9.50/3.41	MAYBE
9.50/3.43	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
9.50/3.43	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.50/3.43	
9.50/3.43	
9.50/3.43	Left Termination of the query pattern
9.50/3.43	
9.50/3.43	subset(a,g)
9.50/3.43	
9.50/3.43	w.r.t. the given Prolog program could not be shown:
9.50/3.43	
9.50/3.43	(0) Prolog
9.50/3.43	(1) PrologToPiTRSProof [SOUND, 0 ms]
9.50/3.43	(2) PiTRS
9.50/3.43	    (3) DependencyPairsProof [EQUIVALENT, 0 ms]
9.50/3.43	    (4) PiDP
9.50/3.43	    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
9.50/3.43	    (6) AND
9.50/3.43	        (7) PiDP
9.50/3.43	            (8) UsableRulesProof [EQUIVALENT, 0 ms]
9.50/3.43	            (9) PiDP
9.50/3.43	            (10) PiDPToQDPProof [SOUND, 0 ms]
9.50/3.43	            (11) QDP
9.50/3.43	            (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.50/3.43	            (13) YES
9.50/3.43	        (14) PiDP
9.50/3.43	            (15) UsableRulesProof [EQUIVALENT, 0 ms]
9.50/3.43	            (16) PiDP
9.50/3.43	            (17) PiDPToQDPProof [SOUND, 1 ms]
9.50/3.43	            (18) QDP
9.50/3.43	            (19) TransformationProof [SOUND, 0 ms]
9.50/3.43	            (20) QDP
9.50/3.43	            (21) TransformationProof [EQUIVALENT, 0 ms]
9.50/3.43	            (22) QDP
9.50/3.43	(23) PrologToPiTRSProof [SOUND, 0 ms]
9.50/3.43	(24) PiTRS
9.50/3.43	    (25) DependencyPairsProof [EQUIVALENT, 0 ms]
9.50/3.43	    (26) PiDP
9.50/3.43	    (27) DependencyGraphProof [EQUIVALENT, 0 ms]
9.50/3.43	    (28) AND
9.50/3.43	        (29) PiDP
9.50/3.43	            (30) UsableRulesProof [EQUIVALENT, 0 ms]
9.50/3.43	            (31) PiDP
9.50/3.43	            (32) PiDPToQDPProof [SOUND, 0 ms]
9.50/3.43	            (33) QDP
9.50/3.43	            (34) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.50/3.43	            (35) YES
9.50/3.43	        (36) PiDP
9.50/3.43	            (37) UsableRulesProof [EQUIVALENT, 0 ms]
9.50/3.43	            (38) PiDP
9.50/3.43	            (39) PiDPToQDPProof [SOUND, 1 ms]
9.50/3.43	            (40) QDP
9.50/3.43	            (41) TransformationProof [SOUND, 0 ms]
9.50/3.43	            (42) QDP
9.50/3.43	            (43) TransformationProof [EQUIVALENT, 0 ms]
9.50/3.43	            (44) QDP
9.50/3.43	(45) PrologToDTProblemTransformerProof [SOUND, 0 ms]
9.50/3.43	(46) TRIPLES
9.50/3.43	    (47) TriplesToPiDPProof [SOUND, 0 ms]
9.50/3.43	    (48) PiDP
9.50/3.43	    (49) DependencyGraphProof [EQUIVALENT, 0 ms]
9.50/3.43	    (50) AND
9.50/3.43	        (51) PiDP
9.50/3.43	            (52) UsableRulesProof [EQUIVALENT, 0 ms]
9.50/3.43	            (53) PiDP
9.50/3.43	            (54) PiDPToQDPProof [SOUND, 0 ms]
9.50/3.43	            (55) QDP
9.50/3.43	            (56) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.50/3.43	            (57) YES
9.50/3.43	        (58) PiDP
9.50/3.43	            (59) PiDPToQDPProof [SOUND, 0 ms]
9.50/3.43	            (60) QDP
9.50/3.43	            (61) TransformationProof [SOUND, 0 ms]
9.50/3.43	            (62) QDP
9.50/3.43	            (63) TransformationProof [EQUIVALENT, 0 ms]
9.50/3.43	            (64) QDP
9.50/3.43	            (65) TransformationProof [EQUIVALENT, 0 ms]
9.50/3.43	            (66) QDP
9.50/3.43	(67) PrologToTRSTransformerProof [SOUND, 0 ms]
9.50/3.43	(68) QTRS
9.50/3.43	    (69) DependencyPairsProof [EQUIVALENT, 0 ms]
9.50/3.43	    (70) QDP
9.50/3.43	    (71) DependencyGraphProof [EQUIVALENT, 0 ms]
9.50/3.43	    (72) AND
9.50/3.43	        (73) QDP
9.50/3.43	            (74) UsableRulesProof [EQUIVALENT, 0 ms]
9.50/3.43	            (75) QDP
9.50/3.43	            (76) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.50/3.43	            (77) YES
9.50/3.43	        (78) QDP
9.50/3.43	            (79) NonTerminationLoopProof [COMPLETE, 11 ms]
9.50/3.43	            (80) NO
9.50/3.43	(81) PrologToIRSwTTransformerProof [SOUND, 0 ms]
9.50/3.43	(82) AND
9.50/3.43	    (83) IRSwT
9.50/3.43	        (84) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
9.50/3.43	        (85) IRSwT
9.50/3.43	        (86) IntTRSCompressionProof [EQUIVALENT, 42 ms]
9.50/3.43	        (87) IRSwT
9.50/3.43	        (88) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
9.50/3.43	        (89) IRSwT
9.50/3.43	        (90) IRSwTTerminationDigraphProof [EQUIVALENT, 6 ms]
9.50/3.43	        (91) IRSwT
9.50/3.43	        (92) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 34 ms]
9.50/3.43	        (93) IRSwT
9.50/3.43	        (94) TempFilterProof [SOUND, 2 ms]
9.50/3.43	        (95) IRSwT
9.50/3.43	        (96) IRSwTToQDPProof [SOUND, 0 ms]
9.50/3.43	        (97) QDP
9.50/3.43	        (98) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.50/3.43	        (99) YES
9.50/3.43	    (100) IRSwT
9.50/3.43	        (101) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
9.50/3.43	        (102) IRSwT
9.50/3.43	        (103) IntTRSCompressionProof [EQUIVALENT, 9 ms]
9.50/3.43	        (104) IRSwT
9.50/3.43	        (105) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
9.50/3.43	        (106) IRSwT
9.50/3.43	        (107) IRSwTTerminationDigraphProof [EQUIVALENT, 9 ms]
9.50/3.43	        (108) IRSwT
9.50/3.43	
9.50/3.43	
9.50/3.43	----------------------------------------
9.50/3.43	
9.50/3.43	(0)
9.50/3.43	Obligation:
9.50/3.43	Clauses:
9.50/3.43	
9.50/3.43	subset([], X1).
9.50/3.43	subset(.(X, Xs), Ys) :- ','(member(X, Ys), subset(Xs, Ys)).
9.50/3.43	member(X, .(X, X2)).
9.50/3.43	member(X, .(X3, Xs)) :- member(X, Xs).
9.50/3.43	
9.50/3.43	
9.50/3.43	Query: subset(a,g)
9.50/3.43	----------------------------------------
9.50/3.43	
9.50/3.43	(1) PrologToPiTRSProof (SOUND)
9.50/3.43	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
9.50/3.43	
9.50/3.43	subset_in_2: (f,b)
9.50/3.43	
9.50/3.43	member_in_2: (f,b)
9.50/3.43	
9.50/3.43	Transforming Prolog into the following Term Rewriting System:
9.50/3.43	
9.50/3.43	Pi-finite rewrite system:
9.50/3.43	The TRS R consists of the following rules:
9.50/3.43	
9.50/3.43	   subset_in_ag([], X1) -> subset_out_ag([], X1)
9.50/3.43	   subset_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, member_in_ag(X, Ys))
9.50/3.43	   member_in_ag(X, .(X, X2)) -> member_out_ag(X, .(X, X2))
9.50/3.43	   member_in_ag(X, .(X3, Xs)) -> U3_ag(X, X3, Xs, member_in_ag(X, Xs))
9.50/3.43	   U3_ag(X, X3, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X3, Xs))
9.50/3.43	   U1_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U2_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
9.50/3.43	   U2_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)
9.50/3.43	
9.50/3.43	The argument filtering Pi contains the following mapping:
9.50/3.43	subset_in_ag(x1, x2)  =  subset_in_ag(x2)
9.50/3.43	
9.50/3.43	subset_out_ag(x1, x2)  =  subset_out_ag(x1)
9.50/3.43	
9.50/3.43	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
9.50/3.43	
9.50/3.43	member_in_ag(x1, x2)  =  member_in_ag(x2)
9.50/3.43	
9.50/3.43	.(x1, x2)  =  .(x1, x2)
9.50/3.43	
9.50/3.43	member_out_ag(x1, x2)  =  member_out_ag(x1)
9.50/3.43	
9.50/3.43	U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
9.50/3.43	
9.50/3.43	U2_ag(x1, x2, x3, x4)  =  U2_ag(x1, x4)
9.50/3.43	
9.50/3.43	
9.50/3.43	
9.50/3.43	
9.50/3.43	
9.50/3.43	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
9.50/3.43	
9.50/3.43	
9.50/3.43	
9.50/3.43	----------------------------------------
9.50/3.43	
9.50/3.43	(2)
9.50/3.43	Obligation:
9.50/3.43	Pi-finite rewrite system:
9.50/3.43	The TRS R consists of the following rules:
9.50/3.43	
9.50/3.43	   subset_in_ag([], X1) -> subset_out_ag([], X1)
9.50/3.43	   subset_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, member_in_ag(X, Ys))
9.50/3.43	   member_in_ag(X, .(X, X2)) -> member_out_ag(X, .(X, X2))
9.50/3.43	   member_in_ag(X, .(X3, Xs)) -> U3_ag(X, X3, Xs, member_in_ag(X, Xs))
9.50/3.43	   U3_ag(X, X3, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X3, Xs))
9.50/3.43	   U1_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U2_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
9.50/3.43	   U2_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)
9.50/3.43	
9.50/3.43	The argument filtering Pi contains the following mapping:
9.50/3.43	subset_in_ag(x1, x2)  =  subset_in_ag(x2)
9.50/3.43	
9.50/3.43	subset_out_ag(x1, x2)  =  subset_out_ag(x1)
9.50/3.43	
9.50/3.43	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
9.50/3.43	
9.50/3.43	member_in_ag(x1, x2)  =  member_in_ag(x2)
9.50/3.43	
9.50/3.43	.(x1, x2)  =  .(x1, x2)
9.50/3.43	
9.50/3.43	member_out_ag(x1, x2)  =  member_out_ag(x1)
9.50/3.43	
9.50/3.43	U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
9.50/3.43	
9.50/3.43	U2_ag(x1, x2, x3, x4)  =  U2_ag(x1, x4)
9.50/3.43	
9.50/3.43	
9.50/3.43	
9.50/3.43	----------------------------------------
9.50/3.43	
9.50/3.43	(3) DependencyPairsProof (EQUIVALENT)
9.50/3.43	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
9.50/3.43	Pi DP problem:
9.50/3.43	The TRS P consists of the following rules:
9.50/3.43	
9.50/3.43	   SUBSET_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, member_in_ag(X, Ys))
9.50/3.43	   SUBSET_IN_AG(.(X, Xs), Ys) -> MEMBER_IN_AG(X, Ys)
9.50/3.43	   MEMBER_IN_AG(X, .(X3, Xs)) -> U3_AG(X, X3, Xs, member_in_ag(X, Xs))
9.50/3.43	   MEMBER_IN_AG(X, .(X3, Xs)) -> MEMBER_IN_AG(X, Xs)
9.50/3.43	   U1_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> U2_AG(X, Xs, Ys, subset_in_ag(Xs, Ys))
9.50/3.43	   U1_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys)
9.50/3.43	
9.50/3.43	The TRS R consists of the following rules:
9.50/3.43	
9.50/3.43	   subset_in_ag([], X1) -> subset_out_ag([], X1)
9.50/3.43	   subset_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, member_in_ag(X, Ys))
9.50/3.43	   member_in_ag(X, .(X, X2)) -> member_out_ag(X, .(X, X2))
9.50/3.43	   member_in_ag(X, .(X3, Xs)) -> U3_ag(X, X3, Xs, member_in_ag(X, Xs))
9.50/3.43	   U3_ag(X, X3, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X3, Xs))
9.50/3.43	   U1_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U2_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
9.50/3.43	   U2_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)
9.50/3.43	
9.50/3.43	The argument filtering Pi contains the following mapping:
9.50/3.43	subset_in_ag(x1, x2)  =  subset_in_ag(x2)
9.50/3.43	
9.50/3.43	subset_out_ag(x1, x2)  =  subset_out_ag(x1)
9.50/3.43	
9.50/3.43	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
9.50/3.43	
9.50/3.43	member_in_ag(x1, x2)  =  member_in_ag(x2)
9.50/3.43	
9.50/3.43	.(x1, x2)  =  .(x1, x2)
9.50/3.43	
9.50/3.43	member_out_ag(x1, x2)  =  member_out_ag(x1)
9.50/3.43	
9.50/3.43	U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
9.50/3.43	
9.50/3.43	U2_ag(x1, x2, x3, x4)  =  U2_ag(x1, x4)
9.50/3.43	
9.50/3.43	SUBSET_IN_AG(x1, x2)  =  SUBSET_IN_AG(x2)
9.50/3.43	
9.50/3.43	U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
9.50/3.43	
9.50/3.43	MEMBER_IN_AG(x1, x2)  =  MEMBER_IN_AG(x2)
9.50/3.43	
9.50/3.43	U3_AG(x1, x2, x3, x4)  =  U3_AG(x4)
9.50/3.43	
9.50/3.43	U2_AG(x1, x2, x3, x4)  =  U2_AG(x1, x4)
9.50/3.43	
9.50/3.43	
9.50/3.43	We have to consider all (P,R,Pi)-chains
9.50/3.43	----------------------------------------
9.50/3.43	
9.50/3.43	(4)
9.50/3.43	Obligation:
9.50/3.43	Pi DP problem:
9.50/3.43	The TRS P consists of the following rules:
9.50/3.43	
9.50/3.43	   SUBSET_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, member_in_ag(X, Ys))
9.50/3.43	   SUBSET_IN_AG(.(X, Xs), Ys) -> MEMBER_IN_AG(X, Ys)
9.50/3.43	   MEMBER_IN_AG(X, .(X3, Xs)) -> U3_AG(X, X3, Xs, member_in_ag(X, Xs))
9.50/3.43	   MEMBER_IN_AG(X, .(X3, Xs)) -> MEMBER_IN_AG(X, Xs)
9.50/3.43	   U1_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> U2_AG(X, Xs, Ys, subset_in_ag(Xs, Ys))
9.50/3.43	   U1_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys)
9.50/3.43	
9.50/3.43	The TRS R consists of the following rules:
9.50/3.43	
9.50/3.43	   subset_in_ag([], X1) -> subset_out_ag([], X1)
9.50/3.43	   subset_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, member_in_ag(X, Ys))
9.50/3.43	   member_in_ag(X, .(X, X2)) -> member_out_ag(X, .(X, X2))
9.50/3.43	   member_in_ag(X, .(X3, Xs)) -> U3_ag(X, X3, Xs, member_in_ag(X, Xs))
9.50/3.43	   U3_ag(X, X3, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X3, Xs))
9.50/3.43	   U1_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U2_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
9.50/3.43	   U2_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)
9.50/3.43	
9.50/3.43	The argument filtering Pi contains the following mapping:
9.50/3.43	subset_in_ag(x1, x2)  =  subset_in_ag(x2)
9.50/3.43	
9.50/3.43	subset_out_ag(x1, x2)  =  subset_out_ag(x1)
9.50/3.43	
9.50/3.43	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
9.50/3.43	
9.50/3.43	member_in_ag(x1, x2)  =  member_in_ag(x2)
9.50/3.43	
9.50/3.43	.(x1, x2)  =  .(x1, x2)
9.50/3.43	
9.50/3.43	member_out_ag(x1, x2)  =  member_out_ag(x1)
9.50/3.43	
9.50/3.43	U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
9.50/3.43	
9.50/3.43	U2_ag(x1, x2, x3, x4)  =  U2_ag(x1, x4)
9.50/3.43	
9.50/3.43	SUBSET_IN_AG(x1, x2)  =  SUBSET_IN_AG(x2)
9.50/3.43	
9.50/3.43	U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
9.50/3.43	
9.50/3.43	MEMBER_IN_AG(x1, x2)  =  MEMBER_IN_AG(x2)
9.50/3.43	
9.50/3.43	U3_AG(x1, x2, x3, x4)  =  U3_AG(x4)
9.50/3.43	
9.50/3.43	U2_AG(x1, x2, x3, x4)  =  U2_AG(x1, x4)
9.50/3.43	
9.50/3.43	
9.50/3.43	We have to consider all (P,R,Pi)-chains
9.50/3.43	----------------------------------------
9.50/3.43	
9.50/3.43	(5) DependencyGraphProof (EQUIVALENT)
9.50/3.43	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes.
9.50/3.43	----------------------------------------
9.50/3.43	
9.50/3.43	(6)
9.50/3.43	Complex Obligation (AND)
9.50/3.43	
9.50/3.43	----------------------------------------
9.50/3.43	
9.50/3.43	(7)
9.50/3.43	Obligation:
9.50/3.43	Pi DP problem:
9.50/3.43	The TRS P consists of the following rules:
9.50/3.43	
9.50/3.43	   MEMBER_IN_AG(X, .(X3, Xs)) -> MEMBER_IN_AG(X, Xs)
9.50/3.43	
9.50/3.43	The TRS R consists of the following rules:
9.50/3.43	
9.50/3.43	   subset_in_ag([], X1) -> subset_out_ag([], X1)
9.50/3.43	   subset_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, member_in_ag(X, Ys))
9.50/3.43	   member_in_ag(X, .(X, X2)) -> member_out_ag(X, .(X, X2))
9.50/3.43	   member_in_ag(X, .(X3, Xs)) -> U3_ag(X, X3, Xs, member_in_ag(X, Xs))
9.50/3.43	   U3_ag(X, X3, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X3, Xs))
9.50/3.43	   U1_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U2_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
9.50/3.43	   U2_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)
9.50/3.43	
9.50/3.43	The argument filtering Pi contains the following mapping:
9.50/3.43	subset_in_ag(x1, x2)  =  subset_in_ag(x2)
9.50/3.43	
9.50/3.43	subset_out_ag(x1, x2)  =  subset_out_ag(x1)
9.50/3.43	
9.50/3.43	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
9.50/3.43	
9.50/3.43	member_in_ag(x1, x2)  =  member_in_ag(x2)
9.50/3.43	
9.50/3.43	.(x1, x2)  =  .(x1, x2)
9.50/3.43	
9.50/3.43	member_out_ag(x1, x2)  =  member_out_ag(x1)
9.50/3.43	
9.50/3.43	U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
9.50/3.43	
9.50/3.43	U2_ag(x1, x2, x3, x4)  =  U2_ag(x1, x4)
9.50/3.43	
9.50/3.43	MEMBER_IN_AG(x1, x2)  =  MEMBER_IN_AG(x2)
9.50/3.43	
9.50/3.43	
9.50/3.43	We have to consider all (P,R,Pi)-chains
9.50/3.43	----------------------------------------
9.50/3.43	
9.50/3.43	(8) UsableRulesProof (EQUIVALENT)
9.50/3.43	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
9.50/3.43	----------------------------------------
9.50/3.43	
9.50/3.43	(9)
9.50/3.43	Obligation:
9.50/3.43	Pi DP problem:
9.50/3.43	The TRS P consists of the following rules:
9.50/3.43	
9.50/3.43	   MEMBER_IN_AG(X, .(X3, Xs)) -> MEMBER_IN_AG(X, Xs)
9.50/3.43	
9.50/3.43	R is empty.
9.50/3.43	The argument filtering Pi contains the following mapping:
9.50/3.43	.(x1, x2)  =  .(x1, x2)
9.50/3.43	
9.50/3.43	MEMBER_IN_AG(x1, x2)  =  MEMBER_IN_AG(x2)
9.50/3.43	
9.50/3.43	
9.50/3.43	We have to consider all (P,R,Pi)-chains
9.50/3.43	----------------------------------------
9.50/3.43	
9.50/3.43	(10) PiDPToQDPProof (SOUND)
9.50/3.43	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
9.50/3.43	----------------------------------------
9.50/3.43	
9.50/3.43	(11)
9.50/3.43	Obligation:
9.50/3.43	Q DP problem:
9.50/3.43	The TRS P consists of the following rules:
9.50/3.43	
9.50/3.43	   MEMBER_IN_AG(.(X3, Xs)) -> MEMBER_IN_AG(Xs)
9.50/3.43	
9.50/3.43	R is empty.
9.50/3.43	Q is empty.
9.50/3.43	We have to consider all (P,Q,R)-chains.
9.50/3.43	----------------------------------------
9.50/3.43	
9.50/3.43	(12) QDPSizeChangeProof (EQUIVALENT)
9.50/3.43	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. 
9.50/3.43	
9.50/3.43	From the DPs we obtained the following set of size-change graphs:
9.50/3.43	*MEMBER_IN_AG(.(X3, Xs)) -> MEMBER_IN_AG(Xs)
9.50/3.43	The graph contains the following edges 1 > 1
9.50/3.43	
9.50/3.43	
9.50/3.43	----------------------------------------
9.50/3.43	
9.50/3.43	(13)
9.50/3.43	YES
9.50/3.43	
9.50/3.43	----------------------------------------
9.50/3.43	
9.50/3.43	(14)
9.50/3.43	Obligation:
9.50/3.43	Pi DP problem:
9.50/3.43	The TRS P consists of the following rules:
9.50/3.43	
9.50/3.43	   U1_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys)
9.50/3.43	   SUBSET_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, member_in_ag(X, Ys))
9.50/3.43	
9.50/3.43	The TRS R consists of the following rules:
9.50/3.43	
9.50/3.43	   subset_in_ag([], X1) -> subset_out_ag([], X1)
9.50/3.43	   subset_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, member_in_ag(X, Ys))
9.50/3.43	   member_in_ag(X, .(X, X2)) -> member_out_ag(X, .(X, X2))
9.50/3.43	   member_in_ag(X, .(X3, Xs)) -> U3_ag(X, X3, Xs, member_in_ag(X, Xs))
9.50/3.43	   U3_ag(X, X3, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X3, Xs))
9.50/3.43	   U1_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U2_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
9.50/3.43	   U2_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)
9.50/3.43	
9.50/3.43	The argument filtering Pi contains the following mapping:
9.50/3.43	subset_in_ag(x1, x2)  =  subset_in_ag(x2)
9.50/3.43	
9.50/3.43	subset_out_ag(x1, x2)  =  subset_out_ag(x1)
9.50/3.43	
9.50/3.43	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
9.50/3.43	
9.50/3.43	member_in_ag(x1, x2)  =  member_in_ag(x2)
9.50/3.43	
9.50/3.43	.(x1, x2)  =  .(x1, x2)
9.50/3.43	
9.50/3.43	member_out_ag(x1, x2)  =  member_out_ag(x1)
9.50/3.43	
9.50/3.43	U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
9.50/3.43	
9.50/3.43	U2_ag(x1, x2, x3, x4)  =  U2_ag(x1, x4)
9.50/3.43	
9.50/3.43	SUBSET_IN_AG(x1, x2)  =  SUBSET_IN_AG(x2)
9.50/3.43	
9.50/3.43	U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
9.50/3.43	
9.50/3.43	
9.50/3.43	We have to consider all (P,R,Pi)-chains
9.50/3.43	----------------------------------------
9.50/3.43	
9.50/3.43	(15) UsableRulesProof (EQUIVALENT)
9.50/3.43	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
9.50/3.43	----------------------------------------
9.50/3.43	
9.50/3.43	(16)
9.50/3.43	Obligation:
9.50/3.43	Pi DP problem:
9.50/3.43	The TRS P consists of the following rules:
9.50/3.43	
9.50/3.43	   U1_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys)
9.50/3.43	   SUBSET_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, member_in_ag(X, Ys))
9.50/3.43	
9.50/3.43	The TRS R consists of the following rules:
9.50/3.43	
9.50/3.43	   member_in_ag(X, .(X, X2)) -> member_out_ag(X, .(X, X2))
9.50/3.43	   member_in_ag(X, .(X3, Xs)) -> U3_ag(X, X3, Xs, member_in_ag(X, Xs))
9.50/3.43	   U3_ag(X, X3, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X3, Xs))
9.50/3.43	
9.50/3.43	The argument filtering Pi contains the following mapping:
9.50/3.43	member_in_ag(x1, x2)  =  member_in_ag(x2)
9.50/3.43	
9.50/3.43	.(x1, x2)  =  .(x1, x2)
9.50/3.43	
9.50/3.43	member_out_ag(x1, x2)  =  member_out_ag(x1)
9.50/3.43	
9.50/3.43	U3_ag(x1, x2, x3, x4)  =  U3_ag(x4)
9.50/3.43	
9.50/3.43	SUBSET_IN_AG(x1, x2)  =  SUBSET_IN_AG(x2)
9.50/3.43	
9.50/3.43	U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
9.50/3.43	
9.50/3.43	
9.50/3.43	We have to consider all (P,R,Pi)-chains
9.50/3.43	----------------------------------------
9.50/3.43	
9.50/3.43	(17) PiDPToQDPProof (SOUND)
9.50/3.43	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
9.50/3.43	----------------------------------------
9.50/3.43	
9.50/3.43	(18)
9.50/3.43	Obligation:
9.50/3.43	Q DP problem:
9.50/3.43	The TRS P consists of the following rules:
9.50/3.43	
9.50/3.43	   U1_AG(Ys, member_out_ag(X)) -> SUBSET_IN_AG(Ys)
9.50/3.43	   SUBSET_IN_AG(Ys) -> U1_AG(Ys, member_in_ag(Ys))
9.50/3.43	
9.50/3.43	The TRS R consists of the following rules:
9.50/3.43	
9.50/3.43	   member_in_ag(.(X, X2)) -> member_out_ag(X)
9.50/3.43	   member_in_ag(.(X3, Xs)) -> U3_ag(member_in_ag(Xs))
9.50/3.43	   U3_ag(member_out_ag(X)) -> member_out_ag(X)
9.50/3.43	
9.50/3.43	The set Q consists of the following terms:
9.50/3.43	
9.50/3.43	   member_in_ag(x0)
9.50/3.43	   U3_ag(x0)
9.50/3.43	
9.50/3.43	We have to consider all (P,Q,R)-chains.
9.50/3.43	----------------------------------------
9.50/3.43	
9.50/3.43	(19) TransformationProof (SOUND)
9.50/3.43	By narrowing [LPAR04] the rule SUBSET_IN_AG(Ys) -> U1_AG(Ys, member_in_ag(Ys)) at position [1] we obtained the following new rules [LPAR04]:
9.50/3.43	
9.50/3.43	   (SUBSET_IN_AG(.(x0, x1)) -> U1_AG(.(x0, x1), member_out_ag(x0)),SUBSET_IN_AG(.(x0, x1)) -> U1_AG(.(x0, x1), member_out_ag(x0)))
9.50/3.43	   (SUBSET_IN_AG(.(x0, x1)) -> U1_AG(.(x0, x1), U3_ag(member_in_ag(x1))),SUBSET_IN_AG(.(x0, x1)) -> U1_AG(.(x0, x1), U3_ag(member_in_ag(x1))))
9.50/3.43	
9.50/3.43	
9.50/3.43	----------------------------------------
9.50/3.43	
9.50/3.43	(20)
9.50/3.43	Obligation:
9.50/3.43	Q DP problem:
9.50/3.43	The TRS P consists of the following rules:
9.50/3.43	
9.50/3.43	   U1_AG(Ys, member_out_ag(X)) -> SUBSET_IN_AG(Ys)
9.50/3.43	   SUBSET_IN_AG(.(x0, x1)) -> U1_AG(.(x0, x1), member_out_ag(x0))
9.50/3.45	   SUBSET_IN_AG(.(x0, x1)) -> U1_AG(.(x0, x1), U3_ag(member_in_ag(x1)))
9.50/3.45	
9.50/3.45	The TRS R consists of the following rules:
9.50/3.45	
9.50/3.45	   member_in_ag(.(X, X2)) -> member_out_ag(X)
9.50/3.45	   member_in_ag(.(X3, Xs)) -> U3_ag(member_in_ag(Xs))
9.50/3.45	   U3_ag(member_out_ag(X)) -> member_out_ag(X)
9.50/3.45	
9.50/3.45	The set Q consists of the following terms:
9.50/3.45	
9.50/3.45	   member_in_ag(x0)
9.50/3.45	   U3_ag(x0)
9.50/3.45	
9.50/3.45	We have to consider all (P,Q,R)-chains.
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(21) TransformationProof (EQUIVALENT)
9.50/3.45	By instantiating [LPAR04] the rule U1_AG(Ys, member_out_ag(X)) -> SUBSET_IN_AG(Ys) we obtained the following new rules [LPAR04]:
9.50/3.45	
9.50/3.45	   (U1_AG(.(z0, z1), member_out_ag(z0)) -> SUBSET_IN_AG(.(z0, z1)),U1_AG(.(z0, z1), member_out_ag(z0)) -> SUBSET_IN_AG(.(z0, z1)))
9.50/3.45	   (U1_AG(.(z0, z1), member_out_ag(x1)) -> SUBSET_IN_AG(.(z0, z1)),U1_AG(.(z0, z1), member_out_ag(x1)) -> SUBSET_IN_AG(.(z0, z1)))
9.50/3.45	
9.50/3.45	
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(22)
9.50/3.45	Obligation:
9.50/3.45	Q DP problem:
9.50/3.45	The TRS P consists of the following rules:
9.50/3.45	
9.50/3.45	   SUBSET_IN_AG(.(x0, x1)) -> U1_AG(.(x0, x1), member_out_ag(x0))
9.50/3.45	   SUBSET_IN_AG(.(x0, x1)) -> U1_AG(.(x0, x1), U3_ag(member_in_ag(x1)))
9.50/3.45	   U1_AG(.(z0, z1), member_out_ag(z0)) -> SUBSET_IN_AG(.(z0, z1))
9.50/3.45	   U1_AG(.(z0, z1), member_out_ag(x1)) -> SUBSET_IN_AG(.(z0, z1))
9.50/3.45	
9.50/3.45	The TRS R consists of the following rules:
9.50/3.45	
9.50/3.45	   member_in_ag(.(X, X2)) -> member_out_ag(X)
9.50/3.45	   member_in_ag(.(X3, Xs)) -> U3_ag(member_in_ag(Xs))
9.50/3.45	   U3_ag(member_out_ag(X)) -> member_out_ag(X)
9.50/3.45	
9.50/3.45	The set Q consists of the following terms:
9.50/3.45	
9.50/3.45	   member_in_ag(x0)
9.50/3.45	   U3_ag(x0)
9.50/3.45	
9.50/3.45	We have to consider all (P,Q,R)-chains.
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(23) PrologToPiTRSProof (SOUND)
9.50/3.45	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
9.50/3.45	
9.50/3.45	subset_in_2: (f,b)
9.50/3.45	
9.50/3.45	member_in_2: (f,b)
9.50/3.45	
9.50/3.45	Transforming Prolog into the following Term Rewriting System:
9.50/3.45	
9.50/3.45	Pi-finite rewrite system:
9.50/3.45	The TRS R consists of the following rules:
9.50/3.45	
9.50/3.45	   subset_in_ag([], X1) -> subset_out_ag([], X1)
9.50/3.45	   subset_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, member_in_ag(X, Ys))
9.50/3.45	   member_in_ag(X, .(X, X2)) -> member_out_ag(X, .(X, X2))
9.50/3.45	   member_in_ag(X, .(X3, Xs)) -> U3_ag(X, X3, Xs, member_in_ag(X, Xs))
9.50/3.45	   U3_ag(X, X3, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X3, Xs))
9.50/3.45	   U1_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U2_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
9.50/3.45	   U2_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)
9.50/3.45	
9.50/3.45	The argument filtering Pi contains the following mapping:
9.50/3.45	subset_in_ag(x1, x2)  =  subset_in_ag(x2)
9.50/3.45	
9.50/3.45	subset_out_ag(x1, x2)  =  subset_out_ag(x1, x2)
9.50/3.45	
9.50/3.45	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
9.50/3.45	
9.50/3.45	member_in_ag(x1, x2)  =  member_in_ag(x2)
9.50/3.45	
9.50/3.45	.(x1, x2)  =  .(x1, x2)
9.50/3.45	
9.50/3.45	member_out_ag(x1, x2)  =  member_out_ag(x1, x2)
9.50/3.45	
9.50/3.45	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
9.50/3.45	
9.50/3.45	U2_ag(x1, x2, x3, x4)  =  U2_ag(x1, x3, x4)
9.50/3.45	
9.50/3.45	
9.50/3.45	
9.50/3.45	
9.50/3.45	
9.50/3.45	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
9.50/3.45	
9.50/3.45	
9.50/3.45	
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(24)
9.50/3.45	Obligation:
9.50/3.45	Pi-finite rewrite system:
9.50/3.45	The TRS R consists of the following rules:
9.50/3.45	
9.50/3.45	   subset_in_ag([], X1) -> subset_out_ag([], X1)
9.50/3.45	   subset_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, member_in_ag(X, Ys))
9.50/3.45	   member_in_ag(X, .(X, X2)) -> member_out_ag(X, .(X, X2))
9.50/3.45	   member_in_ag(X, .(X3, Xs)) -> U3_ag(X, X3, Xs, member_in_ag(X, Xs))
9.50/3.45	   U3_ag(X, X3, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X3, Xs))
9.50/3.45	   U1_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U2_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
9.50/3.45	   U2_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)
9.50/3.45	
9.50/3.45	The argument filtering Pi contains the following mapping:
9.50/3.45	subset_in_ag(x1, x2)  =  subset_in_ag(x2)
9.50/3.45	
9.50/3.45	subset_out_ag(x1, x2)  =  subset_out_ag(x1, x2)
9.50/3.45	
9.50/3.45	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
9.50/3.45	
9.50/3.45	member_in_ag(x1, x2)  =  member_in_ag(x2)
9.50/3.45	
9.50/3.45	.(x1, x2)  =  .(x1, x2)
9.50/3.45	
9.50/3.45	member_out_ag(x1, x2)  =  member_out_ag(x1, x2)
9.50/3.45	
9.50/3.45	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
9.50/3.45	
9.50/3.45	U2_ag(x1, x2, x3, x4)  =  U2_ag(x1, x3, x4)
9.50/3.45	
9.50/3.45	
9.50/3.45	
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(25) DependencyPairsProof (EQUIVALENT)
9.50/3.45	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
9.50/3.45	Pi DP problem:
9.50/3.45	The TRS P consists of the following rules:
9.50/3.45	
9.50/3.45	   SUBSET_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, member_in_ag(X, Ys))
9.50/3.45	   SUBSET_IN_AG(.(X, Xs), Ys) -> MEMBER_IN_AG(X, Ys)
9.50/3.45	   MEMBER_IN_AG(X, .(X3, Xs)) -> U3_AG(X, X3, Xs, member_in_ag(X, Xs))
9.50/3.45	   MEMBER_IN_AG(X, .(X3, Xs)) -> MEMBER_IN_AG(X, Xs)
9.50/3.45	   U1_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> U2_AG(X, Xs, Ys, subset_in_ag(Xs, Ys))
9.50/3.45	   U1_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys)
9.50/3.45	
9.50/3.45	The TRS R consists of the following rules:
9.50/3.45	
9.50/3.45	   subset_in_ag([], X1) -> subset_out_ag([], X1)
9.50/3.45	   subset_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, member_in_ag(X, Ys))
9.50/3.45	   member_in_ag(X, .(X, X2)) -> member_out_ag(X, .(X, X2))
9.50/3.45	   member_in_ag(X, .(X3, Xs)) -> U3_ag(X, X3, Xs, member_in_ag(X, Xs))
9.50/3.45	   U3_ag(X, X3, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X3, Xs))
9.50/3.45	   U1_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U2_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
9.50/3.45	   U2_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)
9.50/3.45	
9.50/3.45	The argument filtering Pi contains the following mapping:
9.50/3.45	subset_in_ag(x1, x2)  =  subset_in_ag(x2)
9.50/3.45	
9.50/3.45	subset_out_ag(x1, x2)  =  subset_out_ag(x1, x2)
9.50/3.45	
9.50/3.45	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
9.50/3.45	
9.50/3.45	member_in_ag(x1, x2)  =  member_in_ag(x2)
9.50/3.45	
9.50/3.45	.(x1, x2)  =  .(x1, x2)
9.50/3.45	
9.50/3.45	member_out_ag(x1, x2)  =  member_out_ag(x1, x2)
9.50/3.45	
9.50/3.45	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
9.50/3.45	
9.50/3.45	U2_ag(x1, x2, x3, x4)  =  U2_ag(x1, x3, x4)
9.50/3.45	
9.50/3.45	SUBSET_IN_AG(x1, x2)  =  SUBSET_IN_AG(x2)
9.50/3.45	
9.50/3.45	U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
9.50/3.45	
9.50/3.45	MEMBER_IN_AG(x1, x2)  =  MEMBER_IN_AG(x2)
9.50/3.45	
9.50/3.45	U3_AG(x1, x2, x3, x4)  =  U3_AG(x2, x3, x4)
9.50/3.45	
9.50/3.45	U2_AG(x1, x2, x3, x4)  =  U2_AG(x1, x3, x4)
9.50/3.45	
9.50/3.45	
9.50/3.45	We have to consider all (P,R,Pi)-chains
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(26)
9.50/3.45	Obligation:
9.50/3.45	Pi DP problem:
9.50/3.45	The TRS P consists of the following rules:
9.50/3.45	
9.50/3.45	   SUBSET_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, member_in_ag(X, Ys))
9.50/3.45	   SUBSET_IN_AG(.(X, Xs), Ys) -> MEMBER_IN_AG(X, Ys)
9.50/3.45	   MEMBER_IN_AG(X, .(X3, Xs)) -> U3_AG(X, X3, Xs, member_in_ag(X, Xs))
9.50/3.45	   MEMBER_IN_AG(X, .(X3, Xs)) -> MEMBER_IN_AG(X, Xs)
9.50/3.45	   U1_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> U2_AG(X, Xs, Ys, subset_in_ag(Xs, Ys))
9.50/3.45	   U1_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys)
9.50/3.45	
9.50/3.45	The TRS R consists of the following rules:
9.50/3.45	
9.50/3.45	   subset_in_ag([], X1) -> subset_out_ag([], X1)
9.50/3.45	   subset_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, member_in_ag(X, Ys))
9.50/3.45	   member_in_ag(X, .(X, X2)) -> member_out_ag(X, .(X, X2))
9.50/3.45	   member_in_ag(X, .(X3, Xs)) -> U3_ag(X, X3, Xs, member_in_ag(X, Xs))
9.50/3.45	   U3_ag(X, X3, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X3, Xs))
9.50/3.45	   U1_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U2_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
9.50/3.45	   U2_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)
9.50/3.45	
9.50/3.45	The argument filtering Pi contains the following mapping:
9.50/3.45	subset_in_ag(x1, x2)  =  subset_in_ag(x2)
9.50/3.45	
9.50/3.45	subset_out_ag(x1, x2)  =  subset_out_ag(x1, x2)
9.50/3.45	
9.50/3.45	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
9.50/3.45	
9.50/3.45	member_in_ag(x1, x2)  =  member_in_ag(x2)
9.50/3.45	
9.50/3.45	.(x1, x2)  =  .(x1, x2)
9.50/3.45	
9.50/3.45	member_out_ag(x1, x2)  =  member_out_ag(x1, x2)
9.50/3.45	
9.50/3.45	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
9.50/3.45	
9.50/3.45	U2_ag(x1, x2, x3, x4)  =  U2_ag(x1, x3, x4)
9.50/3.45	
9.50/3.45	SUBSET_IN_AG(x1, x2)  =  SUBSET_IN_AG(x2)
9.50/3.45	
9.50/3.45	U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
9.50/3.45	
9.50/3.45	MEMBER_IN_AG(x1, x2)  =  MEMBER_IN_AG(x2)
9.50/3.45	
9.50/3.45	U3_AG(x1, x2, x3, x4)  =  U3_AG(x2, x3, x4)
9.50/3.45	
9.50/3.45	U2_AG(x1, x2, x3, x4)  =  U2_AG(x1, x3, x4)
9.50/3.45	
9.50/3.45	
9.50/3.45	We have to consider all (P,R,Pi)-chains
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(27) DependencyGraphProof (EQUIVALENT)
9.50/3.45	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes.
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(28)
9.50/3.45	Complex Obligation (AND)
9.50/3.45	
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(29)
9.50/3.45	Obligation:
9.50/3.45	Pi DP problem:
9.50/3.45	The TRS P consists of the following rules:
9.50/3.45	
9.50/3.45	   MEMBER_IN_AG(X, .(X3, Xs)) -> MEMBER_IN_AG(X, Xs)
9.50/3.45	
9.50/3.45	The TRS R consists of the following rules:
9.50/3.45	
9.50/3.45	   subset_in_ag([], X1) -> subset_out_ag([], X1)
9.50/3.45	   subset_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, member_in_ag(X, Ys))
9.50/3.45	   member_in_ag(X, .(X, X2)) -> member_out_ag(X, .(X, X2))
9.50/3.45	   member_in_ag(X, .(X3, Xs)) -> U3_ag(X, X3, Xs, member_in_ag(X, Xs))
9.50/3.45	   U3_ag(X, X3, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X3, Xs))
9.50/3.45	   U1_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U2_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
9.50/3.45	   U2_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)
9.50/3.45	
9.50/3.45	The argument filtering Pi contains the following mapping:
9.50/3.45	subset_in_ag(x1, x2)  =  subset_in_ag(x2)
9.50/3.45	
9.50/3.45	subset_out_ag(x1, x2)  =  subset_out_ag(x1, x2)
9.50/3.45	
9.50/3.45	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
9.50/3.45	
9.50/3.45	member_in_ag(x1, x2)  =  member_in_ag(x2)
9.50/3.45	
9.50/3.45	.(x1, x2)  =  .(x1, x2)
9.50/3.45	
9.50/3.45	member_out_ag(x1, x2)  =  member_out_ag(x1, x2)
9.50/3.45	
9.50/3.45	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
9.50/3.45	
9.50/3.45	U2_ag(x1, x2, x3, x4)  =  U2_ag(x1, x3, x4)
9.50/3.45	
9.50/3.45	MEMBER_IN_AG(x1, x2)  =  MEMBER_IN_AG(x2)
9.50/3.45	
9.50/3.45	
9.50/3.45	We have to consider all (P,R,Pi)-chains
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(30) UsableRulesProof (EQUIVALENT)
9.50/3.45	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(31)
9.50/3.45	Obligation:
9.50/3.45	Pi DP problem:
9.50/3.45	The TRS P consists of the following rules:
9.50/3.45	
9.50/3.45	   MEMBER_IN_AG(X, .(X3, Xs)) -> MEMBER_IN_AG(X, Xs)
9.50/3.45	
9.50/3.45	R is empty.
9.50/3.45	The argument filtering Pi contains the following mapping:
9.50/3.45	.(x1, x2)  =  .(x1, x2)
9.50/3.45	
9.50/3.45	MEMBER_IN_AG(x1, x2)  =  MEMBER_IN_AG(x2)
9.50/3.45	
9.50/3.45	
9.50/3.45	We have to consider all (P,R,Pi)-chains
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(32) PiDPToQDPProof (SOUND)
9.50/3.45	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(33)
9.50/3.45	Obligation:
9.50/3.45	Q DP problem:
9.50/3.45	The TRS P consists of the following rules:
9.50/3.45	
9.50/3.45	   MEMBER_IN_AG(.(X3, Xs)) -> MEMBER_IN_AG(Xs)
9.50/3.45	
9.50/3.45	R is empty.
9.50/3.45	Q is empty.
9.50/3.45	We have to consider all (P,Q,R)-chains.
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(34) QDPSizeChangeProof (EQUIVALENT)
9.50/3.45	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. 
9.50/3.45	
9.50/3.45	From the DPs we obtained the following set of size-change graphs:
9.50/3.45	*MEMBER_IN_AG(.(X3, Xs)) -> MEMBER_IN_AG(Xs)
9.50/3.45	The graph contains the following edges 1 > 1
9.50/3.45	
9.50/3.45	
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(35)
9.50/3.45	YES
9.50/3.45	
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(36)
9.50/3.45	Obligation:
9.50/3.45	Pi DP problem:
9.50/3.45	The TRS P consists of the following rules:
9.50/3.45	
9.50/3.45	   U1_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys)
9.50/3.45	   SUBSET_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, member_in_ag(X, Ys))
9.50/3.45	
9.50/3.45	The TRS R consists of the following rules:
9.50/3.45	
9.50/3.45	   subset_in_ag([], X1) -> subset_out_ag([], X1)
9.50/3.45	   subset_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, member_in_ag(X, Ys))
9.50/3.45	   member_in_ag(X, .(X, X2)) -> member_out_ag(X, .(X, X2))
9.50/3.45	   member_in_ag(X, .(X3, Xs)) -> U3_ag(X, X3, Xs, member_in_ag(X, Xs))
9.50/3.45	   U3_ag(X, X3, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X3, Xs))
9.50/3.45	   U1_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U2_ag(X, Xs, Ys, subset_in_ag(Xs, Ys))
9.50/3.45	   U2_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys)
9.50/3.45	
9.50/3.45	The argument filtering Pi contains the following mapping:
9.50/3.45	subset_in_ag(x1, x2)  =  subset_in_ag(x2)
9.50/3.45	
9.50/3.45	subset_out_ag(x1, x2)  =  subset_out_ag(x1, x2)
9.50/3.45	
9.50/3.45	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
9.50/3.45	
9.50/3.45	member_in_ag(x1, x2)  =  member_in_ag(x2)
9.50/3.45	
9.50/3.45	.(x1, x2)  =  .(x1, x2)
9.50/3.45	
9.50/3.45	member_out_ag(x1, x2)  =  member_out_ag(x1, x2)
9.50/3.45	
9.50/3.45	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
9.50/3.45	
9.50/3.45	U2_ag(x1, x2, x3, x4)  =  U2_ag(x1, x3, x4)
9.50/3.45	
9.50/3.45	SUBSET_IN_AG(x1, x2)  =  SUBSET_IN_AG(x2)
9.50/3.45	
9.50/3.45	U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
9.50/3.45	
9.50/3.45	
9.50/3.45	We have to consider all (P,R,Pi)-chains
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(37) UsableRulesProof (EQUIVALENT)
9.50/3.45	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(38)
9.50/3.45	Obligation:
9.50/3.45	Pi DP problem:
9.50/3.45	The TRS P consists of the following rules:
9.50/3.45	
9.50/3.45	   U1_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys)
9.50/3.45	   SUBSET_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, member_in_ag(X, Ys))
9.50/3.45	
9.50/3.45	The TRS R consists of the following rules:
9.50/3.45	
9.50/3.45	   member_in_ag(X, .(X, X2)) -> member_out_ag(X, .(X, X2))
9.50/3.45	   member_in_ag(X, .(X3, Xs)) -> U3_ag(X, X3, Xs, member_in_ag(X, Xs))
9.50/3.45	   U3_ag(X, X3, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X3, Xs))
9.50/3.45	
9.50/3.45	The argument filtering Pi contains the following mapping:
9.50/3.45	member_in_ag(x1, x2)  =  member_in_ag(x2)
9.50/3.45	
9.50/3.45	.(x1, x2)  =  .(x1, x2)
9.50/3.45	
9.50/3.45	member_out_ag(x1, x2)  =  member_out_ag(x1, x2)
9.50/3.45	
9.50/3.45	U3_ag(x1, x2, x3, x4)  =  U3_ag(x2, x3, x4)
9.50/3.45	
9.50/3.45	SUBSET_IN_AG(x1, x2)  =  SUBSET_IN_AG(x2)
9.50/3.45	
9.50/3.45	U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
9.50/3.45	
9.50/3.45	
9.50/3.45	We have to consider all (P,R,Pi)-chains
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(39) PiDPToQDPProof (SOUND)
9.50/3.45	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(40)
9.50/3.45	Obligation:
9.50/3.45	Q DP problem:
9.50/3.45	The TRS P consists of the following rules:
9.50/3.45	
9.50/3.45	   U1_AG(Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Ys)
9.50/3.45	   SUBSET_IN_AG(Ys) -> U1_AG(Ys, member_in_ag(Ys))
9.50/3.45	
9.50/3.45	The TRS R consists of the following rules:
9.50/3.45	
9.50/3.45	   member_in_ag(.(X, X2)) -> member_out_ag(X, .(X, X2))
9.50/3.45	   member_in_ag(.(X3, Xs)) -> U3_ag(X3, Xs, member_in_ag(Xs))
9.50/3.45	   U3_ag(X3, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X3, Xs))
9.50/3.45	
9.50/3.45	The set Q consists of the following terms:
9.50/3.45	
9.50/3.45	   member_in_ag(x0)
9.50/3.45	   U3_ag(x0, x1, x2)
9.50/3.45	
9.50/3.45	We have to consider all (P,Q,R)-chains.
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(41) TransformationProof (SOUND)
9.50/3.45	By narrowing [LPAR04] the rule SUBSET_IN_AG(Ys) -> U1_AG(Ys, member_in_ag(Ys)) at position [1] we obtained the following new rules [LPAR04]:
9.50/3.45	
9.50/3.45	   (SUBSET_IN_AG(.(x0, x1)) -> U1_AG(.(x0, x1), member_out_ag(x0, .(x0, x1))),SUBSET_IN_AG(.(x0, x1)) -> U1_AG(.(x0, x1), member_out_ag(x0, .(x0, x1))))
9.50/3.45	   (SUBSET_IN_AG(.(x0, x1)) -> U1_AG(.(x0, x1), U3_ag(x0, x1, member_in_ag(x1))),SUBSET_IN_AG(.(x0, x1)) -> U1_AG(.(x0, x1), U3_ag(x0, x1, member_in_ag(x1))))
9.50/3.45	
9.50/3.45	
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(42)
9.50/3.45	Obligation:
9.50/3.45	Q DP problem:
9.50/3.45	The TRS P consists of the following rules:
9.50/3.45	
9.50/3.45	   U1_AG(Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Ys)
9.50/3.45	   SUBSET_IN_AG(.(x0, x1)) -> U1_AG(.(x0, x1), member_out_ag(x0, .(x0, x1)))
9.50/3.45	   SUBSET_IN_AG(.(x0, x1)) -> U1_AG(.(x0, x1), U3_ag(x0, x1, member_in_ag(x1)))
9.50/3.45	
9.50/3.45	The TRS R consists of the following rules:
9.50/3.45	
9.50/3.45	   member_in_ag(.(X, X2)) -> member_out_ag(X, .(X, X2))
9.50/3.45	   member_in_ag(.(X3, Xs)) -> U3_ag(X3, Xs, member_in_ag(Xs))
9.50/3.45	   U3_ag(X3, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X3, Xs))
9.50/3.45	
9.50/3.45	The set Q consists of the following terms:
9.50/3.45	
9.50/3.45	   member_in_ag(x0)
9.50/3.45	   U3_ag(x0, x1, x2)
9.50/3.45	
9.50/3.45	We have to consider all (P,Q,R)-chains.
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(43) TransformationProof (EQUIVALENT)
9.50/3.45	By instantiating [LPAR04] the rule U1_AG(Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Ys) we obtained the following new rules [LPAR04]:
9.50/3.45	
9.50/3.45	   (U1_AG(.(z0, z1), member_out_ag(z0, .(z0, z1))) -> SUBSET_IN_AG(.(z0, z1)),U1_AG(.(z0, z1), member_out_ag(z0, .(z0, z1))) -> SUBSET_IN_AG(.(z0, z1)))
9.50/3.45	   (U1_AG(.(z0, z1), member_out_ag(x1, .(z0, z1))) -> SUBSET_IN_AG(.(z0, z1)),U1_AG(.(z0, z1), member_out_ag(x1, .(z0, z1))) -> SUBSET_IN_AG(.(z0, z1)))
9.50/3.45	
9.50/3.45	
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(44)
9.50/3.45	Obligation:
9.50/3.45	Q DP problem:
9.50/3.45	The TRS P consists of the following rules:
9.50/3.45	
9.50/3.45	   SUBSET_IN_AG(.(x0, x1)) -> U1_AG(.(x0, x1), member_out_ag(x0, .(x0, x1)))
9.50/3.45	   SUBSET_IN_AG(.(x0, x1)) -> U1_AG(.(x0, x1), U3_ag(x0, x1, member_in_ag(x1)))
9.50/3.45	   U1_AG(.(z0, z1), member_out_ag(z0, .(z0, z1))) -> SUBSET_IN_AG(.(z0, z1))
9.50/3.45	   U1_AG(.(z0, z1), member_out_ag(x1, .(z0, z1))) -> SUBSET_IN_AG(.(z0, z1))
9.50/3.45	
9.50/3.45	The TRS R consists of the following rules:
9.50/3.45	
9.50/3.45	   member_in_ag(.(X, X2)) -> member_out_ag(X, .(X, X2))
9.50/3.45	   member_in_ag(.(X3, Xs)) -> U3_ag(X3, Xs, member_in_ag(Xs))
9.50/3.45	   U3_ag(X3, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(X3, Xs))
9.50/3.45	
9.50/3.45	The set Q consists of the following terms:
9.50/3.45	
9.50/3.45	   member_in_ag(x0)
9.50/3.45	   U3_ag(x0, x1, x2)
9.50/3.45	
9.50/3.45	We have to consider all (P,Q,R)-chains.
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(45) PrologToDTProblemTransformerProof (SOUND)
9.50/3.45	Built DT problem from termination graph DT10.
9.50/3.45	
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9.50/3.45	    "root": 10,
9.50/3.45	    "program": {
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9.50/3.45	        "clauses": [
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9.50/3.45	                "(subset ([]) X1)",
9.50/3.45	                null
9.50/3.45	            ],
9.50/3.45	            [
9.50/3.45	                "(subset (. X Xs) Ys)",
9.50/3.45	                "(',' (member X Ys) (subset Xs Ys))"
9.50/3.45	            ],
9.50/3.45	            [
9.50/3.45	                "(member X (. X X2))",
9.50/3.45	                null
9.50/3.45	            ],
9.50/3.45	            [
9.50/3.45	                "(member X (. X3 Xs))",
9.50/3.45	                "(member X Xs)"
9.50/3.45	            ]
9.50/3.45	        ]
9.50/3.45	    },
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9.50/3.45	                }
9.50/3.45	            },
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9.50/3.45	                    "scope": -1,
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9.50/3.45	                }],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [[
9.50/3.45	                        "(subset T1 (. T106 T107))",
9.50/3.45	                        "(subset ([]) X5)"
9.50/3.45	                    ]],
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9.50/3.45	                    },
9.50/3.45	                    "ground": [
9.50/3.45	                        "T106",
9.50/3.45	                        "T107"
9.50/3.45	                    ],
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9.50/3.45	                }],
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9.50/3.45	                        "T21",
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9.50/3.45	            },
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9.50/3.45	                        "(subset T1 T81)",
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9.50/3.45	                        "(subset T1 T81)",
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9.50/3.45	                        "(subset T1 T81)",
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9.50/3.45	                        "T35",
9.50/3.45	                        "T36"
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9.50/3.45	                        "(subset T1 T81)",
9.50/3.45	                        "(subset ([]) X5)"
9.50/3.45	                    ]],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": ["T81"],
9.50/3.45	                    "free": ["X5"],
9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            },
9.50/3.45	            "164": {
9.50/3.45	                "goal": [],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": [],
9.50/3.45	                    "free": [],
9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            },
9.50/3.45	            "143": {
9.50/3.45	                "goal": [{
9.50/3.45	                    "clause": 1,
9.50/3.45	                    "scope": 1,
9.50/3.45	                    "term": "(subset T1 T2)"
9.50/3.45	                }],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [[
9.50/3.45	                        "(subset T1 T2)",
9.50/3.45	                        "(subset ([]) X5)"
9.50/3.45	                    ]],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": ["T2"],
9.50/3.45	                    "free": ["X5"],
9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            },
9.50/3.45	            "165": {
9.50/3.45	                "goal": [{
9.50/3.45	                    "clause": -1,
9.50/3.45	                    "scope": -1,
9.50/3.45	                    "term": "(member T37 T36)"
9.50/3.45	                }],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": ["T36"],
9.50/3.45	                    "free": [],
9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            },
9.50/3.45	            "166": {
9.50/3.45	                "goal": [{
9.50/3.45	                    "clause": -1,
9.50/3.45	                    "scope": -1,
9.50/3.45	                    "term": "(subset T43 (. T35 T36))"
9.50/3.45	                }],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": [
9.50/3.45	                        "T35",
9.50/3.45	                        "T36"
9.50/3.45	                    ],
9.50/3.45	                    "free": [],
9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            },
9.50/3.45	            "188": {
9.50/3.45	                "goal": [{
9.50/3.45	                    "clause": -1,
9.50/3.45	                    "scope": -1,
9.50/3.45	                    "term": "(subset T94 (. T92 T93))"
9.50/3.45	                }],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [[
9.50/3.45	                        "(subset T1 (. T92 T93))",
9.50/3.45	                        "(subset ([]) X5)"
9.50/3.45	                    ]],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": [
9.50/3.45	                        "T92",
9.50/3.45	                        "T93"
9.50/3.45	                    ],
9.50/3.45	                    "free": ["X5"],
9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            },
9.50/3.45	            "167": {
9.50/3.45	                "goal": [
9.50/3.45	                    {
9.50/3.45	                        "clause": 2,
9.50/3.45	                        "scope": 3,
9.50/3.45	                        "term": "(member T37 T36)"
9.50/3.45	                    },
9.50/3.45	                    {
9.50/3.45	                        "clause": 3,
9.50/3.45	                        "scope": 3,
9.50/3.45	                        "term": "(member T37 T36)"
9.50/3.45	                    }
9.50/3.45	                ],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": ["T36"],
9.50/3.45	                    "free": [],
9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            },
9.50/3.45	            "168": {
9.50/3.45	                "goal": [{
9.50/3.45	                    "clause": 2,
9.50/3.45	                    "scope": 3,
9.50/3.45	                    "term": "(member T37 T36)"
9.50/3.45	                }],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": ["T36"],
9.50/3.45	                    "free": [],
9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            },
9.50/3.45	            "169": {
9.50/3.45	                "goal": [{
9.50/3.45	                    "clause": 3,
9.50/3.45	                    "scope": 3,
9.50/3.45	                    "term": "(member T37 T36)"
9.50/3.45	                }],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": ["T36"],
9.50/3.45	                    "free": [],
9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            },
9.50/3.45	            "21": {
9.50/3.45	                "goal": [
9.50/3.45	                    {
9.50/3.45	                        "clause": 0,
9.50/3.45	                        "scope": 1,
9.50/3.45	                        "term": "(subset T1 T2)"
9.50/3.45	                    },
9.50/3.45	                    {
9.50/3.45	                        "clause": 1,
9.50/3.45	                        "scope": 1,
9.50/3.45	                        "term": "(subset T1 T2)"
9.50/3.45	                    }
9.50/3.45	                ],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": ["T2"],
9.50/3.45	                    "free": [],
9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            }
9.50/3.45	        },
9.50/3.45	        "edges": [
9.50/3.45	            {
9.50/3.45	                "from": 10,
9.50/3.45	                "to": 21,
9.50/3.45	                "label": "CASE"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 21,
9.50/3.45	                "to": 141,
9.50/3.45	                "label": "EVAL with clause\nsubset([], X5).\nand substitutionT1 -> [],\nT2 -> T4,\nX5 -> T4"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 21,
9.50/3.45	                "to": 143,
9.50/3.45	                "label": "EVAL-BACKTRACK"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 141,
9.50/3.45	                "to": 152,
9.50/3.45	                "label": "SUCCESS"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 143,
9.50/3.45	                "to": 181,
9.50/3.45	                "label": "EVAL with clause\nsubset(.(X69, X70), X71) :- ','(member(X69, X71), subset(X70, X71)).\nand substitutionX69 -> T82,\nX70 -> T83,\nT1 -> .(T82, T83),\nT2 -> T81,\nX71 -> T81,\nT79 -> T82,\nT80 -> T83"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 143,
9.50/3.45	                "to": 182,
9.50/3.45	                "label": "EVAL-BACKTRACK"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 152,
9.50/3.45	                "to": 153,
9.50/3.45	                "label": "EVAL with clause\nsubset(.(X9, X10), X11) :- ','(member(X9, X11), subset(X10, X11)).\nand substitutionX9 -> T11,\nX10 -> T12,\nT1 -> .(T11, T12),\nT4 -> T10,\nX11 -> T10,\nT8 -> T11,\nT9 -> T12"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 152,
9.50/3.45	                "to": 154,
9.50/3.45	                "label": "EVAL-BACKTRACK"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 153,
9.50/3.45	                "to": 155,
9.50/3.45	                "label": "CASE"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 155,
9.50/3.45	                "to": 156,
9.50/3.45	                "label": "PARALLEL"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 155,
9.50/3.45	                "to": 157,
9.50/3.45	                "label": "PARALLEL"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 156,
9.50/3.45	                "to": 158,
9.50/3.45	                "label": "EVAL with clause\nmember(X20, .(X20, X21)).\nand substitutionT11 -> T21,\nX20 -> T21,\nX21 -> T22,\nT10 -> .(T21, T22),\nT12 -> T23"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 156,
9.50/3.45	                "to": 159,
9.50/3.45	                "label": "EVAL-BACKTRACK"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 157,
9.50/3.45	                "to": 163,
9.50/3.45	                "label": "EVAL with clause\nmember(X30, .(X31, X32)) :- member(X30, X32).\nand substitutionT11 -> T37,\nX30 -> T37,\nX31 -> T35,\nX32 -> T36,\nT10 -> .(T35, T36),\nT34 -> T37,\nT12 -> T38"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 157,
9.50/3.45	                "to": 164,
9.50/3.45	                "label": "EVAL-BACKTRACK"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 158,
9.50/3.45	                "to": 10,
9.50/3.45	                "label": "INSTANCE with matching:\nT1 -> T23\nT2 -> .(T21, T22)"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 163,
9.50/3.45	                "to": 165,
9.50/3.45	                "label": "SPLIT 1"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 163,
9.50/3.45	                "to": 166,
9.50/3.45	                "label": "SPLIT 2\nnew knowledge:\nT37 is ground\nT36 is ground\nreplacements:T38 -> T43"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 165,
9.50/3.45	                "to": 167,
9.50/3.45	                "label": "CASE"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 166,
9.50/3.45	                "to": 10,
9.50/3.45	                "label": "INSTANCE with matching:\nT1 -> T43\nT2 -> .(T35, T36)"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 167,
9.50/3.45	                "to": 168,
9.50/3.45	                "label": "PARALLEL"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 167,
9.50/3.45	                "to": 169,
9.50/3.45	                "label": "PARALLEL"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 168,
9.50/3.45	                "to": 170,
9.50/3.45	                "label": "EVAL with clause\nmember(X49, .(X49, X50)).\nand substitutionT37 -> T56,\nX49 -> T56,\nX50 -> T57,\nT36 -> .(T56, T57)"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 168,
9.50/3.45	                "to": 171,
9.50/3.45	                "label": "EVAL-BACKTRACK"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 169,
9.50/3.45	                "to": 179,
9.50/3.45	                "label": "EVAL with clause\nmember(X57, .(X58, X59)) :- member(X57, X59).\nand substitutionT37 -> T67,\nX57 -> T67,\nX58 -> T65,\nX59 -> T66,\nT36 -> .(T65, T66),\nT64 -> T67"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 169,
9.50/3.45	                "to": 180,
9.50/3.45	                "label": "EVAL-BACKTRACK"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 170,
9.50/3.45	                "to": 172,
9.50/3.45	                "label": "SUCCESS"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 179,
9.50/3.45	                "to": 165,
9.50/3.45	                "label": "INSTANCE with matching:\nT37 -> T67\nT36 -> T66"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 181,
9.50/3.45	                "to": 183,
9.50/3.45	                "label": "CASE"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 183,
9.50/3.45	                "to": 184,
9.50/3.45	                "label": "PARALLEL"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 183,
9.50/3.45	                "to": 185,
9.50/3.45	                "label": "PARALLEL"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 184,
9.50/3.45	                "to": 188,
9.50/3.45	                "label": "EVAL with clause\nmember(X80, .(X80, X81)).\nand substitutionT82 -> T92,\nX80 -> T92,\nX81 -> T93,\nT81 -> .(T92, T93),\nT83 -> T94"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 184,
9.50/3.45	                "to": 210,
9.50/3.45	                "label": "EVAL-BACKTRACK"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 185,
9.50/3.45	                "to": 211,
9.50/3.45	                "label": "EVAL with clause\nmember(X90, .(X91, X92)) :- member(X90, X92).\nand substitutionT82 -> T108,\nX90 -> T108,\nX91 -> T106,\nX92 -> T107,\nT81 -> .(T106, T107),\nT105 -> T108,\nT83 -> T109"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 185,
9.50/3.45	                "to": 212,
9.50/3.45	                "label": "EVAL-BACKTRACK"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 188,
9.50/3.45	                "to": 10,
9.50/3.45	                "label": "INSTANCE with matching:\nT1 -> T94\nT2 -> .(T92, T93)"
9.50/3.45	            },
9.50/3.45	            {
9.50/3.45	                "from": 211,
9.50/3.45	                "to": 163,
9.50/3.45	                "label": "INSTANCE with matching:\nT37 -> T108\nT36 -> T107\nT38 -> T109\nT35 -> T106"
9.50/3.45	            }
9.50/3.45	        ],
9.50/3.45	        "type": "Graph"
9.50/3.45	    }
9.50/3.45	}
9.50/3.45	
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(46)
9.50/3.45	Obligation:
9.50/3.45	Triples:
9.50/3.45	
9.50/3.45	memberC(X1, .(X2, X3)) :- memberC(X1, X3).
9.50/3.45	pB(X1, X2, X3, X4) :- memberC(X1, X2).
9.50/3.45	pB(X1, X2, X3, X4) :- ','(membercC(X1, X2), subsetA(X3, .(X4, X2))).
9.50/3.45	subsetA(.(X1, X2), .(X1, X3)) :- subsetA(X2, .(X1, X3)).
9.50/3.45	subsetA(.(X1, X2), .(X3, X4)) :- pB(X1, X4, X2, X3).
9.50/3.45	subsetA(.(X1, X2), .(X1, X3)) :- subsetA(X2, .(X1, X3)).
9.50/3.45	subsetA(.(X1, X2), .(X3, X4)) :- pB(X1, X4, X2, X3).
9.50/3.45	
9.50/3.45	Clauses:
9.50/3.45	
9.50/3.45	subsetcA([], X1).
9.50/3.45	subsetcA(.(X1, X2), .(X1, X3)) :- subsetcA(X2, .(X1, X3)).
9.50/3.45	subsetcA(.(X1, X2), .(X3, X4)) :- qcB(X1, X4, X2, X3).
9.50/3.45	subsetcA(.(X1, X2), .(X1, X3)) :- subsetcA(X2, .(X1, X3)).
9.50/3.45	subsetcA(.(X1, X2), .(X3, X4)) :- qcB(X1, X4, X2, X3).
9.50/3.45	membercC(X1, .(X1, X2)).
9.50/3.45	membercC(X1, .(X2, X3)) :- membercC(X1, X3).
9.50/3.45	qcB(X1, X2, X3, X4) :- ','(membercC(X1, X2), subsetcA(X3, .(X4, X2))).
9.50/3.45	
9.50/3.45	Afs:
9.50/3.45	
9.50/3.45	subsetA(x1, x2)  =  subsetA(x2)
9.50/3.45	
9.50/3.45	
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(47) TriplesToPiDPProof (SOUND)
9.50/3.45	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
9.50/3.45	
9.50/3.45	subsetA_in_2: (f,b)
9.50/3.45	
9.50/3.45	pB_in_4: (f,b,f,b)
9.50/3.45	
9.50/3.45	memberC_in_2: (f,b)
9.50/3.45	
9.50/3.45	membercC_in_2: (f,b)
9.50/3.45	
9.50/3.45	Transforming TRIPLES into the following Term Rewriting System:
9.50/3.45	
9.50/3.45	Pi DP problem:
9.50/3.45	The TRS P consists of the following rules:
9.50/3.45	
9.50/3.45	   SUBSETA_IN_AG(.(X1, X2), .(X1, X3)) -> U5_AG(X1, X2, X3, subsetA_in_ag(X2, .(X1, X3)))
9.50/3.45	   SUBSETA_IN_AG(.(X1, X2), .(X1, X3)) -> SUBSETA_IN_AG(X2, .(X1, X3))
9.50/3.45	   SUBSETA_IN_AG(.(X1, X2), .(X3, X4)) -> U6_AG(X1, X2, X3, X4, pB_in_agag(X1, X4, X2, X3))
9.50/3.45	   SUBSETA_IN_AG(.(X1, X2), .(X3, X4)) -> PB_IN_AGAG(X1, X4, X2, X3)
9.50/3.45	   PB_IN_AGAG(X1, X2, X3, X4) -> U2_AGAG(X1, X2, X3, X4, memberC_in_ag(X1, X2))
9.50/3.45	   PB_IN_AGAG(X1, X2, X3, X4) -> MEMBERC_IN_AG(X1, X2)
9.50/3.45	   MEMBERC_IN_AG(X1, .(X2, X3)) -> U1_AG(X1, X2, X3, memberC_in_ag(X1, X3))
9.50/3.45	   MEMBERC_IN_AG(X1, .(X2, X3)) -> MEMBERC_IN_AG(X1, X3)
9.50/3.45	   PB_IN_AGAG(X1, X2, X3, X4) -> U3_AGAG(X1, X2, X3, X4, membercC_in_ag(X1, X2))
9.50/3.45	   U3_AGAG(X1, X2, X3, X4, membercC_out_ag(X1, X2)) -> U4_AGAG(X1, X2, X3, X4, subsetA_in_ag(X3, .(X4, X2)))
9.50/3.45	   U3_AGAG(X1, X2, X3, X4, membercC_out_ag(X1, X2)) -> SUBSETA_IN_AG(X3, .(X4, X2))
9.50/3.45	
9.50/3.45	The TRS R consists of the following rules:
9.50/3.45	
9.50/3.45	   membercC_in_ag(X1, .(X1, X2)) -> membercC_out_ag(X1, .(X1, X2))
9.50/3.45	   membercC_in_ag(X1, .(X2, X3)) -> U10_ag(X1, X2, X3, membercC_in_ag(X1, X3))
9.50/3.45	   U10_ag(X1, X2, X3, membercC_out_ag(X1, X3)) -> membercC_out_ag(X1, .(X2, X3))
9.50/3.45	
9.50/3.45	The argument filtering Pi contains the following mapping:
9.50/3.45	subsetA_in_ag(x1, x2)  =  subsetA_in_ag(x2)
9.50/3.45	
9.50/3.45	.(x1, x2)  =  .(x1, x2)
9.50/3.45	
9.50/3.45	pB_in_agag(x1, x2, x3, x4)  =  pB_in_agag(x2, x4)
9.50/3.45	
9.50/3.45	memberC_in_ag(x1, x2)  =  memberC_in_ag(x2)
9.50/3.45	
9.50/3.45	membercC_in_ag(x1, x2)  =  membercC_in_ag(x2)
9.50/3.45	
9.50/3.45	membercC_out_ag(x1, x2)  =  membercC_out_ag(x1, x2)
9.50/3.45	
9.50/3.45	U10_ag(x1, x2, x3, x4)  =  U10_ag(x2, x3, x4)
9.50/3.45	
9.50/3.45	SUBSETA_IN_AG(x1, x2)  =  SUBSETA_IN_AG(x2)
9.50/3.45	
9.50/3.45	U5_AG(x1, x2, x3, x4)  =  U5_AG(x1, x3, x4)
9.50/3.45	
9.50/3.45	U6_AG(x1, x2, x3, x4, x5)  =  U6_AG(x3, x4, x5)
9.50/3.45	
9.50/3.45	PB_IN_AGAG(x1, x2, x3, x4)  =  PB_IN_AGAG(x2, x4)
9.50/3.45	
9.50/3.45	U2_AGAG(x1, x2, x3, x4, x5)  =  U2_AGAG(x2, x4, x5)
9.50/3.45	
9.50/3.45	MEMBERC_IN_AG(x1, x2)  =  MEMBERC_IN_AG(x2)
9.50/3.45	
9.50/3.45	U1_AG(x1, x2, x3, x4)  =  U1_AG(x2, x3, x4)
9.50/3.45	
9.50/3.45	U3_AGAG(x1, x2, x3, x4, x5)  =  U3_AGAG(x2, x4, x5)
9.50/3.45	
9.50/3.45	U4_AGAG(x1, x2, x3, x4, x5)  =  U4_AGAG(x1, x2, x4, x5)
9.50/3.45	
9.50/3.45	
9.50/3.45	We have to consider all (P,R,Pi)-chains
9.50/3.45	
9.50/3.45	
9.50/3.45	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
9.50/3.45	
9.50/3.45	
9.50/3.45	
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(48)
9.50/3.45	Obligation:
9.50/3.45	Pi DP problem:
9.50/3.45	The TRS P consists of the following rules:
9.50/3.45	
9.50/3.45	   SUBSETA_IN_AG(.(X1, X2), .(X1, X3)) -> U5_AG(X1, X2, X3, subsetA_in_ag(X2, .(X1, X3)))
9.50/3.45	   SUBSETA_IN_AG(.(X1, X2), .(X1, X3)) -> SUBSETA_IN_AG(X2, .(X1, X3))
9.50/3.45	   SUBSETA_IN_AG(.(X1, X2), .(X3, X4)) -> U6_AG(X1, X2, X3, X4, pB_in_agag(X1, X4, X2, X3))
9.50/3.45	   SUBSETA_IN_AG(.(X1, X2), .(X3, X4)) -> PB_IN_AGAG(X1, X4, X2, X3)
9.50/3.45	   PB_IN_AGAG(X1, X2, X3, X4) -> U2_AGAG(X1, X2, X3, X4, memberC_in_ag(X1, X2))
9.50/3.45	   PB_IN_AGAG(X1, X2, X3, X4) -> MEMBERC_IN_AG(X1, X2)
9.50/3.45	   MEMBERC_IN_AG(X1, .(X2, X3)) -> U1_AG(X1, X2, X3, memberC_in_ag(X1, X3))
9.50/3.45	   MEMBERC_IN_AG(X1, .(X2, X3)) -> MEMBERC_IN_AG(X1, X3)
9.50/3.45	   PB_IN_AGAG(X1, X2, X3, X4) -> U3_AGAG(X1, X2, X3, X4, membercC_in_ag(X1, X2))
9.50/3.45	   U3_AGAG(X1, X2, X3, X4, membercC_out_ag(X1, X2)) -> U4_AGAG(X1, X2, X3, X4, subsetA_in_ag(X3, .(X4, X2)))
9.50/3.45	   U3_AGAG(X1, X2, X3, X4, membercC_out_ag(X1, X2)) -> SUBSETA_IN_AG(X3, .(X4, X2))
9.50/3.45	
9.50/3.45	The TRS R consists of the following rules:
9.50/3.45	
9.50/3.45	   membercC_in_ag(X1, .(X1, X2)) -> membercC_out_ag(X1, .(X1, X2))
9.50/3.45	   membercC_in_ag(X1, .(X2, X3)) -> U10_ag(X1, X2, X3, membercC_in_ag(X1, X3))
9.50/3.45	   U10_ag(X1, X2, X3, membercC_out_ag(X1, X3)) -> membercC_out_ag(X1, .(X2, X3))
9.50/3.45	
9.50/3.45	The argument filtering Pi contains the following mapping:
9.50/3.45	subsetA_in_ag(x1, x2)  =  subsetA_in_ag(x2)
9.50/3.45	
9.50/3.45	.(x1, x2)  =  .(x1, x2)
9.50/3.45	
9.50/3.45	pB_in_agag(x1, x2, x3, x4)  =  pB_in_agag(x2, x4)
9.50/3.45	
9.50/3.45	memberC_in_ag(x1, x2)  =  memberC_in_ag(x2)
9.50/3.45	
9.50/3.45	membercC_in_ag(x1, x2)  =  membercC_in_ag(x2)
9.50/3.45	
9.50/3.45	membercC_out_ag(x1, x2)  =  membercC_out_ag(x1, x2)
9.50/3.45	
9.50/3.45	U10_ag(x1, x2, x3, x4)  =  U10_ag(x2, x3, x4)
9.50/3.45	
9.50/3.45	SUBSETA_IN_AG(x1, x2)  =  SUBSETA_IN_AG(x2)
9.50/3.45	
9.50/3.45	U5_AG(x1, x2, x3, x4)  =  U5_AG(x1, x3, x4)
9.50/3.45	
9.50/3.45	U6_AG(x1, x2, x3, x4, x5)  =  U6_AG(x3, x4, x5)
9.50/3.45	
9.50/3.45	PB_IN_AGAG(x1, x2, x3, x4)  =  PB_IN_AGAG(x2, x4)
9.50/3.45	
9.50/3.45	U2_AGAG(x1, x2, x3, x4, x5)  =  U2_AGAG(x2, x4, x5)
9.50/3.45	
9.50/3.45	MEMBERC_IN_AG(x1, x2)  =  MEMBERC_IN_AG(x2)
9.50/3.45	
9.50/3.45	U1_AG(x1, x2, x3, x4)  =  U1_AG(x2, x3, x4)
9.50/3.45	
9.50/3.45	U3_AGAG(x1, x2, x3, x4, x5)  =  U3_AGAG(x2, x4, x5)
9.50/3.45	
9.50/3.45	U4_AGAG(x1, x2, x3, x4, x5)  =  U4_AGAG(x1, x2, x4, x5)
9.50/3.45	
9.50/3.45	
9.50/3.45	We have to consider all (P,R,Pi)-chains
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(49) DependencyGraphProof (EQUIVALENT)
9.50/3.45	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(50)
9.50/3.45	Complex Obligation (AND)
9.50/3.45	
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(51)
9.50/3.45	Obligation:
9.50/3.45	Pi DP problem:
9.50/3.45	The TRS P consists of the following rules:
9.50/3.45	
9.50/3.45	   MEMBERC_IN_AG(X1, .(X2, X3)) -> MEMBERC_IN_AG(X1, X3)
9.50/3.45	
9.50/3.45	The TRS R consists of the following rules:
9.50/3.45	
9.50/3.45	   membercC_in_ag(X1, .(X1, X2)) -> membercC_out_ag(X1, .(X1, X2))
9.50/3.45	   membercC_in_ag(X1, .(X2, X3)) -> U10_ag(X1, X2, X3, membercC_in_ag(X1, X3))
9.50/3.45	   U10_ag(X1, X2, X3, membercC_out_ag(X1, X3)) -> membercC_out_ag(X1, .(X2, X3))
9.50/3.45	
9.50/3.45	The argument filtering Pi contains the following mapping:
9.50/3.45	.(x1, x2)  =  .(x1, x2)
9.50/3.45	
9.50/3.45	membercC_in_ag(x1, x2)  =  membercC_in_ag(x2)
9.50/3.45	
9.50/3.45	membercC_out_ag(x1, x2)  =  membercC_out_ag(x1, x2)
9.50/3.45	
9.50/3.45	U10_ag(x1, x2, x3, x4)  =  U10_ag(x2, x3, x4)
9.50/3.45	
9.50/3.45	MEMBERC_IN_AG(x1, x2)  =  MEMBERC_IN_AG(x2)
9.50/3.45	
9.50/3.45	
9.50/3.45	We have to consider all (P,R,Pi)-chains
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(52) UsableRulesProof (EQUIVALENT)
9.50/3.45	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(53)
9.50/3.45	Obligation:
9.50/3.45	Pi DP problem:
9.50/3.45	The TRS P consists of the following rules:
9.50/3.45	
9.50/3.45	   MEMBERC_IN_AG(X1, .(X2, X3)) -> MEMBERC_IN_AG(X1, X3)
9.50/3.45	
9.50/3.45	R is empty.
9.50/3.45	The argument filtering Pi contains the following mapping:
9.50/3.45	.(x1, x2)  =  .(x1, x2)
9.50/3.45	
9.50/3.45	MEMBERC_IN_AG(x1, x2)  =  MEMBERC_IN_AG(x2)
9.50/3.45	
9.50/3.45	
9.50/3.45	We have to consider all (P,R,Pi)-chains
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(54) PiDPToQDPProof (SOUND)
9.50/3.45	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(55)
9.50/3.45	Obligation:
9.50/3.45	Q DP problem:
9.50/3.45	The TRS P consists of the following rules:
9.50/3.45	
9.50/3.45	   MEMBERC_IN_AG(.(X2, X3)) -> MEMBERC_IN_AG(X3)
9.50/3.45	
9.50/3.45	R is empty.
9.50/3.45	Q is empty.
9.50/3.45	We have to consider all (P,Q,R)-chains.
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(56) QDPSizeChangeProof (EQUIVALENT)
9.50/3.45	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. 
9.50/3.45	
9.50/3.45	From the DPs we obtained the following set of size-change graphs:
9.50/3.45	*MEMBERC_IN_AG(.(X2, X3)) -> MEMBERC_IN_AG(X3)
9.50/3.45	The graph contains the following edges 1 > 1
9.50/3.45	
9.50/3.45	
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(57)
9.50/3.45	YES
9.50/3.45	
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(58)
9.50/3.45	Obligation:
9.50/3.45	Pi DP problem:
9.50/3.45	The TRS P consists of the following rules:
9.50/3.45	
9.50/3.45	   SUBSETA_IN_AG(.(X1, X2), .(X3, X4)) -> PB_IN_AGAG(X1, X4, X2, X3)
9.50/3.45	   PB_IN_AGAG(X1, X2, X3, X4) -> U3_AGAG(X1, X2, X3, X4, membercC_in_ag(X1, X2))
9.50/3.45	   U3_AGAG(X1, X2, X3, X4, membercC_out_ag(X1, X2)) -> SUBSETA_IN_AG(X3, .(X4, X2))
9.50/3.45	   SUBSETA_IN_AG(.(X1, X2), .(X1, X3)) -> SUBSETA_IN_AG(X2, .(X1, X3))
9.50/3.45	
9.50/3.45	The TRS R consists of the following rules:
9.50/3.45	
9.50/3.45	   membercC_in_ag(X1, .(X1, X2)) -> membercC_out_ag(X1, .(X1, X2))
9.50/3.45	   membercC_in_ag(X1, .(X2, X3)) -> U10_ag(X1, X2, X3, membercC_in_ag(X1, X3))
9.50/3.45	   U10_ag(X1, X2, X3, membercC_out_ag(X1, X3)) -> membercC_out_ag(X1, .(X2, X3))
9.50/3.45	
9.50/3.45	The argument filtering Pi contains the following mapping:
9.50/3.45	.(x1, x2)  =  .(x1, x2)
9.50/3.45	
9.50/3.45	membercC_in_ag(x1, x2)  =  membercC_in_ag(x2)
9.50/3.45	
9.50/3.45	membercC_out_ag(x1, x2)  =  membercC_out_ag(x1, x2)
9.50/3.45	
9.50/3.45	U10_ag(x1, x2, x3, x4)  =  U10_ag(x2, x3, x4)
9.50/3.45	
9.50/3.45	SUBSETA_IN_AG(x1, x2)  =  SUBSETA_IN_AG(x2)
9.50/3.45	
9.50/3.45	PB_IN_AGAG(x1, x2, x3, x4)  =  PB_IN_AGAG(x2, x4)
9.50/3.45	
9.50/3.45	U3_AGAG(x1, x2, x3, x4, x5)  =  U3_AGAG(x2, x4, x5)
9.50/3.45	
9.50/3.45	
9.50/3.45	We have to consider all (P,R,Pi)-chains
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(59) PiDPToQDPProof (SOUND)
9.50/3.45	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(60)
9.50/3.45	Obligation:
9.50/3.45	Q DP problem:
9.50/3.45	The TRS P consists of the following rules:
9.50/3.45	
9.50/3.45	   SUBSETA_IN_AG(.(X3, X4)) -> PB_IN_AGAG(X4, X3)
9.50/3.45	   PB_IN_AGAG(X2, X4) -> U3_AGAG(X2, X4, membercC_in_ag(X2))
9.50/3.45	   U3_AGAG(X2, X4, membercC_out_ag(X1, X2)) -> SUBSETA_IN_AG(.(X4, X2))
9.50/3.45	   SUBSETA_IN_AG(.(X1, X3)) -> SUBSETA_IN_AG(.(X1, X3))
9.50/3.45	
9.50/3.45	The TRS R consists of the following rules:
9.50/3.45	
9.50/3.45	   membercC_in_ag(.(X1, X2)) -> membercC_out_ag(X1, .(X1, X2))
9.50/3.45	   membercC_in_ag(.(X2, X3)) -> U10_ag(X2, X3, membercC_in_ag(X3))
9.50/3.45	   U10_ag(X2, X3, membercC_out_ag(X1, X3)) -> membercC_out_ag(X1, .(X2, X3))
9.50/3.45	
9.50/3.45	The set Q consists of the following terms:
9.50/3.45	
9.50/3.45	   membercC_in_ag(x0)
9.50/3.45	   U10_ag(x0, x1, x2)
9.50/3.45	
9.50/3.45	We have to consider all (P,Q,R)-chains.
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(61) TransformationProof (SOUND)
9.50/3.45	By narrowing [LPAR04] the rule PB_IN_AGAG(X2, X4) -> U3_AGAG(X2, X4, membercC_in_ag(X2)) at position [2] we obtained the following new rules [LPAR04]:
9.50/3.45	
9.50/3.45	   (PB_IN_AGAG(.(x0, x1), y1) -> U3_AGAG(.(x0, x1), y1, membercC_out_ag(x0, .(x0, x1))),PB_IN_AGAG(.(x0, x1), y1) -> U3_AGAG(.(x0, x1), y1, membercC_out_ag(x0, .(x0, x1))))
9.50/3.45	   (PB_IN_AGAG(.(x0, x1), y1) -> U3_AGAG(.(x0, x1), y1, U10_ag(x0, x1, membercC_in_ag(x1))),PB_IN_AGAG(.(x0, x1), y1) -> U3_AGAG(.(x0, x1), y1, U10_ag(x0, x1, membercC_in_ag(x1))))
9.50/3.45	
9.50/3.45	
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(62)
9.50/3.45	Obligation:
9.50/3.45	Q DP problem:
9.50/3.45	The TRS P consists of the following rules:
9.50/3.45	
9.50/3.45	   SUBSETA_IN_AG(.(X3, X4)) -> PB_IN_AGAG(X4, X3)
9.50/3.45	   U3_AGAG(X2, X4, membercC_out_ag(X1, X2)) -> SUBSETA_IN_AG(.(X4, X2))
9.50/3.45	   SUBSETA_IN_AG(.(X1, X3)) -> SUBSETA_IN_AG(.(X1, X3))
9.50/3.45	   PB_IN_AGAG(.(x0, x1), y1) -> U3_AGAG(.(x0, x1), y1, membercC_out_ag(x0, .(x0, x1)))
9.50/3.45	   PB_IN_AGAG(.(x0, x1), y1) -> U3_AGAG(.(x0, x1), y1, U10_ag(x0, x1, membercC_in_ag(x1)))
9.50/3.45	
9.50/3.45	The TRS R consists of the following rules:
9.50/3.45	
9.50/3.45	   membercC_in_ag(.(X1, X2)) -> membercC_out_ag(X1, .(X1, X2))
9.50/3.45	   membercC_in_ag(.(X2, X3)) -> U10_ag(X2, X3, membercC_in_ag(X3))
9.50/3.45	   U10_ag(X2, X3, membercC_out_ag(X1, X3)) -> membercC_out_ag(X1, .(X2, X3))
9.50/3.45	
9.50/3.45	The set Q consists of the following terms:
9.50/3.45	
9.50/3.45	   membercC_in_ag(x0)
9.50/3.45	   U10_ag(x0, x1, x2)
9.50/3.45	
9.50/3.45	We have to consider all (P,Q,R)-chains.
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(63) TransformationProof (EQUIVALENT)
9.50/3.45	By instantiating [LPAR04] the rule U3_AGAG(X2, X4, membercC_out_ag(X1, X2)) -> SUBSETA_IN_AG(.(X4, X2)) we obtained the following new rules [LPAR04]:
9.50/3.45	
9.50/3.45	   (U3_AGAG(.(z0, z1), z2, membercC_out_ag(z0, .(z0, z1))) -> SUBSETA_IN_AG(.(z2, .(z0, z1))),U3_AGAG(.(z0, z1), z2, membercC_out_ag(z0, .(z0, z1))) -> SUBSETA_IN_AG(.(z2, .(z0, z1))))
9.50/3.45	   (U3_AGAG(.(z0, z1), z2, membercC_out_ag(x2, .(z0, z1))) -> SUBSETA_IN_AG(.(z2, .(z0, z1))),U3_AGAG(.(z0, z1), z2, membercC_out_ag(x2, .(z0, z1))) -> SUBSETA_IN_AG(.(z2, .(z0, z1))))
9.50/3.45	
9.50/3.45	
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(64)
9.50/3.45	Obligation:
9.50/3.45	Q DP problem:
9.50/3.45	The TRS P consists of the following rules:
9.50/3.45	
9.50/3.45	   SUBSETA_IN_AG(.(X3, X4)) -> PB_IN_AGAG(X4, X3)
9.50/3.45	   SUBSETA_IN_AG(.(X1, X3)) -> SUBSETA_IN_AG(.(X1, X3))
9.50/3.45	   PB_IN_AGAG(.(x0, x1), y1) -> U3_AGAG(.(x0, x1), y1, membercC_out_ag(x0, .(x0, x1)))
9.50/3.45	   PB_IN_AGAG(.(x0, x1), y1) -> U3_AGAG(.(x0, x1), y1, U10_ag(x0, x1, membercC_in_ag(x1)))
9.50/3.45	   U3_AGAG(.(z0, z1), z2, membercC_out_ag(z0, .(z0, z1))) -> SUBSETA_IN_AG(.(z2, .(z0, z1)))
9.50/3.45	   U3_AGAG(.(z0, z1), z2, membercC_out_ag(x2, .(z0, z1))) -> SUBSETA_IN_AG(.(z2, .(z0, z1)))
9.50/3.45	
9.50/3.45	The TRS R consists of the following rules:
9.50/3.45	
9.50/3.45	   membercC_in_ag(.(X1, X2)) -> membercC_out_ag(X1, .(X1, X2))
9.50/3.45	   membercC_in_ag(.(X2, X3)) -> U10_ag(X2, X3, membercC_in_ag(X3))
9.50/3.45	   U10_ag(X2, X3, membercC_out_ag(X1, X3)) -> membercC_out_ag(X1, .(X2, X3))
9.50/3.45	
9.50/3.45	The set Q consists of the following terms:
9.50/3.45	
9.50/3.45	   membercC_in_ag(x0)
9.50/3.45	   U10_ag(x0, x1, x2)
9.50/3.45	
9.50/3.45	We have to consider all (P,Q,R)-chains.
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(65) TransformationProof (EQUIVALENT)
9.50/3.45	By forward instantiating [JAR06] the rule SUBSETA_IN_AG(.(X3, X4)) -> PB_IN_AGAG(X4, X3) we obtained the following new rules [LPAR04]:
9.50/3.45	
9.50/3.45	   (SUBSETA_IN_AG(.(x0, .(y_0, y_1))) -> PB_IN_AGAG(.(y_0, y_1), x0),SUBSETA_IN_AG(.(x0, .(y_0, y_1))) -> PB_IN_AGAG(.(y_0, y_1), x0))
9.50/3.45	
9.50/3.45	
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(66)
9.50/3.45	Obligation:
9.50/3.45	Q DP problem:
9.50/3.45	The TRS P consists of the following rules:
9.50/3.45	
9.50/3.45	   SUBSETA_IN_AG(.(X1, X3)) -> SUBSETA_IN_AG(.(X1, X3))
9.50/3.45	   PB_IN_AGAG(.(x0, x1), y1) -> U3_AGAG(.(x0, x1), y1, membercC_out_ag(x0, .(x0, x1)))
9.50/3.45	   PB_IN_AGAG(.(x0, x1), y1) -> U3_AGAG(.(x0, x1), y1, U10_ag(x0, x1, membercC_in_ag(x1)))
9.50/3.45	   U3_AGAG(.(z0, z1), z2, membercC_out_ag(z0, .(z0, z1))) -> SUBSETA_IN_AG(.(z2, .(z0, z1)))
9.50/3.45	   U3_AGAG(.(z0, z1), z2, membercC_out_ag(x2, .(z0, z1))) -> SUBSETA_IN_AG(.(z2, .(z0, z1)))
9.50/3.45	   SUBSETA_IN_AG(.(x0, .(y_0, y_1))) -> PB_IN_AGAG(.(y_0, y_1), x0)
9.50/3.45	
9.50/3.45	The TRS R consists of the following rules:
9.50/3.45	
9.50/3.45	   membercC_in_ag(.(X1, X2)) -> membercC_out_ag(X1, .(X1, X2))
9.50/3.45	   membercC_in_ag(.(X2, X3)) -> U10_ag(X2, X3, membercC_in_ag(X3))
9.50/3.45	   U10_ag(X2, X3, membercC_out_ag(X1, X3)) -> membercC_out_ag(X1, .(X2, X3))
9.50/3.45	
9.50/3.45	The set Q consists of the following terms:
9.50/3.45	
9.50/3.45	   membercC_in_ag(x0)
9.50/3.45	   U10_ag(x0, x1, x2)
9.50/3.45	
9.50/3.45	We have to consider all (P,Q,R)-chains.
9.50/3.45	----------------------------------------
9.50/3.45	
9.50/3.45	(67) PrologToTRSTransformerProof (SOUND)
9.50/3.45	Transformed Prolog program to TRS.
9.50/3.45	
9.50/3.45	{
9.50/3.45	    "root": 9,
9.50/3.45	    "program": {
9.50/3.45	        "directives": [],
9.50/3.45	        "clauses": [
9.50/3.45	            [
9.50/3.45	                "(subset ([]) X1)",
9.50/3.45	                null
9.50/3.45	            ],
9.50/3.45	            [
9.50/3.45	                "(subset (. X Xs) Ys)",
9.50/3.45	                "(',' (member X Ys) (subset Xs Ys))"
9.50/3.45	            ],
9.50/3.45	            [
9.50/3.45	                "(member X (. X X2))",
9.50/3.45	                null
9.50/3.45	            ],
9.50/3.45	            [
9.50/3.45	                "(member X (. X3 Xs))",
9.50/3.45	                "(member X Xs)"
9.50/3.45	            ]
9.50/3.45	        ]
9.50/3.45	    },
9.50/3.45	    "graph": {
9.50/3.45	        "nodes": {
9.50/3.45	            "24": {
9.50/3.45	                "goal": [
9.50/3.45	                    {
9.50/3.45	                        "clause": 0,
9.50/3.45	                        "scope": 1,
9.50/3.45	                        "term": "(subset T1 T2)"
9.50/3.45	                    },
9.50/3.45	                    {
9.50/3.45	                        "clause": 1,
9.50/3.45	                        "scope": 1,
9.50/3.45	                        "term": "(subset T1 T2)"
9.50/3.45	                    }
9.50/3.45	                ],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": ["T2"],
9.50/3.45	                    "free": [],
9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            },
9.50/3.45	            "25": {
9.50/3.45	                "goal": [{
9.50/3.45	                    "clause": 0,
9.50/3.45	                    "scope": 1,
9.50/3.45	                    "term": "(subset T1 T2)"
9.50/3.45	                }],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": ["T2"],
9.50/3.45	                    "free": [],
9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            },
9.50/3.45	            "26": {
9.50/3.45	                "goal": [{
9.50/3.45	                    "clause": 1,
9.50/3.45	                    "scope": 1,
9.50/3.45	                    "term": "(subset T1 T2)"
9.50/3.45	                }],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [],
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9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": ["T2"],
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9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            },
9.50/3.45	            "type": "Nodes",
9.50/3.45	            "150": {
9.50/3.45	                "goal": [{
9.50/3.45	                    "clause": -1,
9.50/3.45	                    "scope": -1,
9.50/3.45	                    "term": "(',' (member T17 T16) (subset T18 T16))"
9.50/3.45	                }],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": ["T16"],
9.50/3.45	                    "free": [],
9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            },
9.50/3.45	            "140": {
9.50/3.45	                "goal": [{
9.50/3.45	                    "clause": -1,
9.50/3.45	                    "scope": -1,
9.50/3.45	                    "term": "(true)"
9.50/3.45	                }],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": [],
9.50/3.45	                    "free": [],
9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            },
9.50/3.45	            "151": {
9.50/3.45	                "goal": [],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": [],
9.50/3.45	                    "free": [],
9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            },
9.50/3.45	            "142": {
9.50/3.45	                "goal": [],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": [],
9.50/3.45	                    "free": [],
9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            },
9.50/3.45	            "200": {
9.50/3.45	                "goal": [{
9.50/3.45	                    "clause": -1,
9.50/3.45	                    "scope": -1,
9.50/3.45	                    "term": "(member T17 T16)"
9.50/3.45	                }],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": ["T16"],
9.50/3.45	                    "free": [],
9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            },
9.50/3.45	            "201": {
9.50/3.45	                "goal": [{
9.50/3.45	                    "clause": -1,
9.50/3.45	                    "scope": -1,
9.50/3.45	                    "term": "(subset T23 T16)"
9.50/3.45	                }],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": ["T16"],
9.50/3.45	                    "free": [],
9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            },
9.50/3.45	            "147": {
9.50/3.45	                "goal": [],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": [],
9.50/3.45	                    "free": [],
9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            },
9.50/3.45	            "202": {
9.50/3.45	                "goal": [
9.50/3.45	                    {
9.50/3.45	                        "clause": 2,
9.50/3.45	                        "scope": 2,
9.50/3.45	                        "term": "(member T17 T16)"
9.50/3.45	                    },
9.50/3.45	                    {
9.50/3.45	                        "clause": 3,
9.50/3.45	                        "scope": 2,
9.50/3.45	                        "term": "(member T17 T16)"
9.50/3.45	                    }
9.50/3.45	                ],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": ["T16"],
9.50/3.45	                    "free": [],
9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            },
9.50/3.45	            "203": {
9.50/3.45	                "goal": [{
9.50/3.45	                    "clause": 2,
9.50/3.45	                    "scope": 2,
9.50/3.45	                    "term": "(member T17 T16)"
9.50/3.45	                }],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.45	                        "relations": []
9.50/3.45	                    },
9.50/3.45	                    "ground": ["T16"],
9.50/3.45	                    "free": [],
9.50/3.45	                    "exprvars": []
9.50/3.45	                }
9.50/3.45	            },
9.50/3.45	            "204": {
9.50/3.45	                "goal": [{
9.50/3.45	                    "clause": 3,
9.50/3.45	                    "scope": 2,
9.50/3.45	                    "term": "(member T17 T16)"
9.50/3.45	                }],
9.50/3.45	                "kb": {
9.50/3.45	                    "nonunifying": [],
9.50/3.45	                    "intvars": {},
9.50/3.45	                    "arithmetic": {
9.50/3.45	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": ["T16"],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "205": {
9.50/3.46	                "goal": [{
9.50/3.46	                    "clause": -1,
9.50/3.46	                    "scope": -1,
9.50/3.46	                    "term": "(true)"
9.50/3.46	                }],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": [],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "206": {
9.50/3.46	                "goal": [],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": [],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "9": {
9.50/3.46	                "goal": [{
9.50/3.46	                    "clause": -1,
9.50/3.46	                    "scope": -1,
9.50/3.46	                    "term": "(subset T1 T2)"
9.50/3.46	                }],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": ["T2"],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "207": {
9.50/3.46	                "goal": [],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": [],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "208": {
9.50/3.46	                "goal": [{
9.50/3.46	                    "clause": -1,
9.50/3.46	                    "scope": -1,
9.50/3.46	                    "term": "(member T47 T46)"
9.50/3.46	                }],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": ["T46"],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "209": {
9.50/3.46	                "goal": [],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": [],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            }
9.50/3.46	        },
9.50/3.46	        "edges": [
9.50/3.46	            {
9.50/3.46	                "from": 9,
9.50/3.46	                "to": 24,
9.50/3.46	                "label": "CASE"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 24,
9.50/3.46	                "to": 25,
9.50/3.46	                "label": "PARALLEL"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 24,
9.50/3.46	                "to": 26,
9.50/3.46	                "label": "PARALLEL"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 25,
9.50/3.46	                "to": 140,
9.50/3.46	                "label": "EVAL with clause\nsubset([], X8).\nand substitutionT1 -> [],\nT2 -> T7,\nX8 -> T7"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 25,
9.50/3.46	                "to": 142,
9.50/3.46	                "label": "EVAL-BACKTRACK"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 26,
9.50/3.46	                "to": 150,
9.50/3.46	                "label": "EVAL with clause\nsubset(.(X15, X16), X17) :- ','(member(X15, X17), subset(X16, X17)).\nand substitutionX15 -> T17,\nX16 -> T18,\nT1 -> .(T17, T18),\nT2 -> T16,\nX17 -> T16,\nT14 -> T17,\nT15 -> T18"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 26,
9.50/3.46	                "to": 151,
9.50/3.46	                "label": "EVAL-BACKTRACK"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 140,
9.50/3.46	                "to": 147,
9.50/3.46	                "label": "SUCCESS"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 150,
9.50/3.46	                "to": 200,
9.50/3.46	                "label": "SPLIT 1"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 150,
9.50/3.46	                "to": 201,
9.50/3.46	                "label": "SPLIT 2\nnew knowledge:\nT17 is ground\nT16 is ground\nreplacements:T18 -> T23"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 200,
9.50/3.46	                "to": 202,
9.50/3.46	                "label": "CASE"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 201,
9.50/3.46	                "to": 9,
9.50/3.46	                "label": "INSTANCE with matching:\nT1 -> T23\nT2 -> T16"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 202,
9.50/3.46	                "to": 203,
9.50/3.46	                "label": "PARALLEL"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 202,
9.50/3.46	                "to": 204,
9.50/3.46	                "label": "PARALLEL"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 203,
9.50/3.46	                "to": 205,
9.50/3.46	                "label": "EVAL with clause\nmember(X34, .(X34, X35)).\nand substitutionT17 -> T36,\nX34 -> T36,\nX35 -> T37,\nT16 -> .(T36, T37)"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 203,
9.50/3.46	                "to": 206,
9.50/3.46	                "label": "EVAL-BACKTRACK"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 204,
9.50/3.46	                "to": 208,
9.50/3.46	                "label": "EVAL with clause\nmember(X42, .(X43, X44)) :- member(X42, X44).\nand substitutionT17 -> T47,\nX42 -> T47,\nX43 -> T45,\nX44 -> T46,\nT16 -> .(T45, T46),\nT44 -> T47"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 204,
9.50/3.46	                "to": 209,
9.50/3.46	                "label": "EVAL-BACKTRACK"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 205,
9.50/3.46	                "to": 207,
9.50/3.46	                "label": "SUCCESS"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 208,
9.50/3.46	                "to": 200,
9.50/3.46	                "label": "INSTANCE with matching:\nT17 -> T47\nT16 -> T46"
9.50/3.46	            }
9.50/3.46	        ],
9.50/3.46	        "type": "Graph"
9.50/3.46	    }
9.50/3.46	}
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(68)
9.50/3.46	Obligation:
9.50/3.46	Q restricted rewrite system:
9.50/3.46	The TRS R consists of the following rules:
9.50/3.46	
9.50/3.46	   f9_in(T7) -> f9_out1([])
9.50/3.46	   f9_in(T16) -> U1(f150_in(T16), T16)
9.50/3.46	   U1(f150_out1(T17, T18), T16) -> f9_out1(.(T17, T18))
9.50/3.46	   f200_in(.(T36, T37)) -> f200_out1(T36)
9.50/3.46	   f200_in(.(T45, T46)) -> U2(f200_in(T46), .(T45, T46))
9.50/3.46	   U2(f200_out1(T47), .(T45, T46)) -> f200_out1(T47)
9.50/3.46	   f150_in(T16) -> U3(f200_in(T16), T16)
9.50/3.46	   U3(f200_out1(T17), T16) -> U4(f9_in(T16), T16, T17)
9.50/3.46	   U4(f9_out1(T23), T16, T17) -> f150_out1(T17, T23)
9.50/3.46	
9.50/3.46	Q is empty.
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(69) DependencyPairsProof (EQUIVALENT)
9.50/3.46	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(70)
9.50/3.46	Obligation:
9.50/3.46	Q DP problem:
9.50/3.46	The TRS P consists of the following rules:
9.50/3.46	
9.50/3.46	   F9_IN(T16) -> U1^1(f150_in(T16), T16)
9.50/3.46	   F9_IN(T16) -> F150_IN(T16)
9.50/3.46	   F200_IN(.(T45, T46)) -> U2^1(f200_in(T46), .(T45, T46))
9.50/3.46	   F200_IN(.(T45, T46)) -> F200_IN(T46)
9.50/3.46	   F150_IN(T16) -> U3^1(f200_in(T16), T16)
9.50/3.46	   F150_IN(T16) -> F200_IN(T16)
9.50/3.46	   U3^1(f200_out1(T17), T16) -> U4^1(f9_in(T16), T16, T17)
9.50/3.46	   U3^1(f200_out1(T17), T16) -> F9_IN(T16)
9.50/3.46	
9.50/3.46	The TRS R consists of the following rules:
9.50/3.46	
9.50/3.46	   f9_in(T7) -> f9_out1([])
9.50/3.46	   f9_in(T16) -> U1(f150_in(T16), T16)
9.50/3.46	   U1(f150_out1(T17, T18), T16) -> f9_out1(.(T17, T18))
9.50/3.46	   f200_in(.(T36, T37)) -> f200_out1(T36)
9.50/3.46	   f200_in(.(T45, T46)) -> U2(f200_in(T46), .(T45, T46))
9.50/3.46	   U2(f200_out1(T47), .(T45, T46)) -> f200_out1(T47)
9.50/3.46	   f150_in(T16) -> U3(f200_in(T16), T16)
9.50/3.46	   U3(f200_out1(T17), T16) -> U4(f9_in(T16), T16, T17)
9.50/3.46	   U4(f9_out1(T23), T16, T17) -> f150_out1(T17, T23)
9.50/3.46	
9.50/3.46	Q is empty.
9.50/3.46	We have to consider all minimal (P,Q,R)-chains.
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(71) DependencyGraphProof (EQUIVALENT)
9.50/3.46	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes.
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(72)
9.50/3.46	Complex Obligation (AND)
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(73)
9.50/3.46	Obligation:
9.50/3.46	Q DP problem:
9.50/3.46	The TRS P consists of the following rules:
9.50/3.46	
9.50/3.46	   F200_IN(.(T45, T46)) -> F200_IN(T46)
9.50/3.46	
9.50/3.46	The TRS R consists of the following rules:
9.50/3.46	
9.50/3.46	   f9_in(T7) -> f9_out1([])
9.50/3.46	   f9_in(T16) -> U1(f150_in(T16), T16)
9.50/3.46	   U1(f150_out1(T17, T18), T16) -> f9_out1(.(T17, T18))
9.50/3.46	   f200_in(.(T36, T37)) -> f200_out1(T36)
9.50/3.46	   f200_in(.(T45, T46)) -> U2(f200_in(T46), .(T45, T46))
9.50/3.46	   U2(f200_out1(T47), .(T45, T46)) -> f200_out1(T47)
9.50/3.46	   f150_in(T16) -> U3(f200_in(T16), T16)
9.50/3.46	   U3(f200_out1(T17), T16) -> U4(f9_in(T16), T16, T17)
9.50/3.46	   U4(f9_out1(T23), T16, T17) -> f150_out1(T17, T23)
9.50/3.46	
9.50/3.46	Q is empty.
9.50/3.46	We have to consider all minimal (P,Q,R)-chains.
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(74) UsableRulesProof (EQUIVALENT)
9.50/3.46	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(75)
9.50/3.46	Obligation:
9.50/3.46	Q DP problem:
9.50/3.46	The TRS P consists of the following rules:
9.50/3.46	
9.50/3.46	   F200_IN(.(T45, T46)) -> F200_IN(T46)
9.50/3.46	
9.50/3.46	R is empty.
9.50/3.46	Q is empty.
9.50/3.46	We have to consider all minimal (P,Q,R)-chains.
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(76) QDPSizeChangeProof (EQUIVALENT)
9.50/3.46	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. 
9.50/3.46	
9.50/3.46	From the DPs we obtained the following set of size-change graphs:
9.50/3.46	*F200_IN(.(T45, T46)) -> F200_IN(T46)
9.50/3.46	The graph contains the following edges 1 > 1
9.50/3.46	
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(77)
9.50/3.46	YES
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(78)
9.50/3.46	Obligation:
9.50/3.46	Q DP problem:
9.50/3.46	The TRS P consists of the following rules:
9.50/3.46	
9.50/3.46	   F9_IN(T16) -> F150_IN(T16)
9.50/3.46	   F150_IN(T16) -> U3^1(f200_in(T16), T16)
9.50/3.46	   U3^1(f200_out1(T17), T16) -> F9_IN(T16)
9.50/3.46	
9.50/3.46	The TRS R consists of the following rules:
9.50/3.46	
9.50/3.46	   f9_in(T7) -> f9_out1([])
9.50/3.46	   f9_in(T16) -> U1(f150_in(T16), T16)
9.50/3.46	   U1(f150_out1(T17, T18), T16) -> f9_out1(.(T17, T18))
9.50/3.46	   f200_in(.(T36, T37)) -> f200_out1(T36)
9.50/3.46	   f200_in(.(T45, T46)) -> U2(f200_in(T46), .(T45, T46))
9.50/3.46	   U2(f200_out1(T47), .(T45, T46)) -> f200_out1(T47)
9.50/3.46	   f150_in(T16) -> U3(f200_in(T16), T16)
9.50/3.46	   U3(f200_out1(T17), T16) -> U4(f9_in(T16), T16, T17)
9.50/3.46	   U4(f9_out1(T23), T16, T17) -> f150_out1(T17, T23)
9.50/3.46	
9.50/3.46	Q is empty.
9.50/3.46	We have to consider all minimal (P,Q,R)-chains.
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(79) NonTerminationLoopProof (COMPLETE)
9.50/3.46	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
9.50/3.46	Found a loop by narrowing to the left:
9.50/3.46	
9.50/3.46	s = F150_IN(.(T36, T37)) evaluates to  t =F150_IN(.(T36, T37))
9.50/3.46	
9.50/3.46	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
9.50/3.46	* Matcher: [ ]
9.50/3.46	* Semiunifier: [ ]
9.50/3.46	
9.50/3.46	--------------------------------------------------------------------------------
9.50/3.46	Rewriting sequence
9.50/3.46	
9.50/3.46	F150_IN(.(T36, T37)) -> U3^1(f200_in(.(T36, T37)), .(T36, T37))
9.50/3.46	with rule F150_IN(T16) -> U3^1(f200_in(T16), T16) at position [] and matcher [T16 / .(T36, T37)]
9.50/3.46	
9.50/3.46	U3^1(f200_in(.(T36, T37)), .(T36, T37)) -> U3^1(f200_out1(T36), .(T36, T37))
9.50/3.46	with rule f200_in(.(T36', T37')) -> f200_out1(T36') at position [0] and matcher [T36' / T36, T37' / T37]
9.50/3.46	
9.50/3.46	U3^1(f200_out1(T36), .(T36, T37)) -> F9_IN(.(T36, T37))
9.50/3.46	with rule U3^1(f200_out1(T17), T16') -> F9_IN(T16') at position [] and matcher [T17 / T36, T16' / .(T36, T37)]
9.50/3.46	
9.50/3.46	F9_IN(.(T36, T37)) -> F150_IN(.(T36, T37))
9.50/3.46	with rule F9_IN(T16) -> F150_IN(T16)
9.50/3.46	
9.50/3.46	Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
9.50/3.46	
9.50/3.46	
9.50/3.46	All these steps are and every following step will be a correct step w.r.t to Q.
9.50/3.46	
9.50/3.46	
9.50/3.46	
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(80)
9.50/3.46	NO
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(81) PrologToIRSwTTransformerProof (SOUND)
9.50/3.46	Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert
9.50/3.46	
9.50/3.46	{
9.50/3.46	    "root": 11,
9.50/3.46	    "program": {
9.50/3.46	        "directives": [],
9.50/3.46	        "clauses": [
9.50/3.46	            [
9.50/3.46	                "(subset ([]) X1)",
9.50/3.46	                null
9.50/3.46	            ],
9.50/3.46	            [
9.50/3.46	                "(subset (. X Xs) Ys)",
9.50/3.46	                "(',' (member X Ys) (subset Xs Ys))"
9.50/3.46	            ],
9.50/3.46	            [
9.50/3.46	                "(member X (. X X2))",
9.50/3.46	                null
9.50/3.46	            ],
9.50/3.46	            [
9.50/3.46	                "(member X (. X3 Xs))",
9.50/3.46	                "(member X Xs)"
9.50/3.46	            ]
9.50/3.46	        ]
9.50/3.46	    },
9.50/3.46	    "graph": {
9.50/3.46	        "nodes": {
9.50/3.46	            "11": {
9.50/3.46	                "goal": [{
9.50/3.46	                    "clause": -1,
9.50/3.46	                    "scope": -1,
9.50/3.46	                    "term": "(subset T1 T2)"
9.50/3.46	                }],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": ["T2"],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "22": {
9.50/3.46	                "goal": [{
9.50/3.46	                    "clause": 0,
9.50/3.46	                    "scope": 1,
9.50/3.46	                    "term": "(subset T1 T2)"
9.50/3.46	                }],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": ["T2"],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "23": {
9.50/3.46	                "goal": [{
9.50/3.46	                    "clause": 1,
9.50/3.46	                    "scope": 1,
9.50/3.46	                    "term": "(subset T1 T2)"
9.50/3.46	                }],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": ["T2"],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "19": {
9.50/3.46	                "goal": [
9.50/3.46	                    {
9.50/3.46	                        "clause": 0,
9.50/3.46	                        "scope": 1,
9.50/3.46	                        "term": "(subset T1 T2)"
9.50/3.46	                    },
9.50/3.46	                    {
9.50/3.46	                        "clause": 1,
9.50/3.46	                        "scope": 1,
9.50/3.46	                        "term": "(subset T1 T2)"
9.50/3.46	                    }
9.50/3.46	                ],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": ["T2"],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "190": {
9.50/3.46	                "goal": [],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": [],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "191": {
9.50/3.46	                "goal": [],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": [],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "192": {
9.50/3.46	                "goal": [{
9.50/3.46	                    "clause": -1,
9.50/3.46	                    "scope": -1,
9.50/3.46	                    "term": "(member T47 T46)"
9.50/3.46	                }],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": ["T46"],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "160": {
9.50/3.46	                "goal": [{
9.50/3.46	                    "clause": -1,
9.50/3.46	                    "scope": -1,
9.50/3.46	                    "term": "(member T17 T16)"
9.50/3.46	                }],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": ["T16"],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "193": {
9.50/3.46	                "goal": [],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": [],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "type": "Nodes",
9.50/3.46	            "161": {
9.50/3.46	                "goal": [{
9.50/3.46	                    "clause": -1,
9.50/3.46	                    "scope": -1,
9.50/3.46	                    "term": "(subset T23 T16)"
9.50/3.46	                }],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": ["T16"],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "162": {
9.50/3.46	                "goal": [
9.50/3.46	                    {
9.50/3.46	                        "clause": 2,
9.50/3.46	                        "scope": 2,
9.50/3.46	                        "term": "(member T17 T16)"
9.50/3.46	                    },
9.50/3.46	                    {
9.50/3.46	                        "clause": 3,
9.50/3.46	                        "scope": 2,
9.50/3.46	                        "term": "(member T17 T16)"
9.50/3.46	                    }
9.50/3.46	                ],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": ["T16"],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "186": {
9.50/3.46	                "goal": [{
9.50/3.46	                    "clause": 2,
9.50/3.46	                    "scope": 2,
9.50/3.46	                    "term": "(member T17 T16)"
9.50/3.46	                }],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": ["T16"],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "187": {
9.50/3.46	                "goal": [{
9.50/3.46	                    "clause": 3,
9.50/3.46	                    "scope": 2,
9.50/3.46	                    "term": "(member T17 T16)"
9.50/3.46	                }],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": ["T16"],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "144": {
9.50/3.46	                "goal": [{
9.50/3.46	                    "clause": -1,
9.50/3.46	                    "scope": -1,
9.50/3.46	                    "term": "(true)"
9.50/3.46	                }],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": [],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "145": {
9.50/3.46	                "goal": [],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": [],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "189": {
9.50/3.46	                "goal": [{
9.50/3.46	                    "clause": -1,
9.50/3.46	                    "scope": -1,
9.50/3.46	                    "term": "(true)"
9.50/3.46	                }],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": [],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "146": {
9.50/3.46	                "goal": [],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": [],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "148": {
9.50/3.46	                "goal": [{
9.50/3.46	                    "clause": -1,
9.50/3.46	                    "scope": -1,
9.50/3.46	                    "term": "(',' (member T17 T16) (subset T18 T16))"
9.50/3.46	                }],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": ["T16"],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            },
9.50/3.46	            "149": {
9.50/3.46	                "goal": [],
9.50/3.46	                "kb": {
9.50/3.46	                    "nonunifying": [],
9.50/3.46	                    "intvars": {},
9.50/3.46	                    "arithmetic": {
9.50/3.46	                        "type": "PlainIntegerRelationState",
9.50/3.46	                        "relations": []
9.50/3.46	                    },
9.50/3.46	                    "ground": [],
9.50/3.46	                    "free": [],
9.50/3.46	                    "exprvars": []
9.50/3.46	                }
9.50/3.46	            }
9.50/3.46	        },
9.50/3.46	        "edges": [
9.50/3.46	            {
9.50/3.46	                "from": 11,
9.50/3.46	                "to": 19,
9.50/3.46	                "label": "CASE"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 19,
9.50/3.46	                "to": 22,
9.50/3.46	                "label": "PARALLEL"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 19,
9.50/3.46	                "to": 23,
9.50/3.46	                "label": "PARALLEL"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 22,
9.50/3.46	                "to": 144,
9.50/3.46	                "label": "EVAL with clause\nsubset([], X8).\nand substitutionT1 -> [],\nT2 -> T7,\nX8 -> T7"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 22,
9.50/3.46	                "to": 145,
9.50/3.46	                "label": "EVAL-BACKTRACK"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 23,
9.50/3.46	                "to": 148,
9.50/3.46	                "label": "EVAL with clause\nsubset(.(X15, X16), X17) :- ','(member(X15, X17), subset(X16, X17)).\nand substitutionX15 -> T17,\nX16 -> T18,\nT1 -> .(T17, T18),\nT2 -> T16,\nX17 -> T16,\nT14 -> T17,\nT15 -> T18"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 23,
9.50/3.46	                "to": 149,
9.50/3.46	                "label": "EVAL-BACKTRACK"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 144,
9.50/3.46	                "to": 146,
9.50/3.46	                "label": "SUCCESS"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 148,
9.50/3.46	                "to": 160,
9.50/3.46	                "label": "SPLIT 1"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 148,
9.50/3.46	                "to": 161,
9.50/3.46	                "label": "SPLIT 2\nnew knowledge:\nT17 is ground\nT16 is ground\nreplacements:T18 -> T23"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 160,
9.50/3.46	                "to": 162,
9.50/3.46	                "label": "CASE"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 161,
9.50/3.46	                "to": 11,
9.50/3.46	                "label": "INSTANCE with matching:\nT1 -> T23\nT2 -> T16"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 162,
9.50/3.46	                "to": 186,
9.50/3.46	                "label": "PARALLEL"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 162,
9.50/3.46	                "to": 187,
9.50/3.46	                "label": "PARALLEL"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 186,
9.50/3.46	                "to": 189,
9.50/3.46	                "label": "EVAL with clause\nmember(X34, .(X34, X35)).\nand substitutionT17 -> T36,\nX34 -> T36,\nX35 -> T37,\nT16 -> .(T36, T37)"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 186,
9.50/3.46	                "to": 190,
9.50/3.46	                "label": "EVAL-BACKTRACK"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 187,
9.50/3.46	                "to": 192,
9.50/3.46	                "label": "EVAL with clause\nmember(X42, .(X43, X44)) :- member(X42, X44).\nand substitutionT17 -> T47,\nX42 -> T47,\nX43 -> T45,\nX44 -> T46,\nT16 -> .(T45, T46),\nT44 -> T47"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 187,
9.50/3.46	                "to": 193,
9.50/3.46	                "label": "EVAL-BACKTRACK"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 189,
9.50/3.46	                "to": 191,
9.50/3.46	                "label": "SUCCESS"
9.50/3.46	            },
9.50/3.46	            {
9.50/3.46	                "from": 192,
9.50/3.46	                "to": 160,
9.50/3.46	                "label": "INSTANCE with matching:\nT17 -> T47\nT16 -> T46"
9.50/3.46	            }
9.50/3.46	        ],
9.50/3.46	        "type": "Graph"
9.50/3.46	    }
9.50/3.46	}
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(82)
9.50/3.46	Complex Obligation (AND)
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(83)
9.50/3.46	Obligation:
9.50/3.46	Rules:
9.50/3.46	f186_out(T16) -> f162_out(T16) :|: TRUE
9.50/3.46	f162_in(x) -> f187_in(x) :|: TRUE
9.50/3.46	f187_out(x1) -> f162_out(x1) :|: TRUE
9.50/3.46	f162_in(x2) -> f186_in(x2) :|: TRUE
9.50/3.46	f160_out(T46) -> f192_out(T46) :|: TRUE
9.50/3.46	f192_in(x3) -> f160_in(x3) :|: TRUE
9.50/3.46	f193_out -> f187_out(x4) :|: TRUE
9.50/3.46	f187_in(x5) -> f193_in :|: TRUE
9.50/3.46	f192_out(x6) -> f187_out(.(x7, x6)) :|: TRUE
9.50/3.46	f187_in(.(x8, x9)) -> f192_in(x9) :|: TRUE
9.50/3.46	f162_out(x10) -> f160_out(x10) :|: TRUE
9.50/3.46	f160_in(x11) -> f162_in(x11) :|: TRUE
9.50/3.46	f11_in(T2) -> f19_in(T2) :|: TRUE
9.50/3.46	f19_out(x12) -> f11_out(x12) :|: TRUE
9.50/3.46	f19_in(x13) -> f23_in(x13) :|: TRUE
9.50/3.46	f22_out(x14) -> f19_out(x14) :|: TRUE
9.50/3.46	f19_in(x15) -> f22_in(x15) :|: TRUE
9.50/3.46	f23_out(x16) -> f19_out(x16) :|: TRUE
9.50/3.46	f149_out -> f23_out(x17) :|: TRUE
9.50/3.46	f148_out(x18) -> f23_out(x18) :|: TRUE
9.50/3.46	f23_in(x19) -> f149_in :|: TRUE
9.50/3.46	f23_in(x20) -> f148_in(x20) :|: TRUE
9.50/3.46	f161_out(x21) -> f148_out(x21) :|: TRUE
9.50/3.46	f160_out(x22) -> f161_in(x22) :|: TRUE
9.50/3.46	f148_in(x23) -> f160_in(x23) :|: TRUE
9.50/3.46	Start term: f11_in(T2)
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(84) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
9.50/3.46	Constructed simple dependency graph.
9.50/3.46	
9.50/3.46	Simplified to the following IRSwTs:
9.50/3.46	
9.50/3.46	intTRSProblem:
9.50/3.46	f162_in(x) -> f187_in(x) :|: TRUE
9.50/3.46	f192_in(x3) -> f160_in(x3) :|: TRUE
9.50/3.46	f187_in(.(x8, x9)) -> f192_in(x9) :|: TRUE
9.50/3.46	f160_in(x11) -> f162_in(x11) :|: TRUE
9.50/3.46	
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(85)
9.50/3.46	Obligation:
9.50/3.46	Rules:
9.50/3.46	f162_in(x) -> f187_in(x) :|: TRUE
9.50/3.46	f192_in(x3) -> f160_in(x3) :|: TRUE
9.50/3.46	f187_in(.(x8, x9)) -> f192_in(x9) :|: TRUE
9.50/3.46	f160_in(x11) -> f162_in(x11) :|: TRUE
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(86) IntTRSCompressionProof (EQUIVALENT)
9.50/3.46	Compressed rules.
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(87)
9.50/3.46	Obligation:
9.50/3.46	Rules:
9.50/3.46	f162_in(.(x8:0, x9:0)) -> f162_in(x9:0) :|: TRUE
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(88) IRSFormatTransformerProof (EQUIVALENT)
9.50/3.46	Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(89)
9.50/3.46	Obligation:
9.50/3.46	Rules:
9.50/3.46	f162_in(.(x8:0, x9:0)) -> f162_in(x9:0) :|: TRUE
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(90) IRSwTTerminationDigraphProof (EQUIVALENT)
9.50/3.46	Constructed termination digraph!
9.50/3.46	Nodes:
9.50/3.46	(1) f162_in(.(x8:0, x9:0)) -> f162_in(x9:0) :|: TRUE
9.50/3.46	
9.50/3.46	Arcs:
9.50/3.46	(1) -> (1)
9.50/3.46	
9.50/3.46	This digraph is fully evaluated!
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(91)
9.50/3.46	Obligation:
9.50/3.46	
9.50/3.46	Termination digraph:
9.50/3.46	Nodes:
9.50/3.46	(1) f162_in(.(x8:0, x9:0)) -> f162_in(x9:0) :|: TRUE
9.50/3.46	
9.50/3.46	Arcs:
9.50/3.46	(1) -> (1)
9.50/3.46	
9.50/3.46	This digraph is fully evaluated!
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(92) IntTRSUnneededArgumentFilterProof (EQUIVALENT)
9.50/3.46	Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements:
9.50/3.46	
9.50/3.46	   .(x1, x2) -> .(x2)
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(93)
9.50/3.46	Obligation:
9.50/3.46	Rules:
9.50/3.46	f162_in(.(x9:0)) -> f162_in(x9:0) :|: TRUE
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(94) TempFilterProof (SOUND)
9.50/3.46	Used the following sort dictionary for filtering: 
9.50/3.46	f162_in(VARIABLE)
9.50/3.46	.(VARIABLE)
9.50/3.46	Removed predefined arithmetic.
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(95)
9.50/3.46	Obligation:
9.50/3.46	Rules:
9.50/3.46	f162_in(.(x9:0)) -> f162_in(x9:0)
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(96) IRSwTToQDPProof (SOUND)
9.50/3.46	Removed the integers and created a QDP-Problem.
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(97)
9.50/3.46	Obligation:
9.50/3.46	Q DP problem:
9.50/3.46	The TRS P consists of the following rules:
9.50/3.46	
9.50/3.46	   f162_in(.(x9:0)) -> f162_in(x9:0)
9.50/3.46	
9.50/3.46	R is empty.
9.50/3.46	Q is empty.
9.50/3.46	We have to consider all (P,Q,R)-chains.
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(98) QDPSizeChangeProof (EQUIVALENT)
9.50/3.46	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. 
9.50/3.46	
9.50/3.46	From the DPs we obtained the following set of size-change graphs:
9.50/3.46	*f162_in(.(x9:0)) -> f162_in(x9:0)
9.50/3.46	The graph contains the following edges 1 > 1
9.50/3.46	
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(99)
9.50/3.46	YES
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(100)
9.50/3.46	Obligation:
9.50/3.46	Rules:
9.50/3.46	f161_out(T16) -> f148_out(T16) :|: TRUE
9.50/3.46	f160_out(x) -> f161_in(x) :|: TRUE
9.50/3.46	f148_in(x1) -> f160_in(x1) :|: TRUE
9.50/3.46	f189_in -> f189_out :|: TRUE
9.50/3.46	f186_in(x2) -> f190_in :|: TRUE
9.50/3.46	f189_out -> f186_out(.(T36, T37)) :|: TRUE
9.50/3.46	f186_in(.(x3, x4)) -> f189_in :|: TRUE
9.50/3.46	f190_out -> f186_out(x5) :|: TRUE
9.50/3.46	f19_in(T2) -> f23_in(T2) :|: TRUE
9.50/3.46	f22_out(x6) -> f19_out(x6) :|: TRUE
9.50/3.46	f19_in(x7) -> f22_in(x7) :|: TRUE
9.50/3.46	f23_out(x8) -> f19_out(x8) :|: TRUE
9.50/3.46	f11_in(x9) -> f19_in(x9) :|: TRUE
9.50/3.46	f19_out(x10) -> f11_out(x10) :|: TRUE
9.50/3.46	f11_out(x11) -> f161_out(x11) :|: TRUE
9.50/3.46	f161_in(x12) -> f11_in(x12) :|: TRUE
9.50/3.46	f149_out -> f23_out(x13) :|: TRUE
9.50/3.46	f148_out(x14) -> f23_out(x14) :|: TRUE
9.50/3.46	f23_in(x15) -> f149_in :|: TRUE
9.50/3.46	f23_in(x16) -> f148_in(x16) :|: TRUE
9.50/3.46	f186_out(x17) -> f162_out(x17) :|: TRUE
9.50/3.46	f162_in(x18) -> f187_in(x18) :|: TRUE
9.50/3.46	f187_out(x19) -> f162_out(x19) :|: TRUE
9.50/3.46	f162_in(x20) -> f186_in(x20) :|: TRUE
9.50/3.46	f160_out(T46) -> f192_out(T46) :|: TRUE
9.50/3.46	f192_in(x21) -> f160_in(x21) :|: TRUE
9.50/3.46	f193_out -> f187_out(x22) :|: TRUE
9.50/3.46	f187_in(x23) -> f193_in :|: TRUE
9.50/3.46	f192_out(x24) -> f187_out(.(x25, x24)) :|: TRUE
9.50/3.46	f187_in(.(x26, x27)) -> f192_in(x27) :|: TRUE
9.50/3.46	f162_out(x28) -> f160_out(x28) :|: TRUE
9.50/3.46	f160_in(x29) -> f162_in(x29) :|: TRUE
9.50/3.46	Start term: f11_in(T2)
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(101) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
9.50/3.46	Constructed simple dependency graph.
9.50/3.46	
9.50/3.46	Simplified to the following IRSwTs:
9.50/3.46	
9.50/3.46	intTRSProblem:
9.50/3.46	f160_out(x) -> f161_in(x) :|: TRUE
9.50/3.46	f148_in(x1) -> f160_in(x1) :|: TRUE
9.50/3.46	f189_in -> f189_out :|: TRUE
9.50/3.46	f189_out -> f186_out(.(T36, T37)) :|: TRUE
9.50/3.46	f186_in(.(x3, x4)) -> f189_in :|: TRUE
9.50/3.46	f19_in(T2) -> f23_in(T2) :|: TRUE
9.50/3.46	f11_in(x9) -> f19_in(x9) :|: TRUE
9.50/3.46	f161_in(x12) -> f11_in(x12) :|: TRUE
9.50/3.46	f23_in(x16) -> f148_in(x16) :|: TRUE
9.50/3.46	f186_out(x17) -> f162_out(x17) :|: TRUE
9.50/3.46	f162_in(x18) -> f187_in(x18) :|: TRUE
9.50/3.46	f187_out(x19) -> f162_out(x19) :|: TRUE
9.50/3.46	f162_in(x20) -> f186_in(x20) :|: TRUE
9.50/3.46	f160_out(T46) -> f192_out(T46) :|: TRUE
9.50/3.46	f192_in(x21) -> f160_in(x21) :|: TRUE
9.50/3.46	f192_out(x24) -> f187_out(.(x25, x24)) :|: TRUE
9.50/3.46	f187_in(.(x26, x27)) -> f192_in(x27) :|: TRUE
9.50/3.46	f162_out(x28) -> f160_out(x28) :|: TRUE
9.50/3.46	f160_in(x29) -> f162_in(x29) :|: TRUE
9.50/3.46	
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(102)
9.50/3.46	Obligation:
9.50/3.46	Rules:
9.50/3.46	f160_out(x) -> f161_in(x) :|: TRUE
9.50/3.46	f148_in(x1) -> f160_in(x1) :|: TRUE
9.50/3.46	f189_in -> f189_out :|: TRUE
9.50/3.46	f189_out -> f186_out(.(T36, T37)) :|: TRUE
9.50/3.46	f186_in(.(x3, x4)) -> f189_in :|: TRUE
9.50/3.46	f19_in(T2) -> f23_in(T2) :|: TRUE
9.50/3.46	f11_in(x9) -> f19_in(x9) :|: TRUE
9.50/3.46	f161_in(x12) -> f11_in(x12) :|: TRUE
9.50/3.46	f23_in(x16) -> f148_in(x16) :|: TRUE
9.50/3.46	f186_out(x17) -> f162_out(x17) :|: TRUE
9.50/3.46	f162_in(x18) -> f187_in(x18) :|: TRUE
9.50/3.46	f187_out(x19) -> f162_out(x19) :|: TRUE
9.50/3.46	f162_in(x20) -> f186_in(x20) :|: TRUE
9.50/3.46	f160_out(T46) -> f192_out(T46) :|: TRUE
9.50/3.46	f192_in(x21) -> f160_in(x21) :|: TRUE
9.50/3.46	f192_out(x24) -> f187_out(.(x25, x24)) :|: TRUE
9.50/3.46	f187_in(.(x26, x27)) -> f192_in(x27) :|: TRUE
9.50/3.46	f162_out(x28) -> f160_out(x28) :|: TRUE
9.50/3.46	f160_in(x29) -> f162_in(x29) :|: TRUE
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(103) IntTRSCompressionProof (EQUIVALENT)
9.50/3.46	Compressed rules.
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(104)
9.50/3.46	Obligation:
9.50/3.46	Rules:
9.50/3.46	f160_out(T46:0) -> f160_out(.(x25:0, T46:0)) :|: TRUE
9.50/3.46	f160_out(x:0) -> f162_in(x:0) :|: TRUE
9.50/3.46	f162_in(.(x3:0, x4:0)) -> f160_out(.(T36:0, T37:0)) :|: TRUE
9.50/3.46	f162_in(.(x26:0, x27:0)) -> f162_in(x27:0) :|: TRUE
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(105) IRSFormatTransformerProof (EQUIVALENT)
9.50/3.46	Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(106)
9.50/3.46	Obligation:
9.50/3.46	Rules:
9.50/3.46	f160_out(T46:0) -> f160_out(.(x25:0, T46:0)) :|: TRUE
9.50/3.46	f160_out(x:0) -> f162_in(x:0) :|: TRUE
9.50/3.46	f162_in(.(x3:0, x4:0)) -> f160_out(.(T36:0, T37:0)) :|: TRUE
9.50/3.46	f162_in(.(x26:0, x27:0)) -> f162_in(x27:0) :|: TRUE
9.50/3.46	
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(107) IRSwTTerminationDigraphProof (EQUIVALENT)
9.50/3.46	Constructed termination digraph!
9.50/3.46	Nodes:
9.50/3.46	(1) f160_out(T46:0) -> f160_out(.(x25:0, T46:0)) :|: TRUE
9.50/3.46	(2) f160_out(x:0) -> f162_in(x:0) :|: TRUE
9.50/3.46	(3) f162_in(.(x3:0, x4:0)) -> f160_out(.(T36:0, T37:0)) :|: TRUE
9.50/3.46	(4) f162_in(.(x26:0, x27:0)) -> f162_in(x27:0) :|: TRUE
9.50/3.46	
9.50/3.46	Arcs:
9.50/3.46	(1) -> (1), (2)
9.50/3.46	(2) -> (3), (4)
9.50/3.46	(3) -> (1), (2)
9.50/3.46	(4) -> (3), (4)
9.50/3.46	
9.50/3.46	This digraph is fully evaluated!
9.50/3.46	----------------------------------------
9.50/3.46	
9.50/3.46	(108)
9.50/3.46	Obligation:
9.50/3.46	
9.50/3.46	Termination digraph:
9.50/3.46	Nodes:
9.50/3.46	(1) f160_out(T46:0) -> f160_out(.(x25:0, T46:0)) :|: TRUE
9.50/3.46	(2) f162_in(.(x3:0, x4:0)) -> f160_out(.(T36:0, T37:0)) :|: TRUE
9.50/3.46	(3) f162_in(.(x26:0, x27:0)) -> f162_in(x27:0) :|: TRUE
9.50/3.46	(4) f160_out(x:0) -> f162_in(x:0) :|: TRUE
9.50/3.46	
9.50/3.46	Arcs:
9.50/3.46	(1) -> (1), (4)
9.50/3.46	(2) -> (1), (4)
9.50/3.46	(3) -> (2), (3)
9.50/3.46	(4) -> (2), (3)
9.50/3.46	
9.50/3.46	This digraph is fully evaluated!
9.79/3.50	EOF