6.39/2.43	MAYBE
6.39/2.47	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
6.39/2.47	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.39/2.47	
6.39/2.47	
6.39/2.47	Left Termination of the query pattern
6.39/2.47	
6.39/2.47	member(g,a)
6.39/2.47	
6.39/2.47	w.r.t. the given Prolog program could not be shown:
6.39/2.47	
6.39/2.47	(0) Prolog
6.39/2.47	(1) PrologToPiTRSProof [SOUND, 0 ms]
6.39/2.47	(2) PiTRS
6.39/2.47	    (3) DependencyPairsProof [EQUIVALENT, 0 ms]
6.39/2.47	    (4) PiDP
6.39/2.47	    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
6.39/2.47	    (6) PiDP
6.39/2.47	    (7) UsableRulesProof [EQUIVALENT, 0 ms]
6.39/2.47	    (8) PiDP
6.39/2.47	    (9) PiDPToQDPProof [SOUND, 0 ms]
6.39/2.47	    (10) QDP
6.39/2.47	(11) PrologToPiTRSProof [SOUND, 0 ms]
6.39/2.47	(12) PiTRS
6.39/2.47	    (13) DependencyPairsProof [EQUIVALENT, 0 ms]
6.39/2.47	    (14) PiDP
6.39/2.47	    (15) DependencyGraphProof [EQUIVALENT, 0 ms]
6.39/2.47	    (16) PiDP
6.39/2.47	    (17) UsableRulesProof [EQUIVALENT, 0 ms]
6.39/2.47	    (18) PiDP
6.39/2.47	    (19) PiDPToQDPProof [SOUND, 0 ms]
6.39/2.47	    (20) QDP
6.39/2.47	(21) PrologToTRSTransformerProof [SOUND, 0 ms]
6.39/2.47	(22) QTRS
6.39/2.47	    (23) DependencyPairsProof [EQUIVALENT, 0 ms]
6.39/2.47	    (24) QDP
6.39/2.47	    (25) DependencyGraphProof [EQUIVALENT, 0 ms]
6.39/2.47	    (26) QDP
6.39/2.47	    (27) MNOCProof [EQUIVALENT, 0 ms]
6.39/2.47	    (28) QDP
6.39/2.47	    (29) UsableRulesProof [EQUIVALENT, 0 ms]
6.39/2.47	    (30) QDP
6.39/2.47	    (31) QReductionProof [EQUIVALENT, 0 ms]
6.39/2.47	    (32) QDP
6.39/2.47	(33) PrologToIRSwTTransformerProof [SOUND, 0 ms]
6.39/2.47	(34) IRSwT
6.39/2.47	    (35) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
6.39/2.47	    (36) IRSwT
6.39/2.47	    (37) IntTRSCompressionProof [EQUIVALENT, 20 ms]
6.39/2.47	    (38) IRSwT
6.39/2.47	    (39) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
6.39/2.47	    (40) IRSwT
6.39/2.47	    (41) IRSwTTerminationDigraphProof [EQUIVALENT, 7 ms]
6.39/2.47	    (42) IRSwT
6.39/2.47	    (43) FilterProof [EQUIVALENT, 0 ms]
6.39/2.47	    (44) IntTRS
6.39/2.47	    (45) IntTRSPeriodicNontermProof [COMPLETE, 6 ms]
6.39/2.47	    (46) NO
6.39/2.47	(47) PrologToDTProblemTransformerProof [SOUND, 0 ms]
6.39/2.47	(48) TRIPLES
6.39/2.47	    (49) TriplesToPiDPProof [SOUND, 8 ms]
6.39/2.47	    (50) PiDP
6.39/2.47	    (51) DependencyGraphProof [EQUIVALENT, 0 ms]
6.39/2.47	    (52) PiDP
6.39/2.47	    (53) PiDPToQDPProof [SOUND, 0 ms]
6.39/2.47	    (54) QDP
6.39/2.47	
6.39/2.47	
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(0)
6.39/2.47	Obligation:
6.39/2.47	Clauses:
6.39/2.47	
6.39/2.47	member(X, .(X, X1)).
6.39/2.47	member(X, .(X2, Xs)) :- member(X, Xs).
6.39/2.47	
6.39/2.47	
6.39/2.47	Query: member(g,a)
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(1) PrologToPiTRSProof (SOUND)
6.39/2.47	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.39/2.47	
6.39/2.47	member_in_2: (b,f)
6.39/2.47	
6.39/2.47	Transforming Prolog into the following Term Rewriting System:
6.39/2.47	
6.39/2.47	Pi-finite rewrite system:
6.39/2.47	The TRS R consists of the following rules:
6.39/2.47	
6.39/2.47	   member_in_ga(X, .(X, X1)) -> member_out_ga(X, .(X, X1))
6.39/2.47	   member_in_ga(X, .(X2, Xs)) -> U1_ga(X, X2, Xs, member_in_ga(X, Xs))
6.39/2.47	   U1_ga(X, X2, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X2, Xs))
6.39/2.47	
6.39/2.47	The argument filtering Pi contains the following mapping:
6.39/2.47	member_in_ga(x1, x2)  =  member_in_ga(x1)
6.39/2.47	
6.39/2.47	member_out_ga(x1, x2)  =  member_out_ga
6.39/2.47	
6.39/2.47	U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
6.39/2.47	
6.39/2.47	
6.39/2.47	
6.39/2.47	
6.39/2.47	
6.39/2.47	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.39/2.47	
6.39/2.47	
6.39/2.47	
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(2)
6.39/2.47	Obligation:
6.39/2.47	Pi-finite rewrite system:
6.39/2.47	The TRS R consists of the following rules:
6.39/2.47	
6.39/2.47	   member_in_ga(X, .(X, X1)) -> member_out_ga(X, .(X, X1))
6.39/2.47	   member_in_ga(X, .(X2, Xs)) -> U1_ga(X, X2, Xs, member_in_ga(X, Xs))
6.39/2.47	   U1_ga(X, X2, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X2, Xs))
6.39/2.47	
6.39/2.47	The argument filtering Pi contains the following mapping:
6.39/2.47	member_in_ga(x1, x2)  =  member_in_ga(x1)
6.39/2.47	
6.39/2.47	member_out_ga(x1, x2)  =  member_out_ga
6.39/2.47	
6.39/2.47	U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
6.39/2.47	
6.39/2.47	
6.39/2.47	
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(3) DependencyPairsProof (EQUIVALENT)
6.39/2.47	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.39/2.47	Pi DP problem:
6.39/2.47	The TRS P consists of the following rules:
6.39/2.47	
6.39/2.47	   MEMBER_IN_GA(X, .(X2, Xs)) -> U1_GA(X, X2, Xs, member_in_ga(X, Xs))
6.39/2.47	   MEMBER_IN_GA(X, .(X2, Xs)) -> MEMBER_IN_GA(X, Xs)
6.39/2.47	
6.39/2.47	The TRS R consists of the following rules:
6.39/2.47	
6.39/2.47	   member_in_ga(X, .(X, X1)) -> member_out_ga(X, .(X, X1))
6.39/2.47	   member_in_ga(X, .(X2, Xs)) -> U1_ga(X, X2, Xs, member_in_ga(X, Xs))
6.39/2.47	   U1_ga(X, X2, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X2, Xs))
6.39/2.47	
6.39/2.47	The argument filtering Pi contains the following mapping:
6.39/2.47	member_in_ga(x1, x2)  =  member_in_ga(x1)
6.39/2.47	
6.39/2.47	member_out_ga(x1, x2)  =  member_out_ga
6.39/2.47	
6.39/2.47	U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
6.39/2.47	
6.39/2.47	MEMBER_IN_GA(x1, x2)  =  MEMBER_IN_GA(x1)
6.39/2.47	
6.39/2.47	U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
6.39/2.47	
6.39/2.47	
6.39/2.47	We have to consider all (P,R,Pi)-chains
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(4)
6.39/2.47	Obligation:
6.39/2.47	Pi DP problem:
6.39/2.47	The TRS P consists of the following rules:
6.39/2.47	
6.39/2.47	   MEMBER_IN_GA(X, .(X2, Xs)) -> U1_GA(X, X2, Xs, member_in_ga(X, Xs))
6.39/2.47	   MEMBER_IN_GA(X, .(X2, Xs)) -> MEMBER_IN_GA(X, Xs)
6.39/2.47	
6.39/2.47	The TRS R consists of the following rules:
6.39/2.47	
6.39/2.47	   member_in_ga(X, .(X, X1)) -> member_out_ga(X, .(X, X1))
6.39/2.47	   member_in_ga(X, .(X2, Xs)) -> U1_ga(X, X2, Xs, member_in_ga(X, Xs))
6.39/2.47	   U1_ga(X, X2, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X2, Xs))
6.39/2.47	
6.39/2.47	The argument filtering Pi contains the following mapping:
6.39/2.47	member_in_ga(x1, x2)  =  member_in_ga(x1)
6.39/2.47	
6.39/2.47	member_out_ga(x1, x2)  =  member_out_ga
6.39/2.47	
6.39/2.47	U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
6.39/2.47	
6.39/2.47	MEMBER_IN_GA(x1, x2)  =  MEMBER_IN_GA(x1)
6.39/2.47	
6.39/2.47	U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
6.39/2.47	
6.39/2.47	
6.39/2.47	We have to consider all (P,R,Pi)-chains
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(5) DependencyGraphProof (EQUIVALENT)
6.39/2.47	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(6)
6.39/2.47	Obligation:
6.39/2.47	Pi DP problem:
6.39/2.47	The TRS P consists of the following rules:
6.39/2.47	
6.39/2.47	   MEMBER_IN_GA(X, .(X2, Xs)) -> MEMBER_IN_GA(X, Xs)
6.39/2.47	
6.39/2.47	The TRS R consists of the following rules:
6.39/2.47	
6.39/2.47	   member_in_ga(X, .(X, X1)) -> member_out_ga(X, .(X, X1))
6.39/2.47	   member_in_ga(X, .(X2, Xs)) -> U1_ga(X, X2, Xs, member_in_ga(X, Xs))
6.39/2.47	   U1_ga(X, X2, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X2, Xs))
6.39/2.47	
6.39/2.47	The argument filtering Pi contains the following mapping:
6.39/2.47	member_in_ga(x1, x2)  =  member_in_ga(x1)
6.39/2.47	
6.39/2.47	member_out_ga(x1, x2)  =  member_out_ga
6.39/2.47	
6.39/2.47	U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
6.39/2.47	
6.39/2.47	MEMBER_IN_GA(x1, x2)  =  MEMBER_IN_GA(x1)
6.39/2.47	
6.39/2.47	
6.39/2.47	We have to consider all (P,R,Pi)-chains
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(7) UsableRulesProof (EQUIVALENT)
6.39/2.47	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(8)
6.39/2.47	Obligation:
6.39/2.47	Pi DP problem:
6.39/2.47	The TRS P consists of the following rules:
6.39/2.47	
6.39/2.47	   MEMBER_IN_GA(X, .(X2, Xs)) -> MEMBER_IN_GA(X, Xs)
6.39/2.47	
6.39/2.47	R is empty.
6.39/2.47	The argument filtering Pi contains the following mapping:
6.39/2.47	MEMBER_IN_GA(x1, x2)  =  MEMBER_IN_GA(x1)
6.39/2.47	
6.39/2.47	
6.39/2.47	We have to consider all (P,R,Pi)-chains
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(9) PiDPToQDPProof (SOUND)
6.39/2.47	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(10)
6.39/2.47	Obligation:
6.39/2.47	Q DP problem:
6.39/2.47	The TRS P consists of the following rules:
6.39/2.47	
6.39/2.47	   MEMBER_IN_GA(X) -> MEMBER_IN_GA(X)
6.39/2.47	
6.39/2.47	R is empty.
6.39/2.47	Q is empty.
6.39/2.47	We have to consider all (P,Q,R)-chains.
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(11) PrologToPiTRSProof (SOUND)
6.39/2.47	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.39/2.47	
6.39/2.47	member_in_2: (b,f)
6.39/2.47	
6.39/2.47	Transforming Prolog into the following Term Rewriting System:
6.39/2.47	
6.39/2.47	Pi-finite rewrite system:
6.39/2.47	The TRS R consists of the following rules:
6.39/2.47	
6.39/2.47	   member_in_ga(X, .(X, X1)) -> member_out_ga(X, .(X, X1))
6.39/2.47	   member_in_ga(X, .(X2, Xs)) -> U1_ga(X, X2, Xs, member_in_ga(X, Xs))
6.39/2.47	   U1_ga(X, X2, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X2, Xs))
6.39/2.47	
6.39/2.47	The argument filtering Pi contains the following mapping:
6.39/2.47	member_in_ga(x1, x2)  =  member_in_ga(x1)
6.39/2.47	
6.39/2.47	member_out_ga(x1, x2)  =  member_out_ga(x1)
6.39/2.47	
6.39/2.47	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
6.39/2.47	
6.39/2.47	
6.39/2.47	
6.39/2.47	
6.39/2.47	
6.39/2.47	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.39/2.47	
6.39/2.47	
6.39/2.47	
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(12)
6.39/2.47	Obligation:
6.39/2.47	Pi-finite rewrite system:
6.39/2.47	The TRS R consists of the following rules:
6.39/2.47	
6.39/2.47	   member_in_ga(X, .(X, X1)) -> member_out_ga(X, .(X, X1))
6.39/2.47	   member_in_ga(X, .(X2, Xs)) -> U1_ga(X, X2, Xs, member_in_ga(X, Xs))
6.39/2.47	   U1_ga(X, X2, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X2, Xs))
6.39/2.47	
6.39/2.47	The argument filtering Pi contains the following mapping:
6.39/2.47	member_in_ga(x1, x2)  =  member_in_ga(x1)
6.39/2.47	
6.39/2.47	member_out_ga(x1, x2)  =  member_out_ga(x1)
6.39/2.47	
6.39/2.47	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
6.39/2.47	
6.39/2.47	
6.39/2.47	
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(13) DependencyPairsProof (EQUIVALENT)
6.39/2.47	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.39/2.47	Pi DP problem:
6.39/2.47	The TRS P consists of the following rules:
6.39/2.47	
6.39/2.47	   MEMBER_IN_GA(X, .(X2, Xs)) -> U1_GA(X, X2, Xs, member_in_ga(X, Xs))
6.39/2.47	   MEMBER_IN_GA(X, .(X2, Xs)) -> MEMBER_IN_GA(X, Xs)
6.39/2.47	
6.39/2.47	The TRS R consists of the following rules:
6.39/2.47	
6.39/2.47	   member_in_ga(X, .(X, X1)) -> member_out_ga(X, .(X, X1))
6.39/2.47	   member_in_ga(X, .(X2, Xs)) -> U1_ga(X, X2, Xs, member_in_ga(X, Xs))
6.39/2.47	   U1_ga(X, X2, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X2, Xs))
6.39/2.47	
6.39/2.47	The argument filtering Pi contains the following mapping:
6.39/2.47	member_in_ga(x1, x2)  =  member_in_ga(x1)
6.39/2.47	
6.39/2.47	member_out_ga(x1, x2)  =  member_out_ga(x1)
6.39/2.47	
6.39/2.47	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
6.39/2.47	
6.39/2.47	MEMBER_IN_GA(x1, x2)  =  MEMBER_IN_GA(x1)
6.39/2.47	
6.39/2.47	U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
6.39/2.47	
6.39/2.47	
6.39/2.47	We have to consider all (P,R,Pi)-chains
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(14)
6.39/2.47	Obligation:
6.39/2.47	Pi DP problem:
6.39/2.47	The TRS P consists of the following rules:
6.39/2.47	
6.39/2.47	   MEMBER_IN_GA(X, .(X2, Xs)) -> U1_GA(X, X2, Xs, member_in_ga(X, Xs))
6.39/2.47	   MEMBER_IN_GA(X, .(X2, Xs)) -> MEMBER_IN_GA(X, Xs)
6.39/2.47	
6.39/2.47	The TRS R consists of the following rules:
6.39/2.47	
6.39/2.47	   member_in_ga(X, .(X, X1)) -> member_out_ga(X, .(X, X1))
6.39/2.47	   member_in_ga(X, .(X2, Xs)) -> U1_ga(X, X2, Xs, member_in_ga(X, Xs))
6.39/2.47	   U1_ga(X, X2, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X2, Xs))
6.39/2.47	
6.39/2.47	The argument filtering Pi contains the following mapping:
6.39/2.47	member_in_ga(x1, x2)  =  member_in_ga(x1)
6.39/2.47	
6.39/2.47	member_out_ga(x1, x2)  =  member_out_ga(x1)
6.39/2.47	
6.39/2.47	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
6.39/2.47	
6.39/2.47	MEMBER_IN_GA(x1, x2)  =  MEMBER_IN_GA(x1)
6.39/2.47	
6.39/2.47	U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
6.39/2.47	
6.39/2.47	
6.39/2.47	We have to consider all (P,R,Pi)-chains
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(15) DependencyGraphProof (EQUIVALENT)
6.39/2.47	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(16)
6.39/2.47	Obligation:
6.39/2.47	Pi DP problem:
6.39/2.47	The TRS P consists of the following rules:
6.39/2.47	
6.39/2.47	   MEMBER_IN_GA(X, .(X2, Xs)) -> MEMBER_IN_GA(X, Xs)
6.39/2.47	
6.39/2.47	The TRS R consists of the following rules:
6.39/2.47	
6.39/2.47	   member_in_ga(X, .(X, X1)) -> member_out_ga(X, .(X, X1))
6.39/2.47	   member_in_ga(X, .(X2, Xs)) -> U1_ga(X, X2, Xs, member_in_ga(X, Xs))
6.39/2.47	   U1_ga(X, X2, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X2, Xs))
6.39/2.47	
6.39/2.47	The argument filtering Pi contains the following mapping:
6.39/2.47	member_in_ga(x1, x2)  =  member_in_ga(x1)
6.39/2.47	
6.39/2.47	member_out_ga(x1, x2)  =  member_out_ga(x1)
6.39/2.47	
6.39/2.47	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
6.39/2.47	
6.39/2.47	MEMBER_IN_GA(x1, x2)  =  MEMBER_IN_GA(x1)
6.39/2.47	
6.39/2.47	
6.39/2.47	We have to consider all (P,R,Pi)-chains
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(17) UsableRulesProof (EQUIVALENT)
6.39/2.47	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(18)
6.39/2.47	Obligation:
6.39/2.47	Pi DP problem:
6.39/2.47	The TRS P consists of the following rules:
6.39/2.47	
6.39/2.47	   MEMBER_IN_GA(X, .(X2, Xs)) -> MEMBER_IN_GA(X, Xs)
6.39/2.47	
6.39/2.47	R is empty.
6.39/2.47	The argument filtering Pi contains the following mapping:
6.39/2.47	MEMBER_IN_GA(x1, x2)  =  MEMBER_IN_GA(x1)
6.39/2.47	
6.39/2.47	
6.39/2.47	We have to consider all (P,R,Pi)-chains
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(19) PiDPToQDPProof (SOUND)
6.39/2.47	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(20)
6.39/2.47	Obligation:
6.39/2.47	Q DP problem:
6.39/2.47	The TRS P consists of the following rules:
6.39/2.47	
6.39/2.47	   MEMBER_IN_GA(X) -> MEMBER_IN_GA(X)
6.39/2.47	
6.39/2.47	R is empty.
6.39/2.47	Q is empty.
6.39/2.47	We have to consider all (P,Q,R)-chains.
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(21) PrologToTRSTransformerProof (SOUND)
6.39/2.47	Transformed Prolog program to TRS.
6.39/2.47	
6.39/2.47	{
6.39/2.47	    "root": 7,
6.39/2.47	    "program": {
6.39/2.47	        "directives": [],
6.39/2.47	        "clauses": [
6.39/2.47	            [
6.39/2.47	                "(member X (. X X1))",
6.39/2.47	                null
6.39/2.47	            ],
6.39/2.47	            [
6.39/2.47	                "(member X (. X2 Xs))",
6.39/2.47	                "(member X Xs)"
6.39/2.47	            ]
6.39/2.47	        ]
6.39/2.47	    },
6.39/2.47	    "graph": {
6.39/2.47	        "nodes": {
6.39/2.47	            "11": {
6.39/2.47	                "goal": [{
6.39/2.47	                    "clause": 0,
6.39/2.47	                    "scope": 1,
6.39/2.47	                    "term": "(member T1 T2)"
6.39/2.47	                }],
6.39/2.47	                "kb": {
6.39/2.47	                    "nonunifying": [],
6.39/2.47	                    "intvars": {},
6.39/2.47	                    "arithmetic": {
6.39/2.47	                        "type": "PlainIntegerRelationState",
6.39/2.47	                        "relations": []
6.39/2.47	                    },
6.39/2.47	                    "ground": ["T1"],
6.39/2.47	                    "free": [],
6.39/2.47	                    "exprvars": []
6.39/2.47	                }
6.39/2.47	            },
6.39/2.47	            "12": {
6.39/2.47	                "goal": [{
6.39/2.47	                    "clause": 1,
6.39/2.47	                    "scope": 1,
6.39/2.47	                    "term": "(member T1 T2)"
6.39/2.47	                }],
6.39/2.47	                "kb": {
6.39/2.47	                    "nonunifying": [],
6.39/2.47	                    "intvars": {},
6.39/2.47	                    "arithmetic": {
6.39/2.47	                        "type": "PlainIntegerRelationState",
6.39/2.47	                        "relations": []
6.39/2.47	                    },
6.39/2.47	                    "ground": ["T1"],
6.39/2.47	                    "free": [],
6.39/2.47	                    "exprvars": []
6.39/2.47	                }
6.39/2.47	            },
6.39/2.47	            "59": {
6.39/2.47	                "goal": [{
6.39/2.47	                    "clause": -1,
6.39/2.47	                    "scope": -1,
6.39/2.47	                    "term": "(true)"
6.39/2.47	                }],
6.39/2.47	                "kb": {
6.39/2.47	                    "nonunifying": [],
6.39/2.47	                    "intvars": {},
6.39/2.47	                    "arithmetic": {
6.39/2.47	                        "type": "PlainIntegerRelationState",
6.39/2.47	                        "relations": []
6.39/2.47	                    },
6.39/2.47	                    "ground": [],
6.39/2.47	                    "free": [],
6.39/2.47	                    "exprvars": []
6.39/2.47	                }
6.39/2.47	            },
6.39/2.47	            "7": {
6.39/2.47	                "goal": [{
6.39/2.47	                    "clause": -1,
6.39/2.47	                    "scope": -1,
6.39/2.47	                    "term": "(member T1 T2)"
6.39/2.47	                }],
6.39/2.47	                "kb": {
6.39/2.47	                    "nonunifying": [],
6.39/2.47	                    "intvars": {},
6.39/2.47	                    "arithmetic": {
6.39/2.47	                        "type": "PlainIntegerRelationState",
6.39/2.47	                        "relations": []
6.39/2.47	                    },
6.39/2.47	                    "ground": ["T1"],
6.39/2.47	                    "free": [],
6.39/2.47	                    "exprvars": []
6.39/2.47	                }
6.39/2.47	            },
6.39/2.47	            "8": {
6.39/2.47	                "goal": [
6.39/2.47	                    {
6.39/2.47	                        "clause": 0,
6.39/2.47	                        "scope": 1,
6.39/2.47	                        "term": "(member T1 T2)"
6.39/2.47	                    },
6.39/2.47	                    {
6.39/2.47	                        "clause": 1,
6.39/2.47	                        "scope": 1,
6.39/2.47	                        "term": "(member T1 T2)"
6.39/2.47	                    }
6.39/2.47	                ],
6.39/2.47	                "kb": {
6.39/2.47	                    "nonunifying": [],
6.39/2.47	                    "intvars": {},
6.39/2.47	                    "arithmetic": {
6.39/2.47	                        "type": "PlainIntegerRelationState",
6.39/2.47	                        "relations": []
6.39/2.47	                    },
6.39/2.47	                    "ground": ["T1"],
6.39/2.47	                    "free": [],
6.39/2.47	                    "exprvars": []
6.39/2.47	                }
6.39/2.47	            },
6.39/2.47	            "60": {
6.39/2.47	                "goal": [],
6.39/2.47	                "kb": {
6.39/2.47	                    "nonunifying": [],
6.39/2.47	                    "intvars": {},
6.39/2.47	                    "arithmetic": {
6.39/2.47	                        "type": "PlainIntegerRelationState",
6.39/2.47	                        "relations": []
6.39/2.47	                    },
6.39/2.47	                    "ground": [],
6.39/2.47	                    "free": [],
6.39/2.47	                    "exprvars": []
6.39/2.47	                }
6.39/2.47	            },
6.39/2.47	            "61": {
6.39/2.47	                "goal": [],
6.39/2.47	                "kb": {
6.39/2.47	                    "nonunifying": [],
6.39/2.47	                    "intvars": {},
6.39/2.47	                    "arithmetic": {
6.39/2.47	                        "type": "PlainIntegerRelationState",
6.39/2.47	                        "relations": []
6.39/2.47	                    },
6.39/2.47	                    "ground": [],
6.39/2.47	                    "free": [],
6.39/2.47	                    "exprvars": []
6.39/2.47	                }
6.39/2.47	            },
6.39/2.47	            "type": "Nodes",
6.39/2.47	            "63": {
6.39/2.47	                "goal": [{
6.39/2.47	                    "clause": -1,
6.39/2.47	                    "scope": -1,
6.39/2.47	                    "term": "(member T19 T22)"
6.39/2.47	                }],
6.39/2.47	                "kb": {
6.39/2.47	                    "nonunifying": [],
6.39/2.47	                    "intvars": {},
6.39/2.47	                    "arithmetic": {
6.39/2.47	                        "type": "PlainIntegerRelationState",
6.39/2.47	                        "relations": []
6.39/2.47	                    },
6.39/2.47	                    "ground": ["T19"],
6.39/2.47	                    "free": [],
6.39/2.47	                    "exprvars": []
6.39/2.47	                }
6.39/2.47	            },
6.39/2.47	            "64": {
6.39/2.47	                "goal": [],
6.39/2.47	                "kb": {
6.39/2.47	                    "nonunifying": [],
6.39/2.47	                    "intvars": {},
6.39/2.47	                    "arithmetic": {
6.39/2.47	                        "type": "PlainIntegerRelationState",
6.39/2.47	                        "relations": []
6.39/2.47	                    },
6.39/2.47	                    "ground": [],
6.39/2.47	                    "free": [],
6.39/2.47	                    "exprvars": []
6.39/2.47	                }
6.39/2.47	            }
6.39/2.47	        },
6.39/2.47	        "edges": [
6.39/2.47	            {
6.39/2.47	                "from": 7,
6.39/2.47	                "to": 8,
6.39/2.47	                "label": "CASE"
6.39/2.47	            },
6.39/2.47	            {
6.39/2.47	                "from": 8,
6.39/2.47	                "to": 11,
6.39/2.47	                "label": "PARALLEL"
6.39/2.47	            },
6.39/2.47	            {
6.39/2.47	                "from": 8,
6.39/2.47	                "to": 12,
6.39/2.47	                "label": "PARALLEL"
6.39/2.47	            },
6.39/2.47	            {
6.39/2.47	                "from": 11,
6.39/2.47	                "to": 59,
6.39/2.47	                "label": "EVAL with clause\nmember(X11, .(X11, X12)).\nand substitutionT1 -> T11,\nX11 -> T11,\nX12 -> T12,\nT2 -> .(T11, T12)"
6.39/2.47	            },
6.39/2.47	            {
6.39/2.47	                "from": 11,
6.39/2.47	                "to": 60,
6.39/2.47	                "label": "EVAL-BACKTRACK"
6.39/2.47	            },
6.39/2.47	            {
6.39/2.47	                "from": 12,
6.39/2.47	                "to": 63,
6.39/2.47	                "label": "EVAL with clause\nmember(X19, .(X20, X21)) :- member(X19, X21).\nand substitutionT1 -> T19,\nX19 -> T19,\nX20 -> T20,\nX21 -> T22,\nT2 -> .(T20, T22),\nT21 -> T22"
6.39/2.47	            },
6.39/2.47	            {
6.39/2.47	                "from": 12,
6.39/2.47	                "to": 64,
6.39/2.47	                "label": "EVAL-BACKTRACK"
6.39/2.47	            },
6.39/2.47	            {
6.39/2.47	                "from": 59,
6.39/2.47	                "to": 61,
6.39/2.47	                "label": "SUCCESS"
6.39/2.47	            },
6.39/2.47	            {
6.39/2.47	                "from": 63,
6.39/2.47	                "to": 7,
6.39/2.47	                "label": "INSTANCE with matching:\nT1 -> T19\nT2 -> T22"
6.39/2.47	            }
6.39/2.47	        ],
6.39/2.47	        "type": "Graph"
6.39/2.47	    }
6.39/2.47	}
6.39/2.47	
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(22)
6.39/2.47	Obligation:
6.39/2.47	Q restricted rewrite system:
6.39/2.47	The TRS R consists of the following rules:
6.39/2.47	
6.39/2.47	   f7_in(T11) -> f7_out1
6.39/2.47	   f7_in(T19) -> U1(f7_in(T19), T19)
6.39/2.47	   U1(f7_out1, T19) -> f7_out1
6.39/2.47	
6.39/2.47	Q is empty.
6.39/2.47	
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(23) DependencyPairsProof (EQUIVALENT)
6.39/2.47	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(24)
6.39/2.47	Obligation:
6.39/2.47	Q DP problem:
6.39/2.47	The TRS P consists of the following rules:
6.39/2.47	
6.39/2.47	   F7_IN(T19) -> U1^1(f7_in(T19), T19)
6.39/2.47	   F7_IN(T19) -> F7_IN(T19)
6.39/2.47	
6.39/2.47	The TRS R consists of the following rules:
6.39/2.47	
6.39/2.47	   f7_in(T11) -> f7_out1
6.39/2.47	   f7_in(T19) -> U1(f7_in(T19), T19)
6.39/2.47	   U1(f7_out1, T19) -> f7_out1
6.39/2.47	
6.39/2.47	Q is empty.
6.39/2.47	We have to consider all minimal (P,Q,R)-chains.
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(25) DependencyGraphProof (EQUIVALENT)
6.39/2.47	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(26)
6.39/2.47	Obligation:
6.39/2.47	Q DP problem:
6.39/2.47	The TRS P consists of the following rules:
6.39/2.47	
6.39/2.47	   F7_IN(T19) -> F7_IN(T19)
6.39/2.47	
6.39/2.47	The TRS R consists of the following rules:
6.39/2.47	
6.39/2.47	   f7_in(T11) -> f7_out1
6.39/2.47	   f7_in(T19) -> U1(f7_in(T19), T19)
6.39/2.47	   U1(f7_out1, T19) -> f7_out1
6.39/2.47	
6.39/2.47	Q is empty.
6.39/2.47	We have to consider all minimal (P,Q,R)-chains.
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(27) MNOCProof (EQUIVALENT)
6.39/2.47	We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(28)
6.39/2.47	Obligation:
6.39/2.47	Q DP problem:
6.39/2.47	The TRS P consists of the following rules:
6.39/2.47	
6.39/2.47	   F7_IN(T19) -> F7_IN(T19)
6.39/2.47	
6.39/2.47	The TRS R consists of the following rules:
6.39/2.47	
6.39/2.47	   f7_in(T11) -> f7_out1
6.39/2.47	   f7_in(T19) -> U1(f7_in(T19), T19)
6.39/2.47	   U1(f7_out1, T19) -> f7_out1
6.39/2.47	
6.39/2.47	The set Q consists of the following terms:
6.39/2.47	
6.39/2.47	   f7_in(x0)
6.39/2.47	   U1(f7_out1, x0)
6.39/2.47	
6.39/2.47	We have to consider all minimal (P,Q,R)-chains.
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(29) UsableRulesProof (EQUIVALENT)
6.39/2.47	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(30)
6.39/2.47	Obligation:
6.39/2.47	Q DP problem:
6.39/2.47	The TRS P consists of the following rules:
6.39/2.47	
6.39/2.47	   F7_IN(T19) -> F7_IN(T19)
6.39/2.47	
6.39/2.47	R is empty.
6.39/2.47	The set Q consists of the following terms:
6.39/2.47	
6.39/2.47	   f7_in(x0)
6.39/2.47	   U1(f7_out1, x0)
6.39/2.47	
6.39/2.47	We have to consider all minimal (P,Q,R)-chains.
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(31) QReductionProof (EQUIVALENT)
6.39/2.47	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
6.39/2.47	
6.39/2.47	   f7_in(x0)
6.39/2.47	   U1(f7_out1, x0)
6.39/2.47	
6.39/2.47	
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(32)
6.39/2.47	Obligation:
6.39/2.47	Q DP problem:
6.39/2.47	The TRS P consists of the following rules:
6.39/2.47	
6.39/2.47	   F7_IN(T19) -> F7_IN(T19)
6.39/2.47	
6.39/2.47	R is empty.
6.39/2.47	Q is empty.
6.39/2.47	We have to consider all minimal (P,Q,R)-chains.
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(33) PrologToIRSwTTransformerProof (SOUND)
6.39/2.47	Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert
6.39/2.47	
6.39/2.47	{
6.39/2.47	    "root": 3,
6.39/2.47	    "program": {
6.39/2.47	        "directives": [],
6.39/2.47	        "clauses": [
6.39/2.47	            [
6.39/2.47	                "(member X (. X X1))",
6.39/2.47	                null
6.39/2.47	            ],
6.39/2.47	            [
6.39/2.47	                "(member X (. X2 Xs))",
6.39/2.47	                "(member X Xs)"
6.39/2.47	            ]
6.39/2.47	        ]
6.39/2.47	    },
6.39/2.47	    "graph": {
6.39/2.47	        "nodes": {
6.39/2.47	            "3": {
6.39/2.47	                "goal": [{
6.39/2.47	                    "clause": -1,
6.39/2.47	                    "scope": -1,
6.39/2.47	                    "term": "(member T1 T2)"
6.39/2.47	                }],
6.39/2.47	                "kb": {
6.39/2.47	                    "nonunifying": [],
6.39/2.47	                    "intvars": {},
6.39/2.47	                    "arithmetic": {
6.39/2.47	                        "type": "PlainIntegerRelationState",
6.39/2.47	                        "relations": []
6.39/2.47	                    },
6.39/2.47	                    "ground": ["T1"],
6.39/2.47	                    "free": [],
6.39/2.47	                    "exprvars": []
6.39/2.47	                }
6.39/2.47	            },
6.39/2.47	            "6": {
6.39/2.47	                "goal": [
6.39/2.47	                    {
6.39/2.47	                        "clause": 0,
6.39/2.47	                        "scope": 1,
6.39/2.47	                        "term": "(member T1 T2)"
6.39/2.47	                    },
6.39/2.47	                    {
6.39/2.47	                        "clause": 1,
6.39/2.47	                        "scope": 1,
6.39/2.47	                        "term": "(member T1 T2)"
6.39/2.47	                    }
6.39/2.47	                ],
6.39/2.47	                "kb": {
6.39/2.47	                    "nonunifying": [],
6.39/2.47	                    "intvars": {},
6.39/2.47	                    "arithmetic": {
6.39/2.47	                        "type": "PlainIntegerRelationState",
6.39/2.47	                        "relations": []
6.39/2.47	                    },
6.39/2.47	                    "ground": ["T1"],
6.39/2.47	                    "free": [],
6.39/2.47	                    "exprvars": []
6.39/2.47	                }
6.39/2.47	            },
6.39/2.47	            "70": {
6.39/2.47	                "goal": [{
6.39/2.47	                    "clause": 0,
6.39/2.47	                    "scope": 1,
6.39/2.47	                    "term": "(member T1 T2)"
6.39/2.47	                }],
6.39/2.47	                "kb": {
6.39/2.47	                    "nonunifying": [],
6.39/2.47	                    "intvars": {},
6.39/2.47	                    "arithmetic": {
6.39/2.47	                        "type": "PlainIntegerRelationState",
6.39/2.47	                        "relations": []
6.39/2.47	                    },
6.39/2.47	                    "ground": ["T1"],
6.39/2.47	                    "free": [],
6.39/2.47	                    "exprvars": []
6.39/2.47	                }
6.39/2.47	            },
6.39/2.47	            "71": {
6.39/2.47	                "goal": [{
6.39/2.47	                    "clause": 1,
6.39/2.47	                    "scope": 1,
6.39/2.47	                    "term": "(member T1 T2)"
6.39/2.47	                }],
6.39/2.47	                "kb": {
6.39/2.47	                    "nonunifying": [],
6.39/2.47	                    "intvars": {},
6.39/2.47	                    "arithmetic": {
6.39/2.47	                        "type": "PlainIntegerRelationState",
6.39/2.47	                        "relations": []
6.39/2.47	                    },
6.39/2.47	                    "ground": ["T1"],
6.39/2.47	                    "free": [],
6.39/2.47	                    "exprvars": []
6.39/2.47	                }
6.39/2.47	            },
6.39/2.47	            "72": {
6.39/2.47	                "goal": [{
6.39/2.47	                    "clause": -1,
6.39/2.47	                    "scope": -1,
6.39/2.47	                    "term": "(true)"
6.39/2.47	                }],
6.39/2.47	                "kb": {
6.39/2.47	                    "nonunifying": [],
6.39/2.47	                    "intvars": {},
6.39/2.47	                    "arithmetic": {
6.39/2.47	                        "type": "PlainIntegerRelationState",
6.39/2.47	                        "relations": []
6.39/2.47	                    },
6.39/2.47	                    "ground": [],
6.39/2.47	                    "free": [],
6.39/2.47	                    "exprvars": []
6.39/2.47	                }
6.39/2.47	            },
6.39/2.47	            "type": "Nodes",
6.39/2.47	            "73": {
6.39/2.47	                "goal": [],
6.39/2.47	                "kb": {
6.39/2.47	                    "nonunifying": [],
6.39/2.47	                    "intvars": {},
6.39/2.47	                    "arithmetic": {
6.39/2.47	                        "type": "PlainIntegerRelationState",
6.39/2.47	                        "relations": []
6.39/2.47	                    },
6.39/2.47	                    "ground": [],
6.39/2.47	                    "free": [],
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6.39/2.47	                }
6.39/2.47	            },
6.39/2.47	            "74": {
6.39/2.47	                "goal": [],
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6.39/2.47	                    "nonunifying": [],
6.39/2.47	                    "intvars": {},
6.39/2.47	                    "arithmetic": {
6.39/2.47	                        "type": "PlainIntegerRelationState",
6.39/2.47	                        "relations": []
6.39/2.47	                    },
6.39/2.47	                    "ground": [],
6.39/2.47	                    "free": [],
6.39/2.47	                    "exprvars": []
6.39/2.47	                }
6.39/2.47	            },
6.39/2.47	            "75": {
6.39/2.47	                "goal": [{
6.39/2.47	                    "clause": -1,
6.39/2.47	                    "scope": -1,
6.39/2.47	                    "term": "(member T19 T22)"
6.39/2.47	                }],
6.39/2.47	                "kb": {
6.39/2.47	                    "nonunifying": [],
6.39/2.47	                    "intvars": {},
6.39/2.47	                    "arithmetic": {
6.39/2.47	                        "type": "PlainIntegerRelationState",
6.39/2.47	                        "relations": []
6.39/2.47	                    },
6.39/2.47	                    "ground": ["T19"],
6.39/2.47	                    "free": [],
6.39/2.47	                    "exprvars": []
6.39/2.47	                }
6.39/2.47	            },
6.39/2.47	            "76": {
6.39/2.47	                "goal": [],
6.39/2.47	                "kb": {
6.39/2.47	                    "nonunifying": [],
6.39/2.47	                    "intvars": {},
6.39/2.47	                    "arithmetic": {
6.39/2.47	                        "type": "PlainIntegerRelationState",
6.39/2.47	                        "relations": []
6.39/2.47	                    },
6.39/2.47	                    "ground": [],
6.39/2.47	                    "free": [],
6.39/2.47	                    "exprvars": []
6.39/2.47	                }
6.39/2.47	            }
6.39/2.47	        },
6.39/2.47	        "edges": [
6.39/2.47	            {
6.39/2.47	                "from": 3,
6.39/2.47	                "to": 6,
6.39/2.47	                "label": "CASE"
6.39/2.47	            },
6.39/2.47	            {
6.39/2.47	                "from": 6,
6.39/2.47	                "to": 70,
6.39/2.47	                "label": "PARALLEL"
6.39/2.47	            },
6.39/2.47	            {
6.39/2.47	                "from": 6,
6.39/2.47	                "to": 71,
6.39/2.47	                "label": "PARALLEL"
6.39/2.47	            },
6.39/2.47	            {
6.39/2.47	                "from": 70,
6.39/2.47	                "to": 72,
6.39/2.47	                "label": "EVAL with clause\nmember(X11, .(X11, X12)).\nand substitutionT1 -> T11,\nX11 -> T11,\nX12 -> T12,\nT2 -> .(T11, T12)"
6.39/2.47	            },
6.39/2.47	            {
6.39/2.47	                "from": 70,
6.39/2.47	                "to": 73,
6.39/2.47	                "label": "EVAL-BACKTRACK"
6.39/2.47	            },
6.39/2.47	            {
6.39/2.47	                "from": 71,
6.39/2.47	                "to": 75,
6.39/2.47	                "label": "EVAL with clause\nmember(X19, .(X20, X21)) :- member(X19, X21).\nand substitutionT1 -> T19,\nX19 -> T19,\nX20 -> T20,\nX21 -> T22,\nT2 -> .(T20, T22),\nT21 -> T22"
6.39/2.47	            },
6.39/2.47	            {
6.39/2.47	                "from": 71,
6.39/2.47	                "to": 76,
6.39/2.47	                "label": "EVAL-BACKTRACK"
6.39/2.47	            },
6.39/2.47	            {
6.39/2.47	                "from": 72,
6.39/2.47	                "to": 74,
6.39/2.47	                "label": "SUCCESS"
6.39/2.47	            },
6.39/2.47	            {
6.39/2.47	                "from": 75,
6.39/2.47	                "to": 3,
6.39/2.47	                "label": "INSTANCE with matching:\nT1 -> T19\nT2 -> T22"
6.39/2.47	            }
6.39/2.47	        ],
6.39/2.47	        "type": "Graph"
6.39/2.47	    }
6.39/2.47	}
6.39/2.47	
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(34)
6.39/2.47	Obligation:
6.39/2.47	Rules:
6.39/2.47	f3_out(T19) -> f75_out(T19) :|: TRUE
6.39/2.47	f75_in(x) -> f3_in(x) :|: TRUE
6.39/2.47	f6_in(T1) -> f70_in(T1) :|: TRUE
6.39/2.47	f70_out(x1) -> f6_out(x1) :|: TRUE
6.39/2.47	f71_out(x2) -> f6_out(x2) :|: TRUE
6.39/2.47	f6_in(x3) -> f71_in(x3) :|: TRUE
6.39/2.47	f71_in(x4) -> f76_in :|: TRUE
6.39/2.47	f75_out(x5) -> f71_out(x5) :|: TRUE
6.39/2.47	f71_in(x6) -> f75_in(x6) :|: TRUE
6.39/2.47	f76_out -> f71_out(x7) :|: TRUE
6.39/2.47	f6_out(x8) -> f3_out(x8) :|: TRUE
6.39/2.47	f3_in(x9) -> f6_in(x9) :|: TRUE
6.39/2.47	Start term: f3_in(T1)
6.39/2.47	
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(35) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
6.39/2.47	Constructed simple dependency graph.
6.39/2.47	
6.39/2.47	Simplified to the following IRSwTs:
6.39/2.47	
6.39/2.47	intTRSProblem:
6.39/2.47	f75_in(x) -> f3_in(x) :|: TRUE
6.39/2.47	f6_in(x3) -> f71_in(x3) :|: TRUE
6.39/2.47	f71_in(x6) -> f75_in(x6) :|: TRUE
6.39/2.47	f3_in(x9) -> f6_in(x9) :|: TRUE
6.39/2.47	
6.39/2.47	
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(36)
6.39/2.47	Obligation:
6.39/2.47	Rules:
6.39/2.47	f75_in(x) -> f3_in(x) :|: TRUE
6.39/2.47	f6_in(x3) -> f71_in(x3) :|: TRUE
6.39/2.47	f71_in(x6) -> f75_in(x6) :|: TRUE
6.39/2.47	f3_in(x9) -> f6_in(x9) :|: TRUE
6.39/2.47	
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(37) IntTRSCompressionProof (EQUIVALENT)
6.39/2.47	Compressed rules.
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(38)
6.39/2.47	Obligation:
6.39/2.47	Rules:
6.39/2.47	f6_in(x3:0) -> f6_in(x3:0) :|: TRUE
6.39/2.47	
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(39) IRSFormatTransformerProof (EQUIVALENT)
6.39/2.47	Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(40)
6.39/2.47	Obligation:
6.39/2.47	Rules:
6.39/2.47	f6_in(x3:0) -> f6_in(x3:0) :|: TRUE
6.39/2.47	
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(41) IRSwTTerminationDigraphProof (EQUIVALENT)
6.39/2.47	Constructed termination digraph!
6.39/2.47	Nodes:
6.39/2.47	(1) f6_in(x3:0) -> f6_in(x3:0) :|: TRUE
6.39/2.47	
6.39/2.47	Arcs:
6.39/2.47	(1) -> (1)
6.39/2.47	
6.39/2.47	This digraph is fully evaluated!
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(42)
6.39/2.47	Obligation:
6.39/2.47	
6.39/2.47	Termination digraph:
6.39/2.47	Nodes:
6.39/2.47	(1) f6_in(x3:0) -> f6_in(x3:0) :|: TRUE
6.39/2.47	
6.39/2.47	Arcs:
6.39/2.47	(1) -> (1)
6.39/2.47	
6.39/2.47	This digraph is fully evaluated!
6.39/2.47	
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(43) FilterProof (EQUIVALENT)
6.39/2.47	Used the following sort dictionary for filtering: 
6.39/2.47	f6_in(VARIABLE)
6.39/2.47	Replaced non-predefined constructor symbols by 0.
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(44)
6.39/2.47	Obligation:
6.39/2.47	Rules:
6.39/2.47	f6_in(x3:0) -> f6_in(x3:0) :|: TRUE
6.39/2.47	
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(45) IntTRSPeriodicNontermProof (COMPLETE)
6.39/2.47	Normalized system to the following form:
6.39/2.47	f(pc, x3:0) -> f(1, x3:0) :|: pc = 1 && TRUE
6.39/2.47	Witness term starting non-terminating reduction: f(1, -8)
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(46)
6.39/2.47	NO
6.39/2.47	
6.39/2.47	----------------------------------------
6.39/2.47	
6.39/2.47	(47) PrologToDTProblemTransformerProof (SOUND)
6.39/2.47	Built DT problem from termination graph DT10.
6.39/2.47	
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6.39/2.47	    "program": {
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6.39/2.47	        "clauses": [
6.39/2.47	            [
6.39/2.47	                "(member X (. X X1))",
6.39/2.47	                null
6.39/2.47	            ],
6.39/2.47	            [
6.39/2.47	                "(member X (. X2 Xs))",
6.39/2.47	                "(member X Xs)"
6.39/2.47	            ]
6.39/2.47	        ]
6.39/2.47	    },
6.39/2.47	    "graph": {
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6.39/2.47	                    "clause": -1,
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6.39/2.47	                    "term": "(true)"
6.39/2.47	                }],
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6.39/2.47	                }
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6.39/2.47	            "90": {
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6.39/2.47	            },
6.39/2.47	            "93": {
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6.39/2.47	                    "scope": -1,
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6.39/2.47	                }],
6.39/2.47	                "kb": {
6.39/2.47	                    "nonunifying": [[
6.39/2.47	                        "(member T41 T2)",
6.39/2.47	                        "(member X5 (. X5 X6))"
6.39/2.47	                    ]],
6.39/2.47	                    "intvars": {},
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6.39/2.47	                        "relations": []
6.39/2.47	                    },
6.39/2.47	                    "ground": ["T41"],
6.39/2.47	                    "free": [
6.39/2.47	                        "X5",
6.39/2.47	                        "X6"
6.39/2.47	                    ],
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6.39/2.47	                }
6.39/2.47	            },
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6.39/2.47	                        "relations": []
6.39/2.47	                    },
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6.39/2.47	            },
6.39/2.47	            "95": {
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6.39/2.47	                    },
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6.39/2.47	                        "(member T41 T2)",
6.39/2.47	                        "(member X5 (. X5 X6))"
6.39/2.47	                    ]],
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6.39/2.47	                        "X5",
6.39/2.47	                        "X6"
6.39/2.47	                    ],
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6.39/2.47	            },
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6.39/2.47	                }],
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6.39/2.47	                        "(member T41 T2)",
6.39/2.47	                        "(member X5 (. X5 X6))"
6.39/2.47	                    ]],
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6.39/2.47	                        "type": "PlainIntegerRelationState",
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6.39/2.47	                    },
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6.39/2.47	                        "X5",
6.39/2.47	                        "X6"
6.39/2.47	                    ],
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6.39/2.47	                }
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6.39/2.47	                        "(member T41 T2)",
6.39/2.47	                        "(member X5 (. X5 X6))"
6.39/2.47	                    ]],
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6.39/2.47	                    },
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6.39/2.47	                        "X5",
6.39/2.47	                        "X6"
6.39/2.47	                    ],
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6.39/2.47	                }
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6.39/2.47	                        "relations": []
6.39/2.47	                    },
6.39/2.47	                    "ground": ["T1"],
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6.39/2.47	                }
6.39/2.47	            },
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6.39/2.47	                    "scope": -1,
6.39/2.47	                    "term": "(true)"
6.39/2.47	                }],
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6.39/2.47	                        "relations": []
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6.39/2.47	                }
6.39/2.47	            },
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6.39/2.47	                        "(member T1 T2)",
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6.39/2.47	                        "(member T61 T2)",
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6.39/2.47	                        "X6"
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6.39/2.48	            },
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6.39/2.48	            }
6.39/2.48	        ],
6.39/2.48	        "type": "Graph"
6.39/2.48	    }
6.39/2.48	}
6.39/2.48	
6.39/2.48	----------------------------------------
6.39/2.48	
6.39/2.48	(48)
6.39/2.48	Obligation:
6.39/2.48	Triples:
6.39/2.48	
6.39/2.48	memberA(X1, .(X2, .(X3, X4))) :- memberA(X1, X4).
6.39/2.48	memberA(X1, .(X2, .(X3, X4))) :- memberA(X1, X4).
6.39/2.48	
6.39/2.48	Clauses:
6.39/2.48	
6.39/2.48	membercA(X1, .(X1, X2)).
6.39/2.48	membercA(X1, .(X2, .(X1, X3))).
6.39/2.48	membercA(X1, .(X2, .(X3, X4))) :- membercA(X1, X4).
6.39/2.48	membercA(X1, .(X2, .(X1, X3))).
6.39/2.48	membercA(X1, .(X2, .(X3, X4))) :- membercA(X1, X4).
6.39/2.48	
6.39/2.48	Afs:
6.39/2.48	
6.39/2.48	memberA(x1, x2)  =  memberA(x1)
6.39/2.48	
6.39/2.48	
6.39/2.48	----------------------------------------
6.39/2.48	
6.39/2.48	(49) TriplesToPiDPProof (SOUND)
6.39/2.48	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.39/2.48	
6.39/2.48	memberA_in_2: (b,f)
6.39/2.48	
6.39/2.48	Transforming TRIPLES into the following Term Rewriting System:
6.39/2.48	
6.39/2.48	Pi DP problem:
6.39/2.48	The TRS P consists of the following rules:
6.39/2.48	
6.39/2.48	   MEMBERA_IN_GA(X1, .(X2, .(X3, X4))) -> U1_GA(X1, X2, X3, X4, memberA_in_ga(X1, X4))
6.39/2.48	   MEMBERA_IN_GA(X1, .(X2, .(X3, X4))) -> MEMBERA_IN_GA(X1, X4)
6.39/2.48	
6.39/2.48	R is empty.
6.39/2.48	The argument filtering Pi contains the following mapping:
6.39/2.48	memberA_in_ga(x1, x2)  =  memberA_in_ga(x1)
6.39/2.48	
6.39/2.48	.(x1, x2)  =  .(x2)
6.39/2.48	
6.39/2.48	MEMBERA_IN_GA(x1, x2)  =  MEMBERA_IN_GA(x1)
6.39/2.48	
6.39/2.48	U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x5)
6.39/2.48	
6.39/2.48	
6.39/2.48	We have to consider all (P,R,Pi)-chains
6.39/2.48	
6.39/2.48	
6.39/2.48	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
6.39/2.48	
6.39/2.48	
6.39/2.48	
6.39/2.48	----------------------------------------
6.39/2.48	
6.39/2.48	(50)
6.39/2.48	Obligation:
6.39/2.48	Pi DP problem:
6.39/2.48	The TRS P consists of the following rules:
6.39/2.48	
6.39/2.48	   MEMBERA_IN_GA(X1, .(X2, .(X3, X4))) -> U1_GA(X1, X2, X3, X4, memberA_in_ga(X1, X4))
6.39/2.48	   MEMBERA_IN_GA(X1, .(X2, .(X3, X4))) -> MEMBERA_IN_GA(X1, X4)
6.39/2.48	
6.39/2.48	R is empty.
6.39/2.48	The argument filtering Pi contains the following mapping:
6.39/2.48	memberA_in_ga(x1, x2)  =  memberA_in_ga(x1)
6.39/2.48	
6.39/2.48	.(x1, x2)  =  .(x2)
6.39/2.48	
6.39/2.48	MEMBERA_IN_GA(x1, x2)  =  MEMBERA_IN_GA(x1)
6.39/2.48	
6.39/2.48	U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x5)
6.39/2.48	
6.39/2.48	
6.39/2.48	We have to consider all (P,R,Pi)-chains
6.39/2.48	----------------------------------------
6.39/2.48	
6.39/2.48	(51) DependencyGraphProof (EQUIVALENT)
6.39/2.48	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
6.39/2.48	----------------------------------------
6.39/2.48	
6.39/2.48	(52)
6.39/2.48	Obligation:
6.39/2.48	Pi DP problem:
6.39/2.48	The TRS P consists of the following rules:
6.39/2.48	
6.39/2.48	   MEMBERA_IN_GA(X1, .(X2, .(X3, X4))) -> MEMBERA_IN_GA(X1, X4)
6.39/2.48	
6.39/2.48	R is empty.
6.39/2.48	The argument filtering Pi contains the following mapping:
6.39/2.48	.(x1, x2)  =  .(x2)
6.39/2.48	
6.39/2.48	MEMBERA_IN_GA(x1, x2)  =  MEMBERA_IN_GA(x1)
6.39/2.48	
6.39/2.48	
6.39/2.48	We have to consider all (P,R,Pi)-chains
6.39/2.48	----------------------------------------
6.39/2.48	
6.39/2.48	(53) PiDPToQDPProof (SOUND)
6.39/2.48	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.39/2.48	----------------------------------------
6.39/2.48	
6.39/2.48	(54)
6.39/2.48	Obligation:
6.39/2.48	Q DP problem:
6.39/2.48	The TRS P consists of the following rules:
6.39/2.48	
6.39/2.48	   MEMBERA_IN_GA(X1) -> MEMBERA_IN_GA(X1)
6.39/2.48	
6.39/2.48	R is empty.
6.39/2.48	Q is empty.
6.39/2.48	We have to consider all (P,Q,R)-chains.
6.39/2.51	EOF