4.20/2.22	YES
4.28/2.24	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
4.28/2.24	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.28/2.24	
4.28/2.24	
4.28/2.24	Left Termination of the query pattern
4.28/2.24	
4.28/2.24	gopher(g,a)
4.28/2.24	
4.28/2.24	w.r.t. the given Prolog program could successfully be proven:
4.28/2.24	
4.28/2.24	(0) Prolog
4.28/2.24	(1) PrologToPiTRSProof [SOUND, 0 ms]
4.28/2.24	(2) PiTRS
4.28/2.24	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
4.28/2.24	(4) PiDP
4.28/2.24	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
4.28/2.24	(6) PiDP
4.28/2.24	(7) UsableRulesProof [EQUIVALENT, 0 ms]
4.28/2.24	(8) PiDP
4.28/2.24	(9) PiDPToQDPProof [SOUND, 0 ms]
4.28/2.24	(10) QDP
4.28/2.24	(11) MRRProof [EQUIVALENT, 11 ms]
4.28/2.24	(12) QDP
4.28/2.24	(13) PisEmptyProof [EQUIVALENT, 0 ms]
4.28/2.24	(14) YES
4.28/2.24	
4.28/2.24	
4.28/2.24	----------------------------------------
4.28/2.24	
4.28/2.24	(0)
4.28/2.24	Obligation:
4.28/2.24	Clauses:
4.28/2.24	
4.28/2.24	gopher(nil, nil).
4.28/2.24	gopher(cons(nil, Y), cons(nil, Y)).
4.28/2.24	gopher(cons(cons(U, V), W), X) :- gopher(cons(U, cons(V, W)), X).
4.28/2.24	
4.28/2.24	
4.28/2.24	Query: gopher(g,a)
4.28/2.24	----------------------------------------
4.28/2.24	
4.28/2.24	(1) PrologToPiTRSProof (SOUND)
4.28/2.24	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
4.28/2.24	
4.28/2.24	gopher_in_2: (b,f)
4.28/2.24	
4.28/2.24	Transforming Prolog into the following Term Rewriting System:
4.28/2.24	
4.28/2.24	Pi-finite rewrite system:
4.28/2.24	The TRS R consists of the following rules:
4.28/2.24	
4.28/2.24	   gopher_in_ga(nil, nil) -> gopher_out_ga(nil, nil)
4.28/2.24	   gopher_in_ga(cons(nil, Y), cons(nil, Y)) -> gopher_out_ga(cons(nil, Y), cons(nil, Y))
4.28/2.24	   gopher_in_ga(cons(cons(U, V), W), X) -> U1_ga(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X))
4.28/2.24	   U1_ga(U, V, W, X, gopher_out_ga(cons(U, cons(V, W)), X)) -> gopher_out_ga(cons(cons(U, V), W), X)
4.28/2.24	
4.28/2.24	The argument filtering Pi contains the following mapping:
4.28/2.24	gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
4.28/2.24	
4.28/2.24	nil  =  nil
4.28/2.24	
4.28/2.24	gopher_out_ga(x1, x2)  =  gopher_out_ga(x1, x2)
4.28/2.24	
4.28/2.24	cons(x1, x2)  =  cons(x1, x2)
4.28/2.24	
4.28/2.24	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
4.28/2.24	
4.28/2.24	
4.28/2.24	
4.28/2.24	
4.28/2.24	
4.28/2.24	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
4.28/2.24	
4.28/2.24	
4.28/2.24	
4.28/2.24	----------------------------------------
4.28/2.24	
4.28/2.24	(2)
4.28/2.24	Obligation:
4.28/2.24	Pi-finite rewrite system:
4.28/2.24	The TRS R consists of the following rules:
4.28/2.24	
4.28/2.24	   gopher_in_ga(nil, nil) -> gopher_out_ga(nil, nil)
4.28/2.24	   gopher_in_ga(cons(nil, Y), cons(nil, Y)) -> gopher_out_ga(cons(nil, Y), cons(nil, Y))
4.28/2.24	   gopher_in_ga(cons(cons(U, V), W), X) -> U1_ga(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X))
4.28/2.24	   U1_ga(U, V, W, X, gopher_out_ga(cons(U, cons(V, W)), X)) -> gopher_out_ga(cons(cons(U, V), W), X)
4.28/2.24	
4.28/2.24	The argument filtering Pi contains the following mapping:
4.28/2.24	gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
4.28/2.24	
4.28/2.24	nil  =  nil
4.28/2.24	
4.28/2.24	gopher_out_ga(x1, x2)  =  gopher_out_ga(x1, x2)
4.28/2.24	
4.28/2.24	cons(x1, x2)  =  cons(x1, x2)
4.28/2.24	
4.28/2.24	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
4.28/2.24	
4.28/2.24	
4.28/2.24	
4.28/2.24	----------------------------------------
4.28/2.24	
4.28/2.24	(3) DependencyPairsProof (EQUIVALENT)
4.28/2.24	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
4.28/2.24	Pi DP problem:
4.28/2.24	The TRS P consists of the following rules:
4.28/2.24	
4.28/2.24	   GOPHER_IN_GA(cons(cons(U, V), W), X) -> U1_GA(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X))
4.28/2.24	   GOPHER_IN_GA(cons(cons(U, V), W), X) -> GOPHER_IN_GA(cons(U, cons(V, W)), X)
4.28/2.24	
4.28/2.24	The TRS R consists of the following rules:
4.28/2.24	
4.28/2.24	   gopher_in_ga(nil, nil) -> gopher_out_ga(nil, nil)
4.28/2.24	   gopher_in_ga(cons(nil, Y), cons(nil, Y)) -> gopher_out_ga(cons(nil, Y), cons(nil, Y))
4.28/2.24	   gopher_in_ga(cons(cons(U, V), W), X) -> U1_ga(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X))
4.28/2.24	   U1_ga(U, V, W, X, gopher_out_ga(cons(U, cons(V, W)), X)) -> gopher_out_ga(cons(cons(U, V), W), X)
4.28/2.24	
4.28/2.24	The argument filtering Pi contains the following mapping:
4.28/2.24	gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
4.28/2.24	
4.28/2.24	nil  =  nil
4.28/2.24	
4.28/2.24	gopher_out_ga(x1, x2)  =  gopher_out_ga(x1, x2)
4.28/2.24	
4.28/2.24	cons(x1, x2)  =  cons(x1, x2)
4.28/2.24	
4.28/2.24	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
4.28/2.24	
4.28/2.24	GOPHER_IN_GA(x1, x2)  =  GOPHER_IN_GA(x1)
4.28/2.24	
4.28/2.24	U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
4.28/2.24	
4.28/2.24	
4.28/2.24	We have to consider all (P,R,Pi)-chains
4.28/2.24	----------------------------------------
4.28/2.24	
4.28/2.24	(4)
4.28/2.24	Obligation:
4.28/2.24	Pi DP problem:
4.28/2.24	The TRS P consists of the following rules:
4.28/2.24	
4.28/2.24	   GOPHER_IN_GA(cons(cons(U, V), W), X) -> U1_GA(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X))
4.28/2.24	   GOPHER_IN_GA(cons(cons(U, V), W), X) -> GOPHER_IN_GA(cons(U, cons(V, W)), X)
4.28/2.24	
4.28/2.24	The TRS R consists of the following rules:
4.28/2.24	
4.28/2.24	   gopher_in_ga(nil, nil) -> gopher_out_ga(nil, nil)
4.28/2.24	   gopher_in_ga(cons(nil, Y), cons(nil, Y)) -> gopher_out_ga(cons(nil, Y), cons(nil, Y))
4.28/2.24	   gopher_in_ga(cons(cons(U, V), W), X) -> U1_ga(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X))
4.28/2.24	   U1_ga(U, V, W, X, gopher_out_ga(cons(U, cons(V, W)), X)) -> gopher_out_ga(cons(cons(U, V), W), X)
4.28/2.24	
4.28/2.24	The argument filtering Pi contains the following mapping:
4.28/2.24	gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
4.28/2.24	
4.28/2.24	nil  =  nil
4.28/2.24	
4.28/2.24	gopher_out_ga(x1, x2)  =  gopher_out_ga(x1, x2)
4.28/2.24	
4.28/2.24	cons(x1, x2)  =  cons(x1, x2)
4.28/2.24	
4.28/2.24	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
4.28/2.24	
4.28/2.24	GOPHER_IN_GA(x1, x2)  =  GOPHER_IN_GA(x1)
4.28/2.24	
4.28/2.24	U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x3, x5)
4.28/2.24	
4.28/2.24	
4.28/2.24	We have to consider all (P,R,Pi)-chains
4.28/2.24	----------------------------------------
4.28/2.24	
4.28/2.24	(5) DependencyGraphProof (EQUIVALENT)
4.28/2.24	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
4.28/2.24	----------------------------------------
4.28/2.24	
4.28/2.24	(6)
4.28/2.24	Obligation:
4.28/2.24	Pi DP problem:
4.28/2.24	The TRS P consists of the following rules:
4.28/2.24	
4.28/2.24	   GOPHER_IN_GA(cons(cons(U, V), W), X) -> GOPHER_IN_GA(cons(U, cons(V, W)), X)
4.28/2.24	
4.28/2.24	The TRS R consists of the following rules:
4.28/2.24	
4.28/2.24	   gopher_in_ga(nil, nil) -> gopher_out_ga(nil, nil)
4.28/2.24	   gopher_in_ga(cons(nil, Y), cons(nil, Y)) -> gopher_out_ga(cons(nil, Y), cons(nil, Y))
4.28/2.24	   gopher_in_ga(cons(cons(U, V), W), X) -> U1_ga(U, V, W, X, gopher_in_ga(cons(U, cons(V, W)), X))
4.28/2.24	   U1_ga(U, V, W, X, gopher_out_ga(cons(U, cons(V, W)), X)) -> gopher_out_ga(cons(cons(U, V), W), X)
4.28/2.24	
4.28/2.24	The argument filtering Pi contains the following mapping:
4.28/2.24	gopher_in_ga(x1, x2)  =  gopher_in_ga(x1)
4.28/2.24	
4.28/2.24	nil  =  nil
4.28/2.24	
4.28/2.24	gopher_out_ga(x1, x2)  =  gopher_out_ga(x1, x2)
4.28/2.24	
4.28/2.24	cons(x1, x2)  =  cons(x1, x2)
4.28/2.24	
4.28/2.24	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x3, x5)
4.28/2.24	
4.28/2.24	GOPHER_IN_GA(x1, x2)  =  GOPHER_IN_GA(x1)
4.28/2.24	
4.28/2.24	
4.28/2.24	We have to consider all (P,R,Pi)-chains
4.28/2.24	----------------------------------------
4.28/2.24	
4.28/2.24	(7) UsableRulesProof (EQUIVALENT)
4.28/2.24	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
4.28/2.24	----------------------------------------
4.28/2.24	
4.28/2.24	(8)
4.28/2.24	Obligation:
4.28/2.24	Pi DP problem:
4.28/2.24	The TRS P consists of the following rules:
4.28/2.24	
4.28/2.24	   GOPHER_IN_GA(cons(cons(U, V), W), X) -> GOPHER_IN_GA(cons(U, cons(V, W)), X)
4.28/2.24	
4.28/2.24	R is empty.
4.28/2.24	The argument filtering Pi contains the following mapping:
4.28/2.24	cons(x1, x2)  =  cons(x1, x2)
4.28/2.24	
4.28/2.24	GOPHER_IN_GA(x1, x2)  =  GOPHER_IN_GA(x1)
4.28/2.24	
4.28/2.24	
4.28/2.24	We have to consider all (P,R,Pi)-chains
4.28/2.24	----------------------------------------
4.28/2.24	
4.28/2.24	(9) PiDPToQDPProof (SOUND)
4.28/2.24	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
4.28/2.24	----------------------------------------
4.28/2.24	
4.28/2.24	(10)
4.28/2.24	Obligation:
4.28/2.24	Q DP problem:
4.28/2.24	The TRS P consists of the following rules:
4.28/2.24	
4.28/2.24	   GOPHER_IN_GA(cons(cons(U, V), W)) -> GOPHER_IN_GA(cons(U, cons(V, W)))
4.28/2.24	
4.28/2.24	R is empty.
4.28/2.24	Q is empty.
4.28/2.24	We have to consider all (P,Q,R)-chains.
4.28/2.24	----------------------------------------
4.28/2.24	
4.28/2.24	(11) MRRProof (EQUIVALENT)
4.28/2.24	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
4.28/2.24	
4.28/2.24	Strictly oriented dependency pairs:
4.28/2.24	
4.28/2.24	   GOPHER_IN_GA(cons(cons(U, V), W)) -> GOPHER_IN_GA(cons(U, cons(V, W)))
4.28/2.24	
4.28/2.24	
4.28/2.24	Used ordering: Polynomial interpretation [POLO]:
4.28/2.24	
4.28/2.24	   POL(GOPHER_IN_GA(x_1)) = 2*x_1
4.28/2.24	   POL(cons(x_1, x_2)) = 2 + 2*x_1 + x_2
4.28/2.24	
4.28/2.24	
4.28/2.24	----------------------------------------
4.28/2.24	
4.28/2.24	(12)
4.28/2.24	Obligation:
4.28/2.24	Q DP problem:
4.28/2.24	P is empty.
4.28/2.24	R is empty.
4.28/2.24	Q is empty.
4.28/2.24	We have to consider all (P,Q,R)-chains.
4.28/2.24	----------------------------------------
4.28/2.24	
4.28/2.24	(13) PisEmptyProof (EQUIVALENT)
4.28/2.24	The TRS P is empty. Hence, there is no (P,Q,R) chain.
4.28/2.24	----------------------------------------
4.28/2.24	
4.28/2.24	(14)
4.28/2.24	YES
4.28/2.27	EOF