6.64/2.55	MAYBE
6.67/2.57	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
6.67/2.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.67/2.57	
6.67/2.57	
6.67/2.57	Left Termination of the query pattern
6.67/2.57	
6.67/2.57	reverse(a,g)
6.67/2.57	
6.67/2.57	w.r.t. the given Prolog program could not be shown:
6.67/2.57	
6.67/2.57	(0) Prolog
6.67/2.57	(1) PrologToPiTRSProof [SOUND, 0 ms]
6.67/2.57	(2) PiTRS
6.67/2.57	    (3) DependencyPairsProof [EQUIVALENT, 0 ms]
6.67/2.57	    (4) PiDP
6.67/2.57	    (5) DependencyGraphProof [EQUIVALENT, 2 ms]
6.67/2.57	    (6) AND
6.67/2.57	        (7) PiDP
6.67/2.57	            (8) UsableRulesProof [EQUIVALENT, 0 ms]
6.67/2.57	            (9) PiDP
6.67/2.57	            (10) PiDPToQDPProof [SOUND, 0 ms]
6.67/2.57	            (11) QDP
6.67/2.57	            (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.67/2.57	            (13) YES
6.67/2.57	        (14) PiDP
6.67/2.57	            (15) UsableRulesProof [EQUIVALENT, 0 ms]
6.67/2.57	            (16) PiDP
6.67/2.57	            (17) PiDPToQDPProof [SOUND, 0 ms]
6.67/2.57	            (18) QDP
6.67/2.57	            (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.67/2.57	            (20) YES
6.67/2.57	        (21) PiDP
6.67/2.57	            (22) UsableRulesProof [EQUIVALENT, 0 ms]
6.67/2.57	            (23) PiDP
6.67/2.57	            (24) PiDPToQDPProof [SOUND, 0 ms]
6.67/2.57	            (25) QDP
6.67/2.57	(26) PrologToPiTRSProof [SOUND, 0 ms]
6.67/2.57	(27) PiTRS
6.67/2.57	    (28) DependencyPairsProof [EQUIVALENT, 0 ms]
6.67/2.57	    (29) PiDP
6.67/2.57	    (30) DependencyGraphProof [EQUIVALENT, 7 ms]
6.67/2.57	    (31) AND
6.67/2.57	        (32) PiDP
6.67/2.57	            (33) UsableRulesProof [EQUIVALENT, 0 ms]
6.67/2.57	            (34) PiDP
6.67/2.57	            (35) PiDPToQDPProof [SOUND, 0 ms]
6.67/2.57	            (36) QDP
6.67/2.57	            (37) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.67/2.57	            (38) YES
6.67/2.57	        (39) PiDP
6.67/2.57	            (40) UsableRulesProof [EQUIVALENT, 0 ms]
6.67/2.57	            (41) PiDP
6.67/2.57	            (42) PiDPToQDPProof [SOUND, 0 ms]
6.67/2.57	            (43) QDP
6.67/2.57	            (44) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.67/2.57	            (45) YES
6.67/2.57	        (46) PiDP
6.67/2.57	            (47) UsableRulesProof [EQUIVALENT, 0 ms]
6.67/2.57	            (48) PiDP
6.67/2.57	            (49) PiDPToQDPProof [SOUND, 0 ms]
6.67/2.57	            (50) QDP
6.67/2.57	(51) PrologToDTProblemTransformerProof [SOUND, 54 ms]
6.67/2.57	(52) TRIPLES
6.67/2.57	    (53) TriplesToPiDPProof [SOUND, 0 ms]
6.67/2.57	    (54) PiDP
6.67/2.57	    (55) DependencyGraphProof [EQUIVALENT, 0 ms]
6.67/2.57	    (56) AND
6.67/2.57	        (57) PiDP
6.67/2.57	            (58) UsableRulesProof [EQUIVALENT, 0 ms]
6.67/2.57	            (59) PiDP
6.67/2.57	            (60) PiDPToQDPProof [SOUND, 0 ms]
6.67/2.57	            (61) QDP
6.67/2.57	            (62) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.67/2.57	            (63) YES
6.67/2.57	        (64) PiDP
6.67/2.57	            (65) UsableRulesProof [EQUIVALENT, 0 ms]
6.67/2.57	            (66) PiDP
6.67/2.57	            (67) PiDPToQDPProof [SOUND, 0 ms]
6.67/2.57	            (68) QDP
6.67/2.57	            (69) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.67/2.57	            (70) YES
6.67/2.57	        (71) PiDP
6.67/2.57	            (72) UsableRulesProof [EQUIVALENT, 0 ms]
6.67/2.57	            (73) PiDP
6.67/2.57	            (74) PiDPToQDPProof [SOUND, 0 ms]
6.67/2.57	            (75) QDP
6.67/2.57	(76) PrologToIRSwTTransformerProof [SOUND, 35 ms]
6.67/2.57	(77) AND
6.67/2.57	    (78) IRSwT
6.67/2.57	        (79) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
6.67/2.57	        (80) TRUE
6.67/2.57	    (81) IRSwT
6.67/2.57	        (82) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
6.67/2.57	        (83) TRUE
6.67/2.57	    (84) IRSwT
6.67/2.57	        (85) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
6.67/2.57	        (86) IRSwT
6.67/2.57	        (87) IntTRSCompressionProof [EQUIVALENT, 20 ms]
6.67/2.57	        (88) IRSwT
6.67/2.57	        (89) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
6.67/2.57	        (90) IRSwT
6.67/2.57	        (91) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
6.67/2.57	        (92) IRSwT
6.67/2.57	        (93) FilterProof [EQUIVALENT, 0 ms]
6.67/2.57	        (94) IntTRS
6.67/2.57	        (95) IntTRSPeriodicNontermProof [COMPLETE, 7 ms]
6.67/2.57	        (96) NO
6.67/2.57	(97) PrologToTRSTransformerProof [SOUND, 19 ms]
6.67/2.57	(98) QTRS
6.67/2.57	    (99) QTRSRRRProof [EQUIVALENT, 113 ms]
6.67/2.57	    (100) QTRS
6.67/2.57	    (101) QTRSRRRProof [EQUIVALENT, 0 ms]
6.67/2.57	    (102) QTRS
6.67/2.57	    (103) QTRSRRRProof [EQUIVALENT, 11 ms]
6.67/2.57	    (104) QTRS
6.67/2.57	    (105) QTRSRRRProof [EQUIVALENT, 5 ms]
6.67/2.57	    (106) QTRS
6.67/2.57	
6.67/2.57	
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(0)
6.67/2.57	Obligation:
6.67/2.57	Clauses:
6.67/2.57	
6.67/2.57	app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
6.67/2.57	app([], Ys, Ys).
6.67/2.57	reverse(.(X, Xs), Ys) :- ','(reverse(Xs, Zs), app(Zs, .(X, []), Ys)).
6.67/2.57	reverse([], []).
6.67/2.57	
6.67/2.57	
6.67/2.57	Query: reverse(a,g)
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(1) PrologToPiTRSProof (SOUND)
6.67/2.57	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.67/2.57	
6.67/2.57	reverse_in_2: (f,b) (f,f)
6.67/2.57	
6.67/2.57	app_in_3: (b,b,f) (b,b,b)
6.67/2.57	
6.67/2.57	Transforming Prolog into the following Term Rewriting System:
6.67/2.57	
6.67/2.57	Pi-finite rewrite system:
6.67/2.57	The TRS R consists of the following rules:
6.67/2.57	
6.67/2.57	   reverse_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   reverse_in_aa(.(X, Xs), Ys) -> U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   reverse_in_aa([], []) -> reverse_out_aa([], [])
6.67/2.57	   U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.67/2.57	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.67/2.57	   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
6.67/2.57	   U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.67/2.57	   U3_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> reverse_out_aa(.(X, Xs), Ys)
6.67/2.57	   U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.67/2.57	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U1_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.67/2.57	   app_in_ggg([], Ys, Ys) -> app_out_ggg([], Ys, Ys)
6.67/2.57	   U1_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.67/2.57	   U3_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> reverse_out_ag(.(X, Xs), Ys)
6.67/2.57	   reverse_in_ag([], []) -> reverse_out_ag([], [])
6.67/2.57	
6.67/2.57	The argument filtering Pi contains the following mapping:
6.67/2.57	reverse_in_ag(x1, x2)  =  reverse_in_ag(x2)
6.67/2.57	
6.67/2.57	U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
6.67/2.57	
6.67/2.57	reverse_in_aa(x1, x2)  =  reverse_in_aa
6.67/2.57	
6.67/2.57	U2_aa(x1, x2, x3, x4)  =  U2_aa(x4)
6.67/2.57	
6.67/2.57	reverse_out_aa(x1, x2)  =  reverse_out_aa(x1, x2)
6.67/2.57	
6.67/2.57	U3_aa(x1, x2, x3, x4)  =  U3_aa(x2, x4)
6.67/2.57	
6.67/2.57	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.67/2.57	
6.67/2.57	.(x1, x2)  =  .(x2)
6.67/2.57	
6.67/2.57	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
6.67/2.57	
6.67/2.57	[]  =  []
6.67/2.57	
6.67/2.57	app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
6.67/2.57	
6.67/2.57	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x4)
6.67/2.57	
6.67/2.57	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.67/2.57	
6.67/2.57	U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x5)
6.67/2.57	
6.67/2.57	app_out_ggg(x1, x2, x3)  =  app_out_ggg
6.67/2.57	
6.67/2.57	reverse_out_ag(x1, x2)  =  reverse_out_ag(x1)
6.67/2.57	
6.67/2.57	
6.67/2.57	
6.67/2.57	
6.67/2.57	
6.67/2.57	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.67/2.57	
6.67/2.57	
6.67/2.57	
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(2)
6.67/2.57	Obligation:
6.67/2.57	Pi-finite rewrite system:
6.67/2.57	The TRS R consists of the following rules:
6.67/2.57	
6.67/2.57	   reverse_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   reverse_in_aa(.(X, Xs), Ys) -> U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   reverse_in_aa([], []) -> reverse_out_aa([], [])
6.67/2.57	   U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.67/2.57	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.67/2.57	   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
6.67/2.57	   U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.67/2.57	   U3_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> reverse_out_aa(.(X, Xs), Ys)
6.67/2.57	   U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.67/2.57	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U1_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.67/2.57	   app_in_ggg([], Ys, Ys) -> app_out_ggg([], Ys, Ys)
6.67/2.57	   U1_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.67/2.57	   U3_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> reverse_out_ag(.(X, Xs), Ys)
6.67/2.57	   reverse_in_ag([], []) -> reverse_out_ag([], [])
6.67/2.57	
6.67/2.57	The argument filtering Pi contains the following mapping:
6.67/2.57	reverse_in_ag(x1, x2)  =  reverse_in_ag(x2)
6.67/2.57	
6.67/2.57	U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
6.67/2.57	
6.67/2.57	reverse_in_aa(x1, x2)  =  reverse_in_aa
6.67/2.57	
6.67/2.57	U2_aa(x1, x2, x3, x4)  =  U2_aa(x4)
6.67/2.57	
6.67/2.57	reverse_out_aa(x1, x2)  =  reverse_out_aa(x1, x2)
6.67/2.57	
6.67/2.57	U3_aa(x1, x2, x3, x4)  =  U3_aa(x2, x4)
6.67/2.57	
6.67/2.57	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.67/2.57	
6.67/2.57	.(x1, x2)  =  .(x2)
6.67/2.57	
6.67/2.57	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
6.67/2.57	
6.67/2.57	[]  =  []
6.67/2.57	
6.67/2.57	app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
6.67/2.57	
6.67/2.57	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x4)
6.67/2.57	
6.67/2.57	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.67/2.57	
6.67/2.57	U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x5)
6.67/2.57	
6.67/2.57	app_out_ggg(x1, x2, x3)  =  app_out_ggg
6.67/2.57	
6.67/2.57	reverse_out_ag(x1, x2)  =  reverse_out_ag(x1)
6.67/2.57	
6.67/2.57	
6.67/2.57	
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(3) DependencyPairsProof (EQUIVALENT)
6.67/2.57	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.67/2.57	Pi DP problem:
6.67/2.57	The TRS P consists of the following rules:
6.67/2.57	
6.67/2.57	   REVERSE_IN_AG(.(X, Xs), Ys) -> U2_AG(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   REVERSE_IN_AG(.(X, Xs), Ys) -> REVERSE_IN_AA(Xs, Zs)
6.67/2.57	   REVERSE_IN_AA(.(X, Xs), Ys) -> U2_AA(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   REVERSE_IN_AA(.(X, Xs), Ys) -> REVERSE_IN_AA(Xs, Zs)
6.67/2.57	   U2_AA(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_AA(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.67/2.57	   U2_AA(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> APP_IN_GGA(Zs, .(X, []), Ys)
6.67/2.57	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U1_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.67/2.57	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
6.67/2.57	   U2_AG(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_AG(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.67/2.57	   U2_AG(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> APP_IN_GGG(Zs, .(X, []), Ys)
6.67/2.57	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> U1_GGG(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.67/2.57	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.67/2.57	
6.67/2.57	The TRS R consists of the following rules:
6.67/2.57	
6.67/2.57	   reverse_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   reverse_in_aa(.(X, Xs), Ys) -> U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   reverse_in_aa([], []) -> reverse_out_aa([], [])
6.67/2.57	   U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.67/2.57	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.67/2.57	   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
6.67/2.57	   U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.67/2.57	   U3_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> reverse_out_aa(.(X, Xs), Ys)
6.67/2.57	   U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.67/2.57	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U1_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.67/2.57	   app_in_ggg([], Ys, Ys) -> app_out_ggg([], Ys, Ys)
6.67/2.57	   U1_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.67/2.57	   U3_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> reverse_out_ag(.(X, Xs), Ys)
6.67/2.57	   reverse_in_ag([], []) -> reverse_out_ag([], [])
6.67/2.57	
6.67/2.57	The argument filtering Pi contains the following mapping:
6.67/2.57	reverse_in_ag(x1, x2)  =  reverse_in_ag(x2)
6.67/2.57	
6.67/2.57	U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
6.67/2.57	
6.67/2.57	reverse_in_aa(x1, x2)  =  reverse_in_aa
6.67/2.57	
6.67/2.57	U2_aa(x1, x2, x3, x4)  =  U2_aa(x4)
6.67/2.57	
6.67/2.57	reverse_out_aa(x1, x2)  =  reverse_out_aa(x1, x2)
6.67/2.57	
6.67/2.57	U3_aa(x1, x2, x3, x4)  =  U3_aa(x2, x4)
6.67/2.57	
6.67/2.57	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.67/2.57	
6.67/2.57	.(x1, x2)  =  .(x2)
6.67/2.57	
6.67/2.57	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
6.67/2.57	
6.67/2.57	[]  =  []
6.67/2.57	
6.67/2.57	app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
6.67/2.57	
6.67/2.57	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x4)
6.67/2.57	
6.67/2.57	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.67/2.57	
6.67/2.57	U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x5)
6.67/2.57	
6.67/2.57	app_out_ggg(x1, x2, x3)  =  app_out_ggg
6.67/2.57	
6.67/2.57	reverse_out_ag(x1, x2)  =  reverse_out_ag(x1)
6.67/2.57	
6.67/2.57	REVERSE_IN_AG(x1, x2)  =  REVERSE_IN_AG(x2)
6.67/2.57	
6.67/2.57	U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)
6.67/2.57	
6.67/2.57	REVERSE_IN_AA(x1, x2)  =  REVERSE_IN_AA
6.67/2.57	
6.67/2.57	U2_AA(x1, x2, x3, x4)  =  U2_AA(x4)
6.67/2.57	
6.67/2.57	U3_AA(x1, x2, x3, x4)  =  U3_AA(x2, x4)
6.67/2.57	
6.67/2.57	APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
6.67/2.57	
6.67/2.57	U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x5)
6.67/2.57	
6.67/2.57	U3_AG(x1, x2, x3, x4)  =  U3_AG(x2, x4)
6.67/2.57	
6.67/2.57	APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)
6.67/2.57	
6.67/2.57	U1_GGG(x1, x2, x3, x4, x5)  =  U1_GGG(x5)
6.67/2.57	
6.67/2.57	
6.67/2.57	We have to consider all (P,R,Pi)-chains
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(4)
6.67/2.57	Obligation:
6.67/2.57	Pi DP problem:
6.67/2.57	The TRS P consists of the following rules:
6.67/2.57	
6.67/2.57	   REVERSE_IN_AG(.(X, Xs), Ys) -> U2_AG(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   REVERSE_IN_AG(.(X, Xs), Ys) -> REVERSE_IN_AA(Xs, Zs)
6.67/2.57	   REVERSE_IN_AA(.(X, Xs), Ys) -> U2_AA(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   REVERSE_IN_AA(.(X, Xs), Ys) -> REVERSE_IN_AA(Xs, Zs)
6.67/2.57	   U2_AA(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_AA(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.67/2.57	   U2_AA(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> APP_IN_GGA(Zs, .(X, []), Ys)
6.67/2.57	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U1_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.67/2.57	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
6.67/2.57	   U2_AG(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_AG(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.67/2.57	   U2_AG(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> APP_IN_GGG(Zs, .(X, []), Ys)
6.67/2.57	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> U1_GGG(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.67/2.57	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.67/2.57	
6.67/2.57	The TRS R consists of the following rules:
6.67/2.57	
6.67/2.57	   reverse_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   reverse_in_aa(.(X, Xs), Ys) -> U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   reverse_in_aa([], []) -> reverse_out_aa([], [])
6.67/2.57	   U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.67/2.57	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.67/2.57	   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
6.67/2.57	   U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.67/2.57	   U3_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> reverse_out_aa(.(X, Xs), Ys)
6.67/2.57	   U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.67/2.57	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U1_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.67/2.57	   app_in_ggg([], Ys, Ys) -> app_out_ggg([], Ys, Ys)
6.67/2.57	   U1_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.67/2.57	   U3_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> reverse_out_ag(.(X, Xs), Ys)
6.67/2.57	   reverse_in_ag([], []) -> reverse_out_ag([], [])
6.67/2.57	
6.67/2.57	The argument filtering Pi contains the following mapping:
6.67/2.57	reverse_in_ag(x1, x2)  =  reverse_in_ag(x2)
6.67/2.57	
6.67/2.57	U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
6.67/2.57	
6.67/2.57	reverse_in_aa(x1, x2)  =  reverse_in_aa
6.67/2.57	
6.67/2.57	U2_aa(x1, x2, x3, x4)  =  U2_aa(x4)
6.67/2.57	
6.67/2.57	reverse_out_aa(x1, x2)  =  reverse_out_aa(x1, x2)
6.67/2.57	
6.67/2.57	U3_aa(x1, x2, x3, x4)  =  U3_aa(x2, x4)
6.67/2.57	
6.67/2.57	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.67/2.57	
6.67/2.57	.(x1, x2)  =  .(x2)
6.67/2.57	
6.67/2.57	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
6.67/2.57	
6.67/2.57	[]  =  []
6.67/2.57	
6.67/2.57	app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
6.67/2.57	
6.67/2.57	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x4)
6.67/2.57	
6.67/2.57	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.67/2.57	
6.67/2.57	U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x5)
6.67/2.57	
6.67/2.57	app_out_ggg(x1, x2, x3)  =  app_out_ggg
6.67/2.57	
6.67/2.57	reverse_out_ag(x1, x2)  =  reverse_out_ag(x1)
6.67/2.57	
6.67/2.57	REVERSE_IN_AG(x1, x2)  =  REVERSE_IN_AG(x2)
6.67/2.57	
6.67/2.57	U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)
6.67/2.57	
6.67/2.57	REVERSE_IN_AA(x1, x2)  =  REVERSE_IN_AA
6.67/2.57	
6.67/2.57	U2_AA(x1, x2, x3, x4)  =  U2_AA(x4)
6.67/2.57	
6.67/2.57	U3_AA(x1, x2, x3, x4)  =  U3_AA(x2, x4)
6.67/2.57	
6.67/2.57	APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
6.67/2.57	
6.67/2.57	U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x5)
6.67/2.57	
6.67/2.57	U3_AG(x1, x2, x3, x4)  =  U3_AG(x2, x4)
6.67/2.57	
6.67/2.57	APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)
6.67/2.57	
6.67/2.57	U1_GGG(x1, x2, x3, x4, x5)  =  U1_GGG(x5)
6.67/2.57	
6.67/2.57	
6.67/2.57	We have to consider all (P,R,Pi)-chains
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(5) DependencyGraphProof (EQUIVALENT)
6.67/2.57	The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 9 less nodes.
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(6)
6.67/2.57	Complex Obligation (AND)
6.67/2.57	
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(7)
6.67/2.57	Obligation:
6.67/2.57	Pi DP problem:
6.67/2.57	The TRS P consists of the following rules:
6.67/2.57	
6.67/2.57	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.67/2.57	
6.67/2.57	The TRS R consists of the following rules:
6.67/2.57	
6.67/2.57	   reverse_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   reverse_in_aa(.(X, Xs), Ys) -> U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   reverse_in_aa([], []) -> reverse_out_aa([], [])
6.67/2.57	   U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.67/2.57	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.67/2.57	   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
6.67/2.57	   U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.67/2.57	   U3_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> reverse_out_aa(.(X, Xs), Ys)
6.67/2.57	   U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.67/2.57	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U1_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.67/2.57	   app_in_ggg([], Ys, Ys) -> app_out_ggg([], Ys, Ys)
6.67/2.57	   U1_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.67/2.57	   U3_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> reverse_out_ag(.(X, Xs), Ys)
6.67/2.57	   reverse_in_ag([], []) -> reverse_out_ag([], [])
6.67/2.57	
6.67/2.57	The argument filtering Pi contains the following mapping:
6.67/2.57	reverse_in_ag(x1, x2)  =  reverse_in_ag(x2)
6.67/2.57	
6.67/2.57	U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
6.67/2.57	
6.67/2.57	reverse_in_aa(x1, x2)  =  reverse_in_aa
6.67/2.57	
6.67/2.57	U2_aa(x1, x2, x3, x4)  =  U2_aa(x4)
6.67/2.57	
6.67/2.57	reverse_out_aa(x1, x2)  =  reverse_out_aa(x1, x2)
6.67/2.57	
6.67/2.57	U3_aa(x1, x2, x3, x4)  =  U3_aa(x2, x4)
6.67/2.57	
6.67/2.57	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.67/2.57	
6.67/2.57	.(x1, x2)  =  .(x2)
6.67/2.57	
6.67/2.57	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
6.67/2.57	
6.67/2.57	[]  =  []
6.67/2.57	
6.67/2.57	app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
6.67/2.57	
6.67/2.57	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x4)
6.67/2.57	
6.67/2.57	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.67/2.57	
6.67/2.57	U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x5)
6.67/2.57	
6.67/2.57	app_out_ggg(x1, x2, x3)  =  app_out_ggg
6.67/2.57	
6.67/2.57	reverse_out_ag(x1, x2)  =  reverse_out_ag(x1)
6.67/2.57	
6.67/2.57	APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)
6.67/2.57	
6.67/2.57	
6.67/2.57	We have to consider all (P,R,Pi)-chains
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(8) UsableRulesProof (EQUIVALENT)
6.67/2.57	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(9)
6.67/2.57	Obligation:
6.67/2.57	Pi DP problem:
6.67/2.57	The TRS P consists of the following rules:
6.67/2.57	
6.67/2.57	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.67/2.57	
6.67/2.57	R is empty.
6.67/2.57	The argument filtering Pi contains the following mapping:
6.67/2.57	.(x1, x2)  =  .(x2)
6.67/2.57	
6.67/2.57	APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)
6.67/2.57	
6.67/2.57	
6.67/2.57	We have to consider all (P,R,Pi)-chains
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(10) PiDPToQDPProof (SOUND)
6.67/2.57	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(11)
6.67/2.57	Obligation:
6.67/2.57	Q DP problem:
6.67/2.57	The TRS P consists of the following rules:
6.67/2.57	
6.67/2.57	   APP_IN_GGG(.(Xs), Ys, .(Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.67/2.57	
6.67/2.57	R is empty.
6.67/2.57	Q is empty.
6.67/2.57	We have to consider all (P,Q,R)-chains.
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(12) QDPSizeChangeProof (EQUIVALENT)
6.67/2.57	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. 
6.67/2.57	
6.67/2.57	From the DPs we obtained the following set of size-change graphs:
6.67/2.57	*APP_IN_GGG(.(Xs), Ys, .(Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.67/2.57	The graph contains the following edges 1 > 1, 2 >= 2, 3 > 3
6.67/2.57	
6.67/2.57	
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(13)
6.67/2.57	YES
6.67/2.57	
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(14)
6.67/2.57	Obligation:
6.67/2.57	Pi DP problem:
6.67/2.57	The TRS P consists of the following rules:
6.67/2.57	
6.67/2.57	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
6.67/2.57	
6.67/2.57	The TRS R consists of the following rules:
6.67/2.57	
6.67/2.57	   reverse_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   reverse_in_aa(.(X, Xs), Ys) -> U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   reverse_in_aa([], []) -> reverse_out_aa([], [])
6.67/2.57	   U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.67/2.57	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.67/2.57	   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
6.67/2.57	   U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.67/2.57	   U3_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> reverse_out_aa(.(X, Xs), Ys)
6.67/2.57	   U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.67/2.57	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U1_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.67/2.57	   app_in_ggg([], Ys, Ys) -> app_out_ggg([], Ys, Ys)
6.67/2.57	   U1_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.67/2.57	   U3_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> reverse_out_ag(.(X, Xs), Ys)
6.67/2.57	   reverse_in_ag([], []) -> reverse_out_ag([], [])
6.67/2.57	
6.67/2.57	The argument filtering Pi contains the following mapping:
6.67/2.57	reverse_in_ag(x1, x2)  =  reverse_in_ag(x2)
6.67/2.57	
6.67/2.57	U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
6.67/2.57	
6.67/2.57	reverse_in_aa(x1, x2)  =  reverse_in_aa
6.67/2.57	
6.67/2.57	U2_aa(x1, x2, x3, x4)  =  U2_aa(x4)
6.67/2.57	
6.67/2.57	reverse_out_aa(x1, x2)  =  reverse_out_aa(x1, x2)
6.67/2.57	
6.67/2.57	U3_aa(x1, x2, x3, x4)  =  U3_aa(x2, x4)
6.67/2.57	
6.67/2.57	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.67/2.57	
6.67/2.57	.(x1, x2)  =  .(x2)
6.67/2.57	
6.67/2.57	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
6.67/2.57	
6.67/2.57	[]  =  []
6.67/2.57	
6.67/2.57	app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
6.67/2.57	
6.67/2.57	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x4)
6.67/2.57	
6.67/2.57	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.67/2.57	
6.67/2.57	U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x5)
6.67/2.57	
6.67/2.57	app_out_ggg(x1, x2, x3)  =  app_out_ggg
6.67/2.57	
6.67/2.57	reverse_out_ag(x1, x2)  =  reverse_out_ag(x1)
6.67/2.57	
6.67/2.57	APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
6.67/2.57	
6.67/2.57	
6.67/2.57	We have to consider all (P,R,Pi)-chains
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(15) UsableRulesProof (EQUIVALENT)
6.67/2.57	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(16)
6.67/2.57	Obligation:
6.67/2.57	Pi DP problem:
6.67/2.57	The TRS P consists of the following rules:
6.67/2.57	
6.67/2.57	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
6.67/2.57	
6.67/2.57	R is empty.
6.67/2.57	The argument filtering Pi contains the following mapping:
6.67/2.57	.(x1, x2)  =  .(x2)
6.67/2.57	
6.67/2.57	APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
6.67/2.57	
6.67/2.57	
6.67/2.57	We have to consider all (P,R,Pi)-chains
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(17) PiDPToQDPProof (SOUND)
6.67/2.57	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(18)
6.67/2.57	Obligation:
6.67/2.57	Q DP problem:
6.67/2.57	The TRS P consists of the following rules:
6.67/2.57	
6.67/2.57	   APP_IN_GGA(.(Xs), Ys) -> APP_IN_GGA(Xs, Ys)
6.67/2.57	
6.67/2.57	R is empty.
6.67/2.57	Q is empty.
6.67/2.57	We have to consider all (P,Q,R)-chains.
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(19) QDPSizeChangeProof (EQUIVALENT)
6.67/2.57	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. 
6.67/2.57	
6.67/2.57	From the DPs we obtained the following set of size-change graphs:
6.67/2.57	*APP_IN_GGA(.(Xs), Ys) -> APP_IN_GGA(Xs, Ys)
6.67/2.57	The graph contains the following edges 1 > 1, 2 >= 2
6.67/2.57	
6.67/2.57	
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(20)
6.67/2.57	YES
6.67/2.57	
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(21)
6.67/2.57	Obligation:
6.67/2.57	Pi DP problem:
6.67/2.57	The TRS P consists of the following rules:
6.67/2.57	
6.67/2.57	   REVERSE_IN_AA(.(X, Xs), Ys) -> REVERSE_IN_AA(Xs, Zs)
6.67/2.57	
6.67/2.57	The TRS R consists of the following rules:
6.67/2.57	
6.67/2.57	   reverse_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   reverse_in_aa(.(X, Xs), Ys) -> U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   reverse_in_aa([], []) -> reverse_out_aa([], [])
6.67/2.57	   U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.67/2.57	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.67/2.57	   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
6.67/2.57	   U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.67/2.57	   U3_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> reverse_out_aa(.(X, Xs), Ys)
6.67/2.57	   U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.67/2.57	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U1_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.67/2.57	   app_in_ggg([], Ys, Ys) -> app_out_ggg([], Ys, Ys)
6.67/2.57	   U1_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.67/2.57	   U3_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> reverse_out_ag(.(X, Xs), Ys)
6.67/2.57	   reverse_in_ag([], []) -> reverse_out_ag([], [])
6.67/2.57	
6.67/2.57	The argument filtering Pi contains the following mapping:
6.67/2.57	reverse_in_ag(x1, x2)  =  reverse_in_ag(x2)
6.67/2.57	
6.67/2.57	U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
6.67/2.57	
6.67/2.57	reverse_in_aa(x1, x2)  =  reverse_in_aa
6.67/2.57	
6.67/2.57	U2_aa(x1, x2, x3, x4)  =  U2_aa(x4)
6.67/2.57	
6.67/2.57	reverse_out_aa(x1, x2)  =  reverse_out_aa(x1, x2)
6.67/2.57	
6.67/2.57	U3_aa(x1, x2, x3, x4)  =  U3_aa(x2, x4)
6.67/2.57	
6.67/2.57	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.67/2.57	
6.67/2.57	.(x1, x2)  =  .(x2)
6.67/2.57	
6.67/2.57	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x5)
6.67/2.57	
6.67/2.57	[]  =  []
6.67/2.57	
6.67/2.57	app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
6.67/2.57	
6.67/2.57	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x4)
6.67/2.57	
6.67/2.57	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.67/2.57	
6.67/2.57	U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x5)
6.67/2.57	
6.67/2.57	app_out_ggg(x1, x2, x3)  =  app_out_ggg
6.67/2.57	
6.67/2.57	reverse_out_ag(x1, x2)  =  reverse_out_ag(x1)
6.67/2.57	
6.67/2.57	REVERSE_IN_AA(x1, x2)  =  REVERSE_IN_AA
6.67/2.57	
6.67/2.57	
6.67/2.57	We have to consider all (P,R,Pi)-chains
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(22) UsableRulesProof (EQUIVALENT)
6.67/2.57	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(23)
6.67/2.57	Obligation:
6.67/2.57	Pi DP problem:
6.67/2.57	The TRS P consists of the following rules:
6.67/2.57	
6.67/2.57	   REVERSE_IN_AA(.(X, Xs), Ys) -> REVERSE_IN_AA(Xs, Zs)
6.67/2.57	
6.67/2.57	R is empty.
6.67/2.57	The argument filtering Pi contains the following mapping:
6.67/2.57	.(x1, x2)  =  .(x2)
6.67/2.57	
6.67/2.57	REVERSE_IN_AA(x1, x2)  =  REVERSE_IN_AA
6.67/2.57	
6.67/2.57	
6.67/2.57	We have to consider all (P,R,Pi)-chains
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(24) PiDPToQDPProof (SOUND)
6.67/2.57	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(25)
6.67/2.57	Obligation:
6.67/2.57	Q DP problem:
6.67/2.57	The TRS P consists of the following rules:
6.67/2.57	
6.67/2.57	   REVERSE_IN_AA -> REVERSE_IN_AA
6.67/2.57	
6.67/2.57	R is empty.
6.67/2.57	Q is empty.
6.67/2.57	We have to consider all (P,Q,R)-chains.
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(26) PrologToPiTRSProof (SOUND)
6.67/2.57	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.67/2.57	
6.67/2.57	reverse_in_2: (f,b) (f,f)
6.67/2.57	
6.67/2.57	app_in_3: (b,b,f) (b,b,b)
6.67/2.57	
6.67/2.57	Transforming Prolog into the following Term Rewriting System:
6.67/2.57	
6.67/2.57	Pi-finite rewrite system:
6.67/2.57	The TRS R consists of the following rules:
6.67/2.57	
6.67/2.57	   reverse_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   reverse_in_aa(.(X, Xs), Ys) -> U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   reverse_in_aa([], []) -> reverse_out_aa([], [])
6.67/2.57	   U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.67/2.57	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.67/2.57	   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
6.67/2.57	   U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.67/2.57	   U3_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> reverse_out_aa(.(X, Xs), Ys)
6.67/2.57	   U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.67/2.57	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U1_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.67/2.57	   app_in_ggg([], Ys, Ys) -> app_out_ggg([], Ys, Ys)
6.67/2.57	   U1_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.67/2.57	   U3_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> reverse_out_ag(.(X, Xs), Ys)
6.67/2.57	   reverse_in_ag([], []) -> reverse_out_ag([], [])
6.67/2.57	
6.67/2.57	The argument filtering Pi contains the following mapping:
6.67/2.57	reverse_in_ag(x1, x2)  =  reverse_in_ag(x2)
6.67/2.57	
6.67/2.57	U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
6.67/2.57	
6.67/2.57	reverse_in_aa(x1, x2)  =  reverse_in_aa
6.67/2.57	
6.67/2.57	U2_aa(x1, x2, x3, x4)  =  U2_aa(x4)
6.67/2.57	
6.67/2.57	reverse_out_aa(x1, x2)  =  reverse_out_aa(x1, x2)
6.67/2.57	
6.67/2.57	U3_aa(x1, x2, x3, x4)  =  U3_aa(x2, x4)
6.67/2.57	
6.67/2.57	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.67/2.57	
6.67/2.57	.(x1, x2)  =  .(x2)
6.67/2.57	
6.67/2.57	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
6.67/2.57	
6.67/2.57	[]  =  []
6.67/2.57	
6.67/2.57	app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
6.67/2.57	
6.67/2.57	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
6.67/2.57	
6.67/2.57	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.67/2.57	
6.67/2.57	U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x2, x3, x4, x5)
6.67/2.57	
6.67/2.57	app_out_ggg(x1, x2, x3)  =  app_out_ggg(x1, x2, x3)
6.67/2.57	
6.67/2.57	reverse_out_ag(x1, x2)  =  reverse_out_ag(x1, x2)
6.67/2.57	
6.67/2.57	
6.67/2.57	
6.67/2.57	
6.67/2.57	
6.67/2.57	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.67/2.57	
6.67/2.57	
6.67/2.57	
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(27)
6.67/2.57	Obligation:
6.67/2.57	Pi-finite rewrite system:
6.67/2.57	The TRS R consists of the following rules:
6.67/2.57	
6.67/2.57	   reverse_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   reverse_in_aa(.(X, Xs), Ys) -> U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   reverse_in_aa([], []) -> reverse_out_aa([], [])
6.67/2.57	   U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.67/2.57	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.67/2.57	   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
6.67/2.57	   U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.67/2.57	   U3_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> reverse_out_aa(.(X, Xs), Ys)
6.67/2.57	   U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.67/2.57	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U1_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.67/2.57	   app_in_ggg([], Ys, Ys) -> app_out_ggg([], Ys, Ys)
6.67/2.57	   U1_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.67/2.57	   U3_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> reverse_out_ag(.(X, Xs), Ys)
6.67/2.57	   reverse_in_ag([], []) -> reverse_out_ag([], [])
6.67/2.57	
6.67/2.57	The argument filtering Pi contains the following mapping:
6.67/2.57	reverse_in_ag(x1, x2)  =  reverse_in_ag(x2)
6.67/2.57	
6.67/2.57	U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
6.67/2.57	
6.67/2.57	reverse_in_aa(x1, x2)  =  reverse_in_aa
6.67/2.57	
6.67/2.57	U2_aa(x1, x2, x3, x4)  =  U2_aa(x4)
6.67/2.57	
6.67/2.57	reverse_out_aa(x1, x2)  =  reverse_out_aa(x1, x2)
6.67/2.57	
6.67/2.57	U3_aa(x1, x2, x3, x4)  =  U3_aa(x2, x4)
6.67/2.57	
6.67/2.57	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.67/2.57	
6.67/2.57	.(x1, x2)  =  .(x2)
6.67/2.57	
6.67/2.57	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
6.67/2.57	
6.67/2.57	[]  =  []
6.67/2.57	
6.67/2.57	app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
6.67/2.57	
6.67/2.57	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
6.67/2.57	
6.67/2.57	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.67/2.57	
6.67/2.57	U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x2, x3, x4, x5)
6.67/2.57	
6.67/2.57	app_out_ggg(x1, x2, x3)  =  app_out_ggg(x1, x2, x3)
6.67/2.57	
6.67/2.57	reverse_out_ag(x1, x2)  =  reverse_out_ag(x1, x2)
6.67/2.57	
6.67/2.57	
6.67/2.57	
6.67/2.57	----------------------------------------
6.67/2.57	
6.67/2.57	(28) DependencyPairsProof (EQUIVALENT)
6.67/2.57	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.67/2.57	Pi DP problem:
6.67/2.57	The TRS P consists of the following rules:
6.67/2.57	
6.67/2.57	   REVERSE_IN_AG(.(X, Xs), Ys) -> U2_AG(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   REVERSE_IN_AG(.(X, Xs), Ys) -> REVERSE_IN_AA(Xs, Zs)
6.67/2.57	   REVERSE_IN_AA(.(X, Xs), Ys) -> U2_AA(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   REVERSE_IN_AA(.(X, Xs), Ys) -> REVERSE_IN_AA(Xs, Zs)
6.67/2.57	   U2_AA(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_AA(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.67/2.57	   U2_AA(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> APP_IN_GGA(Zs, .(X, []), Ys)
6.67/2.57	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U1_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.67/2.57	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
6.67/2.57	   U2_AG(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_AG(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.67/2.57	   U2_AG(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> APP_IN_GGG(Zs, .(X, []), Ys)
6.67/2.57	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> U1_GGG(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.67/2.57	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.67/2.57	
6.67/2.57	The TRS R consists of the following rules:
6.67/2.57	
6.67/2.57	   reverse_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   reverse_in_aa(.(X, Xs), Ys) -> U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.57	   reverse_in_aa([], []) -> reverse_out_aa([], [])
6.67/2.57	   U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.67/2.57	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.67/2.57	   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
6.67/2.57	   U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.67/2.57	   U3_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> reverse_out_aa(.(X, Xs), Ys)
6.67/2.57	   U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.67/2.57	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U1_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.67/2.57	   app_in_ggg([], Ys, Ys) -> app_out_ggg([], Ys, Ys)
6.67/2.57	   U1_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.67/2.57	   U3_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> reverse_out_ag(.(X, Xs), Ys)
6.67/2.57	   reverse_in_ag([], []) -> reverse_out_ag([], [])
6.67/2.57	
6.67/2.57	The argument filtering Pi contains the following mapping:
6.67/2.57	reverse_in_ag(x1, x2)  =  reverse_in_ag(x2)
6.67/2.57	
6.67/2.57	U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
6.67/2.57	
6.67/2.57	reverse_in_aa(x1, x2)  =  reverse_in_aa
6.67/2.57	
6.67/2.57	U2_aa(x1, x2, x3, x4)  =  U2_aa(x4)
6.67/2.57	
6.67/2.57	reverse_out_aa(x1, x2)  =  reverse_out_aa(x1, x2)
6.67/2.57	
6.67/2.57	U3_aa(x1, x2, x3, x4)  =  U3_aa(x2, x4)
6.67/2.57	
6.67/2.57	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.67/2.57	
6.67/2.57	.(x1, x2)  =  .(x2)
6.67/2.57	
6.67/2.57	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
6.67/2.57	
6.67/2.57	[]  =  []
6.67/2.57	
6.67/2.57	app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
6.67/2.57	
6.67/2.57	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
6.67/2.57	
6.67/2.57	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.67/2.57	
6.67/2.57	U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x2, x3, x4, x5)
6.67/2.57	
6.67/2.57	app_out_ggg(x1, x2, x3)  =  app_out_ggg(x1, x2, x3)
6.67/2.57	
6.67/2.57	reverse_out_ag(x1, x2)  =  reverse_out_ag(x1, x2)
6.67/2.57	
6.67/2.57	REVERSE_IN_AG(x1, x2)  =  REVERSE_IN_AG(x2)
6.67/2.57	
6.67/2.57	U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)
6.67/2.57	
6.67/2.57	REVERSE_IN_AA(x1, x2)  =  REVERSE_IN_AA
6.67/2.57	
6.67/2.57	U2_AA(x1, x2, x3, x4)  =  U2_AA(x4)
6.67/2.57	
6.67/2.57	U3_AA(x1, x2, x3, x4)  =  U3_AA(x2, x4)
6.67/2.57	
6.67/2.57	APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
6.67/2.57	
6.67/2.57	U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x2, x3, x5)
6.67/2.57	
6.67/2.57	U3_AG(x1, x2, x3, x4)  =  U3_AG(x2, x3, x4)
6.67/2.57	
6.67/2.57	APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)
6.67/2.57	
6.67/2.57	U1_GGG(x1, x2, x3, x4, x5)  =  U1_GGG(x2, x3, x4, x5)
6.67/2.59	
6.67/2.59	
6.67/2.59	We have to consider all (P,R,Pi)-chains
6.67/2.59	----------------------------------------
6.67/2.59	
6.67/2.59	(29)
6.67/2.59	Obligation:
6.67/2.59	Pi DP problem:
6.67/2.59	The TRS P consists of the following rules:
6.67/2.59	
6.67/2.59	   REVERSE_IN_AG(.(X, Xs), Ys) -> U2_AG(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.59	   REVERSE_IN_AG(.(X, Xs), Ys) -> REVERSE_IN_AA(Xs, Zs)
6.67/2.59	   REVERSE_IN_AA(.(X, Xs), Ys) -> U2_AA(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.59	   REVERSE_IN_AA(.(X, Xs), Ys) -> REVERSE_IN_AA(Xs, Zs)
6.67/2.59	   U2_AA(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_AA(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.67/2.59	   U2_AA(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> APP_IN_GGA(Zs, .(X, []), Ys)
6.67/2.59	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U1_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.67/2.59	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
6.67/2.59	   U2_AG(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_AG(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.67/2.59	   U2_AG(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> APP_IN_GGG(Zs, .(X, []), Ys)
6.67/2.59	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> U1_GGG(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.67/2.59	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.67/2.60	
6.67/2.60	The TRS R consists of the following rules:
6.67/2.60	
6.67/2.60	   reverse_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.60	   reverse_in_aa(.(X, Xs), Ys) -> U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.60	   reverse_in_aa([], []) -> reverse_out_aa([], [])
6.67/2.60	   U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.67/2.60	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.67/2.60	   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
6.67/2.60	   U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.67/2.60	   U3_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> reverse_out_aa(.(X, Xs), Ys)
6.67/2.60	   U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.67/2.60	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U1_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.67/2.60	   app_in_ggg([], Ys, Ys) -> app_out_ggg([], Ys, Ys)
6.67/2.60	   U1_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.67/2.60	   U3_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> reverse_out_ag(.(X, Xs), Ys)
6.67/2.60	   reverse_in_ag([], []) -> reverse_out_ag([], [])
6.67/2.60	
6.67/2.60	The argument filtering Pi contains the following mapping:
6.67/2.60	reverse_in_ag(x1, x2)  =  reverse_in_ag(x2)
6.67/2.60	
6.67/2.60	U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
6.67/2.60	
6.67/2.60	reverse_in_aa(x1, x2)  =  reverse_in_aa
6.67/2.60	
6.67/2.60	U2_aa(x1, x2, x3, x4)  =  U2_aa(x4)
6.67/2.60	
6.67/2.60	reverse_out_aa(x1, x2)  =  reverse_out_aa(x1, x2)
6.67/2.60	
6.67/2.60	U3_aa(x1, x2, x3, x4)  =  U3_aa(x2, x4)
6.67/2.60	
6.67/2.60	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.67/2.60	
6.67/2.60	.(x1, x2)  =  .(x2)
6.67/2.60	
6.67/2.60	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
6.67/2.60	
6.67/2.60	[]  =  []
6.67/2.60	
6.67/2.60	app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
6.67/2.60	
6.67/2.60	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
6.67/2.60	
6.67/2.60	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.67/2.60	
6.67/2.60	U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x2, x3, x4, x5)
6.67/2.60	
6.67/2.60	app_out_ggg(x1, x2, x3)  =  app_out_ggg(x1, x2, x3)
6.67/2.60	
6.67/2.60	reverse_out_ag(x1, x2)  =  reverse_out_ag(x1, x2)
6.67/2.60	
6.67/2.60	REVERSE_IN_AG(x1, x2)  =  REVERSE_IN_AG(x2)
6.67/2.60	
6.67/2.60	U2_AG(x1, x2, x3, x4)  =  U2_AG(x3, x4)
6.67/2.60	
6.67/2.60	REVERSE_IN_AA(x1, x2)  =  REVERSE_IN_AA
6.67/2.60	
6.67/2.60	U2_AA(x1, x2, x3, x4)  =  U2_AA(x4)
6.67/2.60	
6.67/2.60	U3_AA(x1, x2, x3, x4)  =  U3_AA(x2, x4)
6.67/2.60	
6.67/2.60	APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
6.67/2.60	
6.67/2.60	U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x2, x3, x5)
6.67/2.60	
6.67/2.60	U3_AG(x1, x2, x3, x4)  =  U3_AG(x2, x3, x4)
6.67/2.60	
6.67/2.60	APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)
6.67/2.60	
6.67/2.60	U1_GGG(x1, x2, x3, x4, x5)  =  U1_GGG(x2, x3, x4, x5)
6.67/2.60	
6.67/2.60	
6.67/2.60	We have to consider all (P,R,Pi)-chains
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(30) DependencyGraphProof (EQUIVALENT)
6.67/2.60	The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 9 less nodes.
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(31)
6.67/2.60	Complex Obligation (AND)
6.67/2.60	
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(32)
6.67/2.60	Obligation:
6.67/2.60	Pi DP problem:
6.67/2.60	The TRS P consists of the following rules:
6.67/2.60	
6.67/2.60	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.67/2.60	
6.67/2.60	The TRS R consists of the following rules:
6.67/2.60	
6.67/2.60	   reverse_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.60	   reverse_in_aa(.(X, Xs), Ys) -> U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.60	   reverse_in_aa([], []) -> reverse_out_aa([], [])
6.67/2.60	   U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.67/2.60	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.67/2.60	   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
6.67/2.60	   U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.67/2.60	   U3_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> reverse_out_aa(.(X, Xs), Ys)
6.67/2.60	   U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.67/2.60	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U1_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.67/2.60	   app_in_ggg([], Ys, Ys) -> app_out_ggg([], Ys, Ys)
6.67/2.60	   U1_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.67/2.60	   U3_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> reverse_out_ag(.(X, Xs), Ys)
6.67/2.60	   reverse_in_ag([], []) -> reverse_out_ag([], [])
6.67/2.60	
6.67/2.60	The argument filtering Pi contains the following mapping:
6.67/2.60	reverse_in_ag(x1, x2)  =  reverse_in_ag(x2)
6.67/2.60	
6.67/2.60	U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
6.67/2.60	
6.67/2.60	reverse_in_aa(x1, x2)  =  reverse_in_aa
6.67/2.60	
6.67/2.60	U2_aa(x1, x2, x3, x4)  =  U2_aa(x4)
6.67/2.60	
6.67/2.60	reverse_out_aa(x1, x2)  =  reverse_out_aa(x1, x2)
6.67/2.60	
6.67/2.60	U3_aa(x1, x2, x3, x4)  =  U3_aa(x2, x4)
6.67/2.60	
6.67/2.60	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.67/2.60	
6.67/2.60	.(x1, x2)  =  .(x2)
6.67/2.60	
6.67/2.60	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
6.67/2.60	
6.67/2.60	[]  =  []
6.67/2.60	
6.67/2.60	app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
6.67/2.60	
6.67/2.60	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
6.67/2.60	
6.67/2.60	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.67/2.60	
6.67/2.60	U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x2, x3, x4, x5)
6.67/2.60	
6.67/2.60	app_out_ggg(x1, x2, x3)  =  app_out_ggg(x1, x2, x3)
6.67/2.60	
6.67/2.60	reverse_out_ag(x1, x2)  =  reverse_out_ag(x1, x2)
6.67/2.60	
6.67/2.60	APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)
6.67/2.60	
6.67/2.60	
6.67/2.60	We have to consider all (P,R,Pi)-chains
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(33) UsableRulesProof (EQUIVALENT)
6.67/2.60	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(34)
6.67/2.60	Obligation:
6.67/2.60	Pi DP problem:
6.67/2.60	The TRS P consists of the following rules:
6.67/2.60	
6.67/2.60	   APP_IN_GGG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.67/2.60	
6.67/2.60	R is empty.
6.67/2.60	The argument filtering Pi contains the following mapping:
6.67/2.60	.(x1, x2)  =  .(x2)
6.67/2.60	
6.67/2.60	APP_IN_GGG(x1, x2, x3)  =  APP_IN_GGG(x1, x2, x3)
6.67/2.60	
6.67/2.60	
6.67/2.60	We have to consider all (P,R,Pi)-chains
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(35) PiDPToQDPProof (SOUND)
6.67/2.60	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(36)
6.67/2.60	Obligation:
6.67/2.60	Q DP problem:
6.67/2.60	The TRS P consists of the following rules:
6.67/2.60	
6.67/2.60	   APP_IN_GGG(.(Xs), Ys, .(Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.67/2.60	
6.67/2.60	R is empty.
6.67/2.60	Q is empty.
6.67/2.60	We have to consider all (P,Q,R)-chains.
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(37) QDPSizeChangeProof (EQUIVALENT)
6.67/2.60	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. 
6.67/2.60	
6.67/2.60	From the DPs we obtained the following set of size-change graphs:
6.67/2.60	*APP_IN_GGG(.(Xs), Ys, .(Zs)) -> APP_IN_GGG(Xs, Ys, Zs)
6.67/2.60	The graph contains the following edges 1 > 1, 2 >= 2, 3 > 3
6.67/2.60	
6.67/2.60	
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(38)
6.67/2.60	YES
6.67/2.60	
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(39)
6.67/2.60	Obligation:
6.67/2.60	Pi DP problem:
6.67/2.60	The TRS P consists of the following rules:
6.67/2.60	
6.67/2.60	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
6.67/2.60	
6.67/2.60	The TRS R consists of the following rules:
6.67/2.60	
6.67/2.60	   reverse_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.60	   reverse_in_aa(.(X, Xs), Ys) -> U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.60	   reverse_in_aa([], []) -> reverse_out_aa([], [])
6.67/2.60	   U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.67/2.60	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.67/2.60	   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
6.67/2.60	   U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.67/2.60	   U3_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> reverse_out_aa(.(X, Xs), Ys)
6.67/2.60	   U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.67/2.60	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U1_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.67/2.60	   app_in_ggg([], Ys, Ys) -> app_out_ggg([], Ys, Ys)
6.67/2.60	   U1_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.67/2.60	   U3_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> reverse_out_ag(.(X, Xs), Ys)
6.67/2.60	   reverse_in_ag([], []) -> reverse_out_ag([], [])
6.67/2.60	
6.67/2.60	The argument filtering Pi contains the following mapping:
6.67/2.60	reverse_in_ag(x1, x2)  =  reverse_in_ag(x2)
6.67/2.60	
6.67/2.60	U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
6.67/2.60	
6.67/2.60	reverse_in_aa(x1, x2)  =  reverse_in_aa
6.67/2.60	
6.67/2.60	U2_aa(x1, x2, x3, x4)  =  U2_aa(x4)
6.67/2.60	
6.67/2.60	reverse_out_aa(x1, x2)  =  reverse_out_aa(x1, x2)
6.67/2.60	
6.67/2.60	U3_aa(x1, x2, x3, x4)  =  U3_aa(x2, x4)
6.67/2.60	
6.67/2.60	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.67/2.60	
6.67/2.60	.(x1, x2)  =  .(x2)
6.67/2.60	
6.67/2.60	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
6.67/2.60	
6.67/2.60	[]  =  []
6.67/2.60	
6.67/2.60	app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
6.67/2.60	
6.67/2.60	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
6.67/2.60	
6.67/2.60	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.67/2.60	
6.67/2.60	U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x2, x3, x4, x5)
6.67/2.60	
6.67/2.60	app_out_ggg(x1, x2, x3)  =  app_out_ggg(x1, x2, x3)
6.67/2.60	
6.67/2.60	reverse_out_ag(x1, x2)  =  reverse_out_ag(x1, x2)
6.67/2.60	
6.67/2.60	APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
6.67/2.60	
6.67/2.60	
6.67/2.60	We have to consider all (P,R,Pi)-chains
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(40) UsableRulesProof (EQUIVALENT)
6.67/2.60	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(41)
6.67/2.60	Obligation:
6.67/2.60	Pi DP problem:
6.67/2.60	The TRS P consists of the following rules:
6.67/2.60	
6.67/2.60	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
6.67/2.60	
6.67/2.60	R is empty.
6.67/2.60	The argument filtering Pi contains the following mapping:
6.67/2.60	.(x1, x2)  =  .(x2)
6.67/2.60	
6.67/2.60	APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
6.67/2.60	
6.67/2.60	
6.67/2.60	We have to consider all (P,R,Pi)-chains
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(42) PiDPToQDPProof (SOUND)
6.67/2.60	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(43)
6.67/2.60	Obligation:
6.67/2.60	Q DP problem:
6.67/2.60	The TRS P consists of the following rules:
6.67/2.60	
6.67/2.60	   APP_IN_GGA(.(Xs), Ys) -> APP_IN_GGA(Xs, Ys)
6.67/2.60	
6.67/2.60	R is empty.
6.67/2.60	Q is empty.
6.67/2.60	We have to consider all (P,Q,R)-chains.
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(44) QDPSizeChangeProof (EQUIVALENT)
6.67/2.60	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. 
6.67/2.60	
6.67/2.60	From the DPs we obtained the following set of size-change graphs:
6.67/2.60	*APP_IN_GGA(.(Xs), Ys) -> APP_IN_GGA(Xs, Ys)
6.67/2.60	The graph contains the following edges 1 > 1, 2 >= 2
6.67/2.60	
6.67/2.60	
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(45)
6.67/2.60	YES
6.67/2.60	
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(46)
6.67/2.60	Obligation:
6.67/2.60	Pi DP problem:
6.67/2.60	The TRS P consists of the following rules:
6.67/2.60	
6.67/2.60	   REVERSE_IN_AA(.(X, Xs), Ys) -> REVERSE_IN_AA(Xs, Zs)
6.67/2.60	
6.67/2.60	The TRS R consists of the following rules:
6.67/2.60	
6.67/2.60	   reverse_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.60	   reverse_in_aa(.(X, Xs), Ys) -> U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
6.67/2.60	   reverse_in_aa([], []) -> reverse_out_aa([], [])
6.67/2.60	   U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_aa(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
6.67/2.60	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U1_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
6.67/2.60	   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
6.67/2.60	   U1_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
6.67/2.60	   U3_aa(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> reverse_out_aa(.(X, Xs), Ys)
6.67/2.60	   U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) -> U3_ag(X, Xs, Ys, app_in_ggg(Zs, .(X, []), Ys))
6.67/2.60	   app_in_ggg(.(X, Xs), Ys, .(X, Zs)) -> U1_ggg(X, Xs, Ys, Zs, app_in_ggg(Xs, Ys, Zs))
6.67/2.60	   app_in_ggg([], Ys, Ys) -> app_out_ggg([], Ys, Ys)
6.67/2.60	   U1_ggg(X, Xs, Ys, Zs, app_out_ggg(Xs, Ys, Zs)) -> app_out_ggg(.(X, Xs), Ys, .(X, Zs))
6.67/2.60	   U3_ag(X, Xs, Ys, app_out_ggg(Zs, .(X, []), Ys)) -> reverse_out_ag(.(X, Xs), Ys)
6.67/2.60	   reverse_in_ag([], []) -> reverse_out_ag([], [])
6.67/2.60	
6.67/2.60	The argument filtering Pi contains the following mapping:
6.67/2.60	reverse_in_ag(x1, x2)  =  reverse_in_ag(x2)
6.67/2.60	
6.67/2.60	U2_ag(x1, x2, x3, x4)  =  U2_ag(x3, x4)
6.67/2.60	
6.67/2.60	reverse_in_aa(x1, x2)  =  reverse_in_aa
6.67/2.60	
6.67/2.60	U2_aa(x1, x2, x3, x4)  =  U2_aa(x4)
6.67/2.60	
6.67/2.60	reverse_out_aa(x1, x2)  =  reverse_out_aa(x1, x2)
6.67/2.60	
6.67/2.60	U3_aa(x1, x2, x3, x4)  =  U3_aa(x2, x4)
6.67/2.60	
6.67/2.60	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
6.67/2.60	
6.67/2.60	.(x1, x2)  =  .(x2)
6.67/2.60	
6.67/2.60	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x3, x5)
6.67/2.60	
6.67/2.60	[]  =  []
6.67/2.60	
6.67/2.60	app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)
6.67/2.60	
6.67/2.60	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
6.67/2.60	
6.67/2.60	app_in_ggg(x1, x2, x3)  =  app_in_ggg(x1, x2, x3)
6.67/2.60	
6.67/2.60	U1_ggg(x1, x2, x3, x4, x5)  =  U1_ggg(x2, x3, x4, x5)
6.67/2.60	
6.67/2.60	app_out_ggg(x1, x2, x3)  =  app_out_ggg(x1, x2, x3)
6.67/2.60	
6.67/2.60	reverse_out_ag(x1, x2)  =  reverse_out_ag(x1, x2)
6.67/2.60	
6.67/2.60	REVERSE_IN_AA(x1, x2)  =  REVERSE_IN_AA
6.67/2.60	
6.67/2.60	
6.67/2.60	We have to consider all (P,R,Pi)-chains
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(47) UsableRulesProof (EQUIVALENT)
6.67/2.60	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(48)
6.67/2.60	Obligation:
6.67/2.60	Pi DP problem:
6.67/2.60	The TRS P consists of the following rules:
6.67/2.60	
6.67/2.60	   REVERSE_IN_AA(.(X, Xs), Ys) -> REVERSE_IN_AA(Xs, Zs)
6.67/2.60	
6.67/2.60	R is empty.
6.67/2.60	The argument filtering Pi contains the following mapping:
6.67/2.60	.(x1, x2)  =  .(x2)
6.67/2.60	
6.67/2.60	REVERSE_IN_AA(x1, x2)  =  REVERSE_IN_AA
6.67/2.60	
6.67/2.60	
6.67/2.60	We have to consider all (P,R,Pi)-chains
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(49) PiDPToQDPProof (SOUND)
6.67/2.60	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(50)
6.67/2.60	Obligation:
6.67/2.60	Q DP problem:
6.67/2.60	The TRS P consists of the following rules:
6.67/2.60	
6.67/2.60	   REVERSE_IN_AA -> REVERSE_IN_AA
6.67/2.60	
6.67/2.60	R is empty.
6.67/2.60	Q is empty.
6.67/2.60	We have to consider all (P,Q,R)-chains.
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(51) PrologToDTProblemTransformerProof (SOUND)
6.67/2.60	Built DT problem from termination graph DT10.
6.67/2.60	
6.67/2.60	{
6.67/2.60	    "root": 6,
6.67/2.60	    "program": {
6.67/2.60	        "directives": [],
6.67/2.60	        "clauses": [
6.67/2.60	            [
6.67/2.60	                "(app (. X Xs) Ys (. X Zs))",
6.67/2.60	                "(app Xs Ys Zs)"
6.67/2.60	            ],
6.67/2.60	            [
6.67/2.60	                "(app ([]) Ys Ys)",
6.67/2.60	                null
6.67/2.60	            ],
6.67/2.60	            [
6.67/2.60	                "(reverse (. X Xs) Ys)",
6.67/2.60	                "(',' (reverse Xs Zs) (app Zs (. X ([])) Ys))"
6.67/2.60	            ],
6.67/2.60	            [
6.67/2.60	                "(reverse ([]) ([]))",
6.67/2.60	                null
6.67/2.60	            ]
6.67/2.60	        ]
6.67/2.60	    },
6.67/2.60	    "graph": {
6.67/2.60	        "nodes": {
6.67/2.60	            "190": {
6.67/2.60	                "goal": [{
6.67/2.60	                    "clause": 2,
6.67/2.60	                    "scope": 3,
6.67/2.60	                    "term": "(reverse T21 X29)"
6.67/2.60	                }],
6.67/2.60	                "kb": {
6.67/2.60	                    "nonunifying": [],
6.67/2.60	                    "intvars": {},
6.67/2.60	                    "arithmetic": {
6.67/2.60	                        "type": "PlainIntegerRelationState",
6.67/2.60	                        "relations": []
6.67/2.60	                    },
6.67/2.60	                    "ground": [],
6.67/2.60	                    "free": ["X29"],
6.67/2.60	                    "exprvars": []
6.67/2.60	                }
6.67/2.60	            },
6.67/2.60	            "191": {
6.67/2.60	                "goal": [{
6.67/2.60	                    "clause": 3,
6.67/2.60	                    "scope": 3,
6.67/2.60	                    "term": "(reverse T21 X29)"
6.67/2.60	                }],
6.67/2.60	                "kb": {
6.67/2.60	                    "nonunifying": [],
6.67/2.60	                    "intvars": {},
6.67/2.60	                    "arithmetic": {
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6.67/2.60	                    },
6.67/2.60	                    "ground": [],
6.67/2.60	                    "free": ["X29"],
6.67/2.60	                    "exprvars": []
6.67/2.60	                }
6.67/2.60	            },
6.67/2.60	            "type": "Nodes",
6.67/2.60	            "194": {
6.67/2.60	                "goal": [{
6.67/2.60	                    "clause": -1,
6.67/2.60	                    "scope": -1,
6.67/2.60	                    "term": "(',' (reverse T41 X62) (app X62 (. T42 ([])) X63))"
6.67/2.60	                }],
6.67/2.60	                "kb": {
6.67/2.60	                    "nonunifying": [],
6.67/2.60	                    "intvars": {},
6.67/2.60	                    "arithmetic": {
6.67/2.60	                        "type": "PlainIntegerRelationState",
6.67/2.60	                        "relations": []
6.67/2.60	                    },
6.67/2.60	                    "ground": [],
6.67/2.60	                    "free": [
6.67/2.60	                        "X63",
6.67/2.60	                        "X62"
6.67/2.60	                    ],
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6.67/2.60	                }
6.67/2.60	            },
6.67/2.60	            "196": {
6.67/2.60	                "goal": [],
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6.67/2.60	                    "nonunifying": [],
6.67/2.60	                    "intvars": {},
6.67/2.60	                    "arithmetic": {
6.67/2.60	                        "type": "PlainIntegerRelationState",
6.67/2.60	                        "relations": []
6.67/2.60	                    },
6.67/2.60	                    "ground": [],
6.67/2.60	                    "free": [],
6.67/2.60	                    "exprvars": []
6.67/2.60	                }
6.67/2.60	            },
6.67/2.60	            "274": {
6.67/2.60	                "goal": [{
6.67/2.60	                    "clause": -1,
6.67/2.60	                    "scope": -1,
6.67/2.60	                    "term": "(true)"
6.67/2.60	                }],
6.67/2.60	                "kb": {
6.67/2.60	                    "nonunifying": [],
6.67/2.60	                    "intvars": {},
6.67/2.60	                    "arithmetic": {
6.67/2.60	                        "type": "PlainIntegerRelationState",
6.67/2.60	                        "relations": []
6.67/2.60	                    },
6.67/2.60	                    "ground": [],
6.67/2.60	                    "free": [],
6.67/2.60	                    "exprvars": []
6.67/2.60	                }
6.67/2.60	            },
6.67/2.60	            "110": {
6.67/2.60	                "goal": [{
6.67/2.60	                    "clause": 2,
6.67/2.60	                    "scope": 2,
6.67/2.60	                    "term": "(',' (reverse T9 X7) (app X7 (. T10 ([])) T8))"
6.67/2.60	                }],
6.67/2.60	                "kb": {
6.67/2.60	                    "nonunifying": [],
6.67/2.60	                    "intvars": {},
6.67/2.60	                    "arithmetic": {
6.67/2.60	                        "type": "PlainIntegerRelationState",
6.67/2.60	                        "relations": []
6.67/2.60	                    },
6.67/2.60	                    "ground": ["T8"],
6.67/2.60	                    "free": ["X7"],
6.67/2.60	                    "exprvars": []
6.67/2.60	                }
6.67/2.60	            },
6.67/2.60	            "275": {
6.67/2.60	                "goal": [],
6.67/2.60	                "kb": {
6.67/2.60	                    "nonunifying": [],
6.67/2.60	                    "intvars": {},
6.67/2.60	                    "arithmetic": {
6.67/2.60	                        "type": "PlainIntegerRelationState",
6.67/2.60	                        "relations": []
6.67/2.60	                    },
6.67/2.60	                    "ground": [],
6.67/2.60	                    "free": [],
6.67/2.60	                    "exprvars": []
6.67/2.60	                }
6.67/2.60	            },
6.67/2.60	            "111": {
6.67/2.60	                "goal": [
6.67/2.60	                    {
6.67/2.60	                        "clause": 3,
6.67/2.60	                        "scope": 2,
6.67/2.60	                        "term": "(',' (reverse T9 X7) (app X7 (. T10 ([])) T8))"
6.67/2.60	                    },
6.67/2.60	                    {
6.67/2.60	                        "clause": -1,
6.67/2.60	                        "scope": 2,
6.67/2.60	                        "term": null
6.67/2.60	                    },
6.67/2.60	                    {
6.67/2.60	                        "clause": 3,
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6.67/2.60	            },
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6.67/2.60	            },
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6.67/2.60	                }],
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6.67/2.60	            },
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6.67/2.60	                    },
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6.67/2.60	            },
6.67/2.60	            "257": {
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6.67/2.60	                        "clause": 0,
6.67/2.60	                        "scope": 4,
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6.67/2.60	                    },
6.67/2.60	                    {
6.67/2.60	                        "clause": 1,
6.67/2.60	                        "scope": 4,
6.67/2.60	                        "term": "(app T45 (. T46 ([])) X63)"
6.67/2.60	                    }
6.67/2.60	                ],
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6.67/2.60	            },
6.67/2.60	            "258": {
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6.67/2.60	                }],
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6.67/2.60	                }
6.67/2.60	            },
6.67/2.60	            "259": {
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6.67/2.60	                    "clause": 1,
6.67/2.60	                    "scope": 4,
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6.67/2.60	                }],
6.67/2.60	                "kb": {
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6.67/2.60	                        "type": "PlainIntegerRelationState",
6.67/2.60	                        "relations": []
6.67/2.60	                    },
6.67/2.60	                    "ground": [],
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6.67/2.60	                }
6.67/2.60	            },
6.67/2.60	            "181": {
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6.67/2.60	                    {
6.67/2.60	                        "clause": 2,
6.67/2.60	                        "scope": 3,
6.67/2.60	                        "term": "(reverse T21 X29)"
6.67/2.60	                    },
6.67/2.60	                    {
6.67/2.60	                        "clause": 3,
6.67/2.60	                        "scope": 3,
6.67/2.60	                        "term": "(reverse T21 X29)"
6.67/2.60	                    }
6.67/2.60	                ],
6.67/2.60	                "kb": {
6.67/2.60	                    "nonunifying": [],
6.67/2.60	                    "intvars": {},
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6.67/2.60	                        "type": "PlainIntegerRelationState",
6.67/2.60	                        "relations": []
6.67/2.60	                    },
6.67/2.60	                    "ground": [],
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6.67/2.60	                }
6.67/2.60	            },
6.67/2.60	            "260": {
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6.67/2.60	                    "clause": -1,
6.67/2.60	                    "scope": -1,
6.67/2.60	                    "term": "(app T67 (. T68 ([])) X103)"
6.67/2.60	                }],
6.67/2.60	                "kb": {
6.67/2.60	                    "nonunifying": [],
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6.67/2.60	                    "arithmetic": {
6.67/2.60	                        "type": "PlainIntegerRelationState",
6.67/2.60	                        "relations": []
6.67/2.60	                    },
6.67/2.60	                    "ground": [],
6.67/2.60	                    "free": ["X103"],
6.67/2.60	                    "exprvars": []
6.67/2.60	                }
6.67/2.60	            },
6.67/2.60	            "261": {
6.67/2.60	                "goal": [],
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6.67/2.60	                    "nonunifying": [],
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6.67/2.60	                    "arithmetic": {
6.67/2.60	                        "type": "PlainIntegerRelationState",
6.67/2.60	                        "relations": []
6.67/2.60	                    },
6.67/2.60	                    "ground": [],
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6.67/2.60	                    "exprvars": []
6.67/2.60	                }
6.67/2.60	            },
6.67/2.60	            "300": {
6.67/2.60	                "goal": [],
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6.67/2.60	                    "nonunifying": [],
6.67/2.60	                    "intvars": {},
6.67/2.60	                    "arithmetic": {
6.67/2.60	                        "type": "PlainIntegerRelationState",
6.67/2.60	                        "relations": []
6.67/2.60	                    },
6.67/2.60	                    "ground": [],
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6.67/2.60	                }
6.67/2.60	            },
6.67/2.60	            "107": {
6.67/2.60	                "goal": [
6.67/2.60	                    {
6.67/2.60	                        "clause": -1,
6.67/2.60	                        "scope": -1,
6.67/2.60	                        "term": "(',' (reverse T9 X7) (app X7 (. T10 ([])) T8))"
6.67/2.60	                    },
6.67/2.60	                    {
6.67/2.60	                        "clause": 3,
6.67/2.60	                        "scope": 1,
6.67/2.60	                        "term": "(reverse T1 T8)"
6.67/2.60	                    }
6.67/2.60	                ],
6.67/2.60	                "kb": {
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6.67/2.60	                    "intvars": {},
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6.67/2.60	                        "type": "PlainIntegerRelationState",
6.67/2.60	                        "relations": []
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6.67/2.60	                }
6.67/2.60	            },
6.67/2.60	            "108": {
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6.67/2.60	                    "clause": 3,
6.67/2.60	                    "scope": 1,
6.67/2.60	                    "term": "(reverse T1 T2)"
6.67/2.60	                }],
6.67/2.60	                "kb": {
6.67/2.60	                    "nonunifying": [[
6.67/2.60	                        "(reverse T1 T2)",
6.67/2.60	                        "(reverse (. X4 X5) X6)"
6.67/2.60	                    ]],
6.67/2.60	                    "intvars": {},
6.67/2.60	                    "arithmetic": {
6.67/2.60	                        "type": "PlainIntegerRelationState",
6.67/2.60	                        "relations": []
6.67/2.60	                    },
6.67/2.60	                    "ground": ["T2"],
6.67/2.60	                    "free": [
6.67/2.60	                        "X4",
6.67/2.60	                        "X5",
6.67/2.60	                        "X6"
6.67/2.60	                    ],
6.67/2.60	                    "exprvars": []
6.67/2.60	                }
6.67/2.60	            },
6.67/2.60	            "109": {
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6.67/2.60	                    {
6.67/2.60	                        "clause": 2,
6.67/2.60	                        "scope": 2,
6.67/2.60	                        "term": "(',' (reverse T9 X7) (app X7 (. T10 ([])) T8))"
6.67/2.60	                    },
6.67/2.60	                    {
6.67/2.60	                        "clause": 3,
6.67/2.60	                        "scope": 2,
6.67/2.60	                        "term": "(',' (reverse T9 X7) (app X7 (. T10 ([])) T8))"
6.67/2.60	                    },
6.67/2.60	                    {
6.67/2.60	                        "clause": -1,
6.67/2.60	                        "scope": 2,
6.67/2.60	                        "term": null
6.67/2.60	                    },
6.67/2.60	                    {
6.67/2.60	                        "clause": 3,
6.67/2.60	                        "scope": 1,
6.67/2.60	                        "term": "(reverse T1 T8)"
6.67/2.60	                    }
6.67/2.60	                ],
6.67/2.60	                "kb": {
6.67/2.60	                    "nonunifying": [],
6.67/2.60	                    "intvars": {},
6.67/2.60	                    "arithmetic": {
6.67/2.60	                        "type": "PlainIntegerRelationState",
6.67/2.60	                        "relations": []
6.67/2.60	                    },
6.67/2.60	                    "ground": ["T8"],
6.67/2.60	                    "free": ["X7"],
6.67/2.60	                    "exprvars": []
6.67/2.60	                }
6.67/2.60	            }
6.67/2.60	        },
6.67/2.60	        "edges": [
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6.67/2.60	                "from": 6,
6.67/2.60	                "to": 11,
6.67/2.60	                "label": "CASE"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 11,
6.67/2.60	                "to": 107,
6.67/2.60	                "label": "EVAL with clause\nreverse(.(X4, X5), X6) :- ','(reverse(X5, X7), app(X7, .(X4, []), X6)).\nand substitutionX4 -> T10,\nX5 -> T9,\nT1 -> .(T10, T9),\nT2 -> T8,\nX6 -> T8,\nT7 -> T9,\nT6 -> T10"
6.67/2.60	            },
6.67/2.60	            {
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6.67/2.60	            {
6.67/2.60	                "from": 108,
6.67/2.60	                "to": 298,
6.67/2.60	                "label": "EVAL with clause\nreverse([], []).\nand substitutionT1 -> [],\nT2 -> []"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 108,
6.67/2.60	                "to": 299,
6.67/2.60	                "label": "EVAL-BACKTRACK"
6.67/2.60	            },
6.67/2.60	            {
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6.67/2.60	                "to": 110,
6.67/2.60	                "label": "PARALLEL"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 109,
6.67/2.60	                "to": 111,
6.67/2.60	                "label": "PARALLEL"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 110,
6.67/2.60	                "to": 112,
6.67/2.60	                "label": "EVAL with clause\nreverse(.(X26, X27), X28) :- ','(reverse(X27, X29), app(X29, .(X26, []), X28)).\nand substitutionX26 -> T22,\nX27 -> T21,\nT9 -> .(T22, T21),\nX7 -> X30,\nX28 -> X30,\nT20 -> T21,\nT19 -> T22,\nT10 -> T23"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 110,
6.67/2.60	                "to": 113,
6.67/2.60	                "label": "EVAL-BACKTRACK"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 111,
6.67/2.60	                "to": 290,
6.67/2.60	                "label": "PARALLEL"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 111,
6.67/2.60	                "to": 291,
6.67/2.60	                "label": "PARALLEL"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 112,
6.67/2.60	                "to": 162,
6.67/2.60	                "label": "SPLIT 1"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 112,
6.67/2.60	                "to": 167,
6.67/2.60	                "label": "SPLIT 2\nreplacements:X29 -> T26,\nT22 -> T27,\nT23 -> T28"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 162,
6.67/2.60	                "to": 181,
6.67/2.60	                "label": "CASE"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 167,
6.67/2.60	                "to": 280,
6.67/2.60	                "label": "SPLIT 1"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 167,
6.67/2.60	                "to": 281,
6.67/2.60	                "label": "SPLIT 2\nreplacements:X30 -> T78,\nT28 -> T79"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 181,
6.67/2.60	                "to": 190,
6.67/2.60	                "label": "PARALLEL"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 181,
6.67/2.60	                "to": 191,
6.67/2.60	                "label": "PARALLEL"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 190,
6.67/2.60	                "to": 194,
6.67/2.60	                "label": "EVAL with clause\nreverse(.(X59, X60), X61) :- ','(reverse(X60, X62), app(X62, .(X59, []), X61)).\nand substitutionX59 -> T42,\nX60 -> T41,\nT21 -> .(T42, T41),\nX29 -> X63,\nX61 -> X63,\nT40 -> T41,\nT39 -> T42"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 190,
6.67/2.60	                "to": 196,
6.67/2.60	                "label": "EVAL-BACKTRACK"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 191,
6.67/2.60	                "to": 277,
6.67/2.60	                "label": "EVAL with clause\nreverse([], []).\nand substitutionT21 -> [],\nX29 -> []"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 191,
6.67/2.60	                "to": 278,
6.67/2.60	                "label": "EVAL-BACKTRACK"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 194,
6.67/2.60	                "to": 199,
6.67/2.60	                "label": "SPLIT 1"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 194,
6.67/2.60	                "to": 201,
6.67/2.60	                "label": "SPLIT 2\nreplacements:X62 -> T45,\nT42 -> T46"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 199,
6.67/2.60	                "to": 162,
6.67/2.60	                "label": "INSTANCE with matching:\nT21 -> T41\nX29 -> X62"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 201,
6.67/2.60	                "to": 257,
6.67/2.60	                "label": "CASE"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 257,
6.67/2.60	                "to": 258,
6.67/2.60	                "label": "PARALLEL"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 257,
6.67/2.60	                "to": 259,
6.67/2.60	                "label": "PARALLEL"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 258,
6.67/2.60	                "to": 260,
6.67/2.60	                "label": "EVAL with clause\napp(.(X99, X100), X101, .(X99, X102)) :- app(X100, X101, X102).\nand substitutionX99 -> T64,\nX100 -> T67,\nT45 -> .(T64, T67),\nT46 -> T68,\nX101 -> .(T68, []),\nX102 -> X103,\nX63 -> .(T64, X103),\nT65 -> T67,\nT66 -> T68"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 258,
6.67/2.60	                "to": 261,
6.67/2.60	                "label": "EVAL-BACKTRACK"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 259,
6.67/2.60	                "to": 274,
6.67/2.60	                "label": "EVAL with clause\napp([], X111, X111).\nand substitutionT45 -> [],\nT46 -> T74,\nX111 -> .(T74, []),\nX63 -> .(T74, [])"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 259,
6.67/2.60	                "to": 275,
6.67/2.60	                "label": "EVAL-BACKTRACK"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 260,
6.67/2.60	                "to": 201,
6.67/2.60	                "label": "INSTANCE with matching:\nT45 -> T67\nT46 -> T68\nX63 -> X103"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 274,
6.67/2.60	                "to": 276,
6.67/2.60	                "label": "SUCCESS"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 277,
6.67/2.60	                "to": 279,
6.67/2.60	                "label": "SUCCESS"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 280,
6.67/2.60	                "to": 201,
6.67/2.60	                "label": "INSTANCE with matching:\nT45 -> T26\nT46 -> T27\nX63 -> X30"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 281,
6.67/2.60	                "to": 282,
6.67/2.60	                "label": "CASE"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 282,
6.67/2.60	                "to": 283,
6.67/2.60	                "label": "PARALLEL"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 282,
6.67/2.60	                "to": 284,
6.67/2.60	                "label": "PARALLEL"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 283,
6.67/2.60	                "to": 285,
6.67/2.60	                "label": "EVAL with clause\napp(.(X142, X143), X144, .(X142, X145)) :- app(X143, X144, X145).\nand substitutionX142 -> T103,\nX143 -> T107,\nT78 -> .(T103, T107),\nT79 -> T108,\nX144 -> .(T108, []),\nX145 -> T106,\nT8 -> .(T103, T106),\nT104 -> T107,\nT105 -> T108"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 283,
6.67/2.60	                "to": 286,
6.67/2.60	                "label": "EVAL-BACKTRACK"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 284,
6.67/2.60	                "to": 287,
6.67/2.60	                "label": "EVAL with clause\napp([], X152, X152).\nand substitutionT78 -> [],\nT79 -> T115,\nX152 -> .(T115, []),\nT8 -> .(T115, [])"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 284,
6.67/2.60	                "to": 288,
6.67/2.60	                "label": "EVAL-BACKTRACK"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 285,
6.67/2.60	                "to": 281,
6.67/2.60	                "label": "INSTANCE with matching:\nT78 -> T107\nT79 -> T108\nT8 -> T106"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 287,
6.67/2.60	                "to": 289,
6.67/2.60	                "label": "SUCCESS"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 290,
6.67/2.60	                "to": 292,
6.67/2.60	                "label": "EVAL with clause\nreverse([], []).\nand substitutionT9 -> [],\nX7 -> [],\nT10 -> T116"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 290,
6.67/2.60	                "to": 293,
6.67/2.60	                "label": "EVAL-BACKTRACK"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 291,
6.67/2.60	                "to": 294,
6.67/2.60	                "label": "FAILURE"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 292,
6.67/2.60	                "to": 281,
6.67/2.60	                "label": "INSTANCE with matching:\nT78 -> []\nT79 -> T116"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 294,
6.67/2.60	                "to": 295,
6.67/2.60	                "label": "EVAL with clause\nreverse([], []).\nand substitutionT1 -> [],\nT8 -> []"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 294,
6.67/2.60	                "to": 296,
6.67/2.60	                "label": "EVAL-BACKTRACK"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 295,
6.67/2.60	                "to": 297,
6.67/2.60	                "label": "SUCCESS"
6.67/2.60	            },
6.67/2.60	            {
6.67/2.60	                "from": 298,
6.67/2.60	                "to": 300,
6.67/2.60	                "label": "SUCCESS"
6.67/2.60	            }
6.67/2.60	        ],
6.67/2.60	        "type": "Graph"
6.67/2.60	    }
6.67/2.60	}
6.67/2.60	
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(52)
6.67/2.60	Obligation:
6.67/2.60	Triples:
6.67/2.60	
6.67/2.60	reverseA(.(X1, X2), X3) :- reverseA(X2, X4).
6.67/2.60	reverseA(.(X1, X2), X3) :- ','(reversecA(X2, X4), appB(X4, X1, X3)).
6.67/2.60	appB(.(X1, X2), X3, .(X1, X4)) :- appB(X2, X3, X4).
6.67/2.60	appC(.(X1, X2), X3, .(X1, X4)) :- appC(X2, X3, X4).
6.67/2.60	reverseD(.(X1, .(X2, X3)), X4) :- reverseA(X3, X5).
6.67/2.60	reverseD(.(X1, .(X2, X3)), X4) :- ','(reversecA(X3, X5), appB(X5, X2, X6)).
6.67/2.60	reverseD(.(X1, .(X2, X3)), X4) :- ','(reversecA(X3, X5), ','(appcB(X5, X2, X6), appC(X6, X1, X4))).
6.67/2.60	reverseD(.(X1, []), X2) :- appC([], X1, X2).
6.67/2.60	
6.67/2.60	Clauses:
6.67/2.60	
6.67/2.60	reversecA(.(X1, X2), X3) :- ','(reversecA(X2, X4), appcB(X4, X1, X3)).
6.67/2.60	reversecA([], []).
6.67/2.60	appcB(.(X1, X2), X3, .(X1, X4)) :- appcB(X2, X3, X4).
6.67/2.60	appcB([], X1, .(X1, [])).
6.67/2.60	appcC(.(X1, X2), X3, .(X1, X4)) :- appcC(X2, X3, X4).
6.67/2.60	appcC([], X1, .(X1, [])).
6.67/2.60	
6.67/2.60	Afs:
6.67/2.60	
6.67/2.60	reverseD(x1, x2)  =  reverseD(x2)
6.67/2.60	
6.67/2.60	
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(53) TriplesToPiDPProof (SOUND)
6.67/2.60	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.67/2.60	
6.67/2.60	reverseD_in_2: (f,b)
6.67/2.60	
6.67/2.60	reverseA_in_2: (f,f)
6.67/2.60	
6.67/2.60	reversecA_in_2: (f,f)
6.67/2.60	
6.67/2.60	appcB_in_3: (b,f,f)
6.67/2.60	
6.67/2.60	appB_in_3: (b,f,f)
6.67/2.60	
6.67/2.60	appC_in_3: (b,f,b)
6.67/2.60	
6.67/2.60	Transforming TRIPLES into the following Term Rewriting System:
6.67/2.60	
6.67/2.60	Pi DP problem:
6.67/2.60	The TRS P consists of the following rules:
6.67/2.60	
6.67/2.60	   REVERSED_IN_AG(.(X1, .(X2, X3)), X4) -> U6_AG(X1, X2, X3, X4, reverseA_in_aa(X3, X5))
6.67/2.60	   REVERSED_IN_AG(.(X1, .(X2, X3)), X4) -> REVERSEA_IN_AA(X3, X5)
6.67/2.60	   REVERSEA_IN_AA(.(X1, X2), X3) -> U1_AA(X1, X2, X3, reverseA_in_aa(X2, X4))
6.67/2.60	   REVERSEA_IN_AA(.(X1, X2), X3) -> REVERSEA_IN_AA(X2, X4)
6.67/2.60	   REVERSEA_IN_AA(.(X1, X2), X3) -> U2_AA(X1, X2, X3, reversecA_in_aa(X2, X4))
6.67/2.60	   U2_AA(X1, X2, X3, reversecA_out_aa(X2, X4)) -> U3_AA(X1, X2, X3, appB_in_gaa(X4, X1, X3))
6.67/2.60	   U2_AA(X1, X2, X3, reversecA_out_aa(X2, X4)) -> APPB_IN_GAA(X4, X1, X3)
6.67/2.60	   APPB_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> U4_GAA(X1, X2, X3, X4, appB_in_gaa(X2, X3, X4))
6.67/2.60	   APPB_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPB_IN_GAA(X2, X3, X4)
6.67/2.60	   REVERSED_IN_AG(.(X1, .(X2, X3)), X4) -> U7_AG(X1, X2, X3, X4, reversecA_in_aa(X3, X5))
6.67/2.60	   U7_AG(X1, X2, X3, X4, reversecA_out_aa(X3, X5)) -> U8_AG(X1, X2, X3, X4, appB_in_gaa(X5, X2, X6))
6.67/2.60	   U7_AG(X1, X2, X3, X4, reversecA_out_aa(X3, X5)) -> APPB_IN_GAA(X5, X2, X6)
6.67/2.60	   U7_AG(X1, X2, X3, X4, reversecA_out_aa(X3, X5)) -> U9_AG(X1, X2, X3, X4, appcB_in_gaa(X5, X2, X6))
6.67/2.60	   U9_AG(X1, X2, X3, X4, appcB_out_gaa(X5, X2, X6)) -> U10_AG(X1, X2, X3, X4, appC_in_gag(X6, X1, X4))
6.67/2.60	   U9_AG(X1, X2, X3, X4, appcB_out_gaa(X5, X2, X6)) -> APPC_IN_GAG(X6, X1, X4)
6.67/2.60	   APPC_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> U5_GAG(X1, X2, X3, X4, appC_in_gag(X2, X3, X4))
6.67/2.60	   APPC_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPC_IN_GAG(X2, X3, X4)
6.67/2.60	   REVERSED_IN_AG(.(X1, []), X2) -> U11_AG(X1, X2, appC_in_gag([], X1, X2))
6.67/2.60	   REVERSED_IN_AG(.(X1, []), X2) -> APPC_IN_GAG([], X1, X2)
6.67/2.60	
6.67/2.60	The TRS R consists of the following rules:
6.67/2.60	
6.67/2.60	   reversecA_in_aa(.(X1, X2), X3) -> U13_aa(X1, X2, X3, reversecA_in_aa(X2, X4))
6.67/2.60	   reversecA_in_aa([], []) -> reversecA_out_aa([], [])
6.67/2.60	   U13_aa(X1, X2, X3, reversecA_out_aa(X2, X4)) -> U14_aa(X1, X2, X3, appcB_in_gaa(X4, X1, X3))
6.67/2.60	   appcB_in_gaa(.(X1, X2), X3, .(X1, X4)) -> U15_gaa(X1, X2, X3, X4, appcB_in_gaa(X2, X3, X4))
6.67/2.60	   appcB_in_gaa([], X1, .(X1, [])) -> appcB_out_gaa([], X1, .(X1, []))
6.67/2.60	   U15_gaa(X1, X2, X3, X4, appcB_out_gaa(X2, X3, X4)) -> appcB_out_gaa(.(X1, X2), X3, .(X1, X4))
6.67/2.60	   U14_aa(X1, X2, X3, appcB_out_gaa(X4, X1, X3)) -> reversecA_out_aa(.(X1, X2), X3)
6.67/2.60	
6.67/2.60	The argument filtering Pi contains the following mapping:
6.67/2.60	reverseA_in_aa(x1, x2)  =  reverseA_in_aa
6.67/2.60	
6.67/2.60	reversecA_in_aa(x1, x2)  =  reversecA_in_aa
6.67/2.60	
6.67/2.60	U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
6.67/2.60	
6.67/2.60	reversecA_out_aa(x1, x2)  =  reversecA_out_aa(x1, x2)
6.67/2.60	
6.67/2.60	U14_aa(x1, x2, x3, x4)  =  U14_aa(x2, x4)
6.67/2.60	
6.67/2.60	appcB_in_gaa(x1, x2, x3)  =  appcB_in_gaa(x1)
6.67/2.60	
6.67/2.60	.(x1, x2)  =  .(x2)
6.67/2.60	
6.67/2.60	U15_gaa(x1, x2, x3, x4, x5)  =  U15_gaa(x2, x5)
6.67/2.60	
6.67/2.60	[]  =  []
6.67/2.60	
6.67/2.60	appcB_out_gaa(x1, x2, x3)  =  appcB_out_gaa(x1, x3)
6.67/2.60	
6.67/2.60	appB_in_gaa(x1, x2, x3)  =  appB_in_gaa(x1)
6.67/2.60	
6.67/2.60	appC_in_gag(x1, x2, x3)  =  appC_in_gag(x1, x3)
6.67/2.60	
6.67/2.60	REVERSED_IN_AG(x1, x2)  =  REVERSED_IN_AG(x2)
6.67/2.60	
6.67/2.60	U6_AG(x1, x2, x3, x4, x5)  =  U6_AG(x4, x5)
6.67/2.60	
6.67/2.60	REVERSEA_IN_AA(x1, x2)  =  REVERSEA_IN_AA
6.67/2.60	
6.67/2.60	U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
6.67/2.60	
6.67/2.60	U2_AA(x1, x2, x3, x4)  =  U2_AA(x4)
6.67/2.60	
6.67/2.60	U3_AA(x1, x2, x3, x4)  =  U3_AA(x2, x4)
6.67/2.60	
6.67/2.60	APPB_IN_GAA(x1, x2, x3)  =  APPB_IN_GAA(x1)
6.67/2.60	
6.67/2.60	U4_GAA(x1, x2, x3, x4, x5)  =  U4_GAA(x2, x5)
6.67/2.60	
6.67/2.60	U7_AG(x1, x2, x3, x4, x5)  =  U7_AG(x4, x5)
6.67/2.60	
6.67/2.60	U8_AG(x1, x2, x3, x4, x5)  =  U8_AG(x3, x4, x5)
6.67/2.60	
6.67/2.60	U9_AG(x1, x2, x3, x4, x5)  =  U9_AG(x3, x4, x5)
6.67/2.60	
6.67/2.60	U10_AG(x1, x2, x3, x4, x5)  =  U10_AG(x3, x4, x5)
6.67/2.60	
6.67/2.60	APPC_IN_GAG(x1, x2, x3)  =  APPC_IN_GAG(x1, x3)
6.67/2.60	
6.67/2.60	U5_GAG(x1, x2, x3, x4, x5)  =  U5_GAG(x2, x4, x5)
6.67/2.60	
6.67/2.60	U11_AG(x1, x2, x3)  =  U11_AG(x2, x3)
6.67/2.60	
6.67/2.60	
6.67/2.60	We have to consider all (P,R,Pi)-chains
6.67/2.60	
6.67/2.60	
6.67/2.60	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
6.67/2.60	
6.67/2.60	
6.67/2.60	
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(54)
6.67/2.60	Obligation:
6.67/2.60	Pi DP problem:
6.67/2.60	The TRS P consists of the following rules:
6.67/2.60	
6.67/2.60	   REVERSED_IN_AG(.(X1, .(X2, X3)), X4) -> U6_AG(X1, X2, X3, X4, reverseA_in_aa(X3, X5))
6.67/2.60	   REVERSED_IN_AG(.(X1, .(X2, X3)), X4) -> REVERSEA_IN_AA(X3, X5)
6.67/2.60	   REVERSEA_IN_AA(.(X1, X2), X3) -> U1_AA(X1, X2, X3, reverseA_in_aa(X2, X4))
6.67/2.60	   REVERSEA_IN_AA(.(X1, X2), X3) -> REVERSEA_IN_AA(X2, X4)
6.67/2.60	   REVERSEA_IN_AA(.(X1, X2), X3) -> U2_AA(X1, X2, X3, reversecA_in_aa(X2, X4))
6.67/2.60	   U2_AA(X1, X2, X3, reversecA_out_aa(X2, X4)) -> U3_AA(X1, X2, X3, appB_in_gaa(X4, X1, X3))
6.67/2.60	   U2_AA(X1, X2, X3, reversecA_out_aa(X2, X4)) -> APPB_IN_GAA(X4, X1, X3)
6.67/2.60	   APPB_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> U4_GAA(X1, X2, X3, X4, appB_in_gaa(X2, X3, X4))
6.67/2.60	   APPB_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPB_IN_GAA(X2, X3, X4)
6.67/2.60	   REVERSED_IN_AG(.(X1, .(X2, X3)), X4) -> U7_AG(X1, X2, X3, X4, reversecA_in_aa(X3, X5))
6.67/2.60	   U7_AG(X1, X2, X3, X4, reversecA_out_aa(X3, X5)) -> U8_AG(X1, X2, X3, X4, appB_in_gaa(X5, X2, X6))
6.67/2.60	   U7_AG(X1, X2, X3, X4, reversecA_out_aa(X3, X5)) -> APPB_IN_GAA(X5, X2, X6)
6.67/2.60	   U7_AG(X1, X2, X3, X4, reversecA_out_aa(X3, X5)) -> U9_AG(X1, X2, X3, X4, appcB_in_gaa(X5, X2, X6))
6.67/2.60	   U9_AG(X1, X2, X3, X4, appcB_out_gaa(X5, X2, X6)) -> U10_AG(X1, X2, X3, X4, appC_in_gag(X6, X1, X4))
6.67/2.60	   U9_AG(X1, X2, X3, X4, appcB_out_gaa(X5, X2, X6)) -> APPC_IN_GAG(X6, X1, X4)
6.67/2.60	   APPC_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> U5_GAG(X1, X2, X3, X4, appC_in_gag(X2, X3, X4))
6.67/2.60	   APPC_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPC_IN_GAG(X2, X3, X4)
6.67/2.60	   REVERSED_IN_AG(.(X1, []), X2) -> U11_AG(X1, X2, appC_in_gag([], X1, X2))
6.67/2.60	   REVERSED_IN_AG(.(X1, []), X2) -> APPC_IN_GAG([], X1, X2)
6.67/2.60	
6.67/2.60	The TRS R consists of the following rules:
6.67/2.60	
6.67/2.60	   reversecA_in_aa(.(X1, X2), X3) -> U13_aa(X1, X2, X3, reversecA_in_aa(X2, X4))
6.67/2.60	   reversecA_in_aa([], []) -> reversecA_out_aa([], [])
6.67/2.60	   U13_aa(X1, X2, X3, reversecA_out_aa(X2, X4)) -> U14_aa(X1, X2, X3, appcB_in_gaa(X4, X1, X3))
6.67/2.60	   appcB_in_gaa(.(X1, X2), X3, .(X1, X4)) -> U15_gaa(X1, X2, X3, X4, appcB_in_gaa(X2, X3, X4))
6.67/2.60	   appcB_in_gaa([], X1, .(X1, [])) -> appcB_out_gaa([], X1, .(X1, []))
6.67/2.60	   U15_gaa(X1, X2, X3, X4, appcB_out_gaa(X2, X3, X4)) -> appcB_out_gaa(.(X1, X2), X3, .(X1, X4))
6.67/2.60	   U14_aa(X1, X2, X3, appcB_out_gaa(X4, X1, X3)) -> reversecA_out_aa(.(X1, X2), X3)
6.67/2.60	
6.67/2.60	The argument filtering Pi contains the following mapping:
6.67/2.60	reverseA_in_aa(x1, x2)  =  reverseA_in_aa
6.67/2.60	
6.67/2.60	reversecA_in_aa(x1, x2)  =  reversecA_in_aa
6.67/2.60	
6.67/2.60	U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
6.67/2.60	
6.67/2.60	reversecA_out_aa(x1, x2)  =  reversecA_out_aa(x1, x2)
6.67/2.60	
6.67/2.60	U14_aa(x1, x2, x3, x4)  =  U14_aa(x2, x4)
6.67/2.60	
6.67/2.60	appcB_in_gaa(x1, x2, x3)  =  appcB_in_gaa(x1)
6.67/2.60	
6.67/2.60	.(x1, x2)  =  .(x2)
6.67/2.60	
6.67/2.60	U15_gaa(x1, x2, x3, x4, x5)  =  U15_gaa(x2, x5)
6.67/2.60	
6.67/2.60	[]  =  []
6.67/2.60	
6.67/2.60	appcB_out_gaa(x1, x2, x3)  =  appcB_out_gaa(x1, x3)
6.67/2.60	
6.67/2.60	appB_in_gaa(x1, x2, x3)  =  appB_in_gaa(x1)
6.67/2.60	
6.67/2.60	appC_in_gag(x1, x2, x3)  =  appC_in_gag(x1, x3)
6.67/2.60	
6.67/2.60	REVERSED_IN_AG(x1, x2)  =  REVERSED_IN_AG(x2)
6.67/2.60	
6.67/2.60	U6_AG(x1, x2, x3, x4, x5)  =  U6_AG(x4, x5)
6.67/2.60	
6.67/2.60	REVERSEA_IN_AA(x1, x2)  =  REVERSEA_IN_AA
6.67/2.60	
6.67/2.60	U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)
6.67/2.60	
6.67/2.60	U2_AA(x1, x2, x3, x4)  =  U2_AA(x4)
6.67/2.60	
6.67/2.60	U3_AA(x1, x2, x3, x4)  =  U3_AA(x2, x4)
6.67/2.60	
6.67/2.60	APPB_IN_GAA(x1, x2, x3)  =  APPB_IN_GAA(x1)
6.67/2.60	
6.67/2.60	U4_GAA(x1, x2, x3, x4, x5)  =  U4_GAA(x2, x5)
6.67/2.60	
6.67/2.60	U7_AG(x1, x2, x3, x4, x5)  =  U7_AG(x4, x5)
6.67/2.60	
6.67/2.60	U8_AG(x1, x2, x3, x4, x5)  =  U8_AG(x3, x4, x5)
6.67/2.60	
6.67/2.60	U9_AG(x1, x2, x3, x4, x5)  =  U9_AG(x3, x4, x5)
6.67/2.60	
6.67/2.60	U10_AG(x1, x2, x3, x4, x5)  =  U10_AG(x3, x4, x5)
6.67/2.60	
6.67/2.60	APPC_IN_GAG(x1, x2, x3)  =  APPC_IN_GAG(x1, x3)
6.67/2.60	
6.67/2.60	U5_GAG(x1, x2, x3, x4, x5)  =  U5_GAG(x2, x4, x5)
6.67/2.60	
6.67/2.60	U11_AG(x1, x2, x3)  =  U11_AG(x2, x3)
6.67/2.60	
6.67/2.60	
6.67/2.60	We have to consider all (P,R,Pi)-chains
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(55) DependencyGraphProof (EQUIVALENT)
6.67/2.60	The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 16 less nodes.
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(56)
6.67/2.60	Complex Obligation (AND)
6.67/2.60	
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(57)
6.67/2.60	Obligation:
6.67/2.60	Pi DP problem:
6.67/2.60	The TRS P consists of the following rules:
6.67/2.60	
6.67/2.60	   APPC_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPC_IN_GAG(X2, X3, X4)
6.67/2.60	
6.67/2.60	The TRS R consists of the following rules:
6.67/2.60	
6.67/2.60	   reversecA_in_aa(.(X1, X2), X3) -> U13_aa(X1, X2, X3, reversecA_in_aa(X2, X4))
6.67/2.60	   reversecA_in_aa([], []) -> reversecA_out_aa([], [])
6.67/2.60	   U13_aa(X1, X2, X3, reversecA_out_aa(X2, X4)) -> U14_aa(X1, X2, X3, appcB_in_gaa(X4, X1, X3))
6.67/2.60	   appcB_in_gaa(.(X1, X2), X3, .(X1, X4)) -> U15_gaa(X1, X2, X3, X4, appcB_in_gaa(X2, X3, X4))
6.67/2.60	   appcB_in_gaa([], X1, .(X1, [])) -> appcB_out_gaa([], X1, .(X1, []))
6.67/2.60	   U15_gaa(X1, X2, X3, X4, appcB_out_gaa(X2, X3, X4)) -> appcB_out_gaa(.(X1, X2), X3, .(X1, X4))
6.67/2.60	   U14_aa(X1, X2, X3, appcB_out_gaa(X4, X1, X3)) -> reversecA_out_aa(.(X1, X2), X3)
6.67/2.60	
6.67/2.60	The argument filtering Pi contains the following mapping:
6.67/2.60	reversecA_in_aa(x1, x2)  =  reversecA_in_aa
6.67/2.60	
6.67/2.60	U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
6.67/2.60	
6.67/2.60	reversecA_out_aa(x1, x2)  =  reversecA_out_aa(x1, x2)
6.67/2.60	
6.67/2.60	U14_aa(x1, x2, x3, x4)  =  U14_aa(x2, x4)
6.67/2.60	
6.67/2.60	appcB_in_gaa(x1, x2, x3)  =  appcB_in_gaa(x1)
6.67/2.60	
6.67/2.60	.(x1, x2)  =  .(x2)
6.67/2.60	
6.67/2.60	U15_gaa(x1, x2, x3, x4, x5)  =  U15_gaa(x2, x5)
6.67/2.60	
6.67/2.60	[]  =  []
6.67/2.60	
6.67/2.60	appcB_out_gaa(x1, x2, x3)  =  appcB_out_gaa(x1, x3)
6.67/2.60	
6.67/2.60	APPC_IN_GAG(x1, x2, x3)  =  APPC_IN_GAG(x1, x3)
6.67/2.60	
6.67/2.60	
6.67/2.60	We have to consider all (P,R,Pi)-chains
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(58) UsableRulesProof (EQUIVALENT)
6.67/2.60	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(59)
6.67/2.60	Obligation:
6.67/2.60	Pi DP problem:
6.67/2.60	The TRS P consists of the following rules:
6.67/2.60	
6.67/2.60	   APPC_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPC_IN_GAG(X2, X3, X4)
6.67/2.60	
6.67/2.60	R is empty.
6.67/2.60	The argument filtering Pi contains the following mapping:
6.67/2.60	.(x1, x2)  =  .(x2)
6.67/2.60	
6.67/2.60	APPC_IN_GAG(x1, x2, x3)  =  APPC_IN_GAG(x1, x3)
6.67/2.60	
6.67/2.60	
6.67/2.60	We have to consider all (P,R,Pi)-chains
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(60) PiDPToQDPProof (SOUND)
6.67/2.60	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(61)
6.67/2.60	Obligation:
6.67/2.60	Q DP problem:
6.67/2.60	The TRS P consists of the following rules:
6.67/2.60	
6.67/2.60	   APPC_IN_GAG(.(X2), .(X4)) -> APPC_IN_GAG(X2, X4)
6.67/2.60	
6.67/2.60	R is empty.
6.67/2.60	Q is empty.
6.67/2.60	We have to consider all (P,Q,R)-chains.
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(62) QDPSizeChangeProof (EQUIVALENT)
6.67/2.60	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. 
6.67/2.60	
6.67/2.60	From the DPs we obtained the following set of size-change graphs:
6.67/2.60	*APPC_IN_GAG(.(X2), .(X4)) -> APPC_IN_GAG(X2, X4)
6.67/2.60	The graph contains the following edges 1 > 1, 2 > 2
6.67/2.60	
6.67/2.60	
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(63)
6.67/2.60	YES
6.67/2.60	
6.67/2.60	----------------------------------------
6.67/2.60	
6.67/2.60	(64)
6.67/2.60	Obligation:
6.67/2.60	Pi DP problem:
6.67/2.60	The TRS P consists of the following rules:
6.67/2.61	
6.67/2.61	   APPB_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPB_IN_GAA(X2, X3, X4)
6.67/2.61	
6.67/2.61	The TRS R consists of the following rules:
6.67/2.61	
6.67/2.61	   reversecA_in_aa(.(X1, X2), X3) -> U13_aa(X1, X2, X3, reversecA_in_aa(X2, X4))
6.67/2.61	   reversecA_in_aa([], []) -> reversecA_out_aa([], [])
6.67/2.61	   U13_aa(X1, X2, X3, reversecA_out_aa(X2, X4)) -> U14_aa(X1, X2, X3, appcB_in_gaa(X4, X1, X3))
6.67/2.61	   appcB_in_gaa(.(X1, X2), X3, .(X1, X4)) -> U15_gaa(X1, X2, X3, X4, appcB_in_gaa(X2, X3, X4))
6.67/2.61	   appcB_in_gaa([], X1, .(X1, [])) -> appcB_out_gaa([], X1, .(X1, []))
6.67/2.61	   U15_gaa(X1, X2, X3, X4, appcB_out_gaa(X2, X3, X4)) -> appcB_out_gaa(.(X1, X2), X3, .(X1, X4))
6.67/2.61	   U14_aa(X1, X2, X3, appcB_out_gaa(X4, X1, X3)) -> reversecA_out_aa(.(X1, X2), X3)
6.67/2.61	
6.67/2.61	The argument filtering Pi contains the following mapping:
6.67/2.61	reversecA_in_aa(x1, x2)  =  reversecA_in_aa
6.67/2.61	
6.67/2.61	U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
6.67/2.61	
6.67/2.61	reversecA_out_aa(x1, x2)  =  reversecA_out_aa(x1, x2)
6.67/2.61	
6.67/2.61	U14_aa(x1, x2, x3, x4)  =  U14_aa(x2, x4)
6.67/2.61	
6.67/2.61	appcB_in_gaa(x1, x2, x3)  =  appcB_in_gaa(x1)
6.67/2.61	
6.67/2.61	.(x1, x2)  =  .(x2)
6.67/2.61	
6.67/2.61	U15_gaa(x1, x2, x3, x4, x5)  =  U15_gaa(x2, x5)
6.67/2.61	
6.67/2.61	[]  =  []
6.67/2.61	
6.67/2.61	appcB_out_gaa(x1, x2, x3)  =  appcB_out_gaa(x1, x3)
6.67/2.61	
6.67/2.61	APPB_IN_GAA(x1, x2, x3)  =  APPB_IN_GAA(x1)
6.67/2.61	
6.67/2.61	
6.67/2.61	We have to consider all (P,R,Pi)-chains
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(65) UsableRulesProof (EQUIVALENT)
6.67/2.61	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(66)
6.67/2.61	Obligation:
6.67/2.61	Pi DP problem:
6.67/2.61	The TRS P consists of the following rules:
6.67/2.61	
6.67/2.61	   APPB_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPB_IN_GAA(X2, X3, X4)
6.67/2.61	
6.67/2.61	R is empty.
6.67/2.61	The argument filtering Pi contains the following mapping:
6.67/2.61	.(x1, x2)  =  .(x2)
6.67/2.61	
6.67/2.61	APPB_IN_GAA(x1, x2, x3)  =  APPB_IN_GAA(x1)
6.67/2.61	
6.67/2.61	
6.67/2.61	We have to consider all (P,R,Pi)-chains
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(67) PiDPToQDPProof (SOUND)
6.67/2.61	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(68)
6.67/2.61	Obligation:
6.67/2.61	Q DP problem:
6.67/2.61	The TRS P consists of the following rules:
6.67/2.61	
6.67/2.61	   APPB_IN_GAA(.(X2)) -> APPB_IN_GAA(X2)
6.67/2.61	
6.67/2.61	R is empty.
6.67/2.61	Q is empty.
6.67/2.61	We have to consider all (P,Q,R)-chains.
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(69) QDPSizeChangeProof (EQUIVALENT)
6.67/2.61	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. 
6.67/2.61	
6.67/2.61	From the DPs we obtained the following set of size-change graphs:
6.67/2.61	*APPB_IN_GAA(.(X2)) -> APPB_IN_GAA(X2)
6.67/2.61	The graph contains the following edges 1 > 1
6.67/2.61	
6.67/2.61	
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(70)
6.67/2.61	YES
6.67/2.61	
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(71)
6.67/2.61	Obligation:
6.67/2.61	Pi DP problem:
6.67/2.61	The TRS P consists of the following rules:
6.67/2.61	
6.67/2.61	   REVERSEA_IN_AA(.(X1, X2), X3) -> REVERSEA_IN_AA(X2, X4)
6.67/2.61	
6.67/2.61	The TRS R consists of the following rules:
6.67/2.61	
6.67/2.61	   reversecA_in_aa(.(X1, X2), X3) -> U13_aa(X1, X2, X3, reversecA_in_aa(X2, X4))
6.67/2.61	   reversecA_in_aa([], []) -> reversecA_out_aa([], [])
6.67/2.61	   U13_aa(X1, X2, X3, reversecA_out_aa(X2, X4)) -> U14_aa(X1, X2, X3, appcB_in_gaa(X4, X1, X3))
6.67/2.61	   appcB_in_gaa(.(X1, X2), X3, .(X1, X4)) -> U15_gaa(X1, X2, X3, X4, appcB_in_gaa(X2, X3, X4))
6.67/2.61	   appcB_in_gaa([], X1, .(X1, [])) -> appcB_out_gaa([], X1, .(X1, []))
6.67/2.61	   U15_gaa(X1, X2, X3, X4, appcB_out_gaa(X2, X3, X4)) -> appcB_out_gaa(.(X1, X2), X3, .(X1, X4))
6.67/2.61	   U14_aa(X1, X2, X3, appcB_out_gaa(X4, X1, X3)) -> reversecA_out_aa(.(X1, X2), X3)
6.67/2.61	
6.67/2.61	The argument filtering Pi contains the following mapping:
6.67/2.61	reversecA_in_aa(x1, x2)  =  reversecA_in_aa
6.67/2.61	
6.67/2.61	U13_aa(x1, x2, x3, x4)  =  U13_aa(x4)
6.67/2.61	
6.67/2.61	reversecA_out_aa(x1, x2)  =  reversecA_out_aa(x1, x2)
6.67/2.61	
6.67/2.61	U14_aa(x1, x2, x3, x4)  =  U14_aa(x2, x4)
6.67/2.61	
6.67/2.61	appcB_in_gaa(x1, x2, x3)  =  appcB_in_gaa(x1)
6.67/2.61	
6.67/2.61	.(x1, x2)  =  .(x2)
6.67/2.61	
6.67/2.61	U15_gaa(x1, x2, x3, x4, x5)  =  U15_gaa(x2, x5)
6.67/2.61	
6.67/2.61	[]  =  []
6.67/2.61	
6.67/2.61	appcB_out_gaa(x1, x2, x3)  =  appcB_out_gaa(x1, x3)
6.67/2.61	
6.67/2.61	REVERSEA_IN_AA(x1, x2)  =  REVERSEA_IN_AA
6.67/2.61	
6.67/2.61	
6.67/2.61	We have to consider all (P,R,Pi)-chains
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(72) UsableRulesProof (EQUIVALENT)
6.67/2.61	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(73)
6.67/2.61	Obligation:
6.67/2.61	Pi DP problem:
6.67/2.61	The TRS P consists of the following rules:
6.67/2.61	
6.67/2.61	   REVERSEA_IN_AA(.(X1, X2), X3) -> REVERSEA_IN_AA(X2, X4)
6.67/2.61	
6.67/2.61	R is empty.
6.67/2.61	The argument filtering Pi contains the following mapping:
6.67/2.61	.(x1, x2)  =  .(x2)
6.67/2.61	
6.67/2.61	REVERSEA_IN_AA(x1, x2)  =  REVERSEA_IN_AA
6.67/2.61	
6.67/2.61	
6.67/2.61	We have to consider all (P,R,Pi)-chains
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(74) PiDPToQDPProof (SOUND)
6.67/2.61	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(75)
6.67/2.61	Obligation:
6.67/2.61	Q DP problem:
6.67/2.61	The TRS P consists of the following rules:
6.67/2.61	
6.67/2.61	   REVERSEA_IN_AA -> REVERSEA_IN_AA
6.67/2.61	
6.67/2.61	R is empty.
6.67/2.61	Q is empty.
6.67/2.61	We have to consider all (P,Q,R)-chains.
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(76) PrologToIRSwTTransformerProof (SOUND)
6.67/2.61	Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert
6.67/2.61	
6.67/2.61	{
6.67/2.61	    "root": 98,
6.67/2.61	    "program": {
6.67/2.61	        "directives": [],
6.67/2.61	        "clauses": [
6.67/2.61	            [
6.67/2.61	                "(app (. X Xs) Ys (. X Zs))",
6.67/2.61	                "(app Xs Ys Zs)"
6.67/2.61	            ],
6.67/2.61	            [
6.67/2.61	                "(app ([]) Ys Ys)",
6.67/2.61	                null
6.67/2.61	            ],
6.67/2.61	            [
6.67/2.61	                "(reverse (. X Xs) Ys)",
6.67/2.61	                "(',' (reverse Xs Zs) (app Zs (. X ([])) Ys))"
6.67/2.61	            ],
6.67/2.61	            [
6.67/2.61	                "(reverse ([]) ([]))",
6.67/2.61	                null
6.67/2.61	            ]
6.67/2.61	        ]
6.67/2.61	    },
6.67/2.61	    "graph": {
6.67/2.61	        "nodes": {
6.67/2.61	            "type": "Nodes",
6.67/2.61	            "250": {
6.67/2.61	                "goal": [],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": [],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "251": {
6.67/2.61	                "goal": [{
6.67/2.61	                    "clause": -1,
6.67/2.61	                    "scope": -1,
6.67/2.61	                    "term": "(true)"
6.67/2.61	                }],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": [],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "230": {
6.67/2.61	                "goal": [],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": [],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "252": {
6.67/2.61	                "goal": [],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": [],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "253": {
6.67/2.61	                "goal": [],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": [],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
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6.67/2.61	                    "ground": [],
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6.67/2.61	                        "X51",
6.67/2.61	                        "X50"
6.67/2.61	                    ],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "229": {
6.67/2.61	                "goal": [{
6.67/2.61	                    "clause": -1,
6.67/2.61	                    "scope": -1,
6.67/2.61	                    "term": "(app T62 (. T63 ([])) X91)"
6.67/2.61	                }],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": ["X91"],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "208": {
6.67/2.61	                "goal": [],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": [],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "209": {
6.67/2.61	                "goal": [{
6.67/2.61	                    "clause": -1,
6.67/2.61	                    "scope": -1,
6.67/2.61	                    "term": "(reverse T36 X50)"
6.67/2.61	                }],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": ["X50"],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            }
6.67/2.61	        },
6.67/2.61	        "edges": [
6.67/2.61	            {
6.67/2.61	                "from": 98,
6.67/2.61	                "to": 101,
6.67/2.61	                "label": "CASE"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 101,
6.67/2.61	                "to": 105,
6.67/2.61	                "label": "PARALLEL"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 101,
6.67/2.61	                "to": 106,
6.67/2.61	                "label": "PARALLEL"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 105,
6.67/2.61	                "to": 114,
6.67/2.61	                "label": "EVAL with clause\nreverse(.(X15, X16), X17) :- ','(reverse(X16, X18), app(X18, .(X15, []), X17)).\nand substitutionX15 -> T19,\nX16 -> T18,\nT1 -> .(T19, T18),\nT2 -> T17,\nX17 -> T17,\nT16 -> T18,\nT15 -> T19"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 105,
6.67/2.61	                "to": 115,
6.67/2.61	                "label": "EVAL-BACKTRACK"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 106,
6.67/2.61	                "to": 254,
6.67/2.61	                "label": "EVAL with clause\nreverse([], []).\nand substitutionT1 -> [],\nT2 -> []"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 106,
6.67/2.61	                "to": 255,
6.67/2.61	                "label": "EVAL-BACKTRACK"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 114,
6.67/2.61	                "to": 202,
6.67/2.61	                "label": "SPLIT 1"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 114,
6.67/2.61	                "to": 203,
6.67/2.61	                "label": "SPLIT 2\nreplacements:X18 -> T22,\nT19 -> T23"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 202,
6.67/2.61	                "to": 204,
6.67/2.61	                "label": "CASE"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 203,
6.67/2.61	                "to": 246,
6.67/2.61	                "label": "CASE"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 204,
6.67/2.61	                "to": 205,
6.67/2.61	                "label": "PARALLEL"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 204,
6.67/2.61	                "to": 206,
6.67/2.61	                "label": "PARALLEL"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 205,
6.67/2.61	                "to": 207,
6.67/2.61	                "label": "EVAL with clause\nreverse(.(X47, X48), X49) :- ','(reverse(X48, X50), app(X50, .(X47, []), X49)).\nand substitutionX47 -> T37,\nX48 -> T36,\nT18 -> .(T37, T36),\nX18 -> X51,\nX49 -> X51,\nT35 -> T36,\nT34 -> T37"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 205,
6.67/2.61	                "to": 208,
6.67/2.61	                "label": "EVAL-BACKTRACK"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 206,
6.67/2.61	                "to": 243,
6.67/2.61	                "label": "EVAL with clause\nreverse([], []).\nand substitutionT18 -> [],\nX18 -> []"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 206,
6.67/2.61	                "to": 244,
6.67/2.61	                "label": "EVAL-BACKTRACK"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 207,
6.67/2.61	                "to": 209,
6.67/2.61	                "label": "SPLIT 1"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 207,
6.67/2.61	                "to": 210,
6.67/2.61	                "label": "SPLIT 2\nreplacements:X50 -> T40,\nT37 -> T41"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 209,
6.67/2.61	                "to": 202,
6.67/2.61	                "label": "INSTANCE with matching:\nT18 -> T36\nX18 -> X50"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 210,
6.67/2.61	                "to": 226,
6.67/2.61	                "label": "CASE"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 226,
6.67/2.61	                "to": 227,
6.67/2.61	                "label": "PARALLEL"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 226,
6.67/2.61	                "to": 228,
6.67/2.61	                "label": "PARALLEL"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 227,
6.67/2.61	                "to": 229,
6.67/2.61	                "label": "EVAL with clause\napp(.(X87, X88), X89, .(X87, X90)) :- app(X88, X89, X90).\nand substitutionX87 -> T59,\nX88 -> T62,\nT40 -> .(T59, T62),\nT41 -> T63,\nX89 -> .(T63, []),\nX90 -> X91,\nX51 -> .(T59, X91),\nT60 -> T62,\nT61 -> T63"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 227,
6.67/2.61	                "to": 230,
6.67/2.61	                "label": "EVAL-BACKTRACK"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 228,
6.67/2.61	                "to": 240,
6.67/2.61	                "label": "EVAL with clause\napp([], X99, X99).\nand substitutionT40 -> [],\nT41 -> T69,\nX99 -> .(T69, []),\nX51 -> .(T69, [])"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 228,
6.67/2.61	                "to": 241,
6.67/2.61	                "label": "EVAL-BACKTRACK"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 229,
6.67/2.61	                "to": 210,
6.67/2.61	                "label": "INSTANCE with matching:\nT40 -> T62\nT41 -> T63\nX51 -> X91"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 240,
6.67/2.61	                "to": 242,
6.67/2.61	                "label": "SUCCESS"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 243,
6.67/2.61	                "to": 245,
6.67/2.61	                "label": "SUCCESS"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 246,
6.67/2.61	                "to": 247,
6.67/2.61	                "label": "PARALLEL"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 246,
6.67/2.61	                "to": 248,
6.67/2.61	                "label": "PARALLEL"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 247,
6.67/2.61	                "to": 249,
6.67/2.61	                "label": "EVAL with clause\napp(.(X120, X121), X122, .(X120, X123)) :- app(X121, X122, X123).\nand substitutionX120 -> T90,\nX121 -> T94,\nT22 -> .(T90, T94),\nT23 -> T95,\nX122 -> .(T95, []),\nX123 -> T93,\nT17 -> .(T90, T93),\nT91 -> T94,\nT92 -> T95"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 247,
6.67/2.61	                "to": 250,
6.67/2.61	                "label": "EVAL-BACKTRACK"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 248,
6.67/2.61	                "to": 251,
6.67/2.61	                "label": "EVAL with clause\napp([], X130, X130).\nand substitutionT22 -> [],\nT23 -> T102,\nX130 -> .(T102, []),\nT17 -> .(T102, [])"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 248,
6.67/2.61	                "to": 252,
6.67/2.61	                "label": "EVAL-BACKTRACK"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 249,
6.67/2.61	                "to": 203,
6.67/2.61	                "label": "INSTANCE with matching:\nT22 -> T94\nT23 -> T95\nT17 -> T93"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 251,
6.67/2.61	                "to": 253,
6.67/2.61	                "label": "SUCCESS"
6.67/2.61	            },
6.67/2.61	            {
6.67/2.61	                "from": 254,
6.67/2.61	                "to": 256,
6.67/2.61	                "label": "SUCCESS"
6.67/2.61	            }
6.67/2.61	        ],
6.67/2.61	        "type": "Graph"
6.67/2.61	    }
6.67/2.61	}
6.67/2.61	
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(77)
6.67/2.61	Complex Obligation (AND)
6.67/2.61	
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(78)
6.67/2.61	Obligation:
6.67/2.61	Rules:
6.67/2.61	f227_out -> f226_out :|: TRUE
6.67/2.61	f228_out -> f226_out :|: TRUE
6.67/2.61	f226_in -> f228_in :|: TRUE
6.67/2.61	f226_in -> f227_in :|: TRUE
6.67/2.61	f226_out -> f210_out :|: TRUE
6.67/2.61	f210_in -> f226_in :|: TRUE
6.67/2.61	f227_in -> f230_in :|: TRUE
6.67/2.61	f230_out -> f227_out :|: TRUE
6.67/2.61	f227_in -> f229_in :|: TRUE
6.67/2.61	f229_out -> f227_out :|: TRUE
6.67/2.61	f229_in -> f210_in :|: TRUE
6.67/2.61	f210_out -> f229_out :|: TRUE
6.67/2.61	f101_out(T2) -> f98_out(T2) :|: TRUE
6.67/2.61	f98_in(x) -> f101_in(x) :|: TRUE
6.67/2.61	f105_out(x1) -> f101_out(x1) :|: TRUE
6.67/2.61	f101_in(x2) -> f106_in(x2) :|: TRUE
6.67/2.61	f101_in(x3) -> f105_in(x3) :|: TRUE
6.67/2.61	f106_out(x4) -> f101_out(x4) :|: TRUE
6.67/2.61	f114_out(T17) -> f105_out(T17) :|: TRUE
6.67/2.61	f115_out -> f105_out(x5) :|: TRUE
6.67/2.61	f105_in(x6) -> f114_in(x6) :|: TRUE
6.67/2.61	f105_in(x7) -> f115_in :|: TRUE
6.67/2.61	f202_out -> f203_in(x8) :|: TRUE
6.67/2.61	f114_in(x9) -> f202_in :|: TRUE
6.67/2.61	f203_out(x10) -> f114_out(x10) :|: TRUE
6.67/2.61	f204_out -> f202_out :|: TRUE
6.67/2.61	f202_in -> f204_in :|: TRUE
6.67/2.61	f204_in -> f205_in :|: TRUE
6.67/2.61	f204_in -> f206_in :|: TRUE
6.67/2.61	f205_out -> f204_out :|: TRUE
6.67/2.61	f206_out -> f204_out :|: TRUE
6.67/2.61	f205_in -> f207_in :|: TRUE
6.67/2.61	f208_out -> f205_out :|: TRUE
6.67/2.61	f207_out -> f205_out :|: TRUE
6.67/2.61	f205_in -> f208_in :|: TRUE
6.67/2.61	f207_in -> f209_in :|: TRUE
6.67/2.61	f210_out -> f207_out :|: TRUE
6.67/2.61	f209_out -> f210_in :|: TRUE
6.67/2.61	Start term: f98_in(T2)
6.67/2.61	
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(79) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
6.67/2.61	Constructed simple dependency graph.
6.67/2.61	
6.67/2.61	Simplified to the following IRSwTs:
6.67/2.61	
6.67/2.61	
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(80)
6.67/2.61	TRUE
6.67/2.61	
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(81)
6.67/2.61	Obligation:
6.67/2.61	Rules:
6.67/2.61	f246_in(T17) -> f247_in(T17) :|: TRUE
6.67/2.61	f246_in(x) -> f248_in(x) :|: TRUE
6.67/2.61	f247_out(x1) -> f246_out(x1) :|: TRUE
6.67/2.61	f248_out(x2) -> f246_out(x2) :|: TRUE
6.67/2.61	f203_in(x3) -> f246_in(x3) :|: TRUE
6.67/2.61	f246_out(x4) -> f203_out(x4) :|: TRUE
6.67/2.61	f203_out(T93) -> f249_out(T93) :|: TRUE
6.67/2.61	f249_in(x5) -> f203_in(x5) :|: TRUE
6.67/2.61	f249_out(x6) -> f247_out(.(x7, x6)) :|: TRUE
6.67/2.61	f247_in(.(x8, x9)) -> f249_in(x9) :|: TRUE
6.67/2.61	f250_out -> f247_out(x10) :|: TRUE
6.67/2.61	f247_in(x11) -> f250_in :|: TRUE
6.67/2.61	f101_out(T2) -> f98_out(T2) :|: TRUE
6.67/2.61	f98_in(x12) -> f101_in(x12) :|: TRUE
6.67/2.61	f105_out(x13) -> f101_out(x13) :|: TRUE
6.67/2.61	f101_in(x14) -> f106_in(x14) :|: TRUE
6.67/2.61	f101_in(x15) -> f105_in(x15) :|: TRUE
6.67/2.61	f106_out(x16) -> f101_out(x16) :|: TRUE
6.67/2.61	f114_out(x17) -> f105_out(x17) :|: TRUE
6.67/2.61	f115_out -> f105_out(x18) :|: TRUE
6.67/2.61	f105_in(x19) -> f114_in(x19) :|: TRUE
6.67/2.61	f105_in(x20) -> f115_in :|: TRUE
6.67/2.61	f202_out -> f203_in(x21) :|: TRUE
6.67/2.61	f114_in(x22) -> f202_in :|: TRUE
6.67/2.61	f203_out(x23) -> f114_out(x23) :|: TRUE
6.67/2.61	Start term: f98_in(T2)
6.67/2.61	
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(82) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
6.67/2.61	Constructed simple dependency graph.
6.67/2.61	
6.67/2.61	Simplified to the following IRSwTs:
6.67/2.61	
6.67/2.61	
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(83)
6.67/2.61	TRUE
6.67/2.61	
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(84)
6.67/2.61	Obligation:
6.67/2.61	Rules:
6.67/2.61	f226_out -> f210_out :|: TRUE
6.67/2.61	f210_in -> f226_in :|: TRUE
6.67/2.61	f229_in -> f210_in :|: TRUE
6.67/2.61	f210_out -> f229_out :|: TRUE
6.67/2.61	f240_in -> f240_out :|: TRUE
6.67/2.61	f202_out -> f209_out :|: TRUE
6.67/2.61	f209_in -> f202_in :|: TRUE
6.67/2.61	f227_out -> f226_out :|: TRUE
6.67/2.61	f228_out -> f226_out :|: TRUE
6.67/2.61	f226_in -> f228_in :|: TRUE
6.67/2.61	f226_in -> f227_in :|: TRUE
6.67/2.61	f207_in -> f209_in :|: TRUE
6.67/2.61	f210_out -> f207_out :|: TRUE
6.67/2.61	f209_out -> f210_in :|: TRUE
6.67/2.61	f205_in -> f207_in :|: TRUE
6.67/2.61	f208_out -> f205_out :|: TRUE
6.67/2.61	f207_out -> f205_out :|: TRUE
6.67/2.61	f205_in -> f208_in :|: TRUE
6.67/2.61	f204_out -> f202_out :|: TRUE
6.67/2.61	f202_in -> f204_in :|: TRUE
6.67/2.61	f227_in -> f230_in :|: TRUE
6.67/2.61	f230_out -> f227_out :|: TRUE
6.67/2.61	f227_in -> f229_in :|: TRUE
6.67/2.61	f229_out -> f227_out :|: TRUE
6.67/2.61	f204_in -> f205_in :|: TRUE
6.67/2.61	f204_in -> f206_in :|: TRUE
6.67/2.61	f205_out -> f204_out :|: TRUE
6.67/2.61	f206_out -> f204_out :|: TRUE
6.67/2.61	f240_out -> f228_out :|: TRUE
6.67/2.61	f241_out -> f228_out :|: TRUE
6.67/2.61	f228_in -> f241_in :|: TRUE
6.67/2.61	f228_in -> f240_in :|: TRUE
6.67/2.61	f101_out(T2) -> f98_out(T2) :|: TRUE
6.67/2.61	f98_in(x) -> f101_in(x) :|: TRUE
6.67/2.61	f105_out(x1) -> f101_out(x1) :|: TRUE
6.67/2.61	f101_in(x2) -> f106_in(x2) :|: TRUE
6.67/2.61	f101_in(x3) -> f105_in(x3) :|: TRUE
6.67/2.61	f106_out(x4) -> f101_out(x4) :|: TRUE
6.67/2.61	f114_out(T17) -> f105_out(T17) :|: TRUE
6.67/2.61	f115_out -> f105_out(x5) :|: TRUE
6.67/2.61	f105_in(x6) -> f114_in(x6) :|: TRUE
6.67/2.61	f105_in(x7) -> f115_in :|: TRUE
6.67/2.61	f202_out -> f203_in(x8) :|: TRUE
6.67/2.61	f114_in(x9) -> f202_in :|: TRUE
6.67/2.61	f203_out(x10) -> f114_out(x10) :|: TRUE
6.67/2.61	Start term: f98_in(T2)
6.67/2.61	
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(85) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
6.67/2.61	Constructed simple dependency graph.
6.67/2.61	
6.67/2.61	Simplified to the following IRSwTs:
6.67/2.61	
6.67/2.61	intTRSProblem:
6.67/2.61	f209_in -> f202_in :|: TRUE
6.67/2.61	f207_in -> f209_in :|: TRUE
6.67/2.61	f205_in -> f207_in :|: TRUE
6.67/2.61	f202_in -> f204_in :|: TRUE
6.67/2.61	f204_in -> f205_in :|: TRUE
6.67/2.61	
6.67/2.61	
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(86)
6.67/2.61	Obligation:
6.67/2.61	Rules:
6.67/2.61	f209_in -> f202_in :|: TRUE
6.67/2.61	f207_in -> f209_in :|: TRUE
6.67/2.61	f205_in -> f207_in :|: TRUE
6.67/2.61	f202_in -> f204_in :|: TRUE
6.67/2.61	f204_in -> f205_in :|: TRUE
6.67/2.61	
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(87) IntTRSCompressionProof (EQUIVALENT)
6.67/2.61	Compressed rules.
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(88)
6.67/2.61	Obligation:
6.67/2.61	Rules:
6.67/2.61	f205_in -> f205_in :|: TRUE
6.67/2.61	
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(89) IRSFormatTransformerProof (EQUIVALENT)
6.67/2.61	Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(90)
6.67/2.61	Obligation:
6.67/2.61	Rules:
6.67/2.61	f205_in -> f205_in :|: TRUE
6.67/2.61	
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(91) IRSwTTerminationDigraphProof (EQUIVALENT)
6.67/2.61	Constructed termination digraph!
6.67/2.61	Nodes:
6.67/2.61	(1) f205_in -> f205_in :|: TRUE
6.67/2.61	
6.67/2.61	Arcs:
6.67/2.61	(1) -> (1)
6.67/2.61	
6.67/2.61	This digraph is fully evaluated!
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(92)
6.67/2.61	Obligation:
6.67/2.61	
6.67/2.61	Termination digraph:
6.67/2.61	Nodes:
6.67/2.61	(1) f205_in -> f205_in :|: TRUE
6.67/2.61	
6.67/2.61	Arcs:
6.67/2.61	(1) -> (1)
6.67/2.61	
6.67/2.61	This digraph is fully evaluated!
6.67/2.61	
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(93) FilterProof (EQUIVALENT)
6.67/2.61	Used the following sort dictionary for filtering: 
6.67/2.61	f205_in()
6.67/2.61	Replaced non-predefined constructor symbols by 0.
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(94)
6.67/2.61	Obligation:
6.67/2.61	Rules:
6.67/2.61	f205_in -> f205_in :|: TRUE
6.67/2.61	
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(95) IntTRSPeriodicNontermProof (COMPLETE)
6.67/2.61	Normalized system to the following form:
6.67/2.61	f(pc) -> f(1) :|: pc = 1 && TRUE
6.67/2.61	Witness term starting non-terminating reduction: f(1)
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(96)
6.67/2.61	NO
6.67/2.61	
6.67/2.61	----------------------------------------
6.67/2.61	
6.67/2.61	(97) PrologToTRSTransformerProof (SOUND)
6.67/2.61	Transformed Prolog program to TRS.
6.67/2.61	
6.67/2.61	{
6.67/2.61	    "root": 12,
6.67/2.61	    "program": {
6.67/2.61	        "directives": [],
6.67/2.61	        "clauses": [
6.67/2.61	            [
6.67/2.61	                "(app (. X Xs) Ys (. X Zs))",
6.67/2.61	                "(app Xs Ys Zs)"
6.67/2.61	            ],
6.67/2.61	            [
6.67/2.61	                "(app ([]) Ys Ys)",
6.67/2.61	                null
6.67/2.61	            ],
6.67/2.61	            [
6.67/2.61	                "(reverse (. X Xs) Ys)",
6.67/2.61	                "(',' (reverse Xs Zs) (app Zs (. X ([])) Ys))"
6.67/2.61	            ],
6.67/2.61	            [
6.67/2.61	                "(reverse ([]) ([]))",
6.67/2.61	                null
6.67/2.61	            ]
6.67/2.61	        ]
6.67/2.61	    },
6.67/2.61	    "graph": {
6.67/2.61	        "nodes": {
6.67/2.61	            "192": {
6.67/2.61	                "goal": [{
6.67/2.61	                    "clause": 2,
6.67/2.61	                    "scope": 2,
6.67/2.61	                    "term": "(reverse T18 X18)"
6.67/2.61	                }],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": ["X18"],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "193": {
6.67/2.61	                "goal": [{
6.67/2.61	                    "clause": 3,
6.67/2.61	                    "scope": 2,
6.67/2.61	                    "term": "(reverse T18 X18)"
6.67/2.61	                }],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": ["X18"],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "type": "Nodes",
6.67/2.61	            "195": {
6.67/2.61	                "goal": [{
6.67/2.61	                    "clause": -1,
6.67/2.61	                    "scope": -1,
6.67/2.61	                    "term": "(',' (reverse T36 X50) (app X50 (. T37 ([])) X51))"
6.67/2.61	                }],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
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6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": [
6.67/2.61	                        "X51",
6.67/2.61	                        "X50"
6.67/2.61	                    ],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "197": {
6.67/2.61	                "goal": [],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": [],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "176": {
6.67/2.61	                "goal": [{
6.67/2.61	                    "clause": -1,
6.67/2.61	                    "scope": -1,
6.67/2.61	                    "term": "(reverse T18 X18)"
6.67/2.61	                }],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": ["X18"],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "198": {
6.67/2.61	                "goal": [{
6.67/2.61	                    "clause": -1,
6.67/2.61	                    "scope": -1,
6.67/2.61	                    "term": "(reverse T36 X50)"
6.67/2.61	                }],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": ["X50"],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "231": {
6.67/2.61	                "goal": [{
6.67/2.61	                    "clause": -1,
6.67/2.61	                    "scope": -1,
6.67/2.61	                    "term": "(app T62 (. T63 ([])) X91)"
6.67/2.61	                }],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": ["X91"],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "177": {
6.67/2.61	                "goal": [{
6.67/2.61	                    "clause": -1,
6.67/2.61	                    "scope": -1,
6.67/2.61	                    "term": "(app T22 (. T23 ([])) T17)"
6.67/2.61	                }],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": ["T17"],
6.67/2.61	                    "free": [],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "232": {
6.67/2.61	                "goal": [],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": [],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "211": {
6.67/2.61	                "goal": [
6.67/2.61	                    {
6.67/2.61	                        "clause": 0,
6.67/2.61	                        "scope": 3,
6.67/2.61	                        "term": "(app T40 (. T41 ([])) X51)"
6.67/2.61	                    },
6.67/2.61	                    {
6.67/2.61	                        "clause": 1,
6.67/2.61	                        "scope": 3,
6.67/2.61	                        "term": "(app T40 (. T41 ([])) X51)"
6.67/2.61	                    }
6.67/2.61	                ],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": ["X51"],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "233": {
6.67/2.61	                "goal": [{
6.67/2.61	                    "clause": -1,
6.67/2.61	                    "scope": -1,
6.67/2.61	                    "term": "(true)"
6.67/2.61	                }],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": [],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "310": {
6.67/2.61	                "goal": [],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": [],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "135": {
6.67/2.61	                "goal": [{
6.67/2.61	                    "clause": -1,
6.67/2.61	                    "scope": -1,
6.67/2.61	                    "term": "(',' (reverse T18 X18) (app X18 (. T19 ([])) T17))"
6.67/2.61	                }],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": ["T17"],
6.67/2.61	                    "free": ["X18"],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "212": {
6.67/2.61	                "goal": [{
6.67/2.61	                    "clause": 0,
6.67/2.61	                    "scope": 3,
6.67/2.61	                    "term": "(app T40 (. T41 ([])) X51)"
6.67/2.61	                }],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
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6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "234": {
6.67/2.61	                "goal": [],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": [],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "213": {
6.67/2.61	                "goal": [{
6.67/2.61	                    "clause": 1,
6.67/2.61	                    "scope": 3,
6.67/2.61	                    "term": "(app T40 (. T41 ([])) X51)"
6.67/2.61	                }],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
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6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
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6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "235": {
6.67/2.61	                "goal": [],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": [],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "236": {
6.67/2.61	                "goal": [{
6.67/2.61	                    "clause": -1,
6.67/2.61	                    "scope": -1,
6.67/2.61	                    "term": "(true)"
6.67/2.61	                }],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": [],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "237": {
6.67/2.61	                "goal": [],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": [],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "238": {
6.67/2.61	                "goal": [],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": [],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "239": {
6.67/2.61	                "goal": [
6.67/2.61	                    {
6.67/2.61	                        "clause": 0,
6.67/2.61	                        "scope": 4,
6.67/2.61	                        "term": "(app T22 (. T23 ([])) T17)"
6.67/2.61	                    },
6.67/2.61	                    {
6.67/2.61	                        "clause": 1,
6.67/2.61	                        "scope": 4,
6.67/2.61	                        "term": "(app T22 (. T23 ([])) T17)"
6.67/2.61	                    }
6.67/2.61	                ],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": ["T17"],
6.67/2.61	                    "free": [],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "70": {
6.67/2.61	                "goal": [{
6.67/2.61	                    "clause": 2,
6.67/2.61	                    "scope": 1,
6.67/2.61	                    "term": "(reverse T1 T2)"
6.67/2.61	                }],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": ["T2"],
6.67/2.61	                    "free": [],
6.67/2.61	                    "exprvars": []
6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "71": {
6.67/2.61	                "goal": [{
6.67/2.61	                    "clause": 3,
6.67/2.61	                    "scope": 1,
6.67/2.61	                    "term": "(reverse T1 T2)"
6.67/2.61	                }],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": ["T2"],
6.67/2.61	                    "free": [],
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6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "12": {
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6.67/2.61	                    "clause": -1,
6.67/2.61	                    "scope": -1,
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6.67/2.61	                }],
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6.67/2.61	            },
6.67/2.61	            "13": {
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6.67/2.61	                    {
6.67/2.61	                        "clause": 2,
6.67/2.61	                        "scope": 1,
6.67/2.61	                        "term": "(reverse T1 T2)"
6.67/2.61	                    },
6.67/2.61	                    {
6.67/2.61	                        "clause": 3,
6.67/2.61	                        "scope": 1,
6.67/2.61	                        "term": "(reverse T1 T2)"
6.67/2.61	                    }
6.67/2.61	                ],
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6.67/2.61	                    },
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6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "184": {
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6.67/2.61	                    {
6.67/2.61	                        "clause": 2,
6.67/2.61	                        "scope": 2,
6.67/2.61	                        "term": "(reverse T18 X18)"
6.67/2.61	                    },
6.67/2.61	                    {
6.67/2.61	                        "clause": 3,
6.67/2.61	                        "scope": 2,
6.67/2.61	                        "term": "(reverse T18 X18)"
6.67/2.61	                    }
6.67/2.61	                ],
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6.67/2.61	                    },
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6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "141": {
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6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
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6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "200": {
6.67/2.61	                "goal": [{
6.67/2.61	                    "clause": -1,
6.67/2.61	                    "scope": -1,
6.67/2.61	                    "term": "(app T40 (. T41 ([])) X51)"
6.67/2.61	                }],
6.67/2.61	                "kb": {
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6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
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6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "301": {
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6.67/2.61	                    "clause": 0,
6.67/2.61	                    "scope": 4,
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6.67/2.61	                }],
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6.67/2.61	                    },
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6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "302": {
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6.67/2.61	                    "scope": 4,
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6.67/2.61	                }],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
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6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": ["T17"],
6.67/2.61	                    "free": [],
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6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "303": {
6.67/2.61	                "goal": [{
6.67/2.61	                    "clause": -1,
6.67/2.61	                    "scope": -1,
6.67/2.61	                    "term": "(app T94 (. T95 ([])) T93)"
6.67/2.61	                }],
6.67/2.61	                "kb": {
6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": ["T93"],
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6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "304": {
6.67/2.61	                "goal": [],
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6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
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6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "305": {
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6.67/2.61	                    "clause": -1,
6.67/2.61	                    "scope": -1,
6.67/2.61	                    "term": "(true)"
6.67/2.61	                }],
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6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
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6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "306": {
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6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
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6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "307": {
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6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
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6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "308": {
6.67/2.61	                "goal": [{
6.67/2.61	                    "clause": -1,
6.67/2.61	                    "scope": -1,
6.67/2.61	                    "term": "(true)"
6.67/2.61	                }],
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6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": [],
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6.67/2.61	                }
6.67/2.61	            },
6.67/2.61	            "309": {
6.67/2.61	                "goal": [],
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6.67/2.61	                    "nonunifying": [],
6.67/2.61	                    "intvars": {},
6.67/2.61	                    "arithmetic": {
6.67/2.61	                        "type": "PlainIntegerRelationState",
6.67/2.61	                        "relations": []
6.67/2.61	                    },
6.67/2.61	                    "ground": [],
6.67/2.61	                    "free": [],
6.67/2.62	                    "exprvars": []
6.67/2.62	                }
6.67/2.62	            }
6.67/2.62	        },
6.67/2.62	        "edges": [
6.67/2.62	            {
6.67/2.62	                "from": 12,
6.67/2.62	                "to": 13,
6.67/2.62	                "label": "CASE"
6.67/2.62	            },
6.67/2.62	            {
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6.67/2.62	                "to": 70,
6.67/2.62	                "label": "PARALLEL"
6.67/2.62	            },
6.67/2.62	            {
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6.67/2.62	                "to": 71,
6.67/2.62	                "label": "PARALLEL"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 70,
6.67/2.62	                "to": 135,
6.67/2.62	                "label": "EVAL with clause\nreverse(.(X15, X16), X17) :- ','(reverse(X16, X18), app(X18, .(X15, []), X17)).\nand substitutionX15 -> T19,\nX16 -> T18,\nT1 -> .(T19, T18),\nT2 -> T17,\nX17 -> T17,\nT16 -> T18,\nT15 -> T19"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 70,
6.67/2.62	                "to": 141,
6.67/2.62	                "label": "EVAL-BACKTRACK"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 71,
6.67/2.62	                "to": 308,
6.67/2.62	                "label": "EVAL with clause\nreverse([], []).\nand substitutionT1 -> [],\nT2 -> []"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 71,
6.67/2.62	                "to": 309,
6.67/2.62	                "label": "EVAL-BACKTRACK"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 135,
6.67/2.62	                "to": 176,
6.67/2.62	                "label": "SPLIT 1"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 135,
6.67/2.62	                "to": 177,
6.67/2.62	                "label": "SPLIT 2\nreplacements:X18 -> T22,\nT19 -> T23"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 176,
6.67/2.62	                "to": 184,
6.67/2.62	                "label": "CASE"
6.67/2.62	            },
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6.67/2.62	                "to": 239,
6.67/2.62	                "label": "CASE"
6.67/2.62	            },
6.67/2.62	            {
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6.67/2.62	                "to": 192,
6.67/2.62	                "label": "PARALLEL"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 184,
6.67/2.62	                "to": 193,
6.67/2.62	                "label": "PARALLEL"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 192,
6.67/2.62	                "to": 195,
6.67/2.62	                "label": "EVAL with clause\nreverse(.(X47, X48), X49) :- ','(reverse(X48, X50), app(X50, .(X47, []), X49)).\nand substitutionX47 -> T37,\nX48 -> T36,\nT18 -> .(T37, T36),\nX18 -> X51,\nX49 -> X51,\nT35 -> T36,\nT34 -> T37"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 192,
6.67/2.62	                "to": 197,
6.67/2.62	                "label": "EVAL-BACKTRACK"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 193,
6.67/2.62	                "to": 236,
6.67/2.62	                "label": "EVAL with clause\nreverse([], []).\nand substitutionT18 -> [],\nX18 -> []"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 193,
6.67/2.62	                "to": 237,
6.67/2.62	                "label": "EVAL-BACKTRACK"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 195,
6.67/2.62	                "to": 198,
6.67/2.62	                "label": "SPLIT 1"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 195,
6.67/2.62	                "to": 200,
6.67/2.62	                "label": "SPLIT 2\nreplacements:X50 -> T40,\nT37 -> T41"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 198,
6.67/2.62	                "to": 176,
6.67/2.62	                "label": "INSTANCE with matching:\nT18 -> T36\nX18 -> X50"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 200,
6.67/2.62	                "to": 211,
6.67/2.62	                "label": "CASE"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 211,
6.67/2.62	                "to": 212,
6.67/2.62	                "label": "PARALLEL"
6.67/2.62	            },
6.67/2.62	            {
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6.67/2.62	                "to": 213,
6.67/2.62	                "label": "PARALLEL"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 212,
6.67/2.62	                "to": 231,
6.67/2.62	                "label": "EVAL with clause\napp(.(X87, X88), X89, .(X87, X90)) :- app(X88, X89, X90).\nand substitutionX87 -> T59,\nX88 -> T62,\nT40 -> .(T59, T62),\nT41 -> T63,\nX89 -> .(T63, []),\nX90 -> X91,\nX51 -> .(T59, X91),\nT60 -> T62,\nT61 -> T63"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 212,
6.67/2.62	                "to": 232,
6.67/2.62	                "label": "EVAL-BACKTRACK"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 213,
6.67/2.62	                "to": 233,
6.67/2.62	                "label": "EVAL with clause\napp([], X99, X99).\nand substitutionT40 -> [],\nT41 -> T69,\nX99 -> .(T69, []),\nX51 -> .(T69, [])"
6.67/2.62	            },
6.67/2.62	            {
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6.67/2.62	                "to": 234,
6.67/2.62	                "label": "EVAL-BACKTRACK"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 231,
6.67/2.62	                "to": 200,
6.67/2.62	                "label": "INSTANCE with matching:\nT40 -> T62\nT41 -> T63\nX51 -> X91"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 233,
6.67/2.62	                "to": 235,
6.67/2.62	                "label": "SUCCESS"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 236,
6.67/2.62	                "to": 238,
6.67/2.62	                "label": "SUCCESS"
6.67/2.62	            },
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6.67/2.62	                "to": 301,
6.67/2.62	                "label": "PARALLEL"
6.67/2.62	            },
6.67/2.62	            {
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6.67/2.62	                "to": 302,
6.67/2.62	                "label": "PARALLEL"
6.67/2.62	            },
6.67/2.62	            {
6.67/2.62	                "from": 301,
6.67/2.62	                "to": 303,
6.67/2.62	                "label": "EVAL with clause\napp(.(X120, X121), X122, .(X120, X123)) :- app(X121, X122, X123).\nand substitutionX120 -> T90,\nX121 -> T94,\nT22 -> .(T90, T94),\nT23 -> T95,\nX122 -> .(T95, []),\nX123 -> T93,\nT17 -> .(T90, T93),\nT91 -> T94,\nT92 -> T95"
6.87/2.62	            },
6.87/2.62	            {
6.87/2.62	                "from": 301,
6.87/2.62	                "to": 304,
6.87/2.62	                "label": "EVAL-BACKTRACK"
6.87/2.62	            },
6.87/2.62	            {
6.87/2.62	                "from": 302,
6.87/2.62	                "to": 305,
6.87/2.62	                "label": "EVAL with clause\napp([], X130, X130).\nand substitutionT22 -> [],\nT23 -> T102,\nX130 -> .(T102, []),\nT17 -> .(T102, [])"
6.87/2.62	            },
6.87/2.62	            {
6.87/2.62	                "from": 302,
6.87/2.62	                "to": 306,
6.87/2.62	                "label": "EVAL-BACKTRACK"
6.87/2.62	            },
6.87/2.62	            {
6.87/2.62	                "from": 303,
6.87/2.62	                "to": 177,
6.87/2.62	                "label": "INSTANCE with matching:\nT22 -> T94\nT23 -> T95\nT17 -> T93"
6.87/2.62	            },
6.87/2.62	            {
6.87/2.62	                "from": 305,
6.87/2.62	                "to": 307,
6.87/2.62	                "label": "SUCCESS"
6.87/2.62	            },
6.87/2.62	            {
6.87/2.62	                "from": 308,
6.87/2.62	                "to": 310,
6.87/2.62	                "label": "SUCCESS"
6.87/2.62	            }
6.87/2.62	        ],
6.87/2.62	        "type": "Graph"
6.87/2.62	    }
6.87/2.62	}
6.87/2.62	
6.87/2.62	----------------------------------------
6.87/2.62	
6.87/2.62	(98)
6.87/2.62	Obligation:
6.87/2.62	Q restricted rewrite system:
6.87/2.62	The TRS R consists of the following rules:
6.87/2.62	
6.87/2.62	   f12_in(T17) -> U1(f135_in(T17), T17)
6.87/2.62	   U1(f135_out1(X18, T19), T17) -> f12_out1
6.87/2.62	   f12_in([]) -> f12_out1
6.87/2.62	   f176_in -> U2(f195_in)
6.87/2.62	   U2(f195_out1) -> f176_out1
6.87/2.62	   f176_in -> f176_out1
6.87/2.62	   f200_in -> U3(f200_in)
6.87/2.62	   U3(f200_out1) -> f200_out1
6.87/2.62	   f200_in -> f200_out1
6.87/2.62	   f177_in(.(T90, T93)) -> U4(f177_in(T93), .(T90, T93))
6.87/2.62	   U4(f177_out1(T94, T95), .(T90, T93)) -> f177_out1(.(T90, T94), T95)
6.87/2.62	   f177_in(.(T102, [])) -> f177_out1([], T102)
6.87/2.62	   f135_in(T17) -> U5(f176_in, T17)
6.87/2.62	   U5(f176_out1, T17) -> U6(f177_in(T17), T17)
6.87/2.62	   U6(f177_out1(T22, T23), T17) -> f135_out1(T22, T23)
6.87/2.62	   f195_in -> U7(f176_in)
6.87/2.62	   U7(f176_out1) -> U8(f200_in)
6.87/2.62	   U8(f200_out1) -> f195_out1
6.87/2.62	
6.87/2.62	Q is empty.
6.87/2.62	
6.87/2.62	----------------------------------------
6.87/2.62	
6.87/2.62	(99) QTRSRRRProof (EQUIVALENT)
6.87/2.62	Used ordering:
6.87/2.62	f12_in/1(YES)
6.87/2.62	U1/2(YES,YES)
6.87/2.62	f135_in/1(YES)
6.87/2.62	f135_out1/2(YES,YES)
6.87/2.62	f12_out1/0)
6.87/2.62	[]/0)
6.87/2.62	f176_in/0)
6.87/2.62	U2/1)YES(
6.87/2.62	f195_in/0)
6.87/2.62	f195_out1/0)
6.87/2.62	f176_out1/0)
6.87/2.62	f200_in/0)
6.87/2.62	U3/1)YES(
6.87/2.62	f200_out1/0)
6.87/2.62	f177_in/1(YES)
6.87/2.62	./2(YES,YES)
6.87/2.62	U4/2(YES,YES)
6.87/2.62	f177_out1/2(YES,YES)
6.87/2.62	U5/2(YES,YES)
6.87/2.62	U6/2(YES,YES)
6.87/2.62	U7/1)YES(
6.87/2.62	U8/1)YES(
6.87/2.62	
6.87/2.62	Quasi precedence:
6.87/2.62	f12_in_1 > U1_2 > f12_out1 > f177_out1_2
6.87/2.62	f12_in_1 > [f135_in_1, f176_in, f195_in, f195_out1, f176_out1, f200_in, f200_out1] > U5_2 > [f135_out1_2, U6_2] > f177_out1_2
6.87/2.62	f12_in_1 > [f135_in_1, f176_in, f195_in, f195_out1, f176_out1, f200_in, f200_out1] > U5_2 > f177_in_1 > [[], ._2, U4_2] > f12_out1 > f177_out1_2
6.87/2.62	
6.87/2.62	
6.87/2.62	Status:
6.87/2.62	f12_in_1: multiset status
6.87/2.62	U1_2: multiset status
6.87/2.62	f135_in_1: [1]
6.87/2.62	f135_out1_2: multiset status
6.87/2.62	f12_out1: multiset status
6.87/2.62	[]: multiset status
6.87/2.62	f176_in: multiset status
6.87/2.62	f195_in: multiset status
6.87/2.62	f195_out1: multiset status
6.87/2.62	f176_out1: multiset status
6.87/2.62	f200_in: multiset status
6.87/2.62	f200_out1: multiset status
6.87/2.62	f177_in_1: multiset status
6.87/2.62	._2: [1,2]
6.87/2.62	U4_2: [2,1]
6.87/2.62	f177_out1_2: multiset status
6.87/2.62	U5_2: multiset status
6.87/2.62	U6_2: multiset status
6.87/2.62	
6.87/2.62	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.87/2.62	
6.87/2.62	   f12_in(T17) -> U1(f135_in(T17), T17)
6.87/2.62	   U1(f135_out1(X18, T19), T17) -> f12_out1
6.87/2.62	   f12_in([]) -> f12_out1
6.87/2.62	   f177_in(.(T90, T93)) -> U4(f177_in(T93), .(T90, T93))
6.87/2.62	   U4(f177_out1(T94, T95), .(T90, T93)) -> f177_out1(.(T90, T94), T95)
6.87/2.62	   f177_in(.(T102, [])) -> f177_out1([], T102)
6.87/2.62	   f135_in(T17) -> U5(f176_in, T17)
6.87/2.62	   U5(f176_out1, T17) -> U6(f177_in(T17), T17)
6.87/2.62	   U6(f177_out1(T22, T23), T17) -> f135_out1(T22, T23)
6.87/2.62	
6.87/2.62	
6.87/2.62	
6.87/2.62	
6.87/2.62	----------------------------------------
6.87/2.62	
6.87/2.62	(100)
6.87/2.62	Obligation:
6.87/2.62	Q restricted rewrite system:
6.87/2.62	The TRS R consists of the following rules:
6.87/2.62	
6.87/2.62	   f176_in -> U2(f195_in)
6.87/2.62	   U2(f195_out1) -> f176_out1
6.87/2.62	   f176_in -> f176_out1
6.87/2.62	   f200_in -> U3(f200_in)
6.87/2.62	   U3(f200_out1) -> f200_out1
6.87/2.62	   f200_in -> f200_out1
6.87/2.62	   f195_in -> U7(f176_in)
6.87/2.62	   U7(f176_out1) -> U8(f200_in)
6.87/2.62	   U8(f200_out1) -> f195_out1
6.87/2.62	
6.87/2.62	Q is empty.
6.87/2.62	
6.87/2.62	----------------------------------------
6.87/2.62	
6.87/2.62	(101) QTRSRRRProof (EQUIVALENT)
6.87/2.62	Used ordering:
6.87/2.62	f176_in/0)
6.87/2.62	U2/1)YES(
6.87/2.62	f195_in/0)
6.87/2.62	f195_out1/0)
6.87/2.62	f176_out1/0)
6.87/2.62	f200_in/0)
6.87/2.62	U3/1)YES(
6.87/2.62	f200_out1/0)
6.87/2.62	U7/1)YES(
6.87/2.62	U8/1)YES(
6.87/2.62	
6.87/2.62	Quasi precedence:
6.87/2.62	[f176_in, f195_in] > [f195_out1, f176_out1, f200_in, f200_out1]
6.87/2.62	
6.87/2.62	
6.87/2.62	Status:
6.87/2.62	f176_in: multiset status
6.87/2.62	f195_in: multiset status
6.87/2.62	f195_out1: multiset status
6.87/2.62	f176_out1: multiset status
6.87/2.62	f200_in: multiset status
6.87/2.62	f200_out1: multiset status
6.87/2.62	
6.87/2.62	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.87/2.62	
6.87/2.62	   f176_in -> f176_out1
6.87/2.62	
6.87/2.62	
6.87/2.62	
6.87/2.62	
6.87/2.62	----------------------------------------
6.87/2.62	
6.87/2.62	(102)
6.87/2.62	Obligation:
6.87/2.62	Q restricted rewrite system:
6.87/2.62	The TRS R consists of the following rules:
6.87/2.62	
6.87/2.62	   f176_in -> U2(f195_in)
6.87/2.62	   U2(f195_out1) -> f176_out1
6.87/2.62	   f200_in -> U3(f200_in)
6.87/2.62	   U3(f200_out1) -> f200_out1
6.87/2.62	   f200_in -> f200_out1
6.87/2.62	   f195_in -> U7(f176_in)
6.87/2.62	   U7(f176_out1) -> U8(f200_in)
6.87/2.62	   U8(f200_out1) -> f195_out1
6.87/2.62	
6.87/2.62	Q is empty.
6.87/2.62	
6.87/2.62	----------------------------------------
6.87/2.62	
6.87/2.62	(103) QTRSRRRProof (EQUIVALENT)
6.87/2.62	Used ordering:
6.87/2.62	Polynomial interpretation [POLO]:
6.87/2.62	
6.87/2.62	   POL(U2(x_1)) = 2*x_1
6.87/2.62	   POL(U3(x_1)) = x_1
6.87/2.62	   POL(U7(x_1)) = 2*x_1
6.87/2.62	   POL(U8(x_1)) = 2 + x_1
6.87/2.62	   POL(f176_in) = 0
6.87/2.62	   POL(f176_out1) = 2
6.87/2.62	   POL(f195_in) = 0
6.87/2.62	   POL(f195_out1) = 1
6.87/2.62	   POL(f200_in) = 1
6.87/2.62	   POL(f200_out1) = 0
6.87/2.62	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.87/2.62	
6.87/2.62	   f200_in -> f200_out1
6.87/2.62	   U7(f176_out1) -> U8(f200_in)
6.87/2.62	   U8(f200_out1) -> f195_out1
6.87/2.62	
6.87/2.62	
6.87/2.62	
6.87/2.62	
6.87/2.62	----------------------------------------
6.87/2.62	
6.87/2.62	(104)
6.87/2.62	Obligation:
6.87/2.62	Q restricted rewrite system:
6.87/2.62	The TRS R consists of the following rules:
6.87/2.62	
6.87/2.62	   f176_in -> U2(f195_in)
6.87/2.62	   U2(f195_out1) -> f176_out1
6.87/2.62	   f200_in -> U3(f200_in)
6.87/2.62	   U3(f200_out1) -> f200_out1
6.87/2.62	   f195_in -> U7(f176_in)
6.87/2.62	
6.87/2.62	Q is empty.
6.87/2.62	
6.87/2.62	----------------------------------------
6.87/2.62	
6.87/2.62	(105) QTRSRRRProof (EQUIVALENT)
6.87/2.62	Used ordering:
6.87/2.62	Polynomial interpretation [POLO]:
6.87/2.62	
6.87/2.62	   POL(U2(x_1)) = x_1
6.87/2.62	   POL(U3(x_1)) = 2*x_1
6.87/2.62	   POL(U7(x_1)) = 2*x_1
6.87/2.62	   POL(f176_in) = 0
6.87/2.62	   POL(f176_out1) = 0
6.87/2.62	   POL(f195_in) = 0
6.87/2.62	   POL(f195_out1) = 1
6.87/2.62	   POL(f200_in) = 0
6.87/2.62	   POL(f200_out1) = 0
6.87/2.62	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.87/2.62	
6.87/2.62	   U2(f195_out1) -> f176_out1
6.87/2.62	
6.87/2.62	
6.87/2.62	
6.87/2.62	
6.87/2.62	----------------------------------------
6.87/2.62	
6.87/2.62	(106)
6.87/2.62	Obligation:
6.87/2.62	Q restricted rewrite system:
6.87/2.62	The TRS R consists of the following rules:
6.87/2.62	
6.87/2.62	   f176_in -> U2(f195_in)
6.87/2.62	   f200_in -> U3(f200_in)
6.87/2.62	   U3(f200_out1) -> f200_out1
6.87/2.62	   f195_in -> U7(f176_in)
6.87/2.62	
6.87/2.62	Q is empty.
6.87/2.65	EOF