3.75/1.71	YES
3.75/1.73	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.75/1.73	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.75/1.73	
3.75/1.73	
3.75/1.73	Left Termination of the query pattern
3.75/1.73	
3.75/1.73	f(g,g,a)
3.75/1.73	
3.75/1.73	w.r.t. the given Prolog program could successfully be proven:
3.75/1.73	
3.75/1.73	(0) Prolog
3.75/1.73	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.75/1.73	(2) PiTRS
3.75/1.73	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.75/1.73	(4) PiDP
3.75/1.73	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.75/1.73	(6) PiDP
3.75/1.73	(7) UsableRulesProof [EQUIVALENT, 0 ms]
3.75/1.73	(8) PiDP
3.75/1.73	(9) PiDPToQDPProof [SOUND, 0 ms]
3.75/1.73	(10) QDP
3.75/1.73	(11) QDPSizeChangeProof [EQUIVALENT, 1 ms]
3.75/1.73	(12) YES
3.75/1.73	
3.75/1.73	
3.75/1.73	----------------------------------------
3.75/1.73	
3.75/1.73	(0)
3.75/1.73	Obligation:
3.75/1.73	Clauses:
3.75/1.73	
3.75/1.73	f([], RES, RES).
3.75/1.73	f(.(Head, Tail), X, RES) :- g(Tail, X, .(Head, Tail), RES).
3.75/1.73	g(A, B, C, RES) :- f(A, .(B, C), RES).
3.75/1.73	
3.75/1.73	
3.75/1.73	Query: f(g,g,a)
3.75/1.73	----------------------------------------
3.75/1.73	
3.75/1.73	(1) PrologToPiTRSProof (SOUND)
3.75/1.73	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.75/1.73	
3.75/1.73	f_in_3: (b,b,f)
3.75/1.73	
3.75/1.73	g_in_4: (b,b,b,f)
3.75/1.73	
3.75/1.73	Transforming Prolog into the following Term Rewriting System:
3.75/1.73	
3.75/1.73	Pi-finite rewrite system:
3.75/1.73	The TRS R consists of the following rules:
3.75/1.73	
3.75/1.73	   f_in_gga([], RES, RES) -> f_out_gga([], RES, RES)
3.75/1.73	   f_in_gga(.(Head, Tail), X, RES) -> U1_gga(Head, Tail, X, RES, g_in_ggga(Tail, X, .(Head, Tail), RES))
3.75/1.73	   g_in_ggga(A, B, C, RES) -> U2_ggga(A, B, C, RES, f_in_gga(A, .(B, C), RES))
3.75/1.73	   U2_ggga(A, B, C, RES, f_out_gga(A, .(B, C), RES)) -> g_out_ggga(A, B, C, RES)
3.75/1.73	   U1_gga(Head, Tail, X, RES, g_out_ggga(Tail, X, .(Head, Tail), RES)) -> f_out_gga(.(Head, Tail), X, RES)
3.75/1.73	
3.75/1.73	The argument filtering Pi contains the following mapping:
3.75/1.73	f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
3.75/1.73	
3.75/1.73	[]  =  []
3.75/1.73	
3.75/1.73	f_out_gga(x1, x2, x3)  =  f_out_gga(x1, x2, x3)
3.75/1.73	
3.75/1.73	.(x1, x2)  =  .(x1, x2)
3.75/1.73	
3.75/1.73	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
3.75/1.73	
3.75/1.73	g_in_ggga(x1, x2, x3, x4)  =  g_in_ggga(x1, x2, x3)
3.75/1.73	
3.75/1.73	U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)
3.75/1.73	
3.75/1.73	g_out_ggga(x1, x2, x3, x4)  =  g_out_ggga(x1, x2, x3, x4)
3.75/1.73	
3.75/1.73	
3.75/1.73	
3.75/1.73	
3.75/1.73	
3.75/1.73	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.75/1.73	
3.75/1.73	
3.75/1.73	
3.75/1.73	----------------------------------------
3.75/1.73	
3.75/1.73	(2)
3.75/1.73	Obligation:
3.75/1.73	Pi-finite rewrite system:
3.75/1.73	The TRS R consists of the following rules:
3.75/1.73	
3.75/1.73	   f_in_gga([], RES, RES) -> f_out_gga([], RES, RES)
3.75/1.73	   f_in_gga(.(Head, Tail), X, RES) -> U1_gga(Head, Tail, X, RES, g_in_ggga(Tail, X, .(Head, Tail), RES))
3.75/1.73	   g_in_ggga(A, B, C, RES) -> U2_ggga(A, B, C, RES, f_in_gga(A, .(B, C), RES))
3.75/1.73	   U2_ggga(A, B, C, RES, f_out_gga(A, .(B, C), RES)) -> g_out_ggga(A, B, C, RES)
3.75/1.73	   U1_gga(Head, Tail, X, RES, g_out_ggga(Tail, X, .(Head, Tail), RES)) -> f_out_gga(.(Head, Tail), X, RES)
3.75/1.73	
3.75/1.73	The argument filtering Pi contains the following mapping:
3.75/1.73	f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
3.75/1.73	
3.75/1.73	[]  =  []
3.75/1.73	
3.75/1.73	f_out_gga(x1, x2, x3)  =  f_out_gga(x1, x2, x3)
3.75/1.73	
3.75/1.73	.(x1, x2)  =  .(x1, x2)
3.75/1.73	
3.75/1.73	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
3.75/1.73	
3.75/1.73	g_in_ggga(x1, x2, x3, x4)  =  g_in_ggga(x1, x2, x3)
3.75/1.73	
3.75/1.73	U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)
3.75/1.73	
3.75/1.73	g_out_ggga(x1, x2, x3, x4)  =  g_out_ggga(x1, x2, x3, x4)
3.75/1.73	
3.75/1.73	
3.75/1.73	
3.75/1.73	----------------------------------------
3.75/1.73	
3.75/1.73	(3) DependencyPairsProof (EQUIVALENT)
3.75/1.73	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.75/1.73	Pi DP problem:
3.75/1.73	The TRS P consists of the following rules:
3.75/1.73	
3.75/1.73	   F_IN_GGA(.(Head, Tail), X, RES) -> U1_GGA(Head, Tail, X, RES, g_in_ggga(Tail, X, .(Head, Tail), RES))
3.75/1.73	   F_IN_GGA(.(Head, Tail), X, RES) -> G_IN_GGGA(Tail, X, .(Head, Tail), RES)
3.75/1.73	   G_IN_GGGA(A, B, C, RES) -> U2_GGGA(A, B, C, RES, f_in_gga(A, .(B, C), RES))
3.75/1.73	   G_IN_GGGA(A, B, C, RES) -> F_IN_GGA(A, .(B, C), RES)
3.75/1.73	
3.75/1.73	The TRS R consists of the following rules:
3.75/1.73	
3.75/1.73	   f_in_gga([], RES, RES) -> f_out_gga([], RES, RES)
3.75/1.73	   f_in_gga(.(Head, Tail), X, RES) -> U1_gga(Head, Tail, X, RES, g_in_ggga(Tail, X, .(Head, Tail), RES))
3.75/1.73	   g_in_ggga(A, B, C, RES) -> U2_ggga(A, B, C, RES, f_in_gga(A, .(B, C), RES))
3.75/1.73	   U2_ggga(A, B, C, RES, f_out_gga(A, .(B, C), RES)) -> g_out_ggga(A, B, C, RES)
3.75/1.73	   U1_gga(Head, Tail, X, RES, g_out_ggga(Tail, X, .(Head, Tail), RES)) -> f_out_gga(.(Head, Tail), X, RES)
3.75/1.73	
3.75/1.73	The argument filtering Pi contains the following mapping:
3.75/1.73	f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
3.75/1.73	
3.75/1.73	[]  =  []
3.75/1.73	
3.75/1.73	f_out_gga(x1, x2, x3)  =  f_out_gga(x1, x2, x3)
3.75/1.73	
3.75/1.73	.(x1, x2)  =  .(x1, x2)
3.75/1.73	
3.75/1.73	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
3.75/1.73	
3.75/1.73	g_in_ggga(x1, x2, x3, x4)  =  g_in_ggga(x1, x2, x3)
3.75/1.73	
3.75/1.73	U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)
3.75/1.73	
3.75/1.73	g_out_ggga(x1, x2, x3, x4)  =  g_out_ggga(x1, x2, x3, x4)
3.75/1.73	
3.75/1.73	F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)
3.75/1.73	
3.75/1.73	U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)
3.75/1.73	
3.75/1.73	G_IN_GGGA(x1, x2, x3, x4)  =  G_IN_GGGA(x1, x2, x3)
3.75/1.73	
3.75/1.73	U2_GGGA(x1, x2, x3, x4, x5)  =  U2_GGGA(x1, x2, x3, x5)
3.75/1.73	
3.75/1.73	
3.75/1.73	We have to consider all (P,R,Pi)-chains
3.75/1.73	----------------------------------------
3.75/1.73	
3.75/1.73	(4)
3.75/1.73	Obligation:
3.75/1.73	Pi DP problem:
3.75/1.73	The TRS P consists of the following rules:
3.75/1.73	
3.75/1.73	   F_IN_GGA(.(Head, Tail), X, RES) -> U1_GGA(Head, Tail, X, RES, g_in_ggga(Tail, X, .(Head, Tail), RES))
3.75/1.73	   F_IN_GGA(.(Head, Tail), X, RES) -> G_IN_GGGA(Tail, X, .(Head, Tail), RES)
3.75/1.73	   G_IN_GGGA(A, B, C, RES) -> U2_GGGA(A, B, C, RES, f_in_gga(A, .(B, C), RES))
3.75/1.73	   G_IN_GGGA(A, B, C, RES) -> F_IN_GGA(A, .(B, C), RES)
3.75/1.73	
3.75/1.73	The TRS R consists of the following rules:
3.75/1.73	
3.75/1.73	   f_in_gga([], RES, RES) -> f_out_gga([], RES, RES)
3.75/1.73	   f_in_gga(.(Head, Tail), X, RES) -> U1_gga(Head, Tail, X, RES, g_in_ggga(Tail, X, .(Head, Tail), RES))
3.75/1.73	   g_in_ggga(A, B, C, RES) -> U2_ggga(A, B, C, RES, f_in_gga(A, .(B, C), RES))
3.75/1.73	   U2_ggga(A, B, C, RES, f_out_gga(A, .(B, C), RES)) -> g_out_ggga(A, B, C, RES)
3.75/1.73	   U1_gga(Head, Tail, X, RES, g_out_ggga(Tail, X, .(Head, Tail), RES)) -> f_out_gga(.(Head, Tail), X, RES)
3.75/1.73	
3.75/1.73	The argument filtering Pi contains the following mapping:
3.75/1.73	f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
3.75/1.73	
3.75/1.73	[]  =  []
3.75/1.73	
3.75/1.73	f_out_gga(x1, x2, x3)  =  f_out_gga(x1, x2, x3)
3.75/1.73	
3.75/1.73	.(x1, x2)  =  .(x1, x2)
3.75/1.73	
3.75/1.73	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
3.75/1.73	
3.75/1.73	g_in_ggga(x1, x2, x3, x4)  =  g_in_ggga(x1, x2, x3)
3.75/1.73	
3.75/1.73	U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)
3.75/1.73	
3.75/1.73	g_out_ggga(x1, x2, x3, x4)  =  g_out_ggga(x1, x2, x3, x4)
3.75/1.73	
3.75/1.73	F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)
3.75/1.73	
3.75/1.73	U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)
3.75/1.73	
3.75/1.73	G_IN_GGGA(x1, x2, x3, x4)  =  G_IN_GGGA(x1, x2, x3)
3.75/1.73	
3.75/1.73	U2_GGGA(x1, x2, x3, x4, x5)  =  U2_GGGA(x1, x2, x3, x5)
3.75/1.73	
3.75/1.73	
3.75/1.73	We have to consider all (P,R,Pi)-chains
3.75/1.73	----------------------------------------
3.75/1.73	
3.75/1.73	(5) DependencyGraphProof (EQUIVALENT)
3.75/1.73	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.
3.75/1.73	----------------------------------------
3.75/1.73	
3.75/1.73	(6)
3.75/1.73	Obligation:
3.75/1.73	Pi DP problem:
3.75/1.73	The TRS P consists of the following rules:
3.75/1.73	
3.75/1.73	   F_IN_GGA(.(Head, Tail), X, RES) -> G_IN_GGGA(Tail, X, .(Head, Tail), RES)
3.75/1.73	   G_IN_GGGA(A, B, C, RES) -> F_IN_GGA(A, .(B, C), RES)
3.75/1.73	
3.75/1.73	The TRS R consists of the following rules:
3.75/1.73	
3.75/1.73	   f_in_gga([], RES, RES) -> f_out_gga([], RES, RES)
3.75/1.73	   f_in_gga(.(Head, Tail), X, RES) -> U1_gga(Head, Tail, X, RES, g_in_ggga(Tail, X, .(Head, Tail), RES))
3.75/1.73	   g_in_ggga(A, B, C, RES) -> U2_ggga(A, B, C, RES, f_in_gga(A, .(B, C), RES))
3.75/1.73	   U2_ggga(A, B, C, RES, f_out_gga(A, .(B, C), RES)) -> g_out_ggga(A, B, C, RES)
3.75/1.73	   U1_gga(Head, Tail, X, RES, g_out_ggga(Tail, X, .(Head, Tail), RES)) -> f_out_gga(.(Head, Tail), X, RES)
3.75/1.73	
3.75/1.73	The argument filtering Pi contains the following mapping:
3.75/1.73	f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
3.75/1.73	
3.75/1.73	[]  =  []
3.75/1.73	
3.75/1.73	f_out_gga(x1, x2, x3)  =  f_out_gga(x1, x2, x3)
3.75/1.73	
3.75/1.73	.(x1, x2)  =  .(x1, x2)
3.75/1.73	
3.75/1.73	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x2, x3, x5)
3.75/1.73	
3.75/1.73	g_in_ggga(x1, x2, x3, x4)  =  g_in_ggga(x1, x2, x3)
3.75/1.73	
3.75/1.73	U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)
3.75/1.73	
3.75/1.73	g_out_ggga(x1, x2, x3, x4)  =  g_out_ggga(x1, x2, x3, x4)
3.75/1.73	
3.75/1.73	F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)
3.75/1.73	
3.75/1.73	G_IN_GGGA(x1, x2, x3, x4)  =  G_IN_GGGA(x1, x2, x3)
3.75/1.73	
3.75/1.73	
3.75/1.73	We have to consider all (P,R,Pi)-chains
3.75/1.73	----------------------------------------
3.75/1.73	
3.75/1.73	(7) UsableRulesProof (EQUIVALENT)
3.75/1.73	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.75/1.73	----------------------------------------
3.75/1.73	
3.75/1.73	(8)
3.75/1.73	Obligation:
3.75/1.73	Pi DP problem:
3.75/1.73	The TRS P consists of the following rules:
3.75/1.73	
3.75/1.73	   F_IN_GGA(.(Head, Tail), X, RES) -> G_IN_GGGA(Tail, X, .(Head, Tail), RES)
3.75/1.73	   G_IN_GGGA(A, B, C, RES) -> F_IN_GGA(A, .(B, C), RES)
3.75/1.73	
3.75/1.73	R is empty.
3.75/1.73	The argument filtering Pi contains the following mapping:
3.75/1.73	.(x1, x2)  =  .(x1, x2)
3.75/1.73	
3.75/1.73	F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)
3.75/1.73	
3.75/1.73	G_IN_GGGA(x1, x2, x3, x4)  =  G_IN_GGGA(x1, x2, x3)
3.75/1.73	
3.75/1.73	
3.75/1.73	We have to consider all (P,R,Pi)-chains
3.75/1.73	----------------------------------------
3.75/1.73	
3.75/1.73	(9) PiDPToQDPProof (SOUND)
3.75/1.73	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.75/1.73	----------------------------------------
3.75/1.73	
3.75/1.73	(10)
3.75/1.73	Obligation:
3.75/1.73	Q DP problem:
3.75/1.73	The TRS P consists of the following rules:
3.75/1.73	
3.75/1.73	   F_IN_GGA(.(Head, Tail), X) -> G_IN_GGGA(Tail, X, .(Head, Tail))
3.75/1.73	   G_IN_GGGA(A, B, C) -> F_IN_GGA(A, .(B, C))
3.75/1.73	
3.75/1.73	R is empty.
3.75/1.73	Q is empty.
3.75/1.73	We have to consider all (P,Q,R)-chains.
3.75/1.73	----------------------------------------
3.75/1.73	
3.75/1.73	(11) QDPSizeChangeProof (EQUIVALENT)
3.75/1.73	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. 
3.75/1.73	
3.75/1.73	From the DPs we obtained the following set of size-change graphs:
3.75/1.73	*G_IN_GGGA(A, B, C) -> F_IN_GGA(A, .(B, C))
3.75/1.73	The graph contains the following edges 1 >= 1
3.75/1.73	
3.75/1.73	
3.75/1.73	*F_IN_GGA(.(Head, Tail), X) -> G_IN_GGGA(Tail, X, .(Head, Tail))
3.75/1.73	The graph contains the following edges 1 > 1, 2 >= 2, 1 >= 3
3.75/1.73	
3.75/1.73	
3.75/1.73	----------------------------------------
3.75/1.73	
3.75/1.73	(12)
3.75/1.73	YES
3.98/1.77	EOF