6.98/2.63	MAYBE
6.98/2.64	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
6.98/2.64	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.98/2.64	
6.98/2.64	
6.98/2.64	Left Termination of the query pattern
6.98/2.64	
6.98/2.64	sublist(g,g)
6.98/2.64	
6.98/2.64	w.r.t. the given Prolog program could not be shown:
6.98/2.64	
6.98/2.64	(0) Prolog
6.98/2.64	(1) PrologToPiTRSProof [SOUND, 0 ms]
6.98/2.64	(2) PiTRS
6.98/2.64	    (3) DependencyPairsProof [EQUIVALENT, 0 ms]
6.98/2.64	    (4) PiDP
6.98/2.64	    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
6.98/2.64	    (6) AND
6.98/2.64	        (7) PiDP
6.98/2.64	            (8) UsableRulesProof [EQUIVALENT, 0 ms]
6.98/2.64	            (9) PiDP
6.98/2.64	            (10) PiDPToQDPProof [SOUND, 0 ms]
6.98/2.64	            (11) QDP
6.98/2.64	            (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.98/2.64	            (13) YES
6.98/2.64	        (14) PiDP
6.98/2.64	            (15) UsableRulesProof [EQUIVALENT, 0 ms]
6.98/2.64	            (16) PiDP
6.98/2.64	            (17) PiDPToQDPProof [SOUND, 0 ms]
6.98/2.64	            (18) QDP
6.98/2.64	(19) PrologToPiTRSProof [SOUND, 0 ms]
6.98/2.64	(20) PiTRS
6.98/2.64	    (21) DependencyPairsProof [EQUIVALENT, 0 ms]
6.98/2.64	    (22) PiDP
6.98/2.64	    (23) DependencyGraphProof [EQUIVALENT, 0 ms]
6.98/2.64	    (24) AND
6.98/2.64	        (25) PiDP
6.98/2.64	            (26) UsableRulesProof [EQUIVALENT, 0 ms]
6.98/2.64	            (27) PiDP
6.98/2.64	            (28) PiDPToQDPProof [SOUND, 0 ms]
6.98/2.64	            (29) QDP
6.98/2.64	            (30) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.98/2.64	            (31) YES
6.98/2.64	        (32) PiDP
6.98/2.64	            (33) UsableRulesProof [EQUIVALENT, 0 ms]
6.98/2.64	            (34) PiDP
6.98/2.64	            (35) PiDPToQDPProof [SOUND, 0 ms]
6.98/2.64	            (36) QDP
6.98/2.64	(37) PrologToTRSTransformerProof [SOUND, 0 ms]
6.98/2.64	(38) QTRS
6.98/2.64	    (39) DependencyPairsProof [EQUIVALENT, 0 ms]
6.98/2.64	    (40) QDP
6.98/2.64	    (41) DependencyGraphProof [EQUIVALENT, 0 ms]
6.98/2.64	    (42) AND
6.98/2.64	        (43) QDP
6.98/2.64	            (44) UsableRulesProof [EQUIVALENT, 0 ms]
6.98/2.64	            (45) QDP
6.98/2.64	            (46) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.98/2.64	            (47) YES
6.98/2.64	        (48) QDP
6.98/2.64	            (49) UsableRulesProof [EQUIVALENT, 0 ms]
6.98/2.64	            (50) QDP
6.98/2.64	(51) PrologToIRSwTTransformerProof [SOUND, 0 ms]
6.98/2.64	(52) AND
6.98/2.64	    (53) IRSwT
6.98/2.64	        (54) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
6.98/2.64	        (55) TRUE
6.98/2.64	    (56) IRSwT
6.98/2.64	        (57) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
6.98/2.64	        (58) IRSwT
6.98/2.64	        (59) IntTRSCompressionProof [EQUIVALENT, 21 ms]
6.98/2.64	        (60) IRSwT
6.98/2.64	        (61) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
6.98/2.64	        (62) IRSwT
6.98/2.64	        (63) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
6.98/2.64	        (64) IRSwT
6.98/2.64	        (65) FilterProof [EQUIVALENT, 0 ms]
6.98/2.64	        (66) IntTRS
6.98/2.64	        (67) IntTRSNonPeriodicNontermProof [COMPLETE, 0 ms]
6.98/2.64	        (68) NO
6.98/2.64	(69) PrologToDTProblemTransformerProof [SOUND, 0 ms]
6.98/2.64	(70) TRIPLES
6.98/2.64	    (71) TriplesToPiDPProof [SOUND, 0 ms]
6.98/2.64	    (72) PiDP
6.98/2.64	    (73) DependencyGraphProof [EQUIVALENT, 0 ms]
6.98/2.64	    (74) AND
6.98/2.64	        (75) PiDP
6.98/2.64	            (76) UsableRulesProof [EQUIVALENT, 0 ms]
6.98/2.64	            (77) PiDP
6.98/2.64	            (78) PiDPToQDPProof [SOUND, 0 ms]
6.98/2.64	            (79) QDP
6.98/2.64	            (80) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.98/2.64	            (81) YES
6.98/2.64	        (82) PiDP
6.98/2.64	            (83) UsableRulesProof [EQUIVALENT, 0 ms]
6.98/2.64	            (84) PiDP
6.98/2.64	            (85) PiDPToQDPProof [SOUND, 0 ms]
6.98/2.64	            (86) QDP
6.98/2.64	        (87) PiDP
6.98/2.64	            (88) UsableRulesProof [EQUIVALENT, 0 ms]
6.98/2.64	            (89) PiDP
6.98/2.64	
6.98/2.64	
6.98/2.64	----------------------------------------
6.98/2.64	
6.98/2.64	(0)
6.98/2.64	Obligation:
6.98/2.64	Clauses:
6.98/2.64	
6.98/2.64	append1([], Ys, Ys).
6.98/2.64	append1(.(X, Xs), Ys, .(X, Zs)) :- append1(Xs, Ys, Zs).
6.98/2.64	append2([], Ys, Ys).
6.98/2.64	append2(.(X, Xs), Ys, .(X, Zs)) :- append2(Xs, Ys, Zs).
6.98/2.64	sublist(X, Y) :- ','(append1(U, X, V), append2(V, W, Y)).
6.98/2.64	
6.98/2.64	
6.98/2.64	Query: sublist(g,g)
6.98/2.64	----------------------------------------
6.98/2.64	
6.98/2.64	(1) PrologToPiTRSProof (SOUND)
6.98/2.64	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.98/2.64	
6.98/2.64	sublist_in_2: (b,b)
6.98/2.64	
6.98/2.64	append1_in_3: (f,b,f)
6.98/2.64	
6.98/2.64	append2_in_3: (b,f,b)
6.98/2.64	
6.98/2.64	Transforming Prolog into the following Term Rewriting System:
6.98/2.64	
6.98/2.64	Pi-finite rewrite system:
6.98/2.64	The TRS R consists of the following rules:
6.98/2.64	
6.98/2.64	   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
6.98/2.64	   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
6.98/2.64	   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
6.98/2.64	   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
6.98/2.64	   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
6.98/2.64	   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
6.98/2.64	   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
6.98/2.64	   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
6.98/2.64	   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)
6.98/2.64	
6.98/2.64	The argument filtering Pi contains the following mapping:
6.98/2.64	sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
6.98/2.64	
6.98/2.64	U3_gg(x1, x2, x3)  =  U3_gg(x1, x2, x3)
6.98/2.64	
6.98/2.64	append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
6.98/2.64	
6.98/2.64	append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x2, x3)
6.98/2.64	
6.98/2.64	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
6.98/2.64	
6.98/2.64	.(x1, x2)  =  .(x2)
6.98/2.64	
6.98/2.64	U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
6.98/2.64	
6.98/2.64	append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
6.98/2.64	
6.98/2.64	[]  =  []
6.98/2.64	
6.98/2.64	append2_out_gag(x1, x2, x3)  =  append2_out_gag(x1, x2, x3)
6.98/2.64	
6.98/2.64	U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)
6.98/2.64	
6.98/2.64	sublist_out_gg(x1, x2)  =  sublist_out_gg(x1, x2)
6.98/2.64	
6.98/2.64	
6.98/2.64	
6.98/2.64	
6.98/2.64	
6.98/2.64	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.98/2.64	
6.98/2.64	
6.98/2.64	
6.98/2.64	----------------------------------------
6.98/2.64	
6.98/2.64	(2)
6.98/2.64	Obligation:
6.98/2.64	Pi-finite rewrite system:
6.98/2.64	The TRS R consists of the following rules:
6.98/2.64	
6.98/2.64	   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
6.98/2.64	   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
6.98/2.64	   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
6.98/2.64	   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
6.98/2.64	   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
6.98/2.64	   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
6.98/2.64	   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
6.98/2.64	   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
6.98/2.64	   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)
6.98/2.64	
6.98/2.64	The argument filtering Pi contains the following mapping:
6.98/2.64	sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
6.98/2.64	
6.98/2.64	U3_gg(x1, x2, x3)  =  U3_gg(x1, x2, x3)
6.98/2.64	
6.98/2.64	append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
6.98/2.64	
6.98/2.64	append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x2, x3)
6.98/2.64	
6.98/2.64	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
6.98/2.64	
6.98/2.64	.(x1, x2)  =  .(x2)
6.98/2.64	
6.98/2.64	U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
6.98/2.64	
6.98/2.64	append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
6.98/2.64	
6.98/2.64	[]  =  []
6.98/2.64	
6.98/2.64	append2_out_gag(x1, x2, x3)  =  append2_out_gag(x1, x2, x3)
6.98/2.64	
6.98/2.64	U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)
6.98/2.64	
6.98/2.64	sublist_out_gg(x1, x2)  =  sublist_out_gg(x1, x2)
6.98/2.64	
6.98/2.64	
6.98/2.64	
6.98/2.64	----------------------------------------
6.98/2.64	
6.98/2.64	(3) DependencyPairsProof (EQUIVALENT)
6.98/2.64	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.98/2.64	Pi DP problem:
6.98/2.64	The TRS P consists of the following rules:
6.98/2.64	
6.98/2.64	   SUBLIST_IN_GG(X, Y) -> U3_GG(X, Y, append1_in_aga(U, X, V))
6.98/2.64	   SUBLIST_IN_GG(X, Y) -> APPEND1_IN_AGA(U, X, V)
6.98/2.64	   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U1_AGA(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
6.98/2.64	   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs)
6.98/2.64	   U3_GG(X, Y, append1_out_aga(U, X, V)) -> U4_GG(X, Y, append2_in_gag(V, W, Y))
6.98/2.64	   U3_GG(X, Y, append1_out_aga(U, X, V)) -> APPEND2_IN_GAG(V, W, Y)
6.98/2.64	   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> U2_GAG(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
6.98/2.64	   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs)
6.98/2.64	
6.98/2.64	The TRS R consists of the following rules:
6.98/2.64	
6.98/2.64	   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
6.98/2.64	   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
6.98/2.64	   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
6.98/2.64	   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
6.98/2.64	   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
6.98/2.64	   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
6.98/2.64	   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
6.98/2.64	   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
6.98/2.65	   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)
6.98/2.65	
6.98/2.65	The argument filtering Pi contains the following mapping:
6.98/2.65	sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
6.98/2.65	
6.98/2.65	U3_gg(x1, x2, x3)  =  U3_gg(x1, x2, x3)
6.98/2.65	
6.98/2.65	append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
6.98/2.65	
6.98/2.65	append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x2, x3)
6.98/2.65	
6.98/2.65	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
6.98/2.65	
6.98/2.65	.(x1, x2)  =  .(x2)
6.98/2.65	
6.98/2.65	U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
6.98/2.65	
6.98/2.65	append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
6.98/2.65	
6.98/2.65	[]  =  []
6.98/2.65	
6.98/2.65	append2_out_gag(x1, x2, x3)  =  append2_out_gag(x1, x2, x3)
6.98/2.65	
6.98/2.65	U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)
6.98/2.65	
6.98/2.65	sublist_out_gg(x1, x2)  =  sublist_out_gg(x1, x2)
6.98/2.65	
6.98/2.65	SUBLIST_IN_GG(x1, x2)  =  SUBLIST_IN_GG(x1, x2)
6.98/2.65	
6.98/2.65	U3_GG(x1, x2, x3)  =  U3_GG(x1, x2, x3)
6.98/2.65	
6.98/2.65	APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)
6.98/2.65	
6.98/2.65	U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x3, x5)
6.98/2.65	
6.98/2.65	U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)
6.98/2.65	
6.98/2.65	APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)
6.98/2.65	
6.98/2.65	U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x2, x4, x5)
6.98/2.65	
6.98/2.65	
6.98/2.65	We have to consider all (P,R,Pi)-chains
6.98/2.65	----------------------------------------
6.98/2.65	
6.98/2.65	(4)
6.98/2.65	Obligation:
6.98/2.65	Pi DP problem:
6.98/2.65	The TRS P consists of the following rules:
6.98/2.65	
6.98/2.65	   SUBLIST_IN_GG(X, Y) -> U3_GG(X, Y, append1_in_aga(U, X, V))
6.98/2.65	   SUBLIST_IN_GG(X, Y) -> APPEND1_IN_AGA(U, X, V)
6.98/2.65	   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U1_AGA(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
6.98/2.65	   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs)
6.98/2.65	   U3_GG(X, Y, append1_out_aga(U, X, V)) -> U4_GG(X, Y, append2_in_gag(V, W, Y))
6.98/2.65	   U3_GG(X, Y, append1_out_aga(U, X, V)) -> APPEND2_IN_GAG(V, W, Y)
6.98/2.65	   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> U2_GAG(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
6.98/2.65	   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs)
6.98/2.65	
6.98/2.65	The TRS R consists of the following rules:
6.98/2.65	
6.98/2.65	   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
6.98/2.65	   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
6.98/2.65	   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
6.98/2.65	   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
6.98/2.65	   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
6.98/2.65	   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
6.98/2.65	   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
6.98/2.65	   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
6.98/2.65	   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)
6.98/2.65	
6.98/2.65	The argument filtering Pi contains the following mapping:
6.98/2.65	sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
6.98/2.65	
6.98/2.65	U3_gg(x1, x2, x3)  =  U3_gg(x1, x2, x3)
6.98/2.65	
6.98/2.65	append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
6.98/2.65	
6.98/2.65	append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x2, x3)
6.98/2.65	
6.98/2.65	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
6.98/2.65	
6.98/2.65	.(x1, x2)  =  .(x2)
6.98/2.65	
6.98/2.65	U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
6.98/2.65	
6.98/2.65	append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
6.98/2.65	
6.98/2.65	[]  =  []
6.98/2.65	
6.98/2.65	append2_out_gag(x1, x2, x3)  =  append2_out_gag(x1, x2, x3)
6.98/2.65	
6.98/2.65	U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)
6.98/2.65	
6.98/2.65	sublist_out_gg(x1, x2)  =  sublist_out_gg(x1, x2)
6.98/2.65	
6.98/2.65	SUBLIST_IN_GG(x1, x2)  =  SUBLIST_IN_GG(x1, x2)
6.98/2.65	
6.98/2.65	U3_GG(x1, x2, x3)  =  U3_GG(x1, x2, x3)
6.98/2.65	
6.98/2.65	APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)
6.98/2.65	
6.98/2.65	U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x3, x5)
6.98/2.65	
6.98/2.65	U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)
6.98/2.65	
6.98/2.65	APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)
6.98/2.65	
6.98/2.65	U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x2, x4, x5)
6.98/2.65	
6.98/2.65	
6.98/2.65	We have to consider all (P,R,Pi)-chains
6.98/2.65	----------------------------------------
6.98/2.65	
6.98/2.65	(5) DependencyGraphProof (EQUIVALENT)
6.98/2.65	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.
6.98/2.65	----------------------------------------
6.98/2.65	
6.98/2.65	(6)
6.98/2.65	Complex Obligation (AND)
6.98/2.65	
6.98/2.65	----------------------------------------
6.98/2.65	
6.98/2.65	(7)
6.98/2.65	Obligation:
6.98/2.65	Pi DP problem:
6.98/2.65	The TRS P consists of the following rules:
6.98/2.65	
6.98/2.65	   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs)
6.98/2.65	
6.98/2.65	The TRS R consists of the following rules:
6.98/2.65	
6.98/2.65	   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
6.98/2.65	   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
6.98/2.65	   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
6.98/2.65	   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
6.98/2.65	   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
6.98/2.65	   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
6.98/2.65	   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
6.98/2.65	   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
6.98/2.65	   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)
6.98/2.65	
6.98/2.65	The argument filtering Pi contains the following mapping:
6.98/2.65	sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
6.98/2.65	
6.98/2.65	U3_gg(x1, x2, x3)  =  U3_gg(x1, x2, x3)
6.98/2.65	
6.98/2.65	append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
6.98/2.65	
6.98/2.65	append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x2, x3)
6.98/2.65	
6.98/2.65	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
6.98/2.65	
6.98/2.65	.(x1, x2)  =  .(x2)
6.98/2.65	
6.98/2.65	U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
6.98/2.65	
6.98/2.65	append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
6.98/2.65	
6.98/2.65	[]  =  []
6.98/2.65	
6.98/2.65	append2_out_gag(x1, x2, x3)  =  append2_out_gag(x1, x2, x3)
6.98/2.65	
6.98/2.65	U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)
6.98/2.65	
6.98/2.65	sublist_out_gg(x1, x2)  =  sublist_out_gg(x1, x2)
6.98/2.65	
6.98/2.65	APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)
6.98/2.65	
6.98/2.65	
6.98/2.65	We have to consider all (P,R,Pi)-chains
6.98/2.65	----------------------------------------
6.98/2.65	
6.98/2.65	(8) UsableRulesProof (EQUIVALENT)
6.98/2.65	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.98/2.65	----------------------------------------
6.98/2.65	
6.98/2.65	(9)
6.98/2.65	Obligation:
6.98/2.65	Pi DP problem:
6.98/2.65	The TRS P consists of the following rules:
6.98/2.65	
6.98/2.65	   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs)
6.98/2.65	
6.98/2.65	R is empty.
6.98/2.65	The argument filtering Pi contains the following mapping:
6.98/2.65	.(x1, x2)  =  .(x2)
6.98/2.65	
6.98/2.65	APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)
6.98/2.65	
6.98/2.65	
6.98/2.65	We have to consider all (P,R,Pi)-chains
6.98/2.65	----------------------------------------
6.98/2.65	
6.98/2.65	(10) PiDPToQDPProof (SOUND)
6.98/2.65	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.98/2.65	----------------------------------------
6.98/2.65	
6.98/2.65	(11)
6.98/2.65	Obligation:
6.98/2.65	Q DP problem:
6.98/2.65	The TRS P consists of the following rules:
6.98/2.65	
6.98/2.65	   APPEND2_IN_GAG(.(Xs), .(Zs)) -> APPEND2_IN_GAG(Xs, Zs)
6.98/2.65	
6.98/2.65	R is empty.
6.98/2.65	Q is empty.
6.98/2.65	We have to consider all (P,Q,R)-chains.
6.98/2.65	----------------------------------------
6.98/2.65	
6.98/2.65	(12) QDPSizeChangeProof (EQUIVALENT)
6.98/2.65	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. 
6.98/2.65	
6.98/2.65	From the DPs we obtained the following set of size-change graphs:
6.98/2.65	*APPEND2_IN_GAG(.(Xs), .(Zs)) -> APPEND2_IN_GAG(Xs, Zs)
6.98/2.65	The graph contains the following edges 1 > 1, 2 > 2
6.98/2.65	
6.98/2.65	
6.98/2.65	----------------------------------------
6.98/2.65	
6.98/2.65	(13)
6.98/2.65	YES
6.98/2.65	
6.98/2.65	----------------------------------------
6.98/2.65	
6.98/2.65	(14)
6.98/2.65	Obligation:
6.98/2.65	Pi DP problem:
6.98/2.65	The TRS P consists of the following rules:
6.98/2.65	
6.98/2.65	   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs)
6.98/2.65	
6.98/2.65	The TRS R consists of the following rules:
6.98/2.65	
6.98/2.65	   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
6.98/2.65	   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
6.98/2.65	   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
6.98/2.65	   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
6.98/2.65	   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
6.98/2.65	   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
6.98/2.65	   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
6.98/2.65	   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
6.98/2.65	   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)
6.98/2.65	
6.98/2.65	The argument filtering Pi contains the following mapping:
6.98/2.65	sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
6.98/2.65	
6.98/2.65	U3_gg(x1, x2, x3)  =  U3_gg(x1, x2, x3)
6.98/2.65	
6.98/2.65	append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
6.98/2.65	
6.98/2.65	append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x2, x3)
6.98/2.65	
6.98/2.65	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x3, x5)
6.98/2.65	
6.98/2.65	.(x1, x2)  =  .(x2)
6.98/2.65	
6.98/2.65	U4_gg(x1, x2, x3)  =  U4_gg(x1, x2, x3)
6.98/2.65	
6.98/2.65	append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
6.98/2.67	
6.98/2.67	[]  =  []
6.98/2.67	
6.98/2.67	append2_out_gag(x1, x2, x3)  =  append2_out_gag(x1, x2, x3)
6.98/2.67	
6.98/2.67	U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x2, x4, x5)
6.98/2.67	
6.98/2.67	sublist_out_gg(x1, x2)  =  sublist_out_gg(x1, x2)
6.98/2.67	
6.98/2.67	APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)
6.98/2.67	
6.98/2.67	
6.98/2.67	We have to consider all (P,R,Pi)-chains
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(15) UsableRulesProof (EQUIVALENT)
6.98/2.67	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(16)
6.98/2.67	Obligation:
6.98/2.67	Pi DP problem:
6.98/2.67	The TRS P consists of the following rules:
6.98/2.67	
6.98/2.67	   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs)
6.98/2.67	
6.98/2.67	R is empty.
6.98/2.67	The argument filtering Pi contains the following mapping:
6.98/2.67	.(x1, x2)  =  .(x2)
6.98/2.67	
6.98/2.67	APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)
6.98/2.67	
6.98/2.67	
6.98/2.67	We have to consider all (P,R,Pi)-chains
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(17) PiDPToQDPProof (SOUND)
6.98/2.67	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(18)
6.98/2.67	Obligation:
6.98/2.67	Q DP problem:
6.98/2.67	The TRS P consists of the following rules:
6.98/2.67	
6.98/2.67	   APPEND1_IN_AGA(Ys) -> APPEND1_IN_AGA(Ys)
6.98/2.67	
6.98/2.67	R is empty.
6.98/2.67	Q is empty.
6.98/2.67	We have to consider all (P,Q,R)-chains.
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(19) PrologToPiTRSProof (SOUND)
6.98/2.67	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.98/2.67	
6.98/2.67	sublist_in_2: (b,b)
6.98/2.67	
6.98/2.67	append1_in_3: (f,b,f)
6.98/2.67	
6.98/2.67	append2_in_3: (b,f,b)
6.98/2.67	
6.98/2.67	Transforming Prolog into the following Term Rewriting System:
6.98/2.67	
6.98/2.67	Pi-finite rewrite system:
6.98/2.67	The TRS R consists of the following rules:
6.98/2.67	
6.98/2.67	   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
6.98/2.67	   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
6.98/2.67	   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
6.98/2.67	   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
6.98/2.67	   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
6.98/2.67	   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
6.98/2.67	   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
6.98/2.67	   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
6.98/2.67	   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)
6.98/2.67	
6.98/2.67	The argument filtering Pi contains the following mapping:
6.98/2.67	sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
6.98/2.67	
6.98/2.67	U3_gg(x1, x2, x3)  =  U3_gg(x2, x3)
6.98/2.67	
6.98/2.67	append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
6.98/2.67	
6.98/2.67	append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x3)
6.98/2.67	
6.98/2.67	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
6.98/2.67	
6.98/2.67	.(x1, x2)  =  .(x2)
6.98/2.67	
6.98/2.67	U4_gg(x1, x2, x3)  =  U4_gg(x3)
6.98/2.67	
6.98/2.67	append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
6.98/2.67	
6.98/2.67	[]  =  []
6.98/2.67	
6.98/2.67	append2_out_gag(x1, x2, x3)  =  append2_out_gag(x2)
6.98/2.67	
6.98/2.67	U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
6.98/2.67	
6.98/2.67	sublist_out_gg(x1, x2)  =  sublist_out_gg
6.98/2.67	
6.98/2.67	
6.98/2.67	
6.98/2.67	
6.98/2.67	
6.98/2.67	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.98/2.67	
6.98/2.67	
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(20)
6.98/2.67	Obligation:
6.98/2.67	Pi-finite rewrite system:
6.98/2.67	The TRS R consists of the following rules:
6.98/2.67	
6.98/2.67	   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
6.98/2.67	   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
6.98/2.67	   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
6.98/2.67	   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
6.98/2.67	   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
6.98/2.67	   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
6.98/2.67	   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
6.98/2.67	   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
6.98/2.67	   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)
6.98/2.67	
6.98/2.67	The argument filtering Pi contains the following mapping:
6.98/2.67	sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
6.98/2.67	
6.98/2.67	U3_gg(x1, x2, x3)  =  U3_gg(x2, x3)
6.98/2.67	
6.98/2.67	append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
6.98/2.67	
6.98/2.67	append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x3)
6.98/2.67	
6.98/2.67	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
6.98/2.67	
6.98/2.67	.(x1, x2)  =  .(x2)
6.98/2.67	
6.98/2.67	U4_gg(x1, x2, x3)  =  U4_gg(x3)
6.98/2.67	
6.98/2.67	append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
6.98/2.67	
6.98/2.67	[]  =  []
6.98/2.67	
6.98/2.67	append2_out_gag(x1, x2, x3)  =  append2_out_gag(x2)
6.98/2.67	
6.98/2.67	U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
6.98/2.67	
6.98/2.67	sublist_out_gg(x1, x2)  =  sublist_out_gg
6.98/2.67	
6.98/2.67	
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(21) DependencyPairsProof (EQUIVALENT)
6.98/2.67	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.98/2.67	Pi DP problem:
6.98/2.67	The TRS P consists of the following rules:
6.98/2.67	
6.98/2.67	   SUBLIST_IN_GG(X, Y) -> U3_GG(X, Y, append1_in_aga(U, X, V))
6.98/2.67	   SUBLIST_IN_GG(X, Y) -> APPEND1_IN_AGA(U, X, V)
6.98/2.67	   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U1_AGA(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
6.98/2.67	   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs)
6.98/2.67	   U3_GG(X, Y, append1_out_aga(U, X, V)) -> U4_GG(X, Y, append2_in_gag(V, W, Y))
6.98/2.67	   U3_GG(X, Y, append1_out_aga(U, X, V)) -> APPEND2_IN_GAG(V, W, Y)
6.98/2.67	   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> U2_GAG(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
6.98/2.67	   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs)
6.98/2.67	
6.98/2.67	The TRS R consists of the following rules:
6.98/2.67	
6.98/2.67	   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
6.98/2.67	   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
6.98/2.67	   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
6.98/2.67	   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
6.98/2.67	   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
6.98/2.67	   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
6.98/2.67	   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
6.98/2.67	   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
6.98/2.67	   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)
6.98/2.67	
6.98/2.67	The argument filtering Pi contains the following mapping:
6.98/2.67	sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
6.98/2.67	
6.98/2.67	U3_gg(x1, x2, x3)  =  U3_gg(x2, x3)
6.98/2.67	
6.98/2.67	append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
6.98/2.67	
6.98/2.67	append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x3)
6.98/2.67	
6.98/2.67	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
6.98/2.67	
6.98/2.67	.(x1, x2)  =  .(x2)
6.98/2.67	
6.98/2.67	U4_gg(x1, x2, x3)  =  U4_gg(x3)
6.98/2.67	
6.98/2.67	append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
6.98/2.67	
6.98/2.67	[]  =  []
6.98/2.67	
6.98/2.67	append2_out_gag(x1, x2, x3)  =  append2_out_gag(x2)
6.98/2.67	
6.98/2.67	U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
6.98/2.67	
6.98/2.67	sublist_out_gg(x1, x2)  =  sublist_out_gg
6.98/2.67	
6.98/2.67	SUBLIST_IN_GG(x1, x2)  =  SUBLIST_IN_GG(x1, x2)
6.98/2.67	
6.98/2.67	U3_GG(x1, x2, x3)  =  U3_GG(x2, x3)
6.98/2.67	
6.98/2.67	APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)
6.98/2.67	
6.98/2.67	U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x5)
6.98/2.67	
6.98/2.67	U4_GG(x1, x2, x3)  =  U4_GG(x3)
6.98/2.67	
6.98/2.67	APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)
6.98/2.67	
6.98/2.67	U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x5)
6.98/2.67	
6.98/2.67	
6.98/2.67	We have to consider all (P,R,Pi)-chains
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(22)
6.98/2.67	Obligation:
6.98/2.67	Pi DP problem:
6.98/2.67	The TRS P consists of the following rules:
6.98/2.67	
6.98/2.67	   SUBLIST_IN_GG(X, Y) -> U3_GG(X, Y, append1_in_aga(U, X, V))
6.98/2.67	   SUBLIST_IN_GG(X, Y) -> APPEND1_IN_AGA(U, X, V)
6.98/2.67	   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U1_AGA(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
6.98/2.67	   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs)
6.98/2.67	   U3_GG(X, Y, append1_out_aga(U, X, V)) -> U4_GG(X, Y, append2_in_gag(V, W, Y))
6.98/2.67	   U3_GG(X, Y, append1_out_aga(U, X, V)) -> APPEND2_IN_GAG(V, W, Y)
6.98/2.67	   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> U2_GAG(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
6.98/2.67	   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs)
6.98/2.67	
6.98/2.67	The TRS R consists of the following rules:
6.98/2.67	
6.98/2.67	   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
6.98/2.67	   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
6.98/2.67	   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
6.98/2.67	   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
6.98/2.67	   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
6.98/2.67	   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
6.98/2.67	   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
6.98/2.67	   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
6.98/2.67	   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)
6.98/2.67	
6.98/2.67	The argument filtering Pi contains the following mapping:
6.98/2.67	sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
6.98/2.67	
6.98/2.67	U3_gg(x1, x2, x3)  =  U3_gg(x2, x3)
6.98/2.67	
6.98/2.67	append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
6.98/2.67	
6.98/2.67	append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x3)
6.98/2.67	
6.98/2.67	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
6.98/2.67	
6.98/2.67	.(x1, x2)  =  .(x2)
6.98/2.67	
6.98/2.67	U4_gg(x1, x2, x3)  =  U4_gg(x3)
6.98/2.67	
6.98/2.67	append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
6.98/2.67	
6.98/2.67	[]  =  []
6.98/2.67	
6.98/2.67	append2_out_gag(x1, x2, x3)  =  append2_out_gag(x2)
6.98/2.67	
6.98/2.67	U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
6.98/2.67	
6.98/2.67	sublist_out_gg(x1, x2)  =  sublist_out_gg
6.98/2.67	
6.98/2.67	SUBLIST_IN_GG(x1, x2)  =  SUBLIST_IN_GG(x1, x2)
6.98/2.67	
6.98/2.67	U3_GG(x1, x2, x3)  =  U3_GG(x2, x3)
6.98/2.67	
6.98/2.67	APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)
6.98/2.67	
6.98/2.67	U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x5)
6.98/2.67	
6.98/2.67	U4_GG(x1, x2, x3)  =  U4_GG(x3)
6.98/2.67	
6.98/2.67	APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)
6.98/2.67	
6.98/2.67	U2_GAG(x1, x2, x3, x4, x5)  =  U2_GAG(x5)
6.98/2.67	
6.98/2.67	
6.98/2.67	We have to consider all (P,R,Pi)-chains
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(23) DependencyGraphProof (EQUIVALENT)
6.98/2.67	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(24)
6.98/2.67	Complex Obligation (AND)
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(25)
6.98/2.67	Obligation:
6.98/2.67	Pi DP problem:
6.98/2.67	The TRS P consists of the following rules:
6.98/2.67	
6.98/2.67	   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs)
6.98/2.67	
6.98/2.67	The TRS R consists of the following rules:
6.98/2.67	
6.98/2.67	   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
6.98/2.67	   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
6.98/2.67	   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
6.98/2.67	   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
6.98/2.67	   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
6.98/2.67	   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
6.98/2.67	   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
6.98/2.67	   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
6.98/2.67	   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)
6.98/2.67	
6.98/2.67	The argument filtering Pi contains the following mapping:
6.98/2.67	sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
6.98/2.67	
6.98/2.67	U3_gg(x1, x2, x3)  =  U3_gg(x2, x3)
6.98/2.67	
6.98/2.67	append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
6.98/2.67	
6.98/2.67	append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x3)
6.98/2.67	
6.98/2.67	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
6.98/2.67	
6.98/2.67	.(x1, x2)  =  .(x2)
6.98/2.67	
6.98/2.67	U4_gg(x1, x2, x3)  =  U4_gg(x3)
6.98/2.67	
6.98/2.67	append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
6.98/2.67	
6.98/2.67	[]  =  []
6.98/2.67	
6.98/2.67	append2_out_gag(x1, x2, x3)  =  append2_out_gag(x2)
6.98/2.67	
6.98/2.67	U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
6.98/2.67	
6.98/2.67	sublist_out_gg(x1, x2)  =  sublist_out_gg
6.98/2.67	
6.98/2.67	APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)
6.98/2.67	
6.98/2.67	
6.98/2.67	We have to consider all (P,R,Pi)-chains
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(26) UsableRulesProof (EQUIVALENT)
6.98/2.67	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(27)
6.98/2.67	Obligation:
6.98/2.67	Pi DP problem:
6.98/2.67	The TRS P consists of the following rules:
6.98/2.67	
6.98/2.67	   APPEND2_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_GAG(Xs, Ys, Zs)
6.98/2.67	
6.98/2.67	R is empty.
6.98/2.67	The argument filtering Pi contains the following mapping:
6.98/2.67	.(x1, x2)  =  .(x2)
6.98/2.67	
6.98/2.67	APPEND2_IN_GAG(x1, x2, x3)  =  APPEND2_IN_GAG(x1, x3)
6.98/2.67	
6.98/2.67	
6.98/2.67	We have to consider all (P,R,Pi)-chains
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(28) PiDPToQDPProof (SOUND)
6.98/2.67	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(29)
6.98/2.67	Obligation:
6.98/2.67	Q DP problem:
6.98/2.67	The TRS P consists of the following rules:
6.98/2.67	
6.98/2.67	   APPEND2_IN_GAG(.(Xs), .(Zs)) -> APPEND2_IN_GAG(Xs, Zs)
6.98/2.67	
6.98/2.67	R is empty.
6.98/2.67	Q is empty.
6.98/2.67	We have to consider all (P,Q,R)-chains.
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(30) QDPSizeChangeProof (EQUIVALENT)
6.98/2.67	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. 
6.98/2.67	
6.98/2.67	From the DPs we obtained the following set of size-change graphs:
6.98/2.67	*APPEND2_IN_GAG(.(Xs), .(Zs)) -> APPEND2_IN_GAG(Xs, Zs)
6.98/2.67	The graph contains the following edges 1 > 1, 2 > 2
6.98/2.67	
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(31)
6.98/2.67	YES
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(32)
6.98/2.67	Obligation:
6.98/2.67	Pi DP problem:
6.98/2.67	The TRS P consists of the following rules:
6.98/2.67	
6.98/2.67	   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs)
6.98/2.67	
6.98/2.67	The TRS R consists of the following rules:
6.98/2.67	
6.98/2.67	   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aga(U, X, V))
6.98/2.67	   append1_in_aga([], Ys, Ys) -> append1_out_aga([], Ys, Ys)
6.98/2.67	   append1_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U1_aga(X, Xs, Ys, Zs, append1_in_aga(Xs, Ys, Zs))
6.98/2.67	   U1_aga(X, Xs, Ys, Zs, append1_out_aga(Xs, Ys, Zs)) -> append1_out_aga(.(X, Xs), Ys, .(X, Zs))
6.98/2.67	   U3_gg(X, Y, append1_out_aga(U, X, V)) -> U4_gg(X, Y, append2_in_gag(V, W, Y))
6.98/2.67	   append2_in_gag([], Ys, Ys) -> append2_out_gag([], Ys, Ys)
6.98/2.67	   append2_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U2_gag(X, Xs, Ys, Zs, append2_in_gag(Xs, Ys, Zs))
6.98/2.67	   U2_gag(X, Xs, Ys, Zs, append2_out_gag(Xs, Ys, Zs)) -> append2_out_gag(.(X, Xs), Ys, .(X, Zs))
6.98/2.67	   U4_gg(X, Y, append2_out_gag(V, W, Y)) -> sublist_out_gg(X, Y)
6.98/2.67	
6.98/2.67	The argument filtering Pi contains the following mapping:
6.98/2.67	sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
6.98/2.67	
6.98/2.67	U3_gg(x1, x2, x3)  =  U3_gg(x2, x3)
6.98/2.67	
6.98/2.67	append1_in_aga(x1, x2, x3)  =  append1_in_aga(x2)
6.98/2.67	
6.98/2.67	append1_out_aga(x1, x2, x3)  =  append1_out_aga(x1, x3)
6.98/2.67	
6.98/2.67	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x5)
6.98/2.67	
6.98/2.67	.(x1, x2)  =  .(x2)
6.98/2.67	
6.98/2.67	U4_gg(x1, x2, x3)  =  U4_gg(x3)
6.98/2.67	
6.98/2.67	append2_in_gag(x1, x2, x3)  =  append2_in_gag(x1, x3)
6.98/2.67	
6.98/2.67	[]  =  []
6.98/2.67	
6.98/2.67	append2_out_gag(x1, x2, x3)  =  append2_out_gag(x2)
6.98/2.67	
6.98/2.67	U2_gag(x1, x2, x3, x4, x5)  =  U2_gag(x5)
6.98/2.67	
6.98/2.67	sublist_out_gg(x1, x2)  =  sublist_out_gg
6.98/2.67	
6.98/2.67	APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)
6.98/2.67	
6.98/2.67	
6.98/2.67	We have to consider all (P,R,Pi)-chains
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(33) UsableRulesProof (EQUIVALENT)
6.98/2.67	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(34)
6.98/2.67	Obligation:
6.98/2.67	Pi DP problem:
6.98/2.67	The TRS P consists of the following rules:
6.98/2.67	
6.98/2.67	   APPEND1_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AGA(Xs, Ys, Zs)
6.98/2.67	
6.98/2.67	R is empty.
6.98/2.67	The argument filtering Pi contains the following mapping:
6.98/2.67	.(x1, x2)  =  .(x2)
6.98/2.67	
6.98/2.67	APPEND1_IN_AGA(x1, x2, x3)  =  APPEND1_IN_AGA(x2)
6.98/2.67	
6.98/2.67	
6.98/2.67	We have to consider all (P,R,Pi)-chains
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(35) PiDPToQDPProof (SOUND)
6.98/2.67	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(36)
6.98/2.67	Obligation:
6.98/2.67	Q DP problem:
6.98/2.67	The TRS P consists of the following rules:
6.98/2.67	
6.98/2.67	   APPEND1_IN_AGA(Ys) -> APPEND1_IN_AGA(Ys)
6.98/2.67	
6.98/2.67	R is empty.
6.98/2.67	Q is empty.
6.98/2.67	We have to consider all (P,Q,R)-chains.
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(37) PrologToTRSTransformerProof (SOUND)
6.98/2.67	Transformed Prolog program to TRS.
6.98/2.67	
6.98/2.67	{
6.98/2.67	    "root": 2,
6.98/2.67	    "program": {
6.98/2.67	        "directives": [],
6.98/2.67	        "clauses": [
6.98/2.67	            [
6.98/2.67	                "(append1 ([]) Ys Ys)",
6.98/2.67	                null
6.98/2.67	            ],
6.98/2.67	            [
6.98/2.67	                "(append1 (. X Xs) Ys (. X Zs))",
6.98/2.67	                "(append1 Xs Ys Zs)"
6.98/2.67	            ],
6.98/2.67	            [
6.98/2.67	                "(append2 ([]) Ys Ys)",
6.98/2.67	                null
6.98/2.67	            ],
6.98/2.67	            [
6.98/2.67	                "(append2 (. X Xs) Ys (. X Zs))",
6.98/2.67	                "(append2 Xs Ys Zs)"
6.98/2.67	            ],
6.98/2.67	            [
6.98/2.67	                "(sublist X Y)",
6.98/2.67	                "(',' (append1 U X V) (append2 V W Y))"
6.98/2.67	            ]
6.98/2.67	        ]
6.98/2.67	    },
6.98/2.67	    "graph": {
6.98/2.67	        "nodes": {
6.98/2.67	            "14": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": 4,
6.98/2.67	                    "scope": 1,
6.98/2.67	                    "term": "(sublist T1 T2)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": [
6.98/2.67	                        "T1",
6.98/2.67	                        "T2"
6.98/2.67	                    ],
6.98/2.67	                    "free": [],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "191": {
6.98/2.67	                "goal": [
6.98/2.67	                    {
6.98/2.67	                        "clause": 0,
6.98/2.67	                        "scope": 2,
6.98/2.67	                        "term": "(append1 X13 T9 X14)"
6.98/2.67	                    },
6.98/2.67	                    {
6.98/2.67	                        "clause": 1,
6.98/2.67	                        "scope": 2,
6.98/2.67	                        "term": "(append1 X13 T9 X14)"
6.98/2.67	                    }
6.98/2.67	                ],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": ["T9"],
6.98/2.67	                    "free": [
6.98/2.67	                        "X13",
6.98/2.67	                        "X14"
6.98/2.67	                    ],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "192": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": 0,
6.98/2.67	                    "scope": 2,
6.98/2.67	                    "term": "(append1 X13 T9 X14)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": ["T9"],
6.98/2.67	                    "free": [
6.98/2.67	                        "X13",
6.98/2.67	                        "X14"
6.98/2.67	                    ],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "193": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": 1,
6.98/2.67	                    "scope": 2,
6.98/2.67	                    "term": "(append1 X13 T9 X14)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": ["T9"],
6.98/2.67	                    "free": [
6.98/2.67	                        "X13",
6.98/2.67	                        "X14"
6.98/2.67	                    ],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "type": "Nodes",
6.98/2.67	            "184": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": -1,
6.98/2.67	                    "scope": -1,
6.98/2.67	                    "term": "(',' (append1 X13 T9 X14) (append2 X14 X15 T10))"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": [
6.98/2.67	                        "T9",
6.98/2.67	                        "T10"
6.98/2.67	                    ],
6.98/2.67	                    "free": [
6.98/2.67	                        "X13",
6.98/2.67	                        "X14",
6.98/2.67	                        "X15"
6.98/2.67	                    ],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "195": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": -1,
6.98/2.67	                    "scope": -1,
6.98/2.67	                    "term": "(true)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": [],
6.98/2.67	                    "free": [],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "196": {
6.98/2.67	                "goal": [],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": [],
6.98/2.67	                    "free": [],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "240": {
6.98/2.67	                "goal": [],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": [],
6.98/2.67	                    "free": [],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "197": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": -1,
6.98/2.67	                    "scope": -1,
6.98/2.67	                    "term": "(append1 X45 T24 X46)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": ["T24"],
6.98/2.67	                    "free": [
6.98/2.67	                        "X45",
6.98/2.67	                        "X46"
6.98/2.67	                    ],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "241": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": -1,
6.98/2.67	                    "scope": -1,
6.98/2.67	                    "term": "(append2 T42 X76 T41)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": ["T41"],
6.98/2.67	                    "free": ["X76"],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "187": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": -1,
6.98/2.67	                    "scope": -1,
6.98/2.67	                    "term": "(append1 X13 T9 X14)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": ["T9"],
6.98/2.67	                    "free": [
6.98/2.67	                        "X13",
6.98/2.67	                        "X14"
6.98/2.67	                    ],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "242": {
6.98/2.67	                "goal": [],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": [],
6.98/2.67	                    "free": [],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "188": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": -1,
6.98/2.67	                    "scope": -1,
6.98/2.67	                    "term": "(append2 T14 X15 T10)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": ["T10"],
6.98/2.67	                    "free": ["X15"],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "2": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": -1,
6.98/2.67	                    "scope": -1,
6.98/2.67	                    "term": "(sublist T1 T2)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": [
6.98/2.67	                        "T1",
6.98/2.67	                        "T2"
6.98/2.67	                    ],
6.98/2.67	                    "free": [],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "235": {
6.98/2.67	                "goal": [
6.98/2.67	                    {
6.98/2.67	                        "clause": 2,
6.98/2.67	                        "scope": 3,
6.98/2.67	                        "term": "(append2 T14 X15 T10)"
6.98/2.67	                    },
6.98/2.67	                    {
6.98/2.67	                        "clause": 3,
6.98/2.67	                        "scope": 3,
6.98/2.67	                        "term": "(append2 T14 X15 T10)"
6.98/2.67	                    }
6.98/2.67	                ],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": ["T10"],
6.98/2.67	                    "free": ["X15"],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "236": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": 2,
6.98/2.67	                    "scope": 3,
6.98/2.67	                    "term": "(append2 T14 X15 T10)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": ["T10"],
6.98/2.67	                    "free": ["X15"],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "237": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": 3,
6.98/2.67	                    "scope": 3,
6.98/2.67	                    "term": "(append2 T14 X15 T10)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": ["T10"],
6.98/2.67	                    "free": ["X15"],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "238": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": -1,
6.98/2.67	                    "scope": -1,
6.98/2.67	                    "term": "(true)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": [],
6.98/2.67	                    "free": [],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "239": {
6.98/2.67	                "goal": [],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": [],
6.98/2.67	                    "free": [],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            }
6.98/2.67	        },
6.98/2.67	        "edges": [
6.98/2.67	            {
6.98/2.67	                "from": 2,
6.98/2.67	                "to": 14,
6.98/2.67	                "label": "CASE"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 14,
6.98/2.67	                "to": 184,
6.98/2.67	                "label": "ONLY EVAL with clause\nsublist(X11, X12) :- ','(append1(X13, X11, X14), append2(X14, X15, X12)).\nand substitutionT1 -> T9,\nX11 -> T9,\nT2 -> T10,\nX12 -> T10"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 184,
6.98/2.67	                "to": 187,
6.98/2.67	                "label": "SPLIT 1"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 184,
6.98/2.67	                "to": 188,
6.98/2.67	                "label": "SPLIT 2\nnew knowledge:\nT9 is ground\nreplacements:X13 -> T13,\nX14 -> T14"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 187,
6.98/2.67	                "to": 191,
6.98/2.67	                "label": "CASE"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 188,
6.98/2.67	                "to": 235,
6.98/2.67	                "label": "CASE"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 191,
6.98/2.67	                "to": 192,
6.98/2.67	                "label": "PARALLEL"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 191,
6.98/2.67	                "to": 193,
6.98/2.67	                "label": "PARALLEL"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 192,
6.98/2.67	                "to": 195,
6.98/2.67	                "label": "ONLY EVAL with clause\nappend1([], X24, X24).\nand substitutionX13 -> [],\nT9 -> T20,\nX24 -> T20,\nX14 -> T20"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 193,
6.98/2.67	                "to": 197,
6.98/2.67	                "label": "ONLY EVAL with clause\nappend1(.(X40, X41), X42, .(X40, X43)) :- append1(X41, X42, X43).\nand substitutionX40 -> X44,\nX41 -> X45,\nX13 -> .(X44, X45),\nT9 -> T24,\nX42 -> T24,\nX43 -> X46,\nX14 -> .(X44, X46)"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 195,
6.98/2.67	                "to": 196,
6.98/2.67	                "label": "SUCCESS"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 197,
6.98/2.67	                "to": 187,
6.98/2.67	                "label": "INSTANCE with matching:\nX13 -> X45\nT9 -> T24\nX14 -> X46"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 235,
6.98/2.67	                "to": 236,
6.98/2.67	                "label": "PARALLEL"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 235,
6.98/2.67	                "to": 237,
6.98/2.67	                "label": "PARALLEL"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 236,
6.98/2.67	                "to": 238,
6.98/2.67	                "label": "EVAL with clause\nappend2([], X60, X60).\nand substitutionT14 -> [],\nX15 -> T32,\nX60 -> T32,\nT10 -> T32,\nX61 -> T32"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 236,
6.98/2.67	                "to": 239,
6.98/2.67	                "label": "EVAL-BACKTRACK"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 237,
6.98/2.67	                "to": 241,
6.98/2.67	                "label": "EVAL with clause\nappend2(.(X72, X73), X74, .(X72, X75)) :- append2(X73, X74, X75).\nand substitutionX72 -> T39,\nX73 -> T42,\nT14 -> .(T39, T42),\nX15 -> X76,\nX74 -> X76,\nX75 -> T41,\nT10 -> .(T39, T41),\nT40 -> T42"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 237,
6.98/2.67	                "to": 242,
6.98/2.67	                "label": "EVAL-BACKTRACK"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 238,
6.98/2.67	                "to": 240,
6.98/2.67	                "label": "SUCCESS"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 241,
6.98/2.67	                "to": 188,
6.98/2.67	                "label": "INSTANCE with matching:\nT14 -> T42\nX15 -> X76\nT10 -> T41"
6.98/2.67	            }
6.98/2.67	        ],
6.98/2.67	        "type": "Graph"
6.98/2.67	    }
6.98/2.67	}
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(38)
6.98/2.67	Obligation:
6.98/2.67	Q restricted rewrite system:
6.98/2.67	The TRS R consists of the following rules:
6.98/2.67	
6.98/2.67	   f2_in(T9, T10) -> U1(f184_in(T9, T10), T9, T10)
6.98/2.67	   U1(f184_out1(X14, X15), T9, T10) -> f2_out1
6.98/2.67	   f187_in(T20) -> f187_out1
6.98/2.67	   f187_in(T24) -> U2(f187_in(T24), T24)
6.98/2.67	   U2(f187_out1, T24) -> f187_out1
6.98/2.67	   f188_in(T32) -> f188_out1([], T32)
6.98/2.67	   f188_in(.(T39, T41)) -> U3(f188_in(T41), .(T39, T41))
6.98/2.67	   U3(f188_out1(T42, X76), .(T39, T41)) -> f188_out1(.(T39, T42), X76)
6.98/2.67	   f184_in(T9, T10) -> U4(f187_in(T9), T9, T10)
6.98/2.67	   U4(f187_out1, T9, T10) -> U5(f188_in(T10), T9, T10)
6.98/2.67	   U5(f188_out1(T14, X15), T9, T10) -> f184_out1(T14, X15)
6.98/2.67	
6.98/2.67	Q is empty.
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(39) DependencyPairsProof (EQUIVALENT)
6.98/2.67	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(40)
6.98/2.67	Obligation:
6.98/2.67	Q DP problem:
6.98/2.67	The TRS P consists of the following rules:
6.98/2.67	
6.98/2.67	   F2_IN(T9, T10) -> U1^1(f184_in(T9, T10), T9, T10)
6.98/2.67	   F2_IN(T9, T10) -> F184_IN(T9, T10)
6.98/2.67	   F187_IN(T24) -> U2^1(f187_in(T24), T24)
6.98/2.67	   F187_IN(T24) -> F187_IN(T24)
6.98/2.67	   F188_IN(.(T39, T41)) -> U3^1(f188_in(T41), .(T39, T41))
6.98/2.67	   F188_IN(.(T39, T41)) -> F188_IN(T41)
6.98/2.67	   F184_IN(T9, T10) -> U4^1(f187_in(T9), T9, T10)
6.98/2.67	   F184_IN(T9, T10) -> F187_IN(T9)
6.98/2.67	   U4^1(f187_out1, T9, T10) -> U5^1(f188_in(T10), T9, T10)
6.98/2.67	   U4^1(f187_out1, T9, T10) -> F188_IN(T10)
6.98/2.67	
6.98/2.67	The TRS R consists of the following rules:
6.98/2.67	
6.98/2.67	   f2_in(T9, T10) -> U1(f184_in(T9, T10), T9, T10)
6.98/2.67	   U1(f184_out1(X14, X15), T9, T10) -> f2_out1
6.98/2.67	   f187_in(T20) -> f187_out1
6.98/2.67	   f187_in(T24) -> U2(f187_in(T24), T24)
6.98/2.67	   U2(f187_out1, T24) -> f187_out1
6.98/2.67	   f188_in(T32) -> f188_out1([], T32)
6.98/2.67	   f188_in(.(T39, T41)) -> U3(f188_in(T41), .(T39, T41))
6.98/2.67	   U3(f188_out1(T42, X76), .(T39, T41)) -> f188_out1(.(T39, T42), X76)
6.98/2.67	   f184_in(T9, T10) -> U4(f187_in(T9), T9, T10)
6.98/2.67	   U4(f187_out1, T9, T10) -> U5(f188_in(T10), T9, T10)
6.98/2.67	   U5(f188_out1(T14, X15), T9, T10) -> f184_out1(T14, X15)
6.98/2.67	
6.98/2.67	Q is empty.
6.98/2.67	We have to consider all minimal (P,Q,R)-chains.
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(41) DependencyGraphProof (EQUIVALENT)
6.98/2.67	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 8 less nodes.
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(42)
6.98/2.67	Complex Obligation (AND)
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(43)
6.98/2.67	Obligation:
6.98/2.67	Q DP problem:
6.98/2.67	The TRS P consists of the following rules:
6.98/2.67	
6.98/2.67	   F188_IN(.(T39, T41)) -> F188_IN(T41)
6.98/2.67	
6.98/2.67	The TRS R consists of the following rules:
6.98/2.67	
6.98/2.67	   f2_in(T9, T10) -> U1(f184_in(T9, T10), T9, T10)
6.98/2.67	   U1(f184_out1(X14, X15), T9, T10) -> f2_out1
6.98/2.67	   f187_in(T20) -> f187_out1
6.98/2.67	   f187_in(T24) -> U2(f187_in(T24), T24)
6.98/2.67	   U2(f187_out1, T24) -> f187_out1
6.98/2.67	   f188_in(T32) -> f188_out1([], T32)
6.98/2.67	   f188_in(.(T39, T41)) -> U3(f188_in(T41), .(T39, T41))
6.98/2.67	   U3(f188_out1(T42, X76), .(T39, T41)) -> f188_out1(.(T39, T42), X76)
6.98/2.67	   f184_in(T9, T10) -> U4(f187_in(T9), T9, T10)
6.98/2.67	   U4(f187_out1, T9, T10) -> U5(f188_in(T10), T9, T10)
6.98/2.67	   U5(f188_out1(T14, X15), T9, T10) -> f184_out1(T14, X15)
6.98/2.67	
6.98/2.67	Q is empty.
6.98/2.67	We have to consider all minimal (P,Q,R)-chains.
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(44) UsableRulesProof (EQUIVALENT)
6.98/2.67	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(45)
6.98/2.67	Obligation:
6.98/2.67	Q DP problem:
6.98/2.67	The TRS P consists of the following rules:
6.98/2.67	
6.98/2.67	   F188_IN(.(T39, T41)) -> F188_IN(T41)
6.98/2.67	
6.98/2.67	R is empty.
6.98/2.67	Q is empty.
6.98/2.67	We have to consider all minimal (P,Q,R)-chains.
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(46) QDPSizeChangeProof (EQUIVALENT)
6.98/2.67	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. 
6.98/2.67	
6.98/2.67	From the DPs we obtained the following set of size-change graphs:
6.98/2.67	*F188_IN(.(T39, T41)) -> F188_IN(T41)
6.98/2.67	The graph contains the following edges 1 > 1
6.98/2.67	
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(47)
6.98/2.67	YES
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(48)
6.98/2.67	Obligation:
6.98/2.67	Q DP problem:
6.98/2.67	The TRS P consists of the following rules:
6.98/2.67	
6.98/2.67	   F187_IN(T24) -> F187_IN(T24)
6.98/2.67	
6.98/2.67	The TRS R consists of the following rules:
6.98/2.67	
6.98/2.67	   f2_in(T9, T10) -> U1(f184_in(T9, T10), T9, T10)
6.98/2.67	   U1(f184_out1(X14, X15), T9, T10) -> f2_out1
6.98/2.67	   f187_in(T20) -> f187_out1
6.98/2.67	   f187_in(T24) -> U2(f187_in(T24), T24)
6.98/2.67	   U2(f187_out1, T24) -> f187_out1
6.98/2.67	   f188_in(T32) -> f188_out1([], T32)
6.98/2.67	   f188_in(.(T39, T41)) -> U3(f188_in(T41), .(T39, T41))
6.98/2.67	   U3(f188_out1(T42, X76), .(T39, T41)) -> f188_out1(.(T39, T42), X76)
6.98/2.67	   f184_in(T9, T10) -> U4(f187_in(T9), T9, T10)
6.98/2.67	   U4(f187_out1, T9, T10) -> U5(f188_in(T10), T9, T10)
6.98/2.67	   U5(f188_out1(T14, X15), T9, T10) -> f184_out1(T14, X15)
6.98/2.67	
6.98/2.67	Q is empty.
6.98/2.67	We have to consider all minimal (P,Q,R)-chains.
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(49) UsableRulesProof (EQUIVALENT)
6.98/2.67	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(50)
6.98/2.67	Obligation:
6.98/2.67	Q DP problem:
6.98/2.67	The TRS P consists of the following rules:
6.98/2.67	
6.98/2.67	   F187_IN(T24) -> F187_IN(T24)
6.98/2.67	
6.98/2.67	R is empty.
6.98/2.67	Q is empty.
6.98/2.67	We have to consider all minimal (P,Q,R)-chains.
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(51) PrologToIRSwTTransformerProof (SOUND)
6.98/2.67	Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert
6.98/2.67	
6.98/2.67	{
6.98/2.67	    "root": 128,
6.98/2.67	    "program": {
6.98/2.67	        "directives": [],
6.98/2.67	        "clauses": [
6.98/2.67	            [
6.98/2.67	                "(append1 ([]) Ys Ys)",
6.98/2.67	                null
6.98/2.67	            ],
6.98/2.67	            [
6.98/2.67	                "(append1 (. X Xs) Ys (. X Zs))",
6.98/2.67	                "(append1 Xs Ys Zs)"
6.98/2.67	            ],
6.98/2.67	            [
6.98/2.67	                "(append2 ([]) Ys Ys)",
6.98/2.67	                null
6.98/2.67	            ],
6.98/2.67	            [
6.98/2.67	                "(append2 (. X Xs) Ys (. X Zs))",
6.98/2.67	                "(append2 Xs Ys Zs)"
6.98/2.67	            ],
6.98/2.67	            [
6.98/2.67	                "(sublist X Y)",
6.98/2.67	                "(',' (append1 U X V) (append2 V W Y))"
6.98/2.67	            ]
6.98/2.67	        ]
6.98/2.67	    },
6.98/2.67	    "graph": {
6.98/2.67	        "nodes": {
6.98/2.67	            "190": {
6.98/2.67	                "goal": [],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": [],
6.98/2.67	                    "free": [],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "180": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": -1,
6.98/2.67	                    "scope": -1,
6.98/2.67	                    "term": "(',' (append1 X13 T9 X14) (append2 X14 X15 T10))"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": [
6.98/2.67	                        "T9",
6.98/2.67	                        "T10"
6.98/2.67	                    ],
6.98/2.67	                    "free": [
6.98/2.67	                        "X13",
6.98/2.67	                        "X14",
6.98/2.67	                        "X15"
6.98/2.67	                    ],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "181": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": -1,
6.98/2.67	                    "scope": -1,
6.98/2.67	                    "term": "(append1 X13 T9 X14)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": ["T9"],
6.98/2.67	                    "free": [
6.98/2.67	                        "X13",
6.98/2.67	                        "X14"
6.98/2.67	                    ],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "182": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": -1,
6.98/2.67	                    "scope": -1,
6.98/2.67	                    "term": "(append2 T14 X15 T10)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": ["T10"],
6.98/2.67	                    "free": ["X15"],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "type": "Nodes",
6.98/2.67	            "183": {
6.98/2.67	                "goal": [
6.98/2.67	                    {
6.98/2.67	                        "clause": 0,
6.98/2.67	                        "scope": 2,
6.98/2.67	                        "term": "(append1 X13 T9 X14)"
6.98/2.67	                    },
6.98/2.67	                    {
6.98/2.67	                        "clause": 1,
6.98/2.67	                        "scope": 2,
6.98/2.67	                        "term": "(append1 X13 T9 X14)"
6.98/2.67	                    }
6.98/2.67	                ],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": ["T9"],
6.98/2.67	                    "free": [
6.98/2.67	                        "X13",
6.98/2.67	                        "X14"
6.98/2.67	                    ],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "194": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": -1,
6.98/2.67	                    "scope": -1,
6.98/2.67	                    "term": "(append1 X45 T24 X46)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": ["T24"],
6.98/2.67	                    "free": [
6.98/2.67	                        "X45",
6.98/2.67	                        "X46"
6.98/2.67	                    ],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "185": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": 0,
6.98/2.67	                    "scope": 2,
6.98/2.67	                    "term": "(append1 X13 T9 X14)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": ["T9"],
6.98/2.67	                    "free": [
6.98/2.67	                        "X13",
6.98/2.67	                        "X14"
6.98/2.67	                    ],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "186": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": 1,
6.98/2.67	                    "scope": 2,
6.98/2.67	                    "term": "(append1 X13 T9 X14)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": ["T9"],
6.98/2.67	                    "free": [
6.98/2.67	                        "X13",
6.98/2.67	                        "X14"
6.98/2.67	                    ],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "231": {
6.98/2.67	                "goal": [],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": [],
6.98/2.67	                    "free": [],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "232": {
6.98/2.67	                "goal": [],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": [],
6.98/2.67	                    "free": [],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "134": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": 4,
6.98/2.67	                    "scope": 1,
6.98/2.67	                    "term": "(sublist T1 T2)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": [
6.98/2.67	                        "T1",
6.98/2.67	                        "T2"
6.98/2.67	                    ],
6.98/2.67	                    "free": [],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "189": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": -1,
6.98/2.67	                    "scope": -1,
6.98/2.67	                    "term": "(true)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": [],
6.98/2.67	                    "free": [],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "233": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": -1,
6.98/2.67	                    "scope": -1,
6.98/2.67	                    "term": "(append2 T42 X76 T41)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": ["T41"],
6.98/2.67	                    "free": ["X76"],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "234": {
6.98/2.67	                "goal": [],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": [],
6.98/2.67	                    "free": [],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "213": {
6.98/2.67	                "goal": [
6.98/2.67	                    {
6.98/2.67	                        "clause": 2,
6.98/2.67	                        "scope": 3,
6.98/2.67	                        "term": "(append2 T14 X15 T10)"
6.98/2.67	                    },
6.98/2.67	                    {
6.98/2.67	                        "clause": 3,
6.98/2.67	                        "scope": 3,
6.98/2.67	                        "term": "(append2 T14 X15 T10)"
6.98/2.67	                    }
6.98/2.67	                ],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": ["T10"],
6.98/2.67	                    "free": ["X15"],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "128": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": -1,
6.98/2.67	                    "scope": -1,
6.98/2.67	                    "term": "(sublist T1 T2)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": [
6.98/2.67	                        "T1",
6.98/2.67	                        "T2"
6.98/2.67	                    ],
6.98/2.67	                    "free": [],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "217": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": 2,
6.98/2.67	                    "scope": 3,
6.98/2.67	                    "term": "(append2 T14 X15 T10)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": ["T10"],
6.98/2.67	                    "free": ["X15"],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "228": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": -1,
6.98/2.67	                    "scope": -1,
6.98/2.67	                    "term": "(true)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": [],
6.98/2.67	                    "free": [],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            },
6.98/2.67	            "218": {
6.98/2.67	                "goal": [{
6.98/2.67	                    "clause": 3,
6.98/2.67	                    "scope": 3,
6.98/2.67	                    "term": "(append2 T14 X15 T10)"
6.98/2.67	                }],
6.98/2.67	                "kb": {
6.98/2.67	                    "nonunifying": [],
6.98/2.67	                    "intvars": {},
6.98/2.67	                    "arithmetic": {
6.98/2.67	                        "type": "PlainIntegerRelationState",
6.98/2.67	                        "relations": []
6.98/2.67	                    },
6.98/2.67	                    "ground": ["T10"],
6.98/2.67	                    "free": ["X15"],
6.98/2.67	                    "exprvars": []
6.98/2.67	                }
6.98/2.67	            }
6.98/2.67	        },
6.98/2.67	        "edges": [
6.98/2.67	            {
6.98/2.67	                "from": 128,
6.98/2.67	                "to": 134,
6.98/2.67	                "label": "CASE"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 134,
6.98/2.67	                "to": 180,
6.98/2.67	                "label": "ONLY EVAL with clause\nsublist(X11, X12) :- ','(append1(X13, X11, X14), append2(X14, X15, X12)).\nand substitutionT1 -> T9,\nX11 -> T9,\nT2 -> T10,\nX12 -> T10"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 180,
6.98/2.67	                "to": 181,
6.98/2.67	                "label": "SPLIT 1"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 180,
6.98/2.67	                "to": 182,
6.98/2.67	                "label": "SPLIT 2\nnew knowledge:\nT9 is ground\nreplacements:X13 -> T13,\nX14 -> T14"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 181,
6.98/2.67	                "to": 183,
6.98/2.67	                "label": "CASE"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 182,
6.98/2.67	                "to": 213,
6.98/2.67	                "label": "CASE"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 183,
6.98/2.67	                "to": 185,
6.98/2.67	                "label": "PARALLEL"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 183,
6.98/2.67	                "to": 186,
6.98/2.67	                "label": "PARALLEL"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 185,
6.98/2.67	                "to": 189,
6.98/2.67	                "label": "ONLY EVAL with clause\nappend1([], X24, X24).\nand substitutionX13 -> [],\nT9 -> T20,\nX24 -> T20,\nX14 -> T20"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 186,
6.98/2.67	                "to": 194,
6.98/2.67	                "label": "ONLY EVAL with clause\nappend1(.(X40, X41), X42, .(X40, X43)) :- append1(X41, X42, X43).\nand substitutionX40 -> X44,\nX41 -> X45,\nX13 -> .(X44, X45),\nT9 -> T24,\nX42 -> T24,\nX43 -> X46,\nX14 -> .(X44, X46)"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 189,
6.98/2.67	                "to": 190,
6.98/2.67	                "label": "SUCCESS"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 194,
6.98/2.67	                "to": 181,
6.98/2.67	                "label": "INSTANCE with matching:\nX13 -> X45\nT9 -> T24\nX14 -> X46"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 213,
6.98/2.67	                "to": 217,
6.98/2.67	                "label": "PARALLEL"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 213,
6.98/2.67	                "to": 218,
6.98/2.67	                "label": "PARALLEL"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 217,
6.98/2.67	                "to": 228,
6.98/2.67	                "label": "EVAL with clause\nappend2([], X60, X60).\nand substitutionT14 -> [],\nX15 -> T32,\nX60 -> T32,\nT10 -> T32,\nX61 -> T32"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 217,
6.98/2.67	                "to": 231,
6.98/2.67	                "label": "EVAL-BACKTRACK"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 218,
6.98/2.67	                "to": 233,
6.98/2.67	                "label": "EVAL with clause\nappend2(.(X72, X73), X74, .(X72, X75)) :- append2(X73, X74, X75).\nand substitutionX72 -> T39,\nX73 -> T42,\nT14 -> .(T39, T42),\nX15 -> X76,\nX74 -> X76,\nX75 -> T41,\nT10 -> .(T39, T41),\nT40 -> T42"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 218,
6.98/2.67	                "to": 234,
6.98/2.67	                "label": "EVAL-BACKTRACK"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 228,
6.98/2.67	                "to": 232,
6.98/2.67	                "label": "SUCCESS"
6.98/2.67	            },
6.98/2.67	            {
6.98/2.67	                "from": 233,
6.98/2.67	                "to": 182,
6.98/2.67	                "label": "INSTANCE with matching:\nT14 -> T42\nX15 -> X76\nT10 -> T41"
6.98/2.67	            }
6.98/2.67	        ],
6.98/2.67	        "type": "Graph"
6.98/2.67	    }
6.98/2.67	}
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(52)
6.98/2.67	Complex Obligation (AND)
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(53)
6.98/2.67	Obligation:
6.98/2.67	Rules:
6.98/2.67	f182_in(T10) -> f213_in(T10) :|: TRUE
6.98/2.67	f213_out(x) -> f182_out(x) :|: TRUE
6.98/2.67	f233_in(T41) -> f182_in(T41) :|: TRUE
6.98/2.67	f182_out(x1) -> f233_out(x1) :|: TRUE
6.98/2.67	f213_in(x2) -> f217_in(x2) :|: TRUE
6.98/2.67	f217_out(x3) -> f213_out(x3) :|: TRUE
6.98/2.67	f213_in(x4) -> f218_in(x4) :|: TRUE
6.98/2.67	f218_out(x5) -> f213_out(x5) :|: TRUE
6.98/2.67	f234_out -> f218_out(x6) :|: TRUE
6.98/2.67	f233_out(x7) -> f218_out(.(x8, x7)) :|: TRUE
6.98/2.67	f218_in(.(x9, x10)) -> f233_in(x10) :|: TRUE
6.98/2.67	f218_in(x11) -> f234_in :|: TRUE
6.98/2.67	f134_out(T1, T2) -> f128_out(T1, T2) :|: TRUE
6.98/2.67	f128_in(x12, x13) -> f134_in(x12, x13) :|: TRUE
6.98/2.67	f134_in(x14, x15) -> f180_in(x14, x15) :|: TRUE
6.98/2.67	f180_out(x16, x17) -> f134_out(x16, x17) :|: TRUE
6.98/2.67	f182_out(x18) -> f180_out(x19, x18) :|: TRUE
6.98/2.67	f181_out(x20) -> f182_in(x21) :|: TRUE
6.98/2.67	f180_in(x22, x23) -> f181_in(x22) :|: TRUE
6.98/2.67	Start term: f128_in(T1, T2)
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(54) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
6.98/2.67	Constructed simple dependency graph.
6.98/2.67	
6.98/2.67	Simplified to the following IRSwTs:
6.98/2.67	
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(55)
6.98/2.67	TRUE
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(56)
6.98/2.67	Obligation:
6.98/2.67	Rules:
6.98/2.67	f186_in(T24) -> f194_in(T24) :|: TRUE
6.98/2.67	f194_out(x) -> f186_out(x) :|: TRUE
6.98/2.67	f183_in(T9) -> f185_in(T9) :|: TRUE
6.98/2.67	f186_out(x1) -> f183_out(x1) :|: TRUE
6.98/2.67	f185_out(x2) -> f183_out(x2) :|: TRUE
6.98/2.67	f183_in(x3) -> f186_in(x3) :|: TRUE
6.98/2.67	f181_out(x4) -> f194_out(x4) :|: TRUE
6.98/2.67	f194_in(x5) -> f181_in(x5) :|: TRUE
6.98/2.67	f183_out(x6) -> f181_out(x6) :|: TRUE
6.98/2.67	f181_in(x7) -> f183_in(x7) :|: TRUE
6.98/2.67	f134_out(T1, T2) -> f128_out(T1, T2) :|: TRUE
6.98/2.67	f128_in(x8, x9) -> f134_in(x8, x9) :|: TRUE
6.98/2.67	f134_in(x10, x11) -> f180_in(x10, x11) :|: TRUE
6.98/2.67	f180_out(x12, x13) -> f134_out(x12, x13) :|: TRUE
6.98/2.67	f182_out(x14) -> f180_out(x15, x14) :|: TRUE
6.98/2.67	f181_out(x16) -> f182_in(x17) :|: TRUE
6.98/2.67	f180_in(x18, x19) -> f181_in(x18) :|: TRUE
6.98/2.67	Start term: f128_in(T1, T2)
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(57) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
6.98/2.67	Constructed simple dependency graph.
6.98/2.67	
6.98/2.67	Simplified to the following IRSwTs:
6.98/2.67	
6.98/2.67	intTRSProblem:
6.98/2.67	f186_in(T24) -> f194_in(T24) :|: TRUE
6.98/2.67	f183_in(x3) -> f186_in(x3) :|: TRUE
6.98/2.67	f194_in(x5) -> f181_in(x5) :|: TRUE
6.98/2.67	f181_in(x7) -> f183_in(x7) :|: TRUE
6.98/2.67	
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(58)
6.98/2.67	Obligation:
6.98/2.67	Rules:
6.98/2.67	f186_in(T24) -> f194_in(T24) :|: TRUE
6.98/2.67	f183_in(x3) -> f186_in(x3) :|: TRUE
6.98/2.67	f194_in(x5) -> f181_in(x5) :|: TRUE
6.98/2.67	f181_in(x7) -> f183_in(x7) :|: TRUE
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(59) IntTRSCompressionProof (EQUIVALENT)
6.98/2.67	Compressed rules.
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(60)
6.98/2.67	Obligation:
6.98/2.67	Rules:
6.98/2.67	f183_in(x3:0) -> f183_in(x3:0) :|: TRUE
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(61) IRSFormatTransformerProof (EQUIVALENT)
6.98/2.67	Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(62)
6.98/2.67	Obligation:
6.98/2.67	Rules:
6.98/2.67	f183_in(x3:0) -> f183_in(x3:0) :|: TRUE
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(63) IRSwTTerminationDigraphProof (EQUIVALENT)
6.98/2.67	Constructed termination digraph!
6.98/2.67	Nodes:
6.98/2.67	(1) f183_in(x3:0) -> f183_in(x3:0) :|: TRUE
6.98/2.67	
6.98/2.67	Arcs:
6.98/2.67	(1) -> (1)
6.98/2.67	
6.98/2.67	This digraph is fully evaluated!
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(64)
6.98/2.67	Obligation:
6.98/2.67	
6.98/2.67	Termination digraph:
6.98/2.67	Nodes:
6.98/2.67	(1) f183_in(x3:0) -> f183_in(x3:0) :|: TRUE
6.98/2.67	
6.98/2.67	Arcs:
6.98/2.67	(1) -> (1)
6.98/2.67	
6.98/2.67	This digraph is fully evaluated!
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(65) FilterProof (EQUIVALENT)
6.98/2.67	Used the following sort dictionary for filtering: 
6.98/2.67	f183_in(VARIABLE)
6.98/2.67	Replaced non-predefined constructor symbols by 0.
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(66)
6.98/2.67	Obligation:
6.98/2.67	Rules:
6.98/2.67	f183_in(x3:0) -> f183_in(x3:0) :|: TRUE
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(67) IntTRSNonPeriodicNontermProof (COMPLETE)
6.98/2.67	Normalized system to the following form:
6.98/2.67	f(pc, x3:0) -> f(1, x3:0) :|: pc = 1 && TRUE
6.98/2.67	Proved unsatisfiability of the following formula, indicating that the system is never left after entering:
6.98/2.67	(((run2_0 = ((1 * 1)) and run2_1 = ((run1_1 * 1))) and (((run1_0 * 1)) = ((1 * 1)) and T)) and !(((run2_0 * 1)) = ((1 * 1)) and T))
6.98/2.67	Proved satisfiability of the following formula, indicating that the system is entered at least once:
6.98/2.67	((run2_0 = ((1 * 1)) and run2_1 = ((run1_1 * 1))) and (((run1_0 * 1)) = ((1 * 1)) and T))
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(68)
6.98/2.67	NO
6.98/2.67	
6.98/2.67	----------------------------------------
6.98/2.67	
6.98/2.67	(69) PrologToDTProblemTransformerProof (SOUND)
6.98/2.67	Built DT problem from termination graph DT10.
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6.98/2.68	            "1": {
6.98/2.68	                "goal": [{
6.98/2.68	                    "clause": -1,
6.98/2.68	                    "scope": -1,
6.98/2.68	                    "term": "(sublist T1 T2)"
6.98/2.68	                }],
6.98/2.68	                "kb": {
6.98/2.68	                    "nonunifying": [],
6.98/2.68	                    "intvars": {},
6.98/2.68	                    "arithmetic": {
6.98/2.68	                        "type": "PlainIntegerRelationState",
6.98/2.68	                        "relations": []
6.98/2.68	                    },
6.98/2.68	                    "ground": [
6.98/2.68	                        "T1",
6.98/2.68	                        "T2"
6.98/2.68	                    ],
6.98/2.68	                    "free": [],
6.98/2.68	                    "exprvars": []
6.98/2.68	                }
6.98/2.68	            },
6.98/2.68	            "122": {
6.98/2.68	                "goal": [{
6.98/2.68	                    "clause": 3,
6.98/2.68	                    "scope": 3,
6.98/2.68	                    "term": "(append2 T15 X7 T6)"
6.98/2.68	                }],
6.98/2.68	                "kb": {
6.98/2.68	                    "nonunifying": [],
6.98/2.68	                    "intvars": {},
6.98/2.68	                    "arithmetic": {
6.98/2.68	                        "type": "PlainIntegerRelationState",
6.98/2.68	                        "relations": []
6.98/2.68	                    },
6.98/2.68	                    "ground": [
6.98/2.68	                        "T6",
6.98/2.68	                        "T15"
6.98/2.68	                    ],
6.98/2.68	                    "free": ["X7"],
6.98/2.68	                    "exprvars": []
6.98/2.68	                }
6.98/2.68	            },
6.98/2.68	            "243": {
6.98/2.68	                "goal": [{
6.98/2.68	                    "clause": -1,
6.98/2.68	                    "scope": -1,
6.98/2.68	                    "term": "(',' (append1 X76 T37 X77) (append2 (. X75 X77) X7 T6))"
6.98/2.68	                }],
6.98/2.68	                "kb": {
6.98/2.68	                    "nonunifying": [],
6.98/2.68	                    "intvars": {},
6.98/2.68	                    "arithmetic": {
6.98/2.68	                        "type": "PlainIntegerRelationState",
6.98/2.68	                        "relations": []
6.98/2.68	                    },
6.98/2.68	                    "ground": [
6.98/2.68	                        "T6",
6.98/2.68	                        "T37"
6.98/2.68	                    ],
6.98/2.68	                    "free": [
6.98/2.68	                        "X7",
6.98/2.68	                        "X75",
6.98/2.68	                        "X76",
6.98/2.68	                        "X77"
6.98/2.68	                    ],
6.98/2.68	                    "exprvars": []
6.98/2.68	                }
6.98/2.68	            },
6.98/2.68	            "244": {
6.98/2.68	                "goal": [{
6.98/2.68	                    "clause": -1,
6.98/2.68	                    "scope": -1,
6.98/2.68	                    "term": "(append1 X76 T37 X77)"
6.98/2.68	                }],
6.98/2.68	                "kb": {
6.98/2.68	                    "nonunifying": [],
6.98/2.68	                    "intvars": {},
6.98/2.68	                    "arithmetic": {
6.98/2.68	                        "type": "PlainIntegerRelationState",
6.98/2.68	                        "relations": []
6.98/2.68	                    },
6.98/2.68	                    "ground": ["T37"],
6.98/2.68	                    "free": [
6.98/2.68	                        "X76",
6.98/2.68	                        "X77"
6.98/2.68	                    ],
6.98/2.68	                    "exprvars": []
6.98/2.68	                }
6.98/2.68	            },
6.98/2.68	            "245": {
6.98/2.68	                "goal": [{
6.98/2.68	                    "clause": -1,
6.98/2.68	                    "scope": -1,
6.98/2.68	                    "term": "(append2 (. X75 T40) X7 T6)"
6.98/2.68	                }],
6.98/2.68	                "kb": {
6.98/2.68	                    "nonunifying": [],
6.98/2.68	                    "intvars": {},
6.98/2.68	                    "arithmetic": {
6.98/2.68	                        "type": "PlainIntegerRelationState",
6.98/2.68	                        "relations": []
6.98/2.68	                    },
6.98/2.68	                    "ground": ["T6"],
6.98/2.68	                    "free": [
6.98/2.68	                        "X7",
6.98/2.68	                        "X75"
6.98/2.68	                    ],
6.98/2.68	                    "exprvars": []
6.98/2.68	                }
6.98/2.68	            },
6.98/2.68	            "147": {
6.98/2.68	                "goal": [{
6.98/2.68	                    "clause": -1,
6.98/2.68	                    "scope": -1,
6.98/2.68	                    "term": "(true)"
6.98/2.68	                }],
6.98/2.68	                "kb": {
6.98/2.68	                    "nonunifying": [],
6.98/2.68	                    "intvars": {},
6.98/2.68	                    "arithmetic": {
6.98/2.68	                        "type": "PlainIntegerRelationState",
6.98/2.68	                        "relations": []
6.98/2.68	                    },
6.98/2.68	                    "ground": [],
6.98/2.68	                    "free": [],
6.98/2.68	                    "exprvars": []
6.98/2.68	                }
6.98/2.68	            },
6.98/2.68	            "202": {
6.98/2.68	                "goal": [{
6.98/2.68	                    "clause": -1,
6.98/2.68	                    "scope": -1,
6.98/2.68	                    "term": "(append2 T30 X49 T31)"
6.98/2.68	                }],
6.98/2.68	                "kb": {
6.98/2.68	                    "nonunifying": [],
6.98/2.68	                    "intvars": {},
6.98/2.68	                    "arithmetic": {
6.98/2.68	                        "type": "PlainIntegerRelationState",
6.98/2.68	                        "relations": []
6.98/2.68	                    },
6.98/2.68	                    "ground": [
6.98/2.68	                        "T30",
6.98/2.68	                        "T31"
6.98/2.68	                    ],
6.98/2.68	                    "free": ["X49"],
6.98/2.68	                    "exprvars": []
6.98/2.68	                }
6.98/2.68	            },
6.98/2.68	            "246": {
6.98/2.68	                "goal": [
6.98/2.68	                    {
6.98/2.68	                        "clause": 0,
6.98/2.68	                        "scope": 4,
6.98/2.68	                        "term": "(append1 X76 T37 X77)"
6.98/2.68	                    },
6.98/2.68	                    {
6.98/2.68	                        "clause": 1,
6.98/2.68	                        "scope": 4,
6.98/2.68	                        "term": "(append1 X76 T37 X77)"
6.98/2.68	                    }
6.98/2.68	                ],
6.98/2.68	                "kb": {
6.98/2.68	                    "nonunifying": [],
6.98/2.68	                    "intvars": {},
6.98/2.68	                    "arithmetic": {
6.98/2.68	                        "type": "PlainIntegerRelationState",
6.98/2.68	                        "relations": []
6.98/2.68	                    },
6.98/2.68	                    "ground": ["T37"],
6.98/2.68	                    "free": [
6.98/2.68	                        "X76",
6.98/2.68	                        "X77"
6.98/2.68	                    ],
6.98/2.68	                    "exprvars": []
6.98/2.68	                }
6.98/2.68	            },
6.98/2.68	            "5": {
6.98/2.68	                "goal": [{
6.98/2.68	                    "clause": 4,
6.98/2.68	                    "scope": 1,
6.98/2.68	                    "term": "(sublist T1 T2)"
6.98/2.68	                }],
6.98/2.68	                "kb": {
6.98/2.68	                    "nonunifying": [],
6.98/2.68	                    "intvars": {},
6.98/2.68	                    "arithmetic": {
6.98/2.68	                        "type": "PlainIntegerRelationState",
6.98/2.68	                        "relations": []
6.98/2.68	                    },
6.98/2.68	                    "ground": [
6.98/2.68	                        "T1",
6.98/2.68	                        "T2"
6.98/2.68	                    ],
6.98/2.68	                    "free": [],
6.98/2.68	                    "exprvars": []
6.98/2.68	                }
6.98/2.68	            },
6.98/2.68	            "203": {
6.98/2.68	                "goal": [],
6.98/2.68	                "kb": {
6.98/2.68	                    "nonunifying": [],
6.98/2.68	                    "intvars": {},
6.98/2.68	                    "arithmetic": {
6.98/2.68	                        "type": "PlainIntegerRelationState",
6.98/2.68	                        "relations": []
6.98/2.68	                    },
6.98/2.68	                    "ground": [],
6.98/2.68	                    "free": [],
6.98/2.68	                    "exprvars": []
6.98/2.68	                }
6.98/2.68	            },
6.98/2.68	            "247": {
6.98/2.68	                "goal": [{
6.98/2.68	                    "clause": 0,
6.98/2.68	                    "scope": 4,
6.98/2.68	                    "term": "(append1 X76 T37 X77)"
6.98/2.68	                }],
6.98/2.68	                "kb": {
6.98/2.68	                    "nonunifying": [],
6.98/2.68	                    "intvars": {},
6.98/2.68	                    "arithmetic": {
6.98/2.68	                        "type": "PlainIntegerRelationState",
6.98/2.68	                        "relations": []
6.98/2.68	                    },
6.98/2.68	                    "ground": ["T37"],
6.98/2.68	                    "free": [
6.98/2.68	                        "X76",
6.98/2.68	                        "X77"
6.98/2.68	                    ],
6.98/2.68	                    "exprvars": []
6.98/2.68	                }
6.98/2.68	            },
6.98/2.68	            "248": {
6.98/2.68	                "goal": [{
6.98/2.68	                    "clause": 1,
6.98/2.68	                    "scope": 4,
6.98/2.68	                    "term": "(append1 X76 T37 X77)"
6.98/2.68	                }],
6.98/2.68	                "kb": {
6.98/2.68	                    "nonunifying": [],
6.98/2.68	                    "intvars": {},
6.98/2.68	                    "arithmetic": {
6.98/2.68	                        "type": "PlainIntegerRelationState",
6.98/2.68	                        "relations": []
6.98/2.68	                    },
6.98/2.68	                    "ground": ["T37"],
6.98/2.68	                    "free": [
6.98/2.68	                        "X76",
6.98/2.68	                        "X77"
6.98/2.68	                    ],
6.98/2.68	                    "exprvars": []
6.98/2.68	                }
6.98/2.68	            },
6.98/2.68	            "249": {
6.98/2.68	                "goal": [{
6.98/2.68	                    "clause": -1,
6.98/2.68	                    "scope": -1,
6.98/2.68	                    "term": "(true)"
6.98/2.68	                }],
6.98/2.68	                "kb": {
6.98/2.68	                    "nonunifying": [],
6.98/2.68	                    "intvars": {},
6.98/2.68	                    "arithmetic": {
6.98/2.68	                        "type": "PlainIntegerRelationState",
6.98/2.68	                        "relations": []
6.98/2.68	                    },
6.98/2.68	                    "ground": [],
6.98/2.68	                    "free": [],
6.98/2.68	                    "exprvars": []
6.98/2.68	                }
6.98/2.68	            }
6.98/2.68	        },
6.98/2.68	        "edges": [
6.98/2.68	            {
6.98/2.68	                "from": 1,
6.98/2.68	                "to": 5,
6.98/2.68	                "label": "CASE"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 5,
6.98/2.68	                "to": 25,
6.98/2.68	                "label": "ONLY EVAL with clause\nsublist(X3, X4) :- ','(append1(X5, X3, X6), append2(X6, X7, X4)).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T6,\nX4 -> T6"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 25,
6.98/2.68	                "to": 26,
6.98/2.68	                "label": "CASE"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 26,
6.98/2.68	                "to": 29,
6.98/2.68	                "label": "PARALLEL"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 26,
6.98/2.68	                "to": 30,
6.98/2.68	                "label": "PARALLEL"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 29,
6.98/2.68	                "to": 52,
6.98/2.68	                "label": "ONLY EVAL with clause\nappend1([], X20, X20).\nand substitutionX5 -> [],\nT5 -> T15,\nX20 -> T15,\nX6 -> T15"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 30,
6.98/2.68	                "to": 243,
6.98/2.68	                "label": "ONLY EVAL with clause\nappend1(.(X71, X72), X73, .(X71, X74)) :- append1(X72, X73, X74).\nand substitutionX71 -> X75,\nX72 -> X76,\nX5 -> .(X75, X76),\nT5 -> T37,\nX73 -> T37,\nX74 -> X77,\nX6 -> .(X75, X77)"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 52,
6.98/2.68	                "to": 120,
6.98/2.68	                "label": "CASE"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 120,
6.98/2.68	                "to": 121,
6.98/2.68	                "label": "PARALLEL"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 120,
6.98/2.68	                "to": 122,
6.98/2.68	                "label": "PARALLEL"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 121,
6.98/2.68	                "to": 147,
6.98/2.68	                "label": "EVAL with clause\nappend2([], X33, X33).\nand substitutionT15 -> [],\nX7 -> T22,\nX33 -> T22,\nT6 -> T22,\nX34 -> T22"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 121,
6.98/2.68	                "to": 155,
6.98/2.68	                "label": "EVAL-BACKTRACK"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 122,
6.98/2.68	                "to": 202,
6.98/2.68	                "label": "EVAL with clause\nappend2(.(X45, X46), X47, .(X45, X48)) :- append2(X46, X47, X48).\nand substitutionX45 -> T29,\nX46 -> T30,\nT15 -> .(T29, T30),\nX7 -> X49,\nX47 -> X49,\nX48 -> T31,\nT6 -> .(T29, T31)"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 122,
6.98/2.68	                "to": 203,
6.98/2.68	                "label": "EVAL-BACKTRACK"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 147,
6.98/2.68	                "to": 173,
6.98/2.68	                "label": "SUCCESS"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 202,
6.98/2.68	                "to": 52,
6.98/2.68	                "label": "INSTANCE with matching:\nT15 -> T30\nX7 -> X49\nT6 -> T31"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 243,
6.98/2.68	                "to": 244,
6.98/2.68	                "label": "SPLIT 1"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 243,
6.98/2.68	                "to": 245,
6.98/2.68	                "label": "SPLIT 2\nnew knowledge:\nT37 is ground\nreplacements:X76 -> T39,\nX77 -> T40"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 244,
6.98/2.68	                "to": 246,
6.98/2.68	                "label": "CASE"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 245,
6.98/2.68	                "to": 252,
6.98/2.68	                "label": "CASE"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 246,
6.98/2.68	                "to": 247,
6.98/2.68	                "label": "PARALLEL"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 246,
6.98/2.68	                "to": 248,
6.98/2.68	                "label": "PARALLEL"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 247,
6.98/2.68	                "to": 249,
6.98/2.68	                "label": "ONLY EVAL with clause\nappend1([], X86, X86).\nand substitutionX76 -> [],\nT37 -> T46,\nX86 -> T46,\nX77 -> T46"
6.98/2.68	            },
6.98/2.68	            {
6.98/2.68	                "from": 248,
6.98/2.68	                "to": 251,
6.98/2.68	                "label": "ONLY EVAL with clause\nappend1(.(X102, X103), X104, .(X102, X105)) :- append1(X103, X104, X105).\nand substitutionX102 -> X106,\nX103 -> X107,\nX76 -> .(X106, X107),\nT37 -> T50,\nX104 -> T50,\nX105 -> X108,\nX77 -> .(X106, X108)"
7.21/2.68	            },
7.21/2.68	            {
7.21/2.68	                "from": 249,
7.21/2.68	                "to": 250,
7.21/2.68	                "label": "SUCCESS"
7.21/2.68	            },
7.21/2.68	            {
7.21/2.68	                "from": 251,
7.21/2.68	                "to": 244,
7.21/2.68	                "label": "INSTANCE with matching:\nX76 -> X107\nT37 -> T50\nX77 -> X108"
7.21/2.68	            },
7.21/2.68	            {
7.21/2.68	                "from": 252,
7.21/2.68	                "to": 253,
7.21/2.68	                "label": "BACKTRACK\nfor clause: append2([], Ys, Ys)because of non-unification"
7.21/2.68	            },
7.21/2.68	            {
7.21/2.68	                "from": 253,
7.21/2.68	                "to": 254,
7.21/2.68	                "label": "EVAL with clause\nappend2(.(X125, X126), X127, .(X125, X128)) :- append2(X126, X127, X128).\nand substitutionX75 -> T59,\nX125 -> T59,\nT40 -> T61,\nX126 -> T61,\nX7 -> X130,\nX127 -> X130,\nX129 -> T59,\nX128 -> T60,\nT6 -> .(T59, T60),\nT58 -> T61"
7.21/2.68	            },
7.21/2.68	            {
7.21/2.68	                "from": 253,
7.21/2.68	                "to": 255,
7.21/2.68	                "label": "EVAL-BACKTRACK"
7.21/2.68	            },
7.21/2.68	            {
7.21/2.68	                "from": 254,
7.21/2.68	                "to": 256,
7.21/2.68	                "label": "CASE"
7.21/2.68	            },
7.21/2.68	            {
7.21/2.68	                "from": 256,
7.21/2.68	                "to": 257,
7.21/2.68	                "label": "PARALLEL"
7.21/2.68	            },
7.21/2.68	            {
7.21/2.68	                "from": 256,
7.21/2.68	                "to": 258,
7.21/2.68	                "label": "PARALLEL"
7.21/2.68	            },
7.21/2.68	            {
7.21/2.68	                "from": 257,
7.21/2.68	                "to": 259,
7.21/2.68	                "label": "EVAL with clause\nappend2([], X143, X143).\nand substitutionT61 -> [],\nX130 -> T68,\nX143 -> T68,\nT60 -> T68,\nX144 -> T68"
7.21/2.68	            },
7.21/2.68	            {
7.21/2.68	                "from": 257,
7.21/2.68	                "to": 260,
7.21/2.68	                "label": "EVAL-BACKTRACK"
7.21/2.68	            },
7.21/2.68	            {
7.21/2.68	                "from": 258,
7.21/2.68	                "to": 262,
7.21/2.68	                "label": "EVAL with clause\nappend2(.(X155, X156), X157, .(X155, X158)) :- append2(X156, X157, X158).\nand substitutionX155 -> T75,\nX156 -> T78,\nT61 -> .(T75, T78),\nX130 -> X159,\nX157 -> X159,\nX158 -> T77,\nT60 -> .(T75, T77),\nT76 -> T78"
7.21/2.68	            },
7.21/2.68	            {
7.21/2.68	                "from": 258,
7.21/2.68	                "to": 263,
7.21/2.68	                "label": "EVAL-BACKTRACK"
7.21/2.68	            },
7.21/2.68	            {
7.21/2.68	                "from": 259,
7.21/2.68	                "to": 261,
7.21/2.68	                "label": "SUCCESS"
7.21/2.68	            },
7.21/2.68	            {
7.21/2.68	                "from": 262,
7.21/2.68	                "to": 254,
7.21/2.68	                "label": "INSTANCE with matching:\nT61 -> T78\nX130 -> X159\nT60 -> T77"
7.21/2.68	            }
7.21/2.68	        ],
7.21/2.68	        "type": "Graph"
7.21/2.68	    }
7.21/2.68	}
7.21/2.68	
7.21/2.68	----------------------------------------
7.21/2.68	
7.21/2.68	(70)
7.21/2.68	Obligation:
7.21/2.68	Triples:
7.21/2.68	
7.21/2.68	append2A(.(X1, X2), X3, .(X1, X4)) :- append2A(X2, X3, X4).
7.21/2.68	append1B(.(X1, X2), X3, .(X1, X4)) :- append1B(X2, X3, X4).
7.21/2.68	append2C(.(X1, X2), X3, .(X1, X4)) :- append2C(X2, X3, X4).
7.21/2.68	sublistD(X1, X2) :- append2A(X1, X3, X2).
7.21/2.68	sublistD(X1, X2) :- append1B(X3, X1, X4).
7.21/2.68	sublistD(X1, .(X2, X3)) :- ','(append1cB(X4, X1, X5), append2C(X5, X6, X3)).
7.21/2.68	
7.21/2.68	Clauses:
7.21/2.68	
7.21/2.68	append2cA([], X1, X1).
7.21/2.68	append2cA(.(X1, X2), X3, .(X1, X4)) :- append2cA(X2, X3, X4).
7.21/2.68	append1cB([], X1, X1).
7.21/2.68	append1cB(.(X1, X2), X3, .(X1, X4)) :- append1cB(X2, X3, X4).
7.21/2.68	append2cC([], X1, X1).
7.21/2.68	append2cC(.(X1, X2), X3, .(X1, X4)) :- append2cC(X2, X3, X4).
7.21/2.68	
7.21/2.68	Afs:
7.21/2.68	
7.21/2.68	sublistD(x1, x2)  =  sublistD(x1, x2)
7.21/2.68	
7.21/2.68	
7.21/2.68	----------------------------------------
7.21/2.68	
7.21/2.68	(71) TriplesToPiDPProof (SOUND)
7.21/2.68	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
7.21/2.68	
7.21/2.68	sublistD_in_2: (b,b)
7.21/2.68	
7.21/2.68	append2A_in_3: (b,f,b)
7.21/2.68	
7.21/2.68	append1B_in_3: (f,b,f)
7.21/2.68	
7.21/2.68	append1cB_in_3: (f,b,f)
7.21/2.68	
7.21/2.68	append2C_in_3: (b,f,b)
7.21/2.68	
7.21/2.68	Transforming TRIPLES into the following Term Rewriting System:
7.21/2.68	
7.21/2.68	Pi DP problem:
7.21/2.68	The TRS P consists of the following rules:
7.21/2.68	
7.21/2.68	   SUBLISTD_IN_GG(X1, X2) -> U4_GG(X1, X2, append2A_in_gag(X1, X3, X2))
7.21/2.68	   SUBLISTD_IN_GG(X1, X2) -> APPEND2A_IN_GAG(X1, X3, X2)
7.21/2.68	   APPEND2A_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> U1_GAG(X1, X2, X3, X4, append2A_in_gag(X2, X3, X4))
7.21/2.68	   APPEND2A_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2A_IN_GAG(X2, X3, X4)
7.21/2.68	   SUBLISTD_IN_GG(X1, X2) -> U5_GG(X1, X2, append1B_in_aga(X3, X1, X4))
7.21/2.68	   SUBLISTD_IN_GG(X1, X2) -> APPEND1B_IN_AGA(X3, X1, X4)
7.21/2.68	   APPEND1B_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> U2_AGA(X1, X2, X3, X4, append1B_in_aga(X2, X3, X4))
7.21/2.68	   APPEND1B_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPEND1B_IN_AGA(X2, X3, X4)
7.21/2.68	   SUBLISTD_IN_GG(X1, .(X2, X3)) -> U6_GG(X1, X2, X3, append1cB_in_aga(X4, X1, X5))
7.21/2.68	   U6_GG(X1, X2, X3, append1cB_out_aga(X4, X1, X5)) -> U7_GG(X1, X2, X3, append2C_in_gag(X5, X6, X3))
7.21/2.68	   U6_GG(X1, X2, X3, append1cB_out_aga(X4, X1, X5)) -> APPEND2C_IN_GAG(X5, X6, X3)
7.21/2.68	   APPEND2C_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> U3_GAG(X1, X2, X3, X4, append2C_in_gag(X2, X3, X4))
7.21/2.68	   APPEND2C_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2C_IN_GAG(X2, X3, X4)
7.21/2.68	
7.21/2.68	The TRS R consists of the following rules:
7.21/2.68	
7.21/2.68	   append1cB_in_aga([], X1, X1) -> append1cB_out_aga([], X1, X1)
7.21/2.68	   append1cB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, append1cB_in_aga(X2, X3, X4))
7.21/2.68	   U10_aga(X1, X2, X3, X4, append1cB_out_aga(X2, X3, X4)) -> append1cB_out_aga(.(X1, X2), X3, .(X1, X4))
7.21/2.68	
7.21/2.68	The argument filtering Pi contains the following mapping:
7.21/2.68	append2A_in_gag(x1, x2, x3)  =  append2A_in_gag(x1, x3)
7.21/2.68	
7.21/2.68	.(x1, x2)  =  .(x2)
7.21/2.68	
7.21/2.68	append1B_in_aga(x1, x2, x3)  =  append1B_in_aga(x2)
7.21/2.68	
7.21/2.68	append1cB_in_aga(x1, x2, x3)  =  append1cB_in_aga(x2)
7.21/2.68	
7.21/2.68	append1cB_out_aga(x1, x2, x3)  =  append1cB_out_aga(x1, x2, x3)
7.21/2.68	
7.21/2.68	U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
7.21/2.68	
7.21/2.68	append2C_in_gag(x1, x2, x3)  =  append2C_in_gag(x1, x3)
7.21/2.68	
7.21/2.68	SUBLISTD_IN_GG(x1, x2)  =  SUBLISTD_IN_GG(x1, x2)
7.21/2.68	
7.21/2.68	U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)
7.21/2.68	
7.21/2.68	APPEND2A_IN_GAG(x1, x2, x3)  =  APPEND2A_IN_GAG(x1, x3)
7.21/2.68	
7.21/2.68	U1_GAG(x1, x2, x3, x4, x5)  =  U1_GAG(x2, x4, x5)
7.21/2.68	
7.21/2.68	U5_GG(x1, x2, x3)  =  U5_GG(x1, x2, x3)
7.21/2.68	
7.21/2.68	APPEND1B_IN_AGA(x1, x2, x3)  =  APPEND1B_IN_AGA(x2)
7.21/2.68	
7.21/2.68	U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
7.21/2.68	
7.21/2.68	U6_GG(x1, x2, x3, x4)  =  U6_GG(x1, x3, x4)
7.21/2.68	
7.21/2.68	U7_GG(x1, x2, x3, x4)  =  U7_GG(x1, x3, x4)
7.21/2.68	
7.21/2.68	APPEND2C_IN_GAG(x1, x2, x3)  =  APPEND2C_IN_GAG(x1, x3)
7.21/2.68	
7.21/2.68	U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x2, x4, x5)
7.21/2.68	
7.21/2.68	
7.21/2.68	We have to consider all (P,R,Pi)-chains
7.21/2.68	
7.21/2.68	
7.21/2.68	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
7.21/2.68	
7.21/2.68	
7.21/2.68	
7.21/2.68	----------------------------------------
7.21/2.68	
7.21/2.68	(72)
7.21/2.68	Obligation:
7.21/2.68	Pi DP problem:
7.21/2.68	The TRS P consists of the following rules:
7.21/2.68	
7.21/2.68	   SUBLISTD_IN_GG(X1, X2) -> U4_GG(X1, X2, append2A_in_gag(X1, X3, X2))
7.21/2.68	   SUBLISTD_IN_GG(X1, X2) -> APPEND2A_IN_GAG(X1, X3, X2)
7.21/2.68	   APPEND2A_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> U1_GAG(X1, X2, X3, X4, append2A_in_gag(X2, X3, X4))
7.21/2.68	   APPEND2A_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2A_IN_GAG(X2, X3, X4)
7.21/2.68	   SUBLISTD_IN_GG(X1, X2) -> U5_GG(X1, X2, append1B_in_aga(X3, X1, X4))
7.21/2.68	   SUBLISTD_IN_GG(X1, X2) -> APPEND1B_IN_AGA(X3, X1, X4)
7.21/2.68	   APPEND1B_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> U2_AGA(X1, X2, X3, X4, append1B_in_aga(X2, X3, X4))
7.21/2.68	   APPEND1B_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPEND1B_IN_AGA(X2, X3, X4)
7.21/2.68	   SUBLISTD_IN_GG(X1, .(X2, X3)) -> U6_GG(X1, X2, X3, append1cB_in_aga(X4, X1, X5))
7.21/2.68	   U6_GG(X1, X2, X3, append1cB_out_aga(X4, X1, X5)) -> U7_GG(X1, X2, X3, append2C_in_gag(X5, X6, X3))
7.21/2.68	   U6_GG(X1, X2, X3, append1cB_out_aga(X4, X1, X5)) -> APPEND2C_IN_GAG(X5, X6, X3)
7.21/2.68	   APPEND2C_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> U3_GAG(X1, X2, X3, X4, append2C_in_gag(X2, X3, X4))
7.21/2.68	   APPEND2C_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2C_IN_GAG(X2, X3, X4)
7.21/2.68	
7.21/2.68	The TRS R consists of the following rules:
7.21/2.68	
7.21/2.68	   append1cB_in_aga([], X1, X1) -> append1cB_out_aga([], X1, X1)
7.21/2.68	   append1cB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, append1cB_in_aga(X2, X3, X4))
7.21/2.68	   U10_aga(X1, X2, X3, X4, append1cB_out_aga(X2, X3, X4)) -> append1cB_out_aga(.(X1, X2), X3, .(X1, X4))
7.21/2.68	
7.21/2.68	The argument filtering Pi contains the following mapping:
7.21/2.68	append2A_in_gag(x1, x2, x3)  =  append2A_in_gag(x1, x3)
7.21/2.68	
7.21/2.68	.(x1, x2)  =  .(x2)
7.21/2.68	
7.21/2.68	append1B_in_aga(x1, x2, x3)  =  append1B_in_aga(x2)
7.21/2.68	
7.21/2.68	append1cB_in_aga(x1, x2, x3)  =  append1cB_in_aga(x2)
7.21/2.68	
7.21/2.68	append1cB_out_aga(x1, x2, x3)  =  append1cB_out_aga(x1, x2, x3)
7.21/2.68	
7.21/2.68	U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
7.21/2.68	
7.21/2.68	append2C_in_gag(x1, x2, x3)  =  append2C_in_gag(x1, x3)
7.21/2.68	
7.21/2.68	SUBLISTD_IN_GG(x1, x2)  =  SUBLISTD_IN_GG(x1, x2)
7.21/2.68	
7.21/2.68	U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)
7.21/2.68	
7.21/2.68	APPEND2A_IN_GAG(x1, x2, x3)  =  APPEND2A_IN_GAG(x1, x3)
7.21/2.68	
7.21/2.68	U1_GAG(x1, x2, x3, x4, x5)  =  U1_GAG(x2, x4, x5)
7.21/2.68	
7.21/2.68	U5_GG(x1, x2, x3)  =  U5_GG(x1, x2, x3)
7.21/2.68	
7.21/2.68	APPEND1B_IN_AGA(x1, x2, x3)  =  APPEND1B_IN_AGA(x2)
7.21/2.68	
7.21/2.68	U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
7.21/2.68	
7.21/2.68	U6_GG(x1, x2, x3, x4)  =  U6_GG(x1, x3, x4)
7.21/2.68	
7.21/2.68	U7_GG(x1, x2, x3, x4)  =  U7_GG(x1, x3, x4)
7.21/2.68	
7.21/2.68	APPEND2C_IN_GAG(x1, x2, x3)  =  APPEND2C_IN_GAG(x1, x3)
7.21/2.68	
7.21/2.68	U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x2, x4, x5)
7.21/2.68	
7.21/2.68	
7.21/2.68	We have to consider all (P,R,Pi)-chains
7.21/2.68	----------------------------------------
7.21/2.68	
7.21/2.68	(73) DependencyGraphProof (EQUIVALENT)
7.21/2.68	The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 10 less nodes.
7.21/2.68	----------------------------------------
7.21/2.68	
7.21/2.68	(74)
7.21/2.68	Complex Obligation (AND)
7.21/2.68	
7.21/2.68	----------------------------------------
7.21/2.68	
7.21/2.68	(75)
7.21/2.68	Obligation:
7.21/2.68	Pi DP problem:
7.21/2.68	The TRS P consists of the following rules:
7.21/2.68	
7.21/2.68	   APPEND2C_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2C_IN_GAG(X2, X3, X4)
7.21/2.68	
7.21/2.68	The TRS R consists of the following rules:
7.21/2.68	
7.21/2.68	   append1cB_in_aga([], X1, X1) -> append1cB_out_aga([], X1, X1)
7.21/2.68	   append1cB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, append1cB_in_aga(X2, X3, X4))
7.21/2.68	   U10_aga(X1, X2, X3, X4, append1cB_out_aga(X2, X3, X4)) -> append1cB_out_aga(.(X1, X2), X3, .(X1, X4))
7.21/2.68	
7.21/2.68	The argument filtering Pi contains the following mapping:
7.21/2.68	.(x1, x2)  =  .(x2)
7.21/2.68	
7.21/2.68	append1cB_in_aga(x1, x2, x3)  =  append1cB_in_aga(x2)
7.21/2.68	
7.21/2.68	append1cB_out_aga(x1, x2, x3)  =  append1cB_out_aga(x1, x2, x3)
7.21/2.68	
7.21/2.68	U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
7.21/2.68	
7.21/2.68	APPEND2C_IN_GAG(x1, x2, x3)  =  APPEND2C_IN_GAG(x1, x3)
7.21/2.68	
7.21/2.68	
7.21/2.68	We have to consider all (P,R,Pi)-chains
7.21/2.68	----------------------------------------
7.21/2.68	
7.21/2.68	(76) UsableRulesProof (EQUIVALENT)
7.21/2.68	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
7.21/2.68	----------------------------------------
7.21/2.68	
7.21/2.68	(77)
7.21/2.68	Obligation:
7.21/2.68	Pi DP problem:
7.21/2.68	The TRS P consists of the following rules:
7.21/2.68	
7.21/2.68	   APPEND2C_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2C_IN_GAG(X2, X3, X4)
7.21/2.68	
7.21/2.68	R is empty.
7.21/2.68	The argument filtering Pi contains the following mapping:
7.21/2.68	.(x1, x2)  =  .(x2)
7.21/2.68	
7.21/2.68	APPEND2C_IN_GAG(x1, x2, x3)  =  APPEND2C_IN_GAG(x1, x3)
7.21/2.68	
7.21/2.68	
7.21/2.68	We have to consider all (P,R,Pi)-chains
7.21/2.68	----------------------------------------
7.21/2.68	
7.21/2.68	(78) PiDPToQDPProof (SOUND)
7.21/2.68	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
7.21/2.68	----------------------------------------
7.21/2.68	
7.21/2.68	(79)
7.21/2.68	Obligation:
7.21/2.68	Q DP problem:
7.21/2.68	The TRS P consists of the following rules:
7.21/2.68	
7.21/2.68	   APPEND2C_IN_GAG(.(X2), .(X4)) -> APPEND2C_IN_GAG(X2, X4)
7.21/2.68	
7.21/2.68	R is empty.
7.21/2.68	Q is empty.
7.21/2.68	We have to consider all (P,Q,R)-chains.
7.21/2.68	----------------------------------------
7.21/2.68	
7.21/2.68	(80) QDPSizeChangeProof (EQUIVALENT)
7.21/2.68	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
7.21/2.68	
7.21/2.68	From the DPs we obtained the following set of size-change graphs:
7.21/2.68	*APPEND2C_IN_GAG(.(X2), .(X4)) -> APPEND2C_IN_GAG(X2, X4)
7.21/2.68	The graph contains the following edges 1 > 1, 2 > 2
7.21/2.68	
7.21/2.68	
7.21/2.68	----------------------------------------
7.21/2.68	
7.21/2.68	(81)
7.21/2.68	YES
7.21/2.68	
7.21/2.68	----------------------------------------
7.21/2.68	
7.21/2.68	(82)
7.21/2.68	Obligation:
7.21/2.68	Pi DP problem:
7.21/2.68	The TRS P consists of the following rules:
7.21/2.68	
7.21/2.68	   APPEND1B_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPEND1B_IN_AGA(X2, X3, X4)
7.21/2.68	
7.21/2.68	The TRS R consists of the following rules:
7.21/2.68	
7.21/2.68	   append1cB_in_aga([], X1, X1) -> append1cB_out_aga([], X1, X1)
7.21/2.68	   append1cB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, append1cB_in_aga(X2, X3, X4))
7.21/2.68	   U10_aga(X1, X2, X3, X4, append1cB_out_aga(X2, X3, X4)) -> append1cB_out_aga(.(X1, X2), X3, .(X1, X4))
7.21/2.68	
7.21/2.68	The argument filtering Pi contains the following mapping:
7.21/2.68	.(x1, x2)  =  .(x2)
7.21/2.68	
7.21/2.68	append1cB_in_aga(x1, x2, x3)  =  append1cB_in_aga(x2)
7.21/2.68	
7.21/2.68	append1cB_out_aga(x1, x2, x3)  =  append1cB_out_aga(x1, x2, x3)
7.21/2.68	
7.21/2.68	U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
7.21/2.68	
7.21/2.68	APPEND1B_IN_AGA(x1, x2, x3)  =  APPEND1B_IN_AGA(x2)
7.21/2.68	
7.21/2.68	
7.21/2.68	We have to consider all (P,R,Pi)-chains
7.21/2.68	----------------------------------------
7.21/2.68	
7.21/2.68	(83) UsableRulesProof (EQUIVALENT)
7.21/2.68	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
7.21/2.68	----------------------------------------
7.21/2.68	
7.21/2.68	(84)
7.21/2.68	Obligation:
7.21/2.68	Pi DP problem:
7.21/2.68	The TRS P consists of the following rules:
7.21/2.68	
7.21/2.68	   APPEND1B_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPEND1B_IN_AGA(X2, X3, X4)
7.21/2.68	
7.21/2.68	R is empty.
7.21/2.68	The argument filtering Pi contains the following mapping:
7.21/2.68	.(x1, x2)  =  .(x2)
7.21/2.68	
7.21/2.68	APPEND1B_IN_AGA(x1, x2, x3)  =  APPEND1B_IN_AGA(x2)
7.21/2.68	
7.21/2.68	
7.21/2.68	We have to consider all (P,R,Pi)-chains
7.21/2.68	----------------------------------------
7.21/2.68	
7.21/2.68	(85) PiDPToQDPProof (SOUND)
7.21/2.68	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
7.21/2.68	----------------------------------------
7.21/2.68	
7.21/2.68	(86)
7.21/2.68	Obligation:
7.21/2.68	Q DP problem:
7.21/2.68	The TRS P consists of the following rules:
7.21/2.68	
7.21/2.68	   APPEND1B_IN_AGA(X3) -> APPEND1B_IN_AGA(X3)
7.21/2.68	
7.21/2.68	R is empty.
7.21/2.68	Q is empty.
7.21/2.68	We have to consider all (P,Q,R)-chains.
7.21/2.68	----------------------------------------
7.21/2.68	
7.21/2.68	(87)
7.21/2.68	Obligation:
7.21/2.68	Pi DP problem:
7.21/2.68	The TRS P consists of the following rules:
7.21/2.68	
7.21/2.68	   APPEND2A_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2A_IN_GAG(X2, X3, X4)
7.21/2.68	
7.21/2.68	The TRS R consists of the following rules:
7.21/2.68	
7.21/2.68	   append1cB_in_aga([], X1, X1) -> append1cB_out_aga([], X1, X1)
7.21/2.68	   append1cB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, append1cB_in_aga(X2, X3, X4))
7.21/2.68	   U10_aga(X1, X2, X3, X4, append1cB_out_aga(X2, X3, X4)) -> append1cB_out_aga(.(X1, X2), X3, .(X1, X4))
7.21/2.68	
7.21/2.68	The argument filtering Pi contains the following mapping:
7.21/2.68	.(x1, x2)  =  .(x2)
7.21/2.68	
7.21/2.68	append1cB_in_aga(x1, x2, x3)  =  append1cB_in_aga(x2)
7.21/2.68	
7.21/2.68	append1cB_out_aga(x1, x2, x3)  =  append1cB_out_aga(x1, x2, x3)
7.21/2.68	
7.21/2.68	U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
7.21/2.68	
7.21/2.68	APPEND2A_IN_GAG(x1, x2, x3)  =  APPEND2A_IN_GAG(x1, x3)
7.21/2.68	
7.21/2.68	
7.21/2.68	We have to consider all (P,R,Pi)-chains
7.21/2.68	----------------------------------------
7.21/2.68	
7.21/2.68	(88) UsableRulesProof (EQUIVALENT)
7.21/2.68	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
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7.21/2.68	
7.21/2.68	(89)
7.21/2.68	Obligation:
7.21/2.68	Pi DP problem:
7.21/2.68	The TRS P consists of the following rules:
7.21/2.68	
7.21/2.68	   APPEND2A_IN_GAG(.(X1, X2), X3, .(X1, X4)) -> APPEND2A_IN_GAG(X2, X3, X4)
7.21/2.68	
7.21/2.68	R is empty.
7.21/2.68	The argument filtering Pi contains the following mapping:
7.21/2.68	.(x1, x2)  =  .(x2)
7.21/2.68	
7.21/2.68	APPEND2A_IN_GAG(x1, x2, x3)  =  APPEND2A_IN_GAG(x1, x3)
7.21/2.68	
7.21/2.68	
7.21/2.68	We have to consider all (P,R,Pi)-chains
7.21/2.72	EOF