3.69/1.72	YES
3.69/1.74	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.69/1.74	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.69/1.74	
3.69/1.74	
3.69/1.74	Left Termination of the query pattern
3.69/1.74	
3.69/1.74	app(a,a,g)
3.69/1.74	
3.69/1.74	w.r.t. the given Prolog program could successfully be proven:
3.69/1.74	
3.69/1.74	(0) Prolog
3.69/1.74	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.69/1.74	(2) PiTRS
3.69/1.74	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.69/1.74	(4) PiDP
3.69/1.74	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.69/1.74	(6) PiDP
3.69/1.74	(7) UsableRulesProof [EQUIVALENT, 1 ms]
3.69/1.74	(8) PiDP
3.69/1.74	(9) PiDPToQDPProof [SOUND, 0 ms]
3.69/1.74	(10) QDP
3.69/1.74	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.69/1.74	(12) YES
3.69/1.74	
3.69/1.74	
3.69/1.74	----------------------------------------
3.69/1.74	
3.69/1.74	(0)
3.69/1.74	Obligation:
3.69/1.74	Clauses:
3.69/1.74	
3.69/1.74	app([], X, X).
3.69/1.74	app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
3.69/1.74	
3.69/1.74	
3.69/1.74	Query: app(a,a,g)
3.69/1.74	----------------------------------------
3.69/1.74	
3.69/1.74	(1) PrologToPiTRSProof (SOUND)
3.69/1.74	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.69/1.74	
3.69/1.74	app_in_3: (f,f,b)
3.69/1.74	
3.69/1.74	Transforming Prolog into the following Term Rewriting System:
3.69/1.74	
3.69/1.74	Pi-finite rewrite system:
3.69/1.74	The TRS R consists of the following rules:
3.69/1.74	
3.69/1.74	   app_in_aag([], X, X) -> app_out_aag([], X, X)
3.69/1.74	   app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
3.69/1.74	   U1_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs))
3.69/1.74	
3.69/1.74	The argument filtering Pi contains the following mapping:
3.69/1.74	app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
3.69/1.74	
3.69/1.74	app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
3.69/1.74	
3.69/1.74	.(x1, x2)  =  .(x1, x2)
3.69/1.74	
3.69/1.74	U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
3.69/1.74	
3.69/1.74	
3.69/1.74	
3.69/1.74	
3.69/1.74	
3.69/1.74	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.69/1.74	
3.69/1.74	
3.69/1.74	
3.69/1.74	----------------------------------------
3.69/1.74	
3.69/1.74	(2)
3.69/1.74	Obligation:
3.69/1.74	Pi-finite rewrite system:
3.69/1.74	The TRS R consists of the following rules:
3.69/1.74	
3.69/1.74	   app_in_aag([], X, X) -> app_out_aag([], X, X)
3.69/1.74	   app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
3.69/1.74	   U1_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs))
3.69/1.74	
3.69/1.74	The argument filtering Pi contains the following mapping:
3.69/1.74	app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
3.69/1.74	
3.69/1.74	app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
3.69/1.74	
3.69/1.74	.(x1, x2)  =  .(x1, x2)
3.69/1.74	
3.69/1.74	U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
3.69/1.74	
3.69/1.74	
3.69/1.74	
3.69/1.74	----------------------------------------
3.69/1.74	
3.69/1.74	(3) DependencyPairsProof (EQUIVALENT)
3.69/1.74	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.69/1.74	Pi DP problem:
3.69/1.74	The TRS P consists of the following rules:
3.69/1.74	
3.69/1.74	   APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U1_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
3.69/1.74	   APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs)
3.69/1.74	
3.69/1.74	The TRS R consists of the following rules:
3.69/1.74	
3.69/1.74	   app_in_aag([], X, X) -> app_out_aag([], X, X)
3.69/1.74	   app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
3.69/1.74	   U1_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs))
3.69/1.74	
3.69/1.74	The argument filtering Pi contains the following mapping:
3.69/1.74	app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
3.69/1.74	
3.69/1.74	app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
3.69/1.74	
3.69/1.74	.(x1, x2)  =  .(x1, x2)
3.69/1.74	
3.69/1.74	U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
3.69/1.74	
3.69/1.74	APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
3.69/1.74	
3.69/1.74	U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x1, x5)
3.69/1.74	
3.69/1.74	
3.69/1.74	We have to consider all (P,R,Pi)-chains
3.69/1.74	----------------------------------------
3.69/1.74	
3.69/1.74	(4)
3.69/1.74	Obligation:
3.69/1.74	Pi DP problem:
3.69/1.74	The TRS P consists of the following rules:
3.69/1.74	
3.69/1.74	   APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U1_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
3.69/1.74	   APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs)
3.69/1.74	
3.69/1.74	The TRS R consists of the following rules:
3.69/1.74	
3.69/1.74	   app_in_aag([], X, X) -> app_out_aag([], X, X)
3.69/1.74	   app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
3.69/1.74	   U1_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs))
3.69/1.74	
3.69/1.74	The argument filtering Pi contains the following mapping:
3.69/1.74	app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
3.69/1.74	
3.69/1.74	app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
3.69/1.74	
3.69/1.74	.(x1, x2)  =  .(x1, x2)
3.69/1.74	
3.69/1.74	U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
3.69/1.74	
3.69/1.74	APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
3.69/1.74	
3.69/1.74	U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x1, x5)
3.69/1.74	
3.69/1.74	
3.69/1.74	We have to consider all (P,R,Pi)-chains
3.69/1.74	----------------------------------------
3.69/1.74	
3.69/1.74	(5) DependencyGraphProof (EQUIVALENT)
3.69/1.74	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
3.69/1.74	----------------------------------------
3.69/1.74	
3.69/1.74	(6)
3.69/1.74	Obligation:
3.69/1.74	Pi DP problem:
3.69/1.74	The TRS P consists of the following rules:
3.69/1.74	
3.69/1.74	   APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs)
3.69/1.74	
3.69/1.74	The TRS R consists of the following rules:
3.69/1.74	
3.69/1.74	   app_in_aag([], X, X) -> app_out_aag([], X, X)
3.69/1.74	   app_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
3.69/1.74	   U1_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) -> app_out_aag(.(X, Xs), Ys, .(X, Zs))
3.69/1.74	
3.69/1.74	The argument filtering Pi contains the following mapping:
3.69/1.74	app_in_aag(x1, x2, x3)  =  app_in_aag(x3)
3.69/1.74	
3.69/1.74	app_out_aag(x1, x2, x3)  =  app_out_aag(x1, x2)
3.69/1.74	
3.69/1.74	.(x1, x2)  =  .(x1, x2)
3.69/1.74	
3.69/1.74	U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
3.69/1.74	
3.69/1.74	APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
3.69/1.74	
3.69/1.74	
3.69/1.74	We have to consider all (P,R,Pi)-chains
3.69/1.74	----------------------------------------
3.69/1.74	
3.69/1.74	(7) UsableRulesProof (EQUIVALENT)
3.69/1.74	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.69/1.74	----------------------------------------
3.69/1.74	
3.69/1.74	(8)
3.69/1.74	Obligation:
3.69/1.74	Pi DP problem:
3.69/1.74	The TRS P consists of the following rules:
3.69/1.74	
3.69/1.74	   APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_AAG(Xs, Ys, Zs)
3.69/1.74	
3.69/1.74	R is empty.
3.69/1.74	The argument filtering Pi contains the following mapping:
3.69/1.74	.(x1, x2)  =  .(x1, x2)
3.69/1.74	
3.69/1.74	APP_IN_AAG(x1, x2, x3)  =  APP_IN_AAG(x3)
3.69/1.74	
3.69/1.74	
3.69/1.74	We have to consider all (P,R,Pi)-chains
3.69/1.74	----------------------------------------
3.69/1.74	
3.69/1.74	(9) PiDPToQDPProof (SOUND)
3.69/1.74	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.69/1.74	----------------------------------------
3.69/1.74	
3.69/1.74	(10)
3.69/1.74	Obligation:
3.69/1.74	Q DP problem:
3.69/1.74	The TRS P consists of the following rules:
3.69/1.74	
3.69/1.74	   APP_IN_AAG(.(X, Zs)) -> APP_IN_AAG(Zs)
3.69/1.74	
3.69/1.74	R is empty.
3.69/1.74	Q is empty.
3.69/1.74	We have to consider all (P,Q,R)-chains.
3.69/1.74	----------------------------------------
3.69/1.74	
3.69/1.74	(11) QDPSizeChangeProof (EQUIVALENT)
3.69/1.74	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
3.69/1.74	
3.69/1.74	From the DPs we obtained the following set of size-change graphs:
3.69/1.74	*APP_IN_AAG(.(X, Zs)) -> APP_IN_AAG(Zs)
3.69/1.74	The graph contains the following edges 1 > 1
3.69/1.74	
3.69/1.74	
3.69/1.74	----------------------------------------
3.69/1.74	
3.69/1.74	(12)
3.69/1.74	YES
3.96/1.77	EOF