6.55/2.59	MAYBE
6.55/2.62	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
6.55/2.62	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.55/2.62	
6.55/2.62	
6.55/2.62	Left Termination of the query pattern
6.55/2.62	
6.55/2.62	rev(a,g)
6.55/2.62	
6.55/2.62	w.r.t. the given Prolog program could not be shown:
6.55/2.62	
6.55/2.62	(0) Prolog
6.55/2.62	(1) PrologToPiTRSProof [SOUND, 0 ms]
6.55/2.62	(2) PiTRS
6.55/2.62	    (3) DependencyPairsProof [EQUIVALENT, 0 ms]
6.55/2.62	    (4) PiDP
6.55/2.62	    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
6.55/2.62	    (6) AND
6.55/2.62	        (7) PiDP
6.55/2.62	            (8) UsableRulesProof [EQUIVALENT, 0 ms]
6.55/2.62	            (9) PiDP
6.55/2.62	            (10) PiDPToQDPProof [SOUND, 0 ms]
6.55/2.62	            (11) QDP
6.55/2.62	            (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.55/2.62	            (13) YES
6.55/2.62	        (14) PiDP
6.55/2.62	            (15) UsableRulesProof [EQUIVALENT, 0 ms]
6.55/2.62	            (16) PiDP
6.55/2.62	            (17) PiDPToQDPProof [SOUND, 0 ms]
6.55/2.62	            (18) QDP
6.55/2.62	            (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.55/2.62	            (20) YES
6.55/2.62	        (21) PiDP
6.55/2.62	            (22) UsableRulesProof [EQUIVALENT, 0 ms]
6.55/2.62	            (23) PiDP
6.55/2.62	            (24) PiDPToQDPProof [SOUND, 0 ms]
6.55/2.62	            (25) QDP
6.55/2.62	(26) PrologToTRSTransformerProof [SOUND, 0 ms]
6.55/2.62	(27) QTRS
6.55/2.62	    (28) QTRSRRRProof [EQUIVALENT, 101 ms]
6.55/2.62	    (29) QTRS
6.55/2.62	    (30) QTRSRRRProof [EQUIVALENT, 0 ms]
6.55/2.62	    (31) QTRS
6.55/2.62	    (32) QTRSRRRProof [EQUIVALENT, 3 ms]
6.55/2.62	    (33) QTRS
6.55/2.62	    (34) QTRSRRRProof [EQUIVALENT, 4 ms]
6.55/2.62	    (35) QTRS
6.55/2.62	    (36) Overlay + Local Confluence [EQUIVALENT, 0 ms]
6.55/2.62	    (37) QTRS
6.55/2.62	    (38) DependencyPairsProof [EQUIVALENT, 0 ms]
6.55/2.62	    (39) QDP
6.55/2.62	    (40) DependencyGraphProof [EQUIVALENT, 0 ms]
6.55/2.62	    (41) AND
6.55/2.62	        (42) QDP
6.55/2.62	        (43) QDP
6.55/2.62	(44) PrologToPiTRSProof [SOUND, 3 ms]
6.55/2.62	(45) PiTRS
6.55/2.62	    (46) DependencyPairsProof [EQUIVALENT, 0 ms]
6.55/2.62	    (47) PiDP
6.55/2.62	    (48) DependencyGraphProof [EQUIVALENT, 0 ms]
6.55/2.62	    (49) AND
6.55/2.62	        (50) PiDP
6.55/2.62	            (51) UsableRulesProof [EQUIVALENT, 0 ms]
6.55/2.62	            (52) PiDP
6.55/2.62	            (53) PiDPToQDPProof [SOUND, 0 ms]
6.55/2.62	            (54) QDP
6.55/2.62	            (55) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.55/2.62	            (56) YES
6.55/2.62	        (57) PiDP
6.55/2.62	            (58) UsableRulesProof [EQUIVALENT, 0 ms]
6.55/2.62	            (59) PiDP
6.55/2.62	            (60) PiDPToQDPProof [SOUND, 0 ms]
6.55/2.62	            (61) QDP
6.55/2.62	            (62) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.55/2.62	            (63) YES
6.55/2.62	        (64) PiDP
6.55/2.62	            (65) UsableRulesProof [EQUIVALENT, 0 ms]
6.55/2.62	            (66) PiDP
6.55/2.62	            (67) PiDPToQDPProof [SOUND, 0 ms]
6.55/2.62	            (68) QDP
6.55/2.62	(69) PrologToDTProblemTransformerProof [SOUND, 40 ms]
6.55/2.62	(70) TRIPLES
6.55/2.62	    (71) TriplesToPiDPProof [SOUND, 23 ms]
6.55/2.62	    (72) PiDP
6.55/2.62	    (73) DependencyGraphProof [EQUIVALENT, 0 ms]
6.55/2.62	    (74) AND
6.55/2.62	        (75) PiDP
6.55/2.62	            (76) UsableRulesProof [EQUIVALENT, 0 ms]
6.55/2.62	            (77) PiDP
6.55/2.62	            (78) PiDPToQDPProof [SOUND, 0 ms]
6.55/2.62	            (79) QDP
6.55/2.62	            (80) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.55/2.62	            (81) YES
6.55/2.62	        (82) PiDP
6.55/2.62	            (83) UsableRulesProof [EQUIVALENT, 0 ms]
6.55/2.62	            (84) PiDP
6.55/2.62	            (85) PiDPToQDPProof [SOUND, 0 ms]
6.55/2.62	            (86) QDP
6.55/2.62	            (87) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.55/2.62	            (88) YES
6.55/2.62	        (89) PiDP
6.55/2.62	            (90) UsableRulesProof [EQUIVALENT, 0 ms]
6.55/2.62	            (91) PiDP
6.55/2.62	            (92) PiDPToQDPProof [SOUND, 0 ms]
6.55/2.62	            (93) QDP
6.55/2.62	(94) PrologToIRSwTTransformerProof [SOUND, 48 ms]
6.55/2.62	(95) AND
6.55/2.62	    (96) IRSwT
6.55/2.62	        (97) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
6.55/2.62	        (98) TRUE
6.55/2.62	    (99) IRSwT
6.55/2.62	        (100) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
6.55/2.62	        (101) TRUE
6.55/2.62	    (102) IRSwT
6.55/2.62	        (103) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
6.55/2.62	        (104) IRSwT
6.55/2.62	        (105) IntTRSCompressionProof [EQUIVALENT, 19 ms]
6.55/2.62	        (106) IRSwT
6.55/2.62	        (107) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
6.55/2.62	        (108) IRSwT
6.55/2.62	        (109) IRSwTTerminationDigraphProof [EQUIVALENT, 3 ms]
6.55/2.62	        (110) IRSwT
6.55/2.62	        (111) FilterProof [EQUIVALENT, 0 ms]
6.55/2.62	        (112) IntTRS
6.55/2.62	        (113) IntTRSNonPeriodicNontermProof [COMPLETE, 6 ms]
6.55/2.62	        (114) NO
6.55/2.62	
6.55/2.62	
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(0)
6.55/2.62	Obligation:
6.55/2.62	Clauses:
6.55/2.62	
6.55/2.62	rev([], []).
6.55/2.62	rev(.(X, Xs), Ys) :- ','(rev(Xs, Zs), app(Zs, .(X, []), Ys)).
6.55/2.62	app([], X, X).
6.55/2.62	app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
6.55/2.62	
6.55/2.62	
6.55/2.62	Query: rev(a,g)
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(1) PrologToPiTRSProof (SOUND)
6.55/2.62	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.55/2.62	
6.55/2.62	rev_in_2: (f,b) (f,f)
6.55/2.62	
6.55/2.62	app_in_3: (b,b,f) (b,b,b)
6.55/2.62	
6.55/2.62	Transforming Prolog into the following Term Rewriting System:
6.55/2.62	
6.55/2.62	Pi-finite rewrite system:
6.55/2.62	The TRS R consists of the following rules:
6.55/2.62	
6.55/2.62	   rev_in_ag([], []) -> rev_out_ag([], [])
6.55/2.62	   rev_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.62	   rev_in_aa([], []) -> rev_out_aa([], [])
6.55/2.62	   rev_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.62	   U1_aa(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.55/2.62	   app_in_gga([], X, X) -> app_out_gga([], X, X)
6.55/2.62	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.55/2.62	   U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.55/2.62	   U2_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> rev_out_aa(.(X, Xs), Ys)
6.55/2.62	   U1_ag(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.55/2.62	   app_in_ggg([], X, X) -> app_out_ggg([], X, X)
6.55/2.62	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U3_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.55/2.62	   U3_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.55/2.62	   U2_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> rev_out_ag(.(X, Xs), Ys)
6.55/2.62	
6.55/2.62	The argument filtering Pi contains the following mapping:
6.55/2.62	rev_in_ag(x1, x2)  =  rev_in_ag(x2)
6.55/2.62	
6.55/2.62	[]  =  []
6.55/2.62	
6.55/2.62	rev_out_ag(x1, x2)  =  rev_out_ag(x1)
6.55/2.62	
6.55/2.62	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
6.55/2.62	
6.55/2.62	rev_in_aa(x1, x2)  =  rev_in_aa
6.55/2.62	
6.55/2.62	rev_out_aa(x1, x2)  =  rev_out_aa(x1, x2)
6.55/2.62	
6.55/2.62	U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
6.55/2.62	
6.55/2.62	U2_aa(x1, x2, x3, x4)  =  U2_aa(x2, x4)
6.55/2.62	
6.55/2.62	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.55/2.62	
6.55/2.62	.(x1, x2)  =  .(x2)
6.55/2.62	
6.55/2.62	app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
6.55/2.62	
6.55/2.62	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
6.55/2.62	
6.55/2.62	U2_ag(x1, x2, x3, x4)  =  U2_ag(x2, x4)
6.55/2.62	
6.55/2.62	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.55/2.62	
6.55/2.62	app_out_ggg(x1, x2, x3)  =  app_out_ggg
6.55/2.62	
6.55/2.62	U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x5)
6.55/2.62	
6.55/2.62	
6.55/2.62	
6.55/2.62	
6.55/2.62	
6.55/2.62	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.55/2.62	
6.55/2.62	
6.55/2.62	
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(2)
6.55/2.62	Obligation:
6.55/2.62	Pi-finite rewrite system:
6.55/2.62	The TRS R consists of the following rules:
6.55/2.62	
6.55/2.62	   rev_in_ag([], []) -> rev_out_ag([], [])
6.55/2.62	   rev_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.62	   rev_in_aa([], []) -> rev_out_aa([], [])
6.55/2.62	   rev_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.62	   U1_aa(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.55/2.62	   app_in_gga([], X, X) -> app_out_gga([], X, X)
6.55/2.62	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.55/2.62	   U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.55/2.62	   U2_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> rev_out_aa(.(X, Xs), Ys)
6.55/2.62	   U1_ag(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.55/2.62	   app_in_ggg([], X, X) -> app_out_ggg([], X, X)
6.55/2.62	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U3_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.55/2.62	   U3_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.55/2.62	   U2_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> rev_out_ag(.(X, Xs), Ys)
6.55/2.62	
6.55/2.62	The argument filtering Pi contains the following mapping:
6.55/2.62	rev_in_ag(x1, x2)  =  rev_in_ag(x2)
6.55/2.62	
6.55/2.62	[]  =  []
6.55/2.62	
6.55/2.62	rev_out_ag(x1, x2)  =  rev_out_ag(x1)
6.55/2.62	
6.55/2.62	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
6.55/2.62	
6.55/2.62	rev_in_aa(x1, x2)  =  rev_in_aa
6.55/2.62	
6.55/2.62	rev_out_aa(x1, x2)  =  rev_out_aa(x1, x2)
6.55/2.62	
6.55/2.62	U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
6.55/2.62	
6.55/2.62	U2_aa(x1, x2, x3, x4)  =  U2_aa(x2, x4)
6.55/2.62	
6.55/2.62	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.55/2.62	
6.55/2.62	.(x1, x2)  =  .(x2)
6.55/2.62	
6.55/2.62	app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
6.55/2.62	
6.55/2.62	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
6.55/2.62	
6.55/2.62	U2_ag(x1, x2, x3, x4)  =  U2_ag(x2, x4)
6.55/2.62	
6.55/2.62	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.55/2.62	
6.55/2.62	app_out_ggg(x1, x2, x3)  =  app_out_ggg
6.55/2.62	
6.55/2.62	U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x5)
6.55/2.62	
6.55/2.62	
6.55/2.62	
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(3) DependencyPairsProof (EQUIVALENT)
6.55/2.62	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.55/2.62	Pi DP problem:
6.55/2.62	The TRS P consists of the following rules:
6.55/2.62	
6.55/2.62	   REV_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.62	   REV_IN_AG(.(X, Xs), Ys) -> REV_IN_AA(Xs, Zs)
6.55/2.62	   REV_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.62	   REV_IN_AA(.(X, Xs), Ys) -> REV_IN_AA(Xs, Zs)
6.55/2.62	   U1_AA(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_AA(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.55/2.62	   U1_AA(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> APP_IN_GGA(Zs, .(X, []), Ys)
6.55/2.62	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U3_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.55/2.62	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
6.55/2.62	   U1_AG(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_AG(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.55/2.62	   U1_AG(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> APP_IN_GGG(Zs, .(X, []), Ys)
6.55/2.62	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> U3_GGG(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.55/2.62	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.55/2.62	
6.55/2.62	The TRS R consists of the following rules:
6.55/2.62	
6.55/2.62	   rev_in_ag([], []) -> rev_out_ag([], [])
6.55/2.62	   rev_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.62	   rev_in_aa([], []) -> rev_out_aa([], [])
6.55/2.62	   rev_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.62	   U1_aa(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.55/2.62	   app_in_gga([], X, X) -> app_out_gga([], X, X)
6.55/2.62	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.55/2.62	   U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.55/2.62	   U2_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> rev_out_aa(.(X, Xs), Ys)
6.55/2.62	   U1_ag(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.55/2.62	   app_in_ggg([], X, X) -> app_out_ggg([], X, X)
6.55/2.62	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U3_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.55/2.62	   U3_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.55/2.62	   U2_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> rev_out_ag(.(X, Xs), Ys)
6.55/2.62	
6.55/2.62	The argument filtering Pi contains the following mapping:
6.55/2.62	rev_in_ag(x1, x2)  =  rev_in_ag(x2)
6.55/2.62	
6.55/2.62	[]  =  []
6.55/2.62	
6.55/2.62	rev_out_ag(x1, x2)  =  rev_out_ag(x1)
6.55/2.62	
6.55/2.62	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
6.55/2.62	
6.55/2.62	rev_in_aa(x1, x2)  =  rev_in_aa
6.55/2.62	
6.55/2.62	rev_out_aa(x1, x2)  =  rev_out_aa(x1, x2)
6.55/2.62	
6.55/2.62	U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
6.55/2.62	
6.55/2.62	U2_aa(x1, x2, x3, x4)  =  U2_aa(x2, x4)
6.55/2.62	
6.55/2.62	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.55/2.62	
6.55/2.62	.(x1, x2)  =  .(x2)
6.55/2.62	
6.55/2.62	app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
6.55/2.62	
6.55/2.62	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
6.55/2.62	
6.55/2.62	U2_ag(x1, x2, x3, x4)  =  U2_ag(x2, x4)
6.55/2.62	
6.55/2.62	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.55/2.62	
6.55/2.62	app_out_ggg(x1, x2, x3)  =  app_out_ggg
6.55/2.62	
6.55/2.62	U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x5)
6.55/2.62	
6.55/2.62	REV_IN_AG(x1, x2)  =  REV_IN_AG(x2)
6.55/2.62	
6.55/2.62	U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
6.55/2.62	
6.55/2.62	REV_IN_AA(x1, x2)  =  REV_IN_AA
6.55/2.62	
6.55/2.62	U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
6.55/2.62	
6.55/2.62	U2_AA(x1, x2, x3, x4)  =  U2_AA(x2, x4)
6.55/2.62	
6.55/2.62	APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
6.55/2.62	
6.55/2.62	U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x5)
6.55/2.62	
6.55/2.62	U2_AG(x1, x2, x3, x4)  =  U2_AG(x2, x4)
6.55/2.62	
6.55/2.62	APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)
6.55/2.62	
6.55/2.62	U3_GGG(x1, x2, x3, x4, x5)  =  U3_GGG(x5)
6.55/2.62	
6.55/2.62	
6.55/2.62	We have to consider all (P,R,Pi)-chains
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(4)
6.55/2.62	Obligation:
6.55/2.62	Pi DP problem:
6.55/2.62	The TRS P consists of the following rules:
6.55/2.62	
6.55/2.62	   REV_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.62	   REV_IN_AG(.(X, Xs), Ys) -> REV_IN_AA(Xs, Zs)
6.55/2.62	   REV_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.62	   REV_IN_AA(.(X, Xs), Ys) -> REV_IN_AA(Xs, Zs)
6.55/2.62	   U1_AA(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_AA(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.55/2.62	   U1_AA(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> APP_IN_GGA(Zs, .(X, []), Ys)
6.55/2.62	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U3_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.55/2.62	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
6.55/2.62	   U1_AG(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_AG(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.55/2.62	   U1_AG(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> APP_IN_GGG(Zs, .(X, []), Ys)
6.55/2.62	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> U3_GGG(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.55/2.62	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.55/2.62	
6.55/2.62	The TRS R consists of the following rules:
6.55/2.62	
6.55/2.62	   rev_in_ag([], []) -> rev_out_ag([], [])
6.55/2.62	   rev_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.62	   rev_in_aa([], []) -> rev_out_aa([], [])
6.55/2.62	   rev_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.62	   U1_aa(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.55/2.62	   app_in_gga([], X, X) -> app_out_gga([], X, X)
6.55/2.62	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.55/2.62	   U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.55/2.62	   U2_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> rev_out_aa(.(X, Xs), Ys)
6.55/2.62	   U1_ag(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.55/2.62	   app_in_ggg([], X, X) -> app_out_ggg([], X, X)
6.55/2.62	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U3_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.55/2.62	   U3_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.55/2.62	   U2_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> rev_out_ag(.(X, Xs), Ys)
6.55/2.62	
6.55/2.62	The argument filtering Pi contains the following mapping:
6.55/2.62	rev_in_ag(x1, x2)  =  rev_in_ag(x2)
6.55/2.62	
6.55/2.62	[]  =  []
6.55/2.62	
6.55/2.62	rev_out_ag(x1, x2)  =  rev_out_ag(x1)
6.55/2.62	
6.55/2.62	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
6.55/2.62	
6.55/2.62	rev_in_aa(x1, x2)  =  rev_in_aa
6.55/2.62	
6.55/2.62	rev_out_aa(x1, x2)  =  rev_out_aa(x1, x2)
6.55/2.62	
6.55/2.62	U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
6.55/2.62	
6.55/2.62	U2_aa(x1, x2, x3, x4)  =  U2_aa(x2, x4)
6.55/2.62	
6.55/2.62	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.55/2.62	
6.55/2.62	.(x1, x2)  =  .(x2)
6.55/2.62	
6.55/2.62	app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
6.55/2.62	
6.55/2.62	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
6.55/2.62	
6.55/2.62	U2_ag(x1, x2, x3, x4)  =  U2_ag(x2, x4)
6.55/2.62	
6.55/2.62	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.55/2.62	
6.55/2.62	app_out_ggg(x1, x2, x3)  =  app_out_ggg
6.55/2.62	
6.55/2.62	U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x5)
6.55/2.62	
6.55/2.62	REV_IN_AG(x1, x2)  =  REV_IN_AG(x2)
6.55/2.62	
6.55/2.62	U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
6.55/2.62	
6.55/2.62	REV_IN_AA(x1, x2)  =  REV_IN_AA
6.55/2.62	
6.55/2.62	U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
6.55/2.62	
6.55/2.62	U2_AA(x1, x2, x3, x4)  =  U2_AA(x2, x4)
6.55/2.62	
6.55/2.62	APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
6.55/2.62	
6.55/2.62	U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x5)
6.55/2.62	
6.55/2.62	U2_AG(x1, x2, x3, x4)  =  U2_AG(x2, x4)
6.55/2.62	
6.55/2.62	APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)
6.55/2.62	
6.55/2.62	U3_GGG(x1, x2, x3, x4, x5)  =  U3_GGG(x5)
6.55/2.62	
6.55/2.62	
6.55/2.62	We have to consider all (P,R,Pi)-chains
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(5) DependencyGraphProof (EQUIVALENT)
6.55/2.62	The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 9 less nodes.
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(6)
6.55/2.62	Complex Obligation (AND)
6.55/2.62	
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(7)
6.55/2.62	Obligation:
6.55/2.62	Pi DP problem:
6.55/2.62	The TRS P consists of the following rules:
6.55/2.62	
6.55/2.62	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.55/2.62	
6.55/2.62	The TRS R consists of the following rules:
6.55/2.62	
6.55/2.62	   rev_in_ag([], []) -> rev_out_ag([], [])
6.55/2.62	   rev_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.62	   rev_in_aa([], []) -> rev_out_aa([], [])
6.55/2.62	   rev_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.62	   U1_aa(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.55/2.62	   app_in_gga([], X, X) -> app_out_gga([], X, X)
6.55/2.62	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.55/2.62	   U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.55/2.62	   U2_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> rev_out_aa(.(X, Xs), Ys)
6.55/2.62	   U1_ag(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.55/2.62	   app_in_ggg([], X, X) -> app_out_ggg([], X, X)
6.55/2.62	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U3_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.55/2.62	   U3_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.55/2.62	   U2_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> rev_out_ag(.(X, Xs), Ys)
6.55/2.62	
6.55/2.62	The argument filtering Pi contains the following mapping:
6.55/2.62	rev_in_ag(x1, x2)  =  rev_in_ag(x2)
6.55/2.62	
6.55/2.62	[]  =  []
6.55/2.62	
6.55/2.62	rev_out_ag(x1, x2)  =  rev_out_ag(x1)
6.55/2.62	
6.55/2.62	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
6.55/2.62	
6.55/2.62	rev_in_aa(x1, x2)  =  rev_in_aa
6.55/2.62	
6.55/2.62	rev_out_aa(x1, x2)  =  rev_out_aa(x1, x2)
6.55/2.62	
6.55/2.62	U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
6.55/2.62	
6.55/2.62	U2_aa(x1, x2, x3, x4)  =  U2_aa(x2, x4)
6.55/2.62	
6.55/2.62	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.55/2.62	
6.55/2.62	.(x1, x2)  =  .(x2)
6.55/2.62	
6.55/2.62	app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
6.55/2.62	
6.55/2.62	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
6.55/2.62	
6.55/2.62	U2_ag(x1, x2, x3, x4)  =  U2_ag(x2, x4)
6.55/2.62	
6.55/2.62	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.55/2.62	
6.55/2.62	app_out_ggg(x1, x2, x3)  =  app_out_ggg
6.55/2.62	
6.55/2.62	U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x5)
6.55/2.62	
6.55/2.62	APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)
6.55/2.62	
6.55/2.62	
6.55/2.62	We have to consider all (P,R,Pi)-chains
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(8) UsableRulesProof (EQUIVALENT)
6.55/2.62	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(9)
6.55/2.62	Obligation:
6.55/2.62	Pi DP problem:
6.55/2.62	The TRS P consists of the following rules:
6.55/2.62	
6.55/2.62	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.55/2.62	
6.55/2.62	R is empty.
6.55/2.62	The argument filtering Pi contains the following mapping:
6.55/2.62	.(x1, x2)  =  .(x2)
6.55/2.62	
6.55/2.62	APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)
6.55/2.62	
6.55/2.62	
6.55/2.62	We have to consider all (P,R,Pi)-chains
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(10) PiDPToQDPProof (SOUND)
6.55/2.62	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(11)
6.55/2.62	Obligation:
6.55/2.62	Q DP problem:
6.55/2.62	The TRS P consists of the following rules:
6.55/2.62	
6.55/2.62	   APP_IN_GGG(.(Xs), Ys, .(Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.55/2.62	
6.55/2.62	R is empty.
6.55/2.62	Q is empty.
6.55/2.62	We have to consider all (P,Q,R)-chains.
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(12) QDPSizeChangeProof (EQUIVALENT)
6.55/2.62	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. 
6.55/2.62	
6.55/2.62	From the DPs we obtained the following set of size-change graphs:
6.55/2.62	*APP_IN_GGG(.(Xs), Ys, .(Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.55/2.62	The graph contains the following edges 1 > 1, 2 >= 2, 3 > 3
6.55/2.62	
6.55/2.62	
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(13)
6.55/2.62	YES
6.55/2.62	
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(14)
6.55/2.62	Obligation:
6.55/2.62	Pi DP problem:
6.55/2.62	The TRS P consists of the following rules:
6.55/2.62	
6.55/2.62	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
6.55/2.62	
6.55/2.62	The TRS R consists of the following rules:
6.55/2.62	
6.55/2.62	   rev_in_ag([], []) -> rev_out_ag([], [])
6.55/2.62	   rev_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.62	   rev_in_aa([], []) -> rev_out_aa([], [])
6.55/2.62	   rev_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.62	   U1_aa(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.55/2.62	   app_in_gga([], X, X) -> app_out_gga([], X, X)
6.55/2.62	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.55/2.62	   U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.55/2.62	   U2_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> rev_out_aa(.(X, Xs), Ys)
6.55/2.62	   U1_ag(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.55/2.62	   app_in_ggg([], X, X) -> app_out_ggg([], X, X)
6.55/2.62	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U3_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.55/2.62	   U3_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.55/2.62	   U2_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> rev_out_ag(.(X, Xs), Ys)
6.55/2.62	
6.55/2.62	The argument filtering Pi contains the following mapping:
6.55/2.62	rev_in_ag(x1, x2)  =  rev_in_ag(x2)
6.55/2.62	
6.55/2.62	[]  =  []
6.55/2.62	
6.55/2.62	rev_out_ag(x1, x2)  =  rev_out_ag(x1)
6.55/2.62	
6.55/2.62	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
6.55/2.62	
6.55/2.62	rev_in_aa(x1, x2)  =  rev_in_aa
6.55/2.62	
6.55/2.62	rev_out_aa(x1, x2)  =  rev_out_aa(x1, x2)
6.55/2.62	
6.55/2.62	U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
6.55/2.62	
6.55/2.62	U2_aa(x1, x2, x3, x4)  =  U2_aa(x2, x4)
6.55/2.62	
6.55/2.62	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.55/2.62	
6.55/2.62	.(x1, x2)  =  .(x2)
6.55/2.62	
6.55/2.62	app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
6.55/2.62	
6.55/2.62	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
6.55/2.62	
6.55/2.62	U2_ag(x1, x2, x3, x4)  =  U2_ag(x2, x4)
6.55/2.62	
6.55/2.62	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.55/2.62	
6.55/2.62	app_out_ggg(x1, x2, x3)  =  app_out_ggg
6.55/2.62	
6.55/2.62	U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x5)
6.55/2.62	
6.55/2.62	APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
6.55/2.62	
6.55/2.62	
6.55/2.62	We have to consider all (P,R,Pi)-chains
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(15) UsableRulesProof (EQUIVALENT)
6.55/2.62	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(16)
6.55/2.62	Obligation:
6.55/2.62	Pi DP problem:
6.55/2.62	The TRS P consists of the following rules:
6.55/2.62	
6.55/2.62	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
6.55/2.62	
6.55/2.62	R is empty.
6.55/2.62	The argument filtering Pi contains the following mapping:
6.55/2.62	.(x1, x2)  =  .(x2)
6.55/2.62	
6.55/2.62	APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
6.55/2.62	
6.55/2.62	
6.55/2.62	We have to consider all (P,R,Pi)-chains
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(17) PiDPToQDPProof (SOUND)
6.55/2.62	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(18)
6.55/2.62	Obligation:
6.55/2.62	Q DP problem:
6.55/2.62	The TRS P consists of the following rules:
6.55/2.62	
6.55/2.62	   APP_IN_GGA(.(Xs), Ys) -> APP_IN_GGA(Xs, Ys)
6.55/2.62	
6.55/2.62	R is empty.
6.55/2.62	Q is empty.
6.55/2.62	We have to consider all (P,Q,R)-chains.
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(19) QDPSizeChangeProof (EQUIVALENT)
6.55/2.62	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. 
6.55/2.62	
6.55/2.62	From the DPs we obtained the following set of size-change graphs:
6.55/2.62	*APP_IN_GGA(.(Xs), Ys) -> APP_IN_GGA(Xs, Ys)
6.55/2.62	The graph contains the following edges 1 > 1, 2 >= 2
6.55/2.62	
6.55/2.62	
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(20)
6.55/2.62	YES
6.55/2.62	
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(21)
6.55/2.62	Obligation:
6.55/2.62	Pi DP problem:
6.55/2.62	The TRS P consists of the following rules:
6.55/2.62	
6.55/2.62	   REV_IN_AA(.(X, Xs), Ys) -> REV_IN_AA(Xs, Zs)
6.55/2.62	
6.55/2.62	The TRS R consists of the following rules:
6.55/2.62	
6.55/2.62	   rev_in_ag([], []) -> rev_out_ag([], [])
6.55/2.62	   rev_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.62	   rev_in_aa([], []) -> rev_out_aa([], [])
6.55/2.62	   rev_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.62	   U1_aa(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.55/2.62	   app_in_gga([], X, X) -> app_out_gga([], X, X)
6.55/2.62	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.55/2.62	   U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.55/2.62	   U2_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> rev_out_aa(.(X, Xs), Ys)
6.55/2.62	   U1_ag(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.55/2.62	   app_in_ggg([], X, X) -> app_out_ggg([], X, X)
6.55/2.62	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U3_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.55/2.62	   U3_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.55/2.62	   U2_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> rev_out_ag(.(X, Xs), Ys)
6.55/2.62	
6.55/2.62	The argument filtering Pi contains the following mapping:
6.55/2.62	rev_in_ag(x1, x2)  =  rev_in_ag(x2)
6.55/2.62	
6.55/2.62	[]  =  []
6.55/2.62	
6.55/2.62	rev_out_ag(x1, x2)  =  rev_out_ag(x1)
6.55/2.62	
6.55/2.62	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
6.55/2.62	
6.55/2.62	rev_in_aa(x1, x2)  =  rev_in_aa
6.55/2.62	
6.55/2.62	rev_out_aa(x1, x2)  =  rev_out_aa(x1, x2)
6.55/2.62	
6.55/2.62	U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
6.55/2.62	
6.55/2.62	U2_aa(x1, x2, x3, x4)  =  U2_aa(x2, x4)
6.55/2.62	
6.55/2.62	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.55/2.62	
6.55/2.62	.(x1, x2)  =  .(x2)
6.55/2.62	
6.55/2.62	app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
6.55/2.62	
6.55/2.62	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
6.55/2.62	
6.55/2.62	U2_ag(x1, x2, x3, x4)  =  U2_ag(x2, x4)
6.55/2.62	
6.55/2.62	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.55/2.62	
6.55/2.62	app_out_ggg(x1, x2, x3)  =  app_out_ggg
6.55/2.62	
6.55/2.62	U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x5)
6.55/2.62	
6.55/2.62	REV_IN_AA(x1, x2)  =  REV_IN_AA
6.55/2.62	
6.55/2.62	
6.55/2.62	We have to consider all (P,R,Pi)-chains
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(22) UsableRulesProof (EQUIVALENT)
6.55/2.62	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(23)
6.55/2.62	Obligation:
6.55/2.62	Pi DP problem:
6.55/2.62	The TRS P consists of the following rules:
6.55/2.62	
6.55/2.62	   REV_IN_AA(.(X, Xs), Ys) -> REV_IN_AA(Xs, Zs)
6.55/2.62	
6.55/2.62	R is empty.
6.55/2.62	The argument filtering Pi contains the following mapping:
6.55/2.62	.(x1, x2)  =  .(x2)
6.55/2.62	
6.55/2.62	REV_IN_AA(x1, x2)  =  REV_IN_AA
6.55/2.62	
6.55/2.62	
6.55/2.62	We have to consider all (P,R,Pi)-chains
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(24) PiDPToQDPProof (SOUND)
6.55/2.62	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(25)
6.55/2.62	Obligation:
6.55/2.62	Q DP problem:
6.55/2.62	The TRS P consists of the following rules:
6.55/2.62	
6.55/2.62	   REV_IN_AA -> REV_IN_AA
6.55/2.62	
6.55/2.62	R is empty.
6.55/2.62	Q is empty.
6.55/2.62	We have to consider all (P,Q,R)-chains.
6.55/2.62	----------------------------------------
6.55/2.62	
6.55/2.62	(26) PrologToTRSTransformerProof (SOUND)
6.55/2.62	Transformed Prolog program to TRS.
6.55/2.62	
6.55/2.62	{
6.55/2.62	    "root": 3,
6.55/2.62	    "program": {
6.55/2.62	        "directives": [],
6.55/2.62	        "clauses": [
6.55/2.62	            [
6.55/2.62	                "(rev ([]) ([]))",
6.55/2.62	                null
6.55/2.62	            ],
6.55/2.62	            [
6.55/2.62	                "(rev (. X Xs) Ys)",
6.55/2.62	                "(',' (rev Xs Zs) (app Zs (. X ([])) Ys))"
6.55/2.62	            ],
6.55/2.62	            [
6.55/2.62	                "(app ([]) X X)",
6.55/2.62	                null
6.55/2.62	            ],
6.55/2.62	            [
6.55/2.62	                "(app (. X Xs) Ys (. X Zs))",
6.55/2.62	                "(app Xs Ys Zs)"
6.55/2.62	            ]
6.55/2.62	        ]
6.55/2.62	    },
6.55/2.62	    "graph": {
6.55/2.62	        "nodes": {
6.55/2.62	            "type": "Nodes",
6.55/2.62	            "194": {
6.55/2.62	                "goal": [],
6.55/2.62	                "kb": {
6.55/2.62	                    "nonunifying": [],
6.55/2.62	                    "intvars": {},
6.55/2.62	                    "arithmetic": {
6.55/2.62	                        "type": "PlainIntegerRelationState",
6.55/2.62	                        "relations": []
6.55/2.62	                    },
6.55/2.62	                    "ground": [],
6.55/2.62	                    "free": [],
6.55/2.62	                    "exprvars": []
6.55/2.62	                }
6.55/2.62	            },
6.55/2.62	            "110": {
6.55/2.62	                "goal": [{
6.55/2.62	                    "clause": -1,
6.55/2.62	                    "scope": -1,
6.55/2.62	                    "term": "(rev T12 X11)"
6.55/2.62	                }],
6.55/2.62	                "kb": {
6.55/2.62	                    "nonunifying": [],
6.55/2.62	                    "intvars": {},
6.55/2.62	                    "arithmetic": {
6.55/2.62	                        "type": "PlainIntegerRelationState",
6.55/2.62	                        "relations": []
6.55/2.62	                    },
6.55/2.62	                    "ground": [],
6.55/2.62	                    "free": ["X11"],
6.55/2.62	                    "exprvars": []
6.55/2.62	                }
6.55/2.62	            },
6.55/2.62	            "198": {
6.55/2.62	                "goal": [{
6.55/2.62	                    "clause": -1,
6.55/2.62	                    "scope": -1,
6.55/2.62	                    "term": "(app T42 (. T43 ([])) X47)"
6.55/2.62	                }],
6.55/2.62	                "kb": {
6.55/2.62	                    "nonunifying": [],
6.55/2.62	                    "intvars": {},
6.55/2.62	                    "arithmetic": {
6.55/2.62	                        "type": "PlainIntegerRelationState",
6.55/2.62	                        "relations": []
6.55/2.62	                    },
6.55/2.62	                    "ground": [],
6.55/2.62	                    "free": ["X47"],
6.55/2.62	                    "exprvars": []
6.55/2.62	                }
6.55/2.62	            },
6.55/2.62	            "111": {
6.55/2.62	                "goal": [{
6.55/2.62	                    "clause": -1,
6.55/2.62	                    "scope": -1,
6.55/2.62	                    "term": "(app T14 (. T15 ([])) T11)"
6.55/2.62	                }],
6.55/2.62	                "kb": {
6.55/2.62	                    "nonunifying": [],
6.55/2.62	                    "intvars": {},
6.55/2.62	                    "arithmetic": {
6.55/2.62	                        "type": "PlainIntegerRelationState",
6.55/2.62	                        "relations": []
6.55/2.62	                    },
6.55/2.62	                    "ground": ["T11"],
6.55/2.62	                    "free": [],
6.55/2.62	                    "exprvars": []
6.55/2.62	                }
6.55/2.62	            },
6.55/2.62	            "199": {
6.55/2.62	                "goal": [],
6.55/2.62	                "kb": {
6.55/2.62	                    "nonunifying": [],
6.55/2.62	                    "intvars": {},
6.55/2.62	                    "arithmetic": {
6.55/2.62	                        "type": "PlainIntegerRelationState",
6.55/2.62	                        "relations": []
6.55/2.62	                    },
6.55/2.62	                    "ground": [],
6.55/2.62	                    "free": [],
6.55/2.62	                    "exprvars": []
6.55/2.62	                }
6.55/2.62	            },
6.55/2.62	            "210": {
6.55/2.62	                "goal": [],
6.55/2.62	                "kb": {
6.55/2.62	                    "nonunifying": [],
6.55/2.62	                    "intvars": {},
6.55/2.62	                    "arithmetic": {
6.55/2.62	                        "type": "PlainIntegerRelationState",
6.55/2.62	                        "relations": []
6.55/2.62	                    },
6.55/2.62	                    "ground": [],
6.55/2.62	                    "free": [],
6.55/2.62	                    "exprvars": []
6.55/2.62	                }
6.55/2.62	            },
6.55/2.62	            "211": {
6.55/2.62	                "goal": [],
6.55/2.62	                "kb": {
6.55/2.62	                    "nonunifying": [],
6.55/2.62	                    "intvars": {},
6.55/2.62	                    "arithmetic": {
6.55/2.62	                        "type": "PlainIntegerRelationState",
6.55/2.62	                        "relations": []
6.55/2.62	                    },
6.55/2.62	                    "ground": [],
6.55/2.62	                    "free": [],
6.55/2.62	                    "exprvars": []
6.55/2.62	                }
6.55/2.62	            },
6.55/2.62	            "214": {
6.55/2.62	                "goal": [{
6.55/2.62	                    "clause": -1,
6.55/2.62	                    "scope": -1,
6.55/2.62	                    "term": "(app T65 (. T66 ([])) T64)"
6.55/2.62	                }],
6.55/2.62	                "kb": {
6.55/2.62	                    "nonunifying": [],
6.55/2.62	                    "intvars": {},
6.55/2.62	                    "arithmetic": {
6.55/2.62	                        "type": "PlainIntegerRelationState",
6.55/2.62	                        "relations": []
6.55/2.62	                    },
6.55/2.62	                    "ground": ["T64"],
6.55/2.62	                    "free": [],
6.55/2.62	                    "exprvars": []
6.55/2.62	                }
6.55/2.62	            },
6.55/2.62	            "215": {
6.55/2.62	                "goal": [],
6.55/2.62	                "kb": {
6.55/2.62	                    "nonunifying": [],
6.55/2.62	                    "intvars": {},
6.55/2.62	                    "arithmetic": {
6.55/2.62	                        "type": "PlainIntegerRelationState",
6.55/2.62	                        "relations": []
6.55/2.62	                    },
6.55/2.62	                    "ground": [],
6.55/2.62	                    "free": [],
6.55/2.62	                    "exprvars": []
6.55/2.62	                }
6.55/2.62	            },
6.55/2.62	            "93": {
6.55/2.62	                "goal": [
6.55/2.62	                    {
6.55/2.62	                        "clause": 0,
6.55/2.62	                        "scope": 1,
6.55/2.62	                        "term": "(rev T1 T2)"
6.55/2.62	                    },
6.55/2.62	                    {
6.55/2.62	                        "clause": 1,
6.55/2.62	                        "scope": 1,
6.55/2.62	                        "term": "(rev T1 T2)"
6.55/2.62	                    }
6.55/2.62	                ],
6.55/2.62	                "kb": {
6.55/2.62	                    "nonunifying": [],
6.55/2.62	                    "intvars": {},
6.55/2.62	                    "arithmetic": {
6.55/2.62	                        "type": "PlainIntegerRelationState",
6.55/2.62	                        "relations": []
6.55/2.62	                    },
6.55/2.62	                    "ground": ["T2"],
6.55/2.62	                    "free": [],
6.55/2.62	                    "exprvars": []
6.55/2.62	                }
6.55/2.62	            },
6.55/2.62	            "96": {
6.55/2.62	                "goal": [{
6.55/2.62	                    "clause": 0,
6.55/2.62	                    "scope": 1,
6.55/2.62	                    "term": "(rev T1 T2)"
6.55/2.62	                }],
6.55/2.62	                "kb": {
6.55/2.62	                    "nonunifying": [],
6.55/2.62	                    "intvars": {},
6.55/2.62	                    "arithmetic": {
6.55/2.62	                        "type": "PlainIntegerRelationState",
6.55/2.62	                        "relations": []
6.55/2.62	                    },
6.55/2.62	                    "ground": ["T2"],
6.55/2.62	                    "free": [],
6.55/2.62	                    "exprvars": []
6.55/2.62	                }
6.55/2.62	            },
6.55/2.62	            "97": {
6.55/2.62	                "goal": [{
6.55/2.62	                    "clause": 1,
6.55/2.62	                    "scope": 1,
6.55/2.62	                    "term": "(rev T1 T2)"
6.55/2.62	                }],
6.55/2.62	                "kb": {
6.55/2.62	                    "nonunifying": [],
6.55/2.62	                    "intvars": {},
6.55/2.62	                    "arithmetic": {
6.55/2.62	                        "type": "PlainIntegerRelationState",
6.55/2.62	                        "relations": []
6.55/2.62	                    },
6.55/2.62	                    "ground": ["T2"],
6.55/2.62	                    "free": [],
6.55/2.62	                    "exprvars": []
6.55/2.62	                }
6.55/2.62	            },
6.55/2.62	            "182": {
6.55/2.62	                "goal": [{
6.55/2.62	                    "clause": -1,
6.55/2.62	                    "scope": -1,
6.55/2.62	                    "term": "(true)"
6.55/2.62	                }],
6.55/2.62	                "kb": {
6.55/2.62	                    "nonunifying": [],
6.55/2.62	                    "intvars": {},
6.55/2.62	                    "arithmetic": {
6.55/2.62	                        "type": "PlainIntegerRelationState",
6.55/2.62	                        "relations": []
6.55/2.62	                    },
6.55/2.62	                    "ground": [],
6.55/2.62	                    "free": [],
6.55/2.62	                    "exprvars": []
6.55/2.62	                }
6.55/2.62	            },
6.55/2.62	            "140": {
6.55/2.62	                "goal": [
6.55/2.62	                    {
6.55/2.62	                        "clause": 2,
6.55/2.62	                        "scope": 3,
6.55/2.62	                        "term": "(app T24 (. T25 ([])) X25)"
6.55/2.62	                    },
6.55/2.62	                    {
6.55/2.62	                        "clause": 3,
6.55/2.62	                        "scope": 3,
6.55/2.62	                        "term": "(app T24 (. T25 ([])) X25)"
6.55/2.62	                    }
6.55/2.62	                ],
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6.55/2.62	            {
6.55/2.62	                "from": 123,
6.55/2.62	                "to": 125,
6.55/2.62	                "label": "SUCCESS"
6.55/2.62	            },
6.55/2.62	            {
6.55/2.62	                "from": 126,
6.55/2.62	                "to": 128,
6.55/2.62	                "label": "SPLIT 1"
6.55/2.62	            },
6.55/2.62	            {
6.55/2.62	                "from": 126,
6.55/2.62	                "to": 129,
6.55/2.62	                "label": "SPLIT 2\nreplacements:X24 -> T24,\nT23 -> T25"
6.55/2.62	            },
6.55/2.62	            {
6.55/2.62	                "from": 128,
6.55/2.63	                "to": 110,
6.55/2.63	                "label": "INSTANCE with matching:\nT12 -> T22\nX11 -> X24"
6.55/2.63	            },
6.55/2.63	            {
6.55/2.63	                "from": 129,
6.55/2.63	                "to": 140,
6.55/2.63	                "label": "CASE"
6.55/2.63	            },
6.55/2.63	            {
6.55/2.63	                "from": 140,
6.55/2.63	                "to": 141,
6.55/2.63	                "label": "PARALLEL"
6.55/2.63	            },
6.55/2.63	            {
6.55/2.63	                "from": 140,
6.55/2.63	                "to": 144,
6.55/2.63	                "label": "PARALLEL"
6.55/2.63	            },
6.55/2.63	            {
6.55/2.63	                "from": 141,
6.55/2.63	                "to": 182,
6.55/2.63	                "label": "EVAL with clause\napp([], X32, X32).\nand substitutionT24 -> [],\nT25 -> T32,\nX32 -> .(T32, []),\nX25 -> .(T32, [])"
6.55/2.63	            },
6.55/2.63	            {
6.55/2.63	                "from": 141,
6.55/2.63	                "to": 189,
6.55/2.63	                "label": "EVAL-BACKTRACK"
6.55/2.63	            },
6.55/2.63	            {
6.55/2.63	                "from": 144,
6.55/2.63	                "to": 198,
6.55/2.63	                "label": "EVAL with clause\napp(.(X43, X44), X45, .(X43, X46)) :- app(X44, X45, X46).\nand substitutionX43 -> T39,\nX44 -> T42,\nT24 -> .(T39, T42),\nT25 -> T43,\nX45 -> .(T43, []),\nX46 -> X47,\nX25 -> .(T39, X47),\nT40 -> T42,\nT41 -> T43"
6.55/2.63	            },
6.55/2.63	            {
6.55/2.63	                "from": 144,
6.55/2.63	                "to": 199,
6.55/2.63	                "label": "EVAL-BACKTRACK"
6.55/2.63	            },
6.55/2.63	            {
6.55/2.63	                "from": 182,
6.55/2.63	                "to": 194,
6.55/2.63	                "label": "SUCCESS"
6.55/2.63	            },
6.55/2.63	            {
6.55/2.63	                "from": 198,
6.55/2.63	                "to": 129,
6.55/2.63	                "label": "INSTANCE with matching:\nT24 -> T42\nT25 -> T43\nX25 -> X47"
6.55/2.63	            },
6.55/2.63	            {
6.55/2.63	                "from": 203,
6.55/2.63	                "to": 205,
6.55/2.63	                "label": "PARALLEL"
6.55/2.63	            },
6.55/2.63	            {
6.55/2.63	                "from": 203,
6.55/2.63	                "to": 206,
6.55/2.63	                "label": "PARALLEL"
6.55/2.63	            },
6.55/2.63	            {
6.55/2.63	                "from": 205,
6.55/2.63	                "to": 209,
6.55/2.63	                "label": "EVAL with clause\napp([], X56, X56).\nand substitutionT14 -> [],\nT15 -> T52,\nX56 -> .(T52, []),\nT11 -> .(T52, [])"
6.55/2.63	            },
6.55/2.63	            {
6.55/2.63	                "from": 205,
6.55/2.63	                "to": 210,
6.55/2.63	                "label": "EVAL-BACKTRACK"
6.55/2.63	            },
6.55/2.63	            {
6.55/2.63	                "from": 206,
6.55/2.63	                "to": 214,
6.55/2.63	                "label": "EVAL with clause\napp(.(X65, X66), X67, .(X65, X68)) :- app(X66, X67, X68).\nand substitutionX65 -> T61,\nX66 -> T65,\nT14 -> .(T61, T65),\nT15 -> T66,\nX67 -> .(T66, []),\nX68 -> T64,\nT11 -> .(T61, T64),\nT62 -> T65,\nT63 -> T66"
6.55/2.63	            },
6.55/2.63	            {
6.55/2.63	                "from": 206,
6.55/2.63	                "to": 215,
6.55/2.63	                "label": "EVAL-BACKTRACK"
6.55/2.63	            },
6.55/2.63	            {
6.55/2.63	                "from": 209,
6.55/2.63	                "to": 211,
6.55/2.63	                "label": "SUCCESS"
6.55/2.63	            },
6.55/2.63	            {
6.55/2.63	                "from": 214,
6.55/2.63	                "to": 111,
6.55/2.63	                "label": "INSTANCE with matching:\nT14 -> T65\nT15 -> T66\nT11 -> T64"
6.55/2.63	            }
6.55/2.63	        ],
6.55/2.63	        "type": "Graph"
6.55/2.63	    }
6.55/2.63	}
6.55/2.63	
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(27)
6.55/2.63	Obligation:
6.55/2.63	Q restricted rewrite system:
6.55/2.63	The TRS R consists of the following rules:
6.55/2.63	
6.55/2.63	   f3_in([]) -> f3_out1
6.55/2.63	   f3_in(T11) -> U1(f104_in(T11), T11)
6.55/2.63	   U1(f104_out1(X11, T13), T11) -> f3_out1
6.55/2.63	   f110_in -> f110_out1
6.55/2.63	   f110_in -> U2(f126_in)
6.55/2.63	   U2(f126_out1) -> f110_out1
6.55/2.63	   f129_in -> f129_out1
6.55/2.63	   f129_in -> U3(f129_in)
6.55/2.63	   U3(f129_out1) -> f129_out1
6.55/2.63	   f111_in(.(T52, [])) -> f111_out1([], T52)
6.55/2.63	   f111_in(.(T61, T64)) -> U4(f111_in(T64), .(T61, T64))
6.55/2.63	   U4(f111_out1(T65, T66), .(T61, T64)) -> f111_out1(.(T61, T65), T66)
6.55/2.63	   f104_in(T11) -> U5(f110_in, T11)
6.55/2.63	   U5(f110_out1, T11) -> U6(f111_in(T11), T11)
6.55/2.63	   U6(f111_out1(T14, T15), T11) -> f104_out1(T14, T15)
6.55/2.63	   f126_in -> U7(f110_in)
6.55/2.63	   U7(f110_out1) -> U8(f129_in)
6.55/2.63	   U8(f129_out1) -> f126_out1
6.55/2.63	
6.55/2.63	Q is empty.
6.55/2.63	
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(28) QTRSRRRProof (EQUIVALENT)
6.55/2.63	Used ordering:
6.55/2.63	f3_in/1(YES)
6.55/2.63	[]/0)
6.55/2.63	f3_out1/0)
6.55/2.63	U1/2(YES,YES)
6.55/2.63	f104_in/1(YES)
6.55/2.63	f104_out1/2(YES,YES)
6.55/2.63	f110_in/0)
6.55/2.63	f110_out1/0)
6.55/2.63	U2/1)YES(
6.55/2.63	f126_in/0)
6.55/2.63	f126_out1/0)
6.55/2.63	f129_in/0)
6.55/2.63	f129_out1/0)
6.55/2.63	U3/1)YES(
6.55/2.63	f111_in/1(YES)
6.55/2.63	./2(YES,YES)
6.55/2.63	f111_out1/2(YES,YES)
6.55/2.63	U4/2(YES,YES)
6.55/2.63	U5/2(YES,YES)
6.55/2.63	U6/2(YES,YES)
6.55/2.63	U7/1)YES(
6.55/2.63	U8/1)YES(
6.55/2.63	
6.55/2.63	Quasi precedence:
6.55/2.63	f3_in_1 > U1_2 > f3_out1
6.55/2.63	f3_in_1 > f104_in_1 > [[], f111_in_1, U5_2] > f3_out1
6.55/2.63	f3_in_1 > f104_in_1 > [[], f111_in_1, U5_2] > U4_2 > f111_out1_2 > ._2
6.55/2.63	f3_in_1 > f104_in_1 > [[], f111_in_1, U5_2] > U6_2 > f104_out1_2
6.55/2.63	f3_in_1 > f104_in_1 > [f110_in, f126_in] > [f110_out1, f126_out1, f129_in, f129_out1]
6.55/2.63	
6.55/2.63	
6.55/2.63	Status:
6.55/2.63	f3_in_1: multiset status
6.55/2.63	[]: multiset status
6.55/2.63	f3_out1: multiset status
6.55/2.63	U1_2: multiset status
6.55/2.63	f104_in_1: multiset status
6.55/2.63	f104_out1_2: multiset status
6.55/2.63	f110_in: multiset status
6.55/2.63	f110_out1: multiset status
6.55/2.63	f126_in: multiset status
6.55/2.63	f126_out1: multiset status
6.55/2.63	f129_in: multiset status
6.55/2.63	f129_out1: multiset status
6.55/2.63	f111_in_1: multiset status
6.55/2.63	._2: multiset status
6.55/2.63	f111_out1_2: multiset status
6.55/2.63	U4_2: multiset status
6.55/2.63	U5_2: multiset status
6.55/2.63	U6_2: multiset status
6.55/2.63	
6.55/2.63	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.55/2.63	
6.55/2.63	   f3_in([]) -> f3_out1
6.55/2.63	   f3_in(T11) -> U1(f104_in(T11), T11)
6.55/2.63	   U1(f104_out1(X11, T13), T11) -> f3_out1
6.55/2.63	   f110_in -> f110_out1
6.55/2.63	   f111_in(.(T52, [])) -> f111_out1([], T52)
6.55/2.63	   f111_in(.(T61, T64)) -> U4(f111_in(T64), .(T61, T64))
6.55/2.63	   U4(f111_out1(T65, T66), .(T61, T64)) -> f111_out1(.(T61, T65), T66)
6.55/2.63	   f104_in(T11) -> U5(f110_in, T11)
6.55/2.63	   U5(f110_out1, T11) -> U6(f111_in(T11), T11)
6.55/2.63	   U6(f111_out1(T14, T15), T11) -> f104_out1(T14, T15)
6.55/2.63	
6.55/2.63	
6.55/2.63	
6.55/2.63	
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(29)
6.55/2.63	Obligation:
6.55/2.63	Q restricted rewrite system:
6.55/2.63	The TRS R consists of the following rules:
6.55/2.63	
6.55/2.63	   f110_in -> U2(f126_in)
6.55/2.63	   U2(f126_out1) -> f110_out1
6.55/2.63	   f129_in -> f129_out1
6.55/2.63	   f129_in -> U3(f129_in)
6.55/2.63	   U3(f129_out1) -> f129_out1
6.55/2.63	   f126_in -> U7(f110_in)
6.55/2.63	   U7(f110_out1) -> U8(f129_in)
6.55/2.63	   U8(f129_out1) -> f126_out1
6.55/2.63	
6.55/2.63	Q is empty.
6.55/2.63	
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(30) QTRSRRRProof (EQUIVALENT)
6.55/2.63	Used ordering:
6.55/2.63	Polynomial interpretation [POLO]:
6.55/2.63	
6.55/2.63	   POL(U2(x_1)) = 2*x_1
6.55/2.63	   POL(U3(x_1)) = x_1
6.55/2.63	   POL(U7(x_1)) = 2*x_1
6.55/2.63	   POL(U8(x_1)) = 2 + x_1
6.55/2.63	   POL(f110_in) = 0
6.55/2.63	   POL(f110_out1) = 2
6.55/2.63	   POL(f126_in) = 0
6.55/2.63	   POL(f126_out1) = 1
6.55/2.63	   POL(f129_in) = 1
6.55/2.63	   POL(f129_out1) = 0
6.55/2.63	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.55/2.63	
6.55/2.63	   f129_in -> f129_out1
6.55/2.63	   U7(f110_out1) -> U8(f129_in)
6.55/2.63	   U8(f129_out1) -> f126_out1
6.55/2.63	
6.55/2.63	
6.55/2.63	
6.55/2.63	
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(31)
6.55/2.63	Obligation:
6.55/2.63	Q restricted rewrite system:
6.55/2.63	The TRS R consists of the following rules:
6.55/2.63	
6.55/2.63	   f110_in -> U2(f126_in)
6.55/2.63	   U2(f126_out1) -> f110_out1
6.55/2.63	   f129_in -> U3(f129_in)
6.55/2.63	   U3(f129_out1) -> f129_out1
6.55/2.63	   f126_in -> U7(f110_in)
6.55/2.63	
6.55/2.63	Q is empty.
6.55/2.63	
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(32) QTRSRRRProof (EQUIVALENT)
6.55/2.63	Used ordering:
6.55/2.63	f110_in/0)
6.55/2.63	U2/1)YES(
6.55/2.63	f126_in/0)
6.55/2.63	f126_out1/0)
6.55/2.63	f110_out1/0)
6.55/2.63	f129_in/0)
6.55/2.63	U3/1)YES(
6.55/2.63	f129_out1/0)
6.55/2.63	U7/1)YES(
6.55/2.63	
6.55/2.63	Quasi precedence:
6.55/2.63	[f110_in, f126_in]
6.55/2.63	f126_out1 > f110_out1
6.55/2.63	
6.55/2.63	
6.55/2.63	Status:
6.55/2.63	f110_in: multiset status
6.55/2.63	f126_in: multiset status
6.55/2.63	f126_out1: multiset status
6.55/2.63	f110_out1: multiset status
6.55/2.63	f129_in: multiset status
6.55/2.63	f129_out1: multiset status
6.55/2.63	
6.55/2.63	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.55/2.63	
6.55/2.63	   U2(f126_out1) -> f110_out1
6.55/2.63	
6.55/2.63	
6.55/2.63	
6.55/2.63	
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(33)
6.55/2.63	Obligation:
6.55/2.63	Q restricted rewrite system:
6.55/2.63	The TRS R consists of the following rules:
6.55/2.63	
6.55/2.63	   f110_in -> U2(f126_in)
6.55/2.63	   f129_in -> U3(f129_in)
6.55/2.63	   U3(f129_out1) -> f129_out1
6.55/2.63	   f126_in -> U7(f110_in)
6.55/2.63	
6.55/2.63	Q is empty.
6.55/2.63	
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(34) QTRSRRRProof (EQUIVALENT)
6.55/2.63	Used ordering:
6.55/2.63	Polynomial interpretation [POLO]:
6.55/2.63	
6.55/2.63	   POL(U2(x_1)) = 2*x_1
6.55/2.63	   POL(U3(x_1)) = 2*x_1
6.55/2.63	   POL(U7(x_1)) = 2*x_1
6.55/2.63	   POL(f110_in) = 0
6.55/2.63	   POL(f126_in) = 0
6.55/2.63	   POL(f129_in) = 0
6.55/2.63	   POL(f129_out1) = 1
6.55/2.63	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.55/2.63	
6.55/2.63	   U3(f129_out1) -> f129_out1
6.55/2.63	
6.55/2.63	
6.55/2.63	
6.55/2.63	
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(35)
6.55/2.63	Obligation:
6.55/2.63	Q restricted rewrite system:
6.55/2.63	The TRS R consists of the following rules:
6.55/2.63	
6.55/2.63	   f110_in -> U2(f126_in)
6.55/2.63	   f129_in -> U3(f129_in)
6.55/2.63	   f126_in -> U7(f110_in)
6.55/2.63	
6.55/2.63	Q is empty.
6.55/2.63	
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(36) Overlay + Local Confluence (EQUIVALENT)
6.55/2.63	The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(37)
6.55/2.63	Obligation:
6.55/2.63	Q restricted rewrite system:
6.55/2.63	The TRS R consists of the following rules:
6.55/2.63	
6.55/2.63	   f110_in -> U2(f126_in)
6.55/2.63	   f129_in -> U3(f129_in)
6.55/2.63	   f126_in -> U7(f110_in)
6.55/2.63	
6.55/2.63	The set Q consists of the following terms:
6.55/2.63	
6.55/2.63	   f110_in
6.55/2.63	   f129_in
6.55/2.63	   f126_in
6.55/2.63	
6.55/2.63	
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(38) DependencyPairsProof (EQUIVALENT)
6.55/2.63	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(39)
6.55/2.63	Obligation:
6.55/2.63	Q DP problem:
6.55/2.63	The TRS P consists of the following rules:
6.55/2.63	
6.55/2.63	   F110_IN -> F126_IN
6.55/2.63	   F129_IN -> F129_IN
6.55/2.63	   F126_IN -> F110_IN
6.55/2.63	
6.55/2.63	The TRS R consists of the following rules:
6.55/2.63	
6.55/2.63	   f110_in -> U2(f126_in)
6.55/2.63	   f129_in -> U3(f129_in)
6.55/2.63	   f126_in -> U7(f110_in)
6.55/2.63	
6.55/2.63	The set Q consists of the following terms:
6.55/2.63	
6.55/2.63	   f110_in
6.55/2.63	   f129_in
6.55/2.63	   f126_in
6.55/2.63	
6.55/2.63	We have to consider all minimal (P,Q,R)-chains.
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(40) DependencyGraphProof (EQUIVALENT)
6.55/2.63	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(41)
6.55/2.63	Complex Obligation (AND)
6.55/2.63	
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(42)
6.55/2.63	Obligation:
6.55/2.63	Q DP problem:
6.55/2.63	The TRS P consists of the following rules:
6.55/2.63	
6.55/2.63	   F129_IN -> F129_IN
6.55/2.63	
6.55/2.63	The TRS R consists of the following rules:
6.55/2.63	
6.55/2.63	   f110_in -> U2(f126_in)
6.55/2.63	   f129_in -> U3(f129_in)
6.55/2.63	   f126_in -> U7(f110_in)
6.55/2.63	
6.55/2.63	The set Q consists of the following terms:
6.55/2.63	
6.55/2.63	   f110_in
6.55/2.63	   f129_in
6.55/2.63	   f126_in
6.55/2.63	
6.55/2.63	We have to consider all minimal (P,Q,R)-chains.
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(43)
6.55/2.63	Obligation:
6.55/2.63	Q DP problem:
6.55/2.63	The TRS P consists of the following rules:
6.55/2.63	
6.55/2.63	   F126_IN -> F110_IN
6.55/2.63	   F110_IN -> F126_IN
6.55/2.63	
6.55/2.63	The TRS R consists of the following rules:
6.55/2.63	
6.55/2.63	   f110_in -> U2(f126_in)
6.55/2.63	   f129_in -> U3(f129_in)
6.55/2.63	   f126_in -> U7(f110_in)
6.55/2.63	
6.55/2.63	The set Q consists of the following terms:
6.55/2.63	
6.55/2.63	   f110_in
6.55/2.63	   f129_in
6.55/2.63	   f126_in
6.55/2.63	
6.55/2.63	We have to consider all minimal (P,Q,R)-chains.
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(44) PrologToPiTRSProof (SOUND)
6.55/2.63	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.55/2.63	
6.55/2.63	rev_in_2: (f,b) (f,f)
6.55/2.63	
6.55/2.63	app_in_3: (b,b,f) (b,b,b)
6.55/2.63	
6.55/2.63	Transforming Prolog into the following Term Rewriting System:
6.55/2.63	
6.55/2.63	Pi-finite rewrite system:
6.55/2.63	The TRS R consists of the following rules:
6.55/2.63	
6.55/2.63	   rev_in_ag([], []) -> rev_out_ag([], [])
6.55/2.63	   rev_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.63	   rev_in_aa([], []) -> rev_out_aa([], [])
6.55/2.63	   rev_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.63	   U1_aa(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.55/2.63	   app_in_gga([], X, X) -> app_out_gga([], X, X)
6.55/2.63	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.55/2.63	   U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.55/2.63	   U2_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> rev_out_aa(.(X, Xs), Ys)
6.55/2.63	   U1_ag(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.55/2.63	   app_in_ggg([], X, X) -> app_out_ggg([], X, X)
6.55/2.63	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U3_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.55/2.63	   U3_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.55/2.63	   U2_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> rev_out_ag(.(X, Xs), Ys)
6.55/2.63	
6.55/2.63	The argument filtering Pi contains the following mapping:
6.55/2.63	rev_in_ag(x1, x2)  =  rev_in_ag(x2)
6.55/2.63	
6.55/2.63	[]  =  []
6.55/2.63	
6.55/2.63	rev_out_ag(x1, x2)  =  rev_out_ag(x1, x2)
6.55/2.63	
6.55/2.63	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
6.55/2.63	
6.55/2.63	rev_in_aa(x1, x2)  =  rev_in_aa
6.55/2.63	
6.55/2.63	rev_out_aa(x1, x2)  =  rev_out_aa(x1, x2)
6.55/2.63	
6.55/2.63	U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
6.55/2.63	
6.55/2.63	U2_aa(x1, x2, x3, x4)  =  U2_aa(x2, x4)
6.55/2.63	
6.55/2.63	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.55/2.63	
6.55/2.63	.(x1, x2)  =  .(x2)
6.55/2.63	
6.55/2.63	app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
6.55/2.63	
6.55/2.63	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x2, x3, x5)
6.55/2.63	
6.55/2.63	U2_ag(x1, x2, x3, x4)  =  U2_ag(x2, x3, x4)
6.55/2.63	
6.55/2.63	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.55/2.63	
6.55/2.63	app_out_ggg(x1, x2, x3)  =  app_out_ggg(x1, x2, x3)
6.55/2.63	
6.55/2.63	U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x2, x3, x4, x5)
6.55/2.63	
6.55/2.63	
6.55/2.63	
6.55/2.63	
6.55/2.63	
6.55/2.63	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.55/2.63	
6.55/2.63	
6.55/2.63	
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(45)
6.55/2.63	Obligation:
6.55/2.63	Pi-finite rewrite system:
6.55/2.63	The TRS R consists of the following rules:
6.55/2.63	
6.55/2.63	   rev_in_ag([], []) -> rev_out_ag([], [])
6.55/2.63	   rev_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.63	   rev_in_aa([], []) -> rev_out_aa([], [])
6.55/2.63	   rev_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.63	   U1_aa(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.55/2.63	   app_in_gga([], X, X) -> app_out_gga([], X, X)
6.55/2.63	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.55/2.63	   U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.55/2.63	   U2_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> rev_out_aa(.(X, Xs), Ys)
6.55/2.63	   U1_ag(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.55/2.63	   app_in_ggg([], X, X) -> app_out_ggg([], X, X)
6.55/2.63	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U3_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.55/2.63	   U3_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.55/2.63	   U2_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> rev_out_ag(.(X, Xs), Ys)
6.55/2.63	
6.55/2.63	The argument filtering Pi contains the following mapping:
6.55/2.63	rev_in_ag(x1, x2)  =  rev_in_ag(x2)
6.55/2.63	
6.55/2.63	[]  =  []
6.55/2.63	
6.55/2.63	rev_out_ag(x1, x2)  =  rev_out_ag(x1, x2)
6.55/2.63	
6.55/2.63	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
6.55/2.63	
6.55/2.63	rev_in_aa(x1, x2)  =  rev_in_aa
6.55/2.63	
6.55/2.63	rev_out_aa(x1, x2)  =  rev_out_aa(x1, x2)
6.55/2.63	
6.55/2.63	U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
6.55/2.63	
6.55/2.63	U2_aa(x1, x2, x3, x4)  =  U2_aa(x2, x4)
6.55/2.63	
6.55/2.63	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.55/2.63	
6.55/2.63	.(x1, x2)  =  .(x2)
6.55/2.63	
6.55/2.63	app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
6.55/2.63	
6.55/2.63	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x2, x3, x5)
6.55/2.63	
6.55/2.63	U2_ag(x1, x2, x3, x4)  =  U2_ag(x2, x3, x4)
6.55/2.63	
6.55/2.63	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.55/2.63	
6.55/2.63	app_out_ggg(x1, x2, x3)  =  app_out_ggg(x1, x2, x3)
6.55/2.63	
6.55/2.63	U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x2, x3, x4, x5)
6.55/2.63	
6.55/2.63	
6.55/2.63	
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(46) DependencyPairsProof (EQUIVALENT)
6.55/2.63	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.55/2.63	Pi DP problem:
6.55/2.63	The TRS P consists of the following rules:
6.55/2.63	
6.55/2.63	   REV_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.63	   REV_IN_AG(.(X, Xs), Ys) -> REV_IN_AA(Xs, Zs)
6.55/2.63	   REV_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.63	   REV_IN_AA(.(X, Xs), Ys) -> REV_IN_AA(Xs, Zs)
6.55/2.63	   U1_AA(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_AA(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.55/2.63	   U1_AA(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> APP_IN_GGA(Zs, .(X, []), Ys)
6.55/2.63	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U3_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.55/2.63	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
6.55/2.63	   U1_AG(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_AG(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.55/2.63	   U1_AG(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> APP_IN_GGG(Zs, .(X, []), Ys)
6.55/2.63	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> U3_GGG(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.55/2.63	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.55/2.63	
6.55/2.63	The TRS R consists of the following rules:
6.55/2.63	
6.55/2.63	   rev_in_ag([], []) -> rev_out_ag([], [])
6.55/2.63	   rev_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.63	   rev_in_aa([], []) -> rev_out_aa([], [])
6.55/2.63	   rev_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.63	   U1_aa(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.55/2.63	   app_in_gga([], X, X) -> app_out_gga([], X, X)
6.55/2.63	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.55/2.63	   U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.55/2.63	   U2_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> rev_out_aa(.(X, Xs), Ys)
6.55/2.63	   U1_ag(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.55/2.63	   app_in_ggg([], X, X) -> app_out_ggg([], X, X)
6.55/2.63	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U3_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.55/2.63	   U3_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.55/2.63	   U2_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> rev_out_ag(.(X, Xs), Ys)
6.55/2.63	
6.55/2.63	The argument filtering Pi contains the following mapping:
6.55/2.63	rev_in_ag(x1, x2)  =  rev_in_ag(x2)
6.55/2.63	
6.55/2.63	[]  =  []
6.55/2.63	
6.55/2.63	rev_out_ag(x1, x2)  =  rev_out_ag(x1, x2)
6.55/2.63	
6.55/2.63	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
6.55/2.63	
6.55/2.63	rev_in_aa(x1, x2)  =  rev_in_aa
6.55/2.63	
6.55/2.63	rev_out_aa(x1, x2)  =  rev_out_aa(x1, x2)
6.55/2.63	
6.55/2.63	U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
6.55/2.63	
6.55/2.63	U2_aa(x1, x2, x3, x4)  =  U2_aa(x2, x4)
6.55/2.63	
6.55/2.63	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.55/2.63	
6.55/2.63	.(x1, x2)  =  .(x2)
6.55/2.63	
6.55/2.63	app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
6.55/2.63	
6.55/2.63	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x2, x3, x5)
6.55/2.63	
6.55/2.63	U2_ag(x1, x2, x3, x4)  =  U2_ag(x2, x3, x4)
6.55/2.63	
6.55/2.63	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.55/2.63	
6.55/2.63	app_out_ggg(x1, x2, x3)  =  app_out_ggg(x1, x2, x3)
6.55/2.63	
6.55/2.63	U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x2, x3, x4, x5)
6.55/2.63	
6.55/2.63	REV_IN_AG(x1, x2)  =  REV_IN_AG(x2)
6.55/2.63	
6.55/2.63	U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
6.55/2.63	
6.55/2.63	REV_IN_AA(x1, x2)  =  REV_IN_AA
6.55/2.63	
6.55/2.63	U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
6.55/2.63	
6.55/2.63	U2_AA(x1, x2, x3, x4)  =  U2_AA(x2, x4)
6.55/2.63	
6.55/2.63	APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
6.55/2.63	
6.55/2.63	U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x2, x3, x5)
6.55/2.63	
6.55/2.63	U2_AG(x1, x2, x3, x4)  =  U2_AG(x2, x3, x4)
6.55/2.63	
6.55/2.63	APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)
6.55/2.63	
6.55/2.63	U3_GGG(x1, x2, x3, x4, x5)  =  U3_GGG(x2, x3, x4, x5)
6.55/2.63	
6.55/2.63	
6.55/2.63	We have to consider all (P,R,Pi)-chains
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(47)
6.55/2.63	Obligation:
6.55/2.63	Pi DP problem:
6.55/2.63	The TRS P consists of the following rules:
6.55/2.63	
6.55/2.63	   REV_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.63	   REV_IN_AG(.(X, Xs), Ys) -> REV_IN_AA(Xs, Zs)
6.55/2.63	   REV_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.63	   REV_IN_AA(.(X, Xs), Ys) -> REV_IN_AA(Xs, Zs)
6.55/2.63	   U1_AA(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_AA(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.55/2.63	   U1_AA(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> APP_IN_GGA(Zs, .(X, []), Ys)
6.55/2.63	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U3_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.55/2.63	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
6.55/2.63	   U1_AG(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_AG(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.55/2.63	   U1_AG(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> APP_IN_GGG(Zs, .(X, []), Ys)
6.55/2.63	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> U3_GGG(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.55/2.63	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.55/2.63	
6.55/2.63	The TRS R consists of the following rules:
6.55/2.63	
6.55/2.63	   rev_in_ag([], []) -> rev_out_ag([], [])
6.55/2.63	   rev_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.63	   rev_in_aa([], []) -> rev_out_aa([], [])
6.55/2.63	   rev_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.63	   U1_aa(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.55/2.63	   app_in_gga([], X, X) -> app_out_gga([], X, X)
6.55/2.63	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.55/2.63	   U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.55/2.63	   U2_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> rev_out_aa(.(X, Xs), Ys)
6.55/2.63	   U1_ag(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.55/2.63	   app_in_ggg([], X, X) -> app_out_ggg([], X, X)
6.55/2.63	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U3_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.55/2.63	   U3_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.55/2.63	   U2_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> rev_out_ag(.(X, Xs), Ys)
6.55/2.63	
6.55/2.63	The argument filtering Pi contains the following mapping:
6.55/2.63	rev_in_ag(x1, x2)  =  rev_in_ag(x2)
6.55/2.63	
6.55/2.63	[]  =  []
6.55/2.63	
6.55/2.63	rev_out_ag(x1, x2)  =  rev_out_ag(x1, x2)
6.55/2.63	
6.55/2.63	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
6.55/2.63	
6.55/2.63	rev_in_aa(x1, x2)  =  rev_in_aa
6.55/2.63	
6.55/2.63	rev_out_aa(x1, x2)  =  rev_out_aa(x1, x2)
6.55/2.63	
6.55/2.63	U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
6.55/2.63	
6.55/2.63	U2_aa(x1, x2, x3, x4)  =  U2_aa(x2, x4)
6.55/2.63	
6.55/2.63	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.55/2.63	
6.55/2.63	.(x1, x2)  =  .(x2)
6.55/2.63	
6.55/2.63	app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
6.55/2.63	
6.55/2.63	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x2, x3, x5)
6.55/2.63	
6.55/2.63	U2_ag(x1, x2, x3, x4)  =  U2_ag(x2, x3, x4)
6.55/2.63	
6.55/2.63	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.55/2.63	
6.55/2.63	app_out_ggg(x1, x2, x3)  =  app_out_ggg(x1, x2, x3)
6.55/2.63	
6.55/2.63	U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x2, x3, x4, x5)
6.55/2.63	
6.55/2.63	REV_IN_AG(x1, x2)  =  REV_IN_AG(x2)
6.55/2.63	
6.55/2.63	U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
6.55/2.63	
6.55/2.63	REV_IN_AA(x1, x2)  =  REV_IN_AA
6.55/2.63	
6.55/2.63	U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
6.55/2.63	
6.55/2.63	U2_AA(x1, x2, x3, x4)  =  U2_AA(x2, x4)
6.55/2.63	
6.55/2.63	APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
6.55/2.63	
6.55/2.63	U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x2, x3, x5)
6.55/2.63	
6.55/2.63	U2_AG(x1, x2, x3, x4)  =  U2_AG(x2, x3, x4)
6.55/2.63	
6.55/2.63	APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)
6.55/2.63	
6.55/2.63	U3_GGG(x1, x2, x3, x4, x5)  =  U3_GGG(x2, x3, x4, x5)
6.55/2.63	
6.55/2.63	
6.55/2.63	We have to consider all (P,R,Pi)-chains
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(48) DependencyGraphProof (EQUIVALENT)
6.55/2.63	The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 9 less nodes.
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(49)
6.55/2.63	Complex Obligation (AND)
6.55/2.63	
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(50)
6.55/2.63	Obligation:
6.55/2.63	Pi DP problem:
6.55/2.63	The TRS P consists of the following rules:
6.55/2.63	
6.55/2.63	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.55/2.63	
6.55/2.63	The TRS R consists of the following rules:
6.55/2.63	
6.55/2.63	   rev_in_ag([], []) -> rev_out_ag([], [])
6.55/2.63	   rev_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.63	   rev_in_aa([], []) -> rev_out_aa([], [])
6.55/2.63	   rev_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.63	   U1_aa(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.55/2.63	   app_in_gga([], X, X) -> app_out_gga([], X, X)
6.55/2.63	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.55/2.63	   U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.55/2.63	   U2_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> rev_out_aa(.(X, Xs), Ys)
6.55/2.63	   U1_ag(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.55/2.63	   app_in_ggg([], X, X) -> app_out_ggg([], X, X)
6.55/2.63	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U3_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.55/2.63	   U3_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.55/2.63	   U2_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> rev_out_ag(.(X, Xs), Ys)
6.55/2.63	
6.55/2.63	The argument filtering Pi contains the following mapping:
6.55/2.63	rev_in_ag(x1, x2)  =  rev_in_ag(x2)
6.55/2.63	
6.55/2.63	[]  =  []
6.55/2.63	
6.55/2.63	rev_out_ag(x1, x2)  =  rev_out_ag(x1, x2)
6.55/2.63	
6.55/2.63	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
6.55/2.63	
6.55/2.63	rev_in_aa(x1, x2)  =  rev_in_aa
6.55/2.63	
6.55/2.63	rev_out_aa(x1, x2)  =  rev_out_aa(x1, x2)
6.55/2.63	
6.55/2.63	U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
6.55/2.63	
6.55/2.63	U2_aa(x1, x2, x3, x4)  =  U2_aa(x2, x4)
6.55/2.63	
6.55/2.63	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.55/2.63	
6.55/2.63	.(x1, x2)  =  .(x2)
6.55/2.63	
6.55/2.63	app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
6.55/2.63	
6.55/2.63	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x2, x3, x5)
6.55/2.63	
6.55/2.63	U2_ag(x1, x2, x3, x4)  =  U2_ag(x2, x3, x4)
6.55/2.63	
6.55/2.63	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.55/2.63	
6.55/2.63	app_out_ggg(x1, x2, x3)  =  app_out_ggg(x1, x2, x3)
6.55/2.63	
6.55/2.63	U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x2, x3, x4, x5)
6.55/2.63	
6.55/2.63	APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)
6.55/2.63	
6.55/2.63	
6.55/2.63	We have to consider all (P,R,Pi)-chains
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(51) UsableRulesProof (EQUIVALENT)
6.55/2.63	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(52)
6.55/2.63	Obligation:
6.55/2.63	Pi DP problem:
6.55/2.63	The TRS P consists of the following rules:
6.55/2.63	
6.55/2.63	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.55/2.63	
6.55/2.63	R is empty.
6.55/2.63	The argument filtering Pi contains the following mapping:
6.55/2.63	.(x1, x2)  =  .(x2)
6.55/2.63	
6.55/2.63	APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)
6.55/2.63	
6.55/2.63	
6.55/2.63	We have to consider all (P,R,Pi)-chains
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(53) PiDPToQDPProof (SOUND)
6.55/2.63	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(54)
6.55/2.63	Obligation:
6.55/2.63	Q DP problem:
6.55/2.63	The TRS P consists of the following rules:
6.55/2.63	
6.55/2.63	   APP_IN_GGG(.(Xs), Ys, .(Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.55/2.63	
6.55/2.63	R is empty.
6.55/2.63	Q is empty.
6.55/2.63	We have to consider all (P,Q,R)-chains.
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(55) QDPSizeChangeProof (EQUIVALENT)
6.55/2.63	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. 
6.55/2.63	
6.55/2.63	From the DPs we obtained the following set of size-change graphs:
6.55/2.63	*APP_IN_GGG(.(Xs), Ys, .(Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.55/2.63	The graph contains the following edges 1 > 1, 2 >= 2, 3 > 3
6.55/2.63	
6.55/2.63	
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(56)
6.55/2.63	YES
6.55/2.63	
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(57)
6.55/2.63	Obligation:
6.55/2.63	Pi DP problem:
6.55/2.63	The TRS P consists of the following rules:
6.55/2.63	
6.55/2.63	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
6.55/2.63	
6.55/2.63	The TRS R consists of the following rules:
6.55/2.63	
6.55/2.63	   rev_in_ag([], []) -> rev_out_ag([], [])
6.55/2.63	   rev_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.63	   rev_in_aa([], []) -> rev_out_aa([], [])
6.55/2.63	   rev_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.63	   U1_aa(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.55/2.63	   app_in_gga([], X, X) -> app_out_gga([], X, X)
6.55/2.63	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.55/2.63	   U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.55/2.63	   U2_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> rev_out_aa(.(X, Xs), Ys)
6.55/2.63	   U1_ag(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.55/2.63	   app_in_ggg([], X, X) -> app_out_ggg([], X, X)
6.55/2.63	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U3_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.55/2.63	   U3_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.55/2.63	   U2_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> rev_out_ag(.(X, Xs), Ys)
6.55/2.63	
6.55/2.63	The argument filtering Pi contains the following mapping:
6.55/2.63	rev_in_ag(x1, x2)  =  rev_in_ag(x2)
6.55/2.63	
6.55/2.63	[]  =  []
6.55/2.63	
6.55/2.63	rev_out_ag(x1, x2)  =  rev_out_ag(x1, x2)
6.55/2.63	
6.55/2.63	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
6.55/2.63	
6.55/2.63	rev_in_aa(x1, x2)  =  rev_in_aa
6.55/2.63	
6.55/2.63	rev_out_aa(x1, x2)  =  rev_out_aa(x1, x2)
6.55/2.63	
6.55/2.63	U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
6.55/2.63	
6.55/2.63	U2_aa(x1, x2, x3, x4)  =  U2_aa(x2, x4)
6.55/2.63	
6.55/2.63	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.55/2.63	
6.55/2.63	.(x1, x2)  =  .(x2)
6.55/2.63	
6.55/2.63	app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
6.55/2.63	
6.55/2.63	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x2, x3, x5)
6.55/2.63	
6.55/2.63	U2_ag(x1, x2, x3, x4)  =  U2_ag(x2, x3, x4)
6.55/2.63	
6.55/2.63	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.55/2.63	
6.55/2.63	app_out_ggg(x1, x2, x3)  =  app_out_ggg(x1, x2, x3)
6.55/2.63	
6.55/2.63	U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x2, x3, x4, x5)
6.55/2.63	
6.55/2.63	APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
6.55/2.63	
6.55/2.63	
6.55/2.63	We have to consider all (P,R,Pi)-chains
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(58) UsableRulesProof (EQUIVALENT)
6.55/2.63	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(59)
6.55/2.63	Obligation:
6.55/2.63	Pi DP problem:
6.55/2.63	The TRS P consists of the following rules:
6.55/2.63	
6.55/2.63	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
6.55/2.63	
6.55/2.63	R is empty.
6.55/2.63	The argument filtering Pi contains the following mapping:
6.55/2.63	.(x1, x2)  =  .(x2)
6.55/2.63	
6.55/2.63	APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
6.55/2.63	
6.55/2.63	
6.55/2.63	We have to consider all (P,R,Pi)-chains
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(60) PiDPToQDPProof (SOUND)
6.55/2.63	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(61)
6.55/2.63	Obligation:
6.55/2.63	Q DP problem:
6.55/2.63	The TRS P consists of the following rules:
6.55/2.63	
6.55/2.63	   APP_IN_GGA(.(Xs), Ys) -> APP_IN_GGA(Xs, Ys)
6.55/2.63	
6.55/2.63	R is empty.
6.55/2.63	Q is empty.
6.55/2.63	We have to consider all (P,Q,R)-chains.
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(62) QDPSizeChangeProof (EQUIVALENT)
6.55/2.63	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. 
6.55/2.63	
6.55/2.63	From the DPs we obtained the following set of size-change graphs:
6.55/2.63	*APP_IN_GGA(.(Xs), Ys) -> APP_IN_GGA(Xs, Ys)
6.55/2.63	The graph contains the following edges 1 > 1, 2 >= 2
6.55/2.63	
6.55/2.63	
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(63)
6.55/2.63	YES
6.55/2.63	
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(64)
6.55/2.63	Obligation:
6.55/2.63	Pi DP problem:
6.55/2.63	The TRS P consists of the following rules:
6.55/2.63	
6.55/2.63	   REV_IN_AA(.(X, Xs), Ys) -> REV_IN_AA(Xs, Zs)
6.55/2.63	
6.55/2.63	The TRS R consists of the following rules:
6.55/2.63	
6.55/2.63	   rev_in_ag([], []) -> rev_out_ag([], [])
6.55/2.63	   rev_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.63	   rev_in_aa([], []) -> rev_out_aa([], [])
6.55/2.63	   rev_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, rev_in_aa(Xs, Zs))
6.55/2.63	   U1_aa(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.55/2.63	   app_in_gga([], X, X) -> app_out_gga([], X, X)
6.55/2.63	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.55/2.63	   U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.55/2.63	   U2_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> rev_out_aa(.(X, Xs), Ys)
6.55/2.63	   U1_ag(X, Xs, Ys, rev_out_aa(Xs, Zs)) -> U2_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.55/2.63	   app_in_ggg([], X, X) -> app_out_ggg([], X, X)
6.55/2.63	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U3_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.55/2.63	   U3_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.55/2.63	   U2_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> rev_out_ag(.(X, Xs), Ys)
6.55/2.63	
6.55/2.63	The argument filtering Pi contains the following mapping:
6.55/2.63	rev_in_ag(x1, x2)  =  rev_in_ag(x2)
6.55/2.63	
6.55/2.63	[]  =  []
6.55/2.63	
6.55/2.63	rev_out_ag(x1, x2)  =  rev_out_ag(x1, x2)
6.55/2.63	
6.55/2.63	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
6.55/2.63	
6.55/2.63	rev_in_aa(x1, x2)  =  rev_in_aa
6.55/2.63	
6.55/2.63	rev_out_aa(x1, x2)  =  rev_out_aa(x1, x2)
6.55/2.63	
6.55/2.63	U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)
6.55/2.63	
6.55/2.63	U2_aa(x1, x2, x3, x4)  =  U2_aa(x2, x4)
6.55/2.63	
6.55/2.63	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.55/2.63	
6.55/2.63	.(x1, x2)  =  .(x2)
6.55/2.63	
6.55/2.63	app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
6.55/2.63	
6.55/2.63	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x2, x3, x5)
6.55/2.63	
6.55/2.63	U2_ag(x1, x2, x3, x4)  =  U2_ag(x2, x3, x4)
6.55/2.63	
6.55/2.63	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.55/2.63	
6.55/2.63	app_out_ggg(x1, x2, x3)  =  app_out_ggg(x1, x2, x3)
6.55/2.63	
6.55/2.63	U3_ggg(x1, x2, x3, x4, x5)  =  U3_ggg(x2, x3, x4, x5)
6.55/2.63	
6.55/2.63	REV_IN_AA(x1, x2)  =  REV_IN_AA
6.55/2.63	
6.55/2.63	
6.55/2.63	We have to consider all (P,R,Pi)-chains
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(65) UsableRulesProof (EQUIVALENT)
6.55/2.63	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(66)
6.55/2.63	Obligation:
6.55/2.63	Pi DP problem:
6.55/2.63	The TRS P consists of the following rules:
6.55/2.63	
6.55/2.63	   REV_IN_AA(.(X, Xs), Ys) -> REV_IN_AA(Xs, Zs)
6.55/2.63	
6.55/2.63	R is empty.
6.55/2.63	The argument filtering Pi contains the following mapping:
6.55/2.63	.(x1, x2)  =  .(x2)
6.55/2.63	
6.55/2.63	REV_IN_AA(x1, x2)  =  REV_IN_AA
6.55/2.63	
6.55/2.63	
6.55/2.63	We have to consider all (P,R,Pi)-chains
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(67) PiDPToQDPProof (SOUND)
6.55/2.63	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(68)
6.55/2.63	Obligation:
6.55/2.63	Q DP problem:
6.55/2.63	The TRS P consists of the following rules:
6.55/2.63	
6.55/2.63	   REV_IN_AA -> REV_IN_AA
6.55/2.63	
6.55/2.63	R is empty.
6.55/2.63	Q is empty.
6.55/2.63	We have to consider all (P,Q,R)-chains.
6.55/2.63	----------------------------------------
6.55/2.63	
6.55/2.63	(69) PrologToDTProblemTransformerProof (SOUND)
6.55/2.63	Built DT problem from termination graph DT10.
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6.55/2.64	                }
6.55/2.64	            },
6.55/2.64	            "86": {
6.55/2.64	                "goal": [{
6.55/2.64	                    "clause": 3,
6.55/2.64	                    "scope": 3,
6.55/2.64	                    "term": "(app ([]) (. T9 ([])) ([]))"
6.55/2.64	                }],
6.55/2.64	                "kb": {
6.55/2.64	                    "nonunifying": [],
6.55/2.64	                    "intvars": {},
6.55/2.64	                    "arithmetic": {
6.55/2.64	                        "type": "PlainIntegerRelationState",
6.55/2.64	                        "relations": []
6.55/2.64	                    },
6.55/2.64	                    "ground": [],
6.55/2.64	                    "free": [],
6.55/2.64	                    "exprvars": []
6.55/2.64	                }
6.55/2.64	            },
6.55/2.64	            "87": {
6.55/2.64	                "goal": [],
6.55/2.64	                "kb": {
6.55/2.64	                    "nonunifying": [],
6.55/2.64	                    "intvars": {},
6.55/2.64	                    "arithmetic": {
6.55/2.64	                        "type": "PlainIntegerRelationState",
6.55/2.64	                        "relations": []
6.55/2.64	                    },
6.55/2.64	                    "ground": [],
6.55/2.64	                    "free": [],
6.55/2.64	                    "exprvars": []
6.55/2.64	                }
6.55/2.64	            }
6.55/2.64	        },
6.55/2.64	        "edges": [
6.55/2.64	            {
6.55/2.64	                "from": 2,
6.55/2.64	                "to": 8,
6.55/2.64	                "label": "CASE"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 8,
6.55/2.64	                "to": 16,
6.55/2.64	                "label": "EVAL with clause\nrev([], []).\nand substitutionT1 -> [],\nT2 -> []"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 8,
6.55/2.64	                "to": 17,
6.55/2.64	                "label": "EVAL-BACKTRACK"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 16,
6.55/2.64	                "to": 18,
6.55/2.64	                "label": "SUCCESS"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 17,
6.55/2.64	                "to": 265,
6.55/2.64	                "label": "EVAL with clause\nrev(.(X81, X82), X83) :- ','(rev(X82, X84), app(X84, .(X81, []), X83)).\nand substitutionX81 -> T74,\nX82 -> T73,\nT1 -> .(T74, T73),\nT2 -> T72,\nX83 -> T72,\nT71 -> T73,\nT70 -> T74"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 17,
6.55/2.64	                "to": 266,
6.55/2.64	                "label": "EVAL-BACKTRACK"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 18,
6.55/2.64	                "to": 38,
6.55/2.64	                "label": "EVAL with clause\nrev(.(X4, X5), X6) :- ','(rev(X5, X7), app(X7, .(X4, []), X6)).\nand substitutionX4 -> T8,\nX5 -> T7,\nT1 -> .(T8, T7),\nX6 -> [],\nT6 -> T7,\nT5 -> T8"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 18,
6.55/2.64	                "to": 45,
6.55/2.64	                "label": "EVAL-BACKTRACK"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 38,
6.55/2.64	                "to": 58,
6.55/2.64	                "label": "CASE"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 58,
6.55/2.64	                "to": 73,
6.55/2.64	                "label": "PARALLEL"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 58,
6.55/2.64	                "to": 75,
6.55/2.64	                "label": "PARALLEL"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 73,
6.55/2.64	                "to": 76,
6.55/2.64	                "label": "EVAL with clause\nrev([], []).\nand substitutionT7 -> [],\nX7 -> [],\nT8 -> T9"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 73,
6.55/2.64	                "to": 77,
6.55/2.64	                "label": "EVAL-BACKTRACK"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 75,
6.55/2.64	                "to": 90,
6.55/2.64	                "label": "EVAL with clause\nrev(.(X24, X25), X26) :- ','(rev(X25, X27), app(X27, .(X24, []), X26)).\nand substitutionX24 -> T20,\nX25 -> T19,\nT7 -> .(T20, T19),\nX7 -> X28,\nX26 -> X28,\nT18 -> T19,\nT17 -> T20,\nT8 -> T21"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 75,
6.55/2.64	                "to": 91,
6.55/2.64	                "label": "EVAL-BACKTRACK"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 76,
6.55/2.64	                "to": 85,
6.55/2.64	                "label": "CASE"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 85,
6.55/2.64	                "to": 86,
6.55/2.64	                "label": "BACKTRACK\nfor clause: app([], X, X)because of non-unification"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 86,
6.55/2.64	                "to": 87,
6.55/2.64	                "label": "BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 90,
6.55/2.64	                "to": 112,
6.55/2.64	                "label": "SPLIT 1"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 90,
6.55/2.64	                "to": 113,
6.55/2.64	                "label": "SPLIT 2\nreplacements:X27 -> T22,\nT20 -> T23,\nT21 -> T24"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 112,
6.55/2.64	                "to": 114,
6.55/2.64	                "label": "CASE"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 113,
6.55/2.64	                "to": 260,
6.55/2.64	                "label": "SPLIT 1"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 113,
6.55/2.64	                "to": 261,
6.55/2.64	                "label": "SPLIT 2\nreplacements:X28 -> T57,\nT24 -> T58"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 114,
6.55/2.64	                "to": 115,
6.55/2.64	                "label": "PARALLEL"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 114,
6.55/2.64	                "to": 116,
6.55/2.64	                "label": "PARALLEL"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 115,
6.55/2.64	                "to": 117,
6.55/2.64	                "label": "EVAL with clause\nrev([], []).\nand substitutionT19 -> [],\nX27 -> []"
6.55/2.64	            },
6.55/2.64	            {
6.55/2.64	                "from": 115,
6.55/2.64	                "to": 118,
6.55/2.64	                "label": "EVAL-BACKTRACK"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 116,
6.91/2.64	                "to": 130,
6.91/2.64	                "label": "EVAL with clause\nrev(.(X38, X39), X40) :- ','(rev(X39, X41), app(X41, .(X38, []), X40)).\nand substitutionX38 -> T32,\nX39 -> T31,\nT19 -> .(T32, T31),\nX27 -> X42,\nX40 -> X42,\nT30 -> T31,\nT29 -> T32"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 116,
6.91/2.64	                "to": 131,
6.91/2.64	                "label": "EVAL-BACKTRACK"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 117,
6.91/2.64	                "to": 119,
6.91/2.64	                "label": "SUCCESS"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 130,
6.91/2.64	                "to": 132,
6.91/2.64	                "label": "SPLIT 1"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 130,
6.91/2.64	                "to": 133,
6.91/2.64	                "label": "SPLIT 2\nreplacements:X41 -> T33,\nT32 -> T34"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 132,
6.91/2.64	                "to": 112,
6.91/2.64	                "label": "INSTANCE with matching:\nT19 -> T31\nX27 -> X41"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 133,
6.91/2.64	                "to": 224,
6.91/2.64	                "label": "CASE"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 224,
6.91/2.64	                "to": 225,
6.91/2.64	                "label": "PARALLEL"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 224,
6.91/2.64	                "to": 226,
6.91/2.64	                "label": "PARALLEL"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 225,
6.91/2.64	                "to": 227,
6.91/2.64	                "label": "EVAL with clause\napp([], X49, X49).\nand substitutionT33 -> [],\nT34 -> T41,\nX49 -> .(T41, []),\nX42 -> .(T41, [])"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 225,
6.91/2.64	                "to": 228,
6.91/2.64	                "label": "EVAL-BACKTRACK"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 226,
6.91/2.64	                "to": 258,
6.91/2.64	                "label": "EVAL with clause\napp(.(X60, X61), X62, .(X60, X63)) :- app(X61, X62, X63).\nand substitutionX60 -> T48,\nX61 -> T51,\nT33 -> .(T48, T51),\nT34 -> T52,\nX62 -> .(T52, []),\nX63 -> X64,\nX42 -> .(T48, X64),\nT49 -> T51,\nT50 -> T52"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 226,
6.91/2.64	                "to": 259,
6.91/2.64	                "label": "EVAL-BACKTRACK"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 227,
6.91/2.64	                "to": 229,
6.91/2.64	                "label": "SUCCESS"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 258,
6.91/2.64	                "to": 133,
6.91/2.64	                "label": "INSTANCE with matching:\nT33 -> T51\nT34 -> T52\nX42 -> X64"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 260,
6.91/2.64	                "to": 133,
6.91/2.64	                "label": "INSTANCE with matching:\nT33 -> T22\nT34 -> T23\nX42 -> X28"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 261,
6.91/2.64	                "to": 262,
6.91/2.64	                "label": "CASE"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 262,
6.91/2.64	                "to": 263,
6.91/2.64	                "label": "BACKTRACK\nfor clause: app([], X, X)because of non-unification"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 263,
6.91/2.64	                "to": 264,
6.91/2.64	                "label": "BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 265,
6.91/2.64	                "to": 267,
6.91/2.64	                "label": "CASE"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 267,
6.91/2.64	                "to": 268,
6.91/2.64	                "label": "PARALLEL"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 267,
6.91/2.64	                "to": 269,
6.91/2.64	                "label": "PARALLEL"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 268,
6.91/2.64	                "to": 270,
6.91/2.64	                "label": "EVAL with clause\nrev([], []).\nand substitutionT73 -> [],\nX84 -> [],\nT74 -> T75"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 268,
6.91/2.64	                "to": 271,
6.91/2.64	                "label": "EVAL-BACKTRACK"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 269,
6.91/2.64	                "to": 279,
6.91/2.64	                "label": "EVAL with clause\nrev(.(X105, X106), X107) :- ','(rev(X106, X108), app(X108, .(X105, []), X107)).\nand substitutionX105 -> T90,\nX106 -> T89,\nT73 -> .(T90, T89),\nX84 -> X109,\nX107 -> X109,\nT88 -> T89,\nT87 -> T90,\nT74 -> T91"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 269,
6.91/2.64	                "to": 280,
6.91/2.64	                "label": "EVAL-BACKTRACK"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 270,
6.91/2.64	                "to": 272,
6.91/2.64	                "label": "CASE"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 272,
6.91/2.64	                "to": 273,
6.91/2.64	                "label": "PARALLEL"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 272,
6.91/2.64	                "to": 274,
6.91/2.64	                "label": "PARALLEL"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 273,
6.91/2.64	                "to": 275,
6.91/2.64	                "label": "EVAL with clause\napp([], X91, X91).\nand substitutionT75 -> T82,\nX91 -> .(T82, []),\nT72 -> .(T82, [])"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 273,
6.91/2.64	                "to": 276,
6.91/2.64	                "label": "EVAL-BACKTRACK"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 274,
6.91/2.64	                "to": 278,
6.91/2.64	                "label": "BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 275,
6.91/2.64	                "to": 277,
6.91/2.64	                "label": "SUCCESS"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 279,
6.91/2.64	                "to": 281,
6.91/2.64	                "label": "SPLIT 1"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 279,
6.91/2.64	                "to": 282,
6.91/2.64	                "label": "SPLIT 2\nreplacements:X108 -> T92,\nT90 -> T93,\nT91 -> T94,\nT1 -> T95"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 281,
6.91/2.64	                "to": 112,
6.91/2.64	                "label": "INSTANCE with matching:\nT19 -> T89\nX27 -> X108"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 282,
6.91/2.64	                "to": 283,
6.91/2.64	                "label": "SPLIT 1"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 282,
6.91/2.64	                "to": 284,
6.91/2.64	                "label": "SPLIT 2\nreplacements:X109 -> T98,\nT94 -> T99,\nT95 -> T100"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 283,
6.91/2.64	                "to": 133,
6.91/2.64	                "label": "INSTANCE with matching:\nT33 -> T92\nT34 -> T93\nX42 -> X109"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 284,
6.91/2.64	                "to": 285,
6.91/2.64	                "label": "CASE"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 285,
6.91/2.64	                "to": 291,
6.91/2.64	                "label": "PARALLEL"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 285,
6.91/2.64	                "to": 292,
6.91/2.64	                "label": "PARALLEL"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 291,
6.91/2.64	                "to": 293,
6.91/2.64	                "label": "EVAL with clause\napp([], X120, X120).\nand substitutionT98 -> [],\nT99 -> T109,\nX120 -> .(T109, []),\nT72 -> .(T109, [])"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 291,
6.91/2.64	                "to": 294,
6.91/2.64	                "label": "EVAL-BACKTRACK"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 292,
6.91/2.64	                "to": 305,
6.91/2.64	                "label": "EVAL with clause\napp(.(X129, X130), X131, .(X129, X132)) :- app(X130, X131, X132).\nand substitutionX129 -> T118,\nX130 -> T122,\nT98 -> .(T118, T122),\nT99 -> T123,\nX131 -> .(T123, []),\nX132 -> T121,\nT72 -> .(T118, T121),\nT119 -> T122,\nT120 -> T123"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 292,
6.91/2.64	                "to": 306,
6.91/2.64	                "label": "EVAL-BACKTRACK"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 293,
6.91/2.64	                "to": 295,
6.91/2.64	                "label": "SUCCESS"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 305,
6.91/2.64	                "to": 307,
6.91/2.64	                "label": "CASE"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 307,
6.91/2.64	                "to": 308,
6.91/2.64	                "label": "PARALLEL"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 307,
6.91/2.64	                "to": 309,
6.91/2.64	                "label": "PARALLEL"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 308,
6.91/2.64	                "to": 310,
6.91/2.64	                "label": "EVAL with clause\napp([], X139, X139).\nand substitutionT122 -> [],\nT123 -> T130,\nX139 -> .(T130, []),\nT121 -> .(T130, [])"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 308,
6.91/2.64	                "to": 311,
6.91/2.64	                "label": "EVAL-BACKTRACK"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 309,
6.91/2.64	                "to": 325,
6.91/2.64	                "label": "EVAL with clause\napp(.(X148, X149), X150, .(X148, X151)) :- app(X149, X150, X151).\nand substitutionX148 -> T139,\nX149 -> T143,\nT122 -> .(T139, T143),\nT123 -> T144,\nX150 -> .(T144, []),\nX151 -> T142,\nT121 -> .(T139, T142),\nT140 -> T143,\nT141 -> T144"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 309,
6.91/2.64	                "to": 326,
6.91/2.64	                "label": "EVAL-BACKTRACK"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 310,
6.91/2.64	                "to": 312,
6.91/2.64	                "label": "SUCCESS"
6.91/2.64	            },
6.91/2.64	            {
6.91/2.64	                "from": 325,
6.91/2.64	                "to": 305,
6.91/2.64	                "label": "INSTANCE with matching:\nT122 -> T143\nT123 -> T144\nT121 -> T142"
6.91/2.64	            }
6.91/2.64	        ],
6.91/2.64	        "type": "Graph"
6.91/2.64	    }
6.91/2.64	}
6.91/2.64	
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(70)
6.91/2.64	Obligation:
6.91/2.64	Triples:
6.91/2.64	
6.91/2.64	revA(.(X1, X2), X3) :- revA(X2, X4).
6.91/2.64	revA(.(X1, X2), X3) :- ','(revcA(X2, X4), appB(X4, X1, X3)).
6.91/2.64	appB(.(X1, X2), X3, .(X1, X4)) :- appB(X2, X3, X4).
6.91/2.64	appC(.(X1, X2), X3, .(X1, X4)) :- appC(X2, X3, X4).
6.91/2.64	revD(.(X1, .(X2, X3)), []) :- revA(X3, X4).
6.91/2.64	revD(.(X1, .(X2, X3)), []) :- ','(revcA(X3, X4), appB(X4, X2, X5)).
6.91/2.64	revD(.(X1, .(X2, X3)), X4) :- revA(X3, X5).
6.91/2.64	revD(.(X1, .(X2, X3)), X4) :- ','(revcA(X3, X5), appB(X5, X2, X6)).
6.91/2.64	revD(.(X1, .(X2, X3)), .(X4, X5)) :- ','(revcA(X3, X6), ','(appcB(X6, X2, .(X4, X7)), appC(X7, X1, X5))).
6.91/2.64	
6.91/2.64	Clauses:
6.91/2.64	
6.91/2.64	revcA([], []).
6.91/2.64	revcA(.(X1, X2), X3) :- ','(revcA(X2, X4), appcB(X4, X1, X3)).
6.91/2.64	appcB([], X1, .(X1, [])).
6.91/2.64	appcB(.(X1, X2), X3, .(X1, X4)) :- appcB(X2, X3, X4).
6.91/2.64	appcC([], X1, .(X1, [])).
6.91/2.64	appcC(.(X1, X2), X3, .(X1, X4)) :- appcC(X2, X3, X4).
6.91/2.64	
6.91/2.64	Afs:
6.91/2.64	
6.91/2.64	revD(x1, x2)  =  revD(x2)
6.91/2.64	
6.91/2.64	
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(71) TriplesToPiDPProof (SOUND)
6.91/2.64	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.91/2.64	
6.91/2.64	revD_in_2: (f,b)
6.91/2.64	
6.91/2.64	revA_in_2: (f,f)
6.91/2.64	
6.91/2.64	revcA_in_2: (f,f)
6.91/2.64	
6.91/2.64	appcB_in_3: (b,f,f)
6.91/2.64	
6.91/2.64	appB_in_3: (b,f,f)
6.91/2.64	
6.91/2.64	appC_in_3: (b,f,b)
6.91/2.64	
6.91/2.64	Transforming TRIPLES into the following Term Rewriting System:
6.91/2.64	
6.91/2.64	Pi DP problem:
6.91/2.64	The TRS P consists of the following rules:
6.91/2.64	
6.91/2.64	   REVD_IN_AG(.(X1, .(X2, X3)), []) -> U6_AG(X1, X2, X3, revA_in_aa(X3, X4))
6.91/2.64	   REVD_IN_AG(.(X1, .(X2, X3)), []) -> REVA_IN_AA(X3, X4)
6.91/2.64	   REVA_IN_AA(.(X1, X2), X3) -> U1_AA(X1, X2, X3, revA_in_aa(X2, X4))
6.91/2.64	   REVA_IN_AA(.(X1, X2), X3) -> REVA_IN_AA(X2, X4)
6.91/2.64	   REVA_IN_AA(.(X1, X2), X3) -> U2_AA(X1, X2, X3, revcA_in_aa(X2, X4))
6.91/2.64	   U2_AA(X1, X2, X3, revcA_out_aa(X2, X4)) -> U3_AA(X1, X2, X3, appB_in_gaa(X4, X1, X3))
6.91/2.64	   U2_AA(X1, X2, X3, revcA_out_aa(X2, X4)) -> APPB_IN_GAA(X4, X1, X3)
6.91/2.64	   APPB_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> U4_GAA(X1, X2, X3, X4, appB_in_gaa(X2, X3, X4))
6.91/2.64	   APPB_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPB_IN_GAA(X2, X3, X4)
6.91/2.64	   REVD_IN_AG(.(X1, .(X2, X3)), []) -> U7_AG(X1, X2, X3, revcA_in_aa(X3, X4))
6.91/2.64	   U7_AG(X1, X2, X3, revcA_out_aa(X3, X4)) -> U8_AG(X1, X2, X3, appB_in_gaa(X4, X2, X5))
6.91/2.64	   U7_AG(X1, X2, X3, revcA_out_aa(X3, X4)) -> APPB_IN_GAA(X4, X2, X5)
6.91/2.64	   REVD_IN_AG(.(X1, .(X2, X3)), X4) -> U9_AG(X1, X2, X3, X4, revA_in_aa(X3, X5))
6.91/2.64	   REVD_IN_AG(.(X1, .(X2, X3)), X4) -> REVA_IN_AA(X3, X5)
6.91/2.64	   REVD_IN_AG(.(X1, .(X2, X3)), X4) -> U10_AG(X1, X2, X3, X4, revcA_in_aa(X3, X5))
6.91/2.64	   U10_AG(X1, X2, X3, X4, revcA_out_aa(X3, X5)) -> U11_AG(X1, X2, X3, X4, appB_in_gaa(X5, X2, X6))
6.91/2.64	   U10_AG(X1, X2, X3, X4, revcA_out_aa(X3, X5)) -> APPB_IN_GAA(X5, X2, X6)
6.91/2.64	   REVD_IN_AG(.(X1, .(X2, X3)), .(X4, X5)) -> U12_AG(X1, X2, X3, X4, X5, revcA_in_aa(X3, X6))
6.91/2.64	   U12_AG(X1, X2, X3, X4, X5, revcA_out_aa(X3, X6)) -> U13_AG(X1, X2, X3, X4, X5, appcB_in_gaa(X6, X2, .(X4, X7)))
6.91/2.64	   U13_AG(X1, X2, X3, X4, X5, appcB_out_gaa(X6, X2, .(X4, X7))) -> U14_AG(X1, X2, X3, X4, X5, appC_in_gag(X7, X1, X5))
6.91/2.64	   U13_AG(X1, X2, X3, X4, X5, appcB_out_gaa(X6, X2, .(X4, X7))) -> APPC_IN_GAG(X7, X1, X5)
6.91/2.64	   APPC_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> U5_GAG(X1, X2, X3, X4, appC_in_gag(X2, X3, X4))
6.91/2.64	   APPC_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPC_IN_GAG(X2, X3, X4)
6.91/2.64	
6.91/2.64	The TRS R consists of the following rules:
6.91/2.64	
6.91/2.64	   revcA_in_aa([], []) -> revcA_out_aa([], [])
6.91/2.64	   revcA_in_aa(.(X1, X2), X3) -> U16_aa(X1, X2, X3, revcA_in_aa(X2, X4))
6.91/2.64	   U16_aa(X1, X2, X3, revcA_out_aa(X2, X4)) -> U17_aa(X1, X2, X3, appcB_in_gaa(X4, X1, X3))
6.91/2.64	   appcB_in_gaa([], X1, .(X1, [])) -> appcB_out_gaa([], X1, .(X1, []))
6.91/2.64	   appcB_in_gaa(.(X1, X2), X3, .(X1, X4)) -> U18_gaa(X1, X2, X3, X4, appcB_in_gaa(X2, X3, X4))
6.91/2.64	   U18_gaa(X1, X2, X3, X4, appcB_out_gaa(X2, X3, X4)) -> appcB_out_gaa(.(X1, X2), X3, .(X1, X4))
6.91/2.64	   U17_aa(X1, X2, X3, appcB_out_gaa(X4, X1, X3)) -> revcA_out_aa(.(X1, X2), X3)
6.91/2.64	
6.91/2.64	The argument filtering Pi contains the following mapping:
6.91/2.64	[]  =  []
6.91/2.64	
6.91/2.64	revA_in_aa(x1, x2)  =  revA_in_aa
6.91/2.64	
6.91/2.64	revcA_in_aa(x1, x2)  =  revcA_in_aa
6.91/2.64	
6.91/2.64	revcA_out_aa(x1, x2)  =  revcA_out_aa(x1, x2)
6.91/2.64	
6.91/2.64	U16_aa(x1, x2, x3, x4)  =  U16_aa(x4)
6.91/2.64	
6.91/2.64	U17_aa(x1, x2, x3, x4)  =  U17_aa(x2, x4)
6.91/2.64	
6.91/2.64	appcB_in_gaa(x1, x2, x3)  =  appcB_in_gaa(x1)
6.91/2.64	
6.91/2.64	appcB_out_gaa(x1, x2, x3)  =  appcB_out_gaa(x1, x3)
6.91/2.64	
6.91/2.64	.(x1, x2)  =  .(x2)
6.91/2.64	
6.91/2.64	U18_gaa(x1, x2, x3, x4, x5)  =  U18_gaa(x2, x5)
6.91/2.64	
6.91/2.64	appB_in_gaa(x1, x2, x3)  =  appB_in_gaa(x1)
6.91/2.64	
6.91/2.64	appC_in_gag(x1, x2, x3)  =  appC_in_gag(x1, x3)
6.91/2.64	
6.91/2.64	REVD_IN_AG(x1, x2)  =  REVD_IN_AG(x2)
6.91/2.64	
6.91/2.64	U6_AG(x1, x2, x3, x4)  =  U6_AG(x4)
6.91/2.64	
6.91/2.64	REVA_IN_AA(x1, x2)  =  REVA_IN_AA
6.91/2.64	
6.91/2.64	U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
6.91/2.64	
6.91/2.64	U2_AA(x1, x2, x3, x4)  =  U2_AA(x4)
6.91/2.64	
6.91/2.64	U3_AA(x1, x2, x3, x4)  =  U3_AA(x2, x4)
6.91/2.64	
6.91/2.64	APPB_IN_GAA(x1, x2, x3)  =  APPB_IN_GAA(x1)
6.91/2.64	
6.91/2.64	U4_GAA(x1, x2, x3, x4, x5)  =  U4_GAA(x2, x5)
6.91/2.64	
6.91/2.64	U7_AG(x1, x2, x3, x4)  =  U7_AG(x4)
6.91/2.64	
6.91/2.64	U8_AG(x1, x2, x3, x4)  =  U8_AG(x3, x4)
6.91/2.64	
6.91/2.64	U9_AG(x1, x2, x3, x4, x5)  =  U9_AG(x4, x5)
6.91/2.64	
6.91/2.64	U10_AG(x1, x2, x3, x4, x5)  =  U10_AG(x4, x5)
6.91/2.64	
6.91/2.64	U11_AG(x1, x2, x3, x4, x5)  =  U11_AG(x3, x4, x5)
6.91/2.64	
6.91/2.64	U12_AG(x1, x2, x3, x4, x5, x6)  =  U12_AG(x5, x6)
6.91/2.64	
6.91/2.64	U13_AG(x1, x2, x3, x4, x5, x6)  =  U13_AG(x3, x5, x6)
6.91/2.64	
6.91/2.64	U14_AG(x1, x2, x3, x4, x5, x6)  =  U14_AG(x3, x5, x6)
6.91/2.64	
6.91/2.64	APPC_IN_GAG(x1, x2, x3)  =  APPC_IN_GAG(x1, x3)
6.91/2.64	
6.91/2.64	U5_GAG(x1, x2, x3, x4, x5)  =  U5_GAG(x2, x4, x5)
6.91/2.64	
6.91/2.64	
6.91/2.64	We have to consider all (P,R,Pi)-chains
6.91/2.64	
6.91/2.64	
6.91/2.64	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
6.91/2.64	
6.91/2.64	
6.91/2.64	
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(72)
6.91/2.64	Obligation:
6.91/2.64	Pi DP problem:
6.91/2.64	The TRS P consists of the following rules:
6.91/2.64	
6.91/2.64	   REVD_IN_AG(.(X1, .(X2, X3)), []) -> U6_AG(X1, X2, X3, revA_in_aa(X3, X4))
6.91/2.64	   REVD_IN_AG(.(X1, .(X2, X3)), []) -> REVA_IN_AA(X3, X4)
6.91/2.64	   REVA_IN_AA(.(X1, X2), X3) -> U1_AA(X1, X2, X3, revA_in_aa(X2, X4))
6.91/2.64	   REVA_IN_AA(.(X1, X2), X3) -> REVA_IN_AA(X2, X4)
6.91/2.64	   REVA_IN_AA(.(X1, X2), X3) -> U2_AA(X1, X2, X3, revcA_in_aa(X2, X4))
6.91/2.64	   U2_AA(X1, X2, X3, revcA_out_aa(X2, X4)) -> U3_AA(X1, X2, X3, appB_in_gaa(X4, X1, X3))
6.91/2.64	   U2_AA(X1, X2, X3, revcA_out_aa(X2, X4)) -> APPB_IN_GAA(X4, X1, X3)
6.91/2.64	   APPB_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> U4_GAA(X1, X2, X3, X4, appB_in_gaa(X2, X3, X4))
6.91/2.64	   APPB_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPB_IN_GAA(X2, X3, X4)
6.91/2.64	   REVD_IN_AG(.(X1, .(X2, X3)), []) -> U7_AG(X1, X2, X3, revcA_in_aa(X3, X4))
6.91/2.64	   U7_AG(X1, X2, X3, revcA_out_aa(X3, X4)) -> U8_AG(X1, X2, X3, appB_in_gaa(X4, X2, X5))
6.91/2.64	   U7_AG(X1, X2, X3, revcA_out_aa(X3, X4)) -> APPB_IN_GAA(X4, X2, X5)
6.91/2.64	   REVD_IN_AG(.(X1, .(X2, X3)), X4) -> U9_AG(X1, X2, X3, X4, revA_in_aa(X3, X5))
6.91/2.64	   REVD_IN_AG(.(X1, .(X2, X3)), X4) -> REVA_IN_AA(X3, X5)
6.91/2.64	   REVD_IN_AG(.(X1, .(X2, X3)), X4) -> U10_AG(X1, X2, X3, X4, revcA_in_aa(X3, X5))
6.91/2.64	   U10_AG(X1, X2, X3, X4, revcA_out_aa(X3, X5)) -> U11_AG(X1, X2, X3, X4, appB_in_gaa(X5, X2, X6))
6.91/2.64	   U10_AG(X1, X2, X3, X4, revcA_out_aa(X3, X5)) -> APPB_IN_GAA(X5, X2, X6)
6.91/2.64	   REVD_IN_AG(.(X1, .(X2, X3)), .(X4, X5)) -> U12_AG(X1, X2, X3, X4, X5, revcA_in_aa(X3, X6))
6.91/2.64	   U12_AG(X1, X2, X3, X4, X5, revcA_out_aa(X3, X6)) -> U13_AG(X1, X2, X3, X4, X5, appcB_in_gaa(X6, X2, .(X4, X7)))
6.91/2.64	   U13_AG(X1, X2, X3, X4, X5, appcB_out_gaa(X6, X2, .(X4, X7))) -> U14_AG(X1, X2, X3, X4, X5, appC_in_gag(X7, X1, X5))
6.91/2.64	   U13_AG(X1, X2, X3, X4, X5, appcB_out_gaa(X6, X2, .(X4, X7))) -> APPC_IN_GAG(X7, X1, X5)
6.91/2.64	   APPC_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> U5_GAG(X1, X2, X3, X4, appC_in_gag(X2, X3, X4))
6.91/2.64	   APPC_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPC_IN_GAG(X2, X3, X4)
6.91/2.64	
6.91/2.64	The TRS R consists of the following rules:
6.91/2.64	
6.91/2.64	   revcA_in_aa([], []) -> revcA_out_aa([], [])
6.91/2.64	   revcA_in_aa(.(X1, X2), X3) -> U16_aa(X1, X2, X3, revcA_in_aa(X2, X4))
6.91/2.64	   U16_aa(X1, X2, X3, revcA_out_aa(X2, X4)) -> U17_aa(X1, X2, X3, appcB_in_gaa(X4, X1, X3))
6.91/2.64	   appcB_in_gaa([], X1, .(X1, [])) -> appcB_out_gaa([], X1, .(X1, []))
6.91/2.64	   appcB_in_gaa(.(X1, X2), X3, .(X1, X4)) -> U18_gaa(X1, X2, X3, X4, appcB_in_gaa(X2, X3, X4))
6.91/2.64	   U18_gaa(X1, X2, X3, X4, appcB_out_gaa(X2, X3, X4)) -> appcB_out_gaa(.(X1, X2), X3, .(X1, X4))
6.91/2.64	   U17_aa(X1, X2, X3, appcB_out_gaa(X4, X1, X3)) -> revcA_out_aa(.(X1, X2), X3)
6.91/2.64	
6.91/2.64	The argument filtering Pi contains the following mapping:
6.91/2.64	[]  =  []
6.91/2.64	
6.91/2.64	revA_in_aa(x1, x2)  =  revA_in_aa
6.91/2.64	
6.91/2.64	revcA_in_aa(x1, x2)  =  revcA_in_aa
6.91/2.64	
6.91/2.64	revcA_out_aa(x1, x2)  =  revcA_out_aa(x1, x2)
6.91/2.64	
6.91/2.64	U16_aa(x1, x2, x3, x4)  =  U16_aa(x4)
6.91/2.64	
6.91/2.64	U17_aa(x1, x2, x3, x4)  =  U17_aa(x2, x4)
6.91/2.64	
6.91/2.64	appcB_in_gaa(x1, x2, x3)  =  appcB_in_gaa(x1)
6.91/2.64	
6.91/2.64	appcB_out_gaa(x1, x2, x3)  =  appcB_out_gaa(x1, x3)
6.91/2.64	
6.91/2.64	.(x1, x2)  =  .(x2)
6.91/2.64	
6.91/2.64	U18_gaa(x1, x2, x3, x4, x5)  =  U18_gaa(x2, x5)
6.91/2.64	
6.91/2.64	appB_in_gaa(x1, x2, x3)  =  appB_in_gaa(x1)
6.91/2.64	
6.91/2.64	appC_in_gag(x1, x2, x3)  =  appC_in_gag(x1, x3)
6.91/2.64	
6.91/2.64	REVD_IN_AG(x1, x2)  =  REVD_IN_AG(x2)
6.91/2.64	
6.91/2.64	U6_AG(x1, x2, x3, x4)  =  U6_AG(x4)
6.91/2.64	
6.91/2.64	REVA_IN_AA(x1, x2)  =  REVA_IN_AA
6.91/2.64	
6.91/2.64	U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
6.91/2.64	
6.91/2.64	U2_AA(x1, x2, x3, x4)  =  U2_AA(x4)
6.91/2.64	
6.91/2.64	U3_AA(x1, x2, x3, x4)  =  U3_AA(x2, x4)
6.91/2.64	
6.91/2.64	APPB_IN_GAA(x1, x2, x3)  =  APPB_IN_GAA(x1)
6.91/2.64	
6.91/2.64	U4_GAA(x1, x2, x3, x4, x5)  =  U4_GAA(x2, x5)
6.91/2.64	
6.91/2.64	U7_AG(x1, x2, x3, x4)  =  U7_AG(x4)
6.91/2.64	
6.91/2.64	U8_AG(x1, x2, x3, x4)  =  U8_AG(x3, x4)
6.91/2.64	
6.91/2.64	U9_AG(x1, x2, x3, x4, x5)  =  U9_AG(x4, x5)
6.91/2.64	
6.91/2.64	U10_AG(x1, x2, x3, x4, x5)  =  U10_AG(x4, x5)
6.91/2.64	
6.91/2.64	U11_AG(x1, x2, x3, x4, x5)  =  U11_AG(x3, x4, x5)
6.91/2.64	
6.91/2.64	U12_AG(x1, x2, x3, x4, x5, x6)  =  U12_AG(x5, x6)
6.91/2.64	
6.91/2.64	U13_AG(x1, x2, x3, x4, x5, x6)  =  U13_AG(x3, x5, x6)
6.91/2.64	
6.91/2.64	U14_AG(x1, x2, x3, x4, x5, x6)  =  U14_AG(x3, x5, x6)
6.91/2.64	
6.91/2.64	APPC_IN_GAG(x1, x2, x3)  =  APPC_IN_GAG(x1, x3)
6.91/2.64	
6.91/2.64	U5_GAG(x1, x2, x3, x4, x5)  =  U5_GAG(x2, x4, x5)
6.91/2.64	
6.91/2.64	
6.91/2.64	We have to consider all (P,R,Pi)-chains
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(73) DependencyGraphProof (EQUIVALENT)
6.91/2.64	The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 20 less nodes.
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(74)
6.91/2.64	Complex Obligation (AND)
6.91/2.64	
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(75)
6.91/2.64	Obligation:
6.91/2.64	Pi DP problem:
6.91/2.64	The TRS P consists of the following rules:
6.91/2.64	
6.91/2.64	   APPC_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPC_IN_GAG(X2, X3, X4)
6.91/2.64	
6.91/2.64	The TRS R consists of the following rules:
6.91/2.64	
6.91/2.64	   revcA_in_aa([], []) -> revcA_out_aa([], [])
6.91/2.64	   revcA_in_aa(.(X1, X2), X3) -> U16_aa(X1, X2, X3, revcA_in_aa(X2, X4))
6.91/2.64	   U16_aa(X1, X2, X3, revcA_out_aa(X2, X4)) -> U17_aa(X1, X2, X3, appcB_in_gaa(X4, X1, X3))
6.91/2.64	   appcB_in_gaa([], X1, .(X1, [])) -> appcB_out_gaa([], X1, .(X1, []))
6.91/2.64	   appcB_in_gaa(.(X1, X2), X3, .(X1, X4)) -> U18_gaa(X1, X2, X3, X4, appcB_in_gaa(X2, X3, X4))
6.91/2.64	   U18_gaa(X1, X2, X3, X4, appcB_out_gaa(X2, X3, X4)) -> appcB_out_gaa(.(X1, X2), X3, .(X1, X4))
6.91/2.64	   U17_aa(X1, X2, X3, appcB_out_gaa(X4, X1, X3)) -> revcA_out_aa(.(X1, X2), X3)
6.91/2.64	
6.91/2.64	The argument filtering Pi contains the following mapping:
6.91/2.64	[]  =  []
6.91/2.64	
6.91/2.64	revcA_in_aa(x1, x2)  =  revcA_in_aa
6.91/2.64	
6.91/2.64	revcA_out_aa(x1, x2)  =  revcA_out_aa(x1, x2)
6.91/2.64	
6.91/2.64	U16_aa(x1, x2, x3, x4)  =  U16_aa(x4)
6.91/2.64	
6.91/2.64	U17_aa(x1, x2, x3, x4)  =  U17_aa(x2, x4)
6.91/2.64	
6.91/2.64	appcB_in_gaa(x1, x2, x3)  =  appcB_in_gaa(x1)
6.91/2.64	
6.91/2.64	appcB_out_gaa(x1, x2, x3)  =  appcB_out_gaa(x1, x3)
6.91/2.64	
6.91/2.64	.(x1, x2)  =  .(x2)
6.91/2.64	
6.91/2.64	U18_gaa(x1, x2, x3, x4, x5)  =  U18_gaa(x2, x5)
6.91/2.64	
6.91/2.64	APPC_IN_GAG(x1, x2, x3)  =  APPC_IN_GAG(x1, x3)
6.91/2.64	
6.91/2.64	
6.91/2.64	We have to consider all (P,R,Pi)-chains
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(76) UsableRulesProof (EQUIVALENT)
6.91/2.64	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(77)
6.91/2.64	Obligation:
6.91/2.64	Pi DP problem:
6.91/2.64	The TRS P consists of the following rules:
6.91/2.64	
6.91/2.64	   APPC_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPC_IN_GAG(X2, X3, X4)
6.91/2.64	
6.91/2.64	R is empty.
6.91/2.64	The argument filtering Pi contains the following mapping:
6.91/2.64	.(x1, x2)  =  .(x2)
6.91/2.64	
6.91/2.64	APPC_IN_GAG(x1, x2, x3)  =  APPC_IN_GAG(x1, x3)
6.91/2.64	
6.91/2.64	
6.91/2.64	We have to consider all (P,R,Pi)-chains
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(78) PiDPToQDPProof (SOUND)
6.91/2.64	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(79)
6.91/2.64	Obligation:
6.91/2.64	Q DP problem:
6.91/2.64	The TRS P consists of the following rules:
6.91/2.64	
6.91/2.64	   APPC_IN_GAG(.(X2), .(X4)) -> APPC_IN_GAG(X2, X4)
6.91/2.64	
6.91/2.64	R is empty.
6.91/2.64	Q is empty.
6.91/2.64	We have to consider all (P,Q,R)-chains.
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(80) QDPSizeChangeProof (EQUIVALENT)
6.91/2.64	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. 
6.91/2.64	
6.91/2.64	From the DPs we obtained the following set of size-change graphs:
6.91/2.64	*APPC_IN_GAG(.(X2), .(X4)) -> APPC_IN_GAG(X2, X4)
6.91/2.64	The graph contains the following edges 1 > 1, 2 > 2
6.91/2.64	
6.91/2.64	
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(81)
6.91/2.64	YES
6.91/2.64	
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(82)
6.91/2.64	Obligation:
6.91/2.64	Pi DP problem:
6.91/2.64	The TRS P consists of the following rules:
6.91/2.64	
6.91/2.64	   APPB_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPB_IN_GAA(X2, X3, X4)
6.91/2.64	
6.91/2.64	The TRS R consists of the following rules:
6.91/2.64	
6.91/2.64	   revcA_in_aa([], []) -> revcA_out_aa([], [])
6.91/2.64	   revcA_in_aa(.(X1, X2), X3) -> U16_aa(X1, X2, X3, revcA_in_aa(X2, X4))
6.91/2.64	   U16_aa(X1, X2, X3, revcA_out_aa(X2, X4)) -> U17_aa(X1, X2, X3, appcB_in_gaa(X4, X1, X3))
6.91/2.64	   appcB_in_gaa([], X1, .(X1, [])) -> appcB_out_gaa([], X1, .(X1, []))
6.91/2.64	   appcB_in_gaa(.(X1, X2), X3, .(X1, X4)) -> U18_gaa(X1, X2, X3, X4, appcB_in_gaa(X2, X3, X4))
6.91/2.64	   U18_gaa(X1, X2, X3, X4, appcB_out_gaa(X2, X3, X4)) -> appcB_out_gaa(.(X1, X2), X3, .(X1, X4))
6.91/2.64	   U17_aa(X1, X2, X3, appcB_out_gaa(X4, X1, X3)) -> revcA_out_aa(.(X1, X2), X3)
6.91/2.64	
6.91/2.64	The argument filtering Pi contains the following mapping:
6.91/2.64	[]  =  []
6.91/2.64	
6.91/2.64	revcA_in_aa(x1, x2)  =  revcA_in_aa
6.91/2.64	
6.91/2.64	revcA_out_aa(x1, x2)  =  revcA_out_aa(x1, x2)
6.91/2.64	
6.91/2.64	U16_aa(x1, x2, x3, x4)  =  U16_aa(x4)
6.91/2.64	
6.91/2.64	U17_aa(x1, x2, x3, x4)  =  U17_aa(x2, x4)
6.91/2.64	
6.91/2.64	appcB_in_gaa(x1, x2, x3)  =  appcB_in_gaa(x1)
6.91/2.64	
6.91/2.64	appcB_out_gaa(x1, x2, x3)  =  appcB_out_gaa(x1, x3)
6.91/2.64	
6.91/2.64	.(x1, x2)  =  .(x2)
6.91/2.64	
6.91/2.64	U18_gaa(x1, x2, x3, x4, x5)  =  U18_gaa(x2, x5)
6.91/2.64	
6.91/2.64	APPB_IN_GAA(x1, x2, x3)  =  APPB_IN_GAA(x1)
6.91/2.64	
6.91/2.64	
6.91/2.64	We have to consider all (P,R,Pi)-chains
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(83) UsableRulesProof (EQUIVALENT)
6.91/2.64	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(84)
6.91/2.64	Obligation:
6.91/2.64	Pi DP problem:
6.91/2.64	The TRS P consists of the following rules:
6.91/2.64	
6.91/2.64	   APPB_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPB_IN_GAA(X2, X3, X4)
6.91/2.64	
6.91/2.64	R is empty.
6.91/2.64	The argument filtering Pi contains the following mapping:
6.91/2.64	.(x1, x2)  =  .(x2)
6.91/2.64	
6.91/2.64	APPB_IN_GAA(x1, x2, x3)  =  APPB_IN_GAA(x1)
6.91/2.64	
6.91/2.64	
6.91/2.64	We have to consider all (P,R,Pi)-chains
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(85) PiDPToQDPProof (SOUND)
6.91/2.64	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(86)
6.91/2.64	Obligation:
6.91/2.64	Q DP problem:
6.91/2.64	The TRS P consists of the following rules:
6.91/2.64	
6.91/2.64	   APPB_IN_GAA(.(X2)) -> APPB_IN_GAA(X2)
6.91/2.64	
6.91/2.64	R is empty.
6.91/2.64	Q is empty.
6.91/2.64	We have to consider all (P,Q,R)-chains.
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(87) QDPSizeChangeProof (EQUIVALENT)
6.91/2.64	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. 
6.91/2.64	
6.91/2.64	From the DPs we obtained the following set of size-change graphs:
6.91/2.64	*APPB_IN_GAA(.(X2)) -> APPB_IN_GAA(X2)
6.91/2.64	The graph contains the following edges 1 > 1
6.91/2.64	
6.91/2.64	
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(88)
6.91/2.64	YES
6.91/2.64	
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(89)
6.91/2.64	Obligation:
6.91/2.64	Pi DP problem:
6.91/2.64	The TRS P consists of the following rules:
6.91/2.64	
6.91/2.64	   REVA_IN_AA(.(X1, X2), X3) -> REVA_IN_AA(X2, X4)
6.91/2.64	
6.91/2.64	The TRS R consists of the following rules:
6.91/2.64	
6.91/2.64	   revcA_in_aa([], []) -> revcA_out_aa([], [])
6.91/2.64	   revcA_in_aa(.(X1, X2), X3) -> U16_aa(X1, X2, X3, revcA_in_aa(X2, X4))
6.91/2.64	   U16_aa(X1, X2, X3, revcA_out_aa(X2, X4)) -> U17_aa(X1, X2, X3, appcB_in_gaa(X4, X1, X3))
6.91/2.64	   appcB_in_gaa([], X1, .(X1, [])) -> appcB_out_gaa([], X1, .(X1, []))
6.91/2.64	   appcB_in_gaa(.(X1, X2), X3, .(X1, X4)) -> U18_gaa(X1, X2, X3, X4, appcB_in_gaa(X2, X3, X4))
6.91/2.64	   U18_gaa(X1, X2, X3, X4, appcB_out_gaa(X2, X3, X4)) -> appcB_out_gaa(.(X1, X2), X3, .(X1, X4))
6.91/2.64	   U17_aa(X1, X2, X3, appcB_out_gaa(X4, X1, X3)) -> revcA_out_aa(.(X1, X2), X3)
6.91/2.64	
6.91/2.64	The argument filtering Pi contains the following mapping:
6.91/2.64	[]  =  []
6.91/2.64	
6.91/2.64	revcA_in_aa(x1, x2)  =  revcA_in_aa
6.91/2.64	
6.91/2.64	revcA_out_aa(x1, x2)  =  revcA_out_aa(x1, x2)
6.91/2.64	
6.91/2.64	U16_aa(x1, x2, x3, x4)  =  U16_aa(x4)
6.91/2.64	
6.91/2.64	U17_aa(x1, x2, x3, x4)  =  U17_aa(x2, x4)
6.91/2.64	
6.91/2.64	appcB_in_gaa(x1, x2, x3)  =  appcB_in_gaa(x1)
6.91/2.64	
6.91/2.64	appcB_out_gaa(x1, x2, x3)  =  appcB_out_gaa(x1, x3)
6.91/2.64	
6.91/2.64	.(x1, x2)  =  .(x2)
6.91/2.64	
6.91/2.64	U18_gaa(x1, x2, x3, x4, x5)  =  U18_gaa(x2, x5)
6.91/2.64	
6.91/2.64	REVA_IN_AA(x1, x2)  =  REVA_IN_AA
6.91/2.64	
6.91/2.64	
6.91/2.64	We have to consider all (P,R,Pi)-chains
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(90) UsableRulesProof (EQUIVALENT)
6.91/2.64	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(91)
6.91/2.64	Obligation:
6.91/2.64	Pi DP problem:
6.91/2.64	The TRS P consists of the following rules:
6.91/2.64	
6.91/2.64	   REVA_IN_AA(.(X1, X2), X3) -> REVA_IN_AA(X2, X4)
6.91/2.64	
6.91/2.64	R is empty.
6.91/2.64	The argument filtering Pi contains the following mapping:
6.91/2.64	.(x1, x2)  =  .(x2)
6.91/2.64	
6.91/2.64	REVA_IN_AA(x1, x2)  =  REVA_IN_AA
6.91/2.64	
6.91/2.64	
6.91/2.64	We have to consider all (P,R,Pi)-chains
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(92) PiDPToQDPProof (SOUND)
6.91/2.64	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(93)
6.91/2.64	Obligation:
6.91/2.64	Q DP problem:
6.91/2.64	The TRS P consists of the following rules:
6.91/2.64	
6.91/2.64	   REVA_IN_AA -> REVA_IN_AA
6.91/2.64	
6.91/2.64	R is empty.
6.91/2.64	Q is empty.
6.91/2.64	We have to consider all (P,Q,R)-chains.
6.91/2.64	----------------------------------------
6.91/2.64	
6.91/2.64	(94) PrologToIRSwTTransformerProof (SOUND)
6.91/2.64	Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert
6.91/2.64	
6.91/2.64	{
6.91/2.64	    "root": 1,
6.91/2.64	    "program": {
6.91/2.64	        "directives": [],
6.91/2.64	        "clauses": [
6.91/2.64	            [
6.91/2.64	                "(rev ([]) ([]))",
6.91/2.64	                null
6.91/2.64	            ],
6.91/2.64	            [
6.91/2.64	                "(rev (. X Xs) Ys)",
6.91/2.64	                "(',' (rev Xs Zs) (app Zs (. X ([])) Ys))"
6.91/2.64	            ],
6.91/2.64	            [
6.91/2.64	                "(app ([]) X X)",
6.91/2.64	                null
6.91/2.64	            ],
6.91/2.64	            [
6.91/2.64	                "(app (. X Xs) Ys (. X Zs))",
6.91/2.64	                "(app Xs Ys Zs)"
6.91/2.64	            ]
6.91/2.64	        ]
6.91/2.64	    },
6.91/2.64	    "graph": {
6.91/2.64	        "nodes": {
6.91/2.64	            "88": {
6.91/2.64	                "goal": [{
6.91/2.64	                    "clause": -1,
6.91/2.64	                    "scope": -1,
6.91/2.64	                    "term": "(rev T12 X11)"
6.91/2.64	                }],
6.91/2.64	                "kb": {
6.91/2.64	                    "nonunifying": [],
6.91/2.64	                    "intvars": {},
6.91/2.64	                    "arithmetic": {
6.91/2.64	                        "type": "PlainIntegerRelationState",
6.91/2.64	                        "relations": []
6.91/2.64	                    },
6.91/2.64	                    "ground": [],
6.91/2.64	                    "free": ["X11"],
6.91/2.64	                    "exprvars": []
6.91/2.64	                }
6.91/2.64	            },
6.91/2.64	            "89": {
6.91/2.64	                "goal": [{
6.91/2.64	                    "clause": -1,
6.91/2.64	                    "scope": -1,
6.91/2.64	                    "term": "(app T14 (. T15 ([])) T11)"
6.91/2.64	                }],
6.91/2.64	                "kb": {
6.91/2.64	                    "nonunifying": [],
6.91/2.64	                    "intvars": {},
6.91/2.64	                    "arithmetic": {
6.91/2.64	                        "type": "PlainIntegerRelationState",
6.91/2.64	                        "relations": []
6.91/2.64	                    },
6.91/2.64	                    "ground": ["T11"],
6.91/2.64	                    "free": [],
6.91/2.64	                    "exprvars": []
6.91/2.64	                }
6.91/2.64	            },
6.91/2.64	            "type": "Nodes",
6.91/2.64	            "212": {
6.91/2.64	                "goal": [{
6.91/2.64	                    "clause": -1,
6.91/2.64	                    "scope": -1,
6.91/2.64	                    "term": "(app T42 (. T43 ([])) X47)"
6.91/2.64	                }],
6.91/2.64	                "kb": {
6.91/2.64	                    "nonunifying": [],
6.91/2.64	                    "intvars": {},
6.91/2.64	                    "arithmetic": {
6.91/2.64	                        "type": "PlainIntegerRelationState",
6.91/2.64	                        "relations": []
6.91/2.64	                    },
6.91/2.64	                    "ground": [],
6.91/2.64	                    "free": ["X47"],
6.91/2.64	                    "exprvars": []
6.91/2.64	                }
6.91/2.64	            },
6.91/2.64	            "213": {
6.91/2.64	                "goal": [],
6.91/2.64	                "kb": {
6.91/2.64	                    "nonunifying": [],
6.91/2.64	                    "intvars": {},
6.91/2.64	                    "arithmetic": {
6.91/2.64	                        "type": "PlainIntegerRelationState",
6.91/2.64	                        "relations": []
6.91/2.64	                    },
6.91/2.64	                    "ground": [],
6.91/2.64	                    "free": [],
6.91/2.64	                    "exprvars": []
6.91/2.64	                }
6.91/2.64	            },
6.91/2.64	            "216": {
6.91/2.64	                "goal": [
6.91/2.64	                    {
6.91/2.64	                        "clause": 2,
6.91/2.64	                        "scope": 4,
6.91/2.64	                        "term": "(app T14 (. T15 ([])) T11)"
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6.91/2.65	                    },
6.91/2.65	                    "ground": [],
6.91/2.65	                    "free": ["X25"],
6.91/2.65	                    "exprvars": []
6.91/2.65	                }
6.91/2.65	            },
6.91/2.65	            "208": {
6.91/2.65	                "goal": [],
6.91/2.65	                "kb": {
6.91/2.65	                    "nonunifying": [],
6.91/2.65	                    "intvars": {},
6.91/2.65	                    "arithmetic": {
6.91/2.65	                        "type": "PlainIntegerRelationState",
6.91/2.65	                        "relations": []
6.91/2.65	                    },
6.91/2.65	                    "ground": [],
6.91/2.65	                    "free": [],
6.91/2.65	                    "exprvars": []
6.91/2.65	                }
6.91/2.65	            },
6.91/2.65	            "83": {
6.91/2.65	                "goal": [{
6.91/2.65	                    "clause": -1,
6.91/2.65	                    "scope": -1,
6.91/2.65	                    "term": "(',' (rev T12 X11) (app X11 (. T13 ([])) T11))"
6.91/2.65	                }],
6.91/2.65	                "kb": {
6.91/2.65	                    "nonunifying": [],
6.91/2.65	                    "intvars": {},
6.91/2.65	                    "arithmetic": {
6.91/2.65	                        "type": "PlainIntegerRelationState",
6.91/2.65	                        "relations": []
6.91/2.65	                    },
6.91/2.65	                    "ground": ["T11"],
6.91/2.65	                    "free": ["X11"],
6.91/2.65	                    "exprvars": []
6.91/2.65	                }
6.91/2.65	            },
6.91/2.65	            "84": {
6.91/2.65	                "goal": [],
6.91/2.65	                "kb": {
6.91/2.65	                    "nonunifying": [],
6.91/2.65	                    "intvars": {},
6.91/2.65	                    "arithmetic": {
6.91/2.65	                        "type": "PlainIntegerRelationState",
6.91/2.65	                        "relations": []
6.91/2.65	                    },
6.91/2.65	                    "ground": [],
6.91/2.65	                    "free": [],
6.91/2.65	                    "exprvars": []
6.91/2.65	                }
6.91/2.65	            }
6.91/2.65	        },
6.91/2.65	        "edges": [
6.91/2.65	            {
6.91/2.65	                "from": 1,
6.91/2.65	                "to": 9,
6.91/2.65	                "label": "CASE"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 9,
6.91/2.65	                "to": 78,
6.91/2.65	                "label": "PARALLEL"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 9,
6.91/2.65	                "to": 79,
6.91/2.65	                "label": "PARALLEL"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 78,
6.91/2.65	                "to": 80,
6.91/2.65	                "label": "EVAL with clause\nrev([], []).\nand substitutionT1 -> [],\nT2 -> []"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 78,
6.91/2.65	                "to": 81,
6.91/2.65	                "label": "EVAL-BACKTRACK"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 79,
6.91/2.65	                "to": 83,
6.91/2.65	                "label": "EVAL with clause\nrev(.(X8, X9), X10) :- ','(rev(X9, X11), app(X11, .(X8, []), X10)).\nand substitutionX8 -> T13,\nX9 -> T12,\nT1 -> .(T13, T12),\nT2 -> T11,\nX10 -> T11,\nT10 -> T12,\nT9 -> T13"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 79,
6.91/2.65	                "to": 84,
6.91/2.65	                "label": "EVAL-BACKTRACK"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 80,
6.91/2.65	                "to": 82,
6.91/2.65	                "label": "SUCCESS"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 83,
6.91/2.65	                "to": 88,
6.91/2.65	                "label": "SPLIT 1"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 83,
6.91/2.65	                "to": 89,
6.91/2.65	                "label": "SPLIT 2\nreplacements:X11 -> T14,\nT13 -> T15"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 88,
6.91/2.65	                "to": 92,
6.91/2.65	                "label": "CASE"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 89,
6.91/2.65	                "to": 216,
6.91/2.65	                "label": "CASE"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 92,
6.91/2.65	                "to": 94,
6.91/2.65	                "label": "PARALLEL"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 92,
6.91/2.65	                "to": 95,
6.91/2.65	                "label": "PARALLEL"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 94,
6.91/2.65	                "to": 98,
6.91/2.65	                "label": "EVAL with clause\nrev([], []).\nand substitutionT12 -> [],\nX11 -> []"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 94,
6.91/2.65	                "to": 99,
6.91/2.65	                "label": "EVAL-BACKTRACK"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 95,
6.91/2.65	                "to": 106,
6.91/2.65	                "label": "EVAL with clause\nrev(.(X21, X22), X23) :- ','(rev(X22, X24), app(X24, .(X21, []), X23)).\nand substitutionX21 -> T23,\nX22 -> T22,\nT12 -> .(T23, T22),\nX11 -> X25,\nX23 -> X25,\nT21 -> T22,\nT20 -> T23"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 95,
6.91/2.65	                "to": 107,
6.91/2.65	                "label": "EVAL-BACKTRACK"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 98,
6.91/2.65	                "to": 101,
6.91/2.65	                "label": "SUCCESS"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 106,
6.91/2.65	                "to": 108,
6.91/2.65	                "label": "SPLIT 1"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 106,
6.91/2.65	                "to": 109,
6.91/2.65	                "label": "SPLIT 2\nreplacements:X24 -> T24,\nT23 -> T25"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 108,
6.91/2.65	                "to": 88,
6.91/2.65	                "label": "INSTANCE with matching:\nT12 -> T22\nX11 -> X24"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 109,
6.91/2.65	                "to": 200,
6.91/2.65	                "label": "CASE"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 200,
6.91/2.65	                "to": 201,
6.91/2.65	                "label": "PARALLEL"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 200,
6.91/2.65	                "to": 202,
6.91/2.65	                "label": "PARALLEL"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 201,
6.91/2.65	                "to": 204,
6.91/2.65	                "label": "EVAL with clause\napp([], X32, X32).\nand substitutionT24 -> [],\nT25 -> T32,\nX32 -> .(T32, []),\nX25 -> .(T32, [])"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 201,
6.91/2.65	                "to": 207,
6.91/2.65	                "label": "EVAL-BACKTRACK"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 202,
6.91/2.65	                "to": 212,
6.91/2.65	                "label": "EVAL with clause\napp(.(X43, X44), X45, .(X43, X46)) :- app(X44, X45, X46).\nand substitutionX43 -> T39,\nX44 -> T42,\nT24 -> .(T39, T42),\nT25 -> T43,\nX45 -> .(T43, []),\nX46 -> X47,\nX25 -> .(T39, X47),\nT40 -> T42,\nT41 -> T43"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 202,
6.91/2.65	                "to": 213,
6.91/2.65	                "label": "EVAL-BACKTRACK"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 204,
6.91/2.65	                "to": 208,
6.91/2.65	                "label": "SUCCESS"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 212,
6.91/2.65	                "to": 109,
6.91/2.65	                "label": "INSTANCE with matching:\nT24 -> T42\nT25 -> T43\nX25 -> X47"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 216,
6.91/2.65	                "to": 217,
6.91/2.65	                "label": "PARALLEL"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 216,
6.91/2.65	                "to": 218,
6.91/2.65	                "label": "PARALLEL"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 217,
6.91/2.65	                "to": 219,
6.91/2.65	                "label": "EVAL with clause\napp([], X56, X56).\nand substitutionT14 -> [],\nT15 -> T52,\nX56 -> .(T52, []),\nT11 -> .(T52, [])"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 217,
6.91/2.65	                "to": 220,
6.91/2.65	                "label": "EVAL-BACKTRACK"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 218,
6.91/2.65	                "to": 222,
6.91/2.65	                "label": "EVAL with clause\napp(.(X65, X66), X67, .(X65, X68)) :- app(X66, X67, X68).\nand substitutionX65 -> T61,\nX66 -> T65,\nT14 -> .(T61, T65),\nT15 -> T66,\nX67 -> .(T66, []),\nX68 -> T64,\nT11 -> .(T61, T64),\nT62 -> T65,\nT63 -> T66"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 218,
6.91/2.65	                "to": 223,
6.91/2.65	                "label": "EVAL-BACKTRACK"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 219,
6.91/2.65	                "to": 221,
6.91/2.65	                "label": "SUCCESS"
6.91/2.65	            },
6.91/2.65	            {
6.91/2.65	                "from": 222,
6.91/2.65	                "to": 89,
6.91/2.65	                "label": "INSTANCE with matching:\nT14 -> T65\nT15 -> T66\nT11 -> T64"
6.91/2.65	            }
6.91/2.65	        ],
6.91/2.65	        "type": "Graph"
6.91/2.65	    }
6.91/2.65	}
6.91/2.65	
6.91/2.65	----------------------------------------
6.91/2.65	
6.91/2.65	(95)
6.91/2.65	Complex Obligation (AND)
6.91/2.65	
6.91/2.65	----------------------------------------
6.91/2.65	
6.91/2.65	(96)
6.91/2.65	Obligation:
6.91/2.65	Rules:
6.91/2.65	f200_in -> f202_in :|: TRUE
6.91/2.65	f201_out -> f200_out :|: TRUE
6.91/2.65	f202_out -> f200_out :|: TRUE
6.91/2.65	f200_in -> f201_in :|: TRUE
6.91/2.65	f212_out -> f202_out :|: TRUE
6.91/2.65	f202_in -> f212_in :|: TRUE
6.91/2.65	f213_out -> f202_out :|: TRUE
6.91/2.65	f202_in -> f213_in :|: TRUE
6.91/2.65	f212_in -> f109_in :|: TRUE
6.91/2.65	f109_out -> f212_out :|: TRUE
6.91/2.65	f109_in -> f200_in :|: TRUE
6.91/2.65	f200_out -> f109_out :|: TRUE
6.91/2.65	f1_in(T2) -> f9_in(T2) :|: TRUE
6.91/2.65	f9_out(x) -> f1_out(x) :|: TRUE
6.91/2.65	f9_in(x1) -> f79_in(x1) :|: TRUE
6.91/2.65	f78_out(x2) -> f9_out(x2) :|: TRUE
6.91/2.65	f9_in(x3) -> f78_in(x3) :|: TRUE
6.91/2.65	f79_out(x4) -> f9_out(x4) :|: TRUE
6.91/2.65	f79_in(T11) -> f83_in(T11) :|: TRUE
6.91/2.65	f84_out -> f79_out(x5) :|: TRUE
6.91/2.65	f83_out(x6) -> f79_out(x6) :|: TRUE
6.91/2.65	f79_in(x7) -> f84_in :|: TRUE
6.91/2.65	f83_in(x8) -> f88_in :|: TRUE
6.91/2.65	f89_out(x9) -> f83_out(x9) :|: TRUE
6.91/2.65	f88_out -> f89_in(x10) :|: TRUE
6.91/2.65	f88_in -> f92_in :|: TRUE
6.91/2.65	f92_out -> f88_out :|: TRUE
6.91/2.65	f94_out -> f92_out :|: TRUE
6.91/2.65	f92_in -> f94_in :|: TRUE
6.91/2.65	f92_in -> f95_in :|: TRUE
6.91/2.65	f95_out -> f92_out :|: TRUE
6.91/2.65	f106_out -> f95_out :|: TRUE
6.91/2.65	f95_in -> f106_in :|: TRUE
6.91/2.65	f107_out -> f95_out :|: TRUE
6.91/2.65	f95_in -> f107_in :|: TRUE
6.91/2.65	f106_in -> f108_in :|: TRUE
6.91/2.65	f109_out -> f106_out :|: TRUE
6.91/2.65	f108_out -> f109_in :|: TRUE
6.91/2.65	Start term: f1_in(T2)
6.91/2.65	
6.91/2.65	----------------------------------------
6.91/2.65	
6.91/2.65	(97) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
6.91/2.65	Constructed simple dependency graph.
6.91/2.65	
6.91/2.65	Simplified to the following IRSwTs:
6.91/2.65	
6.91/2.65	
6.91/2.65	----------------------------------------
6.91/2.65	
6.91/2.65	(98)
6.91/2.65	TRUE
6.91/2.65	
6.91/2.65	----------------------------------------
6.91/2.65	
6.91/2.65	(99)
6.91/2.65	Obligation:
6.91/2.65	Rules:
6.91/2.65	f217_out(T11) -> f216_out(T11) :|: TRUE
6.91/2.65	f216_in(x) -> f218_in(x) :|: TRUE
6.91/2.65	f216_in(x1) -> f217_in(x1) :|: TRUE
6.91/2.65	f218_out(x2) -> f216_out(x2) :|: TRUE
6.91/2.65	f222_in(T64) -> f89_in(T64) :|: TRUE
6.91/2.65	f89_out(x3) -> f222_out(x3) :|: TRUE
6.91/2.65	f218_in(.(x4, x5)) -> f222_in(x5) :|: TRUE
6.91/2.65	f218_in(x6) -> f223_in :|: TRUE
6.91/2.65	f223_out -> f218_out(x7) :|: TRUE
6.91/2.65	f222_out(x8) -> f218_out(.(x9, x8)) :|: TRUE
6.91/2.65	f89_in(x10) -> f216_in(x10) :|: TRUE
6.91/2.65	f216_out(x11) -> f89_out(x11) :|: TRUE
6.91/2.65	f1_in(T2) -> f9_in(T2) :|: TRUE
6.91/2.65	f9_out(x12) -> f1_out(x12) :|: TRUE
6.91/2.65	f9_in(x13) -> f79_in(x13) :|: TRUE
6.91/2.65	f78_out(x14) -> f9_out(x14) :|: TRUE
6.91/2.65	f9_in(x15) -> f78_in(x15) :|: TRUE
6.91/2.65	f79_out(x16) -> f9_out(x16) :|: TRUE
6.91/2.65	f79_in(x17) -> f83_in(x17) :|: TRUE
6.91/2.65	f84_out -> f79_out(x18) :|: TRUE
6.91/2.65	f83_out(x19) -> f79_out(x19) :|: TRUE
6.91/2.65	f79_in(x20) -> f84_in :|: TRUE
6.91/2.65	f83_in(x21) -> f88_in :|: TRUE
6.91/2.65	f89_out(x22) -> f83_out(x22) :|: TRUE
6.91/2.65	f88_out -> f89_in(x23) :|: TRUE
6.91/2.65	Start term: f1_in(T2)
6.91/2.65	
6.91/2.65	----------------------------------------
6.91/2.65	
6.91/2.65	(100) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
6.91/2.65	Constructed simple dependency graph.
6.91/2.65	
6.91/2.65	Simplified to the following IRSwTs:
6.91/2.65	
6.91/2.65	
6.91/2.65	----------------------------------------
6.91/2.65	
6.91/2.65	(101)
6.91/2.65	TRUE
6.91/2.65	
6.91/2.65	----------------------------------------
6.91/2.65	
6.91/2.65	(102)
6.91/2.65	Obligation:
6.91/2.65	Rules:
6.91/2.65	f106_in -> f108_in :|: TRUE
6.91/2.65	f109_out -> f106_out :|: TRUE
6.91/2.65	f108_out -> f109_in :|: TRUE
6.91/2.65	f200_in -> f202_in :|: TRUE
6.91/2.65	f201_out -> f200_out :|: TRUE
6.91/2.65	f202_out -> f200_out :|: TRUE
6.91/2.65	f200_in -> f201_in :|: TRUE
6.91/2.65	f88_in -> f92_in :|: TRUE
6.91/2.65	f92_out -> f88_out :|: TRUE
6.91/2.65	f212_in -> f109_in :|: TRUE
6.91/2.65	f109_out -> f212_out :|: TRUE
6.91/2.65	f94_out -> f92_out :|: TRUE
6.91/2.65	f92_in -> f94_in :|: TRUE
6.91/2.65	f92_in -> f95_in :|: TRUE
6.91/2.65	f95_out -> f92_out :|: TRUE
6.91/2.65	f204_out -> f201_out :|: TRUE
6.91/2.65	f207_out -> f201_out :|: TRUE
6.91/2.65	f201_in -> f204_in :|: TRUE
6.91/2.65	f201_in -> f207_in :|: TRUE
6.91/2.65	f204_in -> f204_out :|: TRUE
6.91/2.65	f212_out -> f202_out :|: TRUE
6.91/2.65	f202_in -> f212_in :|: TRUE
6.91/2.65	f213_out -> f202_out :|: TRUE
6.91/2.65	f202_in -> f213_in :|: TRUE
6.91/2.65	f106_out -> f95_out :|: TRUE
6.91/2.65	f95_in -> f106_in :|: TRUE
6.91/2.65	f107_out -> f95_out :|: TRUE
6.91/2.65	f95_in -> f107_in :|: TRUE
6.91/2.65	f108_in -> f88_in :|: TRUE
6.91/2.65	f88_out -> f108_out :|: TRUE
6.91/2.65	f109_in -> f200_in :|: TRUE
6.91/2.65	f200_out -> f109_out :|: TRUE
6.91/2.65	f1_in(T2) -> f9_in(T2) :|: TRUE
6.91/2.65	f9_out(x) -> f1_out(x) :|: TRUE
6.91/2.65	f9_in(x1) -> f79_in(x1) :|: TRUE
6.91/2.65	f78_out(x2) -> f9_out(x2) :|: TRUE
6.91/2.65	f9_in(x3) -> f78_in(x3) :|: TRUE
6.91/2.65	f79_out(x4) -> f9_out(x4) :|: TRUE
6.91/2.65	f79_in(T11) -> f83_in(T11) :|: TRUE
6.91/2.65	f84_out -> f79_out(x5) :|: TRUE
6.91/2.65	f83_out(x6) -> f79_out(x6) :|: TRUE
6.91/2.65	f79_in(x7) -> f84_in :|: TRUE
6.91/2.65	f83_in(x8) -> f88_in :|: TRUE
6.91/2.65	f89_out(x9) -> f83_out(x9) :|: TRUE
6.91/2.65	f88_out -> f89_in(x10) :|: TRUE
6.91/2.65	Start term: f1_in(T2)
6.91/2.65	
6.91/2.65	----------------------------------------
6.91/2.65	
6.91/2.65	(103) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
6.91/2.65	Constructed simple dependency graph.
6.91/2.65	
6.91/2.65	Simplified to the following IRSwTs:
6.91/2.65	
6.91/2.65	intTRSProblem:
6.91/2.65	f106_in -> f108_in :|: TRUE
6.91/2.65	f88_in -> f92_in :|: TRUE
6.91/2.65	f92_in -> f95_in :|: TRUE
6.91/2.65	f95_in -> f106_in :|: TRUE
6.91/2.65	f108_in -> f88_in :|: TRUE
6.91/2.65	
6.91/2.65	
6.91/2.65	----------------------------------------
6.91/2.65	
6.91/2.65	(104)
6.91/2.65	Obligation:
6.91/2.65	Rules:
6.91/2.65	f106_in -> f108_in :|: TRUE
6.91/2.65	f88_in -> f92_in :|: TRUE
6.91/2.65	f92_in -> f95_in :|: TRUE
6.91/2.65	f95_in -> f106_in :|: TRUE
6.91/2.65	f108_in -> f88_in :|: TRUE
6.91/2.65	
6.91/2.65	----------------------------------------
6.91/2.65	
6.91/2.65	(105) IntTRSCompressionProof (EQUIVALENT)
6.91/2.65	Compressed rules.
6.91/2.65	----------------------------------------
6.91/2.65	
6.91/2.65	(106)
6.91/2.65	Obligation:
6.91/2.65	Rules:
6.91/2.65	f88_in -> f88_in :|: TRUE
6.91/2.65	
6.91/2.65	----------------------------------------
6.91/2.65	
6.91/2.65	(107) IRSFormatTransformerProof (EQUIVALENT)
6.91/2.65	Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
6.91/2.65	----------------------------------------
6.91/2.65	
6.91/2.65	(108)
6.91/2.65	Obligation:
6.91/2.65	Rules:
6.91/2.65	f88_in -> f88_in :|: TRUE
6.91/2.65	
6.91/2.65	----------------------------------------
6.91/2.65	
6.91/2.65	(109) IRSwTTerminationDigraphProof (EQUIVALENT)
6.91/2.65	Constructed termination digraph!
6.91/2.65	Nodes:
6.91/2.65	(1) f88_in -> f88_in :|: TRUE
6.91/2.65	
6.91/2.65	Arcs:
6.91/2.65	(1) -> (1)
6.91/2.65	
6.91/2.65	This digraph is fully evaluated!
6.91/2.65	----------------------------------------
6.91/2.65	
6.91/2.65	(110)
6.91/2.65	Obligation:
6.91/2.65	
6.91/2.65	Termination digraph:
6.91/2.65	Nodes:
6.91/2.65	(1) f88_in -> f88_in :|: TRUE
6.91/2.65	
6.91/2.65	Arcs:
6.91/2.65	(1) -> (1)
6.91/2.65	
6.91/2.65	This digraph is fully evaluated!
6.91/2.65	
6.91/2.65	----------------------------------------
6.91/2.65	
6.91/2.65	(111) FilterProof (EQUIVALENT)
6.91/2.65	Used the following sort dictionary for filtering: 
6.91/2.65	f88_in()
6.91/2.65	Replaced non-predefined constructor symbols by 0.
6.91/2.65	----------------------------------------
6.91/2.65	
6.91/2.65	(112)
6.91/2.65	Obligation:
6.91/2.65	Rules:
6.91/2.65	f88_in -> f88_in :|: TRUE
6.91/2.65	
6.91/2.65	----------------------------------------
6.91/2.65	
6.91/2.65	(113) IntTRSNonPeriodicNontermProof (COMPLETE)
6.91/2.65	Normalized system to the following form:
6.91/2.65	f(pc) -> f(1) :|: pc = 1 && TRUE
6.91/2.65	Proved unsatisfiability of the following formula, indicating that the system is never left after entering:
6.91/2.65	((run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T)) and !(((run2_0 * 1)) = ((1 * 1)) and T))
6.91/2.65	Proved satisfiability of the following formula, indicating that the system is entered at least once:
6.91/2.65	(run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T))
6.91/2.65	
6.91/2.65	----------------------------------------
6.91/2.65	
6.91/2.65	(114)
6.91/2.65	NO
6.91/2.69	EOF