3.83/1.75	YES
3.83/1.75	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.83/1.75	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.83/1.75	
3.83/1.75	
3.83/1.75	Left Termination of the query pattern
3.83/1.75	
3.83/1.75	f(g,g,a)
3.83/1.75	
3.83/1.75	w.r.t. the given Prolog program could successfully be proven:
3.83/1.75	
3.83/1.75	(0) Prolog
3.83/1.75	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.83/1.75	(2) PiTRS
3.83/1.75	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.83/1.75	(4) PiDP
3.83/1.75	(5) DependencyGraphProof [EQUIVALENT, 2 ms]
3.83/1.75	(6) PiDP
3.83/1.75	(7) UsableRulesProof [EQUIVALENT, 0 ms]
3.83/1.75	(8) PiDP
3.83/1.75	(9) PiDPToQDPProof [SOUND, 0 ms]
3.83/1.75	(10) QDP
3.83/1.75	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.83/1.75	(12) YES
3.83/1.75	
3.83/1.75	
3.83/1.75	----------------------------------------
3.83/1.75	
3.83/1.75	(0)
3.83/1.75	Obligation:
3.83/1.75	Clauses:
3.83/1.75	
3.83/1.75	f(RES, [], RES).
3.83/1.75	f([], .(Head, Tail), RES) :- f(.(Head, Tail), Tail, RES).
3.83/1.75	f(.(Head, Tail), Y, RES) :- f(Y, Tail, RES).
3.83/1.75	
3.83/1.75	
3.83/1.75	Query: f(g,g,a)
3.83/1.75	----------------------------------------
3.83/1.75	
3.83/1.75	(1) PrologToPiTRSProof (SOUND)
3.83/1.75	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.83/1.75	
3.83/1.75	f_in_3: (b,b,f)
3.83/1.75	
3.83/1.75	Transforming Prolog into the following Term Rewriting System:
3.83/1.76	
3.83/1.76	Pi-finite rewrite system:
3.83/1.76	The TRS R consists of the following rules:
3.83/1.76	
3.83/1.76	   f_in_gga(RES, [], RES) -> f_out_gga(RES, [], RES)
3.83/1.76	   f_in_gga([], .(Head, Tail), RES) -> U1_gga(Head, Tail, RES, f_in_gga(.(Head, Tail), Tail, RES))
3.83/1.76	   f_in_gga(.(Head, Tail), Y, RES) -> U2_gga(Head, Tail, Y, RES, f_in_gga(Y, Tail, RES))
3.83/1.76	   U2_gga(Head, Tail, Y, RES, f_out_gga(Y, Tail, RES)) -> f_out_gga(.(Head, Tail), Y, RES)
3.83/1.76	   U1_gga(Head, Tail, RES, f_out_gga(.(Head, Tail), Tail, RES)) -> f_out_gga([], .(Head, Tail), RES)
3.83/1.76	
3.83/1.76	The argument filtering Pi contains the following mapping:
3.83/1.76	f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
3.83/1.76	
3.83/1.76	[]  =  []
3.83/1.76	
3.83/1.76	f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
3.83/1.76	
3.83/1.76	.(x1, x2)  =  .(x1, x2)
3.83/1.76	
3.83/1.76	U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
3.83/1.76	
3.83/1.76	U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
3.83/1.76	
3.83/1.76	
3.83/1.76	
3.83/1.76	
3.83/1.76	
3.83/1.76	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.83/1.76	
3.83/1.76	
3.83/1.76	
3.83/1.76	----------------------------------------
3.83/1.76	
3.83/1.76	(2)
3.83/1.76	Obligation:
3.83/1.76	Pi-finite rewrite system:
3.83/1.76	The TRS R consists of the following rules:
3.83/1.76	
3.83/1.76	   f_in_gga(RES, [], RES) -> f_out_gga(RES, [], RES)
3.83/1.76	   f_in_gga([], .(Head, Tail), RES) -> U1_gga(Head, Tail, RES, f_in_gga(.(Head, Tail), Tail, RES))
3.83/1.76	   f_in_gga(.(Head, Tail), Y, RES) -> U2_gga(Head, Tail, Y, RES, f_in_gga(Y, Tail, RES))
3.83/1.76	   U2_gga(Head, Tail, Y, RES, f_out_gga(Y, Tail, RES)) -> f_out_gga(.(Head, Tail), Y, RES)
3.83/1.76	   U1_gga(Head, Tail, RES, f_out_gga(.(Head, Tail), Tail, RES)) -> f_out_gga([], .(Head, Tail), RES)
3.83/1.76	
3.83/1.76	The argument filtering Pi contains the following mapping:
3.83/1.76	f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
3.83/1.76	
3.83/1.76	[]  =  []
3.83/1.76	
3.83/1.76	f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
3.83/1.76	
3.83/1.76	.(x1, x2)  =  .(x1, x2)
3.83/1.76	
3.83/1.76	U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
3.83/1.76	
3.83/1.76	U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
3.83/1.76	
3.83/1.76	
3.83/1.76	
3.83/1.76	----------------------------------------
3.83/1.76	
3.83/1.76	(3) DependencyPairsProof (EQUIVALENT)
3.83/1.76	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.83/1.76	Pi DP problem:
3.83/1.76	The TRS P consists of the following rules:
3.83/1.76	
3.83/1.76	   F_IN_GGA([], .(Head, Tail), RES) -> U1_GGA(Head, Tail, RES, f_in_gga(.(Head, Tail), Tail, RES))
3.83/1.76	   F_IN_GGA([], .(Head, Tail), RES) -> F_IN_GGA(.(Head, Tail), Tail, RES)
3.83/1.76	   F_IN_GGA(.(Head, Tail), Y, RES) -> U2_GGA(Head, Tail, Y, RES, f_in_gga(Y, Tail, RES))
3.83/1.76	   F_IN_GGA(.(Head, Tail), Y, RES) -> F_IN_GGA(Y, Tail, RES)
3.83/1.76	
3.83/1.76	The TRS R consists of the following rules:
3.83/1.76	
3.83/1.76	   f_in_gga(RES, [], RES) -> f_out_gga(RES, [], RES)
3.83/1.76	   f_in_gga([], .(Head, Tail), RES) -> U1_gga(Head, Tail, RES, f_in_gga(.(Head, Tail), Tail, RES))
3.83/1.76	   f_in_gga(.(Head, Tail), Y, RES) -> U2_gga(Head, Tail, Y, RES, f_in_gga(Y, Tail, RES))
3.83/1.76	   U2_gga(Head, Tail, Y, RES, f_out_gga(Y, Tail, RES)) -> f_out_gga(.(Head, Tail), Y, RES)
3.83/1.76	   U1_gga(Head, Tail, RES, f_out_gga(.(Head, Tail), Tail, RES)) -> f_out_gga([], .(Head, Tail), RES)
3.83/1.76	
3.83/1.76	The argument filtering Pi contains the following mapping:
3.83/1.76	f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
3.83/1.76	
3.83/1.76	[]  =  []
3.83/1.76	
3.83/1.76	f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
3.83/1.76	
3.83/1.76	.(x1, x2)  =  .(x1, x2)
3.83/1.76	
3.83/1.76	U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
3.83/1.76	
3.83/1.76	U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
3.83/1.76	
3.83/1.76	F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)
3.83/1.76	
3.83/1.76	U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
3.83/1.76	
3.83/1.76	U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x5)
3.83/1.76	
3.83/1.76	
3.83/1.76	We have to consider all (P,R,Pi)-chains
3.83/1.76	----------------------------------------
3.83/1.76	
3.83/1.76	(4)
3.83/1.76	Obligation:
3.83/1.76	Pi DP problem:
3.83/1.76	The TRS P consists of the following rules:
3.83/1.76	
3.83/1.76	   F_IN_GGA([], .(Head, Tail), RES) -> U1_GGA(Head, Tail, RES, f_in_gga(.(Head, Tail), Tail, RES))
3.83/1.76	   F_IN_GGA([], .(Head, Tail), RES) -> F_IN_GGA(.(Head, Tail), Tail, RES)
3.83/1.76	   F_IN_GGA(.(Head, Tail), Y, RES) -> U2_GGA(Head, Tail, Y, RES, f_in_gga(Y, Tail, RES))
3.83/1.76	   F_IN_GGA(.(Head, Tail), Y, RES) -> F_IN_GGA(Y, Tail, RES)
3.83/1.76	
3.83/1.76	The TRS R consists of the following rules:
3.83/1.76	
3.83/1.76	   f_in_gga(RES, [], RES) -> f_out_gga(RES, [], RES)
3.83/1.76	   f_in_gga([], .(Head, Tail), RES) -> U1_gga(Head, Tail, RES, f_in_gga(.(Head, Tail), Tail, RES))
3.83/1.76	   f_in_gga(.(Head, Tail), Y, RES) -> U2_gga(Head, Tail, Y, RES, f_in_gga(Y, Tail, RES))
3.83/1.76	   U2_gga(Head, Tail, Y, RES, f_out_gga(Y, Tail, RES)) -> f_out_gga(.(Head, Tail), Y, RES)
3.83/1.76	   U1_gga(Head, Tail, RES, f_out_gga(.(Head, Tail), Tail, RES)) -> f_out_gga([], .(Head, Tail), RES)
3.83/1.76	
3.83/1.76	The argument filtering Pi contains the following mapping:
3.83/1.76	f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
3.83/1.76	
3.83/1.76	[]  =  []
3.83/1.76	
3.83/1.76	f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
3.83/1.76	
3.83/1.76	.(x1, x2)  =  .(x1, x2)
3.83/1.76	
3.83/1.76	U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
3.83/1.76	
3.83/1.76	U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
3.83/1.76	
3.83/1.76	F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)
3.83/1.76	
3.83/1.76	U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
3.83/1.76	
3.83/1.76	U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x5)
3.83/1.76	
3.83/1.76	
3.83/1.76	We have to consider all (P,R,Pi)-chains
3.83/1.76	----------------------------------------
3.83/1.76	
3.83/1.76	(5) DependencyGraphProof (EQUIVALENT)
3.83/1.76	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.
3.83/1.76	----------------------------------------
3.83/1.76	
3.83/1.76	(6)
3.83/1.76	Obligation:
3.83/1.76	Pi DP problem:
3.83/1.76	The TRS P consists of the following rules:
3.83/1.76	
3.83/1.76	   F_IN_GGA([], .(Head, Tail), RES) -> F_IN_GGA(.(Head, Tail), Tail, RES)
3.83/1.76	   F_IN_GGA(.(Head, Tail), Y, RES) -> F_IN_GGA(Y, Tail, RES)
3.83/1.76	
3.83/1.76	The TRS R consists of the following rules:
3.83/1.76	
3.83/1.76	   f_in_gga(RES, [], RES) -> f_out_gga(RES, [], RES)
3.83/1.76	   f_in_gga([], .(Head, Tail), RES) -> U1_gga(Head, Tail, RES, f_in_gga(.(Head, Tail), Tail, RES))
3.83/1.76	   f_in_gga(.(Head, Tail), Y, RES) -> U2_gga(Head, Tail, Y, RES, f_in_gga(Y, Tail, RES))
3.83/1.76	   U2_gga(Head, Tail, Y, RES, f_out_gga(Y, Tail, RES)) -> f_out_gga(.(Head, Tail), Y, RES)
3.83/1.76	   U1_gga(Head, Tail, RES, f_out_gga(.(Head, Tail), Tail, RES)) -> f_out_gga([], .(Head, Tail), RES)
3.83/1.76	
3.83/1.76	The argument filtering Pi contains the following mapping:
3.83/1.76	f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
3.83/1.76	
3.83/1.76	[]  =  []
3.83/1.76	
3.83/1.76	f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
3.83/1.76	
3.83/1.76	.(x1, x2)  =  .(x1, x2)
3.83/1.76	
3.83/1.76	U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
3.83/1.76	
3.83/1.76	U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
3.83/1.76	
3.83/1.76	F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)
3.83/1.76	
3.83/1.76	
3.83/1.76	We have to consider all (P,R,Pi)-chains
3.83/1.76	----------------------------------------
3.83/1.76	
3.83/1.76	(7) UsableRulesProof (EQUIVALENT)
3.83/1.76	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.83/1.76	----------------------------------------
3.83/1.76	
3.83/1.76	(8)
3.83/1.76	Obligation:
3.83/1.76	Pi DP problem:
3.83/1.76	The TRS P consists of the following rules:
3.83/1.76	
3.83/1.76	   F_IN_GGA([], .(Head, Tail), RES) -> F_IN_GGA(.(Head, Tail), Tail, RES)
3.83/1.76	   F_IN_GGA(.(Head, Tail), Y, RES) -> F_IN_GGA(Y, Tail, RES)
3.83/1.76	
3.83/1.76	R is empty.
3.83/1.76	The argument filtering Pi contains the following mapping:
3.83/1.76	[]  =  []
3.83/1.76	
3.83/1.76	.(x1, x2)  =  .(x1, x2)
3.83/1.76	
3.83/1.76	F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)
3.83/1.76	
3.83/1.76	
3.83/1.76	We have to consider all (P,R,Pi)-chains
3.83/1.76	----------------------------------------
3.83/1.76	
3.83/1.76	(9) PiDPToQDPProof (SOUND)
3.83/1.76	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.83/1.76	----------------------------------------
3.83/1.76	
3.83/1.76	(10)
3.83/1.76	Obligation:
3.83/1.76	Q DP problem:
3.83/1.76	The TRS P consists of the following rules:
3.83/1.76	
3.83/1.76	   F_IN_GGA([], .(Head, Tail)) -> F_IN_GGA(.(Head, Tail), Tail)
3.83/1.76	   F_IN_GGA(.(Head, Tail), Y) -> F_IN_GGA(Y, Tail)
3.83/1.76	
3.83/1.76	R is empty.
3.83/1.76	Q is empty.
3.83/1.76	We have to consider all (P,Q,R)-chains.
3.83/1.76	----------------------------------------
3.83/1.76	
3.83/1.76	(11) QDPSizeChangeProof (EQUIVALENT)
3.83/1.76	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. 
3.83/1.76	
3.83/1.76	From the DPs we obtained the following set of size-change graphs:
3.83/1.76	*F_IN_GGA(.(Head, Tail), Y) -> F_IN_GGA(Y, Tail)
3.83/1.76	The graph contains the following edges 2 >= 1, 1 > 2
3.83/1.76	
3.83/1.76	
3.83/1.76	*F_IN_GGA([], .(Head, Tail)) -> F_IN_GGA(.(Head, Tail), Tail)
3.83/1.76	The graph contains the following edges 2 >= 1, 2 > 2
3.83/1.76	
3.83/1.76	
3.83/1.76	----------------------------------------
3.83/1.76	
3.83/1.76	(12)
3.83/1.76	YES
4.02/1.80	EOF