesProof (EQUIVALENT)
11.13/3.73	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(34)
11.13/3.73	Obligation:
11.13/3.73	Pi DP problem:
11.13/3.73	The TRS P consists of the following rules:
11.13/3.73	
11.13/3.73	   U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.13/3.73	   U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> PERM_IN_AG(Zs, Ys)
11.13/3.73	   PERM_IN_AG(Xs, .(X, Ys)) -> U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.13/3.73	
11.13/3.73	The TRS R consists of the following rules:
11.13/3.73	
11.13/3.73	   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
11.13/3.73	   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.13/3.73	   U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
11.13/3.73	
11.13/3.73	The argument filtering Pi contains the following mapping:
11.13/3.73	[]  =  []
11.13/3.73	
11.13/3.73	.(x1, x2)  =  .(x1, x2)
11.13/3.73	
11.13/3.73	app_in_aaa(x1, x2, x3)  =  app_in_aaa
11.13/3.73	
11.13/3.73	app_out_aaa(x1, x2, x3)  =  app_out_aaa
11.13/3.73	
11.13/3.73	U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
11.13/3.73	
11.13/3.73	PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
11.13/3.73	
11.13/3.73	U3_AG(x1, x2, x3, x4)  =  U3_AG(x3, x4)
11.13/3.73	
11.13/3.73	U4_AG(x1, x2, x3, x4)  =  U4_AG(x3, x4)
11.13/3.73	
11.13/3.73	
11.13/3.73	We have to consider all (P,R,Pi)-chains
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(35) PiDPToQDPProof (SOUND)
11.13/3.73	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(36)
11.13/3.73	Obligation:
11.13/3.73	Q DP problem:
11.13/3.73	The TRS P consists of the following rules:
11.13/3.73	
11.13/3.73	   U3_AG(Ys, app_out_aaa) -> U4_AG(Ys, app_in_aaa)
11.13/3.73	   U4_AG(Ys, app_out_aaa) -> PERM_IN_AG(Ys)
11.13/3.73	   PERM_IN_AG(.(X, Ys)) -> U3_AG(Ys, app_in_aaa)
11.13/3.73	
11.13/3.73	The TRS R consists of the following rules:
11.13/3.73	
11.13/3.73	   app_in_aaa -> app_out_aaa
11.13/3.73	   app_in_aaa -> U6_aaa(app_in_aaa)
11.13/3.73	   U6_aaa(app_out_aaa) -> app_out_aaa
11.13/3.73	
11.13/3.73	The set Q consists of the following terms:
11.13/3.73	
11.13/3.73	   app_in_aaa
11.13/3.73	   U6_aaa(x0)
11.13/3.73	
11.13/3.73	We have to consider all (P,Q,R)-chains.
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(37) QDPSizeChangeProof (EQUIVALENT)
11.13/3.73	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. 
11.13/3.73	
11.13/3.73	From the DPs we obtained the following set of size-change graphs:
11.13/3.73	*U4_AG(Ys, app_out_aaa) -> PERM_IN_AG(Ys)
11.13/3.73	The graph contains the following edges 1 >= 1
11.13/3.73	
11.13/3.73	
11.13/3.73	*PERM_IN_AG(.(X, Ys)) -> U3_AG(Ys, app_in_aaa)
11.13/3.73	The graph contains the following edges 1 > 1
11.13/3.73	
11.13/3.73	
11.13/3.73	*U3_AG(Ys, app_out_aaa) -> U4_AG(Ys, app_in_aaa)
11.13/3.73	The graph contains the following edges 1 >= 1
11.13/3.73	
11.13/3.73	
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(38)
11.13/3.73	YES
11.13/3.73	
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(39) PrologToPiTRSProof (SOUND)
11.13/3.73	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
11.13/3.73	
11.13/3.73	ss_in_2: (f,b)
11.13/3.73	
11.13/3.73	perm_in_2: (f,b)
11.13/3.73	
11.13/3.73	app_in_3: (f,f,f)
11.13/3.73	
11.13/3.73	ordered_in_1: (b)
11.13/3.73	
11.13/3.73	less_in_2: (b,b)
11.13/3.73	
11.13/3.73	Transforming Prolog into the following Term Rewriting System:
11.13/3.73	
11.13/3.73	Pi-finite rewrite system:
11.13/3.73	The TRS R consists of the following rules:
11.13/3.73	
11.13/3.73	   ss_in_ag(Xs, Ys) -> U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
11.13/3.73	   perm_in_ag([], []) -> perm_out_ag([], [])
11.13/3.73	   perm_in_ag(Xs, .(X, Ys)) -> U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.13/3.73	   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
11.13/3.73	   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.13/3.73	   U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
11.13/3.73	   U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.13/3.73	   U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
11.13/3.73	   U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) -> perm_out_ag(Xs, .(X, Ys))
11.13/3.73	   U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) -> U2_ag(Xs, Ys, ordered_in_g(Ys))
11.13/3.73	   ordered_in_g([]) -> ordered_out_g([])
11.13/3.73	   ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, []))
11.13/3.73	   ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	   less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2))
11.13/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y))
11.13/3.73	   U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
11.13/3.73	   U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
11.13/3.73	   U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs)))
11.13/3.73	   U2_ag(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ag(Xs, Ys)
11.13/3.73	
11.13/3.73	The argument filtering Pi contains the following mapping:
11.13/3.73	ss_in_ag(x1, x2)  =  ss_in_ag(x2)
11.13/3.73	
11.13/3.73	U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
11.13/3.73	
11.13/3.73	perm_in_ag(x1, x2)  =  perm_in_ag(x2)
11.13/3.73	
11.13/3.73	[]  =  []
11.13/3.73	
11.13/3.73	perm_out_ag(x1, x2)  =  perm_out_ag(x2)
11.13/3.73	
11.13/3.73	.(x1, x2)  =  .(x1, x2)
11.13/3.73	
11.13/3.73	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
11.13/3.73	
11.13/3.73	app_in_aaa(x1, x2, x3)  =  app_in_aaa
11.13/3.73	
11.13/3.73	app_out_aaa(x1, x2, x3)  =  app_out_aaa
11.13/3.73	
11.13/3.73	U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
11.13/3.73	
11.13/3.73	U4_ag(x1, x2, x3, x4)  =  U4_ag(x2, x3, x4)
11.13/3.73	
11.13/3.73	U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
11.13/3.73	
11.13/3.73	U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
11.13/3.73	
11.13/3.73	ordered_in_g(x1)  =  ordered_in_g(x1)
11.13/3.73	
11.13/3.73	ordered_out_g(x1)  =  ordered_out_g(x1)
11.13/3.73	
11.13/3.73	U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
11.13/3.73	
11.13/3.73	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
11.13/3.73	
11.13/3.73	0  =  0
11.13/3.73	
11.13/3.73	s(x1)  =  s(x1)
11.13/3.73	
11.13/3.73	less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
11.13/3.73	
11.13/3.73	U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
11.13/3.73	
11.13/3.73	U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
11.13/3.73	
11.13/3.73	ss_out_ag(x1, x2)  =  ss_out_ag(x2)
11.13/3.73	
11.13/3.73	
11.13/3.73	
11.13/3.73	
11.13/3.73	
11.13/3.73	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
11.13/3.73	
11.13/3.73	
11.13/3.73	
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(40)
11.13/3.73	Obligation:
11.13/3.73	Pi-finite rewrite system:
11.13/3.73	The TRS R consists of the following rules:
11.13/3.73	
11.13/3.73	   ss_in_ag(Xs, Ys) -> U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
11.13/3.73	   perm_in_ag([], []) -> perm_out_ag([], [])
11.13/3.73	   perm_in_ag(Xs, .(X, Ys)) -> U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.13/3.73	   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
11.13/3.73	   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.13/3.73	   U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
11.13/3.73	   U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.13/3.73	   U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
11.13/3.73	   U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) -> perm_out_ag(Xs, .(X, Ys))
11.13/3.73	   U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) -> U2_ag(Xs, Ys, ordered_in_g(Ys))
11.13/3.73	   ordered_in_g([]) -> ordered_out_g([])
11.13/3.73	   ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, []))
11.13/3.73	   ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	   less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2))
11.13/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y))
11.13/3.73	   U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
11.13/3.73	   U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
11.13/3.73	   U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs)))
11.13/3.73	   U2_ag(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ag(Xs, Ys)
11.13/3.73	
11.13/3.73	The argument filtering Pi contains the following mapping:
11.13/3.73	ss_in_ag(x1, x2)  =  ss_in_ag(x2)
11.13/3.73	
11.13/3.73	U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
11.13/3.73	
11.13/3.73	perm_in_ag(x1, x2)  =  perm_in_ag(x2)
11.13/3.73	
11.13/3.73	[]  =  []
11.13/3.73	
11.13/3.73	perm_out_ag(x1, x2)  =  perm_out_ag(x2)
11.13/3.73	
11.13/3.73	.(x1, x2)  =  .(x1, x2)
11.13/3.73	
11.13/3.73	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
11.13/3.73	
11.13/3.73	app_in_aaa(x1, x2, x3)  =  app_in_aaa
11.13/3.73	
11.13/3.73	app_out_aaa(x1, x2, x3)  =  app_out_aaa
11.13/3.73	
11.13/3.73	U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
11.13/3.73	
11.13/3.73	U4_ag(x1, x2, x3, x4)  =  U4_ag(x2, x3, x4)
11.13/3.73	
11.13/3.73	U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
11.13/3.73	
11.13/3.73	U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
11.13/3.73	
11.13/3.73	ordered_in_g(x1)  =  ordered_in_g(x1)
11.13/3.73	
11.13/3.73	ordered_out_g(x1)  =  ordered_out_g(x1)
11.13/3.73	
11.13/3.73	U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
11.13/3.73	
11.13/3.73	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
11.13/3.73	
11.13/3.73	0  =  0
11.13/3.73	
11.13/3.73	s(x1)  =  s(x1)
11.13/3.73	
11.13/3.73	less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
11.13/3.73	
11.13/3.73	U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
11.13/3.73	
11.13/3.73	U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
11.13/3.73	
11.13/3.73	ss_out_ag(x1, x2)  =  ss_out_ag(x2)
11.13/3.73	
11.13/3.73	
11.13/3.73	
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(41) DependencyPairsProof (EQUIVALENT)
11.13/3.73	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
11.13/3.73	Pi DP problem:
11.13/3.73	The TRS P consists of the following rules:
11.13/3.73	
11.13/3.73	   SS_IN_AG(Xs, Ys) -> U1_AG(Xs, Ys, perm_in_ag(Xs, Ys))
11.13/3.73	   SS_IN_AG(Xs, Ys) -> PERM_IN_AG(Xs, Ys)
11.13/3.73	   PERM_IN_AG(Xs, .(X, Ys)) -> U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.13/3.73	   PERM_IN_AG(Xs, .(X, Ys)) -> APP_IN_AAA(X1s, .(X, X2s), Xs)
11.13/3.73	   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U6_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.13/3.73	   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs)
11.13/3.73	   U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.13/3.73	   U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> APP_IN_AAA(X1s, X2s, Zs)
11.13/3.73	   U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> U5_AG(Xs, X, Ys, perm_in_ag(Zs, Ys))
11.13/3.73	   U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> PERM_IN_AG(Zs, Ys)
11.13/3.73	   U1_AG(Xs, Ys, perm_out_ag(Xs, Ys)) -> U2_AG(Xs, Ys, ordered_in_g(Ys))
11.13/3.73	   U1_AG(Xs, Ys, perm_out_ag(Xs, Ys)) -> ORDERED_IN_G(Ys)
11.13/3.73	   ORDERED_IN_G(.(X, .(Y, Xs))) -> U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	   ORDERED_IN_G(.(X, .(Y, Xs))) -> LESS_IN_GG(X, s(Y))
11.13/3.73	   LESS_IN_GG(s(X), s(Y)) -> U9_GG(X, Y, less_in_gg(X, Y))
11.13/3.73	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
11.13/3.73	   U7_G(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_G(X, Y, Xs, ordered_in_g(.(Y, Xs)))
11.13/3.73	   U7_G(X, Y, Xs, less_out_gg(X, s(Y))) -> ORDERED_IN_G(.(Y, Xs))
11.13/3.73	
11.13/3.73	The TRS R consists of the following rules:
11.13/3.73	
11.13/3.73	   ss_in_ag(Xs, Ys) -> U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
11.13/3.73	   perm_in_ag([], []) -> perm_out_ag([], [])
11.13/3.73	   perm_in_ag(Xs, .(X, Ys)) -> U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.13/3.73	   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
11.13/3.73	   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.13/3.73	   U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
11.13/3.73	   U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.13/3.73	   U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
11.13/3.73	   U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) -> perm_out_ag(Xs, .(X, Ys))
11.13/3.73	   U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) -> U2_ag(Xs, Ys, ordered_in_g(Ys))
11.13/3.73	   ordered_in_g([]) -> ordered_out_g([])
11.13/3.73	   ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, []))
11.13/3.73	   ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	   less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2))
11.13/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y))
11.13/3.73	   U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
11.13/3.73	   U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
11.13/3.73	   U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs)))
11.13/3.73	   U2_ag(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ag(Xs, Ys)
11.13/3.73	
11.13/3.73	The argument filtering Pi contains the following mapping:
11.13/3.73	ss_in_ag(x1, x2)  =  ss_in_ag(x2)
11.13/3.73	
11.13/3.73	U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
11.13/3.73	
11.13/3.73	perm_in_ag(x1, x2)  =  perm_in_ag(x2)
11.13/3.73	
11.13/3.73	[]  =  []
11.13/3.73	
11.13/3.73	perm_out_ag(x1, x2)  =  perm_out_ag(x2)
11.13/3.73	
11.13/3.73	.(x1, x2)  =  .(x1, x2)
11.13/3.73	
11.13/3.73	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
11.13/3.73	
11.13/3.73	app_in_aaa(x1, x2, x3)  =  app_in_aaa
11.13/3.73	
11.13/3.73	app_out_aaa(x1, x2, x3)  =  app_out_aaa
11.13/3.73	
11.13/3.73	U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
11.13/3.73	
11.13/3.73	U4_ag(x1, x2, x3, x4)  =  U4_ag(x2, x3, x4)
11.13/3.73	
11.13/3.73	U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
11.13/3.73	
11.13/3.73	U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
11.13/3.73	
11.13/3.73	ordered_in_g(x1)  =  ordered_in_g(x1)
11.13/3.73	
11.13/3.73	ordered_out_g(x1)  =  ordered_out_g(x1)
11.13/3.73	
11.13/3.73	U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
11.13/3.73	
11.13/3.73	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
11.13/3.73	
11.13/3.73	0  =  0
11.13/3.73	
11.13/3.73	s(x1)  =  s(x1)
11.13/3.73	
11.13/3.73	less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
11.13/3.73	
11.13/3.73	U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
11.13/3.73	
11.13/3.73	U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
11.13/3.73	
11.13/3.73	ss_out_ag(x1, x2)  =  ss_out_ag(x2)
11.13/3.73	
11.13/3.73	SS_IN_AG(x1, x2)  =  SS_IN_AG(x2)
11.13/3.73	
11.13/3.73	U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
11.13/3.73	
11.13/3.73	PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
11.13/3.73	
11.13/3.73	U3_AG(x1, x2, x3, x4)  =  U3_AG(x2, x3, x4)
11.13/3.73	
11.13/3.73	APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
11.13/3.73	
11.13/3.73	U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)
11.13/3.73	
11.13/3.73	U4_AG(x1, x2, x3, x4)  =  U4_AG(x2, x3, x4)
11.13/3.73	
11.13/3.73	U5_AG(x1, x2, x3, x4)  =  U5_AG(x2, x3, x4)
11.13/3.73	
11.13/3.73	U2_AG(x1, x2, x3)  =  U2_AG(x2, x3)
11.13/3.73	
11.13/3.73	ORDERED_IN_G(x1)  =  ORDERED_IN_G(x1)
11.13/3.73	
11.13/3.73	U7_G(x1, x2, x3, x4)  =  U7_G(x1, x2, x3, x4)
11.13/3.73	
11.13/3.73	LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
11.13/3.73	
11.13/3.73	U9_GG(x1, x2, x3)  =  U9_GG(x1, x2, x3)
11.13/3.73	
11.13/3.73	U8_G(x1, x2, x3, x4)  =  U8_G(x1, x2, x3, x4)
11.13/3.73	
11.13/3.73	
11.13/3.73	We have to consider all (P,R,Pi)-chains
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(42)
11.13/3.73	Obligation:
11.13/3.73	Pi DP problem:
11.13/3.73	The TRS P consists of the following rules:
11.13/3.73	
11.13/3.73	   SS_IN_AG(Xs, Ys) -> U1_AG(Xs, Ys, perm_in_ag(Xs, Ys))
11.13/3.73	   SS_IN_AG(Xs, Ys) -> PERM_IN_AG(Xs, Ys)
11.13/3.73	   PERM_IN_AG(Xs, .(X, Ys)) -> U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.13/3.73	   PERM_IN_AG(Xs, .(X, Ys)) -> APP_IN_AAA(X1s, .(X, X2s), Xs)
11.13/3.73	   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> U6_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.13/3.73	   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs)
11.13/3.73	   U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.13/3.73	   U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> APP_IN_AAA(X1s, X2s, Zs)
11.13/3.73	   U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> U5_AG(Xs, X, Ys, perm_in_ag(Zs, Ys))
11.13/3.73	   U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> PERM_IN_AG(Zs, Ys)
11.13/3.73	   U1_AG(Xs, Ys, perm_out_ag(Xs, Ys)) -> U2_AG(Xs, Ys, ordered_in_g(Ys))
11.13/3.73	   U1_AG(Xs, Ys, perm_out_ag(Xs, Ys)) -> ORDERED_IN_G(Ys)
11.13/3.73	   ORDERED_IN_G(.(X, .(Y, Xs))) -> U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	   ORDERED_IN_G(.(X, .(Y, Xs))) -> LESS_IN_GG(X, s(Y))
11.13/3.73	   LESS_IN_GG(s(X), s(Y)) -> U9_GG(X, Y, less_in_gg(X, Y))
11.13/3.73	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
11.13/3.73	   U7_G(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_G(X, Y, Xs, ordered_in_g(.(Y, Xs)))
11.13/3.73	   U7_G(X, Y, Xs, less_out_gg(X, s(Y))) -> ORDERED_IN_G(.(Y, Xs))
11.13/3.73	
11.13/3.73	The TRS R consists of the following rules:
11.13/3.73	
11.13/3.73	   ss_in_ag(Xs, Ys) -> U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
11.13/3.73	   perm_in_ag([], []) -> perm_out_ag([], [])
11.13/3.73	   perm_in_ag(Xs, .(X, Ys)) -> U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.13/3.73	   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
11.13/3.73	   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.13/3.73	   U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
11.13/3.73	   U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.13/3.73	   U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
11.13/3.73	   U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) -> perm_out_ag(Xs, .(X, Ys))
11.13/3.73	   U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) -> U2_ag(Xs, Ys, ordered_in_g(Ys))
11.13/3.73	   ordered_in_g([]) -> ordered_out_g([])
11.13/3.73	   ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, []))
11.13/3.73	   ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	   less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2))
11.13/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y))
11.13/3.73	   U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
11.13/3.73	   U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
11.13/3.73	   U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs)))
11.13/3.73	   U2_ag(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ag(Xs, Ys)
11.13/3.73	
11.13/3.73	The argument filtering Pi contains the following mapping:
11.13/3.73	ss_in_ag(x1, x2)  =  ss_in_ag(x2)
11.13/3.73	
11.13/3.73	U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
11.13/3.73	
11.13/3.73	perm_in_ag(x1, x2)  =  perm_in_ag(x2)
11.13/3.73	
11.13/3.73	[]  =  []
11.13/3.73	
11.13/3.73	perm_out_ag(x1, x2)  =  perm_out_ag(x2)
11.13/3.73	
11.13/3.73	.(x1, x2)  =  .(x1, x2)
11.13/3.73	
11.13/3.73	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
11.13/3.73	
11.13/3.73	app_in_aaa(x1, x2, x3)  =  app_in_aaa
11.13/3.73	
11.13/3.73	app_out_aaa(x1, x2, x3)  =  app_out_aaa
11.13/3.73	
11.13/3.73	U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
11.13/3.73	
11.13/3.73	U4_ag(x1, x2, x3, x4)  =  U4_ag(x2, x3, x4)
11.13/3.73	
11.13/3.73	U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
11.13/3.73	
11.13/3.73	U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
11.13/3.73	
11.13/3.73	ordered_in_g(x1)  =  ordered_in_g(x1)
11.13/3.73	
11.13/3.73	ordered_out_g(x1)  =  ordered_out_g(x1)
11.13/3.73	
11.13/3.73	U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
11.13/3.73	
11.13/3.73	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
11.13/3.73	
11.13/3.73	0  =  0
11.13/3.73	
11.13/3.73	s(x1)  =  s(x1)
11.13/3.73	
11.13/3.73	less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
11.13/3.73	
11.13/3.73	U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
11.13/3.73	
11.13/3.73	U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
11.13/3.73	
11.13/3.73	ss_out_ag(x1, x2)  =  ss_out_ag(x2)
11.13/3.73	
11.13/3.73	SS_IN_AG(x1, x2)  =  SS_IN_AG(x2)
11.13/3.73	
11.13/3.73	U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
11.13/3.73	
11.13/3.73	PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
11.13/3.73	
11.13/3.73	U3_AG(x1, x2, x3, x4)  =  U3_AG(x2, x3, x4)
11.13/3.73	
11.13/3.73	APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
11.13/3.73	
11.13/3.73	U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)
11.13/3.73	
11.13/3.73	U4_AG(x1, x2, x3, x4)  =  U4_AG(x2, x3, x4)
11.13/3.73	
11.13/3.73	U5_AG(x1, x2, x3, x4)  =  U5_AG(x2, x3, x4)
11.13/3.73	
11.13/3.73	U2_AG(x1, x2, x3)  =  U2_AG(x2, x3)
11.13/3.73	
11.13/3.73	ORDERED_IN_G(x1)  =  ORDERED_IN_G(x1)
11.13/3.73	
11.13/3.73	U7_G(x1, x2, x3, x4)  =  U7_G(x1, x2, x3, x4)
11.13/3.73	
11.13/3.73	LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
11.13/3.73	
11.13/3.73	U9_GG(x1, x2, x3)  =  U9_GG(x1, x2, x3)
11.13/3.73	
11.13/3.73	U8_G(x1, x2, x3, x4)  =  U8_G(x1, x2, x3, x4)
11.13/3.73	
11.13/3.73	
11.13/3.73	We have to consider all (P,R,Pi)-chains
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(43) DependencyGraphProof (EQUIVALENT)
11.13/3.73	The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 11 less nodes.
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(44)
11.13/3.73	Complex Obligation (AND)
11.13/3.73	
11.13/3.73	----------------------------------------
11.13/3.73	
11.13/3.73	(45)
11.13/3.73	Obligation:
11.13/3.73	Pi DP problem:
11.13/3.73	The TRS P consists of the following rules:
11.13/3.73	
11.13/3.73	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
11.13/3.73	
11.13/3.73	The TRS R consists of the following rules:
11.13/3.73	
11.13/3.73	   ss_in_ag(Xs, Ys) -> U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
11.13/3.73	   perm_in_ag([], []) -> perm_out_ag([], [])
11.13/3.73	   perm_in_ag(Xs, .(X, Ys)) -> U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.13/3.73	   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
11.13/3.73	   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.13/3.73	   U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
11.13/3.73	   U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.13/3.73	   U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
11.13/3.73	   U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) -> perm_out_ag(Xs, .(X, Ys))
11.13/3.73	   U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) -> U2_ag(Xs, Ys, ordered_in_g(Ys))
11.13/3.73	   ordered_in_g([]) -> ordered_out_g([])
11.13/3.73	   ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, []))
11.13/3.73	   ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
11.13/3.73	   less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2))
11.13/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y))
11.13/3.73	   U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
11.13/3.73	   U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
11.13/3.73	   U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs)))
11.13/3.73	   U2_ag(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ag(Xs, Ys)
11.13/3.73	
11.13/3.73	The argument filtering Pi contains the following mapping:
11.13/3.73	ss_in_ag(x1, x2)  =  ss_in_ag(x2)
11.13/3.73	
11.13/3.73	U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
11.13/3.73	
11.13/3.73	perm_in_ag(x1, x2)  =  perm_in_ag(x2)
11.13/3.73	
11.13/3.73	[]  =  []
11.13/3.73	
11.13/3.73	perm_out_ag(x1, x2)  =  perm_out_ag(x2)
11.13/3.73	
11.13/3.73	.(x1, x2)  =  .(x1, x2)
11.13/3.73	
11.13/3.73	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
11.13/3.73	
11.13/3.73	app_in_aaa(x1, x2, x3)  =  app_in_aaa
11.13/3.73	
11.13/3.73	app_out_aaa(x1, x2, x3)  =  app_out_aaa
11.17/3.73	
11.17/3.73	U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
11.17/3.73	
11.17/3.73	U4_ag(x1, x2, x3, x4)  =  U4_ag(x2, x3, x4)
11.17/3.73	
11.17/3.73	U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
11.17/3.73	
11.17/3.73	U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
11.17/3.73	
11.17/3.73	ordered_in_g(x1)  =  ordered_in_g(x1)
11.17/3.73	
11.17/3.73	ordered_out_g(x1)  =  ordered_out_g(x1)
11.17/3.73	
11.17/3.73	U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
11.17/3.73	
11.17/3.73	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
11.17/3.73	
11.17/3.73	0  =  0
11.17/3.73	
11.17/3.73	s(x1)  =  s(x1)
11.17/3.73	
11.17/3.73	less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
11.17/3.73	
11.17/3.73	U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
11.17/3.73	
11.17/3.73	U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
11.17/3.73	
11.17/3.73	ss_out_ag(x1, x2)  =  ss_out_ag(x2)
11.17/3.73	
11.17/3.73	LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
11.17/3.73	
11.17/3.73	
11.17/3.73	We have to consider all (P,R,Pi)-chains
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(46) UsableRulesProof (EQUIVALENT)
11.17/3.73	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(47)
11.17/3.73	Obligation:
11.17/3.73	Pi DP problem:
11.17/3.73	The TRS P consists of the following rules:
11.17/3.73	
11.17/3.73	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
11.17/3.73	
11.17/3.73	R is empty.
11.17/3.73	Pi is empty.
11.17/3.73	We have to consider all (P,R,Pi)-chains
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(48) PiDPToQDPProof (EQUIVALENT)
11.17/3.73	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(49)
11.17/3.73	Obligation:
11.17/3.73	Q DP problem:
11.17/3.73	The TRS P consists of the following rules:
11.17/3.73	
11.17/3.73	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
11.17/3.73	
11.17/3.73	R is empty.
11.17/3.73	Q is empty.
11.17/3.73	We have to consider all (P,Q,R)-chains.
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(50) QDPSizeChangeProof (EQUIVALENT)
11.17/3.73	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. 
11.17/3.73	
11.17/3.73	From the DPs we obtained the following set of size-change graphs:
11.17/3.73	*LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
11.17/3.73	The graph contains the following edges 1 > 1, 2 > 2
11.17/3.73	
11.17/3.73	
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(51)
11.17/3.73	YES
11.17/3.73	
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(52)
11.17/3.73	Obligation:
11.17/3.73	Pi DP problem:
11.17/3.73	The TRS P consists of the following rules:
11.17/3.73	
11.17/3.73	   U7_G(X, Y, Xs, less_out_gg(X, s(Y))) -> ORDERED_IN_G(.(Y, Xs))
11.17/3.73	   ORDERED_IN_G(.(X, .(Y, Xs))) -> U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
11.17/3.73	
11.17/3.73	The TRS R consists of the following rules:
11.17/3.73	
11.17/3.73	   ss_in_ag(Xs, Ys) -> U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
11.17/3.73	   perm_in_ag([], []) -> perm_out_ag([], [])
11.17/3.73	   perm_in_ag(Xs, .(X, Ys)) -> U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.17/3.73	   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
11.17/3.73	   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.17/3.73	   U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
11.17/3.73	   U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.17/3.73	   U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
11.17/3.73	   U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) -> perm_out_ag(Xs, .(X, Ys))
11.17/3.73	   U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) -> U2_ag(Xs, Ys, ordered_in_g(Ys))
11.17/3.73	   ordered_in_g([]) -> ordered_out_g([])
11.17/3.73	   ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, []))
11.17/3.73	   ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
11.17/3.73	   less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2))
11.17/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y))
11.17/3.73	   U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
11.17/3.73	   U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
11.17/3.73	   U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs)))
11.17/3.73	   U2_ag(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ag(Xs, Ys)
11.17/3.73	
11.17/3.73	The argument filtering Pi contains the following mapping:
11.17/3.73	ss_in_ag(x1, x2)  =  ss_in_ag(x2)
11.17/3.73	
11.17/3.73	U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
11.17/3.73	
11.17/3.73	perm_in_ag(x1, x2)  =  perm_in_ag(x2)
11.17/3.73	
11.17/3.73	[]  =  []
11.17/3.73	
11.17/3.73	perm_out_ag(x1, x2)  =  perm_out_ag(x2)
11.17/3.73	
11.17/3.73	.(x1, x2)  =  .(x1, x2)
11.17/3.73	
11.17/3.73	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
11.17/3.73	
11.17/3.73	app_in_aaa(x1, x2, x3)  =  app_in_aaa
11.17/3.73	
11.17/3.73	app_out_aaa(x1, x2, x3)  =  app_out_aaa
11.17/3.73	
11.17/3.73	U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
11.17/3.73	
11.17/3.73	U4_ag(x1, x2, x3, x4)  =  U4_ag(x2, x3, x4)
11.17/3.73	
11.17/3.73	U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
11.17/3.73	
11.17/3.73	U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
11.17/3.73	
11.17/3.73	ordered_in_g(x1)  =  ordered_in_g(x1)
11.17/3.73	
11.17/3.73	ordered_out_g(x1)  =  ordered_out_g(x1)
11.17/3.73	
11.17/3.73	U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
11.17/3.73	
11.17/3.73	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
11.17/3.73	
11.17/3.73	0  =  0
11.17/3.73	
11.17/3.73	s(x1)  =  s(x1)
11.17/3.73	
11.17/3.73	less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
11.17/3.73	
11.17/3.73	U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
11.17/3.73	
11.17/3.73	U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
11.17/3.73	
11.17/3.73	ss_out_ag(x1, x2)  =  ss_out_ag(x2)
11.17/3.73	
11.17/3.73	ORDERED_IN_G(x1)  =  ORDERED_IN_G(x1)
11.17/3.73	
11.17/3.73	U7_G(x1, x2, x3, x4)  =  U7_G(x1, x2, x3, x4)
11.17/3.73	
11.17/3.73	
11.17/3.73	We have to consider all (P,R,Pi)-chains
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(53) UsableRulesProof (EQUIVALENT)
11.17/3.73	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(54)
11.17/3.73	Obligation:
11.17/3.73	Pi DP problem:
11.17/3.73	The TRS P consists of the following rules:
11.17/3.73	
11.17/3.73	   U7_G(X, Y, Xs, less_out_gg(X, s(Y))) -> ORDERED_IN_G(.(Y, Xs))
11.17/3.73	   ORDERED_IN_G(.(X, .(Y, Xs))) -> U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
11.17/3.73	
11.17/3.73	The TRS R consists of the following rules:
11.17/3.73	
11.17/3.73	   less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2))
11.17/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y))
11.17/3.73	   U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
11.17/3.73	
11.17/3.73	Pi is empty.
11.17/3.73	We have to consider all (P,R,Pi)-chains
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(55) PiDPToQDPProof (EQUIVALENT)
11.17/3.73	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(56)
11.17/3.73	Obligation:
11.17/3.73	Q DP problem:
11.17/3.73	The TRS P consists of the following rules:
11.17/3.73	
11.17/3.73	   U7_G(X, Y, Xs, less_out_gg(X, s(Y))) -> ORDERED_IN_G(.(Y, Xs))
11.17/3.73	   ORDERED_IN_G(.(X, .(Y, Xs))) -> U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
11.17/3.73	
11.17/3.73	The TRS R consists of the following rules:
11.17/3.73	
11.17/3.73	   less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2))
11.17/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y))
11.17/3.73	   U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
11.17/3.73	
11.17/3.73	The set Q consists of the following terms:
11.17/3.73	
11.17/3.73	   less_in_gg(x0, x1)
11.17/3.73	   U9_gg(x0, x1, x2)
11.17/3.73	
11.17/3.73	We have to consider all (P,Q,R)-chains.
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(57) MRRProof (EQUIVALENT)
11.17/3.73	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
11.17/3.73	
11.17/3.73	Strictly oriented dependency pairs:
11.17/3.73	
11.17/3.73	   U7_G(X, Y, Xs, less_out_gg(X, s(Y))) -> ORDERED_IN_G(.(Y, Xs))
11.17/3.73	   ORDERED_IN_G(.(X, .(Y, Xs))) -> U7_G(X, Y, Xs, less_in_gg(X, s(Y)))
11.17/3.73	
11.17/3.73	
11.17/3.73	Used ordering: Polynomial interpretation [POLO]:
11.17/3.73	
11.17/3.73	   POL(.(x_1, x_2)) = 2*x_1 + x_2
11.17/3.73	   POL(0) = 2
11.17/3.73	   POL(ORDERED_IN_G(x_1)) = 1 + 2*x_1
11.17/3.73	   POL(U7_G(x_1, x_2, x_3, x_4)) = 2*x_1 + 2*x_2 + 2*x_3 + x_4
11.17/3.73	   POL(U9_gg(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3
11.17/3.73	   POL(less_in_gg(x_1, x_2)) = 2*x_1 + x_2
11.17/3.73	   POL(less_out_gg(x_1, x_2)) = 2 + x_1 + x_2
11.17/3.73	   POL(s(x_1)) = 2*x_1
11.17/3.73	
11.17/3.73	
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(58)
11.17/3.73	Obligation:
11.17/3.73	Q DP problem:
11.17/3.73	P is empty.
11.17/3.73	The TRS R consists of the following rules:
11.17/3.73	
11.17/3.73	   less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2))
11.17/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y))
11.17/3.73	   U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
11.17/3.73	
11.17/3.73	The set Q consists of the following terms:
11.17/3.73	
11.17/3.73	   less_in_gg(x0, x1)
11.17/3.73	   U9_gg(x0, x1, x2)
11.17/3.73	
11.17/3.73	We have to consider all (P,Q,R)-chains.
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(59) PisEmptyProof (EQUIVALENT)
11.17/3.73	The TRS P is empty. Hence, there is no (P,Q,R) chain.
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(60)
11.17/3.73	YES
11.17/3.73	
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(61)
11.17/3.73	Obligation:
11.17/3.73	Pi DP problem:
11.17/3.73	The TRS P consists of the following rules:
11.17/3.73	
11.17/3.73	   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs)
11.17/3.73	
11.17/3.73	The TRS R consists of the following rules:
11.17/3.73	
11.17/3.73	   ss_in_ag(Xs, Ys) -> U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
11.17/3.73	   perm_in_ag([], []) -> perm_out_ag([], [])
11.17/3.73	   perm_in_ag(Xs, .(X, Ys)) -> U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.17/3.73	   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
11.17/3.73	   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.17/3.73	   U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
11.17/3.73	   U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.17/3.73	   U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
11.17/3.73	   U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) -> perm_out_ag(Xs, .(X, Ys))
11.17/3.73	   U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) -> U2_ag(Xs, Ys, ordered_in_g(Ys))
11.17/3.73	   ordered_in_g([]) -> ordered_out_g([])
11.17/3.73	   ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, []))
11.17/3.73	   ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
11.17/3.73	   less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2))
11.17/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y))
11.17/3.73	   U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
11.17/3.73	   U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
11.17/3.73	   U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs)))
11.17/3.73	   U2_ag(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ag(Xs, Ys)
11.17/3.73	
11.17/3.73	The argument filtering Pi contains the following mapping:
11.17/3.73	ss_in_ag(x1, x2)  =  ss_in_ag(x2)
11.17/3.73	
11.17/3.73	U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
11.17/3.73	
11.17/3.73	perm_in_ag(x1, x2)  =  perm_in_ag(x2)
11.17/3.73	
11.17/3.73	[]  =  []
11.17/3.73	
11.17/3.73	perm_out_ag(x1, x2)  =  perm_out_ag(x2)
11.17/3.73	
11.17/3.73	.(x1, x2)  =  .(x1, x2)
11.17/3.73	
11.17/3.73	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
11.17/3.73	
11.17/3.73	app_in_aaa(x1, x2, x3)  =  app_in_aaa
11.17/3.73	
11.17/3.73	app_out_aaa(x1, x2, x3)  =  app_out_aaa
11.17/3.73	
11.17/3.73	U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
11.17/3.73	
11.17/3.73	U4_ag(x1, x2, x3, x4)  =  U4_ag(x2, x3, x4)
11.17/3.73	
11.17/3.73	U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
11.17/3.73	
11.17/3.73	U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
11.17/3.73	
11.17/3.73	ordered_in_g(x1)  =  ordered_in_g(x1)
11.17/3.73	
11.17/3.73	ordered_out_g(x1)  =  ordered_out_g(x1)
11.17/3.73	
11.17/3.73	U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
11.17/3.73	
11.17/3.73	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
11.17/3.73	
11.17/3.73	0  =  0
11.17/3.73	
11.17/3.73	s(x1)  =  s(x1)
11.17/3.73	
11.17/3.73	less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
11.17/3.73	
11.17/3.73	U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
11.17/3.73	
11.17/3.73	U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
11.17/3.73	
11.17/3.73	ss_out_ag(x1, x2)  =  ss_out_ag(x2)
11.17/3.73	
11.17/3.73	APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
11.17/3.73	
11.17/3.73	
11.17/3.73	We have to consider all (P,R,Pi)-chains
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(62) UsableRulesProof (EQUIVALENT)
11.17/3.73	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(63)
11.17/3.73	Obligation:
11.17/3.73	Pi DP problem:
11.17/3.73	The TRS P consists of the following rules:
11.17/3.73	
11.17/3.73	   APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAA(Xs, Ys, Zs)
11.17/3.73	
11.17/3.73	R is empty.
11.17/3.73	The argument filtering Pi contains the following mapping:
11.17/3.73	.(x1, x2)  =  .(x1, x2)
11.17/3.73	
11.17/3.73	APP_IN_AAA(x1, x2, x3)  =  APP_IN_AAA
11.17/3.73	
11.17/3.73	
11.17/3.73	We have to consider all (P,R,Pi)-chains
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(64) PiDPToQDPProof (SOUND)
11.17/3.73	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(65)
11.17/3.73	Obligation:
11.17/3.73	Q DP problem:
11.17/3.73	The TRS P consists of the following rules:
11.17/3.73	
11.17/3.73	   APP_IN_AAA -> APP_IN_AAA
11.17/3.73	
11.17/3.73	R is empty.
11.17/3.73	Q is empty.
11.17/3.73	We have to consider all (P,Q,R)-chains.
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(66) NonTerminationLoopProof (COMPLETE)
11.17/3.73	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
11.17/3.73	Found a loop by semiunifying a rule from P directly.
11.17/3.73	
11.17/3.73	s = APP_IN_AAA evaluates to  t =APP_IN_AAA
11.17/3.73	
11.17/3.73	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
11.17/3.73	* Matcher: [ ]
11.17/3.73	* Semiunifier: [ ]
11.17/3.73	
11.17/3.73	--------------------------------------------------------------------------------
11.17/3.73	Rewriting sequence
11.17/3.73	
11.17/3.73	The DP semiunifies directly so there is only one rewrite step from APP_IN_AAA to APP_IN_AAA.
11.17/3.73	
11.17/3.73	
11.17/3.73	
11.17/3.73	
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(67)
11.17/3.73	NO
11.17/3.73	
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(68)
11.17/3.73	Obligation:
11.17/3.73	Pi DP problem:
11.17/3.73	The TRS P consists of the following rules:
11.17/3.73	
11.17/3.73	   U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.17/3.73	   U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> PERM_IN_AG(Zs, Ys)
11.17/3.73	   PERM_IN_AG(Xs, .(X, Ys)) -> U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.17/3.73	
11.17/3.73	The TRS R consists of the following rules:
11.17/3.73	
11.17/3.73	   ss_in_ag(Xs, Ys) -> U1_ag(Xs, Ys, perm_in_ag(Xs, Ys))
11.17/3.73	   perm_in_ag([], []) -> perm_out_ag([], [])
11.17/3.73	   perm_in_ag(Xs, .(X, Ys)) -> U3_ag(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.17/3.73	   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
11.17/3.73	   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.17/3.73	   U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
11.17/3.73	   U3_ag(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_ag(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.17/3.73	   U4_ag(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> U5_ag(Xs, X, Ys, perm_in_ag(Zs, Ys))
11.17/3.73	   U5_ag(Xs, X, Ys, perm_out_ag(Zs, Ys)) -> perm_out_ag(Xs, .(X, Ys))
11.17/3.73	   U1_ag(Xs, Ys, perm_out_ag(Xs, Ys)) -> U2_ag(Xs, Ys, ordered_in_g(Ys))
11.17/3.73	   ordered_in_g([]) -> ordered_out_g([])
11.17/3.73	   ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, []))
11.17/3.73	   ordered_in_g(.(X, .(Y, Xs))) -> U7_g(X, Y, Xs, less_in_gg(X, s(Y)))
11.17/3.73	   less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2))
11.17/3.73	   less_in_gg(s(X), s(Y)) -> U9_gg(X, Y, less_in_gg(X, Y))
11.17/3.73	   U9_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
11.17/3.73	   U7_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U8_g(X, Y, Xs, ordered_in_g(.(Y, Xs)))
11.17/3.73	   U8_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs)))
11.17/3.73	   U2_ag(Xs, Ys, ordered_out_g(Ys)) -> ss_out_ag(Xs, Ys)
11.17/3.73	
11.17/3.73	The argument filtering Pi contains the following mapping:
11.17/3.73	ss_in_ag(x1, x2)  =  ss_in_ag(x2)
11.17/3.73	
11.17/3.73	U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
11.17/3.73	
11.17/3.73	perm_in_ag(x1, x2)  =  perm_in_ag(x2)
11.17/3.73	
11.17/3.73	[]  =  []
11.17/3.73	
11.17/3.73	perm_out_ag(x1, x2)  =  perm_out_ag(x2)
11.17/3.73	
11.17/3.73	.(x1, x2)  =  .(x1, x2)
11.17/3.73	
11.17/3.73	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
11.17/3.73	
11.17/3.73	app_in_aaa(x1, x2, x3)  =  app_in_aaa
11.17/3.73	
11.17/3.73	app_out_aaa(x1, x2, x3)  =  app_out_aaa
11.17/3.73	
11.17/3.73	U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
11.17/3.73	
11.17/3.73	U4_ag(x1, x2, x3, x4)  =  U4_ag(x2, x3, x4)
11.17/3.73	
11.17/3.73	U5_ag(x1, x2, x3, x4)  =  U5_ag(x2, x3, x4)
11.17/3.73	
11.17/3.73	U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
11.17/3.73	
11.17/3.73	ordered_in_g(x1)  =  ordered_in_g(x1)
11.17/3.73	
11.17/3.73	ordered_out_g(x1)  =  ordered_out_g(x1)
11.17/3.73	
11.17/3.73	U7_g(x1, x2, x3, x4)  =  U7_g(x1, x2, x3, x4)
11.17/3.73	
11.17/3.73	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
11.17/3.73	
11.17/3.73	0  =  0
11.17/3.73	
11.17/3.73	s(x1)  =  s(x1)
11.17/3.73	
11.17/3.73	less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
11.17/3.73	
11.17/3.73	U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
11.17/3.73	
11.17/3.73	U8_g(x1, x2, x3, x4)  =  U8_g(x1, x2, x3, x4)
11.17/3.73	
11.17/3.73	ss_out_ag(x1, x2)  =  ss_out_ag(x2)
11.17/3.73	
11.17/3.73	PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
11.17/3.73	
11.17/3.73	U3_AG(x1, x2, x3, x4)  =  U3_AG(x2, x3, x4)
11.17/3.73	
11.17/3.73	U4_AG(x1, x2, x3, x4)  =  U4_AG(x2, x3, x4)
11.17/3.73	
11.17/3.73	
11.17/3.73	We have to consider all (P,R,Pi)-chains
11.17/3.73	----------------------------------------
11.17/3.73	
11.17/3.73	(69) UsableRulesProof (EQUIVALENT)
11.17/3.74	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
11.17/3.74	----------------------------------------
11.17/3.74	
11.17/3.74	(70)
11.17/3.74	Obligation:
11.17/3.74	Pi DP problem:
11.17/3.74	The TRS P consists of the following rules:
11.17/3.74	
11.17/3.74	   U3_AG(Xs, X, Ys, app_out_aaa(X1s, .(X, X2s), Xs)) -> U4_AG(Xs, X, Ys, app_in_aaa(X1s, X2s, Zs))
11.17/3.74	   U4_AG(Xs, X, Ys, app_out_aaa(X1s, X2s, Zs)) -> PERM_IN_AG(Zs, Ys)
11.17/3.74	   PERM_IN_AG(Xs, .(X, Ys)) -> U3_AG(Xs, X, Ys, app_in_aaa(X1s, .(X, X2s), Xs))
11.17/3.74	
11.17/3.74	The TRS R consists of the following rules:
11.17/3.74	
11.17/3.74	   app_in_aaa([], X, X) -> app_out_aaa([], X, X)
11.17/3.74	   app_in_aaa(.(X, Xs), Ys, .(X, Zs)) -> U6_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
11.17/3.74	   U6_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) -> app_out_aaa(.(X, Xs), Ys, .(X, Zs))
11.17/3.74	
11.17/3.74	The argument filtering Pi contains the following mapping:
11.17/3.74	[]  =  []
11.17/3.74	
11.17/3.74	.(x1, x2)  =  .(x1, x2)
11.17/3.74	
11.17/3.74	app_in_aaa(x1, x2, x3)  =  app_in_aaa
11.17/3.74	
11.17/3.74	app_out_aaa(x1, x2, x3)  =  app_out_aaa
11.17/3.74	
11.17/3.74	U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
11.17/3.74	
11.17/3.74	PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
11.17/3.74	
11.17/3.74	U3_AG(x1, x2, x3, x4)  =  U3_AG(x2, x3, x4)
11.17/3.74	
11.17/3.74	U4_AG(x1, x2, x3, x4)  =  U4_AG(x2, x3, x4)
11.17/3.74	
11.17/3.74	
11.17/3.74	We have to consider all (P,R,Pi)-chains
11.17/3.74	----------------------------------------
11.17/3.74	
11.17/3.74	(71) PiDPToQDPProof (SOUND)
11.17/3.74	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
11.17/3.74	----------------------------------------
11.17/3.74	
11.17/3.74	(72)
11.17/3.74	Obligation:
11.17/3.74	Q DP problem:
11.17/3.74	The TRS P consists of the following rules:
11.17/3.74	
11.17/3.74	   U3_AG(X, Ys, app_out_aaa) -> U4_AG(X, Ys, app_in_aaa)
11.17/3.74	   U4_AG(X, Ys, app_out_aaa) -> PERM_IN_AG(Ys)
11.17/3.74	   PERM_IN_AG(.(X, Ys)) -> U3_AG(X, Ys, app_in_aaa)
11.17/3.74	
11.17/3.74	The TRS R consists of the following rules:
11.17/3.74	
11.17/3.74	   app_in_aaa -> app_out_aaa
11.17/3.74	   app_in_aaa -> U6_aaa(app_in_aaa)
11.17/3.74	   U6_aaa(app_out_aaa) -> app_out_aaa
11.17/3.74	
11.17/3.74	The set Q consists of the following terms:
11.17/3.74	
11.17/3.74	   app_in_aaa
11.17/3.74	   U6_aaa(x0)
11.17/3.74	
11.17/3.74	We have to consider all (P,Q,R)-chains.
11.17/3.74	----------------------------------------
11.17/3.74	
11.17/3.74	(73) QDPSizeChangeProof (EQUIVALENT)
11.17/3.74	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. 
11.17/3.74	
11.17/3.74	From the DPs we obtained the following set of size-change graphs:
11.17/3.74	*U4_AG(X, Ys, app_out_aaa) -> PERM_IN_AG(Ys)
11.17/3.74	The graph contains the following edges 2 >= 1
11.17/3.74	
11.17/3.74	
11.17/3.74	*PERM_IN_AG(.(X, Ys)) -> U3_AG(X, Ys, app_in_aaa)
11.17/3.74	The graph contains the following edges 1 > 1, 1 > 2
11.17/3.74	
11.17/3.74	
11.17/3.74	*U3_AG(X, Ys, app_out_aaa) -> U4_AG(X, Ys, app_in_aaa)
11.17/3.74	The graph contains the following edges 1 >= 1, 2 >= 2
11.17/3.74	
11.17/3.74	
11.17/3.74	----------------------------------------
11.17/3.74	
11.17/3.74	(74)
11.17/3.74	YES
11.17/3.74	
11.17/3.74	----------------------------------------
11.17/3.74	
11.17/3.74	(75) PrologToDTProblemTransformerProof (SOUND)
11.17/3.74	Built DT problem from termination graph DT10.
11.17/3.74	
11.17/3.74	{
11.17/3.74	    "root": 4,
11.17/3.74	    "program": {
11.17/3.74	        "directives": [],
11.17/3.74	        "clauses": [
11.17/3.74	            [
11.17/3.74	                "(ss Xs Ys)",
11.17/3.74	                "(',' (perm Xs Ys) (ordered Ys))"
11.17/3.74	            ],
11.17/3.74	            [
11.17/3.74	                "(perm ([]) ([]))",
11.17/3.74	                null
11.17/3.74	            ],
11.17/3.74	            [
11.17/3.74	                "(perm Xs (. X Ys))",
11.17/3.74	                "(',' (app X1s (. X X2s) Xs) (',' (app X1s X2s Zs) (perm Zs Ys)))"
11.17/3.74	            ],
11.17/3.74	            [
11.17/3.74	                "(app ([]) X X)",
11.17/3.74	                null
11.17/3.74	            ],
11.17/3.74	            [
11.17/3.74	                "(app (. X Xs) Ys (. X Zs))",
11.17/3.74	                "(app Xs Ys Zs)"
11.17/3.74	            ],
11.17/3.74	            [
11.17/3.74	                "(ordered ([]))",
11.17/3.74	                null
11.17/3.74	            ],
11.17/3.74	            [
11.17/3.74	                "(ordered (. X1 ([])))",
11.17/3.74	                null
11.17/3.74	            ],
11.17/3.74	            [
11.17/3.74	                "(ordered (. X (. Y Xs)))",
11.17/3.74	                "(',' (less X (s Y)) (ordered (. Y Xs)))"
11.17/3.74	            ],
11.17/3.74	            [
11.17/3.74	                "(less (0) (s X2))",
11.17/3.74	                null
11.17/3.74	            ],
11.17/3.74	            [
11.17/3.74	                "(less (s X) (s Y))",
11.17/3.74	                "(less X Y)"
11.17/3.74	            ]
11.17/3.74	        ]
11.17/3.74	    },
11.17/3.74	    "graph": {
11.17/3.74	        "nodes": {
11.17/3.74	            "390": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "391": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(',' (',' (app X23 (. T15 X24) T17) (',' (app X23 X24 X25) (perm X25 T16))) (ordered (. T15 T16)))"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [
11.17/3.74	                        "T15",
11.17/3.74	                        "T16"
11.17/3.74	                    ],
11.17/3.74	                    "free": [
11.17/3.74	                        "X23",
11.17/3.74	                        "X24",
11.17/3.74	                        "X25"
11.17/3.74	                    ],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "type": "Nodes",
11.17/3.74	            "392": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "470": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": 1,
11.17/3.74	                    "scope": 6,
11.17/3.74	                    "term": "(perm T54 T16)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": ["T16"],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "471": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": 2,
11.17/3.74	                    "scope": 6,
11.17/3.74	                    "term": "(perm T54 T16)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": ["T16"],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "395": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(app X23 (. T15 X24) T17)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": ["T15"],
11.17/3.74	                    "free": [
11.17/3.74	                        "X23",
11.17/3.74	                        "X24"
11.17/3.74	                    ],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "472": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(true)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "396": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(',' (',' (app T22 T23 X25) (perm X25 T16)) (ordered (. T15 T16)))"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [
11.17/3.74	                        "T15",
11.17/3.74	                        "T16"
11.17/3.74	                    ],
11.17/3.74	                    "free": ["X25"],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "473": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "550": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": 7,
11.17/3.74	                    "scope": 7,
11.17/3.74	                    "term": "(ordered (. T15 T16))"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [
11.17/3.74	                        "T15",
11.17/3.74	                        "T16"
11.17/3.74	                    ],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "474": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "551": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(true)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "552": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "553": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "554": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(',' (less T111 (s T112)) (ordered (. T112 T113)))"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [
11.17/3.74	                        "T111",
11.17/3.74	                        "T112",
11.17/3.74	                        "T113"
11.17/3.74	                    ],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "434": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(app T22 T23 X25)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": ["X25"],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "555": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "435": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(',' (perm T54 T16) (ordered (. T15 T16)))"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [
11.17/3.74	                        "T15",
11.17/3.74	                        "T16"
11.17/3.74	                    ],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "556": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(less T111 (s T112))"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [
11.17/3.74	                        "T111",
11.17/3.74	                        "T112"
11.17/3.74	                    ],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "557": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(ordered (. T112 T113))"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [
11.17/3.74	                        "T112",
11.17/3.74	                        "T113"
11.17/3.74	                    ],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "558": {
11.17/3.74	                "goal": [
11.17/3.74	                    {
11.17/3.74	                        "clause": 8,
11.17/3.74	                        "scope": 8,
11.17/3.74	                        "term": "(less T111 (s T112))"
11.17/3.74	                    },
11.17/3.74	                    {
11.17/3.74	                        "clause": 9,
11.17/3.74	                        "scope": 8,
11.17/3.74	                        "term": "(less T111 (s T112))"
11.17/3.74	                    }
11.17/3.74	                ],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [
11.17/3.74	                        "T111",
11.17/3.74	                        "T112"
11.17/3.74	                    ],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "438": {
11.17/3.74	                "goal": [
11.17/3.74	                    {
11.17/3.74	                        "clause": 3,
11.17/3.74	                        "scope": 5,
11.17/3.74	                        "term": "(app T22 T23 X25)"
11.17/3.74	                    },
11.17/3.74	                    {
11.17/3.74	                        "clause": 4,
11.17/3.74	                        "scope": 5,
11.17/3.74	                        "term": "(app T22 T23 X25)"
11.17/3.74	                    }
11.17/3.74	                ],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": ["X25"],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "559": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": 8,
11.17/3.74	                    "scope": 8,
11.17/3.74	                    "term": "(less T111 (s T112))"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [
11.17/3.74	                        "T111",
11.17/3.74	                        "T112"
11.17/3.74	                    ],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "481": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(',' (app X107 (. T82 X108) T84) (',' (app X107 X108 X109) (perm X109 T83)))"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [
11.17/3.74	                        "T82",
11.17/3.74	                        "T83"
11.17/3.74	                    ],
11.17/3.74	                    "free": [
11.17/3.74	                        "X107",
11.17/3.74	                        "X108",
11.17/3.74	                        "X109"
11.17/3.74	                    ],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "482": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "560": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": 9,
11.17/3.74	                    "scope": 8,
11.17/3.74	                    "term": "(less T111 (s T112))"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [
11.17/3.74	                        "T111",
11.17/3.74	                        "T112"
11.17/3.74	                    ],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "440": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": 3,
11.17/3.74	                    "scope": 5,
11.17/3.74	                    "term": "(app T22 T23 X25)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": ["X25"],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "561": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(true)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "441": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": 4,
11.17/3.74	                    "scope": 5,
11.17/3.74	                    "term": "(app T22 T23 X25)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": ["X25"],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "562": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "563": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "564": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(less T127 T128)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [
11.17/3.74	                        "T127",
11.17/3.74	                        "T128"
11.17/3.74	                    ],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "4": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(ss T1 T2)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": ["T2"],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "400": {
11.17/3.74	                "goal": [
11.17/3.74	                    {
11.17/3.74	                        "clause": 3,
11.17/3.74	                        "scope": 4,
11.17/3.74	                        "term": "(app X23 (. T15 X24) T17)"
11.17/3.74	                    },
11.17/3.74	                    {
11.17/3.74	                        "clause": 4,
11.17/3.74	                        "scope": 4,
11.17/3.74	                        "term": "(app X23 (. T15 X24) T17)"
11.17/3.74	                    }
11.17/3.74	                ],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": ["T15"],
11.17/3.74	                    "free": [
11.17/3.74	                        "X23",
11.17/3.74	                        "X24"
11.17/3.74	                    ],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "444": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(true)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "488": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(app X107 (. T82 X108) T84)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": ["T82"],
11.17/3.74	                    "free": [
11.17/3.74	                        "X107",
11.17/3.74	                        "X108"
11.17/3.74	                    ],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "565": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "5": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": 0,
11.17/3.74	                    "scope": 1,
11.17/3.74	                    "term": "(ss T1 T2)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": ["T2"],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "445": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "489": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(',' (app T89 T90 X109) (perm X109 T83))"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": ["T83"],
11.17/3.74	                    "free": ["X109"],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "566": {
11.17/3.74	                "goal": [
11.17/3.74	                    {
11.17/3.74	                        "clause": 8,
11.17/3.74	                        "scope": 9,
11.17/3.74	                        "term": "(less T127 T128)"
11.17/3.74	                    },
11.17/3.74	                    {
11.17/3.74	                        "clause": 9,
11.17/3.74	                        "scope": 9,
11.17/3.74	                        "term": "(less T127 T128)"
11.17/3.74	                    }
11.17/3.74	                ],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [
11.17/3.74	                        "T127",
11.17/3.74	                        "T128"
11.17/3.74	                    ],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "446": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "567": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": 8,
11.17/3.74	                    "scope": 9,
11.17/3.74	                    "term": "(less T127 T128)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [
11.17/3.74	                        "T127",
11.17/3.74	                        "T128"
11.17/3.74	                    ],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "568": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": 9,
11.17/3.74	                    "scope": 9,
11.17/3.74	                    "term": "(less T127 T128)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [
11.17/3.74	                        "T127",
11.17/3.74	                        "T128"
11.17/3.74	                    ],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "404": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": 3,
11.17/3.74	                    "scope": 4,
11.17/3.74	                    "term": "(app X23 (. T15 X24) T17)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": ["T15"],
11.17/3.74	                    "free": [
11.17/3.74	                        "X23",
11.17/3.74	                        "X24"
11.17/3.74	                    ],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "569": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(true)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "405": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": 4,
11.17/3.74	                    "scope": 4,
11.17/3.74	                    "term": "(app X23 (. T15 X24) T17)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": ["T15"],
11.17/3.74	                    "free": [
11.17/3.74	                        "X23",
11.17/3.74	                        "X24"
11.17/3.74	                    ],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "408": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(true)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "409": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "570": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "571": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "572": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(less T140 T141)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [
11.17/3.74	                        "T140",
11.17/3.74	                        "T141"
11.17/3.74	                    ],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "573": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "410": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "414": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(app X63 (. T44 X64) T47)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": ["T44"],
11.17/3.74	                    "free": [
11.17/3.74	                        "X63",
11.17/3.74	                        "X64"
11.17/3.74	                    ],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "458": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(app T71 T72 X92)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": ["X92"],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "71": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(',' (perm T7 T6) (ordered T6))"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": ["T6"],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "417": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "72": {
11.17/3.74	                "goal": [
11.17/3.74	                    {
11.17/3.74	                        "clause": 1,
11.17/3.74	                        "scope": 2,
11.17/3.74	                        "term": "(',' (perm T7 T6) (ordered T6))"
11.17/3.74	                    },
11.17/3.74	                    {
11.17/3.74	                        "clause": 2,
11.17/3.74	                        "scope": 2,
11.17/3.74	                        "term": "(',' (perm T7 T6) (ordered T6))"
11.17/3.74	                    }
11.17/3.74	                ],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": ["T6"],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "73": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": 1,
11.17/3.74	                    "scope": 2,
11.17/3.74	                    "term": "(',' (perm T7 T6) (ordered T6))"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": ["T6"],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "74": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": 2,
11.17/3.74	                    "scope": 2,
11.17/3.74	                    "term": "(',' (perm T7 T6) (ordered T6))"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": ["T6"],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "76": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(ordered ([]))"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "77": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "460": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "384": {
11.17/3.74	                "goal": [
11.17/3.74	                    {
11.17/3.74	                        "clause": 5,
11.17/3.74	                        "scope": 3,
11.17/3.74	                        "term": "(ordered ([]))"
11.17/3.74	                    },
11.17/3.74	                    {
11.17/3.74	                        "clause": 6,
11.17/3.74	                        "scope": 3,
11.17/3.74	                        "term": "(ordered ([]))"
11.17/3.74	                    },
11.17/3.74	                    {
11.17/3.74	                        "clause": 7,
11.17/3.74	                        "scope": 3,
11.17/3.74	                        "term": "(ordered ([]))"
11.17/3.74	                    }
11.17/3.74	                ],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "385": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": 5,
11.17/3.74	                    "scope": 3,
11.17/3.74	                    "term": "(ordered ([]))"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "386": {
11.17/3.74	                "goal": [
11.17/3.74	                    {
11.17/3.74	                        "clause": 6,
11.17/3.74	                        "scope": 3,
11.17/3.74	                        "term": "(ordered ([]))"
11.17/3.74	                    },
11.17/3.74	                    {
11.17/3.74	                        "clause": 7,
11.17/3.74	                        "scope": 3,
11.17/3.74	                        "term": "(ordered ([]))"
11.17/3.74	                    }
11.17/3.74	                ],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "540": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(app T89 T90 X109)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": ["X109"],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "387": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(true)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "541": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(perm T97 T83)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": ["T83"],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "388": {
11.17/3.74	                "goal": [],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "465": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(perm T54 T16)"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": ["T16"],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "389": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": 7,
11.17/3.74	                    "scope": 3,
11.17/3.74	                    "term": "(ordered ([]))"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "466": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": -1,
11.17/3.74	                    "scope": -1,
11.17/3.74	                    "term": "(ordered (. T15 T16))"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [
11.17/3.74	                        "T15",
11.17/3.74	                        "T16"
11.17/3.74	                    ],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "469": {
11.17/3.74	                "goal": [
11.17/3.74	                    {
11.17/3.74	                        "clause": 1,
11.17/3.74	                        "scope": 6,
11.17/3.74	                        "term": "(perm T54 T16)"
11.17/3.74	                    },
11.17/3.74	                    {
11.17/3.74	                        "clause": 2,
11.17/3.74	                        "scope": 6,
11.17/3.74	                        "term": "(perm T54 T16)"
11.17/3.74	                    }
11.17/3.74	                ],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": ["T16"],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "547": {
11.17/3.74	                "goal": [
11.17/3.74	                    {
11.17/3.74	                        "clause": 5,
11.17/3.74	                        "scope": 7,
11.17/3.74	                        "term": "(ordered (. T15 T16))"
11.17/3.74	                    },
11.17/3.74	                    {
11.17/3.74	                        "clause": 6,
11.17/3.74	                        "scope": 7,
11.17/3.74	                        "term": "(ordered (. T15 T16))"
11.17/3.74	                    },
11.17/3.74	                    {
11.17/3.74	                        "clause": 7,
11.17/3.74	                        "scope": 7,
11.17/3.74	                        "term": "(ordered (. T15 T16))"
11.17/3.74	                    }
11.17/3.74	                ],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [
11.17/3.74	                        "T15",
11.17/3.74	                        "T16"
11.17/3.74	                    ],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "548": {
11.17/3.74	                "goal": [
11.17/3.74	                    {
11.17/3.74	                        "clause": 6,
11.17/3.74	                        "scope": 7,
11.17/3.74	                        "term": "(ordered (. T15 T16))"
11.17/3.74	                    },
11.17/3.74	                    {
11.17/3.74	                        "clause": 7,
11.17/3.74	                        "scope": 7,
11.17/3.74	                        "term": "(ordered (. T15 T16))"
11.17/3.74	                    }
11.17/3.74	                ],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [
11.17/3.74	                        "T15",
11.17/3.74	                        "T16"
11.17/3.74	                    ],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            },
11.17/3.74	            "549": {
11.17/3.74	                "goal": [{
11.17/3.74	                    "clause": 6,
11.17/3.74	                    "scope": 7,
11.17/3.74	                    "term": "(ordered (. T15 T16))"
11.17/3.74	                }],
11.17/3.74	                "kb": {
11.17/3.74	                    "nonunifying": [],
11.17/3.74	                    "intvars": {},
11.17/3.74	                    "arithmetic": {
11.17/3.74	                        "type": "PlainIntegerRelationState",
11.17/3.74	                        "relations": []
11.17/3.74	                    },
11.17/3.74	                    "ground": [
11.17/3.74	                        "T15",
11.17/3.74	                        "T16"
11.17/3.74	                    ],
11.17/3.74	                    "free": [],
11.17/3.74	                    "exprvars": []
11.17/3.74	                }
11.17/3.74	            }
11.17/3.74	        },
11.17/3.74	        "edges": [
11.17/3.74	            {
11.17/3.74	                "from": 4,
11.17/3.74	                "to": 5,
11.17/3.74	                "label": "CASE"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 5,
11.17/3.74	                "to": 71,
11.17/3.74	                "label": "ONLY EVAL with clause\nss(X5, X6) :- ','(perm(X5, X6), ordered(X6)).\nand substitutionT1 -> T7,\nX5 -> T7,\nT2 -> T6,\nX6 -> T6,\nT5 -> T7"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 71,
11.17/3.74	                "to": 72,
11.17/3.74	                "label": "CASE"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 72,
11.17/3.74	                "to": 73,
11.17/3.74	                "label": "PARALLEL"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 72,
11.17/3.74	                "to": 74,
11.17/3.74	                "label": "PARALLEL"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 73,
11.17/3.74	                "to": 76,
11.17/3.74	                "label": "EVAL with clause\nperm([], []).\nand substitutionT7 -> [],\nT6 -> []"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 73,
11.17/3.74	                "to": 77,
11.17/3.74	                "label": "EVAL-BACKTRACK"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 74,
11.17/3.74	                "to": 391,
11.17/3.74	                "label": "EVAL with clause\nperm(X20, .(X21, X22)) :- ','(app(X23, .(X21, X24), X20), ','(app(X23, X24, X25), perm(X25, X22))).\nand substitutionT7 -> T17,\nX20 -> T17,\nX21 -> T15,\nX22 -> T16,\nT6 -> .(T15, T16),\nT14 -> T17"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 74,
11.17/3.74	                "to": 392,
11.17/3.74	                "label": "EVAL-BACKTRACK"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 76,
11.17/3.74	                "to": 384,
11.17/3.74	                "label": "CASE"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 384,
11.17/3.74	                "to": 385,
11.17/3.74	                "label": "PARALLEL"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 384,
11.17/3.74	                "to": 386,
11.17/3.74	                "label": "PARALLEL"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 385,
11.17/3.74	                "to": 387,
11.17/3.74	                "label": "ONLY EVAL with clause\nordered([]).\nand substitution"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 386,
11.17/3.74	                "to": 389,
11.17/3.74	                "label": "BACKTRACK\nfor clause: ordered(.(X1, []))because of non-unification"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 387,
11.17/3.74	                "to": 388,
11.17/3.74	                "label": "SUCCESS"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 389,
11.17/3.74	                "to": 390,
11.17/3.74	                "label": "BACKTRACK\nfor clause: ordered(.(X, .(Y, Xs))) :- ','(less(X, s(Y)), ordered(.(Y, Xs)))because of non-unification"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 391,
11.17/3.74	                "to": 395,
11.17/3.74	                "label": "SPLIT 1"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 391,
11.17/3.74	                "to": 396,
11.17/3.74	                "label": "SPLIT 2\nreplacements:X23 -> T22,\nX24 -> T23"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 395,
11.17/3.74	                "to": 400,
11.17/3.74	                "label": "CASE"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 396,
11.17/3.74	                "to": 434,
11.17/3.74	                "label": "SPLIT 1"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 396,
11.17/3.74	                "to": 435,
11.17/3.74	                "label": "SPLIT 2\nreplacements:X25 -> T54"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 400,
11.17/3.74	                "to": 404,
11.17/3.74	                "label": "PARALLEL"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 400,
11.17/3.74	                "to": 405,
11.17/3.74	                "label": "PARALLEL"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 404,
11.17/3.74	                "to": 408,
11.17/3.74	                "label": "EVAL with clause\napp([], X42, X42).\nand substitutionX23 -> [],\nT15 -> T36,\nX24 -> T37,\nX42 -> .(T36, T37),\nX43 -> T37,\nT17 -> .(T36, T37)"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 404,
11.17/3.74	                "to": 409,
11.17/3.74	                "label": "EVAL-BACKTRACK"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 405,
11.17/3.74	                "to": 414,
11.17/3.74	                "label": "EVAL with clause\napp(.(X58, X59), X60, .(X58, X61)) :- app(X59, X60, X61).\nand substitutionX58 -> T45,\nX59 -> X63,\nX23 -> .(T45, X63),\nT15 -> T44,\nX24 -> X64,\nX60 -> .(T44, X64),\nX62 -> T45,\nX61 -> T47,\nT17 -> .(T45, T47),\nT46 -> T47"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 405,
11.17/3.74	                "to": 417,
11.17/3.74	                "label": "EVAL-BACKTRACK"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 408,
11.17/3.74	                "to": 410,
11.17/3.74	                "label": "SUCCESS"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 414,
11.17/3.74	                "to": 395,
11.17/3.74	                "label": "INSTANCE with matching:\nX23 -> X63\nT15 -> T44\nX24 -> X64\nT17 -> T47"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 434,
11.17/3.74	                "to": 438,
11.17/3.74	                "label": "CASE"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 435,
11.17/3.74	                "to": 465,
11.17/3.74	                "label": "SPLIT 1"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 435,
11.17/3.74	                "to": 466,
11.17/3.74	                "label": "SPLIT 2\nnew knowledge:\nT16 is ground"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 438,
11.17/3.74	                "to": 440,
11.17/3.74	                "label": "PARALLEL"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 438,
11.17/3.74	                "to": 441,
11.17/3.74	                "label": "PARALLEL"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 440,
11.17/3.74	                "to": 444,
11.17/3.74	                "label": "EVAL with clause\napp([], X77, X77).\nand substitutionT22 -> [],\nT23 -> T61,\nX77 -> T61,\nX25 -> T61"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 440,
11.17/3.74	                "to": 445,
11.17/3.74	                "label": "EVAL-BACKTRACK"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 441,
11.17/3.74	                "to": 458,
11.17/3.74	                "label": "EVAL with clause\napp(.(X88, X89), X90, .(X88, X91)) :- app(X89, X90, X91).\nand substitutionX88 -> T68,\nX89 -> T71,\nT22 -> .(T68, T71),\nT23 -> T72,\nX90 -> T72,\nX91 -> X92,\nX25 -> .(T68, X92),\nT69 -> T71,\nT70 -> T72"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 441,
11.17/3.74	                "to": 460,
11.17/3.74	                "label": "EVAL-BACKTRACK"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 444,
11.17/3.74	                "to": 446,
11.17/3.74	                "label": "SUCCESS"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 458,
11.17/3.74	                "to": 434,
11.17/3.74	                "label": "INSTANCE with matching:\nT22 -> T71\nT23 -> T72\nX25 -> X92"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 465,
11.17/3.74	                "to": 469,
11.17/3.74	                "label": "CASE"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 466,
11.17/3.74	                "to": 547,
11.17/3.74	                "label": "CASE"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 469,
11.17/3.74	                "to": 470,
11.17/3.74	                "label": "PARALLEL"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 469,
11.17/3.74	                "to": 471,
11.17/3.74	                "label": "PARALLEL"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 470,
11.17/3.74	                "to": 472,
11.17/3.74	                "label": "EVAL with clause\nperm([], []).\nand substitutionT54 -> [],\nT16 -> []"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 470,
11.17/3.74	                "to": 473,
11.17/3.74	                "label": "EVAL-BACKTRACK"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 471,
11.17/3.74	                "to": 481,
11.17/3.74	                "label": "EVAL with clause\nperm(X104, .(X105, X106)) :- ','(app(X107, .(X105, X108), X104), ','(app(X107, X108, X109), perm(X109, X106))).\nand substitutionT54 -> T84,\nX104 -> T84,\nX105 -> T82,\nX106 -> T83,\nT16 -> .(T82, T83),\nT81 -> T84"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 471,
11.17/3.74	                "to": 482,
11.17/3.74	                "label": "EVAL-BACKTRACK"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 472,
11.17/3.74	                "to": 474,
11.17/3.74	                "label": "SUCCESS"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 481,
11.17/3.74	                "to": 488,
11.17/3.74	                "label": "SPLIT 1"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 481,
11.17/3.74	                "to": 489,
11.17/3.74	                "label": "SPLIT 2\nreplacements:X107 -> T89,\nX108 -> T90"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 488,
11.17/3.74	                "to": 395,
11.17/3.74	                "label": "INSTANCE with matching:\nX23 -> X107\nT15 -> T82\nX24 -> X108\nT17 -> T84"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 489,
11.17/3.74	                "to": 540,
11.17/3.74	                "label": "SPLIT 1"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 489,
11.17/3.74	                "to": 541,
11.17/3.74	                "label": "SPLIT 2\nreplacements:X109 -> T97"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 540,
11.17/3.74	                "to": 434,
11.17/3.74	                "label": "INSTANCE with matching:\nT22 -> T89\nT23 -> T90\nX25 -> X109"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 541,
11.17/3.74	                "to": 465,
11.17/3.74	                "label": "INSTANCE with matching:\nT54 -> T97\nT16 -> T83"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 547,
11.17/3.74	                "to": 548,
11.17/3.74	                "label": "BACKTRACK\nfor clause: ordered([])because of non-unification"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 548,
11.17/3.74	                "to": 549,
11.17/3.74	                "label": "PARALLEL"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 548,
11.17/3.74	                "to": 550,
11.17/3.74	                "label": "PARALLEL"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 549,
11.17/3.74	                "to": 551,
11.17/3.74	                "label": "EVAL with clause\nordered(.(X126, [])).\nand substitutionT15 -> T104,\nX126 -> T104,\nT16 -> []"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 549,
11.17/3.74	                "to": 552,
11.17/3.74	                "label": "EVAL-BACKTRACK"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 550,
11.17/3.74	                "to": 554,
11.17/3.74	                "label": "EVAL with clause\nordered(.(X133, .(X134, X135))) :- ','(less(X133, s(X134)), ordered(.(X134, X135))).\nand substitutionT15 -> T111,\nX133 -> T111,\nX134 -> T112,\nX135 -> T113,\nT16 -> .(T112, T113)"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 550,
11.17/3.74	                "to": 555,
11.17/3.74	                "label": "EVAL-BACKTRACK"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 551,
11.17/3.74	                "to": 553,
11.17/3.74	                "label": "SUCCESS"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 554,
11.17/3.74	                "to": 556,
11.17/3.74	                "label": "SPLIT 1"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 554,
11.17/3.74	                "to": 557,
11.17/3.74	                "label": "SPLIT 2\nnew knowledge:\nT111 is ground\nT112 is ground"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 556,
11.17/3.74	                "to": 558,
11.17/3.74	                "label": "CASE"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 557,
11.17/3.74	                "to": 466,
11.17/3.74	                "label": "INSTANCE with matching:\nT15 -> T112\nT16 -> T113"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 558,
11.17/3.74	                "to": 559,
11.17/3.74	                "label": "PARALLEL"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 558,
11.17/3.74	                "to": 560,
11.17/3.74	                "label": "PARALLEL"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 559,
11.17/3.74	                "to": 561,
11.17/3.74	                "label": "EVAL with clause\nless(0, s(X144)).\nand substitutionT111 -> 0,\nT112 -> T122,\nX144 -> T122"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 559,
11.17/3.74	                "to": 562,
11.17/3.74	                "label": "EVAL-BACKTRACK"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 560,
11.17/3.74	                "to": 564,
11.17/3.74	                "label": "EVAL with clause\nless(s(X149), s(X150)) :- less(X149, X150).\nand substitutionX149 -> T127,\nT111 -> s(T127),\nT112 -> T128,\nX150 -> T128"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 560,
11.17/3.74	                "to": 565,
11.17/3.74	                "label": "EVAL-BACKTRACK"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 561,
11.17/3.74	                "to": 563,
11.17/3.74	                "label": "SUCCESS"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 564,
11.17/3.74	                "to": 566,
11.17/3.74	                "label": "CASE"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 566,
11.17/3.74	                "to": 567,
11.17/3.74	                "label": "PARALLEL"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 566,
11.17/3.74	                "to": 568,
11.17/3.74	                "label": "PARALLEL"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 567,
11.17/3.74	                "to": 569,
11.17/3.74	                "label": "EVAL with clause\nless(0, s(X157)).\nand substitutionT127 -> 0,\nX157 -> T135,\nT128 -> s(T135)"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 567,
11.17/3.74	                "to": 570,
11.17/3.74	                "label": "EVAL-BACKTRACK"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 568,
11.17/3.74	                "to": 572,
11.17/3.74	                "label": "EVAL with clause\nless(s(X162), s(X163)) :- less(X162, X163).\nand substitutionX162 -> T140,\nT127 -> s(T140),\nX163 -> T141,\nT128 -> s(T141)"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 568,
11.17/3.74	                "to": 573,
11.17/3.74	                "label": "EVAL-BACKTRACK"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 569,
11.17/3.74	                "to": 571,
11.17/3.74	                "label": "SUCCESS"
11.17/3.74	            },
11.17/3.74	            {
11.17/3.74	                "from": 572,
11.17/3.74	                "to": 564,
11.17/3.74	                "label": "INSTANCE with matching:\nT127 -> T140\nT128 -> T141"
11.17/3.74	            }
11.17/3.74	        ],
11.17/3.74	        "type": "Graph"
11.17/3.74	    }
11.17/3.74	}
11.17/3.74	
11.17/3.74	----------------------------------------
11.17/3.74	
11.17/3.74	(76)
11.17/3.74	Obligation:
11.17/3.74	Triples:
11.17/3.74	
11.17/3.74	appA(.(X1, X2), X3, X4, .(X1, X5)) :- appA(X2, X3, X4, X5).
11.17/3.74	appB(.(X1, X2), X3, .(X1, X4)) :- appB(X2, X3, X4).
11.17/3.74	permC(X1, .(X2, X3)) :- appA(X4, X2, X5, X1).
11.17/3.74	permC(X1, .(X2, X3)) :- ','(appcA(X4, X2, X5, X1), appB(X4, X5, X6)).
11.17/3.74	permC(X1, .(X2, X3)) :- ','(appcA(X4, X2, X5, X1), ','(appcB(X4, X5, X6), permC(X6, X3))).
11.17/3.74	orderedD(s(X1), .(X2, X3)) :- lessF(X1, X2).
11.17/3.74	orderedD(X1, .(X2, X3)) :- ','(lesscE(X1, X2), orderedD(X2, X3)).
11.17/3.74	lessF(s(X1), s(X2)) :- lessF(X1, X2).
11.17/3.74	ssG(X1, .(X2, X3)) :- appA(X4, X2, X5, X1).
11.17/3.74	ssG(X1, .(X2, X3)) :- ','(appcA(X4, X2, X5, X1), appB(X4, X5, X6)).
11.17/3.74	ssG(X1, .(X2, X3)) :- ','(appcA(X4, X2, X5, X1), ','(appcB(X4, X5, X6), permC(X6, X3))).
11.17/3.74	ssG(X1, .(X2, X3)) :- ','(appcA(X4, X2, X5, X1), ','(appcB(X4, X5, X6), ','(permcC(X6, X3), orderedD(X2, X3)))).
11.17/3.74	
11.17/3.74	Clauses:
11.17/3.74	
11.17/3.74	appcA([], X1, X2, .(X1, X2)).
11.17/3.74	appcA(.(X1, X2), X3, X4, .(X1, X5)) :- appcA(X2, X3, X4, X5).
11.17/3.74	appcB([], X1, X1).
11.17/3.74	appcB(.(X1, X2), X3, .(X1, X4)) :- appcB(X2, X3, X4).
11.17/3.74	permcC([], []).
11.17/3.74	permcC(X1, .(X2, X3)) :- ','(appcA(X4, X2, X5, X1), ','(appcB(X4, X5, X6), permcC(X6, X3))).
11.17/3.74	orderedcD(X1, []).
11.17/3.74	orderedcD(X1, .(X2, X3)) :- ','(lesscE(X1, X2), orderedcD(X2, X3)).
11.17/3.74	lesscF(0, s(X1)).
11.17/3.74	lesscF(s(X1), s(X2)) :- lesscF(X1, X2).
11.17/3.74	lesscE(0, X1).
11.17/3.74	lesscE(s(X1), X2) :- lesscF(X1, X2).
11.17/3.74	
11.17/3.74	Afs:
11.17/3.74	
11.17/3.74	ssG(x1, x2)  =  ssG(x2)
11.17/3.74	
11.17/3.74	
11.17/3.74	----------------------------------------
11.17/3.74	
11.17/3.74	(77) TriplesToPiDPProof (SOUND)
11.17/3.74	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
11.17/3.74	
11.17/3.74	ssG_in_2: (f,b)
11.17/3.74	
11.17/3.74	appA_in_4: (f,b,f,f)
11.17/3.74	
11.17/3.74	appcA_in_4: (f,b,f,f)
11.17/3.74	
11.17/3.74	appB_in_3: (f,f,f)
11.17/3.74	
11.17/3.74	appcB_in_3: (f,f,f)
11.17/3.74	
11.17/3.74	permC_in_2: (f,b)
11.17/3.74	
11.17/3.74	permcC_in_2: (f,b)
11.17/3.74	
11.17/3.74	orderedD_in_2: (b,b)
11.17/3.74	
11.17/3.74	lessF_in_2: (b,b)
11.17/3.74	
11.17/3.74	lesscE_in_2: (b,b)
11.17/3.74	
11.17/3.74	lesscF_in_2: (b,b)
11.17/3.74	
11.17/3.74	Transforming TRIPLES into the following Term Rewriting System:
11.17/3.74	
11.17/3.74	Pi DP problem:
11.17/3.74	The TRS P consists of the following rules:
11.17/3.74	
11.17/3.74	   SSG_IN_AG(X1, .(X2, X3)) -> U12_AG(X1, X2, X3, appA_in_agaa(X4, X2, X5, X1))
11.17/3.74	   SSG_IN_AG(X1, .(X2, X3)) -> APPA_IN_AGAA(X4, X2, X5, X1)
11.17/3.74	   APPA_IN_AGAA(.(X1, X2), X3, X4, .(X1, X5)) -> U1_AGAA(X1, X2, X3, X4, X5, appA_in_agaa(X2, X3, X4, X5))
11.17/3.74	   APPA_IN_AGAA(.(X1, X2), X3, X4, .(X1, X5)) -> APPA_IN_AGAA(X2, X3, X4, X5)
11.17/3.74	   SSG_IN_AG(X1, .(X2, X3)) -> U13_AG(X1, X2, X3, appcA_in_agaa(X4, X2, X5, X1))
11.17/3.75	   U13_AG(X1, X2, X3, appcA_out_agaa(X4, X2, X5, X1)) -> U14_AG(X1, X2, X3, appB_in_aaa(X4, X5, X6))
11.17/3.75	   U13_AG(X1, X2, X3, appcA_out_agaa(X4, X2, X5, X1)) -> APPB_IN_AAA(X4, X5, X6)
11.17/3.75	   APPB_IN_AAA(.(X1, X2), X3, .(X1, X4)) -> U2_AAA(X1, X2, X3, X4, appB_in_aaa(X2, X3, X4))
11.17/3.75	   APPB_IN_AAA(.(X1, X2), X3, .(X1, X4)) -> APPB_IN_AAA(X2, X3, X4)
11.17/3.75	   U13_AG(X1, X2, X3, appcA_out_agaa(X4, X2, X5, X1)) -> U15_AG(X1, X2, X3, appcB_in_aaa(X4, X5, X6))
11.17/3.75	   U15_AG(X1, X2, X3, appcB_out_aaa(X4, X5, X6)) -> U16_AG(X1, X2, X3, permC_in_ag(X6, X3))
11.17/3.75	   U15_AG(X1, X2, X3, appcB_out_aaa(X4, X5, X6)) -> PERMC_IN_AG(X6, X3)
11.17/3.75	   PERMC_IN_AG(X1, .(X2, X3)) -> U3_AG(X1, X2, X3, appA_in_agaa(X4, X2, X5, X1))
11.17/3.75	   PERMC_IN_AG(X1, .(X2, X3)) -> APPA_IN_AGAA(X4, X2, X5, X1)
11.17/3.75	   PERMC_IN_AG(X1, .(X2, X3)) -> U4_AG(X1, X2, X3, appcA_in_agaa(X4, X2, X5, X1))
11.17/3.75	   U4_AG(X1, X2, X3, appcA_out_agaa(X4, X2, X5, X1)) -> U5_AG(X1, X2, X3, appB_in_aaa(X4, X5, X6))
11.17/3.75	   U4_AG(X1, X2, X3, appcA_out_agaa(X4, X2, X5, X1)) -> APPB_IN_AAA(X4, X5, X6)
11.17/3.75	   U4_AG(X1, X2, X3, appcA_out_agaa(X4, X2, X5, X1)) -> U6_AG(X1, X2, X3, appcB_in_aaa(X4, X5, X6))
11.17/3.75	   U6_AG(X1, X2, X3, appcB_out_aaa(X4, X5, X6)) -> U7_AG(X1, X2, X3, permC_in_ag(X6, X3))
11.17/3.75	   U6_AG(X1, X2, X3, appcB_out_aaa(X4, X5, X6)) -> PERMC_IN_AG(X6, X3)
11.17/3.75	   U15_AG(X1, X2, X3, appcB_out_aaa(X4, X5, X6)) -> U17_AG(X1, X2, X3, permcC_in_ag(X6, X3))
11.17/3.75	   U17_AG(X1, X2, X3, permcC_out_ag(X6, X3)) -> U18_AG(X1, X2, X3, orderedD_in_gg(X2, X3))
11.17/3.75	   U17_AG(X1, X2, X3, permcC_out_ag(X6, X3)) -> ORDEREDD_IN_GG(X2, X3)
11.17/3.75	   ORDEREDD_IN_GG(s(X1), .(X2, X3)) -> U8_GG(X1, X2, X3, lessF_in_gg(X1, X2))
11.17/3.75	   ORDEREDD_IN_GG(s(X1), .(X2, X3)) -> LESSF_IN_GG(X1, X2)
11.17/3.75	   LESSF_IN_GG(s(X1), s(X2)) -> U11_GG(X1, X2, lessF_in_gg(X1, X2))
11.17/3.75	   LESSF_IN_GG(s(X1), s(X2)) -> LESSF_IN_GG(X1, X2)
11.17/3.75	   ORDEREDD_IN_GG(X1, .(X2, X3)) -> U9_GG(X1, X2, X3, lesscE_in_gg(X1, X2))
11.17/3.75	   U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) -> U10_GG(X1, X2, X3, orderedD_in_gg(X2, X3))
11.17/3.75	   U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) -> ORDEREDD_IN_GG(X2, X3)
11.17/3.75	
11.17/3.75	The TRS R consists of the following rules:
11.17/3.75	
11.17/3.75	   appcA_in_agaa([], X1, X2, .(X1, X2)) -> appcA_out_agaa([], X1, X2, .(X1, X2))
11.17/3.75	   appcA_in_agaa(.(X1, X2), X3, X4, .(X1, X5)) -> U20_agaa(X1, X2, X3, X4, X5, appcA_in_agaa(X2, X3, X4, X5))
11.17/3.75	   U20_agaa(X1, X2, X3, X4, X5, appcA_out_agaa(X2, X3, X4, X5)) -> appcA_out_agaa(.(X1, X2), X3, X4, .(X1, X5))
11.17/3.75	   appcB_in_aaa([], X1, X1) -> appcB_out_aaa([], X1, X1)
11.17/3.75	   appcB_in_aaa(.(X1, X2), X3, .(X1, X4)) -> U21_aaa(X1, X2, X3, X4, appcB_in_aaa(X2, X3, X4))
11.17/3.75	   U21_aaa(X1, X2, X3, X4, appcB_out_aaa(X2, X3, X4)) -> appcB_out_aaa(.(X1, X2), X3, .(X1, X4))
11.17/3.75	   permcC_in_ag([], []) -> permcC_out_ag([], [])
11.17/3.75	   permcC_in_ag(X1, .(X2, X3)) -> U22_ag(X1, X2, X3, appcA_in_agaa(X4, X2, X5, X1))
11.17/3.75	   U22_ag(X1, X2, X3, appcA_out_agaa(X4, X2, X5, X1)) -> U23_ag(X1, X2, X3, appcB_in_aaa(X4, X5, X6))
11.17/3.75	   U23_ag(X1, X2, X3, appcB_out_aaa(X4, X5, X6)) -> U24_ag(X1, X2, X3, permcC_in_ag(X6, X3))
11.17/3.75	   U24_ag(X1, X2, X3, permcC_out_ag(X6, X3)) -> permcC_out_ag(X1, .(X2, X3))
11.17/3.75	   lesscE_in_gg(0, X1) -> lesscE_out_gg(0, X1)
11.17/3.75	   lesscE_in_gg(s(X1), X2) -> U28_gg(X1, X2, lesscF_in_gg(X1, X2))
11.17/3.75	   lesscF_in_gg(0, s(X1)) -> lesscF_out_gg(0, s(X1))
11.17/3.75	   lesscF_in_gg(s(X1), s(X2)) -> U27_gg(X1, X2, lesscF_in_gg(X1, X2))
11.17/3.75	   U27_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), s(X2))
11.17/3.75	   U28_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscE_out_gg(s(X1), X2)
11.17/3.75	
11.17/3.75	The argument filtering Pi contains the following mapping:
11.17/3.75	.(x1, x2)  =  .(x1, x2)
11.17/3.75	
11.17/3.75	appA_in_agaa(x1, x2, x3, x4)  =  appA_in_agaa(x2)
11.17/3.75	
11.17/3.75	appcA_in_agaa(x1, x2, x3, x4)  =  appcA_in_agaa(x2)
11.17/3.75	
11.17/3.75	appcA_out_agaa(x1, x2, x3, x4)  =  appcA_out_agaa(x2)
11.17/3.75	
11.17/3.75	U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
11.17/3.75	
11.17/3.75	appB_in_aaa(x1, x2, x3)  =  appB_in_aaa
11.17/3.75	
11.17/3.75	appcB_in_aaa(x1, x2, x3)  =  appcB_in_aaa
11.17/3.75	
11.17/3.75	appcB_out_aaa(x1, x2, x3)  =  appcB_out_aaa
11.17/3.75	
11.17/3.75	U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
11.17/3.75	
11.17/3.75	permC_in_ag(x1, x2)  =  permC_in_ag(x2)
11.17/3.75	
11.17/3.75	permcC_in_ag(x1, x2)  =  permcC_in_ag(x2)
11.17/3.75	
11.17/3.75	[]  =  []
11.17/3.75	
11.17/3.75	permcC_out_ag(x1, x2)  =  permcC_out_ag(x2)
11.17/3.75	
11.17/3.75	U22_ag(x1, x2, x3, x4)  =  U22_ag(x2, x3, x4)
11.17/3.75	
11.17/3.75	U23_ag(x1, x2, x3, x4)  =  U23_ag(x2, x3, x4)
11.17/3.75	
11.17/3.75	U24_ag(x1, x2, x3, x4)  =  U24_ag(x2, x3, x4)
11.17/3.75	
11.17/3.75	orderedD_in_gg(x1, x2)  =  orderedD_in_gg(x1, x2)
11.17/3.75	
11.17/3.75	s(x1)  =  s(x1)
11.17/3.75	
11.17/3.75	lessF_in_gg(x1, x2)  =  lessF_in_gg(x1, x2)
11.17/3.75	
11.17/3.75	lesscE_in_gg(x1, x2)  =  lesscE_in_gg(x1, x2)
11.17/3.75	
11.17/3.75	0  =  0
11.17/3.75	
11.17/3.75	lesscE_out_gg(x1, x2)  =  lesscE_out_gg(x1, x2)
11.17/3.75	
11.17/3.75	U28_gg(x1, x2, x3)  =  U28_gg(x1, x2, x3)
11.17/3.75	
11.17/3.75	lesscF_in_gg(x1, x2)  =  lesscF_in_gg(x1, x2)
11.17/3.75	
11.17/3.75	lesscF_out_gg(x1, x2)  =  lesscF_out_gg(x1, x2)
11.17/3.75	
11.17/3.75	U27_gg(x1, x2, x3)  =  U27_gg(x1, x2, x3)
11.17/3.75	
11.17/3.75	SSG_IN_AG(x1, x2)  =  SSG_IN_AG(x2)
11.17/3.75	
11.17/3.75	U12_AG(x1, x2, x3, x4)  =  U12_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	APPA_IN_AGAA(x1, x2, x3, x4)  =  APPA_IN_AGAA(x2)
11.17/3.75	
11.17/3.75	U1_AGAA(x1, x2, x3, x4, x5, x6)  =  U1_AGAA(x3, x6)
11.17/3.75	
11.17/3.75	U13_AG(x1, x2, x3, x4)  =  U13_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	U14_AG(x1, x2, x3, x4)  =  U14_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	APPB_IN_AAA(x1, x2, x3)  =  APPB_IN_AAA
11.17/3.75	
11.17/3.75	U2_AAA(x1, x2, x3, x4, x5)  =  U2_AAA(x5)
11.17/3.75	
11.17/3.75	U15_AG(x1, x2, x3, x4)  =  U15_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	U16_AG(x1, x2, x3, x4)  =  U16_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	PERMC_IN_AG(x1, x2)  =  PERMC_IN_AG(x2)
11.17/3.75	
11.17/3.75	U3_AG(x1, x2, x3, x4)  =  U3_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	U4_AG(x1, x2, x3, x4)  =  U4_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	U5_AG(x1, x2, x3, x4)  =  U5_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	U6_AG(x1, x2, x3, x4)  =  U6_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	U7_AG(x1, x2, x3, x4)  =  U7_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	U17_AG(x1, x2, x3, x4)  =  U17_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	U18_AG(x1, x2, x3, x4)  =  U18_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	ORDEREDD_IN_GG(x1, x2)  =  ORDEREDD_IN_GG(x1, x2)
11.17/3.75	
11.17/3.75	U8_GG(x1, x2, x3, x4)  =  U8_GG(x1, x2, x3, x4)
11.17/3.75	
11.17/3.75	LESSF_IN_GG(x1, x2)  =  LESSF_IN_GG(x1, x2)
11.17/3.75	
11.17/3.75	U11_GG(x1, x2, x3)  =  U11_GG(x1, x2, x3)
11.17/3.75	
11.17/3.75	U9_GG(x1, x2, x3, x4)  =  U9_GG(x1, x2, x3, x4)
11.17/3.75	
11.17/3.75	U10_GG(x1, x2, x3, x4)  =  U10_GG(x1, x2, x3, x4)
11.17/3.75	
11.17/3.75	
11.17/3.75	We have to consider all (P,R,Pi)-chains
11.17/3.75	
11.17/3.75	
11.17/3.75	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
11.17/3.75	
11.17/3.75	
11.17/3.75	
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(78)
11.17/3.75	Obligation:
11.17/3.75	Pi DP problem:
11.17/3.75	The TRS P consists of the following rules:
11.17/3.75	
11.17/3.75	   SSG_IN_AG(X1, .(X2, X3)) -> U12_AG(X1, X2, X3, appA_in_agaa(X4, X2, X5, X1))
11.17/3.75	   SSG_IN_AG(X1, .(X2, X3)) -> APPA_IN_AGAA(X4, X2, X5, X1)
11.17/3.75	   APPA_IN_AGAA(.(X1, X2), X3, X4, .(X1, X5)) -> U1_AGAA(X1, X2, X3, X4, X5, appA_in_agaa(X2, X3, X4, X5))
11.17/3.75	   APPA_IN_AGAA(.(X1, X2), X3, X4, .(X1, X5)) -> APPA_IN_AGAA(X2, X3, X4, X5)
11.17/3.75	   SSG_IN_AG(X1, .(X2, X3)) -> U13_AG(X1, X2, X3, appcA_in_agaa(X4, X2, X5, X1))
11.17/3.75	   U13_AG(X1, X2, X3, appcA_out_agaa(X4, X2, X5, X1)) -> U14_AG(X1, X2, X3, appB_in_aaa(X4, X5, X6))
11.17/3.75	   U13_AG(X1, X2, X3, appcA_out_agaa(X4, X2, X5, X1)) -> APPB_IN_AAA(X4, X5, X6)
11.17/3.75	   APPB_IN_AAA(.(X1, X2), X3, .(X1, X4)) -> U2_AAA(X1, X2, X3, X4, appB_in_aaa(X2, X3, X4))
11.17/3.75	   APPB_IN_AAA(.(X1, X2), X3, .(X1, X4)) -> APPB_IN_AAA(X2, X3, X4)
11.17/3.75	   U13_AG(X1, X2, X3, appcA_out_agaa(X4, X2, X5, X1)) -> U15_AG(X1, X2, X3, appcB_in_aaa(X4, X5, X6))
11.17/3.75	   U15_AG(X1, X2, X3, appcB_out_aaa(X4, X5, X6)) -> U16_AG(X1, X2, X3, permC_in_ag(X6, X3))
11.17/3.75	   U15_AG(X1, X2, X3, appcB_out_aaa(X4, X5, X6)) -> PERMC_IN_AG(X6, X3)
11.17/3.75	   PERMC_IN_AG(X1, .(X2, X3)) -> U3_AG(X1, X2, X3, appA_in_agaa(X4, X2, X5, X1))
11.17/3.75	   PERMC_IN_AG(X1, .(X2, X3)) -> APPA_IN_AGAA(X4, X2, X5, X1)
11.17/3.75	   PERMC_IN_AG(X1, .(X2, X3)) -> U4_AG(X1, X2, X3, appcA_in_agaa(X4, X2, X5, X1))
11.17/3.75	   U4_AG(X1, X2, X3, appcA_out_agaa(X4, X2, X5, X1)) -> U5_AG(X1, X2, X3, appB_in_aaa(X4, X5, X6))
11.17/3.75	   U4_AG(X1, X2, X3, appcA_out_agaa(X4, X2, X5, X1)) -> APPB_IN_AAA(X4, X5, X6)
11.17/3.75	   U4_AG(X1, X2, X3, appcA_out_agaa(X4, X2, X5, X1)) -> U6_AG(X1, X2, X3, appcB_in_aaa(X4, X5, X6))
11.17/3.75	   U6_AG(X1, X2, X3, appcB_out_aaa(X4, X5, X6)) -> U7_AG(X1, X2, X3, permC_in_ag(X6, X3))
11.17/3.75	   U6_AG(X1, X2, X3, appcB_out_aaa(X4, X5, X6)) -> PERMC_IN_AG(X6, X3)
11.17/3.75	   U15_AG(X1, X2, X3, appcB_out_aaa(X4, X5, X6)) -> U17_AG(X1, X2, X3, permcC_in_ag(X6, X3))
11.17/3.75	   U17_AG(X1, X2, X3, permcC_out_ag(X6, X3)) -> U18_AG(X1, X2, X3, orderedD_in_gg(X2, X3))
11.17/3.75	   U17_AG(X1, X2, X3, permcC_out_ag(X6, X3)) -> ORDEREDD_IN_GG(X2, X3)
11.17/3.75	   ORDEREDD_IN_GG(s(X1), .(X2, X3)) -> U8_GG(X1, X2, X3, lessF_in_gg(X1, X2))
11.17/3.75	   ORDEREDD_IN_GG(s(X1), .(X2, X3)) -> LESSF_IN_GG(X1, X2)
11.17/3.75	   LESSF_IN_GG(s(X1), s(X2)) -> U11_GG(X1, X2, lessF_in_gg(X1, X2))
11.17/3.75	   LESSF_IN_GG(s(X1), s(X2)) -> LESSF_IN_GG(X1, X2)
11.17/3.75	   ORDEREDD_IN_GG(X1, .(X2, X3)) -> U9_GG(X1, X2, X3, lesscE_in_gg(X1, X2))
11.17/3.75	   U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) -> U10_GG(X1, X2, X3, orderedD_in_gg(X2, X3))
11.17/3.75	   U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) -> ORDEREDD_IN_GG(X2, X3)
11.17/3.75	
11.17/3.75	The TRS R consists of the following rules:
11.17/3.75	
11.17/3.75	   appcA_in_agaa([], X1, X2, .(X1, X2)) -> appcA_out_agaa([], X1, X2, .(X1, X2))
11.17/3.75	   appcA_in_agaa(.(X1, X2), X3, X4, .(X1, X5)) -> U20_agaa(X1, X2, X3, X4, X5, appcA_in_agaa(X2, X3, X4, X5))
11.17/3.75	   U20_agaa(X1, X2, X3, X4, X5, appcA_out_agaa(X2, X3, X4, X5)) -> appcA_out_agaa(.(X1, X2), X3, X4, .(X1, X5))
11.17/3.75	   appcB_in_aaa([], X1, X1) -> appcB_out_aaa([], X1, X1)
11.17/3.75	   appcB_in_aaa(.(X1, X2), X3, .(X1, X4)) -> U21_aaa(X1, X2, X3, X4, appcB_in_aaa(X2, X3, X4))
11.17/3.75	   U21_aaa(X1, X2, X3, X4, appcB_out_aaa(X2, X3, X4)) -> appcB_out_aaa(.(X1, X2), X3, .(X1, X4))
11.17/3.75	   permcC_in_ag([], []) -> permcC_out_ag([], [])
11.17/3.75	   permcC_in_ag(X1, .(X2, X3)) -> U22_ag(X1, X2, X3, appcA_in_agaa(X4, X2, X5, X1))
11.17/3.75	   U22_ag(X1, X2, X3, appcA_out_agaa(X4, X2, X5, X1)) -> U23_ag(X1, X2, X3, appcB_in_aaa(X4, X5, X6))
11.17/3.75	   U23_ag(X1, X2, X3, appcB_out_aaa(X4, X5, X6)) -> U24_ag(X1, X2, X3, permcC_in_ag(X6, X3))
11.17/3.75	   U24_ag(X1, X2, X3, permcC_out_ag(X6, X3)) -> permcC_out_ag(X1, .(X2, X3))
11.17/3.75	   lesscE_in_gg(0, X1) -> lesscE_out_gg(0, X1)
11.17/3.75	   lesscE_in_gg(s(X1), X2) -> U28_gg(X1, X2, lesscF_in_gg(X1, X2))
11.17/3.75	   lesscF_in_gg(0, s(X1)) -> lesscF_out_gg(0, s(X1))
11.17/3.75	   lesscF_in_gg(s(X1), s(X2)) -> U27_gg(X1, X2, lesscF_in_gg(X1, X2))
11.17/3.75	   U27_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), s(X2))
11.17/3.75	   U28_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscE_out_gg(s(X1), X2)
11.17/3.75	
11.17/3.75	The argument filtering Pi contains the following mapping:
11.17/3.75	.(x1, x2)  =  .(x1, x2)
11.17/3.75	
11.17/3.75	appA_in_agaa(x1, x2, x3, x4)  =  appA_in_agaa(x2)
11.17/3.75	
11.17/3.75	appcA_in_agaa(x1, x2, x3, x4)  =  appcA_in_agaa(x2)
11.17/3.75	
11.17/3.75	appcA_out_agaa(x1, x2, x3, x4)  =  appcA_out_agaa(x2)
11.17/3.75	
11.17/3.75	U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
11.17/3.75	
11.17/3.75	appB_in_aaa(x1, x2, x3)  =  appB_in_aaa
11.17/3.75	
11.17/3.75	appcB_in_aaa(x1, x2, x3)  =  appcB_in_aaa
11.17/3.75	
11.17/3.75	appcB_out_aaa(x1, x2, x3)  =  appcB_out_aaa
11.17/3.75	
11.17/3.75	U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
11.17/3.75	
11.17/3.75	permC_in_ag(x1, x2)  =  permC_in_ag(x2)
11.17/3.75	
11.17/3.75	permcC_in_ag(x1, x2)  =  permcC_in_ag(x2)
11.17/3.75	
11.17/3.75	[]  =  []
11.17/3.75	
11.17/3.75	permcC_out_ag(x1, x2)  =  permcC_out_ag(x2)
11.17/3.75	
11.17/3.75	U22_ag(x1, x2, x3, x4)  =  U22_ag(x2, x3, x4)
11.17/3.75	
11.17/3.75	U23_ag(x1, x2, x3, x4)  =  U23_ag(x2, x3, x4)
11.17/3.75	
11.17/3.75	U24_ag(x1, x2, x3, x4)  =  U24_ag(x2, x3, x4)
11.17/3.75	
11.17/3.75	orderedD_in_gg(x1, x2)  =  orderedD_in_gg(x1, x2)
11.17/3.75	
11.17/3.75	s(x1)  =  s(x1)
11.17/3.75	
11.17/3.75	lessF_in_gg(x1, x2)  =  lessF_in_gg(x1, x2)
11.17/3.75	
11.17/3.75	lesscE_in_gg(x1, x2)  =  lesscE_in_gg(x1, x2)
11.17/3.75	
11.17/3.75	0  =  0
11.17/3.75	
11.17/3.75	lesscE_out_gg(x1, x2)  =  lesscE_out_gg(x1, x2)
11.17/3.75	
11.17/3.75	U28_gg(x1, x2, x3)  =  U28_gg(x1, x2, x3)
11.17/3.75	
11.17/3.75	lesscF_in_gg(x1, x2)  =  lesscF_in_gg(x1, x2)
11.17/3.75	
11.17/3.75	lesscF_out_gg(x1, x2)  =  lesscF_out_gg(x1, x2)
11.17/3.75	
11.17/3.75	U27_gg(x1, x2, x3)  =  U27_gg(x1, x2, x3)
11.17/3.75	
11.17/3.75	SSG_IN_AG(x1, x2)  =  SSG_IN_AG(x2)
11.17/3.75	
11.17/3.75	U12_AG(x1, x2, x3, x4)  =  U12_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	APPA_IN_AGAA(x1, x2, x3, x4)  =  APPA_IN_AGAA(x2)
11.17/3.75	
11.17/3.75	U1_AGAA(x1, x2, x3, x4, x5, x6)  =  U1_AGAA(x3, x6)
11.17/3.75	
11.17/3.75	U13_AG(x1, x2, x3, x4)  =  U13_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	U14_AG(x1, x2, x3, x4)  =  U14_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	APPB_IN_AAA(x1, x2, x3)  =  APPB_IN_AAA
11.17/3.75	
11.17/3.75	U2_AAA(x1, x2, x3, x4, x5)  =  U2_AAA(x5)
11.17/3.75	
11.17/3.75	U15_AG(x1, x2, x3, x4)  =  U15_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	U16_AG(x1, x2, x3, x4)  =  U16_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	PERMC_IN_AG(x1, x2)  =  PERMC_IN_AG(x2)
11.17/3.75	
11.17/3.75	U3_AG(x1, x2, x3, x4)  =  U3_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	U4_AG(x1, x2, x3, x4)  =  U4_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	U5_AG(x1, x2, x3, x4)  =  U5_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	U6_AG(x1, x2, x3, x4)  =  U6_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	U7_AG(x1, x2, x3, x4)  =  U7_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	U17_AG(x1, x2, x3, x4)  =  U17_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	U18_AG(x1, x2, x3, x4)  =  U18_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	ORDEREDD_IN_GG(x1, x2)  =  ORDEREDD_IN_GG(x1, x2)
11.17/3.75	
11.17/3.75	U8_GG(x1, x2, x3, x4)  =  U8_GG(x1, x2, x3, x4)
11.17/3.75	
11.17/3.75	LESSF_IN_GG(x1, x2)  =  LESSF_IN_GG(x1, x2)
11.17/3.75	
11.17/3.75	U11_GG(x1, x2, x3)  =  U11_GG(x1, x2, x3)
11.17/3.75	
11.17/3.75	U9_GG(x1, x2, x3, x4)  =  U9_GG(x1, x2, x3, x4)
11.17/3.75	
11.17/3.75	U10_GG(x1, x2, x3, x4)  =  U10_GG(x1, x2, x3, x4)
11.17/3.75	
11.17/3.75	
11.17/3.75	We have to consider all (P,R,Pi)-chains
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(79) DependencyGraphProof (EQUIVALENT)
11.17/3.75	The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 22 less nodes.
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(80)
11.17/3.75	Complex Obligation (AND)
11.17/3.75	
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(81)
11.17/3.75	Obligation:
11.17/3.75	Pi DP problem:
11.17/3.75	The TRS P consists of the following rules:
11.17/3.75	
11.17/3.75	   LESSF_IN_GG(s(X1), s(X2)) -> LESSF_IN_GG(X1, X2)
11.17/3.75	
11.17/3.75	The TRS R consists of the following rules:
11.17/3.75	
11.17/3.75	   appcA_in_agaa([], X1, X2, .(X1, X2)) -> appcA_out_agaa([], X1, X2, .(X1, X2))
11.17/3.75	   appcA_in_agaa(.(X1, X2), X3, X4, .(X1, X5)) -> U20_agaa(X1, X2, X3, X4, X5, appcA_in_agaa(X2, X3, X4, X5))
11.17/3.75	   U20_agaa(X1, X2, X3, X4, X5, appcA_out_agaa(X2, X3, X4, X5)) -> appcA_out_agaa(.(X1, X2), X3, X4, .(X1, X5))
11.17/3.75	   appcB_in_aaa([], X1, X1) -> appcB_out_aaa([], X1, X1)
11.17/3.75	   appcB_in_aaa(.(X1, X2), X3, .(X1, X4)) -> U21_aaa(X1, X2, X3, X4, appcB_in_aaa(X2, X3, X4))
11.17/3.75	   U21_aaa(X1, X2, X3, X4, appcB_out_aaa(X2, X3, X4)) -> appcB_out_aaa(.(X1, X2), X3, .(X1, X4))
11.17/3.75	   permcC_in_ag([], []) -> permcC_out_ag([], [])
11.17/3.75	   permcC_in_ag(X1, .(X2, X3)) -> U22_ag(X1, X2, X3, appcA_in_agaa(X4, X2, X5, X1))
11.17/3.75	   U22_ag(X1, X2, X3, appcA_out_agaa(X4, X2, X5, X1)) -> U23_ag(X1, X2, X3, appcB_in_aaa(X4, X5, X6))
11.17/3.75	   U23_ag(X1, X2, X3, appcB_out_aaa(X4, X5, X6)) -> U24_ag(X1, X2, X3, permcC_in_ag(X6, X3))
11.17/3.75	   U24_ag(X1, X2, X3, permcC_out_ag(X6, X3)) -> permcC_out_ag(X1, .(X2, X3))
11.17/3.75	   lesscE_in_gg(0, X1) -> lesscE_out_gg(0, X1)
11.17/3.75	   lesscE_in_gg(s(X1), X2) -> U28_gg(X1, X2, lesscF_in_gg(X1, X2))
11.17/3.75	   lesscF_in_gg(0, s(X1)) -> lesscF_out_gg(0, s(X1))
11.17/3.75	   lesscF_in_gg(s(X1), s(X2)) -> U27_gg(X1, X2, lesscF_in_gg(X1, X2))
11.17/3.75	   U27_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), s(X2))
11.17/3.75	   U28_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscE_out_gg(s(X1), X2)
11.17/3.75	
11.17/3.75	The argument filtering Pi contains the following mapping:
11.17/3.75	.(x1, x2)  =  .(x1, x2)
11.17/3.75	
11.17/3.75	appcA_in_agaa(x1, x2, x3, x4)  =  appcA_in_agaa(x2)
11.17/3.75	
11.17/3.75	appcA_out_agaa(x1, x2, x3, x4)  =  appcA_out_agaa(x2)
11.17/3.75	
11.17/3.75	U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
11.17/3.75	
11.17/3.75	appcB_in_aaa(x1, x2, x3)  =  appcB_in_aaa
11.17/3.75	
11.17/3.75	appcB_out_aaa(x1, x2, x3)  =  appcB_out_aaa
11.17/3.75	
11.17/3.75	U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
11.17/3.75	
11.17/3.75	permcC_in_ag(x1, x2)  =  permcC_in_ag(x2)
11.17/3.75	
11.17/3.75	[]  =  []
11.17/3.75	
11.17/3.75	permcC_out_ag(x1, x2)  =  permcC_out_ag(x2)
11.17/3.75	
11.17/3.75	U22_ag(x1, x2, x3, x4)  =  U22_ag(x2, x3, x4)
11.17/3.75	
11.17/3.75	U23_ag(x1, x2, x3, x4)  =  U23_ag(x2, x3, x4)
11.17/3.75	
11.17/3.75	U24_ag(x1, x2, x3, x4)  =  U24_ag(x2, x3, x4)
11.17/3.75	
11.17/3.75	s(x1)  =  s(x1)
11.17/3.75	
11.17/3.75	lesscE_in_gg(x1, x2)  =  lesscE_in_gg(x1, x2)
11.17/3.75	
11.17/3.75	0  =  0
11.17/3.75	
11.17/3.75	lesscE_out_gg(x1, x2)  =  lesscE_out_gg(x1, x2)
11.17/3.75	
11.17/3.75	U28_gg(x1, x2, x3)  =  U28_gg(x1, x2, x3)
11.17/3.75	
11.17/3.75	lesscF_in_gg(x1, x2)  =  lesscF_in_gg(x1, x2)
11.17/3.75	
11.17/3.75	lesscF_out_gg(x1, x2)  =  lesscF_out_gg(x1, x2)
11.17/3.75	
11.17/3.75	U27_gg(x1, x2, x3)  =  U27_gg(x1, x2, x3)
11.17/3.75	
11.17/3.75	LESSF_IN_GG(x1, x2)  =  LESSF_IN_GG(x1, x2)
11.17/3.75	
11.17/3.75	
11.17/3.75	We have to consider all (P,R,Pi)-chains
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(82) UsableRulesProof (EQUIVALENT)
11.17/3.75	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(83)
11.17/3.75	Obligation:
11.17/3.75	Pi DP problem:
11.17/3.75	The TRS P consists of the following rules:
11.17/3.75	
11.17/3.75	   LESSF_IN_GG(s(X1), s(X2)) -> LESSF_IN_GG(X1, X2)
11.17/3.75	
11.17/3.75	R is empty.
11.17/3.75	Pi is empty.
11.17/3.75	We have to consider all (P,R,Pi)-chains
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(84) PiDPToQDPProof (EQUIVALENT)
11.17/3.75	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(85)
11.17/3.75	Obligation:
11.17/3.75	Q DP problem:
11.17/3.75	The TRS P consists of the following rules:
11.17/3.75	
11.17/3.75	   LESSF_IN_GG(s(X1), s(X2)) -> LESSF_IN_GG(X1, X2)
11.17/3.75	
11.17/3.75	R is empty.
11.17/3.75	Q is empty.
11.17/3.75	We have to consider all (P,Q,R)-chains.
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(86) QDPSizeChangeProof (EQUIVALENT)
11.17/3.75	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. 
11.17/3.75	
11.17/3.75	From the DPs we obtained the following set of size-change graphs:
11.17/3.75	*LESSF_IN_GG(s(X1), s(X2)) -> LESSF_IN_GG(X1, X2)
11.17/3.75	The graph contains the following edges 1 > 1, 2 > 2
11.17/3.75	
11.17/3.75	
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(87)
11.17/3.75	YES
11.17/3.75	
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(88)
11.17/3.75	Obligation:
11.17/3.75	Pi DP problem:
11.17/3.75	The TRS P consists of the following rules:
11.17/3.75	
11.17/3.75	   ORDEREDD_IN_GG(X1, .(X2, X3)) -> U9_GG(X1, X2, X3, lesscE_in_gg(X1, X2))
11.17/3.75	   U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) -> ORDEREDD_IN_GG(X2, X3)
11.17/3.75	
11.17/3.75	The TRS R consists of the following rules:
11.17/3.75	
11.17/3.75	   appcA_in_agaa([], X1, X2, .(X1, X2)) -> appcA_out_agaa([], X1, X2, .(X1, X2))
11.17/3.75	   appcA_in_agaa(.(X1, X2), X3, X4, .(X1, X5)) -> U20_agaa(X1, X2, X3, X4, X5, appcA_in_agaa(X2, X3, X4, X5))
11.17/3.75	   U20_agaa(X1, X2, X3, X4, X5, appcA_out_agaa(X2, X3, X4, X5)) -> appcA_out_agaa(.(X1, X2), X3, X4, .(X1, X5))
11.17/3.75	   appcB_in_aaa([], X1, X1) -> appcB_out_aaa([], X1, X1)
11.17/3.75	   appcB_in_aaa(.(X1, X2), X3, .(X1, X4)) -> U21_aaa(X1, X2, X3, X4, appcB_in_aaa(X2, X3, X4))
11.17/3.75	   U21_aaa(X1, X2, X3, X4, appcB_out_aaa(X2, X3, X4)) -> appcB_out_aaa(.(X1, X2), X3, .(X1, X4))
11.17/3.75	   permcC_in_ag([], []) -> permcC_out_ag([], [])
11.17/3.75	   permcC_in_ag(X1, .(X2, X3)) -> U22_ag(X1, X2, X3, appcA_in_agaa(X4, X2, X5, X1))
11.17/3.75	   U22_ag(X1, X2, X3, appcA_out_agaa(X4, X2, X5, X1)) -> U23_ag(X1, X2, X3, appcB_in_aaa(X4, X5, X6))
11.17/3.75	   U23_ag(X1, X2, X3, appcB_out_aaa(X4, X5, X6)) -> U24_ag(X1, X2, X3, permcC_in_ag(X6, X3))
11.17/3.75	   U24_ag(X1, X2, X3, permcC_out_ag(X6, X3)) -> permcC_out_ag(X1, .(X2, X3))
11.17/3.75	   lesscE_in_gg(0, X1) -> lesscE_out_gg(0, X1)
11.17/3.75	   lesscE_in_gg(s(X1), X2) -> U28_gg(X1, X2, lesscF_in_gg(X1, X2))
11.17/3.75	   lesscF_in_gg(0, s(X1)) -> lesscF_out_gg(0, s(X1))
11.17/3.75	   lesscF_in_gg(s(X1), s(X2)) -> U27_gg(X1, X2, lesscF_in_gg(X1, X2))
11.17/3.75	   U27_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), s(X2))
11.17/3.75	   U28_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscE_out_gg(s(X1), X2)
11.17/3.75	
11.17/3.75	The argument filtering Pi contains the following mapping:
11.17/3.75	.(x1, x2)  =  .(x1, x2)
11.17/3.75	
11.17/3.75	appcA_in_agaa(x1, x2, x3, x4)  =  appcA_in_agaa(x2)
11.17/3.75	
11.17/3.75	appcA_out_agaa(x1, x2, x3, x4)  =  appcA_out_agaa(x2)
11.17/3.75	
11.17/3.75	U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
11.17/3.75	
11.17/3.75	appcB_in_aaa(x1, x2, x3)  =  appcB_in_aaa
11.17/3.75	
11.17/3.75	appcB_out_aaa(x1, x2, x3)  =  appcB_out_aaa
11.17/3.75	
11.17/3.75	U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
11.17/3.75	
11.17/3.75	permcC_in_ag(x1, x2)  =  permcC_in_ag(x2)
11.17/3.75	
11.17/3.75	[]  =  []
11.17/3.75	
11.17/3.75	permcC_out_ag(x1, x2)  =  permcC_out_ag(x2)
11.17/3.75	
11.17/3.75	U22_ag(x1, x2, x3, x4)  =  U22_ag(x2, x3, x4)
11.17/3.75	
11.17/3.75	U23_ag(x1, x2, x3, x4)  =  U23_ag(x2, x3, x4)
11.17/3.75	
11.17/3.75	U24_ag(x1, x2, x3, x4)  =  U24_ag(x2, x3, x4)
11.17/3.75	
11.17/3.75	s(x1)  =  s(x1)
11.17/3.75	
11.17/3.75	lesscE_in_gg(x1, x2)  =  lesscE_in_gg(x1, x2)
11.17/3.75	
11.17/3.75	0  =  0
11.17/3.75	
11.17/3.75	lesscE_out_gg(x1, x2)  =  lesscE_out_gg(x1, x2)
11.17/3.75	
11.17/3.75	U28_gg(x1, x2, x3)  =  U28_gg(x1, x2, x3)
11.17/3.75	
11.17/3.75	lesscF_in_gg(x1, x2)  =  lesscF_in_gg(x1, x2)
11.17/3.75	
11.17/3.75	lesscF_out_gg(x1, x2)  =  lesscF_out_gg(x1, x2)
11.17/3.75	
11.17/3.75	U27_gg(x1, x2, x3)  =  U27_gg(x1, x2, x3)
11.17/3.75	
11.17/3.75	ORDEREDD_IN_GG(x1, x2)  =  ORDEREDD_IN_GG(x1, x2)
11.17/3.75	
11.17/3.75	U9_GG(x1, x2, x3, x4)  =  U9_GG(x1, x2, x3, x4)
11.17/3.75	
11.17/3.75	
11.17/3.75	We have to consider all (P,R,Pi)-chains
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(89) UsableRulesProof (EQUIVALENT)
11.17/3.75	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(90)
11.17/3.75	Obligation:
11.17/3.75	Pi DP problem:
11.17/3.75	The TRS P consists of the following rules:
11.17/3.75	
11.17/3.75	   ORDEREDD_IN_GG(X1, .(X2, X3)) -> U9_GG(X1, X2, X3, lesscE_in_gg(X1, X2))
11.17/3.75	   U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) -> ORDEREDD_IN_GG(X2, X3)
11.17/3.75	
11.17/3.75	The TRS R consists of the following rules:
11.17/3.75	
11.17/3.75	   lesscE_in_gg(0, X1) -> lesscE_out_gg(0, X1)
11.17/3.75	   lesscE_in_gg(s(X1), X2) -> U28_gg(X1, X2, lesscF_in_gg(X1, X2))
11.17/3.75	   U28_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscE_out_gg(s(X1), X2)
11.17/3.75	   lesscF_in_gg(0, s(X1)) -> lesscF_out_gg(0, s(X1))
11.17/3.75	   lesscF_in_gg(s(X1), s(X2)) -> U27_gg(X1, X2, lesscF_in_gg(X1, X2))
11.17/3.75	   U27_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), s(X2))
11.17/3.75	
11.17/3.75	Pi is empty.
11.17/3.75	We have to consider all (P,R,Pi)-chains
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(91) PiDPToQDPProof (EQUIVALENT)
11.17/3.75	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(92)
11.17/3.75	Obligation:
11.17/3.75	Q DP problem:
11.17/3.75	The TRS P consists of the following rules:
11.17/3.75	
11.17/3.75	   ORDEREDD_IN_GG(X1, .(X2, X3)) -> U9_GG(X1, X2, X3, lesscE_in_gg(X1, X2))
11.17/3.75	   U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) -> ORDEREDD_IN_GG(X2, X3)
11.17/3.75	
11.17/3.75	The TRS R consists of the following rules:
11.17/3.75	
11.17/3.75	   lesscE_in_gg(0, X1) -> lesscE_out_gg(0, X1)
11.17/3.75	   lesscE_in_gg(s(X1), X2) -> U28_gg(X1, X2, lesscF_in_gg(X1, X2))
11.17/3.75	   U28_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscE_out_gg(s(X1), X2)
11.17/3.75	   lesscF_in_gg(0, s(X1)) -> lesscF_out_gg(0, s(X1))
11.17/3.75	   lesscF_in_gg(s(X1), s(X2)) -> U27_gg(X1, X2, lesscF_in_gg(X1, X2))
11.17/3.75	   U27_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), s(X2))
11.17/3.75	
11.17/3.75	The set Q consists of the following terms:
11.17/3.75	
11.17/3.75	   lesscE_in_gg(x0, x1)
11.17/3.75	   U28_gg(x0, x1, x2)
11.17/3.75	   lesscF_in_gg(x0, x1)
11.17/3.75	   U27_gg(x0, x1, x2)
11.17/3.75	
11.17/3.75	We have to consider all (P,Q,R)-chains.
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(93) QDPSizeChangeProof (EQUIVALENT)
11.17/3.75	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. 
11.17/3.75	
11.17/3.75	From the DPs we obtained the following set of size-change graphs:
11.17/3.75	*U9_GG(X1, X2, X3, lesscE_out_gg(X1, X2)) -> ORDEREDD_IN_GG(X2, X3)
11.17/3.75	The graph contains the following edges 2 >= 1, 4 > 1, 3 >= 2
11.17/3.75	
11.17/3.75	
11.17/3.75	*ORDEREDD_IN_GG(X1, .(X2, X3)) -> U9_GG(X1, X2, X3, lesscE_in_gg(X1, X2))
11.17/3.75	The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3
11.17/3.75	
11.17/3.75	
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(94)
11.17/3.75	YES
11.17/3.75	
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(95)
11.17/3.75	Obligation:
11.17/3.75	Pi DP problem:
11.17/3.75	The TRS P consists of the following rules:
11.17/3.75	
11.17/3.75	   APPB_IN_AAA(.(X1, X2), X3, .(X1, X4)) -> APPB_IN_AAA(X2, X3, X4)
11.17/3.75	
11.17/3.75	The TRS R consists of the following rules:
11.17/3.75	
11.17/3.75	   appcA_in_agaa([], X1, X2, .(X1, X2)) -> appcA_out_agaa([], X1, X2, .(X1, X2))
11.17/3.75	   appcA_in_agaa(.(X1, X2), X3, X4, .(X1, X5)) -> U20_agaa(X1, X2, X3, X4, X5, appcA_in_agaa(X2, X3, X4, X5))
11.17/3.75	   U20_agaa(X1, X2, X3, X4, X5, appcA_out_agaa(X2, X3, X4, X5)) -> appcA_out_agaa(.(X1, X2), X3, X4, .(X1, X5))
11.17/3.75	   appcB_in_aaa([], X1, X1) -> appcB_out_aaa([], X1, X1)
11.17/3.75	   appcB_in_aaa(.(X1, X2), X3, .(X1, X4)) -> U21_aaa(X1, X2, X3, X4, appcB_in_aaa(X2, X3, X4))
11.17/3.75	   U21_aaa(X1, X2, X3, X4, appcB_out_aaa(X2, X3, X4)) -> appcB_out_aaa(.(X1, X2), X3, .(X1, X4))
11.17/3.75	   permcC_in_ag([], []) -> permcC_out_ag([], [])
11.17/3.75	   permcC_in_ag(X1, .(X2, X3)) -> U22_ag(X1, X2, X3, appcA_in_agaa(X4, X2, X5, X1))
11.17/3.75	   U22_ag(X1, X2, X3, appcA_out_agaa(X4, X2, X5, X1)) -> U23_ag(X1, X2, X3, appcB_in_aaa(X4, X5, X6))
11.17/3.75	   U23_ag(X1, X2, X3, appcB_out_aaa(X4, X5, X6)) -> U24_ag(X1, X2, X3, permcC_in_ag(X6, X3))
11.17/3.75	   U24_ag(X1, X2, X3, permcC_out_ag(X6, X3)) -> permcC_out_ag(X1, .(X2, X3))
11.17/3.75	   lesscE_in_gg(0, X1) -> lesscE_out_gg(0, X1)
11.17/3.75	   lesscE_in_gg(s(X1), X2) -> U28_gg(X1, X2, lesscF_in_gg(X1, X2))
11.17/3.75	   lesscF_in_gg(0, s(X1)) -> lesscF_out_gg(0, s(X1))
11.17/3.75	   lesscF_in_gg(s(X1), s(X2)) -> U27_gg(X1, X2, lesscF_in_gg(X1, X2))
11.17/3.75	   U27_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), s(X2))
11.17/3.75	   U28_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscE_out_gg(s(X1), X2)
11.17/3.75	
11.17/3.75	The argument filtering Pi contains the following mapping:
11.17/3.75	.(x1, x2)  =  .(x1, x2)
11.17/3.75	
11.17/3.75	appcA_in_agaa(x1, x2, x3, x4)  =  appcA_in_agaa(x2)
11.17/3.75	
11.17/3.75	appcA_out_agaa(x1, x2, x3, x4)  =  appcA_out_agaa(x2)
11.17/3.75	
11.17/3.75	U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
11.17/3.75	
11.17/3.75	appcB_in_aaa(x1, x2, x3)  =  appcB_in_aaa
11.17/3.75	
11.17/3.75	appcB_out_aaa(x1, x2, x3)  =  appcB_out_aaa
11.17/3.75	
11.17/3.75	U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
11.17/3.75	
11.17/3.75	permcC_in_ag(x1, x2)  =  permcC_in_ag(x2)
11.17/3.75	
11.17/3.75	[]  =  []
11.17/3.75	
11.17/3.75	permcC_out_ag(x1, x2)  =  permcC_out_ag(x2)
11.17/3.75	
11.17/3.75	U22_ag(x1, x2, x3, x4)  =  U22_ag(x2, x3, x4)
11.17/3.75	
11.17/3.75	U23_ag(x1, x2, x3, x4)  =  U23_ag(x2, x3, x4)
11.17/3.75	
11.17/3.75	U24_ag(x1, x2, x3, x4)  =  U24_ag(x2, x3, x4)
11.17/3.75	
11.17/3.75	s(x1)  =  s(x1)
11.17/3.75	
11.17/3.75	lesscE_in_gg(x1, x2)  =  lesscE_in_gg(x1, x2)
11.17/3.75	
11.17/3.75	0  =  0
11.17/3.75	
11.17/3.75	lesscE_out_gg(x1, x2)  =  lesscE_out_gg(x1, x2)
11.17/3.75	
11.17/3.75	U28_gg(x1, x2, x3)  =  U28_gg(x1, x2, x3)
11.17/3.75	
11.17/3.75	lesscF_in_gg(x1, x2)  =  lesscF_in_gg(x1, x2)
11.17/3.75	
11.17/3.75	lesscF_out_gg(x1, x2)  =  lesscF_out_gg(x1, x2)
11.17/3.75	
11.17/3.75	U27_gg(x1, x2, x3)  =  U27_gg(x1, x2, x3)
11.17/3.75	
11.17/3.75	APPB_IN_AAA(x1, x2, x3)  =  APPB_IN_AAA
11.17/3.75	
11.17/3.75	
11.17/3.75	We have to consider all (P,R,Pi)-chains
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(96) UsableRulesProof (EQUIVALENT)
11.17/3.75	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(97)
11.17/3.75	Obligation:
11.17/3.75	Pi DP problem:
11.17/3.75	The TRS P consists of the following rules:
11.17/3.75	
11.17/3.75	   APPB_IN_AAA(.(X1, X2), X3, .(X1, X4)) -> APPB_IN_AAA(X2, X3, X4)
11.17/3.75	
11.17/3.75	R is empty.
11.17/3.75	The argument filtering Pi contains the following mapping:
11.17/3.75	.(x1, x2)  =  .(x1, x2)
11.17/3.75	
11.17/3.75	APPB_IN_AAA(x1, x2, x3)  =  APPB_IN_AAA
11.17/3.75	
11.17/3.75	
11.17/3.75	We have to consider all (P,R,Pi)-chains
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(98) PiDPToQDPProof (SOUND)
11.17/3.75	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(99)
11.17/3.75	Obligation:
11.17/3.75	Q DP problem:
11.17/3.75	The TRS P consists of the following rules:
11.17/3.75	
11.17/3.75	   APPB_IN_AAA -> APPB_IN_AAA
11.17/3.75	
11.17/3.75	R is empty.
11.17/3.75	Q is empty.
11.17/3.75	We have to consider all (P,Q,R)-chains.
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(100) NonTerminationLoopProof (COMPLETE)
11.17/3.75	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
11.17/3.75	Found a loop by semiunifying a rule from P directly.
11.17/3.75	
11.17/3.75	s = APPB_IN_AAA evaluates to  t =APPB_IN_AAA
11.17/3.75	
11.17/3.75	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
11.17/3.75	* Matcher: [ ]
11.17/3.75	* Semiunifier: [ ]
11.17/3.75	
11.17/3.75	--------------------------------------------------------------------------------
11.17/3.75	Rewriting sequence
11.17/3.75	
11.17/3.75	The DP semiunifies directly so there is only one rewrite step from APPB_IN_AAA to APPB_IN_AAA.
11.17/3.75	
11.17/3.75	
11.17/3.75	
11.17/3.75	
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(101)
11.17/3.75	NO
11.17/3.75	
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(102)
11.17/3.75	Obligation:
11.17/3.75	Pi DP problem:
11.17/3.75	The TRS P consists of the following rules:
11.17/3.75	
11.17/3.75	   APPA_IN_AGAA(.(X1, X2), X3, X4, .(X1, X5)) -> APPA_IN_AGAA(X2, X3, X4, X5)
11.17/3.75	
11.17/3.75	The TRS R consists of the following rules:
11.17/3.75	
11.17/3.75	   appcA_in_agaa([], X1, X2, .(X1, X2)) -> appcA_out_agaa([], X1, X2, .(X1, X2))
11.17/3.75	   appcA_in_agaa(.(X1, X2), X3, X4, .(X1, X5)) -> U20_agaa(X1, X2, X3, X4, X5, appcA_in_agaa(X2, X3, X4, X5))
11.17/3.75	   U20_agaa(X1, X2, X3, X4, X5, appcA_out_agaa(X2, X3, X4, X5)) -> appcA_out_agaa(.(X1, X2), X3, X4, .(X1, X5))
11.17/3.75	   appcB_in_aaa([], X1, X1) -> appcB_out_aaa([], X1, X1)
11.17/3.75	   appcB_in_aaa(.(X1, X2), X3, .(X1, X4)) -> U21_aaa(X1, X2, X3, X4, appcB_in_aaa(X2, X3, X4))
11.17/3.75	   U21_aaa(X1, X2, X3, X4, appcB_out_aaa(X2, X3, X4)) -> appcB_out_aaa(.(X1, X2), X3, .(X1, X4))
11.17/3.75	   permcC_in_ag([], []) -> permcC_out_ag([], [])
11.17/3.75	   permcC_in_ag(X1, .(X2, X3)) -> U22_ag(X1, X2, X3, appcA_in_agaa(X4, X2, X5, X1))
11.17/3.75	   U22_ag(X1, X2, X3, appcA_out_agaa(X4, X2, X5, X1)) -> U23_ag(X1, X2, X3, appcB_in_aaa(X4, X5, X6))
11.17/3.75	   U23_ag(X1, X2, X3, appcB_out_aaa(X4, X5, X6)) -> U24_ag(X1, X2, X3, permcC_in_ag(X6, X3))
11.17/3.75	   U24_ag(X1, X2, X3, permcC_out_ag(X6, X3)) -> permcC_out_ag(X1, .(X2, X3))
11.17/3.75	   lesscE_in_gg(0, X1) -> lesscE_out_gg(0, X1)
11.17/3.75	   lesscE_in_gg(s(X1), X2) -> U28_gg(X1, X2, lesscF_in_gg(X1, X2))
11.17/3.75	   lesscF_in_gg(0, s(X1)) -> lesscF_out_gg(0, s(X1))
11.17/3.75	   lesscF_in_gg(s(X1), s(X2)) -> U27_gg(X1, X2, lesscF_in_gg(X1, X2))
11.17/3.75	   U27_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), s(X2))
11.17/3.75	   U28_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscE_out_gg(s(X1), X2)
11.17/3.75	
11.17/3.75	The argument filtering Pi contains the following mapping:
11.17/3.75	.(x1, x2)  =  .(x1, x2)
11.17/3.75	
11.17/3.75	appcA_in_agaa(x1, x2, x3, x4)  =  appcA_in_agaa(x2)
11.17/3.75	
11.17/3.75	appcA_out_agaa(x1, x2, x3, x4)  =  appcA_out_agaa(x2)
11.17/3.75	
11.17/3.75	U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
11.17/3.75	
11.17/3.75	appcB_in_aaa(x1, x2, x3)  =  appcB_in_aaa
11.17/3.75	
11.17/3.75	appcB_out_aaa(x1, x2, x3)  =  appcB_out_aaa
11.17/3.75	
11.17/3.75	U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
11.17/3.75	
11.17/3.75	permcC_in_ag(x1, x2)  =  permcC_in_ag(x2)
11.17/3.75	
11.17/3.75	[]  =  []
11.17/3.75	
11.17/3.75	permcC_out_ag(x1, x2)  =  permcC_out_ag(x2)
11.17/3.75	
11.17/3.75	U22_ag(x1, x2, x3, x4)  =  U22_ag(x2, x3, x4)
11.17/3.75	
11.17/3.75	U23_ag(x1, x2, x3, x4)  =  U23_ag(x2, x3, x4)
11.17/3.75	
11.17/3.75	U24_ag(x1, x2, x3, x4)  =  U24_ag(x2, x3, x4)
11.17/3.75	
11.17/3.75	s(x1)  =  s(x1)
11.17/3.75	
11.17/3.75	lesscE_in_gg(x1, x2)  =  lesscE_in_gg(x1, x2)
11.17/3.75	
11.17/3.75	0  =  0
11.17/3.75	
11.17/3.75	lesscE_out_gg(x1, x2)  =  lesscE_out_gg(x1, x2)
11.17/3.75	
11.17/3.75	U28_gg(x1, x2, x3)  =  U28_gg(x1, x2, x3)
11.17/3.75	
11.17/3.75	lesscF_in_gg(x1, x2)  =  lesscF_in_gg(x1, x2)
11.17/3.75	
11.17/3.75	lesscF_out_gg(x1, x2)  =  lesscF_out_gg(x1, x2)
11.17/3.75	
11.17/3.75	U27_gg(x1, x2, x3)  =  U27_gg(x1, x2, x3)
11.17/3.75	
11.17/3.75	APPA_IN_AGAA(x1, x2, x3, x4)  =  APPA_IN_AGAA(x2)
11.17/3.75	
11.17/3.75	
11.17/3.75	We have to consider all (P,R,Pi)-chains
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(103) UsableRulesProof (EQUIVALENT)
11.17/3.75	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(104)
11.17/3.75	Obligation:
11.17/3.75	Pi DP problem:
11.17/3.75	The TRS P consists of the following rules:
11.17/3.75	
11.17/3.75	   APPA_IN_AGAA(.(X1, X2), X3, X4, .(X1, X5)) -> APPA_IN_AGAA(X2, X3, X4, X5)
11.17/3.75	
11.17/3.75	R is empty.
11.17/3.75	The argument filtering Pi contains the following mapping:
11.17/3.75	.(x1, x2)  =  .(x1, x2)
11.17/3.75	
11.17/3.75	APPA_IN_AGAA(x1, x2, x3, x4)  =  APPA_IN_AGAA(x2)
11.17/3.75	
11.17/3.75	
11.17/3.75	We have to consider all (P,R,Pi)-chains
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(105) PiDPToQDPProof (SOUND)
11.17/3.75	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(106)
11.17/3.75	Obligation:
11.17/3.75	Q DP problem:
11.17/3.75	The TRS P consists of the following rules:
11.17/3.75	
11.17/3.75	   APPA_IN_AGAA(X3) -> APPA_IN_AGAA(X3)
11.17/3.75	
11.17/3.75	R is empty.
11.17/3.75	Q is empty.
11.17/3.75	We have to consider all (P,Q,R)-chains.
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(107)
11.17/3.75	Obligation:
11.17/3.75	Pi DP problem:
11.17/3.75	The TRS P consists of the following rules:
11.17/3.75	
11.17/3.75	   PERMC_IN_AG(X1, .(X2, X3)) -> U4_AG(X1, X2, X3, appcA_in_agaa(X4, X2, X5, X1))
11.17/3.75	   U4_AG(X1, X2, X3, appcA_out_agaa(X4, X2, X5, X1)) -> U6_AG(X1, X2, X3, appcB_in_aaa(X4, X5, X6))
11.17/3.75	   U6_AG(X1, X2, X3, appcB_out_aaa(X4, X5, X6)) -> PERMC_IN_AG(X6, X3)
11.17/3.75	
11.17/3.75	The TRS R consists of the following rules:
11.17/3.75	
11.17/3.75	   appcA_in_agaa([], X1, X2, .(X1, X2)) -> appcA_out_agaa([], X1, X2, .(X1, X2))
11.17/3.75	   appcA_in_agaa(.(X1, X2), X3, X4, .(X1, X5)) -> U20_agaa(X1, X2, X3, X4, X5, appcA_in_agaa(X2, X3, X4, X5))
11.17/3.75	   U20_agaa(X1, X2, X3, X4, X5, appcA_out_agaa(X2, X3, X4, X5)) -> appcA_out_agaa(.(X1, X2), X3, X4, .(X1, X5))
11.17/3.75	   appcB_in_aaa([], X1, X1) -> appcB_out_aaa([], X1, X1)
11.17/3.75	   appcB_in_aaa(.(X1, X2), X3, .(X1, X4)) -> U21_aaa(X1, X2, X3, X4, appcB_in_aaa(X2, X3, X4))
11.17/3.75	   U21_aaa(X1, X2, X3, X4, appcB_out_aaa(X2, X3, X4)) -> appcB_out_aaa(.(X1, X2), X3, .(X1, X4))
11.17/3.75	   permcC_in_ag([], []) -> permcC_out_ag([], [])
11.17/3.75	   permcC_in_ag(X1, .(X2, X3)) -> U22_ag(X1, X2, X3, appcA_in_agaa(X4, X2, X5, X1))
11.17/3.75	   U22_ag(X1, X2, X3, appcA_out_agaa(X4, X2, X5, X1)) -> U23_ag(X1, X2, X3, appcB_in_aaa(X4, X5, X6))
11.17/3.75	   U23_ag(X1, X2, X3, appcB_out_aaa(X4, X5, X6)) -> U24_ag(X1, X2, X3, permcC_in_ag(X6, X3))
11.17/3.75	   U24_ag(X1, X2, X3, permcC_out_ag(X6, X3)) -> permcC_out_ag(X1, .(X2, X3))
11.17/3.75	   lesscE_in_gg(0, X1) -> lesscE_out_gg(0, X1)
11.17/3.75	   lesscE_in_gg(s(X1), X2) -> U28_gg(X1, X2, lesscF_in_gg(X1, X2))
11.17/3.75	   lesscF_in_gg(0, s(X1)) -> lesscF_out_gg(0, s(X1))
11.17/3.75	   lesscF_in_gg(s(X1), s(X2)) -> U27_gg(X1, X2, lesscF_in_gg(X1, X2))
11.17/3.75	   U27_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscF_out_gg(s(X1), s(X2))
11.17/3.75	   U28_gg(X1, X2, lesscF_out_gg(X1, X2)) -> lesscE_out_gg(s(X1), X2)
11.17/3.75	
11.17/3.75	The argument filtering Pi contains the following mapping:
11.17/3.75	.(x1, x2)  =  .(x1, x2)
11.17/3.75	
11.17/3.75	appcA_in_agaa(x1, x2, x3, x4)  =  appcA_in_agaa(x2)
11.17/3.75	
11.17/3.75	appcA_out_agaa(x1, x2, x3, x4)  =  appcA_out_agaa(x2)
11.17/3.75	
11.17/3.75	U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
11.17/3.75	
11.17/3.75	appcB_in_aaa(x1, x2, x3)  =  appcB_in_aaa
11.17/3.75	
11.17/3.75	appcB_out_aaa(x1, x2, x3)  =  appcB_out_aaa
11.17/3.75	
11.17/3.75	U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
11.17/3.75	
11.17/3.75	permcC_in_ag(x1, x2)  =  permcC_in_ag(x2)
11.17/3.75	
11.17/3.75	[]  =  []
11.17/3.75	
11.17/3.75	permcC_out_ag(x1, x2)  =  permcC_out_ag(x2)
11.17/3.75	
11.17/3.75	U22_ag(x1, x2, x3, x4)  =  U22_ag(x2, x3, x4)
11.17/3.75	
11.17/3.75	U23_ag(x1, x2, x3, x4)  =  U23_ag(x2, x3, x4)
11.17/3.75	
11.17/3.75	U24_ag(x1, x2, x3, x4)  =  U24_ag(x2, x3, x4)
11.17/3.75	
11.17/3.75	s(x1)  =  s(x1)
11.17/3.75	
11.17/3.75	lesscE_in_gg(x1, x2)  =  lesscE_in_gg(x1, x2)
11.17/3.75	
11.17/3.75	0  =  0
11.17/3.75	
11.17/3.75	lesscE_out_gg(x1, x2)  =  lesscE_out_gg(x1, x2)
11.17/3.75	
11.17/3.75	U28_gg(x1, x2, x3)  =  U28_gg(x1, x2, x3)
11.17/3.75	
11.17/3.75	lesscF_in_gg(x1, x2)  =  lesscF_in_gg(x1, x2)
11.17/3.75	
11.17/3.75	lesscF_out_gg(x1, x2)  =  lesscF_out_gg(x1, x2)
11.17/3.75	
11.17/3.75	U27_gg(x1, x2, x3)  =  U27_gg(x1, x2, x3)
11.17/3.75	
11.17/3.75	PERMC_IN_AG(x1, x2)  =  PERMC_IN_AG(x2)
11.17/3.75	
11.17/3.75	U4_AG(x1, x2, x3, x4)  =  U4_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	U6_AG(x1, x2, x3, x4)  =  U6_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	
11.17/3.75	We have to consider all (P,R,Pi)-chains
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(108) UsableRulesProof (EQUIVALENT)
11.17/3.75	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(109)
11.17/3.75	Obligation:
11.17/3.75	Pi DP problem:
11.17/3.75	The TRS P consists of the following rules:
11.17/3.75	
11.17/3.75	   PERMC_IN_AG(X1, .(X2, X3)) -> U4_AG(X1, X2, X3, appcA_in_agaa(X4, X2, X5, X1))
11.17/3.75	   U4_AG(X1, X2, X3, appcA_out_agaa(X4, X2, X5, X1)) -> U6_AG(X1, X2, X3, appcB_in_aaa(X4, X5, X6))
11.17/3.75	   U6_AG(X1, X2, X3, appcB_out_aaa(X4, X5, X6)) -> PERMC_IN_AG(X6, X3)
11.17/3.75	
11.17/3.75	The TRS R consists of the following rules:
11.17/3.75	
11.17/3.75	   appcA_in_agaa([], X1, X2, .(X1, X2)) -> appcA_out_agaa([], X1, X2, .(X1, X2))
11.17/3.75	   appcA_in_agaa(.(X1, X2), X3, X4, .(X1, X5)) -> U20_agaa(X1, X2, X3, X4, X5, appcA_in_agaa(X2, X3, X4, X5))
11.17/3.75	   appcB_in_aaa([], X1, X1) -> appcB_out_aaa([], X1, X1)
11.17/3.75	   appcB_in_aaa(.(X1, X2), X3, .(X1, X4)) -> U21_aaa(X1, X2, X3, X4, appcB_in_aaa(X2, X3, X4))
11.17/3.75	   U20_agaa(X1, X2, X3, X4, X5, appcA_out_agaa(X2, X3, X4, X5)) -> appcA_out_agaa(.(X1, X2), X3, X4, .(X1, X5))
11.17/3.75	   U21_aaa(X1, X2, X3, X4, appcB_out_aaa(X2, X3, X4)) -> appcB_out_aaa(.(X1, X2), X3, .(X1, X4))
11.17/3.75	
11.17/3.75	The argument filtering Pi contains the following mapping:
11.17/3.75	.(x1, x2)  =  .(x1, x2)
11.17/3.75	
11.17/3.75	appcA_in_agaa(x1, x2, x3, x4)  =  appcA_in_agaa(x2)
11.17/3.75	
11.17/3.75	appcA_out_agaa(x1, x2, x3, x4)  =  appcA_out_agaa(x2)
11.17/3.75	
11.17/3.75	U20_agaa(x1, x2, x3, x4, x5, x6)  =  U20_agaa(x3, x6)
11.17/3.75	
11.17/3.75	appcB_in_aaa(x1, x2, x3)  =  appcB_in_aaa
11.17/3.75	
11.17/3.75	appcB_out_aaa(x1, x2, x3)  =  appcB_out_aaa
11.17/3.75	
11.17/3.75	U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
11.17/3.75	
11.17/3.75	[]  =  []
11.17/3.75	
11.17/3.75	PERMC_IN_AG(x1, x2)  =  PERMC_IN_AG(x2)
11.17/3.75	
11.17/3.75	U4_AG(x1, x2, x3, x4)  =  U4_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	U6_AG(x1, x2, x3, x4)  =  U6_AG(x2, x3, x4)
11.17/3.75	
11.17/3.75	
11.17/3.75	We have to consider all (P,R,Pi)-chains
11.17/3.75	----------------------------------------
11.17/3.75	
11.17/3.75	(110) PrologToTRSTransformerProof (SOUND)
11.17/3.75	Transformed Prolog program to TRS.
11.17/3.75	
11.17/3.75	{
11.17/3.75	    "root": 2,
11.17/3.75	    "program": {
11.17/3.75	        "directives": [],
11.17/3.75	        "clauses": [
11.17/3.75	            [
11.17/3.75	                "(ss Xs Ys)",
11.17/3.75	                "(',' (perm Xs Ys) (ordered Ys))"
11.17/3.75	            ],
11.17/3.75	            [
11.17/3.75	                "(perm ([]) ([]))",
11.17/3.75	                null
11.17/3.75	            ],
11.17/3.75	            [
11.17/3.75	                "(perm Xs (. X Ys))",
11.17/3.75	                "(',' (app X1s (. X X2s) Xs) (',' (app X1s X2s Zs) (perm Zs Ys)))"
11.17/3.75	            ],
11.17/3.75	            [
11.17/3.75	                "(app ([]) X X)",
11.17/3.75	                null
11.17/3.75	            ],
11.17/3.75	            [
11.17/3.75	                "(app (. X Xs) Ys (. X Zs))",
11.17/3.75	                "(app Xs Ys Zs)"
11.17/3.75	            ],
11.17/3.75	            [
11.17/3.75	                "(ordered ([]))",
11.17/3.75	                null
11.17/3.75	            ],
11.17/3.75	            [
11.17/3.75	                "(ordered (. X1 ([])))",
11.17/3.75	                null
11.17/3.75	            ],
11.17/3.75	            [
11.17/3.75	                "(ordered (. X (. Y Xs)))",
11.17/3.75	                "(',' (less X (s Y)) (ordered (. Y Xs)))"
11.17/3.75	            ],
11.17/3.75	            [
11.17/3.75	                "(less (0) (s X2))",
11.17/3.75	                null
11.17/3.75	            ],
11.17/3.75	            [
11.17/3.75	                "(less (s X) (s Y))",
11.17/3.75	                "(less X Y)"
11.17/3.75	            ]
11.17/3.75	        ]
11.17/3.75	    },
11.17/3.75	    "graph": {
11.17/3.75	        "nodes": {
11.17/3.75	            "type": "Nodes",
11.17/3.75	            "397": {
11.17/3.75	                "goal": [
11.17/3.75	                    {
11.17/3.75	                        "clause": 1,
11.17/3.75	                        "scope": 2,
11.17/3.75	                        "term": "(perm T10 T9)"
11.17/3.75	                    },
11.17/3.75	                    {
11.17/3.75	                        "clause": 2,
11.17/3.75	                        "scope": 2,
11.17/3.75	                        "term": "(perm T10 T9)"
11.17/3.75	                    }
11.17/3.75	                ],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": ["T9"],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "398": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": 1,
11.17/3.75	                    "scope": 2,
11.17/3.75	                    "term": "(perm T10 T9)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": ["T9"],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "399": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": 2,
11.17/3.75	                    "scope": 2,
11.17/3.75	                    "term": "(perm T10 T9)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": ["T9"],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "510": {
11.17/3.75	                "goal": [],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "512": {
11.17/3.75	                "goal": [],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "515": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": -1,
11.17/3.75	                    "scope": -1,
11.17/3.75	                    "term": "(',' (less T89 (s T90)) (ordered (. T90 T91)))"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [
11.17/3.75	                        "T89",
11.17/3.75	                        "T90",
11.17/3.75	                        "T91"
11.17/3.75	                    ],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "517": {
11.17/3.75	                "goal": [],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "51": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": -1,
11.17/3.75	                    "scope": -1,
11.17/3.75	                    "term": "(',' (perm T10 T9) (ordered T9))"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": ["T9"],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "2": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": -1,
11.17/3.75	                    "scope": -1,
11.17/3.75	                    "term": "(ss T1 T2)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": ["T2"],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "3": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": 0,
11.17/3.75	                    "scope": 1,
11.17/3.75	                    "term": "(ss T1 T2)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": ["T2"],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "520": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": -1,
11.17/3.75	                    "scope": -1,
11.17/3.75	                    "term": "(less T89 (s T90))"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [
11.17/3.75	                        "T89",
11.17/3.75	                        "T90"
11.17/3.75	                    ],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "521": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": -1,
11.17/3.75	                    "scope": -1,
11.17/3.75	                    "term": "(ordered (. T90 T91))"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [
11.17/3.75	                        "T90",
11.17/3.75	                        "T91"
11.17/3.75	                    ],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "401": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": -1,
11.17/3.75	                    "scope": -1,
11.17/3.75	                    "term": "(true)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "402": {
11.17/3.75	                "goal": [],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "403": {
11.17/3.75	                "goal": [],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "525": {
11.17/3.75	                "goal": [
11.17/3.75	                    {
11.17/3.75	                        "clause": 8,
11.17/3.75	                        "scope": 6,
11.17/3.75	                        "term": "(less T89 (s T90))"
11.17/3.75	                    },
11.17/3.75	                    {
11.17/3.75	                        "clause": 9,
11.17/3.75	                        "scope": 6,
11.17/3.75	                        "term": "(less T89 (s T90))"
11.17/3.75	                    }
11.17/3.75	                ],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [
11.17/3.75	                        "T89",
11.17/3.75	                        "T90"
11.17/3.75	                    ],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "449": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": -1,
11.17/3.75	                    "scope": -1,
11.17/3.75	                    "term": "(app T25 T26 X23)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": ["X23"],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "406": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": -1,
11.17/3.75	                    "scope": -1,
11.17/3.75	                    "term": "(',' (app X21 (. T18 X22) T20) (',' (app X21 X22 X23) (perm X23 T19)))"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [
11.17/3.75	                        "T18",
11.17/3.75	                        "T19"
11.17/3.75	                    ],
11.17/3.75	                    "free": [
11.17/3.75	                        "X21",
11.17/3.75	                        "X22",
11.17/3.75	                        "X23"
11.17/3.75	                    ],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "407": {
11.17/3.75	                "goal": [],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "528": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": 8,
11.17/3.75	                    "scope": 6,
11.17/3.75	                    "term": "(less T89 (s T90))"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [
11.17/3.75	                        "T89",
11.17/3.75	                        "T90"
11.17/3.75	                    ],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "529": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": 9,
11.17/3.75	                    "scope": 6,
11.17/3.75	                    "term": "(less T89 (s T90))"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [
11.17/3.75	                        "T89",
11.17/3.75	                        "T90"
11.17/3.75	                    ],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "490": {
11.17/3.75	                "goal": [],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "491": {
11.17/3.75	                "goal": [],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "450": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": -1,
11.17/3.75	                    "scope": -1,
11.17/3.75	                    "term": "(perm T57 T19)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": ["T19"],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "451": {
11.17/3.75	                "goal": [
11.17/3.75	                    {
11.17/3.75	                        "clause": 3,
11.17/3.75	                        "scope": 4,
11.17/3.75	                        "term": "(app T25 T26 X23)"
11.17/3.75	                    },
11.17/3.75	                    {
11.17/3.75	                        "clause": 4,
11.17/3.75	                        "scope": 4,
11.17/3.75	                        "term": "(app T25 T26 X23)"
11.17/3.75	                    }
11.17/3.75	                ],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": ["X23"],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "452": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": 3,
11.17/3.75	                    "scope": 4,
11.17/3.75	                    "term": "(app T25 T26 X23)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": ["X23"],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "496": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": -1,
11.17/3.75	                    "scope": -1,
11.17/3.75	                    "term": "(app T74 T75 X90)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": ["X90"],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "453": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": 4,
11.17/3.75	                    "scope": 4,
11.17/3.75	                    "term": "(app T25 T26 X23)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": ["X23"],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "497": {
11.17/3.75	                "goal": [],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "530": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": -1,
11.17/3.75	                    "scope": -1,
11.17/3.75	                    "term": "(true)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "411": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": -1,
11.17/3.75	                    "scope": -1,
11.17/3.75	                    "term": "(app X21 (. T18 X22) T20)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": ["T18"],
11.17/3.75	                    "free": [
11.17/3.75	                        "X21",
11.17/3.75	                        "X22"
11.17/3.75	                    ],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "532": {
11.17/3.75	                "goal": [],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "412": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": -1,
11.17/3.75	                    "scope": -1,
11.17/3.75	                    "term": "(',' (app T25 T26 X23) (perm X23 T19))"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": ["T19"],
11.17/3.75	                    "free": ["X23"],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "456": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": -1,
11.17/3.75	                    "scope": -1,
11.17/3.75	                    "term": "(true)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "533": {
11.17/3.75	                "goal": [],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "413": {
11.17/3.75	                "goal": [
11.17/3.75	                    {
11.17/3.75	                        "clause": 3,
11.17/3.75	                        "scope": 3,
11.17/3.75	                        "term": "(app X21 (. T18 X22) T20)"
11.17/3.75	                    },
11.17/3.75	                    {
11.17/3.75	                        "clause": 4,
11.17/3.75	                        "scope": 3,
11.17/3.75	                        "term": "(app X21 (. T18 X22) T20)"
11.17/3.75	                    }
11.17/3.75	                ],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": ["T18"],
11.17/3.75	                    "free": [
11.17/3.75	                        "X21",
11.17/3.75	                        "X22"
11.17/3.75	                    ],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "535": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": -1,
11.17/3.75	                    "scope": -1,
11.17/3.75	                    "term": "(less T105 T106)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [
11.17/3.75	                        "T105",
11.17/3.75	                        "T106"
11.17/3.75	                    ],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "415": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": 3,
11.17/3.75	                    "scope": 3,
11.17/3.75	                    "term": "(app X21 (. T18 X22) T20)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": ["T18"],
11.17/3.75	                    "free": [
11.17/3.75	                        "X21",
11.17/3.75	                        "X22"
11.17/3.75	                    ],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "536": {
11.17/3.75	                "goal": [],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "416": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": 4,
11.17/3.75	                    "scope": 3,
11.17/3.75	                    "term": "(app X21 (. T18 X22) T20)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": ["T18"],
11.17/3.75	                    "free": [
11.17/3.75	                        "X21",
11.17/3.75	                        "X22"
11.17/3.75	                    ],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "537": {
11.17/3.75	                "goal": [
11.17/3.75	                    {
11.17/3.75	                        "clause": 8,
11.17/3.75	                        "scope": 7,
11.17/3.75	                        "term": "(less T105 T106)"
11.17/3.75	                    },
11.17/3.75	                    {
11.17/3.75	                        "clause": 9,
11.17/3.75	                        "scope": 7,
11.17/3.75	                        "term": "(less T105 T106)"
11.17/3.75	                    }
11.17/3.75	                ],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [
11.17/3.75	                        "T105",
11.17/3.75	                        "T106"
11.17/3.75	                    ],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "538": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": 8,
11.17/3.75	                    "scope": 7,
11.17/3.75	                    "term": "(less T105 T106)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [
11.17/3.75	                        "T105",
11.17/3.75	                        "T106"
11.17/3.75	                    ],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "418": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": -1,
11.17/3.75	                    "scope": -1,
11.17/3.75	                    "term": "(true)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "539": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": 9,
11.17/3.75	                    "scope": 7,
11.17/3.75	                    "term": "(less T105 T106)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [
11.17/3.75	                        "T105",
11.17/3.75	                        "T106"
11.17/3.75	                    ],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "419": {
11.17/3.75	                "goal": [],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "382": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": -1,
11.17/3.75	                    "scope": -1,
11.17/3.75	                    "term": "(perm T10 T9)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": ["T9"],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "383": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": -1,
11.17/3.75	                    "scope": -1,
11.17/3.75	                    "term": "(ordered T9)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": ["T9"],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "420": {
11.17/3.75	                "goal": [],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "421": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": -1,
11.17/3.75	                    "scope": -1,
11.17/3.75	                    "term": "(app X61 (. T47 X62) T50)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": ["T47"],
11.17/3.75	                    "free": [
11.17/3.75	                        "X61",
11.17/3.75	                        "X62"
11.17/3.75	                    ],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "542": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": -1,
11.17/3.75	                    "scope": -1,
11.17/3.75	                    "term": "(true)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "422": {
11.17/3.75	                "goal": [],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "543": {
11.17/3.75	                "goal": [],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "544": {
11.17/3.75	                "goal": [],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "501": {
11.17/3.75	                "goal": [
11.17/3.75	                    {
11.17/3.75	                        "clause": 5,
11.17/3.75	                        "scope": 5,
11.17/3.75	                        "term": "(ordered T9)"
11.17/3.75	                    },
11.17/3.75	                    {
11.17/3.75	                        "clause": 6,
11.17/3.75	                        "scope": 5,
11.17/3.75	                        "term": "(ordered T9)"
11.17/3.75	                    },
11.17/3.75	                    {
11.17/3.75	                        "clause": 7,
11.17/3.75	                        "scope": 5,
11.17/3.75	                        "term": "(ordered T9)"
11.17/3.75	                    }
11.17/3.75	                ],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": ["T9"],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "545": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": -1,
11.17/3.75	                    "scope": -1,
11.17/3.75	                    "term": "(less T118 T119)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [
11.17/3.75	                        "T118",
11.17/3.75	                        "T119"
11.17/3.75	                    ],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "502": {
11.17/3.75	                "goal": [{
11.17/3.75	                    "clause": 5,
11.17/3.75	                    "scope": 5,
11.17/3.75	                    "term": "(ordered T9)"
11.17/3.75	                }],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": ["T9"],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.75	            },
11.17/3.75	            "546": {
11.17/3.75	                "goal": [],
11.17/3.75	                "kb": {
11.17/3.75	                    "nonunifying": [],
11.17/3.75	                    "intvars": {},
11.17/3.75	                    "arithmetic": {
11.17/3.75	                        "type": "PlainIntegerRelationState",
11.17/3.75	                        "relations": []
11.17/3.75	                    },
11.17/3.75	                    "ground": [],
11.17/3.75	                    "free": [],
11.17/3.75	                    "exprvars": []
11.17/3.75	                }
11.17/3.76	            },
11.17/3.76	            "503": {
11.17/3.76	                "goal": [
11.17/3.76	                    {
11.17/3.76	                        "clause": 6,
11.17/3.76	                        "scope": 5,
11.17/3.76	                        "term": "(ordered T9)"
11.17/3.76	                    },
11.17/3.76	                    {
11.17/3.76	                        "clause": 7,
11.17/3.76	                        "scope": 5,
11.17/3.76	                        "term": "(ordered T9)"
11.17/3.76	                    }
11.17/3.76	                ],
11.17/3.76	                "kb": {
11.17/3.76	                    "nonunifying": [],
11.17/3.76	                    "intvars": {},
11.17/3.76	                    "arithmetic": {
11.17/3.76	                        "type": "PlainIntegerRelationState",
11.17/3.76	                        "relations": []
11.17/3.76	                    },
11.17/3.76	                    "ground": ["T9"],
11.17/3.76	                    "free": [],
11.17/3.76	                    "exprvars": []
11.17/3.76	                }
11.17/3.76	            },
11.17/3.76	            "504": {
11.17/3.76	                "goal": [{
11.17/3.76	                    "clause": -1,
11.17/3.76	                    "scope": -1,
11.17/3.76	                    "term": "(true)"
11.17/3.76	                }],
11.17/3.76	                "kb": {
11.17/3.76	                    "nonunifying": [],
11.17/3.76	                    "intvars": {},
11.17/3.76	                    "arithmetic": {
11.17/3.76	                        "type": "PlainIntegerRelationState",
11.17/3.76	                        "relations": []
11.17/3.76	                    },
11.17/3.76	                    "ground": [],
11.17/3.76	                    "free": [],
11.17/3.76	                    "exprvars": []
11.17/3.76	                }
11.17/3.76	            },
11.17/3.76	            "505": {
11.17/3.76	                "goal": [],
11.17/3.76	                "kb": {
11.17/3.76	                    "nonunifying": [],
11.17/3.76	                    "intvars": {},
11.17/3.76	                    "arithmetic": {
11.17/3.76	                        "type": "PlainIntegerRelationState",
11.17/3.76	                        "relations": []
11.17/3.76	                    },
11.17/3.76	                    "ground": [],
11.17/3.76	                    "free": [],
11.17/3.76	                    "exprvars": []
11.17/3.76	                }
11.17/3.76	            },
11.17/3.76	            "506": {
11.17/3.76	                "goal": [],
11.17/3.76	                "kb": {
11.17/3.76	                    "nonunifying": [],
11.17/3.76	                    "intvars": {},
11.17/3.76	                    "arithmetic": {
11.17/3.76	                        "type": "PlainIntegerRelationState",
11.17/3.76	                        "relations": []
11.17/3.76	                    },
11.17/3.76	                    "ground": [],
11.17/3.76	                    "free": [],
11.17/3.76	                    "exprvars": []
11.17/3.76	                }
11.17/3.76	            },
11.17/3.76	            "507": {
11.17/3.76	                "goal": [{
11.17/3.76	                    "clause": 6,
11.17/3.76	                    "scope": 5,
11.17/3.76	                    "term": "(ordered T9)"
11.17/3.76	                }],
11.17/3.76	                "kb": {
11.17/3.76	                    "nonunifying": [],
11.17/3.76	                    "intvars": {},
11.17/3.76	                    "arithmetic": {
11.17/3.76	                        "type": "PlainIntegerRelationState",
11.17/3.76	                        "relations": []
11.17/3.76	                    },
11.17/3.76	                    "ground": ["T9"],
11.17/3.76	                    "free": [],
11.17/3.76	                    "exprvars": []
11.17/3.76	                }
11.17/3.76	            },
11.17/3.76	            "508": {
11.17/3.76	                "goal": [{
11.17/3.76	                    "clause": 7,
11.17/3.76	                    "scope": 5,
11.17/3.76	                    "term": "(ordered T9)"
11.17/3.76	                }],
11.17/3.76	                "kb": {
11.17/3.76	                    "nonunifying": [],
11.17/3.76	                    "intvars": {},
11.17/3.76	                    "arithmetic": {
11.17/3.76	                        "type": "PlainIntegerRelationState",
11.17/3.76	                        "relations": []
11.17/3.76	                    },
11.17/3.76	                    "ground": ["T9"],
11.17/3.76	                    "free": [],
11.17/3.76	                    "exprvars": []
11.17/3.76	                }
11.17/3.76	            },
11.17/3.76	            "509": {
11.17/3.76	                "goal": [{
11.17/3.76	                    "clause": -1,
11.17/3.76	                    "scope": -1,
11.17/3.76	                    "term": "(true)"
11.17/3.76	                }],
11.17/3.76	                "kb": {
11.17/3.76	                    "nonunifying": [],
11.17/3.76	                    "intvars": {},
11.17/3.76	                    "arithmetic": {
11.17/3.76	                        "type": "PlainIntegerRelationState",
11.17/3.76	                        "relations": []
11.17/3.76	                    },
11.17/3.76	                    "ground": [],
11.17/3.76	                    "free": [],
11.17/3.76	                    "exprvars": []
11.17/3.76	                }
11.17/3.76	            }
11.17/3.76	        },
11.17/3.76	        "edges": [
11.17/3.76	            {
11.17/3.76	                "from": 2,
11.17/3.76	                "to": 3,
11.17/3.76	                "label": "CASE"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 3,
11.17/3.76	                "to": 51,
11.17/3.76	                "label": "ONLY EVAL with clause\nss(X7, X8) :- ','(perm(X7, X8), ordered(X8)).\nand substitutionT1 -> T10,\nX7 -> T10,\nT2 -> T9,\nX8 -> T9,\nT8 -> T10"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 51,
11.17/3.76	                "to": 382,
11.17/3.76	                "label": "SPLIT 1"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 51,
11.17/3.76	                "to": 383,
11.17/3.76	                "label": "SPLIT 2\nnew knowledge:\nT9 is ground"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 382,
11.17/3.76	                "to": 397,
11.17/3.76	                "label": "CASE"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 383,
11.17/3.76	                "to": 501,
11.17/3.76	                "label": "CASE"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 397,
11.17/3.76	                "to": 398,
11.17/3.76	                "label": "PARALLEL"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 397,
11.17/3.76	                "to": 399,
11.17/3.76	                "label": "PARALLEL"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 398,
11.17/3.76	                "to": 401,
11.17/3.76	                "label": "EVAL with clause\nperm([], []).\nand substitutionT10 -> [],\nT9 -> []"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 398,
11.17/3.76	                "to": 402,
11.17/3.76	                "label": "EVAL-BACKTRACK"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 399,
11.17/3.76	                "to": 406,
11.17/3.76	                "label": "EVAL with clause\nperm(X18, .(X19, X20)) :- ','(app(X21, .(X19, X22), X18), ','(app(X21, X22, X23), perm(X23, X20))).\nand substitutionT10 -> T20,\nX18 -> T20,\nX19 -> T18,\nX20 -> T19,\nT9 -> .(T18, T19),\nT17 -> T20"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 399,
11.17/3.76	                "to": 407,
11.17/3.76	                "label": "EVAL-BACKTRACK"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 401,
11.17/3.76	                "to": 403,
11.17/3.76	                "label": "SUCCESS"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 406,
11.17/3.76	                "to": 411,
11.17/3.76	                "label": "SPLIT 1"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 406,
11.17/3.76	                "to": 412,
11.17/3.76	                "label": "SPLIT 2\nreplacements:X21 -> T25,\nX22 -> T26"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 411,
11.17/3.76	                "to": 413,
11.17/3.76	                "label": "CASE"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 412,
11.17/3.76	                "to": 449,
11.17/3.76	                "label": "SPLIT 1"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 412,
11.17/3.76	                "to": 450,
11.17/3.76	                "label": "SPLIT 2\nreplacements:X23 -> T57"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 413,
11.17/3.76	                "to": 415,
11.17/3.76	                "label": "PARALLEL"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 413,
11.17/3.76	                "to": 416,
11.17/3.76	                "label": "PARALLEL"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 415,
11.17/3.76	                "to": 418,
11.17/3.76	                "label": "EVAL with clause\napp([], X40, X40).\nand substitutionX21 -> [],\nT18 -> T39,\nX22 -> T40,\nX40 -> .(T39, T40),\nX41 -> T40,\nT20 -> .(T39, T40)"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 415,
11.17/3.76	                "to": 419,
11.17/3.76	                "label": "EVAL-BACKTRACK"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 416,
11.17/3.76	                "to": 421,
11.17/3.76	                "label": "EVAL with clause\napp(.(X56, X57), X58, .(X56, X59)) :- app(X57, X58, X59).\nand substitutionX56 -> T48,\nX57 -> X61,\nX21 -> .(T48, X61),\nT18 -> T47,\nX22 -> X62,\nX58 -> .(T47, X62),\nX60 -> T48,\nX59 -> T50,\nT20 -> .(T48, T50),\nT49 -> T50"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 416,
11.17/3.76	                "to": 422,
11.17/3.76	                "label": "EVAL-BACKTRACK"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 418,
11.17/3.76	                "to": 420,
11.17/3.76	                "label": "SUCCESS"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 421,
11.17/3.76	                "to": 411,
11.17/3.76	                "label": "INSTANCE with matching:\nX21 -> X61\nT18 -> T47\nX22 -> X62\nT20 -> T50"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 449,
11.17/3.76	                "to": 451,
11.17/3.76	                "label": "CASE"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 450,
11.17/3.76	                "to": 382,
11.17/3.76	                "label": "INSTANCE with matching:\nT10 -> T57\nT9 -> T19"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 451,
11.17/3.76	                "to": 452,
11.17/3.76	                "label": "PARALLEL"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 451,
11.17/3.76	                "to": 453,
11.17/3.76	                "label": "PARALLEL"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 452,
11.17/3.76	                "to": 456,
11.17/3.76	                "label": "EVAL with clause\napp([], X75, X75).\nand substitutionT25 -> [],\nT26 -> T64,\nX75 -> T64,\nX23 -> T64"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 452,
11.17/3.76	                "to": 490,
11.17/3.76	                "label": "EVAL-BACKTRACK"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 453,
11.17/3.76	                "to": 496,
11.17/3.76	                "label": "EVAL with clause\napp(.(X86, X87), X88, .(X86, X89)) :- app(X87, X88, X89).\nand substitutionX86 -> T71,\nX87 -> T74,\nT25 -> .(T71, T74),\nT26 -> T75,\nX88 -> T75,\nX89 -> X90,\nX23 -> .(T71, X90),\nT72 -> T74,\nT73 -> T75"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 453,
11.17/3.76	                "to": 497,
11.17/3.76	                "label": "EVAL-BACKTRACK"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 456,
11.17/3.76	                "to": 491,
11.17/3.76	                "label": "SUCCESS"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 496,
11.17/3.76	                "to": 449,
11.17/3.76	                "label": "INSTANCE with matching:\nT25 -> T74\nT26 -> T75\nX23 -> X90"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 501,
11.17/3.76	                "to": 502,
11.17/3.76	                "label": "PARALLEL"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 501,
11.17/3.76	                "to": 503,
11.17/3.76	                "label": "PARALLEL"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 502,
11.17/3.76	                "to": 504,
11.17/3.76	                "label": "EVAL with clause\nordered([]).\nand substitutionT9 -> []"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 502,
11.17/3.76	                "to": 505,
11.17/3.76	                "label": "EVAL-BACKTRACK"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 503,
11.17/3.76	                "to": 507,
11.17/3.76	                "label": "PARALLEL"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 503,
11.17/3.76	                "to": 508,
11.17/3.76	                "label": "PARALLEL"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 504,
11.17/3.76	                "to": 506,
11.17/3.76	                "label": "SUCCESS"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 507,
11.17/3.76	                "to": 509,
11.17/3.76	                "label": "EVAL with clause\nordered(.(X97, [])).\nand substitutionX97 -> T82,\nT9 -> .(T82, [])"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 507,
11.17/3.76	                "to": 510,
11.17/3.76	                "label": "EVAL-BACKTRACK"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 508,
11.17/3.76	                "to": 515,
11.17/3.76	                "label": "EVAL with clause\nordered(.(X104, .(X105, X106))) :- ','(less(X104, s(X105)), ordered(.(X105, X106))).\nand substitutionX104 -> T89,\nX105 -> T90,\nX106 -> T91,\nT9 -> .(T89, .(T90, T91))"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 508,
11.17/3.76	                "to": 517,
11.17/3.76	                "label": "EVAL-BACKTRACK"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 509,
11.17/3.76	                "to": 512,
11.17/3.76	                "label": "SUCCESS"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 515,
11.17/3.76	                "to": 520,
11.17/3.76	                "label": "SPLIT 1"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 515,
11.17/3.76	                "to": 521,
11.17/3.76	                "label": "SPLIT 2\nnew knowledge:\nT89 is ground\nT90 is ground"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 520,
11.17/3.76	                "to": 525,
11.17/3.76	                "label": "CASE"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 521,
11.17/3.76	                "to": 383,
11.17/3.76	                "label": "INSTANCE with matching:\nT9 -> .(T90, T91)"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 525,
11.17/3.76	                "to": 528,
11.17/3.76	                "label": "PARALLEL"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 525,
11.17/3.76	                "to": 529,
11.17/3.76	                "label": "PARALLEL"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 528,
11.17/3.76	                "to": 530,
11.17/3.76	                "label": "EVAL with clause\nless(0, s(X115)).\nand substitutionT89 -> 0,\nT90 -> T100,\nX115 -> T100"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 528,
11.17/3.76	                "to": 532,
11.17/3.76	                "label": "EVAL-BACKTRACK"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 529,
11.17/3.76	                "to": 535,
11.17/3.76	                "label": "EVAL with clause\nless(s(X120), s(X121)) :- less(X120, X121).\nand substitutionX120 -> T105,\nT89 -> s(T105),\nT90 -> T106,\nX121 -> T106"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 529,
11.17/3.76	                "to": 536,
11.17/3.76	                "label": "EVAL-BACKTRACK"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 530,
11.17/3.76	                "to": 533,
11.17/3.76	                "label": "SUCCESS"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 535,
11.17/3.76	                "to": 537,
11.17/3.76	                "label": "CASE"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 537,
11.17/3.76	                "to": 538,
11.17/3.76	                "label": "PARALLEL"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 537,
11.17/3.76	                "to": 539,
11.17/3.76	                "label": "PARALLEL"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 538,
11.17/3.76	                "to": 542,
11.17/3.76	                "label": "EVAL with clause\nless(0, s(X128)).\nand substitutionT105 -> 0,\nX128 -> T113,\nT106 -> s(T113)"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 538,
11.17/3.76	                "to": 543,
11.17/3.76	                "label": "EVAL-BACKTRACK"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 539,
11.17/3.76	                "to": 545,
11.17/3.76	                "label": "EVAL with clause\nless(s(X133), s(X134)) :- less(X133, X134).\nand substitutionX133 -> T118,\nT105 -> s(T118),\nX134 -> T119,\nT106 -> s(T119)"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 539,
11.17/3.76	                "to": 546,
11.17/3.76	                "label": "EVAL-BACKTRACK"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 542,
11.17/3.76	                "to": 544,
11.17/3.76	                "label": "SUCCESS"
11.17/3.76	            },
11.17/3.76	            {
11.17/3.76	                "from": 545,
11.17/3.76	                "to": 535,
11.17/3.76	                "label": "INSTANCE with matching:\nT105 -> T118\nT106 -> T119"
11.17/3.76	            }
11.17/3.76	        ],
11.17/3.76	        "type": "Graph"
11.17/3.76	    }
11.17/3.76	}
11.17/3.76	
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(111)
11.17/3.76	Obligation:
11.17/3.76	Q restricted rewrite system:
11.17/3.76	The TRS R consists of the following rules:
11.17/3.76	
11.17/3.76	   f2_in(T9) -> U1(f51_in(T9), T9)
11.17/3.76	   U1(f51_out1, T9) -> f2_out1
11.17/3.76	   f411_in(T39) -> f411_out1
11.17/3.76	   f411_in(T47) -> U2(f411_in(T47), T47)
11.17/3.76	   U2(f411_out1, T47) -> f411_out1
11.17/3.76	   f382_in([]) -> f382_out1
11.17/3.76	   f382_in(.(T18, T19)) -> U3(f406_in(T18, T19), .(T18, T19))
11.17/3.76	   U3(f406_out1, .(T18, T19)) -> f382_out1
11.17/3.76	   f449_in -> f449_out1
11.17/3.76	   f449_in -> U4(f449_in)
11.17/3.76	   U4(f449_out1) -> f449_out1
11.17/3.76	   f383_in([]) -> f383_out1
11.17/3.76	   f383_in(.(T82, [])) -> f383_out1
11.17/3.76	   f383_in(.(T89, .(T90, T91))) -> U5(f515_in(T89, T90, T91), .(T89, .(T90, T91)))
11.17/3.76	   U5(f515_out1, .(T89, .(T90, T91))) -> f383_out1
11.17/3.76	   f535_in(0, s(T113)) -> f535_out1
11.17/3.76	   f535_in(s(T118), s(T119)) -> U6(f535_in(T118, T119), s(T118), s(T119))
11.17/3.76	   U6(f535_out1, s(T118), s(T119)) -> f535_out1
11.17/3.76	   f520_in(0, T100) -> f520_out1
11.17/3.76	   f520_in(s(T105), T106) -> U7(f535_in(T105, T106), s(T105), T106)
11.17/3.76	   U7(f535_out1, s(T105), T106) -> f520_out1
11.17/3.76	   f51_in(T9) -> U8(f382_in(T9), T9)
11.17/3.76	   U8(f382_out1, T9) -> U9(f383_in(T9), T9)
11.17/3.76	   U9(f383_out1, T9) -> f51_out1
11.17/3.76	   f406_in(T18, T19) -> U10(f411_in(T18), T18, T19)
11.17/3.76	   U10(f411_out1, T18, T19) -> U11(f412_in(T19), T18, T19)
11.17/3.76	   U11(f412_out1, T18, T19) -> f406_out1
11.17/3.76	   f412_in(T19) -> U12(f449_in, T19)
11.17/3.76	   U12(f449_out1, T19) -> U13(f382_in(T19), T19)
11.17/3.76	   U13(f382_out1, T19) -> f412_out1
11.17/3.76	   f515_in(T89, T90, T91) -> U14(f520_in(T89, T90), T89, T90, T91)
11.17/3.76	   U14(f520_out1, T89, T90, T91) -> U15(f383_in(.(T90, T91)), T89, T90, T91)
11.17/3.76	   U15(f383_out1, T89, T90, T91) -> f515_out1
11.17/3.76	
11.17/3.76	Q is empty.
11.17/3.76	
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(112) DependencyPairsProof (EQUIVALENT)
11.17/3.76	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(113)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F2_IN(T9) -> U1^1(f51_in(T9), T9)
11.17/3.76	   F2_IN(T9) -> F51_IN(T9)
11.17/3.76	   F411_IN(T47) -> U2^1(f411_in(T47), T47)
11.17/3.76	   F411_IN(T47) -> F411_IN(T47)
11.17/3.76	   F382_IN(.(T18, T19)) -> U3^1(f406_in(T18, T19), .(T18, T19))
11.17/3.76	   F382_IN(.(T18, T19)) -> F406_IN(T18, T19)
11.17/3.76	   F449_IN -> U4^1(f449_in)
11.17/3.76	   F449_IN -> F449_IN
11.17/3.76	   F383_IN(.(T89, .(T90, T91))) -> U5^1(f515_in(T89, T90, T91), .(T89, .(T90, T91)))
11.17/3.76	   F383_IN(.(T89, .(T90, T91))) -> F515_IN(T89, T90, T91)
11.17/3.76	   F535_IN(s(T118), s(T119)) -> U6^1(f535_in(T118, T119), s(T118), s(T119))
11.17/3.76	   F535_IN(s(T118), s(T119)) -> F535_IN(T118, T119)
11.17/3.76	   F520_IN(s(T105), T106) -> U7^1(f535_in(T105, T106), s(T105), T106)
11.17/3.76	   F520_IN(s(T105), T106) -> F535_IN(T105, T106)
11.17/3.76	   F51_IN(T9) -> U8^1(f382_in(T9), T9)
11.17/3.76	   F51_IN(T9) -> F382_IN(T9)
11.17/3.76	   U8^1(f382_out1, T9) -> U9^1(f383_in(T9), T9)
11.17/3.76	   U8^1(f382_out1, T9) -> F383_IN(T9)
11.17/3.76	   F406_IN(T18, T19) -> U10^1(f411_in(T18), T18, T19)
11.17/3.76	   F406_IN(T18, T19) -> F411_IN(T18)
11.17/3.76	   U10^1(f411_out1, T18, T19) -> U11^1(f412_in(T19), T18, T19)
11.17/3.76	   U10^1(f411_out1, T18, T19) -> F412_IN(T19)
11.17/3.76	   F412_IN(T19) -> U12^1(f449_in, T19)
11.17/3.76	   F412_IN(T19) -> F449_IN
11.17/3.76	   U12^1(f449_out1, T19) -> U13^1(f382_in(T19), T19)
11.17/3.76	   U12^1(f449_out1, T19) -> F382_IN(T19)
11.17/3.76	   F515_IN(T89, T90, T91) -> U14^1(f520_in(T89, T90), T89, T90, T91)
11.17/3.76	   F515_IN(T89, T90, T91) -> F520_IN(T89, T90)
11.17/3.76	   U14^1(f520_out1, T89, T90, T91) -> U15^1(f383_in(.(T90, T91)), T89, T90, T91)
11.17/3.76	   U14^1(f520_out1, T89, T90, T91) -> F383_IN(.(T90, T91))
11.17/3.76	
11.17/3.76	The TRS R consists of the following rules:
11.17/3.76	
11.17/3.76	   f2_in(T9) -> U1(f51_in(T9), T9)
11.17/3.76	   U1(f51_out1, T9) -> f2_out1
11.17/3.76	   f411_in(T39) -> f411_out1
11.17/3.76	   f411_in(T47) -> U2(f411_in(T47), T47)
11.17/3.76	   U2(f411_out1, T47) -> f411_out1
11.17/3.76	   f382_in([]) -> f382_out1
11.17/3.76	   f382_in(.(T18, T19)) -> U3(f406_in(T18, T19), .(T18, T19))
11.17/3.76	   U3(f406_out1, .(T18, T19)) -> f382_out1
11.17/3.76	   f449_in -> f449_out1
11.17/3.76	   f449_in -> U4(f449_in)
11.17/3.76	   U4(f449_out1) -> f449_out1
11.17/3.76	   f383_in([]) -> f383_out1
11.17/3.76	   f383_in(.(T82, [])) -> f383_out1
11.17/3.76	   f383_in(.(T89, .(T90, T91))) -> U5(f515_in(T89, T90, T91), .(T89, .(T90, T91)))
11.17/3.76	   U5(f515_out1, .(T89, .(T90, T91))) -> f383_out1
11.17/3.76	   f535_in(0, s(T113)) -> f535_out1
11.17/3.76	   f535_in(s(T118), s(T119)) -> U6(f535_in(T118, T119), s(T118), s(T119))
11.17/3.76	   U6(f535_out1, s(T118), s(T119)) -> f535_out1
11.17/3.76	   f520_in(0, T100) -> f520_out1
11.17/3.76	   f520_in(s(T105), T106) -> U7(f535_in(T105, T106), s(T105), T106)
11.17/3.76	   U7(f535_out1, s(T105), T106) -> f520_out1
11.17/3.76	   f51_in(T9) -> U8(f382_in(T9), T9)
11.17/3.76	   U8(f382_out1, T9) -> U9(f383_in(T9), T9)
11.17/3.76	   U9(f383_out1, T9) -> f51_out1
11.17/3.76	   f406_in(T18, T19) -> U10(f411_in(T18), T18, T19)
11.17/3.76	   U10(f411_out1, T18, T19) -> U11(f412_in(T19), T18, T19)
11.17/3.76	   U11(f412_out1, T18, T19) -> f406_out1
11.17/3.76	   f412_in(T19) -> U12(f449_in, T19)
11.17/3.76	   U12(f449_out1, T19) -> U13(f382_in(T19), T19)
11.17/3.76	   U13(f382_out1, T19) -> f412_out1
11.17/3.76	   f515_in(T89, T90, T91) -> U14(f520_in(T89, T90), T89, T90, T91)
11.17/3.76	   U14(f520_out1, T89, T90, T91) -> U15(f383_in(.(T90, T91)), T89, T90, T91)
11.17/3.76	   U15(f383_out1, T89, T90, T91) -> f515_out1
11.17/3.76	
11.17/3.76	Q is empty.
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(114) DependencyGraphProof (EQUIVALENT)
11.17/3.76	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 19 less nodes.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(115)
11.17/3.76	Complex Obligation (AND)
11.17/3.76	
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(116)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F535_IN(s(T118), s(T119)) -> F535_IN(T118, T119)
11.17/3.76	
11.17/3.76	The TRS R consists of the following rules:
11.17/3.76	
11.17/3.76	   f2_in(T9) -> U1(f51_in(T9), T9)
11.17/3.76	   U1(f51_out1, T9) -> f2_out1
11.17/3.76	   f411_in(T39) -> f411_out1
11.17/3.76	   f411_in(T47) -> U2(f411_in(T47), T47)
11.17/3.76	   U2(f411_out1, T47) -> f411_out1
11.17/3.76	   f382_in([]) -> f382_out1
11.17/3.76	   f382_in(.(T18, T19)) -> U3(f406_in(T18, T19), .(T18, T19))
11.17/3.76	   U3(f406_out1, .(T18, T19)) -> f382_out1
11.17/3.76	   f449_in -> f449_out1
11.17/3.76	   f449_in -> U4(f449_in)
11.17/3.76	   U4(f449_out1) -> f449_out1
11.17/3.76	   f383_in([]) -> f383_out1
11.17/3.76	   f383_in(.(T82, [])) -> f383_out1
11.17/3.76	   f383_in(.(T89, .(T90, T91))) -> U5(f515_in(T89, T90, T91), .(T89, .(T90, T91)))
11.17/3.76	   U5(f515_out1, .(T89, .(T90, T91))) -> f383_out1
11.17/3.76	   f535_in(0, s(T113)) -> f535_out1
11.17/3.76	   f535_in(s(T118), s(T119)) -> U6(f535_in(T118, T119), s(T118), s(T119))
11.17/3.76	   U6(f535_out1, s(T118), s(T119)) -> f535_out1
11.17/3.76	   f520_in(0, T100) -> f520_out1
11.17/3.76	   f520_in(s(T105), T106) -> U7(f535_in(T105, T106), s(T105), T106)
11.17/3.76	   U7(f535_out1, s(T105), T106) -> f520_out1
11.17/3.76	   f51_in(T9) -> U8(f382_in(T9), T9)
11.17/3.76	   U8(f382_out1, T9) -> U9(f383_in(T9), T9)
11.17/3.76	   U9(f383_out1, T9) -> f51_out1
11.17/3.76	   f406_in(T18, T19) -> U10(f411_in(T18), T18, T19)
11.17/3.76	   U10(f411_out1, T18, T19) -> U11(f412_in(T19), T18, T19)
11.17/3.76	   U11(f412_out1, T18, T19) -> f406_out1
11.17/3.76	   f412_in(T19) -> U12(f449_in, T19)
11.17/3.76	   U12(f449_out1, T19) -> U13(f382_in(T19), T19)
11.17/3.76	   U13(f382_out1, T19) -> f412_out1
11.17/3.76	   f515_in(T89, T90, T91) -> U14(f520_in(T89, T90), T89, T90, T91)
11.17/3.76	   U14(f520_out1, T89, T90, T91) -> U15(f383_in(.(T90, T91)), T89, T90, T91)
11.17/3.76	   U15(f383_out1, T89, T90, T91) -> f515_out1
11.17/3.76	
11.17/3.76	Q is empty.
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(117) MNOCProof (EQUIVALENT)
11.17/3.76	We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(118)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F535_IN(s(T118), s(T119)) -> F535_IN(T118, T119)
11.17/3.76	
11.17/3.76	The TRS R consists of the following rules:
11.17/3.76	
11.17/3.76	   f2_in(T9) -> U1(f51_in(T9), T9)
11.17/3.76	   U1(f51_out1, T9) -> f2_out1
11.17/3.76	   f411_in(T39) -> f411_out1
11.17/3.76	   f411_in(T47) -> U2(f411_in(T47), T47)
11.17/3.76	   U2(f411_out1, T47) -> f411_out1
11.17/3.76	   f382_in([]) -> f382_out1
11.17/3.76	   f382_in(.(T18, T19)) -> U3(f406_in(T18, T19), .(T18, T19))
11.17/3.76	   U3(f406_out1, .(T18, T19)) -> f382_out1
11.17/3.76	   f449_in -> f449_out1
11.17/3.76	   f449_in -> U4(f449_in)
11.17/3.76	   U4(f449_out1) -> f449_out1
11.17/3.76	   f383_in([]) -> f383_out1
11.17/3.76	   f383_in(.(T82, [])) -> f383_out1
11.17/3.76	   f383_in(.(T89, .(T90, T91))) -> U5(f515_in(T89, T90, T91), .(T89, .(T90, T91)))
11.17/3.76	   U5(f515_out1, .(T89, .(T90, T91))) -> f383_out1
11.17/3.76	   f535_in(0, s(T113)) -> f535_out1
11.17/3.76	   f535_in(s(T118), s(T119)) -> U6(f535_in(T118, T119), s(T118), s(T119))
11.17/3.76	   U6(f535_out1, s(T118), s(T119)) -> f535_out1
11.17/3.76	   f520_in(0, T100) -> f520_out1
11.17/3.76	   f520_in(s(T105), T106) -> U7(f535_in(T105, T106), s(T105), T106)
11.17/3.76	   U7(f535_out1, s(T105), T106) -> f520_out1
11.17/3.76	   f51_in(T9) -> U8(f382_in(T9), T9)
11.17/3.76	   U8(f382_out1, T9) -> U9(f383_in(T9), T9)
11.17/3.76	   U9(f383_out1, T9) -> f51_out1
11.17/3.76	   f406_in(T18, T19) -> U10(f411_in(T18), T18, T19)
11.17/3.76	   U10(f411_out1, T18, T19) -> U11(f412_in(T19), T18, T19)
11.17/3.76	   U11(f412_out1, T18, T19) -> f406_out1
11.17/3.76	   f412_in(T19) -> U12(f449_in, T19)
11.17/3.76	   U12(f449_out1, T19) -> U13(f382_in(T19), T19)
11.17/3.76	   U13(f382_out1, T19) -> f412_out1
11.17/3.76	   f515_in(T89, T90, T91) -> U14(f520_in(T89, T90), T89, T90, T91)
11.17/3.76	   U14(f520_out1, T89, T90, T91) -> U15(f383_in(.(T90, T91)), T89, T90, T91)
11.17/3.76	   U15(f383_out1, T89, T90, T91) -> f515_out1
11.17/3.76	
11.17/3.76	The set Q consists of the following terms:
11.17/3.76	
11.17/3.76	   f2_in(x0)
11.17/3.76	   U1(f51_out1, x0)
11.17/3.76	   f411_in(x0)
11.17/3.76	   U2(f411_out1, x0)
11.17/3.76	   f382_in([])
11.17/3.76	   f382_in(.(x0, x1))
11.17/3.76	   U3(f406_out1, .(x0, x1))
11.17/3.76	   f449_in
11.17/3.76	   U4(f449_out1)
11.17/3.76	   f383_in([])
11.17/3.76	   f383_in(.(x0, []))
11.17/3.76	   f383_in(.(x0, .(x1, x2)))
11.17/3.76	   U5(f515_out1, .(x0, .(x1, x2)))
11.17/3.76	   f535_in(0, s(x0))
11.17/3.76	   f535_in(s(x0), s(x1))
11.17/3.76	   U6(f535_out1, s(x0), s(x1))
11.17/3.76	   f520_in(0, x0)
11.17/3.76	   f520_in(s(x0), x1)
11.17/3.76	   U7(f535_out1, s(x0), x1)
11.17/3.76	   f51_in(x0)
11.17/3.76	   U8(f382_out1, x0)
11.17/3.76	   U9(f383_out1, x0)
11.17/3.76	   f406_in(x0, x1)
11.17/3.76	   U10(f411_out1, x0, x1)
11.17/3.76	   U11(f412_out1, x0, x1)
11.17/3.76	   f412_in(x0)
11.17/3.76	   U12(f449_out1, x0)
11.17/3.76	   U13(f382_out1, x0)
11.17/3.76	   f515_in(x0, x1, x2)
11.17/3.76	   U14(f520_out1, x0, x1, x2)
11.17/3.76	   U15(f383_out1, x0, x1, x2)
11.17/3.76	
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(119) UsableRulesProof (EQUIVALENT)
11.17/3.76	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(120)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F535_IN(s(T118), s(T119)) -> F535_IN(T118, T119)
11.17/3.76	
11.17/3.76	R is empty.
11.17/3.76	The set Q consists of the following terms:
11.17/3.76	
11.17/3.76	   f2_in(x0)
11.17/3.76	   U1(f51_out1, x0)
11.17/3.76	   f411_in(x0)
11.17/3.76	   U2(f411_out1, x0)
11.17/3.76	   f382_in([])
11.17/3.76	   f382_in(.(x0, x1))
11.17/3.76	   U3(f406_out1, .(x0, x1))
11.17/3.76	   f449_in
11.17/3.76	   U4(f449_out1)
11.17/3.76	   f383_in([])
11.17/3.76	   f383_in(.(x0, []))
11.17/3.76	   f383_in(.(x0, .(x1, x2)))
11.17/3.76	   U5(f515_out1, .(x0, .(x1, x2)))
11.17/3.76	   f535_in(0, s(x0))
11.17/3.76	   f535_in(s(x0), s(x1))
11.17/3.76	   U6(f535_out1, s(x0), s(x1))
11.17/3.76	   f520_in(0, x0)
11.17/3.76	   f520_in(s(x0), x1)
11.17/3.76	   U7(f535_out1, s(x0), x1)
11.17/3.76	   f51_in(x0)
11.17/3.76	   U8(f382_out1, x0)
11.17/3.76	   U9(f383_out1, x0)
11.17/3.76	   f406_in(x0, x1)
11.17/3.76	   U10(f411_out1, x0, x1)
11.17/3.76	   U11(f412_out1, x0, x1)
11.17/3.76	   f412_in(x0)
11.17/3.76	   U12(f449_out1, x0)
11.17/3.76	   U13(f382_out1, x0)
11.17/3.76	   f515_in(x0, x1, x2)
11.17/3.76	   U14(f520_out1, x0, x1, x2)
11.17/3.76	   U15(f383_out1, x0, x1, x2)
11.17/3.76	
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(121) QReductionProof (EQUIVALENT)
11.17/3.76	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
11.17/3.76	
11.17/3.76	   f2_in(x0)
11.17/3.76	   U1(f51_out1, x0)
11.17/3.76	   f411_in(x0)
11.17/3.76	   U2(f411_out1, x0)
11.17/3.76	   f382_in([])
11.17/3.76	   f382_in(.(x0, x1))
11.17/3.76	   U3(f406_out1, .(x0, x1))
11.17/3.76	   f449_in
11.17/3.76	   U4(f449_out1)
11.17/3.76	   f383_in([])
11.17/3.76	   f383_in(.(x0, []))
11.17/3.76	   f383_in(.(x0, .(x1, x2)))
11.17/3.76	   U5(f515_out1, .(x0, .(x1, x2)))
11.17/3.76	   f535_in(0, s(x0))
11.17/3.76	   f535_in(s(x0), s(x1))
11.17/3.76	   U6(f535_out1, s(x0), s(x1))
11.17/3.76	   f520_in(0, x0)
11.17/3.76	   f520_in(s(x0), x1)
11.17/3.76	   U7(f535_out1, s(x0), x1)
11.17/3.76	   f51_in(x0)
11.17/3.76	   U8(f382_out1, x0)
11.17/3.76	   U9(f383_out1, x0)
11.17/3.76	   f406_in(x0, x1)
11.17/3.76	   U10(f411_out1, x0, x1)
11.17/3.76	   U11(f412_out1, x0, x1)
11.17/3.76	   f412_in(x0)
11.17/3.76	   U12(f449_out1, x0)
11.17/3.76	   U13(f382_out1, x0)
11.17/3.76	   f515_in(x0, x1, x2)
11.17/3.76	   U14(f520_out1, x0, x1, x2)
11.17/3.76	   U15(f383_out1, x0, x1, x2)
11.17/3.76	
11.17/3.76	
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(122)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F535_IN(s(T118), s(T119)) -> F535_IN(T118, T119)
11.17/3.76	
11.17/3.76	R is empty.
11.17/3.76	Q is empty.
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(123) QDPSizeChangeProof (EQUIVALENT)
11.17/3.76	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. 
11.17/3.76	
11.17/3.76	From the DPs we obtained the following set of size-change graphs:
11.17/3.76	*F535_IN(s(T118), s(T119)) -> F535_IN(T118, T119)
11.17/3.76	The graph contains the following edges 1 > 1, 2 > 2
11.17/3.76	
11.17/3.76	
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(124)
11.17/3.76	YES
11.17/3.76	
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(125)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F383_IN(.(T89, .(T90, T91))) -> F515_IN(T89, T90, T91)
11.17/3.76	   F515_IN(T89, T90, T91) -> U14^1(f520_in(T89, T90), T89, T90, T91)
11.17/3.76	   U14^1(f520_out1, T89, T90, T91) -> F383_IN(.(T90, T91))
11.17/3.76	
11.17/3.76	The TRS R consists of the following rules:
11.17/3.76	
11.17/3.76	   f2_in(T9) -> U1(f51_in(T9), T9)
11.17/3.76	   U1(f51_out1, T9) -> f2_out1
11.17/3.76	   f411_in(T39) -> f411_out1
11.17/3.76	   f411_in(T47) -> U2(f411_in(T47), T47)
11.17/3.76	   U2(f411_out1, T47) -> f411_out1
11.17/3.76	   f382_in([]) -> f382_out1
11.17/3.76	   f382_in(.(T18, T19)) -> U3(f406_in(T18, T19), .(T18, T19))
11.17/3.76	   U3(f406_out1, .(T18, T19)) -> f382_out1
11.17/3.76	   f449_in -> f449_out1
11.17/3.76	   f449_in -> U4(f449_in)
11.17/3.76	   U4(f449_out1) -> f449_out1
11.17/3.76	   f383_in([]) -> f383_out1
11.17/3.76	   f383_in(.(T82, [])) -> f383_out1
11.17/3.76	   f383_in(.(T89, .(T90, T91))) -> U5(f515_in(T89, T90, T91), .(T89, .(T90, T91)))
11.17/3.76	   U5(f515_out1, .(T89, .(T90, T91))) -> f383_out1
11.17/3.76	   f535_in(0, s(T113)) -> f535_out1
11.17/3.76	   f535_in(s(T118), s(T119)) -> U6(f535_in(T118, T119), s(T118), s(T119))
11.17/3.76	   U6(f535_out1, s(T118), s(T119)) -> f535_out1
11.17/3.76	   f520_in(0, T100) -> f520_out1
11.17/3.76	   f520_in(s(T105), T106) -> U7(f535_in(T105, T106), s(T105), T106)
11.17/3.76	   U7(f535_out1, s(T105), T106) -> f520_out1
11.17/3.76	   f51_in(T9) -> U8(f382_in(T9), T9)
11.17/3.76	   U8(f382_out1, T9) -> U9(f383_in(T9), T9)
11.17/3.76	   U9(f383_out1, T9) -> f51_out1
11.17/3.76	   f406_in(T18, T19) -> U10(f411_in(T18), T18, T19)
11.17/3.76	   U10(f411_out1, T18, T19) -> U11(f412_in(T19), T18, T19)
11.17/3.76	   U11(f412_out1, T18, T19) -> f406_out1
11.17/3.76	   f412_in(T19) -> U12(f449_in, T19)
11.17/3.76	   U12(f449_out1, T19) -> U13(f382_in(T19), T19)
11.17/3.76	   U13(f382_out1, T19) -> f412_out1
11.17/3.76	   f515_in(T89, T90, T91) -> U14(f520_in(T89, T90), T89, T90, T91)
11.17/3.76	   U14(f520_out1, T89, T90, T91) -> U15(f383_in(.(T90, T91)), T89, T90, T91)
11.17/3.76	   U15(f383_out1, T89, T90, T91) -> f515_out1
11.17/3.76	
11.17/3.76	Q is empty.
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(126) MNOCProof (EQUIVALENT)
11.17/3.76	We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(127)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F383_IN(.(T89, .(T90, T91))) -> F515_IN(T89, T90, T91)
11.17/3.76	   F515_IN(T89, T90, T91) -> U14^1(f520_in(T89, T90), T89, T90, T91)
11.17/3.76	   U14^1(f520_out1, T89, T90, T91) -> F383_IN(.(T90, T91))
11.17/3.76	
11.17/3.76	The TRS R consists of the following rules:
11.17/3.76	
11.17/3.76	   f2_in(T9) -> U1(f51_in(T9), T9)
11.17/3.76	   U1(f51_out1, T9) -> f2_out1
11.17/3.76	   f411_in(T39) -> f411_out1
11.17/3.76	   f411_in(T47) -> U2(f411_in(T47), T47)
11.17/3.76	   U2(f411_out1, T47) -> f411_out1
11.17/3.76	   f382_in([]) -> f382_out1
11.17/3.76	   f382_in(.(T18, T19)) -> U3(f406_in(T18, T19), .(T18, T19))
11.17/3.76	   U3(f406_out1, .(T18, T19)) -> f382_out1
11.17/3.76	   f449_in -> f449_out1
11.17/3.76	   f449_in -> U4(f449_in)
11.17/3.76	   U4(f449_out1) -> f449_out1
11.17/3.76	   f383_in([]) -> f383_out1
11.17/3.76	   f383_in(.(T82, [])) -> f383_out1
11.17/3.76	   f383_in(.(T89, .(T90, T91))) -> U5(f515_in(T89, T90, T91), .(T89, .(T90, T91)))
11.17/3.76	   U5(f515_out1, .(T89, .(T90, T91))) -> f383_out1
11.17/3.76	   f535_in(0, s(T113)) -> f535_out1
11.17/3.76	   f535_in(s(T118), s(T119)) -> U6(f535_in(T118, T119), s(T118), s(T119))
11.17/3.76	   U6(f535_out1, s(T118), s(T119)) -> f535_out1
11.17/3.76	   f520_in(0, T100) -> f520_out1
11.17/3.76	   f520_in(s(T105), T106) -> U7(f535_in(T105, T106), s(T105), T106)
11.17/3.76	   U7(f535_out1, s(T105), T106) -> f520_out1
11.17/3.76	   f51_in(T9) -> U8(f382_in(T9), T9)
11.17/3.76	   U8(f382_out1, T9) -> U9(f383_in(T9), T9)
11.17/3.76	   U9(f383_out1, T9) -> f51_out1
11.17/3.76	   f406_in(T18, T19) -> U10(f411_in(T18), T18, T19)
11.17/3.76	   U10(f411_out1, T18, T19) -> U11(f412_in(T19), T18, T19)
11.17/3.76	   U11(f412_out1, T18, T19) -> f406_out1
11.17/3.76	   f412_in(T19) -> U12(f449_in, T19)
11.17/3.76	   U12(f449_out1, T19) -> U13(f382_in(T19), T19)
11.17/3.76	   U13(f382_out1, T19) -> f412_out1
11.17/3.76	   f515_in(T89, T90, T91) -> U14(f520_in(T89, T90), T89, T90, T91)
11.17/3.76	   U14(f520_out1, T89, T90, T91) -> U15(f383_in(.(T90, T91)), T89, T90, T91)
11.17/3.76	   U15(f383_out1, T89, T90, T91) -> f515_out1
11.17/3.76	
11.17/3.76	The set Q consists of the following terms:
11.17/3.76	
11.17/3.76	   f2_in(x0)
11.17/3.76	   U1(f51_out1, x0)
11.17/3.76	   f411_in(x0)
11.17/3.76	   U2(f411_out1, x0)
11.17/3.76	   f382_in([])
11.17/3.76	   f382_in(.(x0, x1))
11.17/3.76	   U3(f406_out1, .(x0, x1))
11.17/3.76	   f449_in
11.17/3.76	   U4(f449_out1)
11.17/3.76	   f383_in([])
11.17/3.76	   f383_in(.(x0, []))
11.17/3.76	   f383_in(.(x0, .(x1, x2)))
11.17/3.76	   U5(f515_out1, .(x0, .(x1, x2)))
11.17/3.76	   f535_in(0, s(x0))
11.17/3.76	   f535_in(s(x0), s(x1))
11.17/3.76	   U6(f535_out1, s(x0), s(x1))
11.17/3.76	   f520_in(0, x0)
11.17/3.76	   f520_in(s(x0), x1)
11.17/3.76	   U7(f535_out1, s(x0), x1)
11.17/3.76	   f51_in(x0)
11.17/3.76	   U8(f382_out1, x0)
11.17/3.76	   U9(f383_out1, x0)
11.17/3.76	   f406_in(x0, x1)
11.17/3.76	   U10(f411_out1, x0, x1)
11.17/3.76	   U11(f412_out1, x0, x1)
11.17/3.76	   f412_in(x0)
11.17/3.76	   U12(f449_out1, x0)
11.17/3.76	   U13(f382_out1, x0)
11.17/3.76	   f515_in(x0, x1, x2)
11.17/3.76	   U14(f520_out1, x0, x1, x2)
11.17/3.76	   U15(f383_out1, x0, x1, x2)
11.17/3.76	
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(128) UsableRulesProof (EQUIVALENT)
11.17/3.76	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(129)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F383_IN(.(T89, .(T90, T91))) -> F515_IN(T89, T90, T91)
11.17/3.76	   F515_IN(T89, T90, T91) -> U14^1(f520_in(T89, T90), T89, T90, T91)
11.17/3.76	   U14^1(f520_out1, T89, T90, T91) -> F383_IN(.(T90, T91))
11.17/3.76	
11.17/3.76	The TRS R consists of the following rules:
11.17/3.76	
11.17/3.76	   f520_in(0, T100) -> f520_out1
11.17/3.76	   f520_in(s(T105), T106) -> U7(f535_in(T105, T106), s(T105), T106)
11.17/3.76	   f535_in(0, s(T113)) -> f535_out1
11.17/3.76	   f535_in(s(T118), s(T119)) -> U6(f535_in(T118, T119), s(T118), s(T119))
11.17/3.76	   U7(f535_out1, s(T105), T106) -> f520_out1
11.17/3.76	   U6(f535_out1, s(T118), s(T119)) -> f535_out1
11.17/3.76	
11.17/3.76	The set Q consists of the following terms:
11.17/3.76	
11.17/3.76	   f2_in(x0)
11.17/3.76	   U1(f51_out1, x0)
11.17/3.76	   f411_in(x0)
11.17/3.76	   U2(f411_out1, x0)
11.17/3.76	   f382_in([])
11.17/3.76	   f382_in(.(x0, x1))
11.17/3.76	   U3(f406_out1, .(x0, x1))
11.17/3.76	   f449_in
11.17/3.76	   U4(f449_out1)
11.17/3.76	   f383_in([])
11.17/3.76	   f383_in(.(x0, []))
11.17/3.76	   f383_in(.(x0, .(x1, x2)))
11.17/3.76	   U5(f515_out1, .(x0, .(x1, x2)))
11.17/3.76	   f535_in(0, s(x0))
11.17/3.76	   f535_in(s(x0), s(x1))
11.17/3.76	   U6(f535_out1, s(x0), s(x1))
11.17/3.76	   f520_in(0, x0)
11.17/3.76	   f520_in(s(x0), x1)
11.17/3.76	   U7(f535_out1, s(x0), x1)
11.17/3.76	   f51_in(x0)
11.17/3.76	   U8(f382_out1, x0)
11.17/3.76	   U9(f383_out1, x0)
11.17/3.76	   f406_in(x0, x1)
11.17/3.76	   U10(f411_out1, x0, x1)
11.17/3.76	   U11(f412_out1, x0, x1)
11.17/3.76	   f412_in(x0)
11.17/3.76	   U12(f449_out1, x0)
11.17/3.76	   U13(f382_out1, x0)
11.17/3.76	   f515_in(x0, x1, x2)
11.17/3.76	   U14(f520_out1, x0, x1, x2)
11.17/3.76	   U15(f383_out1, x0, x1, x2)
11.17/3.76	
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(130) QReductionProof (EQUIVALENT)
11.17/3.76	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
11.17/3.76	
11.17/3.76	   f2_in(x0)
11.17/3.76	   U1(f51_out1, x0)
11.17/3.76	   f411_in(x0)
11.17/3.76	   U2(f411_out1, x0)
11.17/3.76	   f382_in([])
11.17/3.76	   f382_in(.(x0, x1))
11.17/3.76	   U3(f406_out1, .(x0, x1))
11.17/3.76	   f449_in
11.17/3.76	   U4(f449_out1)
11.17/3.76	   f383_in([])
11.17/3.76	   f383_in(.(x0, []))
11.17/3.76	   f383_in(.(x0, .(x1, x2)))
11.17/3.76	   U5(f515_out1, .(x0, .(x1, x2)))
11.17/3.76	   f51_in(x0)
11.17/3.76	   U8(f382_out1, x0)
11.17/3.76	   U9(f383_out1, x0)
11.17/3.76	   f406_in(x0, x1)
11.17/3.76	   U10(f411_out1, x0, x1)
11.17/3.76	   U11(f412_out1, x0, x1)
11.17/3.76	   f412_in(x0)
11.17/3.76	   U12(f449_out1, x0)
11.17/3.76	   U13(f382_out1, x0)
11.17/3.76	   f515_in(x0, x1, x2)
11.17/3.76	   U14(f520_out1, x0, x1, x2)
11.17/3.76	   U15(f383_out1, x0, x1, x2)
11.17/3.76	
11.17/3.76	
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(131)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F383_IN(.(T89, .(T90, T91))) -> F515_IN(T89, T90, T91)
11.17/3.76	   F515_IN(T89, T90, T91) -> U14^1(f520_in(T89, T90), T89, T90, T91)
11.17/3.76	   U14^1(f520_out1, T89, T90, T91) -> F383_IN(.(T90, T91))
11.17/3.76	
11.17/3.76	The TRS R consists of the following rules:
11.17/3.76	
11.17/3.76	   f520_in(0, T100) -> f520_out1
11.17/3.76	   f520_in(s(T105), T106) -> U7(f535_in(T105, T106), s(T105), T106)
11.17/3.76	   f535_in(0, s(T113)) -> f535_out1
11.17/3.76	   f535_in(s(T118), s(T119)) -> U6(f535_in(T118, T119), s(T118), s(T119))
11.17/3.76	   U7(f535_out1, s(T105), T106) -> f520_out1
11.17/3.76	   U6(f535_out1, s(T118), s(T119)) -> f535_out1
11.17/3.76	
11.17/3.76	The set Q consists of the following terms:
11.17/3.76	
11.17/3.76	   f535_in(0, s(x0))
11.17/3.76	   f535_in(s(x0), s(x1))
11.17/3.76	   U6(f535_out1, s(x0), s(x1))
11.17/3.76	   f520_in(0, x0)
11.17/3.76	   f520_in(s(x0), x1)
11.17/3.76	   U7(f535_out1, s(x0), x1)
11.17/3.76	
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(132) QDPOrderProof (EQUIVALENT)
11.17/3.76	We use the reduction pair processor [LPAR04,JAR06].
11.17/3.76	
11.17/3.76	
11.17/3.76	The following pairs can be oriented strictly and are deleted.
11.17/3.76	
11.17/3.76	   F383_IN(.(T89, .(T90, T91))) -> F515_IN(T89, T90, T91)
11.17/3.76	   U14^1(f520_out1, T89, T90, T91) -> F383_IN(.(T90, T91))
11.17/3.76	The remaining pairs can at least be oriented weakly.
11.17/3.76	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
11.17/3.76	
11.17/3.76	POL( U14^1_4(x_1, ..., x_4) ) = 2x_1 + 2x_3 + 2x_4 + 2
11.17/3.76	POL( f520_in_2(x_1, x_2) ) = x_1
11.17/3.76	POL( 0 ) = 2
11.17/3.76	POL( f520_out1 ) = 2
11.17/3.76	POL( s_1(x_1) ) = x_1 + 2
11.17/3.76	POL( U7_3(x_1, ..., x_3) ) = x_1 + 1
11.17/3.76	POL( f535_in_2(x_1, x_2) ) = x_1
11.17/3.76	POL( f535_out1 ) = 1
11.17/3.76	POL( U6_3(x_1, ..., x_3) ) = 2
11.17/3.76	POL( F383_IN_1(x_1) ) = 2x_1 + 1
11.17/3.76	POL( ._2(x_1, x_2) ) = x_1 + x_2 + 2
11.17/3.76	POL( F515_IN_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + 2x_3 + 2
11.17/3.76	
11.17/3.76	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
11.17/3.76	
11.17/3.76	   f520_in(0, T100) -> f520_out1
11.17/3.76	   f520_in(s(T105), T106) -> U7(f535_in(T105, T106), s(T105), T106)
11.17/3.76	   f535_in(0, s(T113)) -> f535_out1
11.17/3.76	   f535_in(s(T118), s(T119)) -> U6(f535_in(T118, T119), s(T118), s(T119))
11.17/3.76	   U7(f535_out1, s(T105), T106) -> f520_out1
11.17/3.76	   U6(f535_out1, s(T118), s(T119)) -> f535_out1
11.17/3.76	
11.17/3.76	
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(133)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F515_IN(T89, T90, T91) -> U14^1(f520_in(T89, T90), T89, T90, T91)
11.17/3.76	
11.17/3.76	The TRS R consists of the following rules:
11.17/3.76	
11.17/3.76	   f520_in(0, T100) -> f520_out1
11.17/3.76	   f520_in(s(T105), T106) -> U7(f535_in(T105, T106), s(T105), T106)
11.17/3.76	   f535_in(0, s(T113)) -> f535_out1
11.17/3.76	   f535_in(s(T118), s(T119)) -> U6(f535_in(T118, T119), s(T118), s(T119))
11.17/3.76	   U7(f535_out1, s(T105), T106) -> f520_out1
11.17/3.76	   U6(f535_out1, s(T118), s(T119)) -> f535_out1
11.17/3.76	
11.17/3.76	The set Q consists of the following terms:
11.17/3.76	
11.17/3.76	   f535_in(0, s(x0))
11.17/3.76	   f535_in(s(x0), s(x1))
11.17/3.76	   U6(f535_out1, s(x0), s(x1))
11.17/3.76	   f520_in(0, x0)
11.17/3.76	   f520_in(s(x0), x1)
11.17/3.76	   U7(f535_out1, s(x0), x1)
11.17/3.76	
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(134) DependencyGraphProof (EQUIVALENT)
11.17/3.76	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(135)
11.17/3.76	TRUE
11.17/3.76	
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(136)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F449_IN -> F449_IN
11.17/3.76	
11.17/3.76	The TRS R consists of the following rules:
11.17/3.76	
11.17/3.76	   f2_in(T9) -> U1(f51_in(T9), T9)
11.17/3.76	   U1(f51_out1, T9) -> f2_out1
11.17/3.76	   f411_in(T39) -> f411_out1
11.17/3.76	   f411_in(T47) -> U2(f411_in(T47), T47)
11.17/3.76	   U2(f411_out1, T47) -> f411_out1
11.17/3.76	   f382_in([]) -> f382_out1
11.17/3.76	   f382_in(.(T18, T19)) -> U3(f406_in(T18, T19), .(T18, T19))
11.17/3.76	   U3(f406_out1, .(T18, T19)) -> f382_out1
11.17/3.76	   f449_in -> f449_out1
11.17/3.76	   f449_in -> U4(f449_in)
11.17/3.76	   U4(f449_out1) -> f449_out1
11.17/3.76	   f383_in([]) -> f383_out1
11.17/3.76	   f383_in(.(T82, [])) -> f383_out1
11.17/3.76	   f383_in(.(T89, .(T90, T91))) -> U5(f515_in(T89, T90, T91), .(T89, .(T90, T91)))
11.17/3.76	   U5(f515_out1, .(T89, .(T90, T91))) -> f383_out1
11.17/3.76	   f535_in(0, s(T113)) -> f535_out1
11.17/3.76	   f535_in(s(T118), s(T119)) -> U6(f535_in(T118, T119), s(T118), s(T119))
11.17/3.76	   U6(f535_out1, s(T118), s(T119)) -> f535_out1
11.17/3.76	   f520_in(0, T100) -> f520_out1
11.17/3.76	   f520_in(s(T105), T106) -> U7(f535_in(T105, T106), s(T105), T106)
11.17/3.76	   U7(f535_out1, s(T105), T106) -> f520_out1
11.17/3.76	   f51_in(T9) -> U8(f382_in(T9), T9)
11.17/3.76	   U8(f382_out1, T9) -> U9(f383_in(T9), T9)
11.17/3.76	   U9(f383_out1, T9) -> f51_out1
11.17/3.76	   f406_in(T18, T19) -> U10(f411_in(T18), T18, T19)
11.17/3.76	   U10(f411_out1, T18, T19) -> U11(f412_in(T19), T18, T19)
11.17/3.76	   U11(f412_out1, T18, T19) -> f406_out1
11.17/3.76	   f412_in(T19) -> U12(f449_in, T19)
11.17/3.76	   U12(f449_out1, T19) -> U13(f382_in(T19), T19)
11.17/3.76	   U13(f382_out1, T19) -> f412_out1
11.17/3.76	   f515_in(T89, T90, T91) -> U14(f520_in(T89, T90), T89, T90, T91)
11.17/3.76	   U14(f520_out1, T89, T90, T91) -> U15(f383_in(.(T90, T91)), T89, T90, T91)
11.17/3.76	   U15(f383_out1, T89, T90, T91) -> f515_out1
11.17/3.76	
11.17/3.76	Q is empty.
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(137) MNOCProof (EQUIVALENT)
11.17/3.76	We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(138)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F449_IN -> F449_IN
11.17/3.76	
11.17/3.76	The TRS R consists of the following rules:
11.17/3.76	
11.17/3.76	   f2_in(T9) -> U1(f51_in(T9), T9)
11.17/3.76	   U1(f51_out1, T9) -> f2_out1
11.17/3.76	   f411_in(T39) -> f411_out1
11.17/3.76	   f411_in(T47) -> U2(f411_in(T47), T47)
11.17/3.76	   U2(f411_out1, T47) -> f411_out1
11.17/3.76	   f382_in([]) -> f382_out1
11.17/3.76	   f382_in(.(T18, T19)) -> U3(f406_in(T18, T19), .(T18, T19))
11.17/3.76	   U3(f406_out1, .(T18, T19)) -> f382_out1
11.17/3.76	   f449_in -> f449_out1
11.17/3.76	   f449_in -> U4(f449_in)
11.17/3.76	   U4(f449_out1) -> f449_out1
11.17/3.76	   f383_in([]) -> f383_out1
11.17/3.76	   f383_in(.(T82, [])) -> f383_out1
11.17/3.76	   f383_in(.(T89, .(T90, T91))) -> U5(f515_in(T89, T90, T91), .(T89, .(T90, T91)))
11.17/3.76	   U5(f515_out1, .(T89, .(T90, T91))) -> f383_out1
11.17/3.76	   f535_in(0, s(T113)) -> f535_out1
11.17/3.76	   f535_in(s(T118), s(T119)) -> U6(f535_in(T118, T119), s(T118), s(T119))
11.17/3.76	   U6(f535_out1, s(T118), s(T119)) -> f535_out1
11.17/3.76	   f520_in(0, T100) -> f520_out1
11.17/3.76	   f520_in(s(T105), T106) -> U7(f535_in(T105, T106), s(T105), T106)
11.17/3.76	   U7(f535_out1, s(T105), T106) -> f520_out1
11.17/3.76	   f51_in(T9) -> U8(f382_in(T9), T9)
11.17/3.76	   U8(f382_out1, T9) -> U9(f383_in(T9), T9)
11.17/3.76	   U9(f383_out1, T9) -> f51_out1
11.17/3.76	   f406_in(T18, T19) -> U10(f411_in(T18), T18, T19)
11.17/3.76	   U10(f411_out1, T18, T19) -> U11(f412_in(T19), T18, T19)
11.17/3.76	   U11(f412_out1, T18, T19) -> f406_out1
11.17/3.76	   f412_in(T19) -> U12(f449_in, T19)
11.17/3.76	   U12(f449_out1, T19) -> U13(f382_in(T19), T19)
11.17/3.76	   U13(f382_out1, T19) -> f412_out1
11.17/3.76	   f515_in(T89, T90, T91) -> U14(f520_in(T89, T90), T89, T90, T91)
11.17/3.76	   U14(f520_out1, T89, T90, T91) -> U15(f383_in(.(T90, T91)), T89, T90, T91)
11.17/3.76	   U15(f383_out1, T89, T90, T91) -> f515_out1
11.17/3.76	
11.17/3.76	The set Q consists of the following terms:
11.17/3.76	
11.17/3.76	   f2_in(x0)
11.17/3.76	   U1(f51_out1, x0)
11.17/3.76	   f411_in(x0)
11.17/3.76	   U2(f411_out1, x0)
11.17/3.76	   f382_in([])
11.17/3.76	   f382_in(.(x0, x1))
11.17/3.76	   U3(f406_out1, .(x0, x1))
11.17/3.76	   f449_in
11.17/3.76	   U4(f449_out1)
11.17/3.76	   f383_in([])
11.17/3.76	   f383_in(.(x0, []))
11.17/3.76	   f383_in(.(x0, .(x1, x2)))
11.17/3.76	   U5(f515_out1, .(x0, .(x1, x2)))
11.17/3.76	   f535_in(0, s(x0))
11.17/3.76	   f535_in(s(x0), s(x1))
11.17/3.76	   U6(f535_out1, s(x0), s(x1))
11.17/3.76	   f520_in(0, x0)
11.17/3.76	   f520_in(s(x0), x1)
11.17/3.76	   U7(f535_out1, s(x0), x1)
11.17/3.76	   f51_in(x0)
11.17/3.76	   U8(f382_out1, x0)
11.17/3.76	   U9(f383_out1, x0)
11.17/3.76	   f406_in(x0, x1)
11.17/3.76	   U10(f411_out1, x0, x1)
11.17/3.76	   U11(f412_out1, x0, x1)
11.17/3.76	   f412_in(x0)
11.17/3.76	   U12(f449_out1, x0)
11.17/3.76	   U13(f382_out1, x0)
11.17/3.76	   f515_in(x0, x1, x2)
11.17/3.76	   U14(f520_out1, x0, x1, x2)
11.17/3.76	   U15(f383_out1, x0, x1, x2)
11.17/3.76	
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(139) UsableRulesProof (EQUIVALENT)
11.17/3.76	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(140)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F449_IN -> F449_IN
11.17/3.76	
11.17/3.76	R is empty.
11.17/3.76	The set Q consists of the following terms:
11.17/3.76	
11.17/3.76	   f2_in(x0)
11.17/3.76	   U1(f51_out1, x0)
11.17/3.76	   f411_in(x0)
11.17/3.76	   U2(f411_out1, x0)
11.17/3.76	   f382_in([])
11.17/3.76	   f382_in(.(x0, x1))
11.17/3.76	   U3(f406_out1, .(x0, x1))
11.17/3.76	   f449_in
11.17/3.76	   U4(f449_out1)
11.17/3.76	   f383_in([])
11.17/3.76	   f383_in(.(x0, []))
11.17/3.76	   f383_in(.(x0, .(x1, x2)))
11.17/3.76	   U5(f515_out1, .(x0, .(x1, x2)))
11.17/3.76	   f535_in(0, s(x0))
11.17/3.76	   f535_in(s(x0), s(x1))
11.17/3.76	   U6(f535_out1, s(x0), s(x1))
11.17/3.76	   f520_in(0, x0)
11.17/3.76	   f520_in(s(x0), x1)
11.17/3.76	   U7(f535_out1, s(x0), x1)
11.17/3.76	   f51_in(x0)
11.17/3.76	   U8(f382_out1, x0)
11.17/3.76	   U9(f383_out1, x0)
11.17/3.76	   f406_in(x0, x1)
11.17/3.76	   U10(f411_out1, x0, x1)
11.17/3.76	   U11(f412_out1, x0, x1)
11.17/3.76	   f412_in(x0)
11.17/3.76	   U12(f449_out1, x0)
11.17/3.76	   U13(f382_out1, x0)
11.17/3.76	   f515_in(x0, x1, x2)
11.17/3.76	   U14(f520_out1, x0, x1, x2)
11.17/3.76	   U15(f383_out1, x0, x1, x2)
11.17/3.76	
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(141) QReductionProof (EQUIVALENT)
11.17/3.76	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
11.17/3.76	
11.17/3.76	   f2_in(x0)
11.17/3.76	   U1(f51_out1, x0)
11.17/3.76	   f411_in(x0)
11.17/3.76	   U2(f411_out1, x0)
11.17/3.76	   f382_in([])
11.17/3.76	   f382_in(.(x0, x1))
11.17/3.76	   U3(f406_out1, .(x0, x1))
11.17/3.76	   f449_in
11.17/3.76	   U4(f449_out1)
11.17/3.76	   f383_in([])
11.17/3.76	   f383_in(.(x0, []))
11.17/3.76	   f383_in(.(x0, .(x1, x2)))
11.17/3.76	   U5(f515_out1, .(x0, .(x1, x2)))
11.17/3.76	   f535_in(0, s(x0))
11.17/3.76	   f535_in(s(x0), s(x1))
11.17/3.76	   U6(f535_out1, s(x0), s(x1))
11.17/3.76	   f520_in(0, x0)
11.17/3.76	   f520_in(s(x0), x1)
11.17/3.76	   U7(f535_out1, s(x0), x1)
11.17/3.76	   f51_in(x0)
11.17/3.76	   U8(f382_out1, x0)
11.17/3.76	   U9(f383_out1, x0)
11.17/3.76	   f406_in(x0, x1)
11.17/3.76	   U10(f411_out1, x0, x1)
11.17/3.76	   U11(f412_out1, x0, x1)
11.17/3.76	   f412_in(x0)
11.17/3.76	   U12(f449_out1, x0)
11.17/3.76	   U13(f382_out1, x0)
11.17/3.76	   f515_in(x0, x1, x2)
11.17/3.76	   U14(f520_out1, x0, x1, x2)
11.17/3.76	   U15(f383_out1, x0, x1, x2)
11.17/3.76	
11.17/3.76	
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(142)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F449_IN -> F449_IN
11.17/3.76	
11.17/3.76	R is empty.
11.17/3.76	Q is empty.
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(143) NonTerminationLoopProof (COMPLETE)
11.17/3.76	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
11.17/3.76	Found a loop by semiunifying a rule from P directly.
11.17/3.76	
11.17/3.76	s = F449_IN evaluates to  t =F449_IN
11.17/3.76	
11.17/3.76	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
11.17/3.76	* Matcher: [ ]
11.17/3.76	* Semiunifier: [ ]
11.17/3.76	
11.17/3.76	--------------------------------------------------------------------------------
11.17/3.76	Rewriting sequence
11.17/3.76	
11.17/3.76	The DP semiunifies directly so there is only one rewrite step from F449_IN to F449_IN.
11.17/3.76	
11.17/3.76	
11.17/3.76	
11.17/3.76	
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(144)
11.17/3.76	NO
11.17/3.76	
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(145)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F411_IN(T47) -> F411_IN(T47)
11.17/3.76	
11.17/3.76	The TRS R consists of the following rules:
11.17/3.76	
11.17/3.76	   f2_in(T9) -> U1(f51_in(T9), T9)
11.17/3.76	   U1(f51_out1, T9) -> f2_out1
11.17/3.76	   f411_in(T39) -> f411_out1
11.17/3.76	   f411_in(T47) -> U2(f411_in(T47), T47)
11.17/3.76	   U2(f411_out1, T47) -> f411_out1
11.17/3.76	   f382_in([]) -> f382_out1
11.17/3.76	   f382_in(.(T18, T19)) -> U3(f406_in(T18, T19), .(T18, T19))
11.17/3.76	   U3(f406_out1, .(T18, T19)) -> f382_out1
11.17/3.76	   f449_in -> f449_out1
11.17/3.76	   f449_in -> U4(f449_in)
11.17/3.76	   U4(f449_out1) -> f449_out1
11.17/3.76	   f383_in([]) -> f383_out1
11.17/3.76	   f383_in(.(T82, [])) -> f383_out1
11.17/3.76	   f383_in(.(T89, .(T90, T91))) -> U5(f515_in(T89, T90, T91), .(T89, .(T90, T91)))
11.17/3.76	   U5(f515_out1, .(T89, .(T90, T91))) -> f383_out1
11.17/3.76	   f535_in(0, s(T113)) -> f535_out1
11.17/3.76	   f535_in(s(T118), s(T119)) -> U6(f535_in(T118, T119), s(T118), s(T119))
11.17/3.76	   U6(f535_out1, s(T118), s(T119)) -> f535_out1
11.17/3.76	   f520_in(0, T100) -> f520_out1
11.17/3.76	   f520_in(s(T105), T106) -> U7(f535_in(T105, T106), s(T105), T106)
11.17/3.76	   U7(f535_out1, s(T105), T106) -> f520_out1
11.17/3.76	   f51_in(T9) -> U8(f382_in(T9), T9)
11.17/3.76	   U8(f382_out1, T9) -> U9(f383_in(T9), T9)
11.17/3.76	   U9(f383_out1, T9) -> f51_out1
11.17/3.76	   f406_in(T18, T19) -> U10(f411_in(T18), T18, T19)
11.17/3.76	   U10(f411_out1, T18, T19) -> U11(f412_in(T19), T18, T19)
11.17/3.76	   U11(f412_out1, T18, T19) -> f406_out1
11.17/3.76	   f412_in(T19) -> U12(f449_in, T19)
11.17/3.76	   U12(f449_out1, T19) -> U13(f382_in(T19), T19)
11.17/3.76	   U13(f382_out1, T19) -> f412_out1
11.17/3.76	   f515_in(T89, T90, T91) -> U14(f520_in(T89, T90), T89, T90, T91)
11.17/3.76	   U14(f520_out1, T89, T90, T91) -> U15(f383_in(.(T90, T91)), T89, T90, T91)
11.17/3.76	   U15(f383_out1, T89, T90, T91) -> f515_out1
11.17/3.76	
11.17/3.76	Q is empty.
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(146) MNOCProof (EQUIVALENT)
11.17/3.76	We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(147)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F411_IN(T47) -> F411_IN(T47)
11.17/3.76	
11.17/3.76	The TRS R consists of the following rules:
11.17/3.76	
11.17/3.76	   f2_in(T9) -> U1(f51_in(T9), T9)
11.17/3.76	   U1(f51_out1, T9) -> f2_out1
11.17/3.76	   f411_in(T39) -> f411_out1
11.17/3.76	   f411_in(T47) -> U2(f411_in(T47), T47)
11.17/3.76	   U2(f411_out1, T47) -> f411_out1
11.17/3.76	   f382_in([]) -> f382_out1
11.17/3.76	   f382_in(.(T18, T19)) -> U3(f406_in(T18, T19), .(T18, T19))
11.17/3.76	   U3(f406_out1, .(T18, T19)) -> f382_out1
11.17/3.76	   f449_in -> f449_out1
11.17/3.76	   f449_in -> U4(f449_in)
11.17/3.76	   U4(f449_out1) -> f449_out1
11.17/3.76	   f383_in([]) -> f383_out1
11.17/3.76	   f383_in(.(T82, [])) -> f383_out1
11.17/3.76	   f383_in(.(T89, .(T90, T91))) -> U5(f515_in(T89, T90, T91), .(T89, .(T90, T91)))
11.17/3.76	   U5(f515_out1, .(T89, .(T90, T91))) -> f383_out1
11.17/3.76	   f535_in(0, s(T113)) -> f535_out1
11.17/3.76	   f535_in(s(T118), s(T119)) -> U6(f535_in(T118, T119), s(T118), s(T119))
11.17/3.76	   U6(f535_out1, s(T118), s(T119)) -> f535_out1
11.17/3.76	   f520_in(0, T100) -> f520_out1
11.17/3.76	   f520_in(s(T105), T106) -> U7(f535_in(T105, T106), s(T105), T106)
11.17/3.76	   U7(f535_out1, s(T105), T106) -> f520_out1
11.17/3.76	   f51_in(T9) -> U8(f382_in(T9), T9)
11.17/3.76	   U8(f382_out1, T9) -> U9(f383_in(T9), T9)
11.17/3.76	   U9(f383_out1, T9) -> f51_out1
11.17/3.76	   f406_in(T18, T19) -> U10(f411_in(T18), T18, T19)
11.17/3.76	   U10(f411_out1, T18, T19) -> U11(f412_in(T19), T18, T19)
11.17/3.76	   U11(f412_out1, T18, T19) -> f406_out1
11.17/3.76	   f412_in(T19) -> U12(f449_in, T19)
11.17/3.76	   U12(f449_out1, T19) -> U13(f382_in(T19), T19)
11.17/3.76	   U13(f382_out1, T19) -> f412_out1
11.17/3.76	   f515_in(T89, T90, T91) -> U14(f520_in(T89, T90), T89, T90, T91)
11.17/3.76	   U14(f520_out1, T89, T90, T91) -> U15(f383_in(.(T90, T91)), T89, T90, T91)
11.17/3.76	   U15(f383_out1, T89, T90, T91) -> f515_out1
11.17/3.76	
11.17/3.76	The set Q consists of the following terms:
11.17/3.76	
11.17/3.76	   f2_in(x0)
11.17/3.76	   U1(f51_out1, x0)
11.17/3.76	   f411_in(x0)
11.17/3.76	   U2(f411_out1, x0)
11.17/3.76	   f382_in([])
11.17/3.76	   f382_in(.(x0, x1))
11.17/3.76	   U3(f406_out1, .(x0, x1))
11.17/3.76	   f449_in
11.17/3.76	   U4(f449_out1)
11.17/3.76	   f383_in([])
11.17/3.76	   f383_in(.(x0, []))
11.17/3.76	   f383_in(.(x0, .(x1, x2)))
11.17/3.76	   U5(f515_out1, .(x0, .(x1, x2)))
11.17/3.76	   f535_in(0, s(x0))
11.17/3.76	   f535_in(s(x0), s(x1))
11.17/3.76	   U6(f535_out1, s(x0), s(x1))
11.17/3.76	   f520_in(0, x0)
11.17/3.76	   f520_in(s(x0), x1)
11.17/3.76	   U7(f535_out1, s(x0), x1)
11.17/3.76	   f51_in(x0)
11.17/3.76	   U8(f382_out1, x0)
11.17/3.76	   U9(f383_out1, x0)
11.17/3.76	   f406_in(x0, x1)
11.17/3.76	   U10(f411_out1, x0, x1)
11.17/3.76	   U11(f412_out1, x0, x1)
11.17/3.76	   f412_in(x0)
11.17/3.76	   U12(f449_out1, x0)
11.17/3.76	   U13(f382_out1, x0)
11.17/3.76	   f515_in(x0, x1, x2)
11.17/3.76	   U14(f520_out1, x0, x1, x2)
11.17/3.76	   U15(f383_out1, x0, x1, x2)
11.17/3.76	
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(148) UsableRulesProof (EQUIVALENT)
11.17/3.76	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(149)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F411_IN(T47) -> F411_IN(T47)
11.17/3.76	
11.17/3.76	R is empty.
11.17/3.76	The set Q consists of the following terms:
11.17/3.76	
11.17/3.76	   f2_in(x0)
11.17/3.76	   U1(f51_out1, x0)
11.17/3.76	   f411_in(x0)
11.17/3.76	   U2(f411_out1, x0)
11.17/3.76	   f382_in([])
11.17/3.76	   f382_in(.(x0, x1))
11.17/3.76	   U3(f406_out1, .(x0, x1))
11.17/3.76	   f449_in
11.17/3.76	   U4(f449_out1)
11.17/3.76	   f383_in([])
11.17/3.76	   f383_in(.(x0, []))
11.17/3.76	   f383_in(.(x0, .(x1, x2)))
11.17/3.76	   U5(f515_out1, .(x0, .(x1, x2)))
11.17/3.76	   f535_in(0, s(x0))
11.17/3.76	   f535_in(s(x0), s(x1))
11.17/3.76	   U6(f535_out1, s(x0), s(x1))
11.17/3.76	   f520_in(0, x0)
11.17/3.76	   f520_in(s(x0), x1)
11.17/3.76	   U7(f535_out1, s(x0), x1)
11.17/3.76	   f51_in(x0)
11.17/3.76	   U8(f382_out1, x0)
11.17/3.76	   U9(f383_out1, x0)
11.17/3.76	   f406_in(x0, x1)
11.17/3.76	   U10(f411_out1, x0, x1)
11.17/3.76	   U11(f412_out1, x0, x1)
11.17/3.76	   f412_in(x0)
11.17/3.76	   U12(f449_out1, x0)
11.17/3.76	   U13(f382_out1, x0)
11.17/3.76	   f515_in(x0, x1, x2)
11.17/3.76	   U14(f520_out1, x0, x1, x2)
11.17/3.76	   U15(f383_out1, x0, x1, x2)
11.17/3.76	
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(150) QReductionProof (EQUIVALENT)
11.17/3.76	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
11.17/3.76	
11.17/3.76	   f2_in(x0)
11.17/3.76	   U1(f51_out1, x0)
11.17/3.76	   f411_in(x0)
11.17/3.76	   U2(f411_out1, x0)
11.17/3.76	   f382_in([])
11.17/3.76	   f382_in(.(x0, x1))
11.17/3.76	   U3(f406_out1, .(x0, x1))
11.17/3.76	   f449_in
11.17/3.76	   U4(f449_out1)
11.17/3.76	   f383_in([])
11.17/3.76	   f383_in(.(x0, []))
11.17/3.76	   f383_in(.(x0, .(x1, x2)))
11.17/3.76	   U5(f515_out1, .(x0, .(x1, x2)))
11.17/3.76	   f535_in(0, s(x0))
11.17/3.76	   f535_in(s(x0), s(x1))
11.17/3.76	   U6(f535_out1, s(x0), s(x1))
11.17/3.76	   f520_in(0, x0)
11.17/3.76	   f520_in(s(x0), x1)
11.17/3.76	   U7(f535_out1, s(x0), x1)
11.17/3.76	   f51_in(x0)
11.17/3.76	   U8(f382_out1, x0)
11.17/3.76	   U9(f383_out1, x0)
11.17/3.76	   f406_in(x0, x1)
11.17/3.76	   U10(f411_out1, x0, x1)
11.17/3.76	   U11(f412_out1, x0, x1)
11.17/3.76	   f412_in(x0)
11.17/3.76	   U12(f449_out1, x0)
11.17/3.76	   U13(f382_out1, x0)
11.17/3.76	   f515_in(x0, x1, x2)
11.17/3.76	   U14(f520_out1, x0, x1, x2)
11.17/3.76	   U15(f383_out1, x0, x1, x2)
11.17/3.76	
11.17/3.76	
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(151)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F411_IN(T47) -> F411_IN(T47)
11.17/3.76	
11.17/3.76	R is empty.
11.17/3.76	Q is empty.
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(152) NonTerminationLoopProof (COMPLETE)
11.17/3.76	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
11.17/3.76	Found a loop by semiunifying a rule from P directly.
11.17/3.76	
11.17/3.76	s = F411_IN(T47) evaluates to  t =F411_IN(T47)
11.17/3.76	
11.17/3.76	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
11.17/3.76	* Matcher: [ ]
11.17/3.76	* Semiunifier: [ ]
11.17/3.76	
11.17/3.76	--------------------------------------------------------------------------------
11.17/3.76	Rewriting sequence
11.17/3.76	
11.17/3.76	The DP semiunifies directly so there is only one rewrite step from F411_IN(T47) to F411_IN(T47).
11.17/3.76	
11.17/3.76	
11.17/3.76	
11.17/3.76	
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(153)
11.17/3.76	NO
11.17/3.76	
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(154)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F406_IN(T18, T19) -> U10^1(f411_in(T18), T18, T19)
11.17/3.76	   U10^1(f411_out1, T18, T19) -> F412_IN(T19)
11.17/3.76	   F412_IN(T19) -> U12^1(f449_in, T19)
11.17/3.76	   U12^1(f449_out1, T19) -> F382_IN(T19)
11.17/3.76	   F382_IN(.(T18, T19)) -> F406_IN(T18, T19)
11.17/3.76	
11.17/3.76	The TRS R consists of the following rules:
11.17/3.76	
11.17/3.76	   f2_in(T9) -> U1(f51_in(T9), T9)
11.17/3.76	   U1(f51_out1, T9) -> f2_out1
11.17/3.76	   f411_in(T39) -> f411_out1
11.17/3.76	   f411_in(T47) -> U2(f411_in(T47), T47)
11.17/3.76	   U2(f411_out1, T47) -> f411_out1
11.17/3.76	   f382_in([]) -> f382_out1
11.17/3.76	   f382_in(.(T18, T19)) -> U3(f406_in(T18, T19), .(T18, T19))
11.17/3.76	   U3(f406_out1, .(T18, T19)) -> f382_out1
11.17/3.76	   f449_in -> f449_out1
11.17/3.76	   f449_in -> U4(f449_in)
11.17/3.76	   U4(f449_out1) -> f449_out1
11.17/3.76	   f383_in([]) -> f383_out1
11.17/3.76	   f383_in(.(T82, [])) -> f383_out1
11.17/3.76	   f383_in(.(T89, .(T90, T91))) -> U5(f515_in(T89, T90, T91), .(T89, .(T90, T91)))
11.17/3.76	   U5(f515_out1, .(T89, .(T90, T91))) -> f383_out1
11.17/3.76	   f535_in(0, s(T113)) -> f535_out1
11.17/3.76	   f535_in(s(T118), s(T119)) -> U6(f535_in(T118, T119), s(T118), s(T119))
11.17/3.76	   U6(f535_out1, s(T118), s(T119)) -> f535_out1
11.17/3.76	   f520_in(0, T100) -> f520_out1
11.17/3.76	   f520_in(s(T105), T106) -> U7(f535_in(T105, T106), s(T105), T106)
11.17/3.76	   U7(f535_out1, s(T105), T106) -> f520_out1
11.17/3.76	   f51_in(T9) -> U8(f382_in(T9), T9)
11.17/3.76	   U8(f382_out1, T9) -> U9(f383_in(T9), T9)
11.17/3.76	   U9(f383_out1, T9) -> f51_out1
11.17/3.76	   f406_in(T18, T19) -> U10(f411_in(T18), T18, T19)
11.17/3.76	   U10(f411_out1, T18, T19) -> U11(f412_in(T19), T18, T19)
11.17/3.76	   U11(f412_out1, T18, T19) -> f406_out1
11.17/3.76	   f412_in(T19) -> U12(f449_in, T19)
11.17/3.76	   U12(f449_out1, T19) -> U13(f382_in(T19), T19)
11.17/3.76	   U13(f382_out1, T19) -> f412_out1
11.17/3.76	   f515_in(T89, T90, T91) -> U14(f520_in(T89, T90), T89, T90, T91)
11.17/3.76	   U14(f520_out1, T89, T90, T91) -> U15(f383_in(.(T90, T91)), T89, T90, T91)
11.17/3.76	   U15(f383_out1, T89, T90, T91) -> f515_out1
11.17/3.76	
11.17/3.76	Q is empty.
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(155) MNOCProof (EQUIVALENT)
11.17/3.76	We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(156)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F406_IN(T18, T19) -> U10^1(f411_in(T18), T18, T19)
11.17/3.76	   U10^1(f411_out1, T18, T19) -> F412_IN(T19)
11.17/3.76	   F412_IN(T19) -> U12^1(f449_in, T19)
11.17/3.76	   U12^1(f449_out1, T19) -> F382_IN(T19)
11.17/3.76	   F382_IN(.(T18, T19)) -> F406_IN(T18, T19)
11.17/3.76	
11.17/3.76	The TRS R consists of the following rules:
11.17/3.76	
11.17/3.76	   f2_in(T9) -> U1(f51_in(T9), T9)
11.17/3.76	   U1(f51_out1, T9) -> f2_out1
11.17/3.76	   f411_in(T39) -> f411_out1
11.17/3.76	   f411_in(T47) -> U2(f411_in(T47), T47)
11.17/3.76	   U2(f411_out1, T47) -> f411_out1
11.17/3.76	   f382_in([]) -> f382_out1
11.17/3.76	   f382_in(.(T18, T19)) -> U3(f406_in(T18, T19), .(T18, T19))
11.17/3.76	   U3(f406_out1, .(T18, T19)) -> f382_out1
11.17/3.76	   f449_in -> f449_out1
11.17/3.76	   f449_in -> U4(f449_in)
11.17/3.76	   U4(f449_out1) -> f449_out1
11.17/3.76	   f383_in([]) -> f383_out1
11.17/3.76	   f383_in(.(T82, [])) -> f383_out1
11.17/3.76	   f383_in(.(T89, .(T90, T91))) -> U5(f515_in(T89, T90, T91), .(T89, .(T90, T91)))
11.17/3.76	   U5(f515_out1, .(T89, .(T90, T91))) -> f383_out1
11.17/3.76	   f535_in(0, s(T113)) -> f535_out1
11.17/3.76	   f535_in(s(T118), s(T119)) -> U6(f535_in(T118, T119), s(T118), s(T119))
11.17/3.76	   U6(f535_out1, s(T118), s(T119)) -> f535_out1
11.17/3.76	   f520_in(0, T100) -> f520_out1
11.17/3.76	   f520_in(s(T105), T106) -> U7(f535_in(T105, T106), s(T105), T106)
11.17/3.76	   U7(f535_out1, s(T105), T106) -> f520_out1
11.17/3.76	   f51_in(T9) -> U8(f382_in(T9), T9)
11.17/3.76	   U8(f382_out1, T9) -> U9(f383_in(T9), T9)
11.17/3.76	   U9(f383_out1, T9) -> f51_out1
11.17/3.76	   f406_in(T18, T19) -> U10(f411_in(T18), T18, T19)
11.17/3.76	   U10(f411_out1, T18, T19) -> U11(f412_in(T19), T18, T19)
11.17/3.76	   U11(f412_out1, T18, T19) -> f406_out1
11.17/3.76	   f412_in(T19) -> U12(f449_in, T19)
11.17/3.76	   U12(f449_out1, T19) -> U13(f382_in(T19), T19)
11.17/3.76	   U13(f382_out1, T19) -> f412_out1
11.17/3.76	   f515_in(T89, T90, T91) -> U14(f520_in(T89, T90), T89, T90, T91)
11.17/3.76	   U14(f520_out1, T89, T90, T91) -> U15(f383_in(.(T90, T91)), T89, T90, T91)
11.17/3.76	   U15(f383_out1, T89, T90, T91) -> f515_out1
11.17/3.76	
11.17/3.76	The set Q consists of the following terms:
11.17/3.76	
11.17/3.76	   f2_in(x0)
11.17/3.76	   U1(f51_out1, x0)
11.17/3.76	   f411_in(x0)
11.17/3.76	   U2(f411_out1, x0)
11.17/3.76	   f382_in([])
11.17/3.76	   f382_in(.(x0, x1))
11.17/3.76	   U3(f406_out1, .(x0, x1))
11.17/3.76	   f449_in
11.17/3.76	   U4(f449_out1)
11.17/3.76	   f383_in([])
11.17/3.76	   f383_in(.(x0, []))
11.17/3.76	   f383_in(.(x0, .(x1, x2)))
11.17/3.76	   U5(f515_out1, .(x0, .(x1, x2)))
11.17/3.76	   f535_in(0, s(x0))
11.17/3.76	   f535_in(s(x0), s(x1))
11.17/3.76	   U6(f535_out1, s(x0), s(x1))
11.17/3.76	   f520_in(0, x0)
11.17/3.76	   f520_in(s(x0), x1)
11.17/3.76	   U7(f535_out1, s(x0), x1)
11.17/3.76	   f51_in(x0)
11.17/3.76	   U8(f382_out1, x0)
11.17/3.76	   U9(f383_out1, x0)
11.17/3.76	   f406_in(x0, x1)
11.17/3.76	   U10(f411_out1, x0, x1)
11.17/3.76	   U11(f412_out1, x0, x1)
11.17/3.76	   f412_in(x0)
11.17/3.76	   U12(f449_out1, x0)
11.17/3.76	   U13(f382_out1, x0)
11.17/3.76	   f515_in(x0, x1, x2)
11.17/3.76	   U14(f520_out1, x0, x1, x2)
11.17/3.76	   U15(f383_out1, x0, x1, x2)
11.17/3.76	
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(157) UsableRulesProof (EQUIVALENT)
11.17/3.76	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(158)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F406_IN(T18, T19) -> U10^1(f411_in(T18), T18, T19)
11.17/3.76	   U10^1(f411_out1, T18, T19) -> F412_IN(T19)
11.17/3.76	   F412_IN(T19) -> U12^1(f449_in, T19)
11.17/3.76	   U12^1(f449_out1, T19) -> F382_IN(T19)
11.17/3.76	   F382_IN(.(T18, T19)) -> F406_IN(T18, T19)
11.17/3.76	
11.17/3.76	The TRS R consists of the following rules:
11.17/3.76	
11.17/3.76	   f449_in -> f449_out1
11.17/3.76	   f449_in -> U4(f449_in)
11.17/3.76	   U4(f449_out1) -> f449_out1
11.17/3.76	   f411_in(T39) -> f411_out1
11.17/3.76	   f411_in(T47) -> U2(f411_in(T47), T47)
11.17/3.76	   U2(f411_out1, T47) -> f411_out1
11.17/3.76	
11.17/3.76	The set Q consists of the following terms:
11.17/3.76	
11.17/3.76	   f2_in(x0)
11.17/3.76	   U1(f51_out1, x0)
11.17/3.76	   f411_in(x0)
11.17/3.76	   U2(f411_out1, x0)
11.17/3.76	   f382_in([])
11.17/3.76	   f382_in(.(x0, x1))
11.17/3.76	   U3(f406_out1, .(x0, x1))
11.17/3.76	   f449_in
11.17/3.76	   U4(f449_out1)
11.17/3.76	   f383_in([])
11.17/3.76	   f383_in(.(x0, []))
11.17/3.76	   f383_in(.(x0, .(x1, x2)))
11.17/3.76	   U5(f515_out1, .(x0, .(x1, x2)))
11.17/3.76	   f535_in(0, s(x0))
11.17/3.76	   f535_in(s(x0), s(x1))
11.17/3.76	   U6(f535_out1, s(x0), s(x1))
11.17/3.76	   f520_in(0, x0)
11.17/3.76	   f520_in(s(x0), x1)
11.17/3.76	   U7(f535_out1, s(x0), x1)
11.17/3.76	   f51_in(x0)
11.17/3.76	   U8(f382_out1, x0)
11.17/3.76	   U9(f383_out1, x0)
11.17/3.76	   f406_in(x0, x1)
11.17/3.76	   U10(f411_out1, x0, x1)
11.17/3.76	   U11(f412_out1, x0, x1)
11.17/3.76	   f412_in(x0)
11.17/3.76	   U12(f449_out1, x0)
11.17/3.76	   U13(f382_out1, x0)
11.17/3.76	   f515_in(x0, x1, x2)
11.17/3.76	   U14(f520_out1, x0, x1, x2)
11.17/3.76	   U15(f383_out1, x0, x1, x2)
11.17/3.76	
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(159) QReductionProof (EQUIVALENT)
11.17/3.76	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
11.17/3.76	
11.17/3.76	   f2_in(x0)
11.17/3.76	   U1(f51_out1, x0)
11.17/3.76	   f382_in([])
11.17/3.76	   f382_in(.(x0, x1))
11.17/3.76	   U3(f406_out1, .(x0, x1))
11.17/3.76	   f383_in([])
11.17/3.76	   f383_in(.(x0, []))
11.17/3.76	   f383_in(.(x0, .(x1, x2)))
11.17/3.76	   U5(f515_out1, .(x0, .(x1, x2)))
11.17/3.76	   f535_in(0, s(x0))
11.17/3.76	   f535_in(s(x0), s(x1))
11.17/3.76	   U6(f535_out1, s(x0), s(x1))
11.17/3.76	   f520_in(0, x0)
11.17/3.76	   f520_in(s(x0), x1)
11.17/3.76	   U7(f535_out1, s(x0), x1)
11.17/3.76	   f51_in(x0)
11.17/3.76	   U8(f382_out1, x0)
11.17/3.76	   U9(f383_out1, x0)
11.17/3.76	   f406_in(x0, x1)
11.17/3.76	   U10(f411_out1, x0, x1)
11.17/3.76	   U11(f412_out1, x0, x1)
11.17/3.76	   f412_in(x0)
11.17/3.76	   U12(f449_out1, x0)
11.17/3.76	   U13(f382_out1, x0)
11.17/3.76	   f515_in(x0, x1, x2)
11.17/3.76	   U14(f520_out1, x0, x1, x2)
11.17/3.76	   U15(f383_out1, x0, x1, x2)
11.17/3.76	
11.17/3.76	
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(160)
11.17/3.76	Obligation:
11.17/3.76	Q DP problem:
11.17/3.76	The TRS P consists of the following rules:
11.17/3.76	
11.17/3.76	   F406_IN(T18, T19) -> U10^1(f411_in(T18), T18, T19)
11.17/3.76	   U10^1(f411_out1, T18, T19) -> F412_IN(T19)
11.17/3.76	   F412_IN(T19) -> U12^1(f449_in, T19)
11.17/3.76	   U12^1(f449_out1, T19) -> F382_IN(T19)
11.17/3.76	   F382_IN(.(T18, T19)) -> F406_IN(T18, T19)
11.17/3.76	
11.17/3.76	The TRS R consists of the following rules:
11.17/3.76	
11.17/3.76	   f449_in -> f449_out1
11.17/3.76	   f449_in -> U4(f449_in)
11.17/3.76	   U4(f449_out1) -> f449_out1
11.17/3.76	   f411_in(T39) -> f411_out1
11.17/3.76	   f411_in(T47) -> U2(f411_in(T47), T47)
11.17/3.76	   U2(f411_out1, T47) -> f411_out1
11.17/3.76	
11.17/3.76	The set Q consists of the following terms:
11.17/3.76	
11.17/3.76	   f411_in(x0)
11.17/3.76	   U2(f411_out1, x0)
11.17/3.76	   f449_in
11.17/3.76	   U4(f449_out1)
11.17/3.76	
11.17/3.76	We have to consider all minimal (P,Q,R)-chains.
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(161) QDPSizeChangeProof (EQUIVALENT)
11.17/3.76	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. 
11.17/3.76	
11.17/3.76	From the DPs we obtained the following set of size-change graphs:
11.17/3.76	*U10^1(f411_out1, T18, T19) -> F412_IN(T19)
11.17/3.76	The graph contains the following edges 3 >= 1
11.17/3.76	
11.17/3.76	
11.17/3.76	*F382_IN(.(T18, T19)) -> F406_IN(T18, T19)
11.17/3.76	The graph contains the following edges 1 > 1, 1 > 2
11.17/3.76	
11.17/3.76	
11.17/3.76	*F412_IN(T19) -> U12^1(f449_in, T19)
11.17/3.76	The graph contains the following edges 1 >= 2
11.17/3.76	
11.17/3.76	
11.17/3.76	*F406_IN(T18, T19) -> U10^1(f411_in(T18), T18, T19)
11.17/3.76	The graph contains the following edges 1 >= 2, 2 >= 3
11.17/3.76	
11.17/3.76	
11.17/3.76	*U12^1(f449_out1, T19) -> F382_IN(T19)
11.17/3.76	The graph contains the following edges 2 >= 1
11.17/3.76	
11.17/3.76	
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(162)
11.17/3.76	YES
11.17/3.76	
11.17/3.76	----------------------------------------
11.17/3.76	
11.17/3.76	(163) PrologToIRSwTTransformerProof (SOUND)
11.17/3.76	Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert
11.17/3.76	
11.17/3.76	{
11.17/3.76	    "root": 1,
11.17/3.76	    "program": {
11.17/3.76	        "directives": [],
11.17/3.76	        "clauses": [
11.17/3.76	            [
11.17/3.76	                "(ss Xs Ys)",
11.17/3.76	                "(',' (perm Xs Ys) (ordered Ys))"
11.17/3.76	            ],
11.17/3.76	            [
11.17/3.76	                "(perm ([]) ([]))",
11.17/3.76	                null
11.17/3.76	            ],
11.17/3.76	            [
11.17/3.76	                "(perm Xs (. X Ys))",
11.17/3.76	                "(',' (app X1s (. X X2s) Xs) (',' (app X1s X2s Zs) (perm Zs Ys)))"
11.17/3.76	            ],
11.17/3.76	            [
11.17/3.76	                "(app ([]) X X)",
11.17/3.76	                null
11.17/3.76	            ],
11.17/3.76	            [
11.17/3.76	                "(app (. X Xs) Ys (. X Zs))",
11.17/3.76	                "(app Xs Ys Zs)"
11.17/3.76	            ],
11.17/3.76	            [
11.17/3.76	                "(ordered ([]))",
11.17/3.76	                null
11.17/3.76	            ],
11.17/3.76	            [
11.17/3.76	                "(ordered (. X1 ([])))",
11.17/3.76	                null
11.17/3.76	            ],
11.17/3.76	            [
11.17/3.76	                "(ordered (. X (. Y Xs)))",
11.17/3.76	                "(',' (less X (s Y)) (ordered (. Y Xs)))"
11.17/3.76	            ],
11.17/3.76	            [
11.17/3.76	                "(less (0) (s X2))",
11.17/3.76	                null
11.17/3.76	            ],
11.17/3.76	            [
11.17/3.76	                "(less (s X) (s Y))",
11.17/3.76	                "(less X Y)"
11.17/3.76	            ]
11.17/3.76	        ]
11.17/3.76	    },
11.17/3.76	    "graph": {
11.17/3.76	        "nodes": {
11.17/3.76	            "type": "Nodes",
11.17/3.76	            "393": {
11.17/3.76	                "goal": [{
11.17/3.76	                    "clause": -1,
11.17/3.76	                    "scope": -1,
11.17/3.76	                    "term": "(perm T10 T9)"
11.17/3.76	                }],
11.17/3.76	                "kb": {
11.17/3.76	                    "nonunifying": [],
11.17/3.76	                    "intvars": {},
11.17/3.76	                    "arithmetic": {
11.17/3.76	                        "type": "PlainIntegerRelationState",
11.17/3.76	                        "relations": []
11.17/3.76	                    },
11.17/3.76	                    "ground": ["T9"],
11.17/3.76	                    "free": [],
11.17/3.76	                    "exprvars": []
11.17/3.76	                }
11.17/3.76	            },
11.17/3.76	            "394": {
11.17/3.76	                "goal": [{
11.17/3.76	                    "clause": -1,
11.17/3.76	                    "scope": -1,
11.17/3.76	                    "term": "(ordered T9)"
11.17/3.76	                }],
11.17/3.76	                "kb": {
11.17/3.76	                    "nonunifying": [],
11.17/3.76	                    "intvars": {},
11.17/3.76	                    "arithmetic": {
11.17/3.76	                        "type": "PlainIntegerRelationState",
11.17/3.76	                        "relations": []
11.17/3.76	                    },
11.17/3.76	                    "ground": ["T9"],
11.17/3.76	                    "free": [],
11.17/3.76	                    "exprvars": []
11.17/3.76	                }
11.17/3.76	            },
11.17/3.76	            "430": {
11.17/3.76	                "goal": [],
11.17/3.76	                "kb": {
11.17/3.76	                    "nonunifying": [],
11.17/3.76	                    "intvars": {},
11.17/3.76	                    "arithmetic": {
11.17/3.76	                        "type": "PlainIntegerRelationState",
11.17/3.76	                        "relations": []
11.17/3.76	                    },
11.17/3.76	                    "ground": [],
11.17/3.76	                    "free": [],
11.17/3.76	                    "exprvars": []
11.17/3.76	                }
11.17/3.76	            },
11.17/3.76	            "431": {
11.17/3.76	                "goal": [{
11.17/3.76	                    "clause": -1,
11.17/3.76	                    "scope": -1,
11.17/3.76	                    "term": "(app X21 (. T18 X22) T20)"
11.17/3.76	                }],
11.17/3.76	                "kb": {
11.17/3.76	                    "nonunifying": [],
11.17/3.76	                    "intvars": {},
11.17/3.76	                    "arithmetic": {
11.17/3.76	                        "type": "PlainIntegerRelationState",
11.17/3.76	                        "relations": []
11.17/3.76	                    },
11.17/3.76	                    "ground": ["T18"],
11.17/3.76	                    "free": [
11.17/3.76	                        "X21",
11.17/3.76	                        "X22"
11.17/3.76	                    ],
11.17/3.76	                    "exprvars": []
11.17/3.76	                }
11.17/3.76	            },
11.17/3.76	            "475": {
11.17/3.76	                "goal": [
11.17/3.76	                    {
11.17/3.76	                        "clause": 5,
11.17/3.76	                        "scope": 5,
11.17/3.76	                        "term": "(ordered T9)"
11.17/3.76	                    },
11.17/3.76	                    {
11.17/3.76	                        "clause": 6,
11.17/3.76	                        "scope": 5,
11.17/3.76	                        "term": "(ordered T9)"
11.17/3.76	                    },
11.17/3.76	                    {
11.17/3.76	                        "clause": 7,
11.17/3.76	                        "scope": 5,
11.17/3.76	                        "term": "(ordered T9)"
11.17/3.76	                    }
11.17/3.76	                ],
11.17/3.76	                "kb": {
11.17/3.76	                    "nonunifying": [],
11.17/3.76	                    "intvars": {},
11.17/3.76	                    "arithmetic": {
11.17/3.76	                        "type": "PlainIntegerRelationState",
11.17/3.76	                        "relations": []
11.17/3.76	                    },
11.17/3.77	                    "ground": ["T9"],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "432": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": -1,
11.17/3.77	                    "scope": -1,
11.17/3.77	                    "term": "(',' (app T25 T26 X23) (perm X23 T19))"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": ["T19"],
11.17/3.77	                    "free": ["X23"],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "476": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": 5,
11.17/3.77	                    "scope": 5,
11.17/3.77	                    "term": "(ordered T9)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": ["T9"],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "433": {
11.17/3.77	                "goal": [
11.17/3.77	                    {
11.17/3.77	                        "clause": 3,
11.17/3.77	                        "scope": 3,
11.17/3.77	                        "term": "(app X21 (. T18 X22) T20)"
11.17/3.77	                    },
11.17/3.77	                    {
11.17/3.77	                        "clause": 4,
11.17/3.77	                        "scope": 3,
11.17/3.77	                        "term": "(app X21 (. T18 X22) T20)"
11.17/3.77	                    }
11.17/3.77	                ],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": ["T18"],
11.17/3.77	                    "free": [
11.17/3.77	                        "X21",
11.17/3.77	                        "X22"
11.17/3.77	                    ],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "477": {
11.17/3.77	                "goal": [
11.17/3.77	                    {
11.17/3.77	                        "clause": 6,
11.17/3.77	                        "scope": 5,
11.17/3.77	                        "term": "(ordered T9)"
11.17/3.77	                    },
11.17/3.77	                    {
11.17/3.77	                        "clause": 7,
11.17/3.77	                        "scope": 5,
11.17/3.77	                        "term": "(ordered T9)"
11.17/3.77	                    }
11.17/3.77	                ],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": ["T9"],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "478": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": -1,
11.17/3.77	                    "scope": -1,
11.17/3.77	                    "term": "(true)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "511": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": -1,
11.17/3.77	                    "scope": -1,
11.17/3.77	                    "term": "(true)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "479": {
11.17/3.77	                "goal": [],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "436": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": 3,
11.17/3.77	                    "scope": 3,
11.17/3.77	                    "term": "(app X21 (. T18 X22) T20)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": ["T18"],
11.17/3.77	                    "free": [
11.17/3.77	                        "X21",
11.17/3.77	                        "X22"
11.17/3.77	                    ],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "513": {
11.17/3.77	                "goal": [],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "437": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": 4,
11.17/3.77	                    "scope": 3,
11.17/3.77	                    "term": "(app X21 (. T18 X22) T20)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": ["T18"],
11.17/3.77	                    "free": [
11.17/3.77	                        "X21",
11.17/3.77	                        "X22"
11.17/3.77	                    ],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "514": {
11.17/3.77	                "goal": [],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "439": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": -1,
11.17/3.77	                    "scope": -1,
11.17/3.77	                    "term": "(true)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "516": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": -1,
11.17/3.77	                    "scope": -1,
11.17/3.77	                    "term": "(less T105 T106)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [
11.17/3.77	                        "T105",
11.17/3.77	                        "T106"
11.17/3.77	                    ],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "518": {
11.17/3.77	                "goal": [],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "519": {
11.17/3.77	                "goal": [
11.17/3.77	                    {
11.17/3.77	                        "clause": 8,
11.17/3.77	                        "scope": 7,
11.17/3.77	                        "term": "(less T105 T106)"
11.17/3.77	                    },
11.17/3.77	                    {
11.17/3.77	                        "clause": 9,
11.17/3.77	                        "scope": 7,
11.17/3.77	                        "term": "(less T105 T106)"
11.17/3.77	                    }
11.17/3.77	                ],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [
11.17/3.77	                        "T105",
11.17/3.77	                        "T106"
11.17/3.77	                    ],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "480": {
11.17/3.77	                "goal": [],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "483": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": 6,
11.17/3.77	                    "scope": 5,
11.17/3.77	                    "term": "(ordered T9)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": ["T9"],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "484": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": 7,
11.17/3.77	                    "scope": 5,
11.17/3.77	                    "term": "(ordered T9)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": ["T9"],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "1": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": -1,
11.17/3.77	                    "scope": -1,
11.17/3.77	                    "term": "(ss T1 T2)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": ["T2"],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "485": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": -1,
11.17/3.77	                    "scope": -1,
11.17/3.77	                    "term": "(true)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "442": {
11.17/3.77	                "goal": [],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "486": {
11.17/3.77	                "goal": [],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "443": {
11.17/3.77	                "goal": [],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "487": {
11.17/3.77	                "goal": [],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "522": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": 8,
11.17/3.77	                    "scope": 7,
11.17/3.77	                    "term": "(less T105 T106)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [
11.17/3.77	                        "T105",
11.17/3.77	                        "T106"
11.17/3.77	                    ],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "523": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": 9,
11.17/3.77	                    "scope": 7,
11.17/3.77	                    "term": "(less T105 T106)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [
11.17/3.77	                        "T105",
11.17/3.77	                        "T106"
11.17/3.77	                    ],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "447": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": -1,
11.17/3.77	                    "scope": -1,
11.17/3.77	                    "term": "(app X61 (. T47 X62) T50)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": ["T47"],
11.17/3.77	                    "free": [
11.17/3.77	                        "X61",
11.17/3.77	                        "X62"
11.17/3.77	                    ],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "524": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": -1,
11.17/3.77	                    "scope": -1,
11.17/3.77	                    "term": "(true)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "448": {
11.17/3.77	                "goal": [],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "526": {
11.17/3.77	                "goal": [],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "527": {
11.17/3.77	                "goal": [],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "26": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": 0,
11.17/3.77	                    "scope": 1,
11.17/3.77	                    "term": "(ss T1 T2)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": ["T2"],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "492": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": -1,
11.17/3.77	                    "scope": -1,
11.17/3.77	                    "term": "(',' (less T89 (s T90)) (ordered (. T90 T91)))"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [
11.17/3.77	                        "T89",
11.17/3.77	                        "T90",
11.17/3.77	                        "T91"
11.17/3.77	                    ],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "493": {
11.17/3.77	                "goal": [],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "494": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": -1,
11.17/3.77	                    "scope": -1,
11.17/3.77	                    "term": "(less T89 (s T90))"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [
11.17/3.77	                        "T89",
11.17/3.77	                        "T90"
11.17/3.77	                    ],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "495": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": -1,
11.17/3.77	                    "scope": -1,
11.17/3.77	                    "term": "(ordered (. T90 T91))"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [
11.17/3.77	                        "T90",
11.17/3.77	                        "T91"
11.17/3.77	                    ],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "454": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": -1,
11.17/3.77	                    "scope": -1,
11.17/3.77	                    "term": "(app T25 T26 X23)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": ["X23"],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "498": {
11.17/3.77	                "goal": [
11.17/3.77	                    {
11.17/3.77	                        "clause": 8,
11.17/3.77	                        "scope": 6,
11.17/3.77	                        "term": "(less T89 (s T90))"
11.17/3.77	                    },
11.17/3.77	                    {
11.17/3.77	                        "clause": 9,
11.17/3.77	                        "scope": 6,
11.17/3.77	                        "term": "(less T89 (s T90))"
11.17/3.77	                    }
11.17/3.77	                ],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [
11.17/3.77	                        "T89",
11.17/3.77	                        "T90"
11.17/3.77	                    ],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "531": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": -1,
11.17/3.77	                    "scope": -1,
11.17/3.77	                    "term": "(less T118 T119)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [
11.17/3.77	                        "T118",
11.17/3.77	                        "T119"
11.17/3.77	                    ],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "455": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": -1,
11.17/3.77	                    "scope": -1,
11.17/3.77	                    "term": "(perm T57 T19)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": ["T19"],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "499": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": 8,
11.17/3.77	                    "scope": 6,
11.17/3.77	                    "term": "(less T89 (s T90))"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [
11.17/3.77	                        "T89",
11.17/3.77	                        "T90"
11.17/3.77	                    ],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "457": {
11.17/3.77	                "goal": [
11.17/3.77	                    {
11.17/3.77	                        "clause": 3,
11.17/3.77	                        "scope": 4,
11.17/3.77	                        "term": "(app T25 T26 X23)"
11.17/3.77	                    },
11.17/3.77	                    {
11.17/3.77	                        "clause": 4,
11.17/3.77	                        "scope": 4,
11.17/3.77	                        "term": "(app T25 T26 X23)"
11.17/3.77	                    }
11.17/3.77	                ],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": ["X23"],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "534": {
11.17/3.77	                "goal": [],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "459": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": 3,
11.17/3.77	                    "scope": 4,
11.17/3.77	                    "term": "(app T25 T26 X23)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": ["X23"],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "75": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": -1,
11.17/3.77	                    "scope": -1,
11.17/3.77	                    "term": "(',' (perm T10 T9) (ordered T9))"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": ["T9"],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "461": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": 4,
11.17/3.77	                    "scope": 4,
11.17/3.77	                    "term": "(app T25 T26 X23)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": ["X23"],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "462": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": -1,
11.17/3.77	                    "scope": -1,
11.17/3.77	                    "term": "(true)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "463": {
11.17/3.77	                "goal": [],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "464": {
11.17/3.77	                "goal": [],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "423": {
11.17/3.77	                "goal": [
11.17/3.77	                    {
11.17/3.77	                        "clause": 1,
11.17/3.77	                        "scope": 2,
11.17/3.77	                        "term": "(perm T10 T9)"
11.17/3.77	                    },
11.17/3.77	                    {
11.17/3.77	                        "clause": 2,
11.17/3.77	                        "scope": 2,
11.17/3.77	                        "term": "(perm T10 T9)"
11.17/3.77	                    }
11.17/3.77	                ],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": ["T9"],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "467": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": -1,
11.17/3.77	                    "scope": -1,
11.17/3.77	                    "term": "(app T74 T75 X90)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": ["X90"],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "500": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": 9,
11.17/3.77	                    "scope": 6,
11.17/3.77	                    "term": "(less T89 (s T90))"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [
11.17/3.77	                        "T89",
11.17/3.77	                        "T90"
11.17/3.77	                    ],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "424": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": 1,
11.17/3.77	                    "scope": 2,
11.17/3.77	                    "term": "(perm T10 T9)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": ["T9"],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "468": {
11.17/3.77	                "goal": [],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "425": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": 2,
11.17/3.77	                    "scope": 2,
11.17/3.77	                    "term": "(perm T10 T9)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": ["T9"],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "426": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": -1,
11.17/3.77	                    "scope": -1,
11.17/3.77	                    "term": "(true)"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "427": {
11.17/3.77	                "goal": [],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "428": {
11.17/3.77	                "goal": [],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [],
11.17/3.77	                    "free": [],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            },
11.17/3.77	            "429": {
11.17/3.77	                "goal": [{
11.17/3.77	                    "clause": -1,
11.17/3.77	                    "scope": -1,
11.17/3.77	                    "term": "(',' (app X21 (. T18 X22) T20) (',' (app X21 X22 X23) (perm X23 T19)))"
11.17/3.77	                }],
11.17/3.77	                "kb": {
11.17/3.77	                    "nonunifying": [],
11.17/3.77	                    "intvars": {},
11.17/3.77	                    "arithmetic": {
11.17/3.77	                        "type": "PlainIntegerRelationState",
11.17/3.77	                        "relations": []
11.17/3.77	                    },
11.17/3.77	                    "ground": [
11.17/3.77	                        "T18",
11.17/3.77	                        "T19"
11.17/3.77	                    ],
11.17/3.77	                    "free": [
11.17/3.77	                        "X21",
11.17/3.77	                        "X22",
11.17/3.77	                        "X23"
11.17/3.77	                    ],
11.17/3.77	                    "exprvars": []
11.17/3.77	                }
11.17/3.77	            }
11.17/3.77	        },
11.17/3.77	        "edges": [
11.17/3.77	            {
11.17/3.77	                "from": 1,
11.17/3.77	                "to": 26,
11.17/3.77	                "label": "CASE"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 26,
11.17/3.77	                "to": 75,
11.17/3.77	                "label": "ONLY EVAL with clause\nss(X7, X8) :- ','(perm(X7, X8), ordered(X8)).\nand substitutionT1 -> T10,\nX7 -> T10,\nT2 -> T9,\nX8 -> T9,\nT8 -> T10"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 75,
11.17/3.77	                "to": 393,
11.17/3.77	                "label": "SPLIT 1"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 75,
11.17/3.77	                "to": 394,
11.17/3.77	                "label": "SPLIT 2\nnew knowledge:\nT9 is ground"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 393,
11.17/3.77	                "to": 423,
11.17/3.77	                "label": "CASE"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 394,
11.17/3.77	                "to": 475,
11.17/3.77	                "label": "CASE"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 423,
11.17/3.77	                "to": 424,
11.17/3.77	                "label": "PARALLEL"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 423,
11.17/3.77	                "to": 425,
11.17/3.77	                "label": "PARALLEL"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 424,
11.17/3.77	                "to": 426,
11.17/3.77	                "label": "EVAL with clause\nperm([], []).\nand substitutionT10 -> [],\nT9 -> []"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 424,
11.17/3.77	                "to": 427,
11.17/3.77	                "label": "EVAL-BACKTRACK"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 425,
11.17/3.77	                "to": 429,
11.17/3.77	                "label": "EVAL with clause\nperm(X18, .(X19, X20)) :- ','(app(X21, .(X19, X22), X18), ','(app(X21, X22, X23), perm(X23, X20))).\nand substitutionT10 -> T20,\nX18 -> T20,\nX19 -> T18,\nX20 -> T19,\nT9 -> .(T18, T19),\nT17 -> T20"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 425,
11.17/3.77	                "to": 430,
11.17/3.77	                "label": "EVAL-BACKTRACK"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 426,
11.17/3.77	                "to": 428,
11.17/3.77	                "label": "SUCCESS"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 429,
11.17/3.77	                "to": 431,
11.17/3.77	                "label": "SPLIT 1"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 429,
11.17/3.77	                "to": 432,
11.17/3.77	                "label": "SPLIT 2\nreplacements:X21 -> T25,\nX22 -> T26"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 431,
11.17/3.77	                "to": 433,
11.17/3.77	                "label": "CASE"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 432,
11.17/3.77	                "to": 454,
11.17/3.77	                "label": "SPLIT 1"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 432,
11.17/3.77	                "to": 455,
11.17/3.77	                "label": "SPLIT 2\nreplacements:X23 -> T57"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 433,
11.17/3.77	                "to": 436,
11.17/3.77	                "label": "PARALLEL"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 433,
11.17/3.77	                "to": 437,
11.17/3.77	                "label": "PARALLEL"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 436,
11.17/3.77	                "to": 439,
11.17/3.77	                "label": "EVAL with clause\napp([], X40, X40).\nand substitutionX21 -> [],\nT18 -> T39,\nX22 -> T40,\nX40 -> .(T39, T40),\nX41 -> T40,\nT20 -> .(T39, T40)"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 436,
11.17/3.77	                "to": 442,
11.17/3.77	                "label": "EVAL-BACKTRACK"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 437,
11.17/3.77	                "to": 447,
11.17/3.77	                "label": "EVAL with clause\napp(.(X56, X57), X58, .(X56, X59)) :- app(X57, X58, X59).\nand substitutionX56 -> T48,\nX57 -> X61,\nX21 -> .(T48, X61),\nT18 -> T47,\nX22 -> X62,\nX58 -> .(T47, X62),\nX60 -> T48,\nX59 -> T50,\nT20 -> .(T48, T50),\nT49 -> T50"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 437,
11.17/3.77	                "to": 448,
11.17/3.77	                "label": "EVAL-BACKTRACK"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 439,
11.17/3.77	                "to": 443,
11.17/3.77	                "label": "SUCCESS"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 447,
11.17/3.77	                "to": 431,
11.17/3.77	                "label": "INSTANCE with matching:\nX21 -> X61\nT18 -> T47\nX22 -> X62\nT20 -> T50"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 454,
11.17/3.77	                "to": 457,
11.17/3.77	                "label": "CASE"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 455,
11.17/3.77	                "to": 393,
11.17/3.77	                "label": "INSTANCE with matching:\nT10 -> T57\nT9 -> T19"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 457,
11.17/3.77	                "to": 459,
11.17/3.77	                "label": "PARALLEL"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 457,
11.17/3.77	                "to": 461,
11.17/3.77	                "label": "PARALLEL"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 459,
11.17/3.77	                "to": 462,
11.17/3.77	                "label": "EVAL with clause\napp([], X75, X75).\nand substitutionT25 -> [],\nT26 -> T64,\nX75 -> T64,\nX23 -> T64"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 459,
11.17/3.77	                "to": 463,
11.17/3.77	                "label": "EVAL-BACKTRACK"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 461,
11.17/3.77	                "to": 467,
11.17/3.77	                "label": "EVAL with clause\napp(.(X86, X87), X88, .(X86, X89)) :- app(X87, X88, X89).\nand substitutionX86 -> T71,\nX87 -> T74,\nT25 -> .(T71, T74),\nT26 -> T75,\nX88 -> T75,\nX89 -> X90,\nX23 -> .(T71, X90),\nT72 -> T74,\nT73 -> T75"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 461,
11.17/3.77	                "to": 468,
11.17/3.77	                "label": "EVAL-BACKTRACK"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 462,
11.17/3.77	                "to": 464,
11.17/3.77	                "label": "SUCCESS"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 467,
11.17/3.77	                "to": 454,
11.17/3.77	                "label": "INSTANCE with matching:\nT25 -> T74\nT26 -> T75\nX23 -> X90"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 475,
11.17/3.77	                "to": 476,
11.17/3.77	                "label": "PARALLEL"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 475,
11.17/3.77	                "to": 477,
11.17/3.77	                "label": "PARALLEL"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 476,
11.17/3.77	                "to": 478,
11.17/3.77	                "label": "EVAL with clause\nordered([]).\nand substitutionT9 -> []"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 476,
11.17/3.77	                "to": 479,
11.17/3.77	                "label": "EVAL-BACKTRACK"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 477,
11.17/3.77	                "to": 483,
11.17/3.77	                "label": "PARALLEL"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 477,
11.17/3.77	                "to": 484,
11.17/3.77	                "label": "PARALLEL"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 478,
11.17/3.77	                "to": 480,
11.17/3.77	                "label": "SUCCESS"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 483,
11.17/3.77	                "to": 485,
11.17/3.77	                "label": "EVAL with clause\nordered(.(X97, [])).\nand substitutionX97 -> T82,\nT9 -> .(T82, [])"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 483,
11.17/3.77	                "to": 486,
11.17/3.77	                "label": "EVAL-BACKTRACK"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 484,
11.17/3.77	                "to": 492,
11.17/3.77	                "label": "EVAL with clause\nordered(.(X104, .(X105, X106))) :- ','(less(X104, s(X105)), ordered(.(X105, X106))).\nand substitutionX104 -> T89,\nX105 -> T90,\nX106 -> T91,\nT9 -> .(T89, .(T90, T91))"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 484,
11.17/3.77	                "to": 493,
11.17/3.77	                "label": "EVAL-BACKTRACK"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 485,
11.17/3.77	                "to": 487,
11.17/3.77	                "label": "SUCCESS"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 492,
11.17/3.77	                "to": 494,
11.17/3.77	                "label": "SPLIT 1"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 492,
11.17/3.77	                "to": 495,
11.17/3.77	                "label": "SPLIT 2\nnew knowledge:\nT89 is ground\nT90 is ground"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 494,
11.17/3.77	                "to": 498,
11.17/3.77	                "label": "CASE"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 495,
11.17/3.77	                "to": 394,
11.17/3.77	                "label": "INSTANCE with matching:\nT9 -> .(T90, T91)"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 498,
11.17/3.77	                "to": 499,
11.17/3.77	                "label": "PARALLEL"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 498,
11.17/3.77	                "to": 500,
11.17/3.77	                "label": "PARALLEL"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 499,
11.17/3.77	                "to": 511,
11.17/3.77	                "label": "EVAL with clause\nless(0, s(X115)).\nand substitutionT89 -> 0,\nT90 -> T100,\nX115 -> T100"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 499,
11.17/3.77	                "to": 513,
11.17/3.77	                "label": "EVAL-BACKTRACK"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 500,
11.17/3.77	                "to": 516,
11.17/3.77	                "label": "EVAL with clause\nless(s(X120), s(X121)) :- less(X120, X121).\nand substitutionX120 -> T105,\nT89 -> s(T105),\nT90 -> T106,\nX121 -> T106"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 500,
11.17/3.77	                "to": 518,
11.17/3.77	                "label": "EVAL-BACKTRACK"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 511,
11.17/3.77	                "to": 514,
11.17/3.77	                "label": "SUCCESS"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 516,
11.17/3.77	                "to": 519,
11.17/3.77	                "label": "CASE"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 519,
11.17/3.77	                "to": 522,
11.17/3.77	                "label": "PARALLEL"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 519,
11.17/3.77	                "to": 523,
11.17/3.77	                "label": "PARALLEL"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 522,
11.17/3.77	                "to": 524,
11.17/3.77	                "label": "EVAL with clause\nless(0, s(X128)).\nand substitutionT105 -> 0,\nX128 -> T113,\nT106 -> s(T113)"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 522,
11.17/3.77	                "to": 526,
11.17/3.77	                "label": "EVAL-BACKTRACK"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 523,
11.17/3.77	                "to": 531,
11.17/3.77	                "label": "EVAL with clause\nless(s(X133), s(X134)) :- less(X133, X134).\nand substitutionX133 -> T118,\nT105 -> s(T118),\nX134 -> T119,\nT106 -> s(T119)"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 523,
11.17/3.77	                "to": 534,
11.17/3.77	                "label": "EVAL-BACKTRACK"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 524,
11.17/3.77	                "to": 527,
11.17/3.77	                "label": "SUCCESS"
11.17/3.77	            },
11.17/3.77	            {
11.17/3.77	                "from": 531,
11.17/3.77	                "to": 516,
11.17/3.77	                "label": "INSTANCE with matching:\nT105 -> T118\nT106 -> T119"
11.17/3.77	            }
11.17/3.77	        ],
11.17/3.77	        "type": "Graph"
11.17/3.77	    }
11.17/3.77	}
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(164)
11.17/3.77	Complex Obligation (AND)
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(165)
11.17/3.77	Obligation:
11.17/3.77	Rules:
11.17/3.77	f519_out(T105, T106) -> f516_out(T105, T106) :|: TRUE
11.17/3.77	f516_in(x, x1) -> f519_in(x, x1) :|: TRUE
11.17/3.77	f523_in(x2, x3) -> f534_in :|: TRUE
11.17/3.77	f523_in(s(T118), s(T119)) -> f531_in(T118, T119) :|: TRUE
11.17/3.77	f534_out -> f523_out(x4, x5) :|: TRUE
11.17/3.77	f531_out(x6, x7) -> f523_out(s(x6), s(x7)) :|: TRUE
11.17/3.77	f519_in(x8, x9) -> f523_in(x8, x9) :|: TRUE
11.17/3.77	f523_out(x10, x11) -> f519_out(x10, x11) :|: TRUE
11.17/3.77	f519_in(x12, x13) -> f522_in(x12, x13) :|: TRUE
11.17/3.77	f522_out(x14, x15) -> f519_out(x14, x15) :|: TRUE
11.17/3.77	f531_in(x16, x17) -> f516_in(x16, x17) :|: TRUE
11.17/3.77	f516_out(x18, x19) -> f531_out(x18, x19) :|: TRUE
11.17/3.77	f26_out(T2) -> f1_out(T2) :|: TRUE
11.17/3.77	f1_in(x20) -> f26_in(x20) :|: TRUE
11.17/3.77	f75_out(T9) -> f26_out(T9) :|: TRUE
11.17/3.77	f26_in(x21) -> f75_in(x21) :|: TRUE
11.17/3.77	f393_out(x22) -> f394_in(x22) :|: TRUE
11.17/3.77	f75_in(x23) -> f393_in(x23) :|: TRUE
11.17/3.77	f394_out(x24) -> f75_out(x24) :|: TRUE
11.17/3.77	f475_out(x25) -> f394_out(x25) :|: TRUE
11.17/3.77	f394_in(x26) -> f475_in(x26) :|: TRUE
11.17/3.77	f475_in(x27) -> f477_in(x27) :|: TRUE
11.17/3.77	f475_in(x28) -> f476_in(x28) :|: TRUE
11.17/3.77	f476_out(x29) -> f475_out(x29) :|: TRUE
11.17/3.77	f477_out(x30) -> f475_out(x30) :|: TRUE
11.17/3.77	f484_out(x31) -> f477_out(x31) :|: TRUE
11.17/3.77	f477_in(x32) -> f483_in(x32) :|: TRUE
11.17/3.77	f483_out(x33) -> f477_out(x33) :|: TRUE
11.17/3.77	f477_in(x34) -> f484_in(x34) :|: TRUE
11.17/3.77	f492_out(T89, T90, T91) -> f484_out(.(T89, .(T90, T91))) :|: TRUE
11.17/3.77	f493_out -> f484_out(x35) :|: TRUE
11.17/3.77	f484_in(x36) -> f493_in :|: TRUE
11.17/3.77	f484_in(.(x37, .(x38, x39))) -> f492_in(x37, x38, x39) :|: TRUE
11.17/3.77	f495_out(x40, x41) -> f492_out(x42, x40, x41) :|: TRUE
11.17/3.77	f494_out(x43, x44) -> f495_in(x44, x45) :|: TRUE
11.17/3.77	f492_in(x46, x47, x48) -> f494_in(x46, x47) :|: TRUE
11.17/3.77	f494_in(x49, x50) -> f498_in(x49, x50) :|: TRUE
11.17/3.77	f498_out(x51, x52) -> f494_out(x51, x52) :|: TRUE
11.17/3.77	f500_out(x53, x54) -> f498_out(x53, x54) :|: TRUE
11.17/3.77	f498_in(x55, x56) -> f499_in(x55, x56) :|: TRUE
11.17/3.77	f498_in(x57, x58) -> f500_in(x57, x58) :|: TRUE
11.17/3.77	f499_out(x59, x60) -> f498_out(x59, x60) :|: TRUE
11.17/3.77	f516_out(x61, x62) -> f500_out(s(x61), x62) :|: TRUE
11.17/3.77	f500_in(s(x63), x64) -> f516_in(x63, x64) :|: TRUE
11.17/3.77	f500_in(x65, x66) -> f518_in :|: TRUE
11.17/3.77	f518_out -> f500_out(x67, x68) :|: TRUE
11.17/3.77	Start term: f1_in(T2)
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(166) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
11.17/3.77	Constructed simple dependency graph.
11.17/3.77	
11.17/3.77	Simplified to the following IRSwTs:
11.17/3.77	
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(167)
11.17/3.77	TRUE
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(168)
11.17/3.77	Obligation:
11.17/3.77	Rules:
11.17/3.77	f461_in -> f468_in :|: TRUE
11.17/3.77	f467_out -> f461_out :|: TRUE
11.17/3.77	f468_out -> f461_out :|: TRUE
11.17/3.77	f461_in -> f467_in :|: TRUE
11.17/3.77	f454_in -> f457_in :|: TRUE
11.17/3.77	f457_out -> f454_out :|: TRUE
11.17/3.77	f461_out -> f457_out :|: TRUE
11.17/3.77	f457_in -> f459_in :|: TRUE
11.17/3.77	f459_out -> f457_out :|: TRUE
11.17/3.77	f457_in -> f461_in :|: TRUE
11.17/3.77	f454_out -> f467_out :|: TRUE
11.17/3.77	f467_in -> f454_in :|: TRUE
11.17/3.77	f26_out(T2) -> f1_out(T2) :|: TRUE
11.17/3.77	f1_in(x) -> f26_in(x) :|: TRUE
11.17/3.77	f75_out(T9) -> f26_out(T9) :|: TRUE
11.17/3.77	f26_in(x1) -> f75_in(x1) :|: TRUE
11.17/3.77	f393_out(x2) -> f394_in(x2) :|: TRUE
11.17/3.77	f75_in(x3) -> f393_in(x3) :|: TRUE
11.17/3.77	f394_out(x4) -> f75_out(x4) :|: TRUE
11.17/3.77	f423_out(x5) -> f393_out(x5) :|: TRUE
11.17/3.77	f393_in(x6) -> f423_in(x6) :|: TRUE
11.17/3.77	f423_in(x7) -> f424_in(x7) :|: TRUE
11.17/3.77	f423_in(x8) -> f425_in(x8) :|: TRUE
11.17/3.77	f425_out(x9) -> f423_out(x9) :|: TRUE
11.17/3.77	f424_out(x10) -> f423_out(x10) :|: TRUE
11.17/3.77	f425_in(x11) -> f430_in :|: TRUE
11.17/3.77	f430_out -> f425_out(x12) :|: TRUE
11.17/3.77	f425_in(.(T18, T19)) -> f429_in(T18, T19) :|: TRUE
11.17/3.77	f429_out(x13, x14) -> f425_out(.(x13, x14)) :|: TRUE
11.17/3.77	f431_out(x15) -> f432_in(x16) :|: TRUE
11.17/3.77	f432_out(x17) -> f429_out(x18, x17) :|: TRUE
11.17/3.77	f429_in(x19, x20) -> f431_in(x19) :|: TRUE
11.17/3.77	f454_out -> f455_in(x21) :|: TRUE
11.17/3.77	f432_in(x22) -> f454_in :|: TRUE
11.17/3.77	f455_out(x23) -> f432_out(x23) :|: TRUE
11.17/3.77	Start term: f1_in(T2)
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(169) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
11.17/3.77	Constructed simple dependency graph.
11.17/3.77	
11.17/3.77	Simplified to the following IRSwTs:
11.17/3.77	
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(170)
11.17/3.77	TRUE
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(171)
11.17/3.77	Obligation:
11.17/3.77	Rules:
11.17/3.77	f448_out -> f437_out(T18) :|: TRUE
11.17/3.77	f437_in(x) -> f448_in :|: TRUE
11.17/3.77	f437_in(T47) -> f447_in(T47) :|: TRUE
11.17/3.77	f447_out(x1) -> f437_out(x1) :|: TRUE
11.17/3.77	f431_in(x2) -> f433_in(x2) :|: TRUE
11.17/3.77	f433_out(x3) -> f431_out(x3) :|: TRUE
11.17/3.77	f433_in(x4) -> f437_in(x4) :|: TRUE
11.17/3.77	f433_in(x5) -> f436_in(x5) :|: TRUE
11.17/3.77	f437_out(x6) -> f433_out(x6) :|: TRUE
11.17/3.77	f436_out(x7) -> f433_out(x7) :|: TRUE
11.17/3.77	f447_in(x8) -> f431_in(x8) :|: TRUE
11.17/3.77	f431_out(x9) -> f447_out(x9) :|: TRUE
11.17/3.77	f26_out(T2) -> f1_out(T2) :|: TRUE
11.17/3.77	f1_in(x10) -> f26_in(x10) :|: TRUE
11.17/3.77	f75_out(T9) -> f26_out(T9) :|: TRUE
11.17/3.77	f26_in(x11) -> f75_in(x11) :|: TRUE
11.17/3.77	f393_out(x12) -> f394_in(x12) :|: TRUE
11.17/3.77	f75_in(x13) -> f393_in(x13) :|: TRUE
11.17/3.77	f394_out(x14) -> f75_out(x14) :|: TRUE
11.17/3.77	f423_out(x15) -> f393_out(x15) :|: TRUE
11.17/3.77	f393_in(x16) -> f423_in(x16) :|: TRUE
11.17/3.77	f423_in(x17) -> f424_in(x17) :|: TRUE
11.17/3.77	f423_in(x18) -> f425_in(x18) :|: TRUE
11.17/3.77	f425_out(x19) -> f423_out(x19) :|: TRUE
11.17/3.77	f424_out(x20) -> f423_out(x20) :|: TRUE
11.17/3.77	f425_in(x21) -> f430_in :|: TRUE
11.17/3.77	f430_out -> f425_out(x22) :|: TRUE
11.17/3.77	f425_in(.(x23, x24)) -> f429_in(x23, x24) :|: TRUE
11.17/3.77	f429_out(x25, x26) -> f425_out(.(x25, x26)) :|: TRUE
11.17/3.77	f431_out(x27) -> f432_in(x28) :|: TRUE
11.17/3.77	f432_out(x29) -> f429_out(x30, x29) :|: TRUE
11.17/3.77	f429_in(x31, x32) -> f431_in(x31) :|: TRUE
11.17/3.77	Start term: f1_in(T2)
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(172) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
11.17/3.77	Constructed simple dependency graph.
11.17/3.77	
11.17/3.77	Simplified to the following IRSwTs:
11.17/3.77	
11.17/3.77	intTRSProblem:
11.17/3.77	f437_in(T47) -> f447_in(T47) :|: TRUE
11.17/3.77	f431_in(x2) -> f433_in(x2) :|: TRUE
11.17/3.77	f433_in(x4) -> f437_in(x4) :|: TRUE
11.17/3.77	f447_in(x8) -> f431_in(x8) :|: TRUE
11.17/3.77	
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(173)
11.17/3.77	Obligation:
11.17/3.77	Rules:
11.17/3.77	f437_in(T47) -> f447_in(T47) :|: TRUE
11.17/3.77	f431_in(x2) -> f433_in(x2) :|: TRUE
11.17/3.77	f433_in(x4) -> f437_in(x4) :|: TRUE
11.17/3.77	f447_in(x8) -> f431_in(x8) :|: TRUE
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(174) IntTRSCompressionProof (EQUIVALENT)
11.17/3.77	Compressed rules.
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(175)
11.17/3.77	Obligation:
11.17/3.77	Rules:
11.17/3.77	f431_in(x2:0) -> f431_in(x2:0) :|: TRUE
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(176) IRSFormatTransformerProof (EQUIVALENT)
11.17/3.77	Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(177)
11.17/3.77	Obligation:
11.17/3.77	Rules:
11.17/3.77	f431_in(x2:0) -> f431_in(x2:0) :|: TRUE
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(178) IRSwTTerminationDigraphProof (EQUIVALENT)
11.17/3.77	Constructed termination digraph!
11.17/3.77	Nodes:
11.17/3.77	(1) f431_in(x2:0) -> f431_in(x2:0) :|: TRUE
11.17/3.77	
11.17/3.77	Arcs:
11.17/3.77	(1) -> (1)
11.17/3.77	
11.17/3.77	This digraph is fully evaluated!
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(179)
11.17/3.77	Obligation:
11.17/3.77	
11.17/3.77	Termination digraph:
11.17/3.77	Nodes:
11.17/3.77	(1) f431_in(x2:0) -> f431_in(x2:0) :|: TRUE
11.17/3.77	
11.17/3.77	Arcs:
11.17/3.77	(1) -> (1)
11.17/3.77	
11.17/3.77	This digraph is fully evaluated!
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(180) FilterProof (EQUIVALENT)
11.17/3.77	Used the following sort dictionary for filtering: 
11.17/3.77	f431_in(VARIABLE)
11.17/3.77	Replaced non-predefined constructor symbols by 0.
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(181)
11.17/3.77	Obligation:
11.17/3.77	Rules:
11.17/3.77	f431_in(x2:0) -> f431_in(x2:0) :|: TRUE
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(182) IntTRSPeriodicNontermProof (COMPLETE)
11.17/3.77	Normalized system to the following form:
11.17/3.77	f(pc, x2:0) -> f(1, x2:0) :|: pc = 1 && TRUE
11.17/3.77	Witness term starting non-terminating reduction: f(1, -8)
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(183)
11.17/3.77	NO
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(184)
11.17/3.77	Obligation:
11.17/3.77	Rules:
11.17/3.77	f492_out(T89, T90, T91) -> f484_out(.(T89, .(T90, T91))) :|: TRUE
11.17/3.77	f493_out -> f484_out(T9) :|: TRUE
11.17/3.77	f484_in(x) -> f493_in :|: TRUE
11.17/3.77	f484_in(.(x1, .(x2, x3))) -> f492_in(x1, x2, x3) :|: TRUE
11.17/3.77	f475_in(x4) -> f477_in(x4) :|: TRUE
11.17/3.77	f475_in(x5) -> f476_in(x5) :|: TRUE
11.17/3.77	f476_out(x6) -> f475_out(x6) :|: TRUE
11.17/3.77	f477_out(x7) -> f475_out(x7) :|: TRUE
11.17/3.77	f495_out(x8, x9) -> f492_out(x10, x8, x9) :|: TRUE
11.17/3.77	f494_out(x11, x12) -> f495_in(x12, x13) :|: TRUE
11.17/3.77	f492_in(x14, x15, x16) -> f494_in(x14, x15) :|: TRUE
11.17/3.77	f519_in(T105, T106) -> f523_in(T105, T106) :|: TRUE
11.17/3.77	f523_out(x17, x18) -> f519_out(x17, x18) :|: TRUE
11.17/3.77	f519_in(x19, x20) -> f522_in(x19, x20) :|: TRUE
11.17/3.77	f522_out(x21, x22) -> f519_out(x21, x22) :|: TRUE
11.17/3.77	f516_out(x23, x24) -> f500_out(s(x23), x24) :|: TRUE
11.17/3.77	f500_in(s(x25), x26) -> f516_in(x25, x26) :|: TRUE
11.17/3.77	f500_in(x27, x28) -> f518_in :|: TRUE
11.17/3.77	f518_out -> f500_out(x29, x30) :|: TRUE
11.17/3.77	f519_out(x31, x32) -> f516_out(x31, x32) :|: TRUE
11.17/3.77	f516_in(x33, x34) -> f519_in(x33, x34) :|: TRUE
11.17/3.77	f484_out(x35) -> f477_out(x35) :|: TRUE
11.17/3.77	f477_in(x36) -> f483_in(x36) :|: TRUE
11.17/3.77	f483_out(x37) -> f477_out(x37) :|: TRUE
11.17/3.77	f477_in(x38) -> f484_in(x38) :|: TRUE
11.17/3.77	f522_in(0, s(T113)) -> f524_in :|: TRUE
11.17/3.77	f524_out -> f522_out(0, s(x39)) :|: TRUE
11.17/3.77	f526_out -> f522_out(x40, x41) :|: TRUE
11.17/3.77	f522_in(x42, x43) -> f526_in :|: TRUE
11.17/3.77	f523_in(x44, x45) -> f534_in :|: TRUE
11.17/3.77	f523_in(s(T118), s(T119)) -> f531_in(T118, T119) :|: TRUE
11.17/3.77	f534_out -> f523_out(x46, x47) :|: TRUE
11.17/3.77	f531_out(x48, x49) -> f523_out(s(x48), s(x49)) :|: TRUE
11.17/3.77	f511_in -> f511_out :|: TRUE
11.17/3.77	f475_out(x50) -> f394_out(x50) :|: TRUE
11.17/3.77	f394_in(x51) -> f475_in(x51) :|: TRUE
11.17/3.77	f500_out(x52, x53) -> f498_out(x52, x53) :|: TRUE
11.17/3.77	f498_in(x54, x55) -> f499_in(x54, x55) :|: TRUE
11.17/3.77	f498_in(x56, x57) -> f500_in(x56, x57) :|: TRUE
11.17/3.77	f499_out(x58, x59) -> f498_out(x58, x59) :|: TRUE
11.17/3.77	f531_in(x60, x61) -> f516_in(x60, x61) :|: TRUE
11.17/3.77	f516_out(x62, x63) -> f531_out(x62, x63) :|: TRUE
11.17/3.77	f494_in(x64, x65) -> f498_in(x64, x65) :|: TRUE
11.17/3.77	f498_out(x66, x67) -> f494_out(x66, x67) :|: TRUE
11.17/3.77	f394_out(.(x68, x69)) -> f495_out(x68, x69) :|: TRUE
11.17/3.77	f495_in(x70, x71) -> f394_in(.(x70, x71)) :|: TRUE
11.17/3.77	f524_in -> f524_out :|: TRUE
11.17/3.77	f511_out -> f499_out(0, T100) :|: TRUE
11.17/3.77	f499_in(x72, x73) -> f513_in :|: TRUE
11.17/3.77	f499_in(0, x74) -> f511_in :|: TRUE
11.17/3.77	f513_out -> f499_out(x75, x76) :|: TRUE
11.17/3.77	f26_out(T2) -> f1_out(T2) :|: TRUE
11.17/3.77	f1_in(x77) -> f26_in(x77) :|: TRUE
11.17/3.77	f75_out(x78) -> f26_out(x78) :|: TRUE
11.17/3.77	f26_in(x79) -> f75_in(x79) :|: TRUE
11.17/3.77	f393_out(x80) -> f394_in(x80) :|: TRUE
11.17/3.77	f75_in(x81) -> f393_in(x81) :|: TRUE
11.17/3.77	f394_out(x82) -> f75_out(x82) :|: TRUE
11.17/3.77	Start term: f1_in(T2)
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(185) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
11.17/3.77	Constructed simple dependency graph.
11.17/3.77	
11.17/3.77	Simplified to the following IRSwTs:
11.17/3.77	
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(186)
11.17/3.77	TRUE
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(187)
11.17/3.77	Obligation:
11.17/3.77	Rules:
11.17/3.77	f431_out(T18) -> f432_in(T19) :|: TRUE
11.17/3.77	f432_out(x) -> f429_out(x1, x) :|: TRUE
11.17/3.77	f429_in(x2, x3) -> f431_in(x2) :|: TRUE
11.17/3.77	f455_in(x4) -> f393_in(x4) :|: TRUE
11.17/3.77	f393_out(x5) -> f455_out(x5) :|: TRUE
11.17/3.77	f439_out -> f436_out(T39) :|: TRUE
11.17/3.77	f442_out -> f436_out(x6) :|: TRUE
11.17/3.77	f436_in(x7) -> f439_in :|: TRUE
11.17/3.77	f436_in(x8) -> f442_in :|: TRUE
11.17/3.77	f448_out -> f437_out(x9) :|: TRUE
11.17/3.77	f437_in(x10) -> f448_in :|: TRUE
11.17/3.77	f437_in(T47) -> f447_in(T47) :|: TRUE
11.17/3.77	f447_out(x11) -> f437_out(x11) :|: TRUE
11.17/3.77	f459_in -> f463_in :|: TRUE
11.17/3.77	f463_out -> f459_out :|: TRUE
11.17/3.77	f459_in -> f462_in :|: TRUE
11.17/3.77	f462_out -> f459_out :|: TRUE
11.17/3.77	f433_in(x12) -> f437_in(x12) :|: TRUE
11.17/3.77	f433_in(x13) -> f436_in(x13) :|: TRUE
11.17/3.77	f437_out(x14) -> f433_out(x14) :|: TRUE
11.17/3.77	f436_out(x15) -> f433_out(x15) :|: TRUE
11.17/3.77	f462_in -> f462_out :|: TRUE
11.17/3.77	f423_in(T9) -> f424_in(T9) :|: TRUE
11.17/3.77	f423_in(x16) -> f425_in(x16) :|: TRUE
11.17/3.77	f425_out(x17) -> f423_out(x17) :|: TRUE
11.17/3.77	f424_out(x18) -> f423_out(x18) :|: TRUE
11.17/3.77	f439_in -> f439_out :|: TRUE
11.17/3.77	f461_in -> f468_in :|: TRUE
11.17/3.77	f467_out -> f461_out :|: TRUE
11.17/3.77	f468_out -> f461_out :|: TRUE
11.17/3.77	f461_in -> f467_in :|: TRUE
11.17/3.77	f454_in -> f457_in :|: TRUE
11.17/3.77	f457_out -> f454_out :|: TRUE
11.17/3.77	f423_out(x19) -> f393_out(x19) :|: TRUE
11.17/3.77	f393_in(x20) -> f423_in(x20) :|: TRUE
11.17/3.77	f425_in(x21) -> f430_in :|: TRUE
11.17/3.77	f430_out -> f425_out(x22) :|: TRUE
11.17/3.77	f425_in(.(x23, x24)) -> f429_in(x23, x24) :|: TRUE
11.17/3.77	f429_out(x25, x26) -> f425_out(.(x25, x26)) :|: TRUE
11.17/3.77	f454_out -> f455_in(x27) :|: TRUE
11.17/3.77	f432_in(x28) -> f454_in :|: TRUE
11.17/3.77	f455_out(x29) -> f432_out(x29) :|: TRUE
11.17/3.77	f431_in(x30) -> f433_in(x30) :|: TRUE
11.17/3.77	f433_out(x31) -> f431_out(x31) :|: TRUE
11.17/3.77	f447_in(x32) -> f431_in(x32) :|: TRUE
11.17/3.77	f431_out(x33) -> f447_out(x33) :|: TRUE
11.17/3.77	f461_out -> f457_out :|: TRUE
11.17/3.77	f457_in -> f459_in :|: TRUE
11.17/3.77	f459_out -> f457_out :|: TRUE
11.17/3.77	f457_in -> f461_in :|: TRUE
11.17/3.77	f454_out -> f467_out :|: TRUE
11.17/3.77	f467_in -> f454_in :|: TRUE
11.17/3.77	f26_out(T2) -> f1_out(T2) :|: TRUE
11.17/3.77	f1_in(x34) -> f26_in(x34) :|: TRUE
11.17/3.77	f75_out(x35) -> f26_out(x35) :|: TRUE
11.17/3.77	f26_in(x36) -> f75_in(x36) :|: TRUE
11.17/3.77	f393_out(x37) -> f394_in(x37) :|: TRUE
11.17/3.77	f75_in(x38) -> f393_in(x38) :|: TRUE
11.17/3.77	f394_out(x39) -> f75_out(x39) :|: TRUE
11.17/3.77	Start term: f1_in(T2)
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(188) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
11.17/3.77	Constructed simple dependency graph.
11.17/3.77	
11.17/3.77	Simplified to the following IRSwTs:
11.17/3.77	
11.17/3.77	intTRSProblem:
11.17/3.77	f431_out(T18) -> f432_in(T19) :|: TRUE
11.17/3.77	f429_in(x2, x3) -> f431_in(x2) :|: TRUE
11.17/3.77	f455_in(x4) -> f393_in(x4) :|: TRUE
11.17/3.77	f439_out -> f436_out(T39) :|: TRUE
11.17/3.77	f436_in(x7) -> f439_in :|: TRUE
11.17/3.77	f437_in(T47) -> f447_in(T47) :|: TRUE
11.17/3.77	f447_out(x11) -> f437_out(x11) :|: TRUE
11.17/3.77	f459_in -> f462_in :|: TRUE
11.17/3.77	f462_out -> f459_out :|: TRUE
11.17/3.77	f433_in(x12) -> f437_in(x12) :|: TRUE
11.17/3.77	f433_in(x13) -> f436_in(x13) :|: TRUE
11.17/3.77	f437_out(x14) -> f433_out(x14) :|: TRUE
11.17/3.77	f436_out(x15) -> f433_out(x15) :|: TRUE
11.17/3.77	f462_in -> f462_out :|: TRUE
11.17/3.77	f423_in(x16) -> f425_in(x16) :|: TRUE
11.17/3.77	f439_in -> f439_out :|: TRUE
11.17/3.77	f467_out -> f461_out :|: TRUE
11.17/3.77	f461_in -> f467_in :|: TRUE
11.17/3.77	f454_in -> f457_in :|: TRUE
11.17/3.77	f457_out -> f454_out :|: TRUE
11.17/3.77	f393_in(x20) -> f423_in(x20) :|: TRUE
11.17/3.77	f425_in(.(x23, x24)) -> f429_in(x23, x24) :|: TRUE
11.17/3.77	f454_out -> f455_in(x27) :|: TRUE
11.17/3.77	f432_in(x28) -> f454_in :|: TRUE
11.17/3.77	f431_in(x30) -> f433_in(x30) :|: TRUE
11.17/3.77	f433_out(x31) -> f431_out(x31) :|: TRUE
11.17/3.77	f447_in(x32) -> f431_in(x32) :|: TRUE
11.17/3.77	f431_out(x33) -> f447_out(x33) :|: TRUE
11.17/3.77	f461_out -> f457_out :|: TRUE
11.17/3.77	f457_in -> f459_in :|: TRUE
11.17/3.77	f459_out -> f457_out :|: TRUE
11.17/3.77	f457_in -> f461_in :|: TRUE
11.17/3.77	f454_out -> f467_out :|: TRUE
11.17/3.77	f467_in -> f454_in :|: TRUE
11.17/3.77	
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(189)
11.17/3.77	Obligation:
11.17/3.77	Rules:
11.17/3.77	f431_out(T18) -> f432_in(T19) :|: TRUE
11.17/3.77	f429_in(x2, x3) -> f431_in(x2) :|: TRUE
11.17/3.77	f455_in(x4) -> f393_in(x4) :|: TRUE
11.17/3.77	f439_out -> f436_out(T39) :|: TRUE
11.17/3.77	f436_in(x7) -> f439_in :|: TRUE
11.17/3.77	f437_in(T47) -> f447_in(T47) :|: TRUE
11.17/3.77	f447_out(x11) -> f437_out(x11) :|: TRUE
11.17/3.77	f459_in -> f462_in :|: TRUE
11.17/3.77	f462_out -> f459_out :|: TRUE
11.17/3.77	f433_in(x12) -> f437_in(x12) :|: TRUE
11.17/3.77	f433_in(x13) -> f436_in(x13) :|: TRUE
11.17/3.77	f437_out(x14) -> f433_out(x14) :|: TRUE
11.17/3.77	f436_out(x15) -> f433_out(x15) :|: TRUE
11.17/3.77	f462_in -> f462_out :|: TRUE
11.17/3.77	f423_in(x16) -> f425_in(x16) :|: TRUE
11.17/3.77	f439_in -> f439_out :|: TRUE
11.17/3.77	f467_out -> f461_out :|: TRUE
11.17/3.77	f461_in -> f467_in :|: TRUE
11.17/3.77	f454_in -> f457_in :|: TRUE
11.17/3.77	f457_out -> f454_out :|: TRUE
11.17/3.77	f393_in(x20) -> f423_in(x20) :|: TRUE
11.17/3.77	f425_in(.(x23, x24)) -> f429_in(x23, x24) :|: TRUE
11.17/3.77	f454_out -> f455_in(x27) :|: TRUE
11.17/3.77	f432_in(x28) -> f454_in :|: TRUE
11.17/3.77	f431_in(x30) -> f433_in(x30) :|: TRUE
11.17/3.77	f433_out(x31) -> f431_out(x31) :|: TRUE
11.17/3.77	f447_in(x32) -> f431_in(x32) :|: TRUE
11.17/3.77	f431_out(x33) -> f447_out(x33) :|: TRUE
11.17/3.77	f461_out -> f457_out :|: TRUE
11.17/3.77	f457_in -> f459_in :|: TRUE
11.17/3.77	f459_out -> f457_out :|: TRUE
11.17/3.77	f457_in -> f461_in :|: TRUE
11.17/3.77	f454_out -> f467_out :|: TRUE
11.17/3.77	f467_in -> f454_in :|: TRUE
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(190) IntTRSCompressionProof (EQUIVALENT)
11.17/3.77	Compressed rules.
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(191)
11.17/3.77	Obligation:
11.17/3.77	Rules:
11.17/3.77	f431_out(x33:0) -> f431_out(x33:0) :|: TRUE
11.17/3.77	f433_in(x13:0) -> f431_out(T39:0) :|: TRUE
11.17/3.77	f457_out -> f433_in(x23:0) :|: TRUE
11.17/3.77	f454_in -> f454_in :|: TRUE
11.17/3.77	f454_in -> f457_out :|: TRUE
11.17/3.77	f457_out -> f457_out :|: TRUE
11.17/3.77	f431_out(T18:0) -> f454_in :|: TRUE
11.17/3.77	f433_in(x12:0) -> f433_in(x12:0) :|: TRUE
11.17/3.77	
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(192) IRSFormatTransformerProof (EQUIVALENT)
11.17/3.77	Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
11.17/3.77	----------------------------------------
11.17/3.77	
11.17/3.77	(193)
11.17/3.77	Obligation:
11.17/3.77	Rules:
11.17/3.77	f431_out(x33:0) -> f431_out(x33:0) :|: TRUE
11.17/3.77	f433_in(x13:0) -> f431_out(T39:0) :|: TRUE
11.17/3.77	f457_out -> f433_in(x23:0) :|: TRUE
11.17/3.77	f454_in -> f454_in :|: TRUE
11.17/3.77	f454_in -> f457_out :|: TRUE
11.17/3.77	f457_out -> f457_out :|: TRUE
11.17/3.77	f431_out(T18:0) -> f454_in :|: TRUE
11.17/3.77	f433_in(x12:0) -> f433_in(x12:0) :|: TRUE
11.17/3.81	EOF