6.30/2.47	MAYBE
6.50/2.49	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
6.50/2.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.50/2.49	
6.50/2.49	
6.50/2.49	Left Termination of the query pattern
6.50/2.49	
6.50/2.49	sublist(g,a)
6.50/2.49	
6.50/2.49	w.r.t. the given Prolog program could not be shown:
6.50/2.49	
6.50/2.49	(0) Prolog
6.50/2.49	(1) PrologToPiTRSProof [SOUND, 0 ms]
6.50/2.49	(2) PiTRS
6.50/2.49	    (3) DependencyPairsProof [EQUIVALENT, 0 ms]
6.50/2.49	    (4) PiDP
6.50/2.49	    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
6.50/2.49	    (6) AND
6.50/2.49	        (7) PiDP
6.50/2.49	            (8) UsableRulesProof [EQUIVALENT, 0 ms]
6.50/2.49	            (9) PiDP
6.50/2.49	            (10) PiDPToQDPProof [SOUND, 6 ms]
6.50/2.49	            (11) QDP
6.50/2.49	            (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.50/2.49	            (13) YES
6.50/2.49	        (14) PiDP
6.50/2.49	            (15) UsableRulesProof [EQUIVALENT, 0 ms]
6.50/2.49	            (16) PiDP
6.50/2.49	            (17) PiDPToQDPProof [SOUND, 0 ms]
6.50/2.49	            (18) QDP
6.50/2.49	(19) PrologToTRSTransformerProof [SOUND, 0 ms]
6.50/2.49	(20) QTRS
6.50/2.49	    (21) DependencyPairsProof [EQUIVALENT, 0 ms]
6.50/2.49	    (22) QDP
6.50/2.49	    (23) DependencyGraphProof [EQUIVALENT, 0 ms]
6.50/2.49	    (24) AND
6.50/2.49	        (25) QDP
6.50/2.49	            (26) MNOCProof [EQUIVALENT, 0 ms]
6.50/2.49	            (27) QDP
6.50/2.49	            (28) UsableRulesProof [EQUIVALENT, 0 ms]
6.50/2.49	            (29) QDP
6.50/2.49	            (30) QReductionProof [EQUIVALENT, 0 ms]
6.50/2.49	            (31) QDP
6.50/2.49	        (32) QDP
6.50/2.49	            (33) MNOCProof [EQUIVALENT, 0 ms]
6.50/2.49	            (34) QDP
6.50/2.49	            (35) UsableRulesProof [EQUIVALENT, 0 ms]
6.50/2.49	            (36) QDP
6.50/2.49	            (37) QReductionProof [EQUIVALENT, 0 ms]
6.50/2.49	            (38) QDP
6.50/2.49	(39) PrologToPiTRSProof [SOUND, 0 ms]
6.50/2.49	(40) PiTRS
6.50/2.49	    (41) DependencyPairsProof [EQUIVALENT, 0 ms]
6.50/2.49	    (42) PiDP
6.50/2.49	    (43) DependencyGraphProof [EQUIVALENT, 0 ms]
6.50/2.49	    (44) AND
6.50/2.49	        (45) PiDP
6.50/2.49	            (46) UsableRulesProof [EQUIVALENT, 0 ms]
6.50/2.49	            (47) PiDP
6.50/2.49	            (48) PiDPToQDPProof [SOUND, 1 ms]
6.50/2.49	            (49) QDP
6.50/2.49	            (50) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.50/2.49	            (51) YES
6.50/2.49	        (52) PiDP
6.50/2.49	            (53) UsableRulesProof [EQUIVALENT, 0 ms]
6.50/2.49	            (54) PiDP
6.50/2.49	            (55) PiDPToQDPProof [SOUND, 0 ms]
6.50/2.49	            (56) QDP
6.50/2.49	(57) PrologToDTProblemTransformerProof [SOUND, 0 ms]
6.50/2.49	(58) TRIPLES
6.50/2.49	    (59) TriplesToPiDPProof [SOUND, 0 ms]
6.50/2.49	    (60) PiDP
6.50/2.49	    (61) DependencyGraphProof [EQUIVALENT, 0 ms]
6.50/2.49	    (62) AND
6.50/2.49	        (63) PiDP
6.50/2.49	            (64) UsableRulesProof [EQUIVALENT, 0 ms]
6.50/2.49	            (65) PiDP
6.50/2.49	            (66) PiDPToQDPProof [SOUND, 0 ms]
6.50/2.49	            (67) QDP
6.50/2.49	            (68) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.50/2.49	            (69) YES
6.50/2.49	        (70) PiDP
6.50/2.49	            (71) UsableRulesProof [EQUIVALENT, 0 ms]
6.50/2.49	            (72) PiDP
6.50/2.49	            (73) PiDPToQDPProof [SOUND, 0 ms]
6.50/2.49	            (74) QDP
6.50/2.49	        (75) PiDP
6.50/2.49	            (76) UsableRulesProof [EQUIVALENT, 0 ms]
6.50/2.49	            (77) PiDP
6.50/2.49	(78) PrologToIRSwTTransformerProof [SOUND, 44 ms]
6.50/2.49	(79) AND
6.50/2.49	    (80) IRSwT
6.50/2.49	        (81) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
6.50/2.49	        (82) TRUE
6.50/2.49	    (83) IRSwT
6.50/2.49	        (84) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
6.50/2.49	        (85) IRSwT
6.50/2.49	        (86) IntTRSCompressionProof [EQUIVALENT, 19 ms]
6.50/2.49	        (87) IRSwT
6.50/2.49	        (88) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
6.50/2.49	        (89) IRSwT
6.50/2.49	        (90) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
6.50/2.49	        (91) IRSwT
6.50/2.49	        (92) FilterProof [EQUIVALENT, 0 ms]
6.50/2.49	        (93) IntTRS
6.50/2.49	        (94) IntTRSPeriodicNontermProof [COMPLETE, 7 ms]
6.50/2.49	        (95) NO
6.50/2.49	
6.50/2.49	
6.50/2.49	----------------------------------------
6.50/2.49	
6.50/2.49	(0)
6.50/2.49	Obligation:
6.50/2.49	Clauses:
6.50/2.49	
6.50/2.49	sublist(X, Y) :- ','(append(U, X, V), append(V, W, Y)).
6.50/2.49	append([], Ys, Ys).
6.50/2.49	append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs).
6.50/2.49	
6.50/2.49	
6.50/2.49	Query: sublist(g,a)
6.50/2.49	----------------------------------------
6.50/2.49	
6.50/2.49	(1) PrologToPiTRSProof (SOUND)
6.50/2.49	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.50/2.49	
6.50/2.49	sublist_in_2: (b,f)
6.50/2.49	
6.50/2.49	append_in_3: (f,b,f) (b,f,f)
6.50/2.49	
6.50/2.49	Transforming Prolog into the following Term Rewriting System:
6.50/2.49	
6.50/2.49	Pi-finite rewrite system:
6.50/2.49	The TRS R consists of the following rules:
6.50/2.49	
6.50/2.49	   sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V))
6.50/2.49	   append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys)
6.50/2.49	   append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs))
6.50/2.49	   U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs))
6.50/2.49	   U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y))
6.50/2.49	   append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys)
6.50/2.49	   append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs))
6.50/2.49	   U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs))
6.50/2.49	   U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y)
6.50/2.49	
6.50/2.49	The argument filtering Pi contains the following mapping:
6.50/2.49	sublist_in_ga(x1, x2)  =  sublist_in_ga(x1)
6.50/2.49	
6.50/2.49	U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
6.50/2.49	
6.50/2.49	append_in_aga(x1, x2, x3)  =  append_in_aga(x2)
6.50/2.49	
6.50/2.49	append_out_aga(x1, x2, x3)  =  append_out_aga(x1, x2, x3)
6.50/2.49	
6.50/2.49	U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x3, x5)
6.50/2.49	
6.50/2.49	.(x1, x2)  =  .(x2)
6.50/2.49	
6.50/2.49	U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
6.50/2.49	
6.50/2.49	append_in_gaa(x1, x2, x3)  =  append_in_gaa(x1)
6.50/2.49	
6.50/2.49	[]  =  []
6.50/2.49	
6.50/2.49	append_out_gaa(x1, x2, x3)  =  append_out_gaa(x1)
6.50/2.49	
6.50/2.49	U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)
6.50/2.49	
6.50/2.49	sublist_out_ga(x1, x2)  =  sublist_out_ga(x1)
6.50/2.49	
6.50/2.49	
6.50/2.49	
6.50/2.49	
6.50/2.49	
6.50/2.49	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.50/2.49	
6.50/2.49	
6.50/2.49	
6.50/2.49	----------------------------------------
6.50/2.49	
6.50/2.49	(2)
6.50/2.49	Obligation:
6.50/2.49	Pi-finite rewrite system:
6.50/2.49	The TRS R consists of the following rules:
6.50/2.49	
6.50/2.49	   sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V))
6.50/2.49	   append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys)
6.50/2.49	   append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs))
6.50/2.49	   U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs))
6.50/2.49	   U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y))
6.50/2.49	   append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys)
6.50/2.49	   append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs))
6.50/2.49	   U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs))
6.50/2.49	   U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y)
6.50/2.49	
6.50/2.49	The argument filtering Pi contains the following mapping:
6.50/2.49	sublist_in_ga(x1, x2)  =  sublist_in_ga(x1)
6.50/2.49	
6.50/2.49	U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
6.50/2.49	
6.50/2.49	append_in_aga(x1, x2, x3)  =  append_in_aga(x2)
6.50/2.49	
6.50/2.49	append_out_aga(x1, x2, x3)  =  append_out_aga(x1, x2, x3)
6.50/2.49	
6.50/2.49	U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x3, x5)
6.50/2.49	
6.50/2.49	.(x1, x2)  =  .(x2)
6.50/2.49	
6.50/2.49	U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
6.50/2.49	
6.50/2.49	append_in_gaa(x1, x2, x3)  =  append_in_gaa(x1)
6.50/2.49	
6.50/2.49	[]  =  []
6.50/2.49	
6.50/2.49	append_out_gaa(x1, x2, x3)  =  append_out_gaa(x1)
6.50/2.49	
6.50/2.49	U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)
6.50/2.49	
6.50/2.49	sublist_out_ga(x1, x2)  =  sublist_out_ga(x1)
6.50/2.49	
6.50/2.49	
6.50/2.49	
6.50/2.49	----------------------------------------
6.50/2.49	
6.50/2.49	(3) DependencyPairsProof (EQUIVALENT)
6.50/2.49	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.50/2.49	Pi DP problem:
6.50/2.49	The TRS P consists of the following rules:
6.50/2.49	
6.50/2.49	   SUBLIST_IN_GA(X, Y) -> U1_GA(X, Y, append_in_aga(U, X, V))
6.50/2.49	   SUBLIST_IN_GA(X, Y) -> APPEND_IN_AGA(U, X, V)
6.50/2.49	   APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U3_AGA(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs))
6.50/2.49	   APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AGA(Xs, Ys, Zs)
6.50/2.49	   U1_GA(X, Y, append_out_aga(U, X, V)) -> U2_GA(X, Y, append_in_gaa(V, W, Y))
6.50/2.49	   U1_GA(X, Y, append_out_aga(U, X, V)) -> APPEND_IN_GAA(V, W, Y)
6.50/2.49	   APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> U3_GAA(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs))
6.50/2.49	   APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GAA(Xs, Ys, Zs)
6.50/2.49	
6.50/2.49	The TRS R consists of the following rules:
6.50/2.49	
6.50/2.49	   sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V))
6.50/2.49	   append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys)
6.50/2.49	   append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs))
6.50/2.49	   U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs))
6.50/2.49	   U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y))
6.50/2.49	   append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys)
6.50/2.49	   append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs))
6.50/2.49	   U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs))
6.50/2.49	   U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y)
6.50/2.52	
6.50/2.52	The argument filtering Pi contains the following mapping:
6.50/2.52	sublist_in_ga(x1, x2)  =  sublist_in_ga(x1)
6.50/2.52	
6.50/2.52	U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
6.50/2.52	
6.50/2.52	append_in_aga(x1, x2, x3)  =  append_in_aga(x2)
6.50/2.52	
6.50/2.52	append_out_aga(x1, x2, x3)  =  append_out_aga(x1, x2, x3)
6.50/2.52	
6.50/2.52	U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x3, x5)
6.50/2.52	
6.50/2.52	.(x1, x2)  =  .(x2)
6.50/2.52	
6.50/2.52	U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
6.50/2.52	
6.50/2.52	append_in_gaa(x1, x2, x3)  =  append_in_gaa(x1)
6.50/2.52	
6.50/2.52	[]  =  []
6.50/2.52	
6.50/2.52	append_out_gaa(x1, x2, x3)  =  append_out_gaa(x1)
6.50/2.52	
6.50/2.52	U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)
6.50/2.52	
6.50/2.52	sublist_out_ga(x1, x2)  =  sublist_out_ga(x1)
6.50/2.52	
6.50/2.52	SUBLIST_IN_GA(x1, x2)  =  SUBLIST_IN_GA(x1)
6.50/2.52	
6.50/2.52	U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
6.50/2.52	
6.50/2.52	APPEND_IN_AGA(x1, x2, x3)  =  APPEND_IN_AGA(x2)
6.50/2.52	
6.50/2.52	U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x3, x5)
6.50/2.52	
6.50/2.52	U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
6.50/2.52	
6.50/2.52	APPEND_IN_GAA(x1, x2, x3)  =  APPEND_IN_GAA(x1)
6.50/2.52	
6.50/2.52	U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x2, x5)
6.50/2.52	
6.50/2.52	
6.50/2.52	We have to consider all (P,R,Pi)-chains
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(4)
6.50/2.52	Obligation:
6.50/2.52	Pi DP problem:
6.50/2.52	The TRS P consists of the following rules:
6.50/2.52	
6.50/2.52	   SUBLIST_IN_GA(X, Y) -> U1_GA(X, Y, append_in_aga(U, X, V))
6.50/2.52	   SUBLIST_IN_GA(X, Y) -> APPEND_IN_AGA(U, X, V)
6.50/2.52	   APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U3_AGA(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs))
6.50/2.52	   APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AGA(Xs, Ys, Zs)
6.50/2.52	   U1_GA(X, Y, append_out_aga(U, X, V)) -> U2_GA(X, Y, append_in_gaa(V, W, Y))
6.50/2.52	   U1_GA(X, Y, append_out_aga(U, X, V)) -> APPEND_IN_GAA(V, W, Y)
6.50/2.52	   APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> U3_GAA(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs))
6.50/2.52	   APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GAA(Xs, Ys, Zs)
6.50/2.52	
6.50/2.52	The TRS R consists of the following rules:
6.50/2.52	
6.50/2.52	   sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V))
6.50/2.52	   append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys)
6.50/2.52	   append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs))
6.50/2.52	   U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs))
6.50/2.52	   U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y))
6.50/2.52	   append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys)
6.50/2.52	   append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs))
6.50/2.52	   U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs))
6.50/2.52	   U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y)
6.50/2.52	
6.50/2.52	The argument filtering Pi contains the following mapping:
6.50/2.52	sublist_in_ga(x1, x2)  =  sublist_in_ga(x1)
6.50/2.52	
6.50/2.52	U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
6.50/2.52	
6.50/2.52	append_in_aga(x1, x2, x3)  =  append_in_aga(x2)
6.50/2.52	
6.50/2.52	append_out_aga(x1, x2, x3)  =  append_out_aga(x1, x2, x3)
6.50/2.52	
6.50/2.52	U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x3, x5)
6.50/2.52	
6.50/2.52	.(x1, x2)  =  .(x2)
6.50/2.52	
6.50/2.52	U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
6.50/2.52	
6.50/2.52	append_in_gaa(x1, x2, x3)  =  append_in_gaa(x1)
6.50/2.52	
6.50/2.52	[]  =  []
6.50/2.52	
6.50/2.52	append_out_gaa(x1, x2, x3)  =  append_out_gaa(x1)
6.50/2.52	
6.50/2.52	U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)
6.50/2.52	
6.50/2.52	sublist_out_ga(x1, x2)  =  sublist_out_ga(x1)
6.50/2.52	
6.50/2.52	SUBLIST_IN_GA(x1, x2)  =  SUBLIST_IN_GA(x1)
6.50/2.52	
6.50/2.52	U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
6.50/2.52	
6.50/2.52	APPEND_IN_AGA(x1, x2, x3)  =  APPEND_IN_AGA(x2)
6.50/2.52	
6.50/2.52	U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x3, x5)
6.50/2.52	
6.50/2.52	U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
6.50/2.52	
6.50/2.52	APPEND_IN_GAA(x1, x2, x3)  =  APPEND_IN_GAA(x1)
6.50/2.52	
6.50/2.52	U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x2, x5)
6.50/2.52	
6.50/2.52	
6.50/2.52	We have to consider all (P,R,Pi)-chains
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(5) DependencyGraphProof (EQUIVALENT)
6.50/2.52	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(6)
6.50/2.52	Complex Obligation (AND)
6.50/2.52	
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(7)
6.50/2.52	Obligation:
6.50/2.52	Pi DP problem:
6.50/2.52	The TRS P consists of the following rules:
6.50/2.52	
6.50/2.52	   APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GAA(Xs, Ys, Zs)
6.50/2.52	
6.50/2.52	The TRS R consists of the following rules:
6.50/2.52	
6.50/2.52	   sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V))
6.50/2.52	   append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys)
6.50/2.52	   append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs))
6.50/2.52	   U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs))
6.50/2.52	   U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y))
6.50/2.52	   append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys)
6.50/2.52	   append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs))
6.50/2.52	   U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs))
6.50/2.52	   U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y)
6.50/2.52	
6.50/2.52	The argument filtering Pi contains the following mapping:
6.50/2.52	sublist_in_ga(x1, x2)  =  sublist_in_ga(x1)
6.50/2.52	
6.50/2.52	U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
6.50/2.52	
6.50/2.52	append_in_aga(x1, x2, x3)  =  append_in_aga(x2)
6.50/2.52	
6.50/2.52	append_out_aga(x1, x2, x3)  =  append_out_aga(x1, x2, x3)
6.50/2.52	
6.50/2.52	U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x3, x5)
6.50/2.52	
6.50/2.52	.(x1, x2)  =  .(x2)
6.50/2.52	
6.50/2.52	U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
6.50/2.52	
6.50/2.52	append_in_gaa(x1, x2, x3)  =  append_in_gaa(x1)
6.50/2.52	
6.50/2.52	[]  =  []
6.50/2.52	
6.50/2.52	append_out_gaa(x1, x2, x3)  =  append_out_gaa(x1)
6.50/2.52	
6.50/2.52	U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)
6.50/2.52	
6.50/2.52	sublist_out_ga(x1, x2)  =  sublist_out_ga(x1)
6.50/2.52	
6.50/2.52	APPEND_IN_GAA(x1, x2, x3)  =  APPEND_IN_GAA(x1)
6.50/2.52	
6.50/2.52	
6.50/2.52	We have to consider all (P,R,Pi)-chains
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(8) UsableRulesProof (EQUIVALENT)
6.50/2.52	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(9)
6.50/2.52	Obligation:
6.50/2.52	Pi DP problem:
6.50/2.52	The TRS P consists of the following rules:
6.50/2.52	
6.50/2.52	   APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GAA(Xs, Ys, Zs)
6.50/2.52	
6.50/2.52	R is empty.
6.50/2.52	The argument filtering Pi contains the following mapping:
6.50/2.52	.(x1, x2)  =  .(x2)
6.50/2.52	
6.50/2.52	APPEND_IN_GAA(x1, x2, x3)  =  APPEND_IN_GAA(x1)
6.50/2.52	
6.50/2.52	
6.50/2.52	We have to consider all (P,R,Pi)-chains
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(10) PiDPToQDPProof (SOUND)
6.50/2.52	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(11)
6.50/2.52	Obligation:
6.50/2.52	Q DP problem:
6.50/2.52	The TRS P consists of the following rules:
6.50/2.52	
6.50/2.52	   APPEND_IN_GAA(.(Xs)) -> APPEND_IN_GAA(Xs)
6.50/2.52	
6.50/2.52	R is empty.
6.50/2.52	Q is empty.
6.50/2.52	We have to consider all (P,Q,R)-chains.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(12) QDPSizeChangeProof (EQUIVALENT)
6.50/2.52	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. 
6.50/2.52	
6.50/2.52	From the DPs we obtained the following set of size-change graphs:
6.50/2.52	*APPEND_IN_GAA(.(Xs)) -> APPEND_IN_GAA(Xs)
6.50/2.52	The graph contains the following edges 1 > 1
6.50/2.52	
6.50/2.52	
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(13)
6.50/2.52	YES
6.50/2.52	
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(14)
6.50/2.52	Obligation:
6.50/2.52	Pi DP problem:
6.50/2.52	The TRS P consists of the following rules:
6.50/2.52	
6.50/2.52	   APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AGA(Xs, Ys, Zs)
6.50/2.52	
6.50/2.52	The TRS R consists of the following rules:
6.50/2.52	
6.50/2.52	   sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V))
6.50/2.52	   append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys)
6.50/2.52	   append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs))
6.50/2.52	   U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs))
6.50/2.52	   U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y))
6.50/2.52	   append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys)
6.50/2.52	   append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs))
6.50/2.52	   U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs))
6.50/2.52	   U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y)
6.50/2.52	
6.50/2.52	The argument filtering Pi contains the following mapping:
6.50/2.52	sublist_in_ga(x1, x2)  =  sublist_in_ga(x1)
6.50/2.52	
6.50/2.52	U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
6.50/2.52	
6.50/2.52	append_in_aga(x1, x2, x3)  =  append_in_aga(x2)
6.50/2.52	
6.50/2.52	append_out_aga(x1, x2, x3)  =  append_out_aga(x1, x2, x3)
6.50/2.52	
6.50/2.52	U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x3, x5)
6.50/2.52	
6.50/2.52	.(x1, x2)  =  .(x2)
6.50/2.52	
6.50/2.52	U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
6.50/2.52	
6.50/2.52	append_in_gaa(x1, x2, x3)  =  append_in_gaa(x1)
6.50/2.52	
6.50/2.52	[]  =  []
6.50/2.52	
6.50/2.52	append_out_gaa(x1, x2, x3)  =  append_out_gaa(x1)
6.50/2.52	
6.50/2.52	U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x2, x5)
6.50/2.52	
6.50/2.52	sublist_out_ga(x1, x2)  =  sublist_out_ga(x1)
6.50/2.52	
6.50/2.52	APPEND_IN_AGA(x1, x2, x3)  =  APPEND_IN_AGA(x2)
6.50/2.52	
6.50/2.52	
6.50/2.52	We have to consider all (P,R,Pi)-chains
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(15) UsableRulesProof (EQUIVALENT)
6.50/2.52	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(16)
6.50/2.52	Obligation:
6.50/2.52	Pi DP problem:
6.50/2.52	The TRS P consists of the following rules:
6.50/2.52	
6.50/2.52	   APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AGA(Xs, Ys, Zs)
6.50/2.52	
6.50/2.52	R is empty.
6.50/2.52	The argument filtering Pi contains the following mapping:
6.50/2.52	.(x1, x2)  =  .(x2)
6.50/2.52	
6.50/2.52	APPEND_IN_AGA(x1, x2, x3)  =  APPEND_IN_AGA(x2)
6.50/2.52	
6.50/2.52	
6.50/2.52	We have to consider all (P,R,Pi)-chains
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(17) PiDPToQDPProof (SOUND)
6.50/2.52	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(18)
6.50/2.52	Obligation:
6.50/2.52	Q DP problem:
6.50/2.52	The TRS P consists of the following rules:
6.50/2.52	
6.50/2.52	   APPEND_IN_AGA(Ys) -> APPEND_IN_AGA(Ys)
6.50/2.52	
6.50/2.52	R is empty.
6.50/2.52	Q is empty.
6.50/2.52	We have to consider all (P,Q,R)-chains.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(19) PrologToTRSTransformerProof (SOUND)
6.50/2.52	Transformed Prolog program to TRS.
6.50/2.52	
6.50/2.52	{
6.50/2.52	    "root": 4,
6.50/2.52	    "program": {
6.50/2.52	        "directives": [],
6.50/2.52	        "clauses": [
6.50/2.52	            [
6.50/2.52	                "(sublist X Y)",
6.50/2.52	                "(',' (append U X V) (append V W Y))"
6.50/2.52	            ],
6.50/2.52	            [
6.50/2.52	                "(append ([]) Ys Ys)",
6.50/2.52	                null
6.50/2.52	            ],
6.50/2.52	            [
6.50/2.52	                "(append (. X Xs) Ys (. X Zs))",
6.50/2.52	                "(append Xs Ys Zs)"
6.50/2.52	            ]
6.50/2.52	        ]
6.50/2.52	    },
6.50/2.52	    "graph": {
6.50/2.52	        "nodes": {
6.50/2.52	            "170": {
6.50/2.52	                "goal": [],
6.50/2.52	                "kb": {
6.50/2.52	                    "nonunifying": [],
6.50/2.52	                    "intvars": {},
6.50/2.52	                    "arithmetic": {
6.50/2.52	                        "type": "PlainIntegerRelationState",
6.50/2.52	                        "relations": []
6.50/2.52	                    },
6.50/2.52	                    "ground": [],
6.50/2.52	                    "free": [],
6.50/2.52	                    "exprvars": []
6.50/2.52	                }
6.50/2.52	            },
6.50/2.52	            "171": {
6.50/2.52	                "goal": [],
6.50/2.52	                "kb": {
6.50/2.52	                    "nonunifying": [],
6.50/2.52	                    "intvars": {},
6.50/2.52	                    "arithmetic": {
6.50/2.52	                        "type": "PlainIntegerRelationState",
6.50/2.52	                        "relations": []
6.50/2.52	                    },
6.50/2.52	                    "ground": [],
6.50/2.52	                    "free": [],
6.50/2.52	                    "exprvars": []
6.50/2.52	                }
6.50/2.52	            },
6.50/2.52	            "type": "Nodes",
6.50/2.52	            "161": {
6.50/2.52	                "goal": [
6.50/2.52	                    {
6.50/2.52	                        "clause": 1,
6.50/2.52	                        "scope": 3,
6.50/2.52	                        "term": "(append T16 X15 T12)"
6.50/2.52	                    },
6.50/2.52	                    {
6.50/2.52	                        "clause": 2,
6.50/2.52	                        "scope": 3,
6.50/2.52	                        "term": "(append T16 X15 T12)"
6.50/2.52	                    }
6.50/2.52	                ],
6.50/2.52	                "kb": {
6.50/2.52	                    "nonunifying": [],
6.50/2.52	                    "intvars": {},
6.50/2.52	                    "arithmetic": {
6.50/2.52	                        "type": "PlainIntegerRelationState",
6.50/2.52	                        "relations": []
6.50/2.52	                    },
6.50/2.52	                    "ground": [],
6.50/2.52	                    "free": ["X15"],
6.50/2.52	                    "exprvars": []
6.50/2.52	                }
6.50/2.52	            },
6.50/2.52	            "151": {
6.50/2.52	                "goal": [{
6.50/2.52	                    "clause": -1,
6.50/2.52	                    "scope": -1,
6.50/2.52	                    "term": "(append X13 T10 X14)"
6.50/2.52	                }],
6.50/2.52	                "kb": {
6.50/2.52	                    "nonunifying": [],
6.50/2.52	                    "intvars": {},
6.50/2.52	                    "arithmetic": {
6.50/2.52	                        "type": "PlainIntegerRelationState",
6.50/2.52	                        "relations": []
6.50/2.52	                    },
6.50/2.52	                    "ground": ["T10"],
6.50/2.52	                    "free": [
6.50/2.52	                        "X13",
6.50/2.52	                        "X14"
6.50/2.52	                    ],
6.50/2.52	                    "exprvars": []
6.50/2.52	                }
6.50/2.52	            },
6.50/2.52	            "152": {
6.50/2.52	                "goal": [{
6.50/2.52	                    "clause": -1,
6.50/2.52	                    "scope": -1,
6.50/2.52	                    "term": "(append T16 X15 T12)"
6.50/2.52	                }],
6.50/2.52	                "kb": {
6.50/2.52	                    "nonunifying": [],
6.50/2.52	                    "intvars": {},
6.50/2.52	                    "arithmetic": {
6.50/2.52	                        "type": "PlainIntegerRelationState",
6.50/2.52	                        "relations": []
6.50/2.52	                    },
6.50/2.52	                    "ground": [],
6.50/2.52	                    "free": ["X15"],
6.50/2.52	                    "exprvars": []
6.50/2.52	                }
6.50/2.52	            },
6.50/2.52	            "163": {
6.50/2.52	                "goal": [{
6.50/2.52	                    "clause": 1,
6.50/2.52	                    "scope": 3,
6.50/2.52	                    "term": "(append T16 X15 T12)"
6.50/2.52	                }],
6.50/2.52	                "kb": {
6.50/2.52	                    "nonunifying": [],
6.50/2.52	                    "intvars": {},
6.50/2.52	                    "arithmetic": {
6.50/2.52	                        "type": "PlainIntegerRelationState",
6.50/2.52	                        "relations": []
6.50/2.52	                    },
6.50/2.52	                    "ground": [],
6.50/2.52	                    "free": ["X15"],
6.50/2.52	                    "exprvars": []
6.50/2.52	                }
6.50/2.52	            },
6.50/2.52	            "185": {
6.50/2.52	                "goal": [{
6.50/2.52	                    "clause": -1,
6.50/2.52	                    "scope": -1,
6.50/2.52	                    "term": "(append T44 X76 T45)"
6.50/2.52	                }],
6.50/2.52	                "kb": {
6.50/2.52	                    "nonunifying": [],
6.50/2.52	                    "intvars": {},
6.50/2.52	                    "arithmetic": {
6.50/2.52	                        "type": "PlainIntegerRelationState",
6.50/2.52	                        "relations": []
6.50/2.52	                    },
6.50/2.52	                    "ground": [],
6.50/2.52	                    "free": ["X76"],
6.50/2.52	                    "exprvars": []
6.50/2.52	                }
6.50/2.52	            },
6.50/2.52	            "153": {
6.50/2.52	                "goal": [
6.50/2.52	                    {
6.50/2.52	                        "clause": 1,
6.50/2.52	                        "scope": 2,
6.50/2.52	                        "term": "(append X13 T10 X14)"
6.50/2.52	                    },
6.50/2.52	                    {
6.50/2.52	                        "clause": 2,
6.50/2.52	                        "scope": 2,
6.50/2.52	                        "term": "(append X13 T10 X14)"
6.50/2.52	                    }
6.50/2.52	                ],
6.50/2.52	                "kb": {
6.50/2.52	                    "nonunifying": [],
6.50/2.52	                    "intvars": {},
6.50/2.52	                    "arithmetic": {
6.50/2.52	                        "type": "PlainIntegerRelationState",
6.50/2.52	                        "relations": []
6.50/2.52	                    },
6.50/2.52	                    "ground": ["T10"],
6.50/2.52	                    "free": [
6.50/2.52	                        "X13",
6.50/2.52	                        "X14"
6.50/2.52	                    ],
6.50/2.52	                    "exprvars": []
6.50/2.52	                }
6.50/2.52	            },
6.50/2.52	            "154": {
6.50/2.52	                "goal": [{
6.50/2.52	                    "clause": 1,
6.50/2.52	                    "scope": 2,
6.50/2.52	                    "term": "(append X13 T10 X14)"
6.50/2.52	                }],
6.50/2.52	                "kb": {
6.50/2.52	                    "nonunifying": [],
6.50/2.52	                    "intvars": {},
6.50/2.52	                    "arithmetic": {
6.50/2.52	                        "type": "PlainIntegerRelationState",
6.50/2.52	                        "relations": []
6.50/2.52	                    },
6.50/2.52	                    "ground": ["T10"],
6.50/2.52	                    "free": [
6.50/2.52	                        "X13",
6.50/2.52	                        "X14"
6.50/2.52	                    ],
6.50/2.52	                    "exprvars": []
6.50/2.52	                }
6.50/2.52	            },
6.50/2.52	            "165": {
6.50/2.52	                "goal": [{
6.50/2.52	                    "clause": 2,
6.50/2.52	                    "scope": 3,
6.50/2.52	                    "term": "(append T16 X15 T12)"
6.50/2.52	                }],
6.50/2.52	                "kb": {
6.50/2.52	                    "nonunifying": [],
6.50/2.52	                    "intvars": {},
6.50/2.52	                    "arithmetic": {
6.50/2.52	                        "type": "PlainIntegerRelationState",
6.50/2.52	                        "relations": []
6.50/2.52	                    },
6.50/2.52	                    "ground": [],
6.50/2.52	                    "free": ["X15"],
6.50/2.52	                    "exprvars": []
6.50/2.52	                }
6.50/2.52	            },
6.50/2.52	            "187": {
6.50/2.52	                "goal": [],
6.50/2.52	                "kb": {
6.50/2.52	                    "nonunifying": [],
6.50/2.52	                    "intvars": {},
6.50/2.52	                    "arithmetic": {
6.50/2.52	                        "type": "PlainIntegerRelationState",
6.50/2.52	                        "relations": []
6.50/2.52	                    },
6.50/2.52	                    "ground": [],
6.50/2.52	                    "free": [],
6.50/2.52	                    "exprvars": []
6.50/2.52	                }
6.50/2.52	            },
6.50/2.52	            "155": {
6.50/2.52	                "goal": [{
6.50/2.52	                    "clause": 2,
6.50/2.52	                    "scope": 2,
6.50/2.52	                    "term": "(append X13 T10 X14)"
6.50/2.52	                }],
6.50/2.52	                "kb": {
6.50/2.52	                    "nonunifying": [],
6.50/2.52	                    "intvars": {},
6.50/2.52	                    "arithmetic": {
6.50/2.52	                        "type": "PlainIntegerRelationState",
6.50/2.52	                        "relations": []
6.50/2.52	                    },
6.50/2.52	                    "ground": ["T10"],
6.50/2.52	                    "free": [
6.50/2.52	                        "X13",
6.50/2.52	                        "X14"
6.50/2.52	                    ],
6.50/2.52	                    "exprvars": []
6.50/2.52	                }
6.50/2.52	            },
6.50/2.52	            "156": {
6.50/2.52	                "goal": [{
6.50/2.52	                    "clause": -1,
6.50/2.52	                    "scope": -1,
6.50/2.52	                    "term": "(true)"
6.50/2.52	                }],
6.50/2.52	                "kb": {
6.50/2.52	                    "nonunifying": [],
6.50/2.52	                    "intvars": {},
6.50/2.52	                    "arithmetic": {
6.50/2.52	                        "type": "PlainIntegerRelationState",
6.50/2.52	                        "relations": []
6.50/2.52	                    },
6.50/2.52	                    "ground": [],
6.50/2.52	                    "free": [],
6.50/2.52	                    "exprvars": []
6.50/2.52	                }
6.50/2.52	            },
6.50/2.52	            "157": {
6.50/2.52	                "goal": [],
6.50/2.52	                "kb": {
6.50/2.52	                    "nonunifying": [],
6.50/2.52	                    "intvars": {},
6.50/2.52	                    "arithmetic": {
6.50/2.52	                        "type": "PlainIntegerRelationState",
6.50/2.52	                        "relations": []
6.50/2.52	                    },
6.50/2.52	                    "ground": [],
6.50/2.52	                    "free": [],
6.50/2.52	                    "exprvars": []
6.50/2.52	                }
6.50/2.52	            },
6.50/2.52	            "4": {
6.50/2.52	                "goal": [{
6.50/2.52	                    "clause": -1,
6.50/2.52	                    "scope": -1,
6.50/2.52	                    "term": "(sublist T1 T2)"
6.50/2.52	                }],
6.50/2.52	                "kb": {
6.50/2.52	                    "nonunifying": [],
6.50/2.52	                    "intvars": {},
6.50/2.52	                    "arithmetic": {
6.50/2.52	                        "type": "PlainIntegerRelationState",
6.50/2.52	                        "relations": []
6.50/2.52	                    },
6.50/2.52	                    "ground": ["T1"],
6.50/2.52	                    "free": [],
6.50/2.52	                    "exprvars": []
6.50/2.52	                }
6.50/2.52	            },
6.50/2.52	            "158": {
6.50/2.52	                "goal": [{
6.50/2.52	                    "clause": -1,
6.50/2.52	                    "scope": -1,
6.50/2.52	                    "term": "(append X45 T26 X46)"
6.50/2.52	                }],
6.50/2.52	                "kb": {
6.50/2.52	                    "nonunifying": [],
6.50/2.52	                    "intvars": {},
6.50/2.52	                    "arithmetic": {
6.50/2.52	                        "type": "PlainIntegerRelationState",
6.50/2.52	                        "relations": []
6.50/2.52	                    },
6.50/2.52	                    "ground": ["T26"],
6.50/2.52	                    "free": [
6.50/2.52	                        "X45",
6.50/2.52	                        "X46"
6.50/2.52	                    ],
6.50/2.52	                    "exprvars": []
6.50/2.52	                }
6.50/2.52	            },
6.50/2.52	            "169": {
6.50/2.52	                "goal": [{
6.50/2.52	                    "clause": -1,
6.50/2.52	                    "scope": -1,
6.50/2.52	                    "term": "(true)"
6.50/2.52	                }],
6.50/2.52	                "kb": {
6.50/2.52	                    "nonunifying": [],
6.50/2.52	                    "intvars": {},
6.50/2.52	                    "arithmetic": {
6.50/2.52	                        "type": "PlainIntegerRelationState",
6.50/2.52	                        "relations": []
6.50/2.52	                    },
6.50/2.52	                    "ground": [],
6.50/2.52	                    "free": [],
6.50/2.52	                    "exprvars": []
6.50/2.52	                }
6.50/2.52	            },
6.50/2.52	            "5": {
6.50/2.52	                "goal": [{
6.50/2.52	                    "clause": 0,
6.50/2.52	                    "scope": 1,
6.50/2.52	                    "term": "(sublist T1 T2)"
6.50/2.52	                }],
6.50/2.52	                "kb": {
6.50/2.52	                    "nonunifying": [],
6.50/2.52	                    "intvars": {},
6.50/2.52	                    "arithmetic": {
6.50/2.52	                        "type": "PlainIntegerRelationState",
6.50/2.52	                        "relations": []
6.50/2.52	                    },
6.50/2.52	                    "ground": ["T1"],
6.50/2.52	                    "free": [],
6.50/2.52	                    "exprvars": []
6.50/2.52	                }
6.50/2.52	            },
6.50/2.52	            "20": {
6.50/2.52	                "goal": [{
6.50/2.52	                    "clause": -1,
6.50/2.52	                    "scope": -1,
6.50/2.52	                    "term": "(',' (append X13 T10 X14) (append X14 X15 T12))"
6.50/2.52	                }],
6.50/2.52	                "kb": {
6.50/2.52	                    "nonunifying": [],
6.50/2.52	                    "intvars": {},
6.50/2.52	                    "arithmetic": {
6.50/2.52	                        "type": "PlainIntegerRelationState",
6.50/2.52	                        "relations": []
6.50/2.52	                    },
6.50/2.52	                    "ground": ["T10"],
6.50/2.52	                    "free": [
6.50/2.52	                        "X13",
6.50/2.52	                        "X14",
6.50/2.52	                        "X15"
6.50/2.52	                    ],
6.50/2.52	                    "exprvars": []
6.50/2.52	                }
6.50/2.52	            }
6.50/2.52	        },
6.50/2.52	        "edges": [
6.50/2.52	            {
6.50/2.52	                "from": 4,
6.50/2.52	                "to": 5,
6.50/2.52	                "label": "CASE"
6.50/2.52	            },
6.50/2.52	            {
6.50/2.52	                "from": 5,
6.50/2.52	                "to": 20,
6.50/2.52	                "label": "ONLY EVAL with clause\nsublist(X11, X12) :- ','(append(X13, X11, X14), append(X14, X15, X12)).\nand substitutionT1 -> T10,\nX11 -> T10,\nT2 -> T12,\nX12 -> T12,\nT11 -> T12"
6.50/2.52	            },
6.50/2.52	            {
6.50/2.52	                "from": 20,
6.50/2.52	                "to": 151,
6.50/2.52	                "label": "SPLIT 1"
6.50/2.52	            },
6.50/2.52	            {
6.50/2.52	                "from": 20,
6.50/2.52	                "to": 152,
6.50/2.52	                "label": "SPLIT 2\nnew knowledge:\nT10 is ground\nreplacements:X13 -> T15,\nX14 -> T16"
6.50/2.52	            },
6.50/2.52	            {
6.50/2.52	                "from": 151,
6.50/2.52	                "to": 153,
6.50/2.52	                "label": "CASE"
6.50/2.52	            },
6.50/2.52	            {
6.50/2.52	                "from": 152,
6.50/2.52	                "to": 161,
6.50/2.52	                "label": "CASE"
6.50/2.52	            },
6.50/2.52	            {
6.50/2.52	                "from": 153,
6.50/2.52	                "to": 154,
6.50/2.52	                "label": "PARALLEL"
6.50/2.52	            },
6.50/2.52	            {
6.50/2.52	                "from": 153,
6.50/2.52	                "to": 155,
6.50/2.52	                "label": "PARALLEL"
6.50/2.52	            },
6.50/2.52	            {
6.50/2.52	                "from": 154,
6.50/2.52	                "to": 156,
6.50/2.52	                "label": "ONLY EVAL with clause\nappend([], X24, X24).\nand substitutionX13 -> [],\nT10 -> T22,\nX24 -> T22,\nX14 -> T22"
6.50/2.52	            },
6.50/2.52	            {
6.50/2.52	                "from": 155,
6.50/2.52	                "to": 158,
6.50/2.52	                "label": "ONLY EVAL with clause\nappend(.(X40, X41), X42, .(X40, X43)) :- append(X41, X42, X43).\nand substitutionX40 -> X44,\nX41 -> X45,\nX13 -> .(X44, X45),\nT10 -> T26,\nX42 -> T26,\nX43 -> X46,\nX14 -> .(X44, X46)"
6.50/2.52	            },
6.50/2.52	            {
6.50/2.52	                "from": 156,
6.50/2.52	                "to": 157,
6.50/2.52	                "label": "SUCCESS"
6.50/2.52	            },
6.50/2.52	            {
6.50/2.52	                "from": 158,
6.50/2.52	                "to": 151,
6.50/2.52	                "label": "INSTANCE with matching:\nX13 -> X45\nT10 -> T26\nX14 -> X46"
6.50/2.52	            },
6.50/2.52	            {
6.50/2.52	                "from": 161,
6.50/2.52	                "to": 163,
6.50/2.52	                "label": "PARALLEL"
6.50/2.52	            },
6.50/2.52	            {
6.50/2.52	                "from": 161,
6.50/2.52	                "to": 165,
6.50/2.52	                "label": "PARALLEL"
6.50/2.52	            },
6.50/2.52	            {
6.50/2.52	                "from": 163,
6.50/2.52	                "to": 169,
6.50/2.52	                "label": "EVAL with clause\nappend([], X60, X60).\nand substitutionT16 -> [],\nX15 -> T34,\nX60 -> T34,\nT12 -> T34,\nX61 -> T34"
6.50/2.52	            },
6.50/2.52	            {
6.50/2.52	                "from": 163,
6.50/2.52	                "to": 170,
6.50/2.52	                "label": "EVAL-BACKTRACK"
6.50/2.52	            },
6.50/2.52	            {
6.50/2.52	                "from": 165,
6.50/2.52	                "to": 185,
6.50/2.52	                "label": "EVAL with clause\nappend(.(X72, X73), X74, .(X72, X75)) :- append(X73, X74, X75).\nand substitutionX72 -> T41,\nX73 -> T44,\nT16 -> .(T41, T44),\nX15 -> X76,\nX74 -> X76,\nX75 -> T45,\nT12 -> .(T41, T45),\nT42 -> T44,\nT43 -> T45"
6.50/2.52	            },
6.50/2.52	            {
6.50/2.52	                "from": 165,
6.50/2.52	                "to": 187,
6.50/2.52	                "label": "EVAL-BACKTRACK"
6.50/2.52	            },
6.50/2.52	            {
6.50/2.52	                "from": 169,
6.50/2.52	                "to": 171,
6.50/2.52	                "label": "SUCCESS"
6.50/2.52	            },
6.50/2.52	            {
6.50/2.52	                "from": 185,
6.50/2.52	                "to": 152,
6.50/2.52	                "label": "INSTANCE with matching:\nT16 -> T44\nX15 -> X76\nT12 -> T45"
6.50/2.52	            }
6.50/2.52	        ],
6.50/2.52	        "type": "Graph"
6.50/2.52	    }
6.50/2.52	}
6.50/2.52	
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(20)
6.50/2.52	Obligation:
6.50/2.52	Q restricted rewrite system:
6.50/2.52	The TRS R consists of the following rules:
6.50/2.52	
6.50/2.52	   f4_in(T10) -> U1(f20_in(T10), T10)
6.50/2.52	   U1(f20_out1, T10) -> f4_out1
6.50/2.52	   f151_in(T22) -> f151_out1
6.50/2.52	   f151_in(T26) -> U2(f151_in(T26), T26)
6.50/2.52	   U2(f151_out1, T26) -> f151_out1
6.50/2.52	   f152_in -> f152_out1
6.50/2.52	   f152_in -> U3(f152_in)
6.50/2.52	   U3(f152_out1) -> f152_out1
6.50/2.52	   f20_in(T10) -> U4(f151_in(T10), T10)
6.50/2.52	   U4(f151_out1, T10) -> U5(f152_in, T10)
6.50/2.52	   U5(f152_out1, T10) -> f20_out1
6.50/2.52	
6.50/2.52	Q is empty.
6.50/2.52	
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(21) DependencyPairsProof (EQUIVALENT)
6.50/2.52	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(22)
6.50/2.52	Obligation:
6.50/2.52	Q DP problem:
6.50/2.52	The TRS P consists of the following rules:
6.50/2.52	
6.50/2.52	   F4_IN(T10) -> U1^1(f20_in(T10), T10)
6.50/2.52	   F4_IN(T10) -> F20_IN(T10)
6.50/2.52	   F151_IN(T26) -> U2^1(f151_in(T26), T26)
6.50/2.52	   F151_IN(T26) -> F151_IN(T26)
6.50/2.52	   F152_IN -> U3^1(f152_in)
6.50/2.52	   F152_IN -> F152_IN
6.50/2.52	   F20_IN(T10) -> U4^1(f151_in(T10), T10)
6.50/2.52	   F20_IN(T10) -> F151_IN(T10)
6.50/2.52	   U4^1(f151_out1, T10) -> U5^1(f152_in, T10)
6.50/2.52	   U4^1(f151_out1, T10) -> F152_IN
6.50/2.52	
6.50/2.52	The TRS R consists of the following rules:
6.50/2.52	
6.50/2.52	   f4_in(T10) -> U1(f20_in(T10), T10)
6.50/2.52	   U1(f20_out1, T10) -> f4_out1
6.50/2.52	   f151_in(T22) -> f151_out1
6.50/2.52	   f151_in(T26) -> U2(f151_in(T26), T26)
6.50/2.52	   U2(f151_out1, T26) -> f151_out1
6.50/2.52	   f152_in -> f152_out1
6.50/2.52	   f152_in -> U3(f152_in)
6.50/2.52	   U3(f152_out1) -> f152_out1
6.50/2.52	   f20_in(T10) -> U4(f151_in(T10), T10)
6.50/2.52	   U4(f151_out1, T10) -> U5(f152_in, T10)
6.50/2.52	   U5(f152_out1, T10) -> f20_out1
6.50/2.52	
6.50/2.52	Q is empty.
6.50/2.52	We have to consider all minimal (P,Q,R)-chains.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(23) DependencyGraphProof (EQUIVALENT)
6.50/2.52	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 8 less nodes.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(24)
6.50/2.52	Complex Obligation (AND)
6.50/2.52	
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(25)
6.50/2.52	Obligation:
6.50/2.52	Q DP problem:
6.50/2.52	The TRS P consists of the following rules:
6.50/2.52	
6.50/2.52	   F152_IN -> F152_IN
6.50/2.52	
6.50/2.52	The TRS R consists of the following rules:
6.50/2.52	
6.50/2.52	   f4_in(T10) -> U1(f20_in(T10), T10)
6.50/2.52	   U1(f20_out1, T10) -> f4_out1
6.50/2.52	   f151_in(T22) -> f151_out1
6.50/2.52	   f151_in(T26) -> U2(f151_in(T26), T26)
6.50/2.52	   U2(f151_out1, T26) -> f151_out1
6.50/2.52	   f152_in -> f152_out1
6.50/2.52	   f152_in -> U3(f152_in)
6.50/2.52	   U3(f152_out1) -> f152_out1
6.50/2.52	   f20_in(T10) -> U4(f151_in(T10), T10)
6.50/2.52	   U4(f151_out1, T10) -> U5(f152_in, T10)
6.50/2.52	   U5(f152_out1, T10) -> f20_out1
6.50/2.52	
6.50/2.52	Q is empty.
6.50/2.52	We have to consider all minimal (P,Q,R)-chains.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(26) MNOCProof (EQUIVALENT)
6.50/2.52	We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(27)
6.50/2.52	Obligation:
6.50/2.52	Q DP problem:
6.50/2.52	The TRS P consists of the following rules:
6.50/2.52	
6.50/2.52	   F152_IN -> F152_IN
6.50/2.52	
6.50/2.52	The TRS R consists of the following rules:
6.50/2.52	
6.50/2.52	   f4_in(T10) -> U1(f20_in(T10), T10)
6.50/2.52	   U1(f20_out1, T10) -> f4_out1
6.50/2.52	   f151_in(T22) -> f151_out1
6.50/2.52	   f151_in(T26) -> U2(f151_in(T26), T26)
6.50/2.52	   U2(f151_out1, T26) -> f151_out1
6.50/2.52	   f152_in -> f152_out1
6.50/2.52	   f152_in -> U3(f152_in)
6.50/2.52	   U3(f152_out1) -> f152_out1
6.50/2.52	   f20_in(T10) -> U4(f151_in(T10), T10)
6.50/2.52	   U4(f151_out1, T10) -> U5(f152_in, T10)
6.50/2.52	   U5(f152_out1, T10) -> f20_out1
6.50/2.52	
6.50/2.52	The set Q consists of the following terms:
6.50/2.52	
6.50/2.52	   f4_in(x0)
6.50/2.52	   U1(f20_out1, x0)
6.50/2.52	   f151_in(x0)
6.50/2.52	   U2(f151_out1, x0)
6.50/2.52	   f152_in
6.50/2.52	   U3(f152_out1)
6.50/2.52	   f20_in(x0)
6.50/2.52	   U4(f151_out1, x0)
6.50/2.52	   U5(f152_out1, x0)
6.50/2.52	
6.50/2.52	We have to consider all minimal (P,Q,R)-chains.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(28) UsableRulesProof (EQUIVALENT)
6.50/2.52	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(29)
6.50/2.52	Obligation:
6.50/2.52	Q DP problem:
6.50/2.52	The TRS P consists of the following rules:
6.50/2.52	
6.50/2.52	   F152_IN -> F152_IN
6.50/2.52	
6.50/2.52	R is empty.
6.50/2.52	The set Q consists of the following terms:
6.50/2.52	
6.50/2.52	   f4_in(x0)
6.50/2.52	   U1(f20_out1, x0)
6.50/2.52	   f151_in(x0)
6.50/2.52	   U2(f151_out1, x0)
6.50/2.52	   f152_in
6.50/2.52	   U3(f152_out1)
6.50/2.52	   f20_in(x0)
6.50/2.52	   U4(f151_out1, x0)
6.50/2.52	   U5(f152_out1, x0)
6.50/2.52	
6.50/2.52	We have to consider all minimal (P,Q,R)-chains.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(30) QReductionProof (EQUIVALENT)
6.50/2.52	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
6.50/2.52	
6.50/2.52	   f4_in(x0)
6.50/2.52	   U1(f20_out1, x0)
6.50/2.52	   f151_in(x0)
6.50/2.52	   U2(f151_out1, x0)
6.50/2.52	   f152_in
6.50/2.52	   U3(f152_out1)
6.50/2.52	   f20_in(x0)
6.50/2.52	   U4(f151_out1, x0)
6.50/2.52	   U5(f152_out1, x0)
6.50/2.52	
6.50/2.52	
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(31)
6.50/2.52	Obligation:
6.50/2.52	Q DP problem:
6.50/2.52	The TRS P consists of the following rules:
6.50/2.52	
6.50/2.52	   F152_IN -> F152_IN
6.50/2.52	
6.50/2.52	R is empty.
6.50/2.52	Q is empty.
6.50/2.52	We have to consider all minimal (P,Q,R)-chains.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(32)
6.50/2.52	Obligation:
6.50/2.52	Q DP problem:
6.50/2.52	The TRS P consists of the following rules:
6.50/2.52	
6.50/2.52	   F151_IN(T26) -> F151_IN(T26)
6.50/2.52	
6.50/2.52	The TRS R consists of the following rules:
6.50/2.52	
6.50/2.52	   f4_in(T10) -> U1(f20_in(T10), T10)
6.50/2.52	   U1(f20_out1, T10) -> f4_out1
6.50/2.52	   f151_in(T22) -> f151_out1
6.50/2.52	   f151_in(T26) -> U2(f151_in(T26), T26)
6.50/2.52	   U2(f151_out1, T26) -> f151_out1
6.50/2.52	   f152_in -> f152_out1
6.50/2.52	   f152_in -> U3(f152_in)
6.50/2.52	   U3(f152_out1) -> f152_out1
6.50/2.52	   f20_in(T10) -> U4(f151_in(T10), T10)
6.50/2.52	   U4(f151_out1, T10) -> U5(f152_in, T10)
6.50/2.52	   U5(f152_out1, T10) -> f20_out1
6.50/2.52	
6.50/2.52	Q is empty.
6.50/2.52	We have to consider all minimal (P,Q,R)-chains.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(33) MNOCProof (EQUIVALENT)
6.50/2.52	We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(34)
6.50/2.52	Obligation:
6.50/2.52	Q DP problem:
6.50/2.52	The TRS P consists of the following rules:
6.50/2.52	
6.50/2.52	   F151_IN(T26) -> F151_IN(T26)
6.50/2.52	
6.50/2.52	The TRS R consists of the following rules:
6.50/2.52	
6.50/2.52	   f4_in(T10) -> U1(f20_in(T10), T10)
6.50/2.52	   U1(f20_out1, T10) -> f4_out1
6.50/2.52	   f151_in(T22) -> f151_out1
6.50/2.52	   f151_in(T26) -> U2(f151_in(T26), T26)
6.50/2.52	   U2(f151_out1, T26) -> f151_out1
6.50/2.52	   f152_in -> f152_out1
6.50/2.52	   f152_in -> U3(f152_in)
6.50/2.52	   U3(f152_out1) -> f152_out1
6.50/2.52	   f20_in(T10) -> U4(f151_in(T10), T10)
6.50/2.52	   U4(f151_out1, T10) -> U5(f152_in, T10)
6.50/2.52	   U5(f152_out1, T10) -> f20_out1
6.50/2.52	
6.50/2.52	The set Q consists of the following terms:
6.50/2.52	
6.50/2.52	   f4_in(x0)
6.50/2.52	   U1(f20_out1, x0)
6.50/2.52	   f151_in(x0)
6.50/2.52	   U2(f151_out1, x0)
6.50/2.52	   f152_in
6.50/2.52	   U3(f152_out1)
6.50/2.52	   f20_in(x0)
6.50/2.52	   U4(f151_out1, x0)
6.50/2.52	   U5(f152_out1, x0)
6.50/2.52	
6.50/2.52	We have to consider all minimal (P,Q,R)-chains.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(35) UsableRulesProof (EQUIVALENT)
6.50/2.52	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(36)
6.50/2.52	Obligation:
6.50/2.52	Q DP problem:
6.50/2.52	The TRS P consists of the following rules:
6.50/2.52	
6.50/2.52	   F151_IN(T26) -> F151_IN(T26)
6.50/2.52	
6.50/2.52	R is empty.
6.50/2.52	The set Q consists of the following terms:
6.50/2.52	
6.50/2.52	   f4_in(x0)
6.50/2.52	   U1(f20_out1, x0)
6.50/2.52	   f151_in(x0)
6.50/2.52	   U2(f151_out1, x0)
6.50/2.52	   f152_in
6.50/2.52	   U3(f152_out1)
6.50/2.52	   f20_in(x0)
6.50/2.52	   U4(f151_out1, x0)
6.50/2.52	   U5(f152_out1, x0)
6.50/2.52	
6.50/2.52	We have to consider all minimal (P,Q,R)-chains.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(37) QReductionProof (EQUIVALENT)
6.50/2.52	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
6.50/2.52	
6.50/2.52	   f4_in(x0)
6.50/2.52	   U1(f20_out1, x0)
6.50/2.52	   f151_in(x0)
6.50/2.52	   U2(f151_out1, x0)
6.50/2.52	   f152_in
6.50/2.52	   U3(f152_out1)
6.50/2.52	   f20_in(x0)
6.50/2.52	   U4(f151_out1, x0)
6.50/2.52	   U5(f152_out1, x0)
6.50/2.52	
6.50/2.52	
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(38)
6.50/2.52	Obligation:
6.50/2.52	Q DP problem:
6.50/2.52	The TRS P consists of the following rules:
6.50/2.52	
6.50/2.52	   F151_IN(T26) -> F151_IN(T26)
6.50/2.52	
6.50/2.52	R is empty.
6.50/2.52	Q is empty.
6.50/2.52	We have to consider all minimal (P,Q,R)-chains.
6.50/2.52	----------------------------------------
6.50/2.52	
6.50/2.52	(39) PrologToPiTRSProof (SOUND)
6.50/2.52	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.50/2.52	
6.50/2.52	sublist_in_2: (b,f)
6.50/2.52	
6.50/2.52	append_in_3: (f,b,f) (b,f,f)
6.50/2.52	
6.50/2.52	Transforming Prolog into the following Term Rewriting System:
6.50/2.52	
6.50/2.52	Pi-finite rewrite system:
6.50/2.52	The TRS R consists of the following rules:
6.50/2.52	
6.50/2.52	   sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V))
6.50/2.52	   append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys)
6.50/2.52	   append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs))
6.50/2.52	   U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs))
6.50/2.52	   U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y))
6.50/2.52	   append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys)
6.50/2.52	   append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs))
6.50/2.52	   U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs))
6.50/2.52	   U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y)
6.50/2.52	
6.50/2.52	The argument filtering Pi contains the following mapping:
6.50/2.52	sublist_in_ga(x1, x2)  =  sublist_in_ga(x1)
6.50/2.52	
6.50/2.52	U1_ga(x1, x2, x3)  =  U1_ga(x3)
6.50/2.52	
6.50/2.52	append_in_aga(x1, x2, x3)  =  append_in_aga(x2)
6.50/2.52	
6.50/2.52	append_out_aga(x1, x2, x3)  =  append_out_aga(x1, x3)
6.50/2.52	
6.50/2.52	U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
6.50/2.52	
6.50/2.52	.(x1, x2)  =  .(x2)
6.50/2.52	
6.50/2.52	U2_ga(x1, x2, x3)  =  U2_ga(x3)
6.50/2.52	
6.50/2.52	append_in_gaa(x1, x2, x3)  =  append_in_gaa(x1)
6.50/2.52	
6.50/2.52	[]  =  []
6.50/2.52	
6.50/2.52	append_out_gaa(x1, x2, x3)  =  append_out_gaa
6.50/2.52	
6.50/2.52	U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
6.50/2.52	
6.50/2.52	sublist_out_ga(x1, x2)  =  sublist_out_ga
6.50/2.52	
6.50/2.52	
6.50/2.52	
6.50/2.52	
6.50/2.52	
6.50/2.52	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.50/2.52	
6.50/2.52	
6.50/2.52	
6.50/2.52	----------------------------------------
6.50/2.53	
6.50/2.53	(40)
6.50/2.53	Obligation:
6.50/2.53	Pi-finite rewrite system:
6.50/2.53	The TRS R consists of the following rules:
6.50/2.53	
6.50/2.53	   sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V))
6.50/2.53	   append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys)
6.50/2.53	   append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs))
6.50/2.53	   U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs))
6.50/2.53	   U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y))
6.50/2.53	   append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys)
6.50/2.53	   append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs))
6.50/2.53	   U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs))
6.50/2.53	   U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y)
6.50/2.53	
6.50/2.53	The argument filtering Pi contains the following mapping:
6.50/2.53	sublist_in_ga(x1, x2)  =  sublist_in_ga(x1)
6.50/2.53	
6.50/2.53	U1_ga(x1, x2, x3)  =  U1_ga(x3)
6.50/2.53	
6.50/2.53	append_in_aga(x1, x2, x3)  =  append_in_aga(x2)
6.50/2.53	
6.50/2.53	append_out_aga(x1, x2, x3)  =  append_out_aga(x1, x3)
6.50/2.53	
6.50/2.53	U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
6.50/2.53	
6.50/2.53	.(x1, x2)  =  .(x2)
6.50/2.53	
6.50/2.53	U2_ga(x1, x2, x3)  =  U2_ga(x3)
6.50/2.53	
6.50/2.53	append_in_gaa(x1, x2, x3)  =  append_in_gaa(x1)
6.50/2.53	
6.50/2.53	[]  =  []
6.50/2.53	
6.50/2.53	append_out_gaa(x1, x2, x3)  =  append_out_gaa
6.50/2.53	
6.50/2.53	U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
6.50/2.53	
6.50/2.53	sublist_out_ga(x1, x2)  =  sublist_out_ga
6.50/2.53	
6.50/2.53	
6.50/2.53	
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(41) DependencyPairsProof (EQUIVALENT)
6.50/2.53	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.50/2.53	Pi DP problem:
6.50/2.53	The TRS P consists of the following rules:
6.50/2.53	
6.50/2.53	   SUBLIST_IN_GA(X, Y) -> U1_GA(X, Y, append_in_aga(U, X, V))
6.50/2.53	   SUBLIST_IN_GA(X, Y) -> APPEND_IN_AGA(U, X, V)
6.50/2.53	   APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U3_AGA(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs))
6.50/2.53	   APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AGA(Xs, Ys, Zs)
6.50/2.53	   U1_GA(X, Y, append_out_aga(U, X, V)) -> U2_GA(X, Y, append_in_gaa(V, W, Y))
6.50/2.53	   U1_GA(X, Y, append_out_aga(U, X, V)) -> APPEND_IN_GAA(V, W, Y)
6.50/2.53	   APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> U3_GAA(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs))
6.50/2.53	   APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GAA(Xs, Ys, Zs)
6.50/2.53	
6.50/2.53	The TRS R consists of the following rules:
6.50/2.53	
6.50/2.53	   sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V))
6.50/2.53	   append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys)
6.50/2.53	   append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs))
6.50/2.53	   U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs))
6.50/2.53	   U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y))
6.50/2.53	   append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys)
6.50/2.53	   append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs))
6.50/2.53	   U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs))
6.50/2.53	   U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y)
6.50/2.53	
6.50/2.53	The argument filtering Pi contains the following mapping:
6.50/2.53	sublist_in_ga(x1, x2)  =  sublist_in_ga(x1)
6.50/2.53	
6.50/2.53	U1_ga(x1, x2, x3)  =  U1_ga(x3)
6.50/2.53	
6.50/2.53	append_in_aga(x1, x2, x3)  =  append_in_aga(x2)
6.50/2.53	
6.50/2.53	append_out_aga(x1, x2, x3)  =  append_out_aga(x1, x3)
6.50/2.53	
6.50/2.53	U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
6.50/2.53	
6.50/2.53	.(x1, x2)  =  .(x2)
6.50/2.53	
6.50/2.53	U2_ga(x1, x2, x3)  =  U2_ga(x3)
6.50/2.53	
6.50/2.53	append_in_gaa(x1, x2, x3)  =  append_in_gaa(x1)
6.50/2.53	
6.50/2.53	[]  =  []
6.50/2.53	
6.50/2.53	append_out_gaa(x1, x2, x3)  =  append_out_gaa
6.50/2.53	
6.50/2.53	U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
6.50/2.53	
6.50/2.53	sublist_out_ga(x1, x2)  =  sublist_out_ga
6.50/2.53	
6.50/2.53	SUBLIST_IN_GA(x1, x2)  =  SUBLIST_IN_GA(x1)
6.50/2.53	
6.50/2.53	U1_GA(x1, x2, x3)  =  U1_GA(x3)
6.50/2.53	
6.50/2.53	APPEND_IN_AGA(x1, x2, x3)  =  APPEND_IN_AGA(x2)
6.50/2.53	
6.50/2.53	U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x5)
6.50/2.53	
6.50/2.53	U2_GA(x1, x2, x3)  =  U2_GA(x3)
6.50/2.53	
6.50/2.53	APPEND_IN_GAA(x1, x2, x3)  =  APPEND_IN_GAA(x1)
6.50/2.53	
6.50/2.53	U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x5)
6.50/2.53	
6.50/2.53	
6.50/2.53	We have to consider all (P,R,Pi)-chains
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(42)
6.50/2.53	Obligation:
6.50/2.53	Pi DP problem:
6.50/2.53	The TRS P consists of the following rules:
6.50/2.53	
6.50/2.53	   SUBLIST_IN_GA(X, Y) -> U1_GA(X, Y, append_in_aga(U, X, V))
6.50/2.53	   SUBLIST_IN_GA(X, Y) -> APPEND_IN_AGA(U, X, V)
6.50/2.53	   APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U3_AGA(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs))
6.50/2.53	   APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AGA(Xs, Ys, Zs)
6.50/2.53	   U1_GA(X, Y, append_out_aga(U, X, V)) -> U2_GA(X, Y, append_in_gaa(V, W, Y))
6.50/2.53	   U1_GA(X, Y, append_out_aga(U, X, V)) -> APPEND_IN_GAA(V, W, Y)
6.50/2.53	   APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> U3_GAA(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs))
6.50/2.53	   APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GAA(Xs, Ys, Zs)
6.50/2.53	
6.50/2.53	The TRS R consists of the following rules:
6.50/2.53	
6.50/2.53	   sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V))
6.50/2.53	   append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys)
6.50/2.53	   append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs))
6.50/2.53	   U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs))
6.50/2.53	   U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y))
6.50/2.53	   append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys)
6.50/2.53	   append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs))
6.50/2.53	   U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs))
6.50/2.53	   U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y)
6.50/2.53	
6.50/2.53	The argument filtering Pi contains the following mapping:
6.50/2.53	sublist_in_ga(x1, x2)  =  sublist_in_ga(x1)
6.50/2.53	
6.50/2.53	U1_ga(x1, x2, x3)  =  U1_ga(x3)
6.50/2.53	
6.50/2.53	append_in_aga(x1, x2, x3)  =  append_in_aga(x2)
6.50/2.53	
6.50/2.53	append_out_aga(x1, x2, x3)  =  append_out_aga(x1, x3)
6.50/2.53	
6.50/2.53	U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
6.50/2.53	
6.50/2.53	.(x1, x2)  =  .(x2)
6.50/2.53	
6.50/2.53	U2_ga(x1, x2, x3)  =  U2_ga(x3)
6.50/2.53	
6.50/2.53	append_in_gaa(x1, x2, x3)  =  append_in_gaa(x1)
6.50/2.53	
6.50/2.53	[]  =  []
6.50/2.53	
6.50/2.53	append_out_gaa(x1, x2, x3)  =  append_out_gaa
6.50/2.53	
6.50/2.53	U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
6.50/2.53	
6.50/2.53	sublist_out_ga(x1, x2)  =  sublist_out_ga
6.50/2.53	
6.50/2.53	SUBLIST_IN_GA(x1, x2)  =  SUBLIST_IN_GA(x1)
6.50/2.53	
6.50/2.53	U1_GA(x1, x2, x3)  =  U1_GA(x3)
6.50/2.53	
6.50/2.53	APPEND_IN_AGA(x1, x2, x3)  =  APPEND_IN_AGA(x2)
6.50/2.53	
6.50/2.53	U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x5)
6.50/2.53	
6.50/2.53	U2_GA(x1, x2, x3)  =  U2_GA(x3)
6.50/2.53	
6.50/2.53	APPEND_IN_GAA(x1, x2, x3)  =  APPEND_IN_GAA(x1)
6.50/2.53	
6.50/2.53	U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x5)
6.50/2.53	
6.50/2.53	
6.50/2.53	We have to consider all (P,R,Pi)-chains
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(43) DependencyGraphProof (EQUIVALENT)
6.50/2.53	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(44)
6.50/2.53	Complex Obligation (AND)
6.50/2.53	
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(45)
6.50/2.53	Obligation:
6.50/2.53	Pi DP problem:
6.50/2.53	The TRS P consists of the following rules:
6.50/2.53	
6.50/2.53	   APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GAA(Xs, Ys, Zs)
6.50/2.53	
6.50/2.53	The TRS R consists of the following rules:
6.50/2.53	
6.50/2.53	   sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V))
6.50/2.53	   append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys)
6.50/2.53	   append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs))
6.50/2.53	   U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs))
6.50/2.53	   U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y))
6.50/2.53	   append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys)
6.50/2.53	   append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs))
6.50/2.53	   U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs))
6.50/2.53	   U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y)
6.50/2.53	
6.50/2.53	The argument filtering Pi contains the following mapping:
6.50/2.53	sublist_in_ga(x1, x2)  =  sublist_in_ga(x1)
6.50/2.53	
6.50/2.53	U1_ga(x1, x2, x3)  =  U1_ga(x3)
6.50/2.53	
6.50/2.53	append_in_aga(x1, x2, x3)  =  append_in_aga(x2)
6.50/2.53	
6.50/2.53	append_out_aga(x1, x2, x3)  =  append_out_aga(x1, x3)
6.50/2.53	
6.50/2.53	U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
6.50/2.53	
6.50/2.53	.(x1, x2)  =  .(x2)
6.50/2.53	
6.50/2.53	U2_ga(x1, x2, x3)  =  U2_ga(x3)
6.50/2.53	
6.50/2.53	append_in_gaa(x1, x2, x3)  =  append_in_gaa(x1)
6.50/2.53	
6.50/2.53	[]  =  []
6.50/2.53	
6.50/2.53	append_out_gaa(x1, x2, x3)  =  append_out_gaa
6.50/2.53	
6.50/2.53	U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
6.50/2.53	
6.50/2.53	sublist_out_ga(x1, x2)  =  sublist_out_ga
6.50/2.53	
6.50/2.53	APPEND_IN_GAA(x1, x2, x3)  =  APPEND_IN_GAA(x1)
6.50/2.53	
6.50/2.53	
6.50/2.53	We have to consider all (P,R,Pi)-chains
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(46) UsableRulesProof (EQUIVALENT)
6.50/2.53	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(47)
6.50/2.53	Obligation:
6.50/2.53	Pi DP problem:
6.50/2.53	The TRS P consists of the following rules:
6.50/2.53	
6.50/2.53	   APPEND_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GAA(Xs, Ys, Zs)
6.50/2.53	
6.50/2.53	R is empty.
6.50/2.53	The argument filtering Pi contains the following mapping:
6.50/2.53	.(x1, x2)  =  .(x2)
6.50/2.53	
6.50/2.53	APPEND_IN_GAA(x1, x2, x3)  =  APPEND_IN_GAA(x1)
6.50/2.53	
6.50/2.53	
6.50/2.53	We have to consider all (P,R,Pi)-chains
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(48) PiDPToQDPProof (SOUND)
6.50/2.53	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(49)
6.50/2.53	Obligation:
6.50/2.53	Q DP problem:
6.50/2.53	The TRS P consists of the following rules:
6.50/2.53	
6.50/2.53	   APPEND_IN_GAA(.(Xs)) -> APPEND_IN_GAA(Xs)
6.50/2.53	
6.50/2.53	R is empty.
6.50/2.53	Q is empty.
6.50/2.53	We have to consider all (P,Q,R)-chains.
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(50) QDPSizeChangeProof (EQUIVALENT)
6.50/2.53	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. 
6.50/2.53	
6.50/2.53	From the DPs we obtained the following set of size-change graphs:
6.50/2.53	*APPEND_IN_GAA(.(Xs)) -> APPEND_IN_GAA(Xs)
6.50/2.53	The graph contains the following edges 1 > 1
6.50/2.53	
6.50/2.53	
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(51)
6.50/2.53	YES
6.50/2.53	
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(52)
6.50/2.53	Obligation:
6.50/2.53	Pi DP problem:
6.50/2.53	The TRS P consists of the following rules:
6.50/2.53	
6.50/2.53	   APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AGA(Xs, Ys, Zs)
6.50/2.53	
6.50/2.53	The TRS R consists of the following rules:
6.50/2.53	
6.50/2.53	   sublist_in_ga(X, Y) -> U1_ga(X, Y, append_in_aga(U, X, V))
6.50/2.53	   append_in_aga([], Ys, Ys) -> append_out_aga([], Ys, Ys)
6.50/2.53	   append_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U3_aga(X, Xs, Ys, Zs, append_in_aga(Xs, Ys, Zs))
6.50/2.53	   U3_aga(X, Xs, Ys, Zs, append_out_aga(Xs, Ys, Zs)) -> append_out_aga(.(X, Xs), Ys, .(X, Zs))
6.50/2.53	   U1_ga(X, Y, append_out_aga(U, X, V)) -> U2_ga(X, Y, append_in_gaa(V, W, Y))
6.50/2.53	   append_in_gaa([], Ys, Ys) -> append_out_gaa([], Ys, Ys)
6.50/2.53	   append_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U3_gaa(X, Xs, Ys, Zs, append_in_gaa(Xs, Ys, Zs))
6.50/2.53	   U3_gaa(X, Xs, Ys, Zs, append_out_gaa(Xs, Ys, Zs)) -> append_out_gaa(.(X, Xs), Ys, .(X, Zs))
6.50/2.53	   U2_ga(X, Y, append_out_gaa(V, W, Y)) -> sublist_out_ga(X, Y)
6.50/2.53	
6.50/2.53	The argument filtering Pi contains the following mapping:
6.50/2.53	sublist_in_ga(x1, x2)  =  sublist_in_ga(x1)
6.50/2.53	
6.50/2.53	U1_ga(x1, x2, x3)  =  U1_ga(x3)
6.50/2.53	
6.50/2.53	append_in_aga(x1, x2, x3)  =  append_in_aga(x2)
6.50/2.53	
6.50/2.53	append_out_aga(x1, x2, x3)  =  append_out_aga(x1, x3)
6.50/2.53	
6.50/2.53	U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x5)
6.50/2.53	
6.50/2.53	.(x1, x2)  =  .(x2)
6.50/2.53	
6.50/2.53	U2_ga(x1, x2, x3)  =  U2_ga(x3)
6.50/2.53	
6.50/2.53	append_in_gaa(x1, x2, x3)  =  append_in_gaa(x1)
6.50/2.53	
6.50/2.53	[]  =  []
6.50/2.53	
6.50/2.53	append_out_gaa(x1, x2, x3)  =  append_out_gaa
6.50/2.53	
6.50/2.53	U3_gaa(x1, x2, x3, x4, x5)  =  U3_gaa(x5)
6.50/2.53	
6.50/2.53	sublist_out_ga(x1, x2)  =  sublist_out_ga
6.50/2.53	
6.50/2.53	APPEND_IN_AGA(x1, x2, x3)  =  APPEND_IN_AGA(x2)
6.50/2.53	
6.50/2.53	
6.50/2.53	We have to consider all (P,R,Pi)-chains
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(53) UsableRulesProof (EQUIVALENT)
6.50/2.53	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(54)
6.50/2.53	Obligation:
6.50/2.53	Pi DP problem:
6.50/2.53	The TRS P consists of the following rules:
6.50/2.53	
6.50/2.53	   APPEND_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AGA(Xs, Ys, Zs)
6.50/2.53	
6.50/2.53	R is empty.
6.50/2.53	The argument filtering Pi contains the following mapping:
6.50/2.53	.(x1, x2)  =  .(x2)
6.50/2.53	
6.50/2.53	APPEND_IN_AGA(x1, x2, x3)  =  APPEND_IN_AGA(x2)
6.50/2.53	
6.50/2.53	
6.50/2.53	We have to consider all (P,R,Pi)-chains
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(55) PiDPToQDPProof (SOUND)
6.50/2.53	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(56)
6.50/2.53	Obligation:
6.50/2.53	Q DP problem:
6.50/2.53	The TRS P consists of the following rules:
6.50/2.53	
6.50/2.53	   APPEND_IN_AGA(Ys) -> APPEND_IN_AGA(Ys)
6.50/2.53	
6.50/2.53	R is empty.
6.50/2.53	Q is empty.
6.50/2.53	We have to consider all (P,Q,R)-chains.
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(57) PrologToDTProblemTransformerProof (SOUND)
6.50/2.53	Built DT problem from termination graph DT10.
6.50/2.53	
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6.50/2.53	    "root": 2,
6.50/2.53	    "program": {
6.50/2.53	        "directives": [],
6.50/2.53	        "clauses": [
6.50/2.53	            [
6.50/2.53	                "(sublist X Y)",
6.50/2.53	                "(',' (append U X V) (append V W Y))"
6.50/2.53	            ],
6.50/2.53	            [
6.50/2.53	                "(append ([]) Ys Ys)",
6.50/2.53	                null
6.50/2.53	            ],
6.50/2.53	            [
6.50/2.53	                "(append (. X Xs) Ys (. X Zs))",
6.50/2.53	                "(append Xs Ys Zs)"
6.50/2.53	            ]
6.50/2.53	        ]
6.50/2.53	    },
6.50/2.53	    "graph": {
6.50/2.53	        "nodes": {
6.50/2.53	            "89": {
6.50/2.53	                "goal": [{
6.50/2.53	                    "clause": -1,
6.50/2.53	                    "scope": -1,
6.50/2.53	                    "term": "(true)"
6.50/2.53	                }],
6.50/2.53	                "kb": {
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6.50/2.53	                    "intvars": {},
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6.50/2.53	                    "ground": ["T39"],
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6.50/2.53	                        "X76",
6.50/2.53	                        "X77"
6.50/2.53	                    ],
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6.50/2.53	                    "term": "(append (. X75 T42) X7 T7)"
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6.50/2.53	                    "free": [
6.50/2.53	                        "X7",
6.50/2.53	                        "X75"
6.50/2.53	                    ],
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6.50/2.53	                }
6.50/2.53	            },
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6.50/2.53	                    },
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6.50/2.53	                }
6.50/2.53	            },
6.50/2.53	            "199": {
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6.50/2.53	                    "clause": -1,
6.50/2.53	                    "scope": -1,
6.50/2.53	                    "term": "(append X107 T52 X108)"
6.50/2.53	                }],
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6.50/2.53	                    "ground": ["T16"],
6.50/2.53	                    "free": ["X7"],
6.50/2.53	                    "exprvars": []
6.50/2.53	                }
6.50/2.53	            },
6.50/2.53	            "208": {
6.50/2.53	                "goal": [{
6.50/2.53	                    "clause": 2,
6.50/2.53	                    "scope": 6,
6.50/2.53	                    "term": "(append T63 X130 T64)"
6.50/2.53	                }],
6.50/2.53	                "kb": {
6.50/2.53	                    "nonunifying": [],
6.50/2.53	                    "intvars": {},
6.50/2.53	                    "arithmetic": {
6.50/2.53	                        "type": "PlainIntegerRelationState",
6.50/2.53	                        "relations": []
6.50/2.53	                    },
6.50/2.53	                    "ground": [],
6.50/2.53	                    "free": ["X130"],
6.50/2.53	                    "exprvars": []
6.50/2.53	                }
6.50/2.53	            },
6.50/2.53	            "209": {
6.50/2.53	                "goal": [{
6.50/2.53	                    "clause": -1,
6.50/2.53	                    "scope": -1,
6.50/2.53	                    "term": "(true)"
6.50/2.53	                }],
6.50/2.53	                "kb": {
6.50/2.53	                    "nonunifying": [],
6.50/2.53	                    "intvars": {},
6.50/2.53	                    "arithmetic": {
6.50/2.53	                        "type": "PlainIntegerRelationState",
6.50/2.53	                        "relations": []
6.50/2.53	                    },
6.50/2.53	                    "ground": [],
6.50/2.53	                    "free": [],
6.50/2.53	                    "exprvars": []
6.50/2.53	                }
6.50/2.53	            },
6.50/2.53	            "84": {
6.50/2.53	                "goal": [{
6.50/2.53	                    "clause": 1,
6.50/2.53	                    "scope": 3,
6.50/2.53	                    "term": "(append T16 X7 T7)"
6.50/2.53	                }],
6.50/2.53	                "kb": {
6.50/2.53	                    "nonunifying": [],
6.50/2.53	                    "intvars": {},
6.50/2.53	                    "arithmetic": {
6.50/2.53	                        "type": "PlainIntegerRelationState",
6.50/2.53	                        "relations": []
6.50/2.53	                    },
6.50/2.53	                    "ground": ["T16"],
6.50/2.53	                    "free": ["X7"],
6.50/2.53	                    "exprvars": []
6.50/2.53	                }
6.50/2.53	            },
6.50/2.53	            "85": {
6.50/2.53	                "goal": [{
6.50/2.53	                    "clause": 2,
6.50/2.53	                    "scope": 3,
6.50/2.53	                    "term": "(append T16 X7 T7)"
6.50/2.53	                }],
6.50/2.53	                "kb": {
6.50/2.53	                    "nonunifying": [],
6.50/2.53	                    "intvars": {},
6.50/2.53	                    "arithmetic": {
6.50/2.53	                        "type": "PlainIntegerRelationState",
6.50/2.53	                        "relations": []
6.50/2.53	                    },
6.50/2.53	                    "ground": ["T16"],
6.50/2.53	                    "free": ["X7"],
6.50/2.53	                    "exprvars": []
6.50/2.53	                }
6.50/2.53	            }
6.50/2.53	        },
6.50/2.53	        "edges": [
6.50/2.53	            {
6.50/2.53	                "from": 2,
6.50/2.53	                "to": 3,
6.50/2.53	                "label": "CASE"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 3,
6.50/2.53	                "to": 8,
6.50/2.53	                "label": "ONLY EVAL with clause\nsublist(X3, X4) :- ','(append(X5, X3, X6), append(X6, X7, X4)).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T7,\nX4 -> T7,\nT6 -> T7"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 8,
6.50/2.53	                "to": 50,
6.50/2.53	                "label": "CASE"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 50,
6.50/2.53	                "to": 74,
6.50/2.53	                "label": "PARALLEL"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 50,
6.50/2.53	                "to": 75,
6.50/2.53	                "label": "PARALLEL"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 74,
6.50/2.53	                "to": 77,
6.50/2.53	                "label": "ONLY EVAL with clause\nappend([], X20, X20).\nand substitutionX5 -> [],\nT5 -> T16,\nX20 -> T16,\nX6 -> T16"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 75,
6.50/2.53	                "to": 164,
6.50/2.53	                "label": "ONLY EVAL with clause\nappend(.(X71, X72), X73, .(X71, X74)) :- append(X72, X73, X74).\nand substitutionX71 -> X75,\nX72 -> X76,\nX5 -> .(X75, X76),\nT5 -> T39,\nX73 -> T39,\nX74 -> X77,\nX6 -> .(X75, X77)"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 77,
6.50/2.53	                "to": 82,
6.50/2.53	                "label": "CASE"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 82,
6.50/2.53	                "to": 84,
6.50/2.53	                "label": "PARALLEL"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 82,
6.50/2.53	                "to": 85,
6.50/2.53	                "label": "PARALLEL"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 84,
6.50/2.53	                "to": 89,
6.50/2.53	                "label": "EVAL with clause\nappend([], X33, X33).\nand substitutionT16 -> [],\nX7 -> T23,\nX33 -> T23,\nT7 -> T23,\nX34 -> T23"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 84,
6.50/2.53	                "to": 91,
6.50/2.53	                "label": "EVAL-BACKTRACK"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 85,
6.50/2.53	                "to": 99,
6.50/2.53	                "label": "EVAL with clause\nappend(.(X45, X46), X47, .(X45, X48)) :- append(X46, X47, X48).\nand substitutionX45 -> T30,\nX46 -> T31,\nT16 -> .(T30, T31),\nX7 -> X49,\nX47 -> X49,\nX48 -> T33,\nT7 -> .(T30, T33),\nT32 -> T33"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 85,
6.50/2.53	                "to": 100,
6.50/2.53	                "label": "EVAL-BACKTRACK"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 89,
6.50/2.53	                "to": 96,
6.50/2.53	                "label": "SUCCESS"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 99,
6.50/2.53	                "to": 77,
6.50/2.53	                "label": "INSTANCE with matching:\nT16 -> T31\nX7 -> X49\nT7 -> T33"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 164,
6.50/2.53	                "to": 172,
6.50/2.53	                "label": "SPLIT 1"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 164,
6.50/2.53	                "to": 173,
6.50/2.53	                "label": "SPLIT 2\nnew knowledge:\nT39 is ground\nreplacements:X76 -> T41,\nX77 -> T42"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 172,
6.50/2.53	                "to": 178,
6.50/2.53	                "label": "CASE"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 173,
6.50/2.53	                "to": 200,
6.50/2.53	                "label": "CASE"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 178,
6.50/2.53	                "to": 182,
6.50/2.53	                "label": "PARALLEL"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 178,
6.50/2.53	                "to": 183,
6.50/2.53	                "label": "PARALLEL"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 182,
6.50/2.53	                "to": 193,
6.50/2.53	                "label": "ONLY EVAL with clause\nappend([], X86, X86).\nand substitutionX76 -> [],\nT39 -> T48,\nX86 -> T48,\nX77 -> T48"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 183,
6.50/2.53	                "to": 199,
6.50/2.53	                "label": "ONLY EVAL with clause\nappend(.(X102, X103), X104, .(X102, X105)) :- append(X103, X104, X105).\nand substitutionX102 -> X106,\nX103 -> X107,\nX76 -> .(X106, X107),\nT39 -> T52,\nX104 -> T52,\nX105 -> X108,\nX77 -> .(X106, X108)"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 193,
6.50/2.53	                "to": 195,
6.50/2.53	                "label": "SUCCESS"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 199,
6.50/2.53	                "to": 172,
6.50/2.53	                "label": "INSTANCE with matching:\nX76 -> X107\nT39 -> T52\nX77 -> X108"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 200,
6.50/2.53	                "to": 201,
6.50/2.53	                "label": "BACKTRACK\nfor clause: append([], Ys, Ys)because of non-unification"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 201,
6.50/2.53	                "to": 202,
6.50/2.53	                "label": "EVAL with clause\nappend(.(X125, X126), X127, .(X125, X128)) :- append(X126, X127, X128).\nand substitutionX75 -> T61,\nX125 -> T61,\nT42 -> T63,\nX126 -> T63,\nX7 -> X130,\nX127 -> X130,\nX129 -> T61,\nX128 -> T64,\nT7 -> .(T61, T64),\nT60 -> T63,\nT62 -> T64"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 201,
6.50/2.53	                "to": 205,
6.50/2.53	                "label": "EVAL-BACKTRACK"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 202,
6.50/2.53	                "to": 206,
6.50/2.53	                "label": "CASE"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 206,
6.50/2.53	                "to": 207,
6.50/2.53	                "label": "PARALLEL"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 206,
6.50/2.53	                "to": 208,
6.50/2.53	                "label": "PARALLEL"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 207,
6.50/2.53	                "to": 209,
6.50/2.53	                "label": "EVAL with clause\nappend([], X143, X143).\nand substitutionT63 -> [],\nX130 -> T71,\nX143 -> T71,\nT64 -> T71,\nX144 -> T71"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 207,
6.50/2.53	                "to": 210,
6.50/2.53	                "label": "EVAL-BACKTRACK"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 208,
6.50/2.53	                "to": 212,
6.50/2.53	                "label": "EVAL with clause\nappend(.(X155, X156), X157, .(X155, X158)) :- append(X156, X157, X158).\nand substitutionX155 -> T78,\nX156 -> T81,\nT63 -> .(T78, T81),\nX130 -> X159,\nX157 -> X159,\nX158 -> T82,\nT64 -> .(T78, T82),\nT79 -> T81,\nT80 -> T82"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 208,
6.50/2.53	                "to": 213,
6.50/2.53	                "label": "EVAL-BACKTRACK"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 209,
6.50/2.53	                "to": 211,
6.50/2.53	                "label": "SUCCESS"
6.50/2.53	            },
6.50/2.53	            {
6.50/2.53	                "from": 212,
6.50/2.53	                "to": 202,
6.50/2.53	                "label": "INSTANCE with matching:\nT63 -> T81\nX130 -> X159\nT64 -> T82"
6.50/2.53	            }
6.50/2.53	        ],
6.50/2.53	        "type": "Graph"
6.50/2.53	    }
6.50/2.53	}
6.50/2.53	
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(58)
6.50/2.53	Obligation:
6.50/2.53	Triples:
6.50/2.53	
6.50/2.53	appendA(.(X1, X2), X3, .(X1, X4)) :- appendA(X2, X3, X4).
6.50/2.53	appendB(.(X1, X2), X3, .(X1, X4)) :- appendB(X2, X3, X4).
6.50/2.53	appendC(.(X1, X2), X3, .(X1, X4)) :- appendC(X2, X3, X4).
6.50/2.53	sublistD(X1, X2) :- appendA(X1, X3, X2).
6.50/2.53	sublistD(X1, X2) :- appendB(X3, X1, X4).
6.50/2.53	sublistD(X1, .(X2, X3)) :- ','(appendcB(X4, X1, X5), appendC(X5, X6, X3)).
6.50/2.53	
6.50/2.53	Clauses:
6.50/2.53	
6.50/2.53	appendcA([], X1, X1).
6.50/2.53	appendcA(.(X1, X2), X3, .(X1, X4)) :- appendcA(X2, X3, X4).
6.50/2.53	appendcB([], X1, X1).
6.50/2.53	appendcB(.(X1, X2), X3, .(X1, X4)) :- appendcB(X2, X3, X4).
6.50/2.53	appendcC([], X1, X1).
6.50/2.53	appendcC(.(X1, X2), X3, .(X1, X4)) :- appendcC(X2, X3, X4).
6.50/2.53	
6.50/2.53	Afs:
6.50/2.53	
6.50/2.53	sublistD(x1, x2)  =  sublistD(x1)
6.50/2.53	
6.50/2.53	
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(59) TriplesToPiDPProof (SOUND)
6.50/2.53	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.50/2.53	
6.50/2.53	sublistD_in_2: (b,f)
6.50/2.53	
6.50/2.53	appendA_in_3: (b,f,f)
6.50/2.53	
6.50/2.53	appendB_in_3: (f,b,f)
6.50/2.53	
6.50/2.53	appendcB_in_3: (f,b,f)
6.50/2.53	
6.50/2.53	appendC_in_3: (b,f,f)
6.50/2.53	
6.50/2.53	Transforming TRIPLES into the following Term Rewriting System:
6.50/2.53	
6.50/2.53	Pi DP problem:
6.50/2.53	The TRS P consists of the following rules:
6.50/2.53	
6.50/2.53	   SUBLISTD_IN_GA(X1, X2) -> U4_GA(X1, X2, appendA_in_gaa(X1, X3, X2))
6.50/2.53	   SUBLISTD_IN_GA(X1, X2) -> APPENDA_IN_GAA(X1, X3, X2)
6.50/2.53	   APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> U1_GAA(X1, X2, X3, X4, appendA_in_gaa(X2, X3, X4))
6.50/2.53	   APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPENDA_IN_GAA(X2, X3, X4)
6.50/2.53	   SUBLISTD_IN_GA(X1, X2) -> U5_GA(X1, X2, appendB_in_aga(X3, X1, X4))
6.50/2.53	   SUBLISTD_IN_GA(X1, X2) -> APPENDB_IN_AGA(X3, X1, X4)
6.50/2.53	   APPENDB_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> U2_AGA(X1, X2, X3, X4, appendB_in_aga(X2, X3, X4))
6.50/2.53	   APPENDB_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPENDB_IN_AGA(X2, X3, X4)
6.50/2.53	   SUBLISTD_IN_GA(X1, .(X2, X3)) -> U6_GA(X1, X2, X3, appendcB_in_aga(X4, X1, X5))
6.50/2.53	   U6_GA(X1, X2, X3, appendcB_out_aga(X4, X1, X5)) -> U7_GA(X1, X2, X3, appendC_in_gaa(X5, X6, X3))
6.50/2.53	   U6_GA(X1, X2, X3, appendcB_out_aga(X4, X1, X5)) -> APPENDC_IN_GAA(X5, X6, X3)
6.50/2.53	   APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> U3_GAA(X1, X2, X3, X4, appendC_in_gaa(X2, X3, X4))
6.50/2.53	   APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPENDC_IN_GAA(X2, X3, X4)
6.50/2.53	
6.50/2.53	The TRS R consists of the following rules:
6.50/2.53	
6.50/2.53	   appendcB_in_aga([], X1, X1) -> appendcB_out_aga([], X1, X1)
6.50/2.53	   appendcB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, appendcB_in_aga(X2, X3, X4))
6.50/2.53	   U10_aga(X1, X2, X3, X4, appendcB_out_aga(X2, X3, X4)) -> appendcB_out_aga(.(X1, X2), X3, .(X1, X4))
6.50/2.53	
6.50/2.53	The argument filtering Pi contains the following mapping:
6.50/2.53	appendA_in_gaa(x1, x2, x3)  =  appendA_in_gaa(x1)
6.50/2.53	
6.50/2.53	.(x1, x2)  =  .(x2)
6.50/2.53	
6.50/2.53	appendB_in_aga(x1, x2, x3)  =  appendB_in_aga(x2)
6.50/2.53	
6.50/2.53	appendcB_in_aga(x1, x2, x3)  =  appendcB_in_aga(x2)
6.50/2.53	
6.50/2.53	appendcB_out_aga(x1, x2, x3)  =  appendcB_out_aga(x1, x2, x3)
6.50/2.53	
6.50/2.53	U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
6.50/2.53	
6.50/2.53	appendC_in_gaa(x1, x2, x3)  =  appendC_in_gaa(x1)
6.50/2.53	
6.50/2.53	SUBLISTD_IN_GA(x1, x2)  =  SUBLISTD_IN_GA(x1)
6.50/2.53	
6.50/2.53	U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
6.50/2.53	
6.50/2.53	APPENDA_IN_GAA(x1, x2, x3)  =  APPENDA_IN_GAA(x1)
6.50/2.53	
6.50/2.53	U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x2, x5)
6.50/2.53	
6.50/2.53	U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
6.50/2.53	
6.50/2.53	APPENDB_IN_AGA(x1, x2, x3)  =  APPENDB_IN_AGA(x2)
6.50/2.53	
6.50/2.53	U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
6.50/2.53	
6.50/2.53	U6_GA(x1, x2, x3, x4)  =  U6_GA(x1, x4)
6.50/2.53	
6.50/2.53	U7_GA(x1, x2, x3, x4)  =  U7_GA(x1, x4)
6.50/2.53	
6.50/2.53	APPENDC_IN_GAA(x1, x2, x3)  =  APPENDC_IN_GAA(x1)
6.50/2.53	
6.50/2.53	U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x2, x5)
6.50/2.53	
6.50/2.53	
6.50/2.53	We have to consider all (P,R,Pi)-chains
6.50/2.53	
6.50/2.53	
6.50/2.53	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
6.50/2.53	
6.50/2.53	
6.50/2.53	
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(60)
6.50/2.53	Obligation:
6.50/2.53	Pi DP problem:
6.50/2.53	The TRS P consists of the following rules:
6.50/2.53	
6.50/2.53	   SUBLISTD_IN_GA(X1, X2) -> U4_GA(X1, X2, appendA_in_gaa(X1, X3, X2))
6.50/2.53	   SUBLISTD_IN_GA(X1, X2) -> APPENDA_IN_GAA(X1, X3, X2)
6.50/2.53	   APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> U1_GAA(X1, X2, X3, X4, appendA_in_gaa(X2, X3, X4))
6.50/2.53	   APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPENDA_IN_GAA(X2, X3, X4)
6.50/2.53	   SUBLISTD_IN_GA(X1, X2) -> U5_GA(X1, X2, appendB_in_aga(X3, X1, X4))
6.50/2.53	   SUBLISTD_IN_GA(X1, X2) -> APPENDB_IN_AGA(X3, X1, X4)
6.50/2.53	   APPENDB_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> U2_AGA(X1, X2, X3, X4, appendB_in_aga(X2, X3, X4))
6.50/2.53	   APPENDB_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPENDB_IN_AGA(X2, X3, X4)
6.50/2.53	   SUBLISTD_IN_GA(X1, .(X2, X3)) -> U6_GA(X1, X2, X3, appendcB_in_aga(X4, X1, X5))
6.50/2.53	   U6_GA(X1, X2, X3, appendcB_out_aga(X4, X1, X5)) -> U7_GA(X1, X2, X3, appendC_in_gaa(X5, X6, X3))
6.50/2.53	   U6_GA(X1, X2, X3, appendcB_out_aga(X4, X1, X5)) -> APPENDC_IN_GAA(X5, X6, X3)
6.50/2.53	   APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> U3_GAA(X1, X2, X3, X4, appendC_in_gaa(X2, X3, X4))
6.50/2.53	   APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPENDC_IN_GAA(X2, X3, X4)
6.50/2.53	
6.50/2.53	The TRS R consists of the following rules:
6.50/2.53	
6.50/2.53	   appendcB_in_aga([], X1, X1) -> appendcB_out_aga([], X1, X1)
6.50/2.53	   appendcB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, appendcB_in_aga(X2, X3, X4))
6.50/2.53	   U10_aga(X1, X2, X3, X4, appendcB_out_aga(X2, X3, X4)) -> appendcB_out_aga(.(X1, X2), X3, .(X1, X4))
6.50/2.53	
6.50/2.53	The argument filtering Pi contains the following mapping:
6.50/2.53	appendA_in_gaa(x1, x2, x3)  =  appendA_in_gaa(x1)
6.50/2.53	
6.50/2.53	.(x1, x2)  =  .(x2)
6.50/2.53	
6.50/2.53	appendB_in_aga(x1, x2, x3)  =  appendB_in_aga(x2)
6.50/2.53	
6.50/2.53	appendcB_in_aga(x1, x2, x3)  =  appendcB_in_aga(x2)
6.50/2.53	
6.50/2.53	appendcB_out_aga(x1, x2, x3)  =  appendcB_out_aga(x1, x2, x3)
6.50/2.53	
6.50/2.53	U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
6.50/2.53	
6.50/2.53	appendC_in_gaa(x1, x2, x3)  =  appendC_in_gaa(x1)
6.50/2.53	
6.50/2.53	SUBLISTD_IN_GA(x1, x2)  =  SUBLISTD_IN_GA(x1)
6.50/2.53	
6.50/2.53	U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
6.50/2.53	
6.50/2.53	APPENDA_IN_GAA(x1, x2, x3)  =  APPENDA_IN_GAA(x1)
6.50/2.53	
6.50/2.53	U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x2, x5)
6.50/2.53	
6.50/2.53	U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
6.50/2.53	
6.50/2.53	APPENDB_IN_AGA(x1, x2, x3)  =  APPENDB_IN_AGA(x2)
6.50/2.53	
6.50/2.53	U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
6.50/2.53	
6.50/2.53	U6_GA(x1, x2, x3, x4)  =  U6_GA(x1, x4)
6.50/2.53	
6.50/2.53	U7_GA(x1, x2, x3, x4)  =  U7_GA(x1, x4)
6.50/2.53	
6.50/2.53	APPENDC_IN_GAA(x1, x2, x3)  =  APPENDC_IN_GAA(x1)
6.50/2.53	
6.50/2.53	U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x2, x5)
6.50/2.53	
6.50/2.53	
6.50/2.53	We have to consider all (P,R,Pi)-chains
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(61) DependencyGraphProof (EQUIVALENT)
6.50/2.53	The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 10 less nodes.
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(62)
6.50/2.53	Complex Obligation (AND)
6.50/2.53	
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(63)
6.50/2.53	Obligation:
6.50/2.53	Pi DP problem:
6.50/2.53	The TRS P consists of the following rules:
6.50/2.53	
6.50/2.53	   APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPENDC_IN_GAA(X2, X3, X4)
6.50/2.53	
6.50/2.53	The TRS R consists of the following rules:
6.50/2.53	
6.50/2.53	   appendcB_in_aga([], X1, X1) -> appendcB_out_aga([], X1, X1)
6.50/2.53	   appendcB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, appendcB_in_aga(X2, X3, X4))
6.50/2.53	   U10_aga(X1, X2, X3, X4, appendcB_out_aga(X2, X3, X4)) -> appendcB_out_aga(.(X1, X2), X3, .(X1, X4))
6.50/2.53	
6.50/2.53	The argument filtering Pi contains the following mapping:
6.50/2.53	.(x1, x2)  =  .(x2)
6.50/2.53	
6.50/2.53	appendcB_in_aga(x1, x2, x3)  =  appendcB_in_aga(x2)
6.50/2.53	
6.50/2.53	appendcB_out_aga(x1, x2, x3)  =  appendcB_out_aga(x1, x2, x3)
6.50/2.53	
6.50/2.53	U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
6.50/2.53	
6.50/2.53	APPENDC_IN_GAA(x1, x2, x3)  =  APPENDC_IN_GAA(x1)
6.50/2.53	
6.50/2.53	
6.50/2.53	We have to consider all (P,R,Pi)-chains
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(64) UsableRulesProof (EQUIVALENT)
6.50/2.53	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(65)
6.50/2.53	Obligation:
6.50/2.53	Pi DP problem:
6.50/2.53	The TRS P consists of the following rules:
6.50/2.53	
6.50/2.53	   APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPENDC_IN_GAA(X2, X3, X4)
6.50/2.53	
6.50/2.53	R is empty.
6.50/2.53	The argument filtering Pi contains the following mapping:
6.50/2.53	.(x1, x2)  =  .(x2)
6.50/2.53	
6.50/2.53	APPENDC_IN_GAA(x1, x2, x3)  =  APPENDC_IN_GAA(x1)
6.50/2.53	
6.50/2.53	
6.50/2.53	We have to consider all (P,R,Pi)-chains
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(66) PiDPToQDPProof (SOUND)
6.50/2.53	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(67)
6.50/2.53	Obligation:
6.50/2.53	Q DP problem:
6.50/2.53	The TRS P consists of the following rules:
6.50/2.53	
6.50/2.53	   APPENDC_IN_GAA(.(X2)) -> APPENDC_IN_GAA(X2)
6.50/2.53	
6.50/2.53	R is empty.
6.50/2.53	Q is empty.
6.50/2.53	We have to consider all (P,Q,R)-chains.
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(68) QDPSizeChangeProof (EQUIVALENT)
6.50/2.53	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. 
6.50/2.53	
6.50/2.53	From the DPs we obtained the following set of size-change graphs:
6.50/2.53	*APPENDC_IN_GAA(.(X2)) -> APPENDC_IN_GAA(X2)
6.50/2.53	The graph contains the following edges 1 > 1
6.50/2.53	
6.50/2.53	
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(69)
6.50/2.53	YES
6.50/2.53	
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(70)
6.50/2.53	Obligation:
6.50/2.53	Pi DP problem:
6.50/2.53	The TRS P consists of the following rules:
6.50/2.53	
6.50/2.53	   APPENDB_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPENDB_IN_AGA(X2, X3, X4)
6.50/2.53	
6.50/2.53	The TRS R consists of the following rules:
6.50/2.53	
6.50/2.53	   appendcB_in_aga([], X1, X1) -> appendcB_out_aga([], X1, X1)
6.50/2.53	   appendcB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, appendcB_in_aga(X2, X3, X4))
6.50/2.53	   U10_aga(X1, X2, X3, X4, appendcB_out_aga(X2, X3, X4)) -> appendcB_out_aga(.(X1, X2), X3, .(X1, X4))
6.50/2.53	
6.50/2.53	The argument filtering Pi contains the following mapping:
6.50/2.53	.(x1, x2)  =  .(x2)
6.50/2.53	
6.50/2.53	appendcB_in_aga(x1, x2, x3)  =  appendcB_in_aga(x2)
6.50/2.53	
6.50/2.53	appendcB_out_aga(x1, x2, x3)  =  appendcB_out_aga(x1, x2, x3)
6.50/2.53	
6.50/2.53	U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
6.50/2.53	
6.50/2.53	APPENDB_IN_AGA(x1, x2, x3)  =  APPENDB_IN_AGA(x2)
6.50/2.53	
6.50/2.53	
6.50/2.53	We have to consider all (P,R,Pi)-chains
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(71) UsableRulesProof (EQUIVALENT)
6.50/2.53	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(72)
6.50/2.53	Obligation:
6.50/2.53	Pi DP problem:
6.50/2.53	The TRS P consists of the following rules:
6.50/2.53	
6.50/2.53	   APPENDB_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPENDB_IN_AGA(X2, X3, X4)
6.50/2.53	
6.50/2.53	R is empty.
6.50/2.53	The argument filtering Pi contains the following mapping:
6.50/2.53	.(x1, x2)  =  .(x2)
6.50/2.53	
6.50/2.53	APPENDB_IN_AGA(x1, x2, x3)  =  APPENDB_IN_AGA(x2)
6.50/2.53	
6.50/2.53	
6.50/2.53	We have to consider all (P,R,Pi)-chains
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(73) PiDPToQDPProof (SOUND)
6.50/2.53	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(74)
6.50/2.53	Obligation:
6.50/2.53	Q DP problem:
6.50/2.53	The TRS P consists of the following rules:
6.50/2.53	
6.50/2.53	   APPENDB_IN_AGA(X3) -> APPENDB_IN_AGA(X3)
6.50/2.53	
6.50/2.53	R is empty.
6.50/2.53	Q is empty.
6.50/2.53	We have to consider all (P,Q,R)-chains.
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(75)
6.50/2.53	Obligation:
6.50/2.53	Pi DP problem:
6.50/2.53	The TRS P consists of the following rules:
6.50/2.53	
6.50/2.53	   APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPENDA_IN_GAA(X2, X3, X4)
6.50/2.53	
6.50/2.53	The TRS R consists of the following rules:
6.50/2.53	
6.50/2.53	   appendcB_in_aga([], X1, X1) -> appendcB_out_aga([], X1, X1)
6.50/2.53	   appendcB_in_aga(.(X1, X2), X3, .(X1, X4)) -> U10_aga(X1, X2, X3, X4, appendcB_in_aga(X2, X3, X4))
6.50/2.53	   U10_aga(X1, X2, X3, X4, appendcB_out_aga(X2, X3, X4)) -> appendcB_out_aga(.(X1, X2), X3, .(X1, X4))
6.50/2.53	
6.50/2.53	The argument filtering Pi contains the following mapping:
6.50/2.53	.(x1, x2)  =  .(x2)
6.50/2.53	
6.50/2.53	appendcB_in_aga(x1, x2, x3)  =  appendcB_in_aga(x2)
6.50/2.53	
6.50/2.53	appendcB_out_aga(x1, x2, x3)  =  appendcB_out_aga(x1, x2, x3)
6.50/2.53	
6.50/2.53	U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
6.50/2.53	
6.50/2.53	APPENDA_IN_GAA(x1, x2, x3)  =  APPENDA_IN_GAA(x1)
6.50/2.53	
6.50/2.53	
6.50/2.53	We have to consider all (P,R,Pi)-chains
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(76) UsableRulesProof (EQUIVALENT)
6.50/2.53	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(77)
6.50/2.53	Obligation:
6.50/2.53	Pi DP problem:
6.50/2.53	The TRS P consists of the following rules:
6.50/2.53	
6.50/2.53	   APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) -> APPENDA_IN_GAA(X2, X3, X4)
6.50/2.53	
6.50/2.53	R is empty.
6.50/2.53	The argument filtering Pi contains the following mapping:
6.50/2.53	.(x1, x2)  =  .(x2)
6.50/2.53	
6.50/2.53	APPENDA_IN_GAA(x1, x2, x3)  =  APPENDA_IN_GAA(x1)
6.50/2.53	
6.50/2.53	
6.50/2.53	We have to consider all (P,R,Pi)-chains
6.50/2.53	----------------------------------------
6.50/2.53	
6.50/2.53	(78) PrologToIRSwTTransformerProof (SOUND)
6.50/2.53	Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert
6.50/2.53	
6.50/2.53	{
6.50/2.53	    "root": 1,
6.50/2.53	    "program": {
6.50/2.53	        "directives": [],
6.50/2.53	        "clauses": [
6.50/2.53	            [
6.50/2.53	                "(sublist X Y)",
6.50/2.53	                "(',' (append U X V) (append V W Y))"
6.50/2.53	            ],
6.50/2.53	            [
6.50/2.53	                "(append ([]) Ys Ys)",
6.50/2.53	                null
6.50/2.53	            ],
6.50/2.53	            [
6.50/2.53	                "(append (. X Xs) Ys (. X Zs))",
6.50/2.53	                "(append Xs Ys Zs)"
6.50/2.53	            ]
6.50/2.53	        ]
6.50/2.53	    },
6.50/2.53	    "graph": {
6.50/2.53	        "nodes": {
6.50/2.53	            "88": {
6.50/2.53	                "goal": [
6.50/2.53	                    {
6.50/2.53	                        "clause": 1,
6.50/2.53	                        "scope": 2,
6.50/2.53	                        "term": "(append X13 T10 X14)"
6.50/2.53	                    },
6.50/2.53	                    {
6.50/2.53	                        "clause": 2,
6.50/2.53	                        "scope": 2,
6.50/2.53	                        "term": "(append X13 T10 X14)"
6.50/2.53	                    }
6.50/2.53	                ],
6.50/2.53	                "kb": {
6.50/2.53	                    "nonunifying": [],
6.50/2.53	                    "intvars": {},
6.50/2.53	                    "arithmetic": {
6.50/2.53	                        "type": "PlainIntegerRelationState",
6.50/2.53	                        "relations": []
6.50/2.53	                    },
6.50/2.53	                    "ground": ["T10"],
6.50/2.53	                    "free": [
6.50/2.53	                        "X13",
6.50/2.53	                        "X14"
6.50/2.53	                    ],
6.50/2.53	                    "exprvars": []
6.50/2.53	                }
6.50/2.53	            },
6.50/2.53	            "78": {
6.50/2.53	                "goal": [{
6.50/2.53	                    "clause": -1,
6.50/2.53	                    "scope": -1,
6.50/2.53	                    "term": "(append X13 T10 X14)"
6.50/2.53	                }],
6.50/2.53	                "kb": {
6.50/2.53	                    "nonunifying": [],
6.50/2.53	                    "intvars": {},
6.50/2.53	                    "arithmetic": {
6.50/2.53	                        "type": "PlainIntegerRelationState",
6.50/2.53	                        "relations": []
6.50/2.53	                    },
6.50/2.53	                    "ground": ["T10"],
6.50/2.53	                    "free": [
6.50/2.53	                        "X13",
6.50/2.53	                        "X14"
6.50/2.53	                    ],
6.50/2.53	                    "exprvars": []
6.50/2.53	                }
6.50/2.53	            },
6.50/2.53	            "79": {
6.50/2.53	                "goal": [{
6.50/2.53	                    "clause": -1,
6.50/2.53	                    "scope": -1,
6.50/2.53	                    "term": "(append T16 X15 T12)"
6.50/2.53	                }],
6.50/2.53	                "kb": {
6.50/2.53	                    "nonunifying": [],
6.50/2.53	                    "intvars": {},
6.50/2.53	                    "arithmetic": {
6.50/2.53	                        "type": "PlainIntegerRelationState",
6.50/2.53	                        "relations": []
6.50/2.53	                    },
6.50/2.53	                    "ground": [],
6.50/2.53	                    "free": ["X15"],
6.50/2.53	                    "exprvars": []
6.50/2.53	                }
6.50/2.53	            },
6.50/2.53	            "160": {
6.50/2.53	                "goal": [{
6.50/2.53	                    "clause": 1,
6.50/2.53	                    "scope": 3,
6.50/2.53	                    "term": "(append T16 X15 T12)"
6.50/2.53	                }],
6.50/2.53	                "kb": {
6.50/2.53	                    "nonunifying": [],
6.50/2.53	                    "intvars": {},
6.50/2.53	                    "arithmetic": {
6.50/2.53	                        "type": "PlainIntegerRelationState",
6.50/2.53	                        "relations": []
6.50/2.53	                    },
6.50/2.53	                    "ground": [],
6.50/2.53	                    "free": ["X15"],
6.50/2.53	                    "exprvars": []
6.50/2.53	                }
6.50/2.53	            },
6.50/2.53	            "type": "Nodes",
6.50/2.53	            "150": {
6.50/2.53	                "goal": [{
6.50/2.53	                    "clause": -1,
6.50/2.53	                    "scope": -1,
6.50/2.53	                    "term": "(append X45 T26 X46)"
6.50/2.53	                }],
6.50/2.53	                "kb": {
6.50/2.53	                    "nonunifying": [],
6.50/2.53	                    "intvars": {},
6.50/2.53	                    "arithmetic": {
6.50/2.53	                        "type": "PlainIntegerRelationState",
6.50/2.53	                        "relations": []
6.50/2.53	                    },
6.50/2.53	                    "ground": ["T26"],
6.50/2.53	                    "free": [
6.50/2.53	                        "X45",
6.50/2.53	                        "X46"
6.50/2.53	                    ],
6.50/2.53	                    "exprvars": []
6.50/2.53	                }
6.50/2.53	            },
6.50/2.53	            "162": {
6.50/2.53	                "goal": [{
6.50/2.53	                    "clause": 2,
6.50/2.53	                    "scope": 3,
6.50/2.53	                    "term": "(append T16 X15 T12)"
6.50/2.53	                }],
6.50/2.53	                "kb": {
6.50/2.53	                    "nonunifying": [],
6.50/2.53	                    "intvars": {},
6.50/2.53	                    "arithmetic": {
6.50/2.53	                        "type": "PlainIntegerRelationState",
6.50/2.53	                        "relations": []
6.50/2.53	                    },
6.50/2.53	                    "ground": [],
6.50/2.53	                    "free": ["X15"],
6.50/2.53	                    "exprvars": []
6.50/2.53	                }
6.50/2.53	            },
6.50/2.53	            "1": {
6.50/2.53	                "goal": [{
6.50/2.53	                    "clause": -1,
6.50/2.53	                    "scope": -1,
6.50/2.53	                    "term": "(sublist T1 T2)"
6.50/2.53	                }],
6.50/2.53	                "kb": {
6.50/2.53	                    "nonunifying": [],
6.50/2.53	                    "intvars": {},
6.50/2.53	                    "arithmetic": {
6.50/2.53	                        "type": "PlainIntegerRelationState",
6.50/2.53	                        "relations": []
6.50/2.53	                    },
6.50/2.53	                    "ground": ["T1"],
6.50/2.53	                    "free": [],
6.50/2.53	                    "exprvars": []
6.50/2.53	                }
6.50/2.53	            },
6.50/2.53	            "166": {
6.50/2.53	                "goal": [{
6.50/2.53	                    "clause": -1,
6.50/2.53	                    "scope": -1,
6.50/2.53	                    "term": "(true)"
6.50/2.53	                }],
6.50/2.53	                "kb": {
6.50/2.53	                    "nonunifying": [],
6.50/2.53	                    "intvars": {},
6.50/2.53	                    "arithmetic": {
6.50/2.53	                        "type": "PlainIntegerRelationState",
6.50/2.53	                        "relations": []
6.50/2.53	                    },
6.50/2.53	                    "ground": [],
6.50/2.53	                    "free": [],
6.50/2.53	                    "exprvars": []
6.50/2.53	                }
6.50/2.53	            },
6.50/2.53	            "167": {
6.50/2.53	                "goal": [],
6.50/2.53	                "kb": {
6.50/2.53	                    "nonunifying": [],
6.50/2.53	                    "intvars": {},
6.50/2.53	                    "arithmetic": {
6.50/2.53	                        "type": "PlainIntegerRelationState",
6.50/2.53	                        "relations": []
6.50/2.53	                    },
6.50/2.53	                    "ground": [],
6.50/2.53	                    "free": [],
6.50/2.53	                    "exprvars": []
6.50/2.53	                }
6.50/2.53	            },
6.50/2.53	            "168": {
6.50/2.53	                "goal": [],
6.50/2.53	                "kb": {
6.50/2.53	                    "nonunifying": [],
6.50/2.53	                    "intvars": {},
6.50/2.53	                    "arithmetic": {
6.50/2.53	                        "type": "PlainIntegerRelationState",
6.50/2.53	                        "relations": []
6.50/2.53	                    },
6.50/2.53	                    "ground": [],
6.50/2.53	                    "free": [],
6.50/2.53	                    "exprvars": []
6.50/2.53	                }
6.50/2.53	            },
6.50/2.53	            "159": {
6.50/2.53	                "goal": [
6.50/2.53	                    {
6.50/2.53	                        "clause": 1,
6.50/2.53	                        "scope": 3,
6.50/2.53	                        "term": "(append T16 X15 T12)"
6.50/2.53	                    },
6.50/2.53	                    {
6.50/2.53	                        "clause": 2,
6.50/2.53	                        "scope": 3,
6.50/2.53	                        "term": "(append T16 X15 T12)"
6.50/2.53	                    }
6.50/2.53	                ],
6.50/2.53	                "kb": {
6.50/2.53	                    "nonunifying": [],
6.50/2.53	                    "intvars": {},
6.50/2.53	                    "arithmetic": {
6.50/2.53	                        "type": "PlainIntegerRelationState",
6.50/2.53	                        "relations": []
6.50/2.53	                    },
6.50/2.53	                    "ground": [],
6.50/2.53	                    "free": ["X15"],
6.50/2.53	                    "exprvars": []
6.50/2.53	                }
6.50/2.53	            },
6.50/2.53	            "203": {
6.50/2.53	                "goal": [{
6.50/2.53	                    "clause": -1,
6.50/2.53	                    "scope": -1,
6.50/2.53	                    "term": "(append T44 X76 T45)"
6.50/2.53	                }],
6.50/2.53	                "kb": {
6.50/2.53	                    "nonunifying": [],
6.50/2.53	                    "intvars": {},
6.50/2.53	                    "arithmetic": {
6.50/2.53	                        "type": "PlainIntegerRelationState",
6.50/2.53	                        "relations": []
6.50/2.53	                    },
6.50/2.53	                    "ground": [],
6.50/2.53	                    "free": ["X76"],
6.50/2.53	                    "exprvars": []
6.50/2.53	                }
6.50/2.53	            },
6.50/2.53	            "6": {
6.50/2.53	                "goal": [{
6.50/2.53	                    "clause": 0,
6.50/2.53	                    "scope": 1,
6.50/2.53	                    "term": "(sublist T1 T2)"
6.50/2.53	                }],
6.50/2.53	                "kb": {
6.50/2.53	                    "nonunifying": [],
6.50/2.53	                    "intvars": {},
6.50/2.53	                    "arithmetic": {
6.50/2.53	                        "type": "PlainIntegerRelationState",
6.50/2.53	                        "relations": []
6.50/2.53	                    },
6.50/2.53	                    "ground": ["T1"],
6.50/2.53	                    "free": [],
6.50/2.53	                    "exprvars": []
6.50/2.53	                }
6.50/2.53	            },
6.50/2.53	            "204": {
6.50/2.53	                "goal": [],
6.50/2.53	                "kb": {
6.50/2.53	                    "nonunifying": [],
6.50/2.53	                    "intvars": {},
6.50/2.53	                    "arithmetic": {
6.50/2.53	                        "type": "PlainIntegerRelationState",
6.50/2.53	                        "relations": []
6.50/2.53	                    },
6.50/2.53	                    "ground": [],
6.50/2.53	                    "free": [],
6.50/2.53	                    "exprvars": []
6.50/2.53	                }
6.50/2.53	            },
6.50/2.53	            "94": {
6.50/2.53	                "goal": [{
6.50/2.53	                    "clause": 1,
6.50/2.53	                    "scope": 2,
6.50/2.53	                    "term": "(append X13 T10 X14)"
6.50/2.53	                }],
6.50/2.53	                "kb": {
6.50/2.53	                    "nonunifying": [],
6.50/2.53	                    "intvars": {},
6.50/2.53	                    "arithmetic": {
6.50/2.53	                        "type": "PlainIntegerRelationState",
6.50/2.53	                        "relations": []
6.50/2.53	                    },
6.50/2.53	                    "ground": ["T10"],
6.50/2.53	                    "free": [
6.50/2.53	                        "X13",
6.50/2.53	                        "X14"
6.50/2.53	                    ],
6.50/2.53	                    "exprvars": []
6.50/2.53	                }
6.50/2.53	            },
6.50/2.53	            "95": {
6.50/2.53	                "goal": [{
6.50/2.53	                    "clause": 2,
6.50/2.53	                    "scope": 2,
6.50/2.53	                    "term": "(append X13 T10 X14)"
6.50/2.53	                }],
6.50/2.53	                "kb": {
6.50/2.53	                    "nonunifying": [],
6.50/2.53	                    "intvars": {},
6.50/2.53	                    "arithmetic": {
6.50/2.53	                        "type": "PlainIntegerRelationState",
6.50/2.53	                        "relations": []
6.50/2.53	                    },
6.50/2.53	                    "ground": ["T10"],
6.50/2.53	                    "free": [
6.50/2.53	                        "X13",
6.50/2.53	                        "X14"
6.50/2.53	                    ],
6.50/2.53	                    "exprvars": []
6.50/2.53	                }
6.50/2.53	            },
6.50/2.53	            "53": {
6.50/2.53	                "goal": [{
6.50/2.53	                    "clause": -1,
6.50/2.53	                    "scope": -1,
6.50/2.53	                    "term": "(',' (append X13 T10 X14) (append X14 X15 T12))"
6.50/2.53	                }],
6.50/2.54	                "kb": {
6.50/2.54	                    "nonunifying": [],
6.50/2.54	                    "intvars": {},
6.50/2.54	                    "arithmetic": {
6.50/2.54	                        "type": "PlainIntegerRelationState",
6.50/2.54	                        "relations": []
6.50/2.54	                    },
6.50/2.54	                    "ground": ["T10"],
6.50/2.54	                    "free": [
6.50/2.54	                        "X13",
6.50/2.54	                        "X14",
6.50/2.54	                        "X15"
6.50/2.54	                    ],
6.50/2.54	                    "exprvars": []
6.50/2.54	                }
6.50/2.54	            },
6.50/2.54	            "97": {
6.50/2.54	                "goal": [{
6.50/2.54	                    "clause": -1,
6.50/2.54	                    "scope": -1,
6.50/2.54	                    "term": "(true)"
6.50/2.54	                }],
6.50/2.54	                "kb": {
6.50/2.54	                    "nonunifying": [],
6.50/2.54	                    "intvars": {},
6.50/2.54	                    "arithmetic": {
6.50/2.54	                        "type": "PlainIntegerRelationState",
6.50/2.54	                        "relations": []
6.50/2.54	                    },
6.50/2.54	                    "ground": [],
6.50/2.54	                    "free": [],
6.50/2.54	                    "exprvars": []
6.50/2.54	                }
6.50/2.54	            },
6.50/2.54	            "98": {
6.50/2.54	                "goal": [],
6.50/2.54	                "kb": {
6.50/2.54	                    "nonunifying": [],
6.50/2.54	                    "intvars": {},
6.50/2.54	                    "arithmetic": {
6.50/2.54	                        "type": "PlainIntegerRelationState",
6.50/2.54	                        "relations": []
6.50/2.54	                    },
6.50/2.54	                    "ground": [],
6.50/2.54	                    "free": [],
6.50/2.54	                    "exprvars": []
6.50/2.54	                }
6.50/2.54	            }
6.50/2.54	        },
6.50/2.54	        "edges": [
6.50/2.54	            {
6.50/2.54	                "from": 1,
6.50/2.54	                "to": 6,
6.50/2.54	                "label": "CASE"
6.50/2.54	            },
6.50/2.54	            {
6.50/2.54	                "from": 6,
6.50/2.54	                "to": 53,
6.50/2.54	                "label": "ONLY EVAL with clause\nsublist(X11, X12) :- ','(append(X13, X11, X14), append(X14, X15, X12)).\nand substitutionT1 -> T10,\nX11 -> T10,\nT2 -> T12,\nX12 -> T12,\nT11 -> T12"
6.50/2.54	            },
6.50/2.54	            {
6.50/2.54	                "from": 53,
6.50/2.54	                "to": 78,
6.50/2.54	                "label": "SPLIT 1"
6.50/2.54	            },
6.50/2.54	            {
6.50/2.54	                "from": 53,
6.50/2.54	                "to": 79,
6.50/2.54	                "label": "SPLIT 2\nnew knowledge:\nT10 is ground\nreplacements:X13 -> T15,\nX14 -> T16"
6.50/2.54	            },
6.50/2.54	            {
6.50/2.54	                "from": 78,
6.50/2.54	                "to": 88,
6.50/2.54	                "label": "CASE"
6.50/2.54	            },
6.50/2.54	            {
6.50/2.54	                "from": 79,
6.50/2.54	                "to": 159,
6.50/2.54	                "label": "CASE"
6.50/2.54	            },
6.50/2.54	            {
6.50/2.54	                "from": 88,
6.50/2.54	                "to": 94,
6.50/2.54	                "label": "PARALLEL"
6.50/2.54	            },
6.50/2.54	            {
6.50/2.54	                "from": 88,
6.50/2.54	                "to": 95,
6.50/2.54	                "label": "PARALLEL"
6.50/2.54	            },
6.50/2.54	            {
6.50/2.54	                "from": 94,
6.50/2.54	                "to": 97,
6.50/2.54	                "label": "ONLY EVAL with clause\nappend([], X24, X24).\nand substitutionX13 -> [],\nT10 -> T22,\nX24 -> T22,\nX14 -> T22"
6.50/2.54	            },
6.50/2.54	            {
6.50/2.54	                "from": 95,
6.50/2.54	                "to": 150,
6.50/2.54	                "label": "ONLY EVAL with clause\nappend(.(X40, X41), X42, .(X40, X43)) :- append(X41, X42, X43).\nand substitutionX40 -> X44,\nX41 -> X45,\nX13 -> .(X44, X45),\nT10 -> T26,\nX42 -> T26,\nX43 -> X46,\nX14 -> .(X44, X46)"
6.50/2.54	            },
6.50/2.54	            {
6.50/2.54	                "from": 97,
6.50/2.54	                "to": 98,
6.50/2.54	                "label": "SUCCESS"
6.50/2.54	            },
6.50/2.54	            {
6.50/2.54	                "from": 150,
6.50/2.54	                "to": 78,
6.50/2.54	                "label": "INSTANCE with matching:\nX13 -> X45\nT10 -> T26\nX14 -> X46"
6.50/2.54	            },
6.50/2.54	            {
6.50/2.54	                "from": 159,
6.50/2.54	                "to": 160,
6.50/2.54	                "label": "PARALLEL"
6.50/2.54	            },
6.50/2.54	            {
6.50/2.54	                "from": 159,
6.50/2.54	                "to": 162,
6.50/2.54	                "label": "PARALLEL"
6.50/2.54	            },
6.50/2.54	            {
6.50/2.54	                "from": 160,
6.50/2.54	                "to": 166,
6.50/2.54	                "label": "EVAL with clause\nappend([], X60, X60).\nand substitutionT16 -> [],\nX15 -> T34,\nX60 -> T34,\nT12 -> T34,\nX61 -> T34"
6.50/2.54	            },
6.50/2.54	            {
6.50/2.54	                "from": 160,
6.50/2.54	                "to": 167,
6.50/2.54	                "label": "EVAL-BACKTRACK"
6.50/2.54	            },
6.50/2.54	            {
6.50/2.54	                "from": 162,
6.50/2.54	                "to": 203,
6.50/2.54	                "label": "EVAL with clause\nappend(.(X72, X73), X74, .(X72, X75)) :- append(X73, X74, X75).\nand substitutionX72 -> T41,\nX73 -> T44,\nT16 -> .(T41, T44),\nX15 -> X76,\nX74 -> X76,\nX75 -> T45,\nT12 -> .(T41, T45),\nT42 -> T44,\nT43 -> T45"
6.50/2.54	            },
6.50/2.54	            {
6.50/2.54	                "from": 162,
6.50/2.54	                "to": 204,
6.50/2.54	                "label": "EVAL-BACKTRACK"
6.50/2.54	            },
6.50/2.54	            {
6.50/2.54	                "from": 166,
6.50/2.54	                "to": 168,
6.50/2.54	                "label": "SUCCESS"
6.50/2.54	            },
6.50/2.54	            {
6.50/2.54	                "from": 203,
6.50/2.54	                "to": 79,
6.50/2.54	                "label": "INSTANCE with matching:\nT16 -> T44\nX15 -> X76\nT12 -> T45"
6.50/2.54	            }
6.50/2.54	        ],
6.50/2.54	        "type": "Graph"
6.50/2.54	    }
6.50/2.54	}
6.50/2.54	
6.50/2.54	----------------------------------------
6.50/2.54	
6.50/2.54	(79)
6.50/2.54	Complex Obligation (AND)
6.50/2.54	
6.50/2.54	----------------------------------------
6.50/2.54	
6.50/2.54	(80)
6.50/2.54	Obligation:
6.50/2.54	Rules:
6.50/2.54	f159_out -> f79_out :|: TRUE
6.50/2.54	f79_in -> f159_in :|: TRUE
6.50/2.54	f159_in -> f160_in :|: TRUE
6.50/2.54	f159_in -> f162_in :|: TRUE
6.50/2.54	f160_out -> f159_out :|: TRUE
6.50/2.54	f162_out -> f159_out :|: TRUE
6.50/2.54	f203_in -> f79_in :|: TRUE
6.50/2.54	f79_out -> f203_out :|: TRUE
6.50/2.54	f162_in -> f203_in :|: TRUE
6.50/2.54	f203_out -> f162_out :|: TRUE
6.50/2.54	f204_out -> f162_out :|: TRUE
6.50/2.54	f162_in -> f204_in :|: TRUE
6.50/2.54	f1_in(T1) -> f6_in(T1) :|: TRUE
6.50/2.54	f6_out(x) -> f1_out(x) :|: TRUE
6.50/2.54	f53_out(T10) -> f6_out(T10) :|: TRUE
6.50/2.54	f6_in(x1) -> f53_in(x1) :|: TRUE
6.50/2.54	f53_in(x2) -> f78_in(x2) :|: TRUE
6.50/2.54	f79_out -> f53_out(x3) :|: TRUE
6.50/2.54	f78_out(x4) -> f79_in :|: TRUE
6.50/2.54	Start term: f1_in(T1)
6.50/2.54	
6.50/2.54	----------------------------------------
6.50/2.54	
6.50/2.54	(81) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
6.50/2.54	Constructed simple dependency graph.
6.50/2.54	
6.50/2.54	Simplified to the following IRSwTs:
6.50/2.54	
6.50/2.54	
6.50/2.54	----------------------------------------
6.50/2.54	
6.50/2.54	(82)
6.50/2.54	TRUE
6.50/2.54	
6.50/2.54	----------------------------------------
6.50/2.54	
6.50/2.54	(83)
6.50/2.54	Obligation:
6.50/2.54	Rules:
6.50/2.54	f78_out(T26) -> f150_out(T26) :|: TRUE
6.50/2.54	f150_in(x) -> f78_in(x) :|: TRUE
6.50/2.54	f95_in(x1) -> f150_in(x1) :|: TRUE
6.50/2.54	f150_out(x2) -> f95_out(x2) :|: TRUE
6.50/2.54	f78_in(T10) -> f88_in(T10) :|: TRUE
6.50/2.54	f88_out(x3) -> f78_out(x3) :|: TRUE
6.50/2.54	f95_out(x4) -> f88_out(x4) :|: TRUE
6.50/2.54	f88_in(x5) -> f94_in(x5) :|: TRUE
6.50/2.54	f88_in(x6) -> f95_in(x6) :|: TRUE
6.50/2.54	f94_out(x7) -> f88_out(x7) :|: TRUE
6.50/2.54	f1_in(T1) -> f6_in(T1) :|: TRUE
6.50/2.54	f6_out(x8) -> f1_out(x8) :|: TRUE
6.50/2.54	f53_out(x9) -> f6_out(x9) :|: TRUE
6.50/2.54	f6_in(x10) -> f53_in(x10) :|: TRUE
6.50/2.54	f53_in(x11) -> f78_in(x11) :|: TRUE
6.50/2.54	f79_out -> f53_out(x12) :|: TRUE
6.50/2.54	f78_out(x13) -> f79_in :|: TRUE
6.50/2.54	Start term: f1_in(T1)
6.50/2.54	
6.50/2.54	----------------------------------------
6.50/2.54	
6.50/2.54	(84) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
6.50/2.54	Constructed simple dependency graph.
6.50/2.54	
6.50/2.54	Simplified to the following IRSwTs:
6.50/2.54	
6.50/2.54	intTRSProblem:
6.50/2.54	f150_in(x) -> f78_in(x) :|: TRUE
6.50/2.54	f95_in(x1) -> f150_in(x1) :|: TRUE
6.50/2.54	f78_in(T10) -> f88_in(T10) :|: TRUE
6.50/2.54	f88_in(x6) -> f95_in(x6) :|: TRUE
6.50/2.54	
6.50/2.54	
6.50/2.54	----------------------------------------
6.50/2.54	
6.50/2.54	(85)
6.50/2.54	Obligation:
6.50/2.54	Rules:
6.50/2.54	f150_in(x) -> f78_in(x) :|: TRUE
6.50/2.54	f95_in(x1) -> f150_in(x1) :|: TRUE
6.50/2.54	f78_in(T10) -> f88_in(T10) :|: TRUE
6.50/2.54	f88_in(x6) -> f95_in(x6) :|: TRUE
6.50/2.54	
6.50/2.54	----------------------------------------
6.50/2.54	
6.50/2.54	(86) IntTRSCompressionProof (EQUIVALENT)
6.50/2.54	Compressed rules.
6.50/2.54	----------------------------------------
6.50/2.54	
6.50/2.54	(87)
6.50/2.54	Obligation:
6.50/2.54	Rules:
6.50/2.54	f95_in(x1:0) -> f95_in(x1:0) :|: TRUE
6.50/2.54	
6.50/2.54	----------------------------------------
6.50/2.54	
6.50/2.54	(88) IRSFormatTransformerProof (EQUIVALENT)
6.50/2.54	Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
6.50/2.54	----------------------------------------
6.50/2.54	
6.50/2.54	(89)
6.50/2.54	Obligation:
6.50/2.54	Rules:
6.50/2.54	f95_in(x1:0) -> f95_in(x1:0) :|: TRUE
6.50/2.54	
6.50/2.54	----------------------------------------
6.50/2.54	
6.50/2.54	(90) IRSwTTerminationDigraphProof (EQUIVALENT)
6.50/2.54	Constructed termination digraph!
6.50/2.54	Nodes:
6.50/2.54	(1) f95_in(x1:0) -> f95_in(x1:0) :|: TRUE
6.50/2.54	
6.50/2.54	Arcs:
6.50/2.54	(1) -> (1)
6.50/2.54	
6.50/2.54	This digraph is fully evaluated!
6.50/2.54	----------------------------------------
6.50/2.54	
6.50/2.54	(91)
6.50/2.54	Obligation:
6.50/2.54	
6.50/2.54	Termination digraph:
6.50/2.54	Nodes:
6.50/2.54	(1) f95_in(x1:0) -> f95_in(x1:0) :|: TRUE
6.50/2.54	
6.50/2.54	Arcs:
6.50/2.54	(1) -> (1)
6.50/2.54	
6.50/2.54	This digraph is fully evaluated!
6.50/2.54	
6.50/2.54	----------------------------------------
6.50/2.54	
6.50/2.54	(92) FilterProof (EQUIVALENT)
6.50/2.54	Used the following sort dictionary for filtering: 
6.50/2.54	f95_in(VARIABLE)
6.50/2.54	Replaced non-predefined constructor symbols by 0.
6.50/2.54	----------------------------------------
6.50/2.54	
6.50/2.54	(93)
6.50/2.54	Obligation:
6.50/2.54	Rules:
6.50/2.54	f95_in(x1:0) -> f95_in(x1:0) :|: TRUE
6.50/2.54	
6.50/2.54	----------------------------------------
6.50/2.54	
6.50/2.54	(94) IntTRSPeriodicNontermProof (COMPLETE)
6.50/2.54	Normalized system to the following form:
6.50/2.54	f(pc, x1:0) -> f(1, x1:0) :|: pc = 1 && TRUE
6.50/2.54	Witness term starting non-terminating reduction: f(1, -8)
6.50/2.54	----------------------------------------
6.50/2.54	
6.50/2.54	(95)
6.50/2.54	NO
6.69/2.58	EOF