6.00/2.42	MAYBE
6.00/2.45	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
6.00/2.45	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.00/2.45	
6.00/2.45	
6.00/2.45	Left Termination of the query pattern
6.00/2.45	
6.00/2.45	suffix(g,a)
6.00/2.45	
6.00/2.45	w.r.t. the given Prolog program could not be shown:
6.00/2.45	
6.00/2.45	(0) Prolog
6.00/2.45	(1) PrologToPiTRSProof [SOUND, 0 ms]
6.00/2.45	(2) PiTRS
6.00/2.45	    (3) DependencyPairsProof [EQUIVALENT, 0 ms]
6.00/2.45	    (4) PiDP
6.00/2.45	    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
6.00/2.45	    (6) PiDP
6.00/2.45	    (7) UsableRulesProof [EQUIVALENT, 0 ms]
6.00/2.45	    (8) PiDP
6.00/2.45	    (9) PiDPToQDPProof [SOUND, 0 ms]
6.00/2.45	    (10) QDP
6.00/2.45	(11) PrologToPiTRSProof [SOUND, 0 ms]
6.00/2.45	(12) PiTRS
6.00/2.45	    (13) DependencyPairsProof [EQUIVALENT, 0 ms]
6.00/2.45	    (14) PiDP
6.00/2.45	    (15) DependencyGraphProof [EQUIVALENT, 0 ms]
6.00/2.45	    (16) PiDP
6.00/2.45	    (17) UsableRulesProof [EQUIVALENT, 0 ms]
6.00/2.45	    (18) PiDP
6.00/2.45	    (19) PiDPToQDPProof [SOUND, 0 ms]
6.00/2.45	    (20) QDP
6.00/2.45	(21) PrologToTRSTransformerProof [SOUND, 0 ms]
6.00/2.45	(22) QTRS
6.00/2.45	    (23) DependencyPairsProof [EQUIVALENT, 0 ms]
6.00/2.45	    (24) QDP
6.00/2.45	    (25) DependencyGraphProof [EQUIVALENT, 0 ms]
6.00/2.45	    (26) QDP
6.00/2.45	    (27) MNOCProof [EQUIVALENT, 0 ms]
6.00/2.45	    (28) QDP
6.00/2.45	    (29) UsableRulesProof [EQUIVALENT, 0 ms]
6.00/2.45	    (30) QDP
6.00/2.45	    (31) QReductionProof [EQUIVALENT, 0 ms]
6.00/2.45	    (32) QDP
6.00/2.45	(33) PrologToDTProblemTransformerProof [SOUND, 0 ms]
6.00/2.45	(34) TRIPLES
6.00/2.45	    (35) TriplesToPiDPProof [SOUND, 0 ms]
6.00/2.45	    (36) PiDP
6.00/2.45	    (37) DependencyGraphProof [EQUIVALENT, 0 ms]
6.00/2.45	    (38) PiDP
6.00/2.45	    (39) PiDPToQDPProof [SOUND, 0 ms]
6.00/2.45	    (40) QDP
6.00/2.45	(41) PrologToIRSwTTransformerProof [SOUND, 0 ms]
6.00/2.45	(42) IRSwT
6.00/2.45	    (43) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
6.00/2.45	    (44) IRSwT
6.00/2.45	    (45) IntTRSCompressionProof [EQUIVALENT, 1 ms]
6.00/2.45	    (46) IRSwT
6.00/2.45	    (47) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
6.00/2.45	    (48) IRSwT
6.00/2.45	    (49) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
6.00/2.45	    (50) IRSwT
6.00/2.45	    (51) FilterProof [EQUIVALENT, 0 ms]
6.00/2.45	    (52) IntTRS
6.00/2.45	    (53) IntTRSPeriodicNontermProof [COMPLETE, 0 ms]
6.00/2.45	    (54) NO
6.00/2.45	
6.00/2.45	
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(0)
6.00/2.45	Obligation:
6.00/2.45	Clauses:
6.00/2.45	
6.00/2.45	suffix(Xs, Ys) :- app(X1, Xs, Ys).
6.00/2.45	app([], X, X).
6.00/2.45	app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
6.00/2.45	
6.00/2.45	
6.00/2.45	Query: suffix(g,a)
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(1) PrologToPiTRSProof (SOUND)
6.00/2.45	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.00/2.45	
6.00/2.45	suffix_in_2: (b,f)
6.00/2.45	
6.00/2.45	app_in_3: (f,b,f)
6.00/2.45	
6.00/2.45	Transforming Prolog into the following Term Rewriting System:
6.00/2.45	
6.00/2.45	Pi-finite rewrite system:
6.00/2.45	The TRS R consists of the following rules:
6.00/2.45	
6.00/2.45	   suffix_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys))
6.00/2.45	   app_in_aga([], X, X) -> app_out_aga([], X, X)
6.00/2.45	   app_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
6.00/2.45	   U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) -> app_out_aga(.(X, Xs), Ys, .(X, Zs))
6.00/2.45	   U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) -> suffix_out_ga(Xs, Ys)
6.00/2.45	
6.00/2.45	The argument filtering Pi contains the following mapping:
6.00/2.45	suffix_in_ga(x1, x2)  =  suffix_in_ga(x1)
6.00/2.45	
6.00/2.45	U1_ga(x1, x2, x3)  =  U1_ga(x3)
6.00/2.45	
6.00/2.45	app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
6.00/2.45	
6.00/2.45	app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x3)
6.00/2.45	
6.00/2.45	U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
6.00/2.45	
6.00/2.45	.(x1, x2)  =  .(x2)
6.00/2.45	
6.00/2.45	suffix_out_ga(x1, x2)  =  suffix_out_ga(x2)
6.00/2.45	
6.00/2.45	
6.00/2.45	
6.00/2.45	
6.00/2.45	
6.00/2.45	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.00/2.45	
6.00/2.45	
6.00/2.45	
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(2)
6.00/2.45	Obligation:
6.00/2.45	Pi-finite rewrite system:
6.00/2.45	The TRS R consists of the following rules:
6.00/2.45	
6.00/2.45	   suffix_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys))
6.00/2.45	   app_in_aga([], X, X) -> app_out_aga([], X, X)
6.00/2.45	   app_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
6.00/2.45	   U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) -> app_out_aga(.(X, Xs), Ys, .(X, Zs))
6.00/2.45	   U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) -> suffix_out_ga(Xs, Ys)
6.00/2.45	
6.00/2.45	The argument filtering Pi contains the following mapping:
6.00/2.45	suffix_in_ga(x1, x2)  =  suffix_in_ga(x1)
6.00/2.45	
6.00/2.45	U1_ga(x1, x2, x3)  =  U1_ga(x3)
6.00/2.45	
6.00/2.45	app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
6.00/2.45	
6.00/2.45	app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x3)
6.00/2.45	
6.00/2.45	U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
6.00/2.45	
6.00/2.45	.(x1, x2)  =  .(x2)
6.00/2.45	
6.00/2.45	suffix_out_ga(x1, x2)  =  suffix_out_ga(x2)
6.00/2.45	
6.00/2.45	
6.00/2.45	
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(3) DependencyPairsProof (EQUIVALENT)
6.00/2.45	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.00/2.45	Pi DP problem:
6.00/2.45	The TRS P consists of the following rules:
6.00/2.45	
6.00/2.45	   SUFFIX_IN_GA(Xs, Ys) -> U1_GA(Xs, Ys, app_in_aga(X1, Xs, Ys))
6.00/2.45	   SUFFIX_IN_GA(Xs, Ys) -> APP_IN_AGA(X1, Xs, Ys)
6.00/2.45	   APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U2_AGA(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
6.00/2.45	   APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AGA(Xs, Ys, Zs)
6.00/2.45	
6.00/2.45	The TRS R consists of the following rules:
6.00/2.45	
6.00/2.45	   suffix_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys))
6.00/2.45	   app_in_aga([], X, X) -> app_out_aga([], X, X)
6.00/2.45	   app_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
6.00/2.45	   U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) -> app_out_aga(.(X, Xs), Ys, .(X, Zs))
6.00/2.45	   U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) -> suffix_out_ga(Xs, Ys)
6.00/2.45	
6.00/2.45	The argument filtering Pi contains the following mapping:
6.00/2.45	suffix_in_ga(x1, x2)  =  suffix_in_ga(x1)
6.00/2.45	
6.00/2.45	U1_ga(x1, x2, x3)  =  U1_ga(x3)
6.00/2.45	
6.00/2.45	app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
6.00/2.45	
6.00/2.45	app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x3)
6.00/2.45	
6.00/2.45	U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
6.00/2.45	
6.00/2.45	.(x1, x2)  =  .(x2)
6.00/2.45	
6.00/2.45	suffix_out_ga(x1, x2)  =  suffix_out_ga(x2)
6.00/2.45	
6.00/2.45	SUFFIX_IN_GA(x1, x2)  =  SUFFIX_IN_GA(x1)
6.00/2.45	
6.00/2.45	U1_GA(x1, x2, x3)  =  U1_GA(x3)
6.00/2.45	
6.00/2.45	APP_IN_AGA(x1, x2, x3)  =  APP_IN_AGA(x2)
6.00/2.45	
6.00/2.45	U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x5)
6.00/2.45	
6.00/2.45	
6.00/2.45	We have to consider all (P,R,Pi)-chains
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(4)
6.00/2.45	Obligation:
6.00/2.45	Pi DP problem:
6.00/2.45	The TRS P consists of the following rules:
6.00/2.45	
6.00/2.45	   SUFFIX_IN_GA(Xs, Ys) -> U1_GA(Xs, Ys, app_in_aga(X1, Xs, Ys))
6.00/2.45	   SUFFIX_IN_GA(Xs, Ys) -> APP_IN_AGA(X1, Xs, Ys)
6.00/2.45	   APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U2_AGA(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
6.00/2.45	   APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AGA(Xs, Ys, Zs)
6.00/2.45	
6.00/2.45	The TRS R consists of the following rules:
6.00/2.45	
6.00/2.45	   suffix_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys))
6.00/2.45	   app_in_aga([], X, X) -> app_out_aga([], X, X)
6.00/2.45	   app_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
6.00/2.45	   U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) -> app_out_aga(.(X, Xs), Ys, .(X, Zs))
6.00/2.45	   U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) -> suffix_out_ga(Xs, Ys)
6.00/2.45	
6.00/2.45	The argument filtering Pi contains the following mapping:
6.00/2.45	suffix_in_ga(x1, x2)  =  suffix_in_ga(x1)
6.00/2.45	
6.00/2.45	U1_ga(x1, x2, x3)  =  U1_ga(x3)
6.00/2.45	
6.00/2.45	app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
6.00/2.45	
6.00/2.45	app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x3)
6.00/2.45	
6.00/2.45	U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
6.00/2.45	
6.00/2.45	.(x1, x2)  =  .(x2)
6.00/2.45	
6.00/2.45	suffix_out_ga(x1, x2)  =  suffix_out_ga(x2)
6.00/2.45	
6.00/2.45	SUFFIX_IN_GA(x1, x2)  =  SUFFIX_IN_GA(x1)
6.00/2.45	
6.00/2.45	U1_GA(x1, x2, x3)  =  U1_GA(x3)
6.00/2.45	
6.00/2.45	APP_IN_AGA(x1, x2, x3)  =  APP_IN_AGA(x2)
6.00/2.45	
6.00/2.45	U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x5)
6.00/2.45	
6.00/2.45	
6.00/2.45	We have to consider all (P,R,Pi)-chains
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(5) DependencyGraphProof (EQUIVALENT)
6.00/2.45	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(6)
6.00/2.45	Obligation:
6.00/2.45	Pi DP problem:
6.00/2.45	The TRS P consists of the following rules:
6.00/2.45	
6.00/2.45	   APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AGA(Xs, Ys, Zs)
6.00/2.45	
6.00/2.45	The TRS R consists of the following rules:
6.00/2.45	
6.00/2.45	   suffix_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys))
6.00/2.45	   app_in_aga([], X, X) -> app_out_aga([], X, X)
6.00/2.45	   app_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
6.00/2.45	   U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) -> app_out_aga(.(X, Xs), Ys, .(X, Zs))
6.00/2.45	   U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) -> suffix_out_ga(Xs, Ys)
6.00/2.45	
6.00/2.45	The argument filtering Pi contains the following mapping:
6.00/2.45	suffix_in_ga(x1, x2)  =  suffix_in_ga(x1)
6.00/2.45	
6.00/2.45	U1_ga(x1, x2, x3)  =  U1_ga(x3)
6.00/2.45	
6.00/2.45	app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
6.00/2.45	
6.00/2.45	app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x3)
6.00/2.45	
6.00/2.45	U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x5)
6.00/2.45	
6.00/2.45	.(x1, x2)  =  .(x2)
6.00/2.45	
6.00/2.45	suffix_out_ga(x1, x2)  =  suffix_out_ga(x2)
6.00/2.45	
6.00/2.45	APP_IN_AGA(x1, x2, x3)  =  APP_IN_AGA(x2)
6.00/2.45	
6.00/2.45	
6.00/2.45	We have to consider all (P,R,Pi)-chains
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(7) UsableRulesProof (EQUIVALENT)
6.00/2.45	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(8)
6.00/2.45	Obligation:
6.00/2.45	Pi DP problem:
6.00/2.45	The TRS P consists of the following rules:
6.00/2.45	
6.00/2.45	   APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AGA(Xs, Ys, Zs)
6.00/2.45	
6.00/2.45	R is empty.
6.00/2.45	The argument filtering Pi contains the following mapping:
6.00/2.45	.(x1, x2)  =  .(x2)
6.00/2.45	
6.00/2.45	APP_IN_AGA(x1, x2, x3)  =  APP_IN_AGA(x2)
6.00/2.45	
6.00/2.45	
6.00/2.45	We have to consider all (P,R,Pi)-chains
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(9) PiDPToQDPProof (SOUND)
6.00/2.45	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(10)
6.00/2.45	Obligation:
6.00/2.45	Q DP problem:
6.00/2.45	The TRS P consists of the following rules:
6.00/2.45	
6.00/2.45	   APP_IN_AGA(Ys) -> APP_IN_AGA(Ys)
6.00/2.45	
6.00/2.45	R is empty.
6.00/2.45	Q is empty.
6.00/2.45	We have to consider all (P,Q,R)-chains.
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(11) PrologToPiTRSProof (SOUND)
6.00/2.45	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.00/2.45	
6.00/2.45	suffix_in_2: (b,f)
6.00/2.45	
6.00/2.45	app_in_3: (f,b,f)
6.00/2.45	
6.00/2.45	Transforming Prolog into the following Term Rewriting System:
6.00/2.45	
6.00/2.45	Pi-finite rewrite system:
6.00/2.45	The TRS R consists of the following rules:
6.00/2.45	
6.00/2.45	   suffix_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys))
6.00/2.45	   app_in_aga([], X, X) -> app_out_aga([], X, X)
6.00/2.45	   app_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
6.00/2.45	   U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) -> app_out_aga(.(X, Xs), Ys, .(X, Zs))
6.00/2.45	   U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) -> suffix_out_ga(Xs, Ys)
6.00/2.45	
6.00/2.45	The argument filtering Pi contains the following mapping:
6.00/2.45	suffix_in_ga(x1, x2)  =  suffix_in_ga(x1)
6.00/2.45	
6.00/2.45	U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
6.00/2.45	
6.00/2.45	app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
6.00/2.45	
6.00/2.45	app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x2, x3)
6.00/2.45	
6.00/2.45	U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
6.00/2.45	
6.00/2.45	.(x1, x2)  =  .(x2)
6.00/2.45	
6.00/2.45	suffix_out_ga(x1, x2)  =  suffix_out_ga(x1, x2)
6.00/2.45	
6.00/2.45	
6.00/2.45	
6.00/2.45	
6.00/2.45	
6.00/2.45	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.00/2.45	
6.00/2.45	
6.00/2.45	
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(12)
6.00/2.45	Obligation:
6.00/2.45	Pi-finite rewrite system:
6.00/2.45	The TRS R consists of the following rules:
6.00/2.45	
6.00/2.45	   suffix_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys))
6.00/2.45	   app_in_aga([], X, X) -> app_out_aga([], X, X)
6.00/2.45	   app_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
6.00/2.45	   U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) -> app_out_aga(.(X, Xs), Ys, .(X, Zs))
6.00/2.45	   U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) -> suffix_out_ga(Xs, Ys)
6.00/2.45	
6.00/2.45	The argument filtering Pi contains the following mapping:
6.00/2.45	suffix_in_ga(x1, x2)  =  suffix_in_ga(x1)
6.00/2.45	
6.00/2.45	U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
6.00/2.45	
6.00/2.45	app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
6.00/2.45	
6.00/2.45	app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x2, x3)
6.00/2.45	
6.00/2.45	U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
6.00/2.45	
6.00/2.45	.(x1, x2)  =  .(x2)
6.00/2.45	
6.00/2.45	suffix_out_ga(x1, x2)  =  suffix_out_ga(x1, x2)
6.00/2.45	
6.00/2.45	
6.00/2.45	
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(13) DependencyPairsProof (EQUIVALENT)
6.00/2.45	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.00/2.45	Pi DP problem:
6.00/2.45	The TRS P consists of the following rules:
6.00/2.45	
6.00/2.45	   SUFFIX_IN_GA(Xs, Ys) -> U1_GA(Xs, Ys, app_in_aga(X1, Xs, Ys))
6.00/2.45	   SUFFIX_IN_GA(Xs, Ys) -> APP_IN_AGA(X1, Xs, Ys)
6.00/2.45	   APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U2_AGA(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
6.00/2.45	   APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AGA(Xs, Ys, Zs)
6.00/2.45	
6.00/2.45	The TRS R consists of the following rules:
6.00/2.45	
6.00/2.45	   suffix_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys))
6.00/2.45	   app_in_aga([], X, X) -> app_out_aga([], X, X)
6.00/2.45	   app_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
6.00/2.45	   U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) -> app_out_aga(.(X, Xs), Ys, .(X, Zs))
6.00/2.45	   U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) -> suffix_out_ga(Xs, Ys)
6.00/2.45	
6.00/2.45	The argument filtering Pi contains the following mapping:
6.00/2.45	suffix_in_ga(x1, x2)  =  suffix_in_ga(x1)
6.00/2.45	
6.00/2.45	U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
6.00/2.45	
6.00/2.45	app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
6.00/2.45	
6.00/2.45	app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x2, x3)
6.00/2.45	
6.00/2.45	U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
6.00/2.45	
6.00/2.45	.(x1, x2)  =  .(x2)
6.00/2.45	
6.00/2.45	suffix_out_ga(x1, x2)  =  suffix_out_ga(x1, x2)
6.00/2.45	
6.00/2.45	SUFFIX_IN_GA(x1, x2)  =  SUFFIX_IN_GA(x1)
6.00/2.45	
6.00/2.45	U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
6.00/2.45	
6.00/2.45	APP_IN_AGA(x1, x2, x3)  =  APP_IN_AGA(x2)
6.00/2.45	
6.00/2.45	U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
6.00/2.45	
6.00/2.45	
6.00/2.45	We have to consider all (P,R,Pi)-chains
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(14)
6.00/2.45	Obligation:
6.00/2.45	Pi DP problem:
6.00/2.45	The TRS P consists of the following rules:
6.00/2.45	
6.00/2.45	   SUFFIX_IN_GA(Xs, Ys) -> U1_GA(Xs, Ys, app_in_aga(X1, Xs, Ys))
6.00/2.45	   SUFFIX_IN_GA(Xs, Ys) -> APP_IN_AGA(X1, Xs, Ys)
6.00/2.45	   APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> U2_AGA(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
6.00/2.45	   APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AGA(Xs, Ys, Zs)
6.00/2.45	
6.00/2.45	The TRS R consists of the following rules:
6.00/2.45	
6.00/2.45	   suffix_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys))
6.00/2.45	   app_in_aga([], X, X) -> app_out_aga([], X, X)
6.00/2.45	   app_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
6.00/2.45	   U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) -> app_out_aga(.(X, Xs), Ys, .(X, Zs))
6.00/2.45	   U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) -> suffix_out_ga(Xs, Ys)
6.00/2.45	
6.00/2.45	The argument filtering Pi contains the following mapping:
6.00/2.45	suffix_in_ga(x1, x2)  =  suffix_in_ga(x1)
6.00/2.45	
6.00/2.45	U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
6.00/2.45	
6.00/2.45	app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
6.00/2.45	
6.00/2.45	app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x2, x3)
6.00/2.45	
6.00/2.45	U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
6.00/2.45	
6.00/2.45	.(x1, x2)  =  .(x2)
6.00/2.45	
6.00/2.45	suffix_out_ga(x1, x2)  =  suffix_out_ga(x1, x2)
6.00/2.45	
6.00/2.45	SUFFIX_IN_GA(x1, x2)  =  SUFFIX_IN_GA(x1)
6.00/2.45	
6.00/2.45	U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
6.00/2.45	
6.00/2.45	APP_IN_AGA(x1, x2, x3)  =  APP_IN_AGA(x2)
6.00/2.45	
6.00/2.45	U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
6.00/2.45	
6.00/2.45	
6.00/2.45	We have to consider all (P,R,Pi)-chains
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(15) DependencyGraphProof (EQUIVALENT)
6.00/2.45	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(16)
6.00/2.45	Obligation:
6.00/2.45	Pi DP problem:
6.00/2.45	The TRS P consists of the following rules:
6.00/2.45	
6.00/2.45	   APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AGA(Xs, Ys, Zs)
6.00/2.45	
6.00/2.45	The TRS R consists of the following rules:
6.00/2.45	
6.00/2.45	   suffix_in_ga(Xs, Ys) -> U1_ga(Xs, Ys, app_in_aga(X1, Xs, Ys))
6.00/2.45	   app_in_aga([], X, X) -> app_out_aga([], X, X)
6.00/2.45	   app_in_aga(.(X, Xs), Ys, .(X, Zs)) -> U2_aga(X, Xs, Ys, Zs, app_in_aga(Xs, Ys, Zs))
6.00/2.45	   U2_aga(X, Xs, Ys, Zs, app_out_aga(Xs, Ys, Zs)) -> app_out_aga(.(X, Xs), Ys, .(X, Zs))
6.00/2.45	   U1_ga(Xs, Ys, app_out_aga(X1, Xs, Ys)) -> suffix_out_ga(Xs, Ys)
6.00/2.45	
6.00/2.45	The argument filtering Pi contains the following mapping:
6.00/2.45	suffix_in_ga(x1, x2)  =  suffix_in_ga(x1)
6.00/2.45	
6.00/2.45	U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
6.00/2.45	
6.00/2.45	app_in_aga(x1, x2, x3)  =  app_in_aga(x2)
6.00/2.45	
6.00/2.45	app_out_aga(x1, x2, x3)  =  app_out_aga(x1, x2, x3)
6.00/2.45	
6.00/2.45	U2_aga(x1, x2, x3, x4, x5)  =  U2_aga(x3, x5)
6.00/2.45	
6.00/2.45	.(x1, x2)  =  .(x2)
6.00/2.45	
6.00/2.45	suffix_out_ga(x1, x2)  =  suffix_out_ga(x1, x2)
6.00/2.45	
6.00/2.45	APP_IN_AGA(x1, x2, x3)  =  APP_IN_AGA(x2)
6.00/2.45	
6.00/2.45	
6.00/2.45	We have to consider all (P,R,Pi)-chains
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(17) UsableRulesProof (EQUIVALENT)
6.00/2.45	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(18)
6.00/2.45	Obligation:
6.00/2.45	Pi DP problem:
6.00/2.45	The TRS P consists of the following rules:
6.00/2.45	
6.00/2.45	   APP_IN_AGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AGA(Xs, Ys, Zs)
6.00/2.45	
6.00/2.45	R is empty.
6.00/2.45	The argument filtering Pi contains the following mapping:
6.00/2.45	.(x1, x2)  =  .(x2)
6.00/2.45	
6.00/2.45	APP_IN_AGA(x1, x2, x3)  =  APP_IN_AGA(x2)
6.00/2.45	
6.00/2.45	
6.00/2.45	We have to consider all (P,R,Pi)-chains
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(19) PiDPToQDPProof (SOUND)
6.00/2.45	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(20)
6.00/2.45	Obligation:
6.00/2.45	Q DP problem:
6.00/2.45	The TRS P consists of the following rules:
6.00/2.45	
6.00/2.45	   APP_IN_AGA(Ys) -> APP_IN_AGA(Ys)
6.00/2.45	
6.00/2.45	R is empty.
6.00/2.45	Q is empty.
6.00/2.45	We have to consider all (P,Q,R)-chains.
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(21) PrologToTRSTransformerProof (SOUND)
6.00/2.45	Transformed Prolog program to TRS.
6.00/2.45	
6.00/2.45	{
6.00/2.45	    "root": 6,
6.00/2.45	    "program": {
6.00/2.45	        "directives": [],
6.00/2.45	        "clauses": [
6.00/2.45	            [
6.00/2.45	                "(suffix Xs Ys)",
6.00/2.45	                "(app X1 Xs Ys)"
6.00/2.45	            ],
6.00/2.45	            [
6.00/2.45	                "(app ([]) X X)",
6.00/2.45	                null
6.00/2.45	            ],
6.00/2.45	            [
6.00/2.45	                "(app (. X Xs) Ys (. X Zs))",
6.00/2.45	                "(app Xs Ys Zs)"
6.00/2.45	            ]
6.00/2.45	        ]
6.00/2.45	    },
6.00/2.45	    "graph": {
6.00/2.45	        "nodes": {
6.00/2.45	            "122": {
6.00/2.45	                "goal": [
6.00/2.45	                    {
6.00/2.45	                        "clause": 1,
6.00/2.45	                        "scope": 2,
6.00/2.45	                        "term": "(app X11 T10 T12)"
6.00/2.45	                    },
6.00/2.45	                    {
6.00/2.45	                        "clause": 2,
6.00/2.45	                        "scope": 2,
6.00/2.45	                        "term": "(app X11 T10 T12)"
6.00/2.45	                    }
6.00/2.45	                ],
6.00/2.45	                "kb": {
6.00/2.45	                    "nonunifying": [],
6.00/2.45	                    "intvars": {},
6.00/2.45	                    "arithmetic": {
6.00/2.45	                        "type": "PlainIntegerRelationState",
6.00/2.45	                        "relations": []
6.00/2.45	                    },
6.00/2.45	                    "ground": ["T10"],
6.00/2.45	                    "free": ["X11"],
6.00/2.45	                    "exprvars": []
6.00/2.45	                }
6.00/2.45	            },
6.00/2.45	            "124": {
6.00/2.45	                "goal": [{
6.00/2.45	                    "clause": 1,
6.00/2.45	                    "scope": 2,
6.00/2.45	                    "term": "(app X11 T10 T12)"
6.00/2.45	                }],
6.00/2.45	                "kb": {
6.00/2.45	                    "nonunifying": [],
6.00/2.45	                    "intvars": {},
6.00/2.45	                    "arithmetic": {
6.00/2.45	                        "type": "PlainIntegerRelationState",
6.00/2.45	                        "relations": []
6.00/2.45	                    },
6.00/2.45	                    "ground": ["T10"],
6.00/2.45	                    "free": ["X11"],
6.00/2.45	                    "exprvars": []
6.00/2.45	                }
6.00/2.45	            },
6.00/2.45	            "125": {
6.00/2.45	                "goal": [{
6.00/2.45	                    "clause": 2,
6.00/2.45	                    "scope": 2,
6.00/2.45	                    "term": "(app X11 T10 T12)"
6.00/2.45	                }],
6.00/2.45	                "kb": {
6.00/2.45	                    "nonunifying": [],
6.00/2.45	                    "intvars": {},
6.00/2.45	                    "arithmetic": {
6.00/2.45	                        "type": "PlainIntegerRelationState",
6.00/2.45	                        "relations": []
6.00/2.45	                    },
6.00/2.45	                    "ground": ["T10"],
6.00/2.45	                    "free": ["X11"],
6.00/2.45	                    "exprvars": []
6.00/2.45	                }
6.00/2.45	            },
6.00/2.45	            "49": {
6.00/2.45	                "goal": [{
6.00/2.45	                    "clause": 0,
6.00/2.45	                    "scope": 1,
6.00/2.45	                    "term": "(suffix T1 T2)"
6.00/2.45	                }],
6.00/2.45	                "kb": {
6.00/2.45	                    "nonunifying": [],
6.00/2.45	                    "intvars": {},
6.00/2.45	                    "arithmetic": {
6.00/2.45	                        "type": "PlainIntegerRelationState",
6.00/2.45	                        "relations": []
6.00/2.45	                    },
6.00/2.45	                    "ground": ["T1"],
6.00/2.45	                    "free": [],
6.00/2.45	                    "exprvars": []
6.00/2.45	                }
6.00/2.45	            },
6.00/2.45	            "126": {
6.00/2.45	                "goal": [{
6.00/2.45	                    "clause": -1,
6.00/2.45	                    "scope": -1,
6.00/2.45	                    "term": "(true)"
6.00/2.45	                }],
6.00/2.45	                "kb": {
6.00/2.45	                    "nonunifying": [],
6.00/2.45	                    "intvars": {},
6.00/2.45	                    "arithmetic": {
6.00/2.45	                        "type": "PlainIntegerRelationState",
6.00/2.45	                        "relations": []
6.00/2.45	                    },
6.00/2.45	                    "ground": [],
6.00/2.45	                    "free": [],
6.00/2.45	                    "exprvars": []
6.00/2.45	                }
6.00/2.45	            },
6.00/2.45	            "6": {
6.00/2.45	                "goal": [{
6.00/2.45	                    "clause": -1,
6.00/2.45	                    "scope": -1,
6.00/2.45	                    "term": "(suffix T1 T2)"
6.00/2.45	                }],
6.00/2.45	                "kb": {
6.00/2.45	                    "nonunifying": [],
6.00/2.45	                    "intvars": {},
6.00/2.45	                    "arithmetic": {
6.00/2.45	                        "type": "PlainIntegerRelationState",
6.00/2.45	                        "relations": []
6.00/2.45	                    },
6.00/2.45	                    "ground": ["T1"],
6.00/2.45	                    "free": [],
6.00/2.45	                    "exprvars": []
6.00/2.45	                }
6.00/2.45	            },
6.00/2.45	            "127": {
6.00/2.45	                "goal": [],
6.00/2.45	                "kb": {
6.00/2.45	                    "nonunifying": [],
6.00/2.45	                    "intvars": {},
6.00/2.45	                    "arithmetic": {
6.00/2.45	                        "type": "PlainIntegerRelationState",
6.00/2.45	                        "relations": []
6.00/2.45	                    },
6.00/2.45	                    "ground": [],
6.00/2.45	                    "free": [],
6.00/2.45	                    "exprvars": []
6.00/2.45	                }
6.00/2.45	            },
6.00/2.45	            "128": {
6.00/2.45	                "goal": [],
6.00/2.45	                "kb": {
6.00/2.45	                    "nonunifying": [],
6.00/2.45	                    "intvars": {},
6.00/2.45	                    "arithmetic": {
6.00/2.45	                        "type": "PlainIntegerRelationState",
6.00/2.45	                        "relations": []
6.00/2.45	                    },
6.00/2.45	                    "ground": [],
6.00/2.45	                    "free": [],
6.00/2.45	                    "exprvars": []
6.00/2.45	                }
6.00/2.45	            },
6.00/2.45	            "129": {
6.00/2.45	                "goal": [{
6.00/2.45	                    "clause": -1,
6.00/2.45	                    "scope": -1,
6.00/2.45	                    "term": "(app X36 T26 T29)"
6.00/2.45	                }],
6.00/2.45	                "kb": {
6.00/2.45	                    "nonunifying": [],
6.00/2.45	                    "intvars": {},
6.00/2.45	                    "arithmetic": {
6.00/2.45	                        "type": "PlainIntegerRelationState",
6.00/2.45	                        "relations": []
6.00/2.45	                    },
6.00/2.45	                    "ground": ["T26"],
6.00/2.45	                    "free": ["X36"],
6.00/2.45	                    "exprvars": []
6.00/2.45	                }
6.00/2.45	            },
6.00/2.45	            "type": "Nodes",
6.00/2.45	            "130": {
6.00/2.45	                "goal": [],
6.00/2.45	                "kb": {
6.00/2.45	                    "nonunifying": [],
6.00/2.45	                    "intvars": {},
6.00/2.45	                    "arithmetic": {
6.00/2.45	                        "type": "PlainIntegerRelationState",
6.00/2.45	                        "relations": []
6.00/2.45	                    },
6.00/2.45	                    "ground": [],
6.00/2.45	                    "free": [],
6.00/2.45	                    "exprvars": []
6.00/2.45	                }
6.00/2.45	            },
6.00/2.45	            "120": {
6.00/2.45	                "goal": [{
6.00/2.45	                    "clause": -1,
6.00/2.45	                    "scope": -1,
6.00/2.45	                    "term": "(app X11 T10 T12)"
6.00/2.45	                }],
6.00/2.45	                "kb": {
6.00/2.45	                    "nonunifying": [],
6.00/2.45	                    "intvars": {},
6.00/2.45	                    "arithmetic": {
6.00/2.45	                        "type": "PlainIntegerRelationState",
6.00/2.45	                        "relations": []
6.00/2.45	                    },
6.00/2.45	                    "ground": ["T10"],
6.00/2.45	                    "free": ["X11"],
6.00/2.45	                    "exprvars": []
6.00/2.45	                }
6.00/2.45	            }
6.00/2.45	        },
6.00/2.45	        "edges": [
6.00/2.45	            {
6.00/2.45	                "from": 6,
6.00/2.45	                "to": 49,
6.00/2.45	                "label": "CASE"
6.00/2.45	            },
6.00/2.45	            {
6.00/2.45	                "from": 49,
6.00/2.45	                "to": 120,
6.00/2.45	                "label": "ONLY EVAL with clause\nsuffix(X9, X10) :- app(X11, X9, X10).\nand substitutionT1 -> T10,\nX9 -> T10,\nT2 -> T12,\nX10 -> T12,\nT11 -> T12"
6.00/2.45	            },
6.00/2.45	            {
6.00/2.45	                "from": 120,
6.00/2.45	                "to": 122,
6.00/2.45	                "label": "CASE"
6.00/2.45	            },
6.00/2.45	            {
6.00/2.45	                "from": 122,
6.00/2.45	                "to": 124,
6.00/2.45	                "label": "PARALLEL"
6.00/2.45	            },
6.00/2.45	            {
6.00/2.45	                "from": 122,
6.00/2.45	                "to": 125,
6.00/2.45	                "label": "PARALLEL"
6.00/2.45	            },
6.00/2.45	            {
6.00/2.45	                "from": 124,
6.00/2.45	                "to": 126,
6.00/2.45	                "label": "EVAL with clause\napp([], X18, X18).\nand substitutionX11 -> [],\nT10 -> T19,\nX18 -> T19,\nT12 -> T19"
6.00/2.45	            },
6.00/2.45	            {
6.00/2.45	                "from": 124,
6.00/2.45	                "to": 127,
6.00/2.45	                "label": "EVAL-BACKTRACK"
6.00/2.45	            },
6.00/2.45	            {
6.00/2.45	                "from": 125,
6.00/2.45	                "to": 129,
6.00/2.45	                "label": "EVAL with clause\napp(.(X31, X32), X33, .(X31, X34)) :- app(X32, X33, X34).\nand substitutionX31 -> T27,\nX32 -> X36,\nX11 -> .(T27, X36),\nT10 -> T26,\nX33 -> T26,\nX35 -> T27,\nX34 -> T29,\nT12 -> .(T27, T29),\nT28 -> T29"
6.00/2.45	            },
6.00/2.45	            {
6.00/2.45	                "from": 125,
6.00/2.45	                "to": 130,
6.00/2.45	                "label": "EVAL-BACKTRACK"
6.00/2.45	            },
6.00/2.45	            {
6.00/2.45	                "from": 126,
6.00/2.45	                "to": 128,
6.00/2.45	                "label": "SUCCESS"
6.00/2.45	            },
6.00/2.45	            {
6.00/2.45	                "from": 129,
6.00/2.45	                "to": 120,
6.00/2.45	                "label": "INSTANCE with matching:\nX11 -> X36\nT10 -> T26\nT12 -> T29"
6.00/2.45	            }
6.00/2.45	        ],
6.00/2.45	        "type": "Graph"
6.00/2.45	    }
6.00/2.45	}
6.00/2.45	
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(22)
6.00/2.45	Obligation:
6.00/2.45	Q restricted rewrite system:
6.00/2.45	The TRS R consists of the following rules:
6.00/2.45	
6.00/2.45	   f6_in(T10) -> U1(f120_in(T10), T10)
6.00/2.45	   U1(f120_out1, T10) -> f6_out1
6.00/2.45	   f120_in(T19) -> f120_out1
6.00/2.45	   f120_in(T26) -> U2(f120_in(T26), T26)
6.00/2.45	   U2(f120_out1, T26) -> f120_out1
6.00/2.45	
6.00/2.45	Q is empty.
6.00/2.45	
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(23) DependencyPairsProof (EQUIVALENT)
6.00/2.45	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(24)
6.00/2.45	Obligation:
6.00/2.45	Q DP problem:
6.00/2.45	The TRS P consists of the following rules:
6.00/2.45	
6.00/2.45	   F6_IN(T10) -> U1^1(f120_in(T10), T10)
6.00/2.45	   F6_IN(T10) -> F120_IN(T10)
6.00/2.45	   F120_IN(T26) -> U2^1(f120_in(T26), T26)
6.00/2.45	   F120_IN(T26) -> F120_IN(T26)
6.00/2.45	
6.00/2.45	The TRS R consists of the following rules:
6.00/2.45	
6.00/2.45	   f6_in(T10) -> U1(f120_in(T10), T10)
6.00/2.45	   U1(f120_out1, T10) -> f6_out1
6.00/2.45	   f120_in(T19) -> f120_out1
6.00/2.45	   f120_in(T26) -> U2(f120_in(T26), T26)
6.00/2.45	   U2(f120_out1, T26) -> f120_out1
6.00/2.45	
6.00/2.45	Q is empty.
6.00/2.45	We have to consider all minimal (P,Q,R)-chains.
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(25) DependencyGraphProof (EQUIVALENT)
6.00/2.45	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(26)
6.00/2.45	Obligation:
6.00/2.45	Q DP problem:
6.00/2.45	The TRS P consists of the following rules:
6.00/2.45	
6.00/2.45	   F120_IN(T26) -> F120_IN(T26)
6.00/2.45	
6.00/2.45	The TRS R consists of the following rules:
6.00/2.45	
6.00/2.45	   f6_in(T10) -> U1(f120_in(T10), T10)
6.00/2.45	   U1(f120_out1, T10) -> f6_out1
6.00/2.45	   f120_in(T19) -> f120_out1
6.00/2.45	   f120_in(T26) -> U2(f120_in(T26), T26)
6.00/2.45	   U2(f120_out1, T26) -> f120_out1
6.00/2.45	
6.00/2.45	Q is empty.
6.00/2.45	We have to consider all minimal (P,Q,R)-chains.
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(27) MNOCProof (EQUIVALENT)
6.00/2.45	We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(28)
6.00/2.45	Obligation:
6.00/2.45	Q DP problem:
6.00/2.45	The TRS P consists of the following rules:
6.00/2.45	
6.00/2.45	   F120_IN(T26) -> F120_IN(T26)
6.00/2.45	
6.00/2.45	The TRS R consists of the following rules:
6.00/2.45	
6.00/2.45	   f6_in(T10) -> U1(f120_in(T10), T10)
6.00/2.45	   U1(f120_out1, T10) -> f6_out1
6.00/2.45	   f120_in(T19) -> f120_out1
6.00/2.45	   f120_in(T26) -> U2(f120_in(T26), T26)
6.00/2.45	   U2(f120_out1, T26) -> f120_out1
6.00/2.45	
6.00/2.45	The set Q consists of the following terms:
6.00/2.45	
6.00/2.45	   f6_in(x0)
6.00/2.45	   U1(f120_out1, x0)
6.00/2.45	   f120_in(x0)
6.00/2.45	   U2(f120_out1, x0)
6.00/2.45	
6.00/2.45	We have to consider all minimal (P,Q,R)-chains.
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(29) UsableRulesProof (EQUIVALENT)
6.00/2.45	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.
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(30)
6.00/2.45	Obligation:
6.00/2.45	Q DP problem:
6.00/2.45	The TRS P consists of the following rules:
6.00/2.45	
6.00/2.45	   F120_IN(T26) -> F120_IN(T26)
6.00/2.45	
6.00/2.45	R is empty.
6.00/2.45	The set Q consists of the following terms:
6.00/2.45	
6.00/2.45	   f6_in(x0)
6.00/2.45	   U1(f120_out1, x0)
6.00/2.45	   f120_in(x0)
6.00/2.45	   U2(f120_out1, x0)
6.00/2.45	
6.00/2.45	We have to consider all minimal (P,Q,R)-chains.
6.00/2.45	----------------------------------------
6.00/2.45	
6.00/2.45	(31) QReductionProof (EQUIVALENT)
6.00/2.45	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
6.00/2.45	
6.00/2.45	   f6_in(x0)
6.00/2.45	   U1(f120_out1, x0)
6.00/2.45	   f120_in(x0)
6.00/2.46	   U2(f120_out1, x0)
6.00/2.46	
6.00/2.46	
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(32)
6.00/2.46	Obligation:
6.00/2.46	Q DP problem:
6.00/2.46	The TRS P consists of the following rules:
6.00/2.46	
6.00/2.46	   F120_IN(T26) -> F120_IN(T26)
6.00/2.46	
6.00/2.46	R is empty.
6.00/2.46	Q is empty.
6.00/2.46	We have to consider all minimal (P,Q,R)-chains.
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(33) PrologToDTProblemTransformerProof (SOUND)
6.00/2.46	Built DT problem from termination graph DT10.
6.00/2.46	
6.00/2.46	{
6.00/2.46	    "root": 2,
6.00/2.46	    "program": {
6.00/2.46	        "directives": [],
6.00/2.46	        "clauses": [
6.00/2.46	            [
6.00/2.46	                "(suffix Xs Ys)",
6.00/2.46	                "(app X1 Xs Ys)"
6.00/2.46	            ],
6.00/2.46	            [
6.00/2.46	                "(app ([]) X X)",
6.00/2.46	                null
6.00/2.46	            ],
6.00/2.46	            [
6.00/2.46	                "(app (. X Xs) Ys (. X Zs))",
6.00/2.46	                "(app Xs Ys Zs)"
6.00/2.46	            ]
6.00/2.46	        ]
6.00/2.46	    },
6.00/2.46	    "graph": {
6.00/2.46	        "nodes": {
6.00/2.46	            "121": {
6.00/2.46	                "goal": [{
6.00/2.46	                    "clause": -1,
6.00/2.46	                    "scope": -1,
6.00/2.46	                    "term": "(app X29 T19 T22)"
6.00/2.46	                }],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": ["T19"],
6.00/2.46	                    "free": ["X29"],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            },
6.00/2.46	            "2": {
6.00/2.46	                "goal": [{
6.00/2.46	                    "clause": -1,
6.00/2.46	                    "scope": -1,
6.00/2.46	                    "term": "(suffix T1 T2)"
6.00/2.46	                }],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": ["T1"],
6.00/2.46	                    "free": [],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            },
6.00/2.46	            "112": {
6.00/2.46	                "goal": [{
6.00/2.46	                    "clause": 1,
6.00/2.46	                    "scope": 2,
6.00/2.46	                    "term": "(app X6 T5 T7)"
6.00/2.46	                }],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": ["T5"],
6.00/2.46	                    "free": ["X6"],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            },
6.00/2.46	            "123": {
6.00/2.46	                "goal": [],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": [],
6.00/2.46	                    "free": [],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            },
6.00/2.46	            "103": {
6.00/2.46	                "goal": [{
6.00/2.46	                    "clause": -1,
6.00/2.46	                    "scope": -1,
6.00/2.46	                    "term": "(app X6 T5 T7)"
6.00/2.46	                }],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": ["T5"],
6.00/2.46	                    "free": ["X6"],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            },
6.00/2.46	            "114": {
6.00/2.46	                "goal": [{
6.00/2.46	                    "clause": 2,
6.00/2.46	                    "scope": 2,
6.00/2.46	                    "term": "(app X6 T5 T7)"
6.00/2.46	                }],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": ["T5"],
6.00/2.46	                    "free": ["X6"],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            },
6.00/2.46	            "16": {
6.00/2.46	                "goal": [{
6.00/2.46	                    "clause": 0,
6.00/2.46	                    "scope": 1,
6.00/2.46	                    "term": "(suffix T1 T2)"
6.00/2.46	                }],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": ["T1"],
6.00/2.46	                    "free": [],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            },
6.00/2.46	            "116": {
6.00/2.46	                "goal": [{
6.00/2.46	                    "clause": -1,
6.00/2.46	                    "scope": -1,
6.00/2.46	                    "term": "(true)"
6.00/2.46	                }],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": [],
6.00/2.46	                    "free": [],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            },
6.00/2.46	            "117": {
6.00/2.46	                "goal": [],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": [],
6.00/2.46	                    "free": [],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            },
6.00/2.46	            "107": {
6.00/2.46	                "goal": [
6.00/2.46	                    {
6.00/2.46	                        "clause": 1,
6.00/2.46	                        "scope": 2,
6.00/2.46	                        "term": "(app X6 T5 T7)"
6.00/2.46	                    },
6.00/2.46	                    {
6.00/2.46	                        "clause": 2,
6.00/2.46	                        "scope": 2,
6.00/2.46	                        "term": "(app X6 T5 T7)"
6.00/2.46	                    }
6.00/2.46	                ],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": ["T5"],
6.00/2.46	                    "free": ["X6"],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            },
6.00/2.46	            "118": {
6.00/2.46	                "goal": [],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": [],
6.00/2.46	                    "free": [],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            },
6.00/2.46	            "type": "Nodes"
6.00/2.46	        },
6.00/2.46	        "edges": [
6.00/2.46	            {
6.00/2.46	                "from": 2,
6.00/2.46	                "to": 16,
6.00/2.46	                "label": "CASE"
6.00/2.46	            },
6.00/2.46	            {
6.00/2.46	                "from": 16,
6.00/2.46	                "to": 103,
6.00/2.46	                "label": "ONLY EVAL with clause\nsuffix(X4, X5) :- app(X6, X4, X5).\nand substitutionT1 -> T5,\nX4 -> T5,\nT2 -> T7,\nX5 -> T7,\nT6 -> T7"
6.00/2.46	            },
6.00/2.46	            {
6.00/2.46	                "from": 103,
6.00/2.46	                "to": 107,
6.00/2.46	                "label": "CASE"
6.00/2.46	            },
6.00/2.46	            {
6.00/2.46	                "from": 107,
6.00/2.46	                "to": 112,
6.00/2.46	                "label": "PARALLEL"
6.00/2.46	            },
6.00/2.46	            {
6.00/2.46	                "from": 107,
6.00/2.46	                "to": 114,
6.00/2.46	                "label": "PARALLEL"
6.00/2.46	            },
6.00/2.46	            {
6.00/2.46	                "from": 112,
6.00/2.46	                "to": 116,
6.00/2.46	                "label": "EVAL with clause\napp([], X11, X11).\nand substitutionX6 -> [],\nT5 -> T12,\nX11 -> T12,\nT7 -> T12"
6.00/2.46	            },
6.00/2.46	            {
6.00/2.46	                "from": 112,
6.00/2.46	                "to": 117,
6.00/2.46	                "label": "EVAL-BACKTRACK"
6.00/2.46	            },
6.00/2.46	            {
6.00/2.46	                "from": 114,
6.00/2.46	                "to": 121,
6.00/2.46	                "label": "EVAL with clause\napp(.(X24, X25), X26, .(X24, X27)) :- app(X25, X26, X27).\nand substitutionX24 -> T20,\nX25 -> X29,\nX6 -> .(T20, X29),\nT5 -> T19,\nX26 -> T19,\nX28 -> T20,\nX27 -> T22,\nT7 -> .(T20, T22),\nT21 -> T22"
6.00/2.46	            },
6.00/2.46	            {
6.00/2.46	                "from": 114,
6.00/2.46	                "to": 123,
6.00/2.46	                "label": "EVAL-BACKTRACK"
6.00/2.46	            },
6.00/2.46	            {
6.00/2.46	                "from": 116,
6.00/2.46	                "to": 118,
6.00/2.46	                "label": "SUCCESS"
6.00/2.46	            },
6.00/2.46	            {
6.00/2.46	                "from": 121,
6.00/2.46	                "to": 103,
6.00/2.46	                "label": "INSTANCE with matching:\nX6 -> X29\nT5 -> T19\nT7 -> T22"
6.00/2.46	            }
6.00/2.46	        ],
6.00/2.46	        "type": "Graph"
6.00/2.46	    }
6.00/2.46	}
6.00/2.46	
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(34)
6.00/2.46	Obligation:
6.00/2.46	Triples:
6.00/2.46	
6.00/2.46	appA(.(X1, X2), X3, .(X1, X4)) :- appA(X2, X3, X4).
6.00/2.46	suffixB(X1, X2) :- appA(X3, X1, X2).
6.00/2.46	
6.00/2.46	Clauses:
6.00/2.46	
6.00/2.46	appcA([], X1, X1).
6.00/2.46	appcA(.(X1, X2), X3, .(X1, X4)) :- appcA(X2, X3, X4).
6.00/2.46	
6.00/2.46	Afs:
6.00/2.46	
6.00/2.46	suffixB(x1, x2)  =  suffixB(x1)
6.00/2.46	
6.00/2.46	
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(35) TriplesToPiDPProof (SOUND)
6.00/2.46	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.00/2.46	
6.00/2.46	suffixB_in_2: (b,f)
6.00/2.46	
6.00/2.46	appA_in_3: (f,b,f)
6.00/2.46	
6.00/2.46	Transforming TRIPLES into the following Term Rewriting System:
6.00/2.46	
6.00/2.46	Pi DP problem:
6.00/2.46	The TRS P consists of the following rules:
6.00/2.46	
6.00/2.46	   SUFFIXB_IN_GA(X1, X2) -> U2_GA(X1, X2, appA_in_aga(X3, X1, X2))
6.00/2.46	   SUFFIXB_IN_GA(X1, X2) -> APPA_IN_AGA(X3, X1, X2)
6.00/2.46	   APPA_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> U1_AGA(X1, X2, X3, X4, appA_in_aga(X2, X3, X4))
6.00/2.46	   APPA_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPA_IN_AGA(X2, X3, X4)
6.00/2.46	
6.00/2.46	R is empty.
6.00/2.46	The argument filtering Pi contains the following mapping:
6.00/2.46	appA_in_aga(x1, x2, x3)  =  appA_in_aga(x2)
6.00/2.46	
6.00/2.46	.(x1, x2)  =  .(x2)
6.00/2.46	
6.00/2.46	SUFFIXB_IN_GA(x1, x2)  =  SUFFIXB_IN_GA(x1)
6.00/2.46	
6.00/2.46	U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
6.00/2.46	
6.00/2.46	APPA_IN_AGA(x1, x2, x3)  =  APPA_IN_AGA(x2)
6.00/2.46	
6.00/2.46	U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x3, x5)
6.00/2.46	
6.00/2.46	
6.00/2.46	We have to consider all (P,R,Pi)-chains
6.00/2.46	
6.00/2.46	
6.00/2.46	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
6.00/2.46	
6.00/2.46	
6.00/2.46	
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(36)
6.00/2.46	Obligation:
6.00/2.46	Pi DP problem:
6.00/2.46	The TRS P consists of the following rules:
6.00/2.46	
6.00/2.46	   SUFFIXB_IN_GA(X1, X2) -> U2_GA(X1, X2, appA_in_aga(X3, X1, X2))
6.00/2.46	   SUFFIXB_IN_GA(X1, X2) -> APPA_IN_AGA(X3, X1, X2)
6.00/2.46	   APPA_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> U1_AGA(X1, X2, X3, X4, appA_in_aga(X2, X3, X4))
6.00/2.46	   APPA_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPA_IN_AGA(X2, X3, X4)
6.00/2.46	
6.00/2.46	R is empty.
6.00/2.46	The argument filtering Pi contains the following mapping:
6.00/2.46	appA_in_aga(x1, x2, x3)  =  appA_in_aga(x2)
6.00/2.46	
6.00/2.46	.(x1, x2)  =  .(x2)
6.00/2.46	
6.00/2.46	SUFFIXB_IN_GA(x1, x2)  =  SUFFIXB_IN_GA(x1)
6.00/2.46	
6.00/2.46	U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
6.00/2.46	
6.00/2.46	APPA_IN_AGA(x1, x2, x3)  =  APPA_IN_AGA(x2)
6.00/2.46	
6.00/2.46	U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x3, x5)
6.00/2.46	
6.00/2.46	
6.00/2.46	We have to consider all (P,R,Pi)-chains
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(37) DependencyGraphProof (EQUIVALENT)
6.00/2.46	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(38)
6.00/2.46	Obligation:
6.00/2.46	Pi DP problem:
6.00/2.46	The TRS P consists of the following rules:
6.00/2.46	
6.00/2.46	   APPA_IN_AGA(.(X1, X2), X3, .(X1, X4)) -> APPA_IN_AGA(X2, X3, X4)
6.00/2.46	
6.00/2.46	R is empty.
6.00/2.46	The argument filtering Pi contains the following mapping:
6.00/2.46	.(x1, x2)  =  .(x2)
6.00/2.46	
6.00/2.46	APPA_IN_AGA(x1, x2, x3)  =  APPA_IN_AGA(x2)
6.00/2.46	
6.00/2.46	
6.00/2.46	We have to consider all (P,R,Pi)-chains
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(39) PiDPToQDPProof (SOUND)
6.00/2.46	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(40)
6.00/2.46	Obligation:
6.00/2.46	Q DP problem:
6.00/2.46	The TRS P consists of the following rules:
6.00/2.46	
6.00/2.46	   APPA_IN_AGA(X3) -> APPA_IN_AGA(X3)
6.00/2.46	
6.00/2.46	R is empty.
6.00/2.46	Q is empty.
6.00/2.46	We have to consider all (P,Q,R)-chains.
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(41) PrologToIRSwTTransformerProof (SOUND)
6.00/2.46	Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert
6.00/2.46	
6.00/2.46	{
6.00/2.46	    "root": 15,
6.00/2.46	    "program": {
6.00/2.46	        "directives": [],
6.00/2.46	        "clauses": [
6.00/2.46	            [
6.00/2.46	                "(suffix Xs Ys)",
6.00/2.46	                "(app X1 Xs Ys)"
6.00/2.46	            ],
6.00/2.46	            [
6.00/2.46	                "(app ([]) X X)",
6.00/2.46	                null
6.00/2.46	            ],
6.00/2.46	            [
6.00/2.46	                "(app (. X Xs) Ys (. X Zs))",
6.00/2.46	                "(app Xs Ys Zs)"
6.00/2.46	            ]
6.00/2.46	        ]
6.00/2.46	    },
6.00/2.46	    "graph": {
6.00/2.46	        "nodes": {
6.00/2.46	            "132": {
6.00/2.46	                "goal": [{
6.00/2.46	                    "clause": 1,
6.00/2.46	                    "scope": 2,
6.00/2.46	                    "term": "(app X11 T10 T12)"
6.00/2.46	                }],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": ["T10"],
6.00/2.46	                    "free": ["X11"],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            },
6.00/2.46	            "133": {
6.00/2.46	                "goal": [{
6.00/2.46	                    "clause": 2,
6.00/2.46	                    "scope": 2,
6.00/2.46	                    "term": "(app X11 T10 T12)"
6.00/2.46	                }],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": ["T10"],
6.00/2.46	                    "free": ["X11"],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            },
6.00/2.46	            "134": {
6.00/2.46	                "goal": [{
6.00/2.46	                    "clause": -1,
6.00/2.46	                    "scope": -1,
6.00/2.46	                    "term": "(true)"
6.00/2.46	                }],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": [],
6.00/2.46	                    "free": [],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            },
6.00/2.46	            "135": {
6.00/2.46	                "goal": [],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": [],
6.00/2.46	                    "free": [],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            },
6.00/2.46	            "15": {
6.00/2.46	                "goal": [{
6.00/2.46	                    "clause": -1,
6.00/2.46	                    "scope": -1,
6.00/2.46	                    "term": "(suffix T1 T2)"
6.00/2.46	                }],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": ["T1"],
6.00/2.46	                    "free": [],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            },
6.00/2.46	            "136": {
6.00/2.46	                "goal": [],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": [],
6.00/2.46	                    "free": [],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            },
6.00/2.46	            "137": {
6.00/2.46	                "goal": [{
6.00/2.46	                    "clause": -1,
6.00/2.46	                    "scope": -1,
6.00/2.46	                    "term": "(app X36 T26 T29)"
6.00/2.46	                }],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": ["T26"],
6.00/2.46	                    "free": ["X36"],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            },
6.00/2.46	            "17": {
6.00/2.46	                "goal": [{
6.00/2.46	                    "clause": 0,
6.00/2.46	                    "scope": 1,
6.00/2.46	                    "term": "(suffix T1 T2)"
6.00/2.46	                }],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": ["T1"],
6.00/2.46	                    "free": [],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            },
6.00/2.46	            "138": {
6.00/2.46	                "goal": [],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": [],
6.00/2.46	                    "free": [],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            },
6.00/2.46	            "119": {
6.00/2.46	                "goal": [{
6.00/2.46	                    "clause": -1,
6.00/2.46	                    "scope": -1,
6.00/2.46	                    "term": "(app X11 T10 T12)"
6.00/2.46	                }],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": ["T10"],
6.00/2.46	                    "free": ["X11"],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            },
6.00/2.46	            "type": "Nodes",
6.00/2.46	            "131": {
6.00/2.46	                "goal": [
6.00/2.46	                    {
6.00/2.46	                        "clause": 1,
6.00/2.46	                        "scope": 2,
6.00/2.46	                        "term": "(app X11 T10 T12)"
6.00/2.46	                    },
6.00/2.46	                    {
6.00/2.46	                        "clause": 2,
6.00/2.46	                        "scope": 2,
6.00/2.46	                        "term": "(app X11 T10 T12)"
6.00/2.46	                    }
6.00/2.46	                ],
6.00/2.46	                "kb": {
6.00/2.46	                    "nonunifying": [],
6.00/2.46	                    "intvars": {},
6.00/2.46	                    "arithmetic": {
6.00/2.46	                        "type": "PlainIntegerRelationState",
6.00/2.46	                        "relations": []
6.00/2.46	                    },
6.00/2.46	                    "ground": ["T10"],
6.00/2.46	                    "free": ["X11"],
6.00/2.46	                    "exprvars": []
6.00/2.46	                }
6.00/2.46	            }
6.00/2.46	        },
6.00/2.46	        "edges": [
6.00/2.46	            {
6.00/2.46	                "from": 15,
6.00/2.46	                "to": 17,
6.00/2.46	                "label": "CASE"
6.00/2.46	            },
6.00/2.46	            {
6.00/2.46	                "from": 17,
6.00/2.46	                "to": 119,
6.00/2.46	                "label": "ONLY EVAL with clause\nsuffix(X9, X10) :- app(X11, X9, X10).\nand substitutionT1 -> T10,\nX9 -> T10,\nT2 -> T12,\nX10 -> T12,\nT11 -> T12"
6.00/2.46	            },
6.00/2.46	            {
6.00/2.46	                "from": 119,
6.00/2.46	                "to": 131,
6.00/2.46	                "label": "CASE"
6.00/2.46	            },
6.00/2.46	            {
6.00/2.46	                "from": 131,
6.00/2.46	                "to": 132,
6.00/2.46	                "label": "PARALLEL"
6.00/2.46	            },
6.00/2.46	            {
6.00/2.46	                "from": 131,
6.00/2.46	                "to": 133,
6.00/2.46	                "label": "PARALLEL"
6.00/2.46	            },
6.00/2.46	            {
6.00/2.46	                "from": 132,
6.00/2.46	                "to": 134,
6.00/2.46	                "label": "EVAL with clause\napp([], X18, X18).\nand substitutionX11 -> [],\nT10 -> T19,\nX18 -> T19,\nT12 -> T19"
6.00/2.46	            },
6.00/2.46	            {
6.00/2.46	                "from": 132,
6.00/2.46	                "to": 135,
6.00/2.46	                "label": "EVAL-BACKTRACK"
6.00/2.46	            },
6.00/2.46	            {
6.00/2.46	                "from": 133,
6.00/2.46	                "to": 137,
6.00/2.46	                "label": "EVAL with clause\napp(.(X31, X32), X33, .(X31, X34)) :- app(X32, X33, X34).\nand substitutionX31 -> T27,\nX32 -> X36,\nX11 -> .(T27, X36),\nT10 -> T26,\nX33 -> T26,\nX35 -> T27,\nX34 -> T29,\nT12 -> .(T27, T29),\nT28 -> T29"
6.00/2.46	            },
6.00/2.46	            {
6.00/2.46	                "from": 133,
6.00/2.46	                "to": 138,
6.00/2.46	                "label": "EVAL-BACKTRACK"
6.00/2.46	            },
6.00/2.46	            {
6.00/2.46	                "from": 134,
6.00/2.46	                "to": 136,
6.00/2.46	                "label": "SUCCESS"
6.00/2.46	            },
6.00/2.46	            {
6.00/2.46	                "from": 137,
6.00/2.46	                "to": 119,
6.00/2.46	                "label": "INSTANCE with matching:\nX11 -> X36\nT10 -> T26\nT12 -> T29"
6.00/2.46	            }
6.00/2.46	        ],
6.00/2.46	        "type": "Graph"
6.00/2.46	    }
6.00/2.46	}
6.00/2.46	
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(42)
6.00/2.46	Obligation:
6.00/2.46	Rules:
6.00/2.46	f131_in(T10) -> f133_in(T10) :|: TRUE
6.00/2.46	f133_out(x) -> f131_out(x) :|: TRUE
6.00/2.46	f131_in(x1) -> f132_in(x1) :|: TRUE
6.00/2.46	f132_out(x2) -> f131_out(x2) :|: TRUE
6.00/2.46	f119_in(x3) -> f131_in(x3) :|: TRUE
6.00/2.46	f131_out(x4) -> f119_out(x4) :|: TRUE
6.00/2.46	f138_out -> f133_out(x5) :|: TRUE
6.00/2.46	f133_in(T26) -> f137_in(T26) :|: TRUE
6.00/2.46	f137_out(x6) -> f133_out(x6) :|: TRUE
6.00/2.46	f133_in(x7) -> f138_in :|: TRUE
6.00/2.46	f119_out(x8) -> f137_out(x8) :|: TRUE
6.00/2.46	f137_in(x9) -> f119_in(x9) :|: TRUE
6.00/2.46	f17_out(T1) -> f15_out(T1) :|: TRUE
6.00/2.46	f15_in(x10) -> f17_in(x10) :|: TRUE
6.00/2.46	f17_in(x11) -> f119_in(x11) :|: TRUE
6.00/2.46	f119_out(x12) -> f17_out(x12) :|: TRUE
6.00/2.46	Start term: f15_in(T1)
6.00/2.46	
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(43) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
6.00/2.46	Constructed simple dependency graph.
6.00/2.46	
6.00/2.46	Simplified to the following IRSwTs:
6.00/2.46	
6.00/2.46	intTRSProblem:
6.00/2.46	f131_in(T10) -> f133_in(T10) :|: TRUE
6.00/2.46	f119_in(x3) -> f131_in(x3) :|: TRUE
6.00/2.46	f133_in(T26) -> f137_in(T26) :|: TRUE
6.00/2.46	f137_in(x9) -> f119_in(x9) :|: TRUE
6.00/2.46	
6.00/2.46	
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(44)
6.00/2.46	Obligation:
6.00/2.46	Rules:
6.00/2.46	f131_in(T10) -> f133_in(T10) :|: TRUE
6.00/2.46	f119_in(x3) -> f131_in(x3) :|: TRUE
6.00/2.46	f133_in(T26) -> f137_in(T26) :|: TRUE
6.00/2.46	f137_in(x9) -> f119_in(x9) :|: TRUE
6.00/2.46	
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(45) IntTRSCompressionProof (EQUIVALENT)
6.00/2.46	Compressed rules.
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(46)
6.00/2.46	Obligation:
6.00/2.46	Rules:
6.00/2.46	f119_in(x3:0) -> f119_in(x3:0) :|: TRUE
6.00/2.46	
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(47) IRSFormatTransformerProof (EQUIVALENT)
6.00/2.46	Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(48)
6.00/2.46	Obligation:
6.00/2.46	Rules:
6.00/2.46	f119_in(x3:0) -> f119_in(x3:0) :|: TRUE
6.00/2.46	
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(49) IRSwTTerminationDigraphProof (EQUIVALENT)
6.00/2.46	Constructed termination digraph!
6.00/2.46	Nodes:
6.00/2.46	(1) f119_in(x3:0) -> f119_in(x3:0) :|: TRUE
6.00/2.46	
6.00/2.46	Arcs:
6.00/2.46	(1) -> (1)
6.00/2.46	
6.00/2.46	This digraph is fully evaluated!
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(50)
6.00/2.46	Obligation:
6.00/2.46	
6.00/2.46	Termination digraph:
6.00/2.46	Nodes:
6.00/2.46	(1) f119_in(x3:0) -> f119_in(x3:0) :|: TRUE
6.00/2.46	
6.00/2.46	Arcs:
6.00/2.46	(1) -> (1)
6.00/2.46	
6.00/2.46	This digraph is fully evaluated!
6.00/2.46	
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(51) FilterProof (EQUIVALENT)
6.00/2.46	Used the following sort dictionary for filtering: 
6.00/2.46	f119_in(VARIABLE)
6.00/2.46	Replaced non-predefined constructor symbols by 0.
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(52)
6.00/2.46	Obligation:
6.00/2.46	Rules:
6.00/2.46	f119_in(x3:0) -> f119_in(x3:0) :|: TRUE
6.00/2.46	
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(53) IntTRSPeriodicNontermProof (COMPLETE)
6.00/2.46	Normalized system to the following form:
6.00/2.46	f(pc, x3:0) -> f(1, x3:0) :|: pc = 1 && TRUE
6.00/2.46	Witness term starting non-terminating reduction: f(1, -8)
6.00/2.46	----------------------------------------
6.00/2.46	
6.00/2.46	(54)
6.00/2.46	NO
6.32/2.47	EOF