3.71/1.70	YES
3.71/1.71	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.71/1.71	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.71/1.71	
3.71/1.71	
3.71/1.71	Left Termination of the query pattern
3.71/1.71	
3.71/1.71	select(a,g,a)
3.71/1.71	
3.71/1.71	w.r.t. the given Prolog program could successfully be proven:
3.71/1.71	
3.71/1.71	(0) Prolog
3.71/1.71	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.71/1.71	(2) PiTRS
3.71/1.71	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.71/1.71	(4) PiDP
3.71/1.71	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.71/1.71	(6) PiDP
3.71/1.71	(7) UsableRulesProof [EQUIVALENT, 0 ms]
3.71/1.71	(8) PiDP
3.71/1.71	(9) PiDPToQDPProof [SOUND, 9 ms]
3.71/1.71	(10) QDP
3.71/1.71	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.71/1.71	(12) YES
3.71/1.71	
3.71/1.71	
3.71/1.71	----------------------------------------
3.71/1.71	
3.71/1.71	(0)
3.71/1.71	Obligation:
3.71/1.71	Clauses:
3.71/1.71	
3.71/1.71	select(X, .(X, Xs), Xs).
3.71/1.71	select(X, .(Y, Xs), .(Y, Zs)) :- select(X, Xs, Zs).
3.71/1.71	
3.71/1.71	
3.71/1.71	Query: select(a,g,a)
3.71/1.71	----------------------------------------
3.71/1.71	
3.71/1.71	(1) PrologToPiTRSProof (SOUND)
3.71/1.71	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.71/1.71	
3.71/1.71	select_in_3: (f,b,f)
3.71/1.71	
3.71/1.71	Transforming Prolog into the following Term Rewriting System:
3.71/1.71	
3.71/1.71	Pi-finite rewrite system:
3.71/1.71	The TRS R consists of the following rules:
3.71/1.71	
3.71/1.71	   select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs)
3.71/1.71	   select_in_aga(X, .(Y, Xs), .(Y, Zs)) -> U1_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
3.71/1.71	   U1_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) -> select_out_aga(X, .(Y, Xs), .(Y, Zs))
3.71/1.71	
3.71/1.71	The argument filtering Pi contains the following mapping:
3.71/1.71	select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
3.71/1.71	
3.71/1.71	.(x1, x2)  =  .(x1, x2)
3.71/1.71	
3.71/1.71	select_out_aga(x1, x2, x3)  =  select_out_aga(x1, x3)
3.71/1.71	
3.71/1.71	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x2, x5)
3.71/1.71	
3.71/1.71	
3.71/1.71	
3.71/1.71	
3.71/1.71	
3.71/1.71	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.71/1.71	
3.71/1.71	
3.71/1.71	
3.71/1.71	----------------------------------------
3.71/1.71	
3.71/1.71	(2)
3.71/1.71	Obligation:
3.71/1.71	Pi-finite rewrite system:
3.71/1.71	The TRS R consists of the following rules:
3.71/1.71	
3.71/1.71	   select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs)
3.71/1.71	   select_in_aga(X, .(Y, Xs), .(Y, Zs)) -> U1_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
3.71/1.71	   U1_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) -> select_out_aga(X, .(Y, Xs), .(Y, Zs))
3.71/1.71	
3.71/1.71	The argument filtering Pi contains the following mapping:
3.71/1.71	select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
3.71/1.71	
3.71/1.71	.(x1, x2)  =  .(x1, x2)
3.71/1.71	
3.71/1.71	select_out_aga(x1, x2, x3)  =  select_out_aga(x1, x3)
3.71/1.71	
3.71/1.71	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x2, x5)
3.71/1.71	
3.71/1.71	
3.71/1.71	
3.71/1.71	----------------------------------------
3.71/1.71	
3.71/1.71	(3) DependencyPairsProof (EQUIVALENT)
3.71/1.71	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.71/1.71	Pi DP problem:
3.71/1.71	The TRS P consists of the following rules:
3.71/1.71	
3.71/1.71	   SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) -> U1_AGA(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
3.71/1.71	   SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_AGA(X, Xs, Zs)
3.71/1.71	
3.71/1.71	The TRS R consists of the following rules:
3.71/1.71	
3.71/1.71	   select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs)
3.71/1.71	   select_in_aga(X, .(Y, Xs), .(Y, Zs)) -> U1_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
3.71/1.71	   U1_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) -> select_out_aga(X, .(Y, Xs), .(Y, Zs))
3.71/1.71	
3.71/1.71	The argument filtering Pi contains the following mapping:
3.71/1.71	select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
3.71/1.71	
3.71/1.71	.(x1, x2)  =  .(x1, x2)
3.71/1.71	
3.71/1.71	select_out_aga(x1, x2, x3)  =  select_out_aga(x1, x3)
3.71/1.71	
3.71/1.71	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x2, x5)
3.71/1.71	
3.71/1.71	SELECT_IN_AGA(x1, x2, x3)  =  SELECT_IN_AGA(x2)
3.71/1.71	
3.71/1.71	U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x2, x5)
3.71/1.71	
3.71/1.71	
3.71/1.71	We have to consider all (P,R,Pi)-chains
3.71/1.71	----------------------------------------
3.71/1.71	
3.71/1.71	(4)
3.71/1.71	Obligation:
3.71/1.71	Pi DP problem:
3.71/1.71	The TRS P consists of the following rules:
3.71/1.71	
3.71/1.71	   SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) -> U1_AGA(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
3.71/1.71	   SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_AGA(X, Xs, Zs)
3.71/1.71	
3.71/1.71	The TRS R consists of the following rules:
3.71/1.71	
3.71/1.71	   select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs)
3.71/1.71	   select_in_aga(X, .(Y, Xs), .(Y, Zs)) -> U1_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
3.71/1.71	   U1_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) -> select_out_aga(X, .(Y, Xs), .(Y, Zs))
3.71/1.71	
3.71/1.71	The argument filtering Pi contains the following mapping:
3.71/1.71	select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
3.71/1.71	
3.71/1.71	.(x1, x2)  =  .(x1, x2)
3.71/1.71	
3.71/1.71	select_out_aga(x1, x2, x3)  =  select_out_aga(x1, x3)
3.71/1.71	
3.71/1.71	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x2, x5)
3.71/1.71	
3.71/1.71	SELECT_IN_AGA(x1, x2, x3)  =  SELECT_IN_AGA(x2)
3.71/1.71	
3.71/1.71	U1_AGA(x1, x2, x3, x4, x5)  =  U1_AGA(x2, x5)
3.71/1.71	
3.71/1.71	
3.71/1.71	We have to consider all (P,R,Pi)-chains
3.71/1.71	----------------------------------------
3.71/1.71	
3.71/1.71	(5) DependencyGraphProof (EQUIVALENT)
3.71/1.71	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
3.71/1.71	----------------------------------------
3.71/1.71	
3.71/1.71	(6)
3.71/1.71	Obligation:
3.71/1.71	Pi DP problem:
3.71/1.71	The TRS P consists of the following rules:
3.71/1.71	
3.71/1.71	   SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_AGA(X, Xs, Zs)
3.71/1.71	
3.71/1.71	The TRS R consists of the following rules:
3.71/1.71	
3.71/1.71	   select_in_aga(X, .(X, Xs), Xs) -> select_out_aga(X, .(X, Xs), Xs)
3.71/1.71	   select_in_aga(X, .(Y, Xs), .(Y, Zs)) -> U1_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
3.71/1.71	   U1_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) -> select_out_aga(X, .(Y, Xs), .(Y, Zs))
3.71/1.71	
3.71/1.71	The argument filtering Pi contains the following mapping:
3.71/1.71	select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
3.71/1.71	
3.71/1.71	.(x1, x2)  =  .(x1, x2)
3.71/1.71	
3.71/1.71	select_out_aga(x1, x2, x3)  =  select_out_aga(x1, x3)
3.71/1.71	
3.71/1.71	U1_aga(x1, x2, x3, x4, x5)  =  U1_aga(x2, x5)
3.71/1.71	
3.71/1.71	SELECT_IN_AGA(x1, x2, x3)  =  SELECT_IN_AGA(x2)
3.71/1.71	
3.71/1.71	
3.71/1.71	We have to consider all (P,R,Pi)-chains
3.71/1.71	----------------------------------------
3.71/1.71	
3.71/1.71	(7) UsableRulesProof (EQUIVALENT)
3.71/1.71	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.71/1.71	----------------------------------------
3.71/1.71	
3.71/1.71	(8)
3.71/1.71	Obligation:
3.71/1.71	Pi DP problem:
3.71/1.71	The TRS P consists of the following rules:
3.71/1.71	
3.71/1.71	   SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_AGA(X, Xs, Zs)
3.71/1.71	
3.71/1.71	R is empty.
3.71/1.71	The argument filtering Pi contains the following mapping:
3.71/1.71	.(x1, x2)  =  .(x1, x2)
3.71/1.71	
3.71/1.71	SELECT_IN_AGA(x1, x2, x3)  =  SELECT_IN_AGA(x2)
3.71/1.71	
3.71/1.71	
3.71/1.71	We have to consider all (P,R,Pi)-chains
3.71/1.71	----------------------------------------
3.71/1.71	
3.71/1.71	(9) PiDPToQDPProof (SOUND)
3.71/1.71	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.71/1.71	----------------------------------------
3.71/1.71	
3.71/1.71	(10)
3.71/1.71	Obligation:
3.71/1.71	Q DP problem:
3.71/1.71	The TRS P consists of the following rules:
3.71/1.71	
3.71/1.71	   SELECT_IN_AGA(.(Y, Xs)) -> SELECT_IN_AGA(Xs)
3.71/1.71	
3.71/1.71	R is empty.
3.71/1.71	Q is empty.
3.71/1.71	We have to consider all (P,Q,R)-chains.
3.71/1.71	----------------------------------------
3.71/1.71	
3.71/1.71	(11) QDPSizeChangeProof (EQUIVALENT)
3.71/1.71	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. 
3.71/1.71	
3.71/1.71	From the DPs we obtained the following set of size-change graphs:
3.71/1.71	*SELECT_IN_AGA(.(Y, Xs)) -> SELECT_IN_AGA(Xs)
3.71/1.71	The graph contains the following edges 1 > 1
3.71/1.71	
3.71/1.71	
3.71/1.71	----------------------------------------
3.71/1.71	
3.71/1.71	(12)
3.71/1.71	YES
3.71/1.73	EOF