3.63/1.70	YES
3.63/1.71	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.63/1.71	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.63/1.71	
3.63/1.71	
3.63/1.71	Left Termination of the query pattern
3.63/1.71	
3.63/1.71	p(g,g,g,a)
3.63/1.71	
3.63/1.71	w.r.t. the given Prolog program could successfully be proven:
3.63/1.71	
3.63/1.71	(0) Prolog
3.63/1.71	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.63/1.71	(2) PiTRS
3.63/1.71	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.63/1.71	(4) PiDP
3.63/1.71	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.63/1.71	(6) PiDP
3.63/1.71	(7) UsableRulesProof [EQUIVALENT, 0 ms]
3.63/1.71	(8) PiDP
3.63/1.71	(9) PiDPToQDPProof [SOUND, 0 ms]
3.63/1.71	(10) QDP
3.63/1.71	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.63/1.71	(12) YES
3.63/1.71	
3.63/1.71	
3.63/1.71	----------------------------------------
3.63/1.71	
3.63/1.71	(0)
3.63/1.71	Obligation:
3.63/1.71	Clauses:
3.63/1.71	
3.63/1.71	p(M, N, s(R), RES) :- p(M, R, N, RES).
3.63/1.71	p(M, s(N), R, RES) :- p(R, N, M, RES).
3.63/1.71	p(M, X1, X2, M).
3.63/1.71	
3.63/1.71	
3.63/1.71	Query: p(g,g,g,a)
3.63/1.71	----------------------------------------
3.63/1.71	
3.63/1.71	(1) PrologToPiTRSProof (SOUND)
3.63/1.71	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.63/1.71	
3.63/1.71	p_in_4: (b,b,b,f)
3.63/1.71	
3.63/1.71	Transforming Prolog into the following Term Rewriting System:
3.63/1.71	
3.63/1.71	Pi-finite rewrite system:
3.63/1.71	The TRS R consists of the following rules:
3.63/1.71	
3.63/1.71	   p_in_ggga(M, N, s(R), RES) -> U1_ggga(M, N, R, RES, p_in_ggga(M, R, N, RES))
3.63/1.71	   p_in_ggga(M, s(N), R, RES) -> U2_ggga(M, N, R, RES, p_in_ggga(R, N, M, RES))
3.63/1.71	   p_in_ggga(M, X1, X2, M) -> p_out_ggga(M, X1, X2, M)
3.63/1.71	   U2_ggga(M, N, R, RES, p_out_ggga(R, N, M, RES)) -> p_out_ggga(M, s(N), R, RES)
3.63/1.71	   U1_ggga(M, N, R, RES, p_out_ggga(M, R, N, RES)) -> p_out_ggga(M, N, s(R), RES)
3.63/1.71	
3.63/1.71	The argument filtering Pi contains the following mapping:
3.63/1.71	p_in_ggga(x1, x2, x3, x4)  =  p_in_ggga(x1, x2, x3)
3.63/1.71	
3.63/1.71	s(x1)  =  s(x1)
3.63/1.71	
3.63/1.71	U1_ggga(x1, x2, x3, x4, x5)  =  U1_ggga(x5)
3.63/1.71	
3.63/1.71	U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x5)
3.63/1.71	
3.63/1.71	p_out_ggga(x1, x2, x3, x4)  =  p_out_ggga(x4)
3.63/1.71	
3.63/1.71	
3.63/1.71	
3.63/1.71	
3.63/1.71	
3.63/1.71	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.63/1.71	
3.63/1.71	
3.63/1.71	
3.63/1.71	----------------------------------------
3.63/1.71	
3.63/1.71	(2)
3.63/1.71	Obligation:
3.63/1.71	Pi-finite rewrite system:
3.63/1.71	The TRS R consists of the following rules:
3.63/1.71	
3.63/1.71	   p_in_ggga(M, N, s(R), RES) -> U1_ggga(M, N, R, RES, p_in_ggga(M, R, N, RES))
3.63/1.71	   p_in_ggga(M, s(N), R, RES) -> U2_ggga(M, N, R, RES, p_in_ggga(R, N, M, RES))
3.63/1.71	   p_in_ggga(M, X1, X2, M) -> p_out_ggga(M, X1, X2, M)
3.63/1.71	   U2_ggga(M, N, R, RES, p_out_ggga(R, N, M, RES)) -> p_out_ggga(M, s(N), R, RES)
3.63/1.71	   U1_ggga(M, N, R, RES, p_out_ggga(M, R, N, RES)) -> p_out_ggga(M, N, s(R), RES)
3.63/1.71	
3.63/1.71	The argument filtering Pi contains the following mapping:
3.63/1.71	p_in_ggga(x1, x2, x3, x4)  =  p_in_ggga(x1, x2, x3)
3.63/1.71	
3.63/1.71	s(x1)  =  s(x1)
3.63/1.71	
3.63/1.71	U1_ggga(x1, x2, x3, x4, x5)  =  U1_ggga(x5)
3.63/1.71	
3.63/1.71	U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x5)
3.63/1.71	
3.63/1.71	p_out_ggga(x1, x2, x3, x4)  =  p_out_ggga(x4)
3.63/1.71	
3.63/1.71	
3.63/1.71	
3.63/1.71	----------------------------------------
3.63/1.71	
3.63/1.71	(3) DependencyPairsProof (EQUIVALENT)
3.63/1.71	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.63/1.71	Pi DP problem:
3.63/1.71	The TRS P consists of the following rules:
3.63/1.71	
3.63/1.71	   P_IN_GGGA(M, N, s(R), RES) -> U1_GGGA(M, N, R, RES, p_in_ggga(M, R, N, RES))
3.63/1.71	   P_IN_GGGA(M, N, s(R), RES) -> P_IN_GGGA(M, R, N, RES)
3.63/1.71	   P_IN_GGGA(M, s(N), R, RES) -> U2_GGGA(M, N, R, RES, p_in_ggga(R, N, M, RES))
3.63/1.71	   P_IN_GGGA(M, s(N), R, RES) -> P_IN_GGGA(R, N, M, RES)
3.63/1.71	
3.63/1.71	The TRS R consists of the following rules:
3.63/1.71	
3.63/1.71	   p_in_ggga(M, N, s(R), RES) -> U1_ggga(M, N, R, RES, p_in_ggga(M, R, N, RES))
3.63/1.71	   p_in_ggga(M, s(N), R, RES) -> U2_ggga(M, N, R, RES, p_in_ggga(R, N, M, RES))
3.63/1.71	   p_in_ggga(M, X1, X2, M) -> p_out_ggga(M, X1, X2, M)
3.63/1.71	   U2_ggga(M, N, R, RES, p_out_ggga(R, N, M, RES)) -> p_out_ggga(M, s(N), R, RES)
3.63/1.71	   U1_ggga(M, N, R, RES, p_out_ggga(M, R, N, RES)) -> p_out_ggga(M, N, s(R), RES)
3.63/1.71	
3.63/1.71	The argument filtering Pi contains the following mapping:
3.63/1.71	p_in_ggga(x1, x2, x3, x4)  =  p_in_ggga(x1, x2, x3)
3.63/1.71	
3.63/1.71	s(x1)  =  s(x1)
3.63/1.71	
3.63/1.71	U1_ggga(x1, x2, x3, x4, x5)  =  U1_ggga(x5)
3.63/1.71	
3.63/1.71	U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x5)
3.63/1.71	
3.63/1.71	p_out_ggga(x1, x2, x3, x4)  =  p_out_ggga(x4)
3.63/1.71	
3.63/1.71	P_IN_GGGA(x1, x2, x3, x4)  =  P_IN_GGGA(x1, x2, x3)
3.63/1.71	
3.63/1.71	U1_GGGA(x1, x2, x3, x4, x5)  =  U1_GGGA(x5)
3.63/1.71	
3.63/1.71	U2_GGGA(x1, x2, x3, x4, x5)  =  U2_GGGA(x5)
3.63/1.71	
3.63/1.71	
3.63/1.71	We have to consider all (P,R,Pi)-chains
3.63/1.71	----------------------------------------
3.63/1.71	
3.63/1.71	(4)
3.63/1.71	Obligation:
3.63/1.71	Pi DP problem:
3.63/1.71	The TRS P consists of the following rules:
3.63/1.71	
3.63/1.71	   P_IN_GGGA(M, N, s(R), RES) -> U1_GGGA(M, N, R, RES, p_in_ggga(M, R, N, RES))
3.63/1.71	   P_IN_GGGA(M, N, s(R), RES) -> P_IN_GGGA(M, R, N, RES)
3.63/1.71	   P_IN_GGGA(M, s(N), R, RES) -> U2_GGGA(M, N, R, RES, p_in_ggga(R, N, M, RES))
3.63/1.71	   P_IN_GGGA(M, s(N), R, RES) -> P_IN_GGGA(R, N, M, RES)
3.63/1.71	
3.63/1.71	The TRS R consists of the following rules:
3.63/1.71	
3.63/1.71	   p_in_ggga(M, N, s(R), RES) -> U1_ggga(M, N, R, RES, p_in_ggga(M, R, N, RES))
3.63/1.71	   p_in_ggga(M, s(N), R, RES) -> U2_ggga(M, N, R, RES, p_in_ggga(R, N, M, RES))
3.63/1.71	   p_in_ggga(M, X1, X2, M) -> p_out_ggga(M, X1, X2, M)
3.63/1.71	   U2_ggga(M, N, R, RES, p_out_ggga(R, N, M, RES)) -> p_out_ggga(M, s(N), R, RES)
3.63/1.71	   U1_ggga(M, N, R, RES, p_out_ggga(M, R, N, RES)) -> p_out_ggga(M, N, s(R), RES)
3.63/1.71	
3.63/1.71	The argument filtering Pi contains the following mapping:
3.63/1.71	p_in_ggga(x1, x2, x3, x4)  =  p_in_ggga(x1, x2, x3)
3.63/1.71	
3.63/1.71	s(x1)  =  s(x1)
3.63/1.71	
3.63/1.71	U1_ggga(x1, x2, x3, x4, x5)  =  U1_ggga(x5)
3.63/1.71	
3.63/1.71	U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x5)
3.63/1.71	
3.63/1.71	p_out_ggga(x1, x2, x3, x4)  =  p_out_ggga(x4)
3.63/1.71	
3.63/1.71	P_IN_GGGA(x1, x2, x3, x4)  =  P_IN_GGGA(x1, x2, x3)
3.63/1.71	
3.63/1.71	U1_GGGA(x1, x2, x3, x4, x5)  =  U1_GGGA(x5)
3.63/1.71	
3.63/1.71	U2_GGGA(x1, x2, x3, x4, x5)  =  U2_GGGA(x5)
3.63/1.71	
3.63/1.71	
3.63/1.71	We have to consider all (P,R,Pi)-chains
3.63/1.71	----------------------------------------
3.63/1.71	
3.63/1.71	(5) DependencyGraphProof (EQUIVALENT)
3.63/1.71	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.
3.63/1.71	----------------------------------------
3.63/1.71	
3.63/1.71	(6)
3.63/1.71	Obligation:
3.63/1.71	Pi DP problem:
3.63/1.71	The TRS P consists of the following rules:
3.63/1.71	
3.63/1.71	   P_IN_GGGA(M, s(N), R, RES) -> P_IN_GGGA(R, N, M, RES)
3.63/1.71	   P_IN_GGGA(M, N, s(R), RES) -> P_IN_GGGA(M, R, N, RES)
3.63/1.71	
3.63/1.71	The TRS R consists of the following rules:
3.63/1.71	
3.63/1.71	   p_in_ggga(M, N, s(R), RES) -> U1_ggga(M, N, R, RES, p_in_ggga(M, R, N, RES))
3.63/1.71	   p_in_ggga(M, s(N), R, RES) -> U2_ggga(M, N, R, RES, p_in_ggga(R, N, M, RES))
3.63/1.71	   p_in_ggga(M, X1, X2, M) -> p_out_ggga(M, X1, X2, M)
3.63/1.71	   U2_ggga(M, N, R, RES, p_out_ggga(R, N, M, RES)) -> p_out_ggga(M, s(N), R, RES)
3.63/1.71	   U1_ggga(M, N, R, RES, p_out_ggga(M, R, N, RES)) -> p_out_ggga(M, N, s(R), RES)
3.63/1.71	
3.63/1.71	The argument filtering Pi contains the following mapping:
3.63/1.71	p_in_ggga(x1, x2, x3, x4)  =  p_in_ggga(x1, x2, x3)
3.63/1.71	
3.63/1.71	s(x1)  =  s(x1)
3.63/1.71	
3.63/1.71	U1_ggga(x1, x2, x3, x4, x5)  =  U1_ggga(x5)
3.63/1.71	
3.63/1.71	U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x5)
3.63/1.71	
3.63/1.71	p_out_ggga(x1, x2, x3, x4)  =  p_out_ggga(x4)
3.63/1.71	
3.63/1.71	P_IN_GGGA(x1, x2, x3, x4)  =  P_IN_GGGA(x1, x2, x3)
3.63/1.71	
3.63/1.71	
3.63/1.71	We have to consider all (P,R,Pi)-chains
3.63/1.71	----------------------------------------
3.63/1.71	
3.63/1.71	(7) UsableRulesProof (EQUIVALENT)
3.63/1.71	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.63/1.71	----------------------------------------
3.63/1.71	
3.63/1.71	(8)
3.63/1.71	Obligation:
3.63/1.71	Pi DP problem:
3.63/1.71	The TRS P consists of the following rules:
3.63/1.71	
3.63/1.71	   P_IN_GGGA(M, s(N), R, RES) -> P_IN_GGGA(R, N, M, RES)
3.63/1.71	   P_IN_GGGA(M, N, s(R), RES) -> P_IN_GGGA(M, R, N, RES)
3.63/1.71	
3.63/1.71	R is empty.
3.63/1.71	The argument filtering Pi contains the following mapping:
3.63/1.71	s(x1)  =  s(x1)
3.63/1.71	
3.63/1.71	P_IN_GGGA(x1, x2, x3, x4)  =  P_IN_GGGA(x1, x2, x3)
3.63/1.71	
3.63/1.71	
3.63/1.71	We have to consider all (P,R,Pi)-chains
3.63/1.71	----------------------------------------
3.63/1.71	
3.63/1.71	(9) PiDPToQDPProof (SOUND)
3.63/1.71	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.63/1.71	----------------------------------------
3.63/1.71	
3.63/1.71	(10)
3.63/1.71	Obligation:
3.63/1.71	Q DP problem:
3.63/1.71	The TRS P consists of the following rules:
3.63/1.71	
3.63/1.71	   P_IN_GGGA(M, s(N), R) -> P_IN_GGGA(R, N, M)
3.63/1.71	   P_IN_GGGA(M, N, s(R)) -> P_IN_GGGA(M, R, N)
3.63/1.71	
3.63/1.71	R is empty.
3.63/1.71	Q is empty.
3.63/1.71	We have to consider all (P,Q,R)-chains.
3.63/1.71	----------------------------------------
3.63/1.71	
3.63/1.71	(11) QDPSizeChangeProof (EQUIVALENT)
3.63/1.71	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. 
3.63/1.71	
3.63/1.71	From the DPs we obtained the following set of size-change graphs:
3.63/1.71	*P_IN_GGGA(M, s(N), R) -> P_IN_GGGA(R, N, M)
3.63/1.71	The graph contains the following edges 3 >= 1, 2 > 2, 1 >= 3
3.63/1.71	
3.63/1.71	
3.63/1.71	*P_IN_GGGA(M, N, s(R)) -> P_IN_GGGA(M, R, N)
3.63/1.71	The graph contains the following edges 1 >= 1, 3 > 2, 2 >= 3
3.63/1.71	
3.63/1.71	
3.63/1.71	----------------------------------------
3.63/1.71	
3.63/1.71	(12)
3.63/1.71	YES
3.63/1.75	EOF