3.57/1.69	YES
3.57/1.70	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.57/1.70	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.57/1.70	
3.57/1.70	
3.57/1.70	Left Termination of the query pattern
3.57/1.70	
3.57/1.70	select(a,g,a)
3.57/1.70	
3.57/1.70	w.r.t. the given Prolog program could successfully be proven:
3.57/1.70	
3.57/1.70	(0) Prolog
3.57/1.70	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.57/1.70	(2) PiTRS
3.57/1.70	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.57/1.70	(4) PiDP
3.57/1.70	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.57/1.70	(6) PiDP
3.57/1.70	(7) UsableRulesProof [EQUIVALENT, 0 ms]
3.57/1.70	(8) PiDP
3.57/1.70	(9) PiDPToQDPProof [SOUND, 0 ms]
3.57/1.70	(10) QDP
3.57/1.70	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.57/1.70	(12) YES
3.57/1.70	
3.57/1.70	
3.57/1.70	----------------------------------------
3.57/1.70	
3.57/1.70	(0)
3.57/1.70	Obligation:
3.57/1.70	Clauses:
3.57/1.70	
3.57/1.70	select(X, .(X, Xs), Xs).
3.57/1.70	select(X, .(Y, Xs), .(Y, Zs)) :- select(X, Xs, Zs).
3.57/1.70	
3.57/1.70	
3.57/1.70	Query: select(a,g,a)
3.57/1.70	----------------------------------------
3.57/1.70	
3.57/1.70	(1) PrologToPiTRSProof (SOUND)
3.57/1.70	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.57/1.70	
3.57/1.70	select_in_3: (f,b,f)
3.57/1.70	
3.57/1.70	Transforming Prolog into the following Term Rewriting System:
3.57/1.70	
3.57/1.70	Pi-finite rewrite system:
3.57/1.70	The TRS R consists of the following rules:
3.57/1.70	
3.57/1.70	   select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs)
3.57/1.70	   select_in_aga(X, .(Y, Xs), .(Y, Zs)) -> U1_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
3.57/1.70	   U1_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) -> select_out_aga(X, .(Y, Xs), .(Y, Zs))
3.57/1.70	
3.57/1.70	The argument filtering Pi contains the following mapping:
3.57/1.70	select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
3.57/1.70	
3.57/1.70	.(x1, x2)  =  .(x1, x2)
3.57/1.70	
3.57/1.70	select_out_aga(x1, x2, x3)  =  select_out_aga(x1, x3)
3.57/1.70	
3.57/1.70	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x2, x5)
3.57/1.70	
3.57/1.70	
3.57/1.70	
3.57/1.70	
3.57/1.70	
3.57/1.70	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.57/1.70	
3.57/1.70	
3.57/1.70	
3.57/1.70	----------------------------------------
3.57/1.70	
3.57/1.70	(2)
3.57/1.70	Obligation:
3.57/1.70	Pi-finite rewrite system:
3.57/1.70	The TRS R consists of the following rules:
3.57/1.70	
3.57/1.70	   select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs)
3.57/1.70	   select_in_aga(X, .(Y, Xs), .(Y, Zs)) -> U1_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
3.57/1.70	   U1_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) -> select_out_aga(X, .(Y, Xs), .(Y, Zs))
3.57/1.70	
3.57/1.70	The argument filtering Pi contains the following mapping:
3.57/1.70	select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
3.57/1.70	
3.57/1.70	.(x1, x2)  =  .(x1, x2)
3.57/1.70	
3.57/1.70	select_out_aga(x1, x2, x3)  =  select_out_aga(x1, x3)
3.57/1.70	
3.57/1.70	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x2, x5)
3.57/1.70	
3.57/1.70	
3.57/1.70	
3.57/1.70	----------------------------------------
3.57/1.70	
3.57/1.70	(3) DependencyPairsProof (EQUIVALENT)
3.57/1.70	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.57/1.70	Pi DP problem:
3.57/1.70	The TRS P consists of the following rules:
3.57/1.70	
3.57/1.70	   SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) -> U1_AGA(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
3.57/1.70	   SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_AGA(X, Xs, Zs)
3.57/1.70	
3.57/1.70	The TRS R consists of the following rules:
3.57/1.70	
3.57/1.70	   select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs)
3.57/1.70	   select_in_aga(X, .(Y, Xs), .(Y, Zs)) -> U1_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
3.57/1.70	   U1_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) -> select_out_aga(X, .(Y, Xs), .(Y, Zs))
3.57/1.70	
3.57/1.70	The argument filtering Pi contains the following mapping:
3.57/1.70	select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
3.57/1.70	
3.57/1.70	.(x1, x2)  =  .(x1, x2)
3.57/1.70	
3.57/1.70	select_out_aga(x1, x2, x3)  =  select_out_aga(x1, x3)
3.57/1.70	
3.57/1.70	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x2, x5)
3.57/1.70	
3.57/1.70	SELECT_IN_AGA(x1, x2, x3)  =  SELECT_IN_AGA(x2)
3.57/1.70	
3.57/1.70	U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x2, x5)
3.57/1.70	
3.57/1.70	
3.57/1.70	We have to consider all (P,R,Pi)-chains
3.57/1.70	----------------------------------------
3.57/1.70	
3.57/1.70	(4)
3.57/1.70	Obligation:
3.57/1.70	Pi DP problem:
3.57/1.70	The TRS P consists of the following rules:
3.57/1.70	
3.57/1.70	   SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) -> U1_AGA(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
3.57/1.70	   SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_AGA(X, Xs, Zs)
3.57/1.70	
3.57/1.70	The TRS R consists of the following rules:
3.57/1.70	
3.57/1.70	   select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs)
3.57/1.70	   select_in_aga(X, .(Y, Xs), .(Y, Zs)) -> U1_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
3.57/1.70	   U1_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) -> select_out_aga(X, .(Y, Xs), .(Y, Zs))
3.57/1.70	
3.57/1.70	The argument filtering Pi contains the following mapping:
3.57/1.70	select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
3.57/1.70	
3.57/1.70	.(x1, x2)  =  .(x1, x2)
3.57/1.70	
3.57/1.70	select_out_aga(x1, x2, x3)  =  select_out_aga(x1, x3)
3.57/1.70	
3.57/1.70	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x2, x5)
3.57/1.70	
3.57/1.70	SELECT_IN_AGA(x1, x2, x3)  =  SELECT_IN_AGA(x2)
3.57/1.70	
3.57/1.70	U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x2, x5)
3.57/1.70	
3.57/1.70	
3.57/1.70	We have to consider all (P,R,Pi)-chains
3.57/1.70	----------------------------------------
3.57/1.70	
3.57/1.70	(5) DependencyGraphProof (EQUIVALENT)
3.57/1.70	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
3.57/1.70	----------------------------------------
3.57/1.70	
3.57/1.70	(6)
3.57/1.70	Obligation:
3.57/1.70	Pi DP problem:
3.57/1.70	The TRS P consists of the following rules:
3.57/1.70	
3.57/1.70	   SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_AGA(X, Xs, Zs)
3.57/1.70	
3.57/1.70	The TRS R consists of the following rules:
3.57/1.70	
3.57/1.70	   select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs)
3.57/1.70	   select_in_aga(X, .(Y, Xs), .(Y, Zs)) -> U1_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
3.57/1.70	   U1_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) -> select_out_aga(X, .(Y, Xs), .(Y, Zs))
3.57/1.70	
3.57/1.70	The argument filtering Pi contains the following mapping:
3.57/1.70	select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
3.57/1.70	
3.57/1.70	.(x1, x2)  =  .(x1, x2)
3.57/1.70	
3.57/1.70	select_out_aga(x1, x2, x3)  =  select_out_aga(x1, x3)
3.57/1.70	
3.57/1.70	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x2, x5)
3.57/1.70	
3.57/1.70	SELECT_IN_AGA(x1, x2, x3)  =  SELECT_IN_AGA(x2)
3.57/1.70	
3.57/1.70	
3.57/1.70	We have to consider all (P,R,Pi)-chains
3.57/1.70	----------------------------------------
3.57/1.70	
3.57/1.70	(7) UsableRulesProof (EQUIVALENT)
3.57/1.70	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.57/1.70	----------------------------------------
3.57/1.70	
3.57/1.70	(8)
3.57/1.70	Obligation:
3.57/1.70	Pi DP problem:
3.57/1.70	The TRS P consists of the following rules:
3.57/1.70	
3.57/1.70	   SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_AGA(X, Xs, Zs)
3.57/1.70	
3.57/1.70	R is empty.
3.57/1.70	The argument filtering Pi contains the following mapping:
3.57/1.70	.(x1, x2)  =  .(x1, x2)
3.57/1.70	
3.57/1.70	SELECT_IN_AGA(x1, x2, x3)  =  SELECT_IN_AGA(x2)
3.57/1.70	
3.57/1.70	
3.57/1.70	We have to consider all (P,R,Pi)-chains
3.57/1.70	----------------------------------------
3.57/1.70	
3.57/1.70	(9) PiDPToQDPProof (SOUND)
3.57/1.70	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.57/1.70	----------------------------------------
3.57/1.70	
3.57/1.70	(10)
3.57/1.70	Obligation:
3.57/1.70	Q DP problem:
3.57/1.70	The TRS P consists of the following rules:
3.57/1.70	
3.57/1.70	   SELECT_IN_AGA(.(Y, Xs)) -> SELECT_IN_AGA(Xs)
3.57/1.70	
3.57/1.70	R is empty.
3.57/1.70	Q is empty.
3.57/1.70	We have to consider all (P,Q,R)-chains.
3.57/1.70	----------------------------------------
3.57/1.70	
3.57/1.70	(11) QDPSizeChangeProof (EQUIVALENT)
3.57/1.70	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. 
3.57/1.70	
3.57/1.70	From the DPs we obtained the following set of size-change graphs:
3.57/1.70	*SELECT_IN_AGA(.(Y, Xs)) -> SELECT_IN_AGA(Xs)
3.57/1.70	The graph contains the following edges 1 > 1
3.57/1.70	
3.57/1.70	
3.57/1.70	----------------------------------------
3.57/1.70	
3.57/1.70	(12)
3.57/1.70	YES
3.82/1.74	EOF