3.68/2.40	YES
3.68/2.40	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.68/2.40	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.68/2.40	
3.68/2.40	
3.68/2.40	Left Termination of the query pattern
3.68/2.40	
3.68/2.40	select(a,a,g)
3.68/2.40	
3.68/2.40	w.r.t. the given Prolog program could successfully be proven:
3.68/2.40	
3.68/2.40	(0) Prolog
3.68/2.40	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.68/2.40	(2) PiTRS
3.68/2.40	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.68/2.40	(4) PiDP
3.68/2.40	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.68/2.40	(6) PiDP
3.68/2.40	(7) UsableRulesProof [EQUIVALENT, 0 ms]
3.68/2.40	(8) PiDP
3.68/2.40	(9) PiDPToQDPProof [SOUND, 0 ms]
3.68/2.40	(10) QDP
3.68/2.40	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.68/2.40	(12) YES
3.68/2.40	
3.68/2.40	
3.68/2.40	----------------------------------------
3.68/2.40	
3.68/2.40	(0)
3.68/2.40	Obligation:
3.68/2.40	Clauses:
3.68/2.40	
3.68/2.40	select(X, .(X, Xs), Xs).
3.68/2.40	select(X, .(Y, Xs), .(Y, Zs)) :- select(X, Xs, Zs).
3.68/2.40	
3.68/2.40	
3.68/2.40	Query: select(a,a,g)
3.68/2.40	----------------------------------------
3.68/2.40	
3.68/2.40	(1) PrologToPiTRSProof (SOUND)
3.68/2.40	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.68/2.40	
3.68/2.40	select_in_3: (f,f,b)
3.68/2.40	
3.68/2.40	Transforming Prolog into the following Term Rewriting System:
3.68/2.40	
3.68/2.40	Pi-finite rewrite system:
3.68/2.40	The TRS R consists of the following rules:
3.68/2.40	
3.68/2.40	   select_in_aag(X, .(X, Xs), Xs) -> select_out_aag(X, .(X, Xs), Xs)
3.68/2.40	   select_in_aag(X, .(Y, Xs), .(Y, Zs)) -> U1_aag(X, Y, Xs, Zs, select_in_aag(X, Xs, Zs))
3.68/2.40	   U1_aag(X, Y, Xs, Zs, select_out_aag(X, Xs, Zs)) -> select_out_aag(X, .(Y, Xs), .(Y, Zs))
3.68/2.40	
3.68/2.40	The argument filtering Pi contains the following mapping:
3.68/2.40	select_in_aag(x1, x2, x3)  =  select_in_aag(x3)
3.68/2.40	
3.68/2.40	select_out_aag(x1, x2, x3)  =  select_out_aag(x2)
3.68/2.40	
3.68/2.40	.(x1, x2)  =  .(x2)
3.68/2.40	
3.68/2.40	U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x5)
3.68/2.40	
3.68/2.40	
3.68/2.40	
3.68/2.40	
3.68/2.40	
3.68/2.40	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.68/2.40	
3.68/2.40	
3.68/2.40	
3.68/2.40	----------------------------------------
3.68/2.40	
3.68/2.40	(2)
3.68/2.40	Obligation:
3.68/2.40	Pi-finite rewrite system:
3.68/2.40	The TRS R consists of the following rules:
3.68/2.40	
3.68/2.40	   select_in_aag(X, .(X, Xs), Xs) -> select_out_aag(X, .(X, Xs), Xs)
3.68/2.40	   select_in_aag(X, .(Y, Xs), .(Y, Zs)) -> U1_aag(X, Y, Xs, Zs, select_in_aag(X, Xs, Zs))
3.68/2.40	   U1_aag(X, Y, Xs, Zs, select_out_aag(X, Xs, Zs)) -> select_out_aag(X, .(Y, Xs), .(Y, Zs))
3.68/2.40	
3.68/2.40	The argument filtering Pi contains the following mapping:
3.68/2.40	select_in_aag(x1, x2, x3)  =  select_in_aag(x3)
3.68/2.40	
3.68/2.40	select_out_aag(x1, x2, x3)  =  select_out_aag(x2)
3.68/2.40	
3.68/2.40	.(x1, x2)  =  .(x2)
3.68/2.40	
3.68/2.40	U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x5)
3.68/2.40	
3.68/2.40	
3.68/2.40	
3.68/2.40	----------------------------------------
3.68/2.40	
3.68/2.40	(3) DependencyPairsProof (EQUIVALENT)
3.68/2.40	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.68/2.40	Pi DP problem:
3.68/2.40	The TRS P consists of the following rules:
3.68/2.40	
3.68/2.40	   SELECT_IN_AAG(X, .(Y, Xs), .(Y, Zs)) -> U1_AAG(X, Y, Xs, Zs, select_in_aag(X, Xs, Zs))
3.68/2.40	   SELECT_IN_AAG(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_AAG(X, Xs, Zs)
3.68/2.40	
3.68/2.40	The TRS R consists of the following rules:
3.68/2.40	
3.68/2.40	   select_in_aag(X, .(X, Xs), Xs) -> select_out_aag(X, .(X, Xs), Xs)
3.68/2.40	   select_in_aag(X, .(Y, Xs), .(Y, Zs)) -> U1_aag(X, Y, Xs, Zs, select_in_aag(X, Xs, Zs))
3.68/2.40	   U1_aag(X, Y, Xs, Zs, select_out_aag(X, Xs, Zs)) -> select_out_aag(X, .(Y, Xs), .(Y, Zs))
3.68/2.40	
3.68/2.40	The argument filtering Pi contains the following mapping:
3.68/2.40	select_in_aag(x1, x2, x3)  =  select_in_aag(x3)
3.68/2.40	
3.68/2.40	select_out_aag(x1, x2, x3)  =  select_out_aag(x2)
3.68/2.40	
3.68/2.40	.(x1, x2)  =  .(x2)
3.68/2.40	
3.68/2.40	U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x5)
3.68/2.40	
3.68/2.40	SELECT_IN_AAG(x1, x2, x3)  =  SELECT_IN_AAG(x3)
3.68/2.40	
3.68/2.40	U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x5)
3.68/2.40	
3.68/2.40	
3.68/2.40	We have to consider all (P,R,Pi)-chains
3.68/2.40	----------------------------------------
3.68/2.40	
3.68/2.40	(4)
3.68/2.40	Obligation:
3.68/2.40	Pi DP problem:
3.68/2.40	The TRS P consists of the following rules:
3.68/2.40	
3.68/2.40	   SELECT_IN_AAG(X, .(Y, Xs), .(Y, Zs)) -> U1_AAG(X, Y, Xs, Zs, select_in_aag(X, Xs, Zs))
3.68/2.40	   SELECT_IN_AAG(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_AAG(X, Xs, Zs)
3.68/2.40	
3.68/2.40	The TRS R consists of the following rules:
3.68/2.40	
3.68/2.40	   select_in_aag(X, .(X, Xs), Xs) -> select_out_aag(X, .(X, Xs), Xs)
3.68/2.40	   select_in_aag(X, .(Y, Xs), .(Y, Zs)) -> U1_aag(X, Y, Xs, Zs, select_in_aag(X, Xs, Zs))
3.68/2.40	   U1_aag(X, Y, Xs, Zs, select_out_aag(X, Xs, Zs)) -> select_out_aag(X, .(Y, Xs), .(Y, Zs))
3.68/2.40	
3.68/2.40	The argument filtering Pi contains the following mapping:
3.68/2.40	select_in_aag(x1, x2, x3)  =  select_in_aag(x3)
3.68/2.40	
3.68/2.40	select_out_aag(x1, x2, x3)  =  select_out_aag(x2)
3.68/2.40	
3.68/2.40	.(x1, x2)  =  .(x2)
3.68/2.40	
3.68/2.40	U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x5)
3.68/2.40	
3.68/2.40	SELECT_IN_AAG(x1, x2, x3)  =  SELECT_IN_AAG(x3)
3.68/2.40	
3.68/2.40	U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x5)
3.68/2.40	
3.68/2.40	
3.68/2.40	We have to consider all (P,R,Pi)-chains
3.68/2.40	----------------------------------------
3.68/2.40	
3.68/2.40	(5) DependencyGraphProof (EQUIVALENT)
3.68/2.40	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
3.68/2.40	----------------------------------------
3.68/2.40	
3.68/2.40	(6)
3.68/2.40	Obligation:
3.68/2.40	Pi DP problem:
3.68/2.40	The TRS P consists of the following rules:
3.68/2.40	
3.68/2.40	   SELECT_IN_AAG(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_AAG(X, Xs, Zs)
3.68/2.40	
3.68/2.40	The TRS R consists of the following rules:
3.68/2.40	
3.68/2.40	   select_in_aag(X, .(X, Xs), Xs) -> select_out_aag(X, .(X, Xs), Xs)
3.68/2.40	   select_in_aag(X, .(Y, Xs), .(Y, Zs)) -> U1_aag(X, Y, Xs, Zs, select_in_aag(X, Xs, Zs))
3.68/2.40	   U1_aag(X, Y, Xs, Zs, select_out_aag(X, Xs, Zs)) -> select_out_aag(X, .(Y, Xs), .(Y, Zs))
3.68/2.40	
3.68/2.40	The argument filtering Pi contains the following mapping:
3.68/2.40	select_in_aag(x1, x2, x3)  =  select_in_aag(x3)
3.68/2.40	
3.68/2.40	select_out_aag(x1, x2, x3)  =  select_out_aag(x2)
3.68/2.40	
3.68/2.40	.(x1, x2)  =  .(x2)
3.68/2.40	
3.68/2.40	U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x5)
3.68/2.40	
3.68/2.40	SELECT_IN_AAG(x1, x2, x3)  =  SELECT_IN_AAG(x3)
3.68/2.40	
3.68/2.40	
3.68/2.40	We have to consider all (P,R,Pi)-chains
3.68/2.40	----------------------------------------
3.68/2.40	
3.68/2.40	(7) UsableRulesProof (EQUIVALENT)
3.68/2.40	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.68/2.40	----------------------------------------
3.68/2.40	
3.68/2.40	(8)
3.68/2.40	Obligation:
3.68/2.40	Pi DP problem:
3.68/2.40	The TRS P consists of the following rules:
3.68/2.40	
3.68/2.40	   SELECT_IN_AAG(X, .(Y, Xs), .(Y, Zs)) -> SELECT_IN_AAG(X, Xs, Zs)
3.68/2.40	
3.68/2.40	R is empty.
3.68/2.40	The argument filtering Pi contains the following mapping:
3.68/2.40	.(x1, x2)  =  .(x2)
3.68/2.40	
3.68/2.40	SELECT_IN_AAG(x1, x2, x3)  =  SELECT_IN_AAG(x3)
3.68/2.40	
3.68/2.40	
3.68/2.40	We have to consider all (P,R,Pi)-chains
3.68/2.40	----------------------------------------
3.68/2.40	
3.68/2.40	(9) PiDPToQDPProof (SOUND)
3.68/2.40	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.68/2.40	----------------------------------------
3.68/2.40	
3.68/2.40	(10)
3.68/2.40	Obligation:
3.68/2.40	Q DP problem:
3.68/2.40	The TRS P consists of the following rules:
3.68/2.40	
3.68/2.40	   SELECT_IN_AAG(.(Zs)) -> SELECT_IN_AAG(Zs)
3.68/2.40	
3.68/2.40	R is empty.
3.68/2.40	Q is empty.
3.68/2.40	We have to consider all (P,Q,R)-chains.
3.68/2.40	----------------------------------------
3.68/2.40	
3.68/2.40	(11) QDPSizeChangeProof (EQUIVALENT)
3.68/2.40	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. 
3.68/2.40	
3.68/2.40	From the DPs we obtained the following set of size-change graphs:
3.68/2.40	*SELECT_IN_AAG(.(Zs)) -> SELECT_IN_AAG(Zs)
3.68/2.40	The graph contains the following edges 1 > 1
3.68/2.40	
3.68/2.40	
3.68/2.40	----------------------------------------
3.68/2.40	
3.68/2.40	(12)
3.68/2.40	YES
4.07/2.49	EOF