3.67/1.80	YES
3.67/1.81	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.67/1.81	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.67/1.81	
3.67/1.81	
3.67/1.81	Left Termination of the query pattern
3.67/1.81	
3.67/1.81	avg(g,a,g)
3.67/1.81	
3.67/1.81	w.r.t. the given Prolog program could successfully be proven:
3.67/1.81	
3.67/1.81	(0) Prolog
3.67/1.81	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.67/1.81	(2) PiTRS
3.67/1.81	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.67/1.81	(4) PiDP
3.67/1.81	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.67/1.81	(6) PiDP
3.67/1.81	(7) UsableRulesProof [EQUIVALENT, 0 ms]
3.67/1.81	(8) PiDP
3.67/1.81	(9) PiDPToQDPProof [SOUND, 0 ms]
3.67/1.81	(10) QDP
3.67/1.81	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.67/1.81	(12) YES
3.67/1.81	
3.67/1.81	
3.67/1.81	----------------------------------------
3.67/1.81	
3.67/1.81	(0)
3.67/1.81	Obligation:
3.67/1.81	Clauses:
3.67/1.81	
3.67/1.81	avg(s(X), Y, Z) :- avg(X, s(Y), Z).
3.67/1.81	avg(X, s(s(s(Y))), s(Z)) :- avg(s(X), Y, Z).
3.67/1.81	avg(0, 0, 0).
3.67/1.81	avg(0, s(0), 0).
3.67/1.81	avg(0, s(s(0)), s(0)).
3.67/1.81	
3.67/1.81	
3.67/1.81	Query: avg(g,a,g)
3.67/1.81	----------------------------------------
3.67/1.81	
3.67/1.81	(1) PrologToPiTRSProof (SOUND)
3.67/1.81	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.67/1.81	
3.67/1.81	avg_in_3: (b,f,b)
3.67/1.81	
3.67/1.81	Transforming Prolog into the following Term Rewriting System:
3.67/1.81	
3.67/1.81	Pi-finite rewrite system:
3.67/1.81	The TRS R consists of the following rules:
3.67/1.81	
3.67/1.81	   avg_in_gag(s(X), Y, Z) -> U1_gag(X, Y, Z, avg_in_gag(X, s(Y), Z))
3.67/1.81	   avg_in_gag(X, s(s(s(Y))), s(Z)) -> U2_gag(X, Y, Z, avg_in_gag(s(X), Y, Z))
3.67/1.81	   avg_in_gag(0, 0, 0) -> avg_out_gag(0, 0, 0)
3.67/1.81	   avg_in_gag(0, s(0), 0) -> avg_out_gag(0, s(0), 0)
3.67/1.81	   avg_in_gag(0, s(s(0)), s(0)) -> avg_out_gag(0, s(s(0)), s(0))
3.67/1.81	   U2_gag(X, Y, Z, avg_out_gag(s(X), Y, Z)) -> avg_out_gag(X, s(s(s(Y))), s(Z))
3.67/1.81	   U1_gag(X, Y, Z, avg_out_gag(X, s(Y), Z)) -> avg_out_gag(s(X), Y, Z)
3.67/1.81	
3.67/1.81	The argument filtering Pi contains the following mapping:
3.67/1.81	avg_in_gag(x1, x2, x3)  =  avg_in_gag(x1, x3)
3.67/1.81	
3.67/1.81	s(x1)  =  s(x1)
3.67/1.81	
3.67/1.81	U1_gag(x1, x2, x3, x4)  =  U1_gag(x4)
3.67/1.81	
3.67/1.81	U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
3.67/1.81	
3.67/1.81	0  =  0
3.67/1.81	
3.67/1.81	avg_out_gag(x1, x2, x3)  =  avg_out_gag(x2)
3.67/1.81	
3.67/1.81	
3.67/1.81	
3.67/1.81	
3.67/1.81	
3.67/1.81	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.67/1.81	
3.67/1.81	
3.67/1.81	
3.67/1.81	----------------------------------------
3.67/1.81	
3.67/1.81	(2)
3.67/1.81	Obligation:
3.67/1.81	Pi-finite rewrite system:
3.67/1.81	The TRS R consists of the following rules:
3.67/1.81	
3.67/1.81	   avg_in_gag(s(X), Y, Z) -> U1_gag(X, Y, Z, avg_in_gag(X, s(Y), Z))
3.67/1.81	   avg_in_gag(X, s(s(s(Y))), s(Z)) -> U2_gag(X, Y, Z, avg_in_gag(s(X), Y, Z))
3.67/1.81	   avg_in_gag(0, 0, 0) -> avg_out_gag(0, 0, 0)
3.67/1.81	   avg_in_gag(0, s(0), 0) -> avg_out_gag(0, s(0), 0)
3.67/1.81	   avg_in_gag(0, s(s(0)), s(0)) -> avg_out_gag(0, s(s(0)), s(0))
3.67/1.81	   U2_gag(X, Y, Z, avg_out_gag(s(X), Y, Z)) -> avg_out_gag(X, s(s(s(Y))), s(Z))
3.67/1.81	   U1_gag(X, Y, Z, avg_out_gag(X, s(Y), Z)) -> avg_out_gag(s(X), Y, Z)
3.67/1.81	
3.67/1.81	The argument filtering Pi contains the following mapping:
3.67/1.81	avg_in_gag(x1, x2, x3)  =  avg_in_gag(x1, x3)
3.67/1.81	
3.67/1.81	s(x1)  =  s(x1)
3.67/1.81	
3.67/1.81	U1_gag(x1, x2, x3, x4)  =  U1_gag(x4)
3.67/1.81	
3.67/1.81	U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
3.67/1.81	
3.67/1.81	0  =  0
3.67/1.81	
3.67/1.81	avg_out_gag(x1, x2, x3)  =  avg_out_gag(x2)
3.67/1.81	
3.67/1.81	
3.67/1.81	
3.67/1.81	----------------------------------------
3.67/1.81	
3.67/1.81	(3) DependencyPairsProof (EQUIVALENT)
3.67/1.81	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.67/1.81	Pi DP problem:
3.67/1.81	The TRS P consists of the following rules:
3.67/1.81	
3.67/1.81	   AVG_IN_GAG(s(X), Y, Z) -> U1_GAG(X, Y, Z, avg_in_gag(X, s(Y), Z))
3.67/1.81	   AVG_IN_GAG(s(X), Y, Z) -> AVG_IN_GAG(X, s(Y), Z)
3.67/1.81	   AVG_IN_GAG(X, s(s(s(Y))), s(Z)) -> U2_GAG(X, Y, Z, avg_in_gag(s(X), Y, Z))
3.67/1.81	   AVG_IN_GAG(X, s(s(s(Y))), s(Z)) -> AVG_IN_GAG(s(X), Y, Z)
3.67/1.81	
3.67/1.81	The TRS R consists of the following rules:
3.67/1.81	
3.67/1.81	   avg_in_gag(s(X), Y, Z) -> U1_gag(X, Y, Z, avg_in_gag(X, s(Y), Z))
3.67/1.81	   avg_in_gag(X, s(s(s(Y))), s(Z)) -> U2_gag(X, Y, Z, avg_in_gag(s(X), Y, Z))
3.67/1.81	   avg_in_gag(0, 0, 0) -> avg_out_gag(0, 0, 0)
3.67/1.81	   avg_in_gag(0, s(0), 0) -> avg_out_gag(0, s(0), 0)
3.67/1.81	   avg_in_gag(0, s(s(0)), s(0)) -> avg_out_gag(0, s(s(0)), s(0))
3.67/1.81	   U2_gag(X, Y, Z, avg_out_gag(s(X), Y, Z)) -> avg_out_gag(X, s(s(s(Y))), s(Z))
3.67/1.81	   U1_gag(X, Y, Z, avg_out_gag(X, s(Y), Z)) -> avg_out_gag(s(X), Y, Z)
3.67/1.81	
3.67/1.81	The argument filtering Pi contains the following mapping:
3.67/1.81	avg_in_gag(x1, x2, x3)  =  avg_in_gag(x1, x3)
3.67/1.81	
3.67/1.81	s(x1)  =  s(x1)
3.67/1.81	
3.67/1.81	U1_gag(x1, x2, x3, x4)  =  U1_gag(x4)
3.67/1.81	
3.67/1.81	U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
3.67/1.81	
3.67/1.81	0  =  0
3.67/1.81	
3.67/1.81	avg_out_gag(x1, x2, x3)  =  avg_out_gag(x2)
3.67/1.81	
3.67/1.81	AVG_IN_GAG(x1, x2, x3)  =  AVG_IN_GAG(x1, x3)
3.67/1.81	
3.67/1.81	U1_GAG(x1, x2, x3, x4)  =  U1_GAG(x4)
3.67/1.81	
3.67/1.81	U2_GAG(x1, x2, x3, x4)  =  U2_GAG(x4)
3.67/1.81	
3.67/1.81	
3.67/1.81	We have to consider all (P,R,Pi)-chains
3.67/1.81	----------------------------------------
3.67/1.81	
3.67/1.81	(4)
3.67/1.81	Obligation:
3.67/1.81	Pi DP problem:
3.67/1.81	The TRS P consists of the following rules:
3.67/1.81	
3.67/1.81	   AVG_IN_GAG(s(X), Y, Z) -> U1_GAG(X, Y, Z, avg_in_gag(X, s(Y), Z))
3.67/1.81	   AVG_IN_GAG(s(X), Y, Z) -> AVG_IN_GAG(X, s(Y), Z)
3.67/1.81	   AVG_IN_GAG(X, s(s(s(Y))), s(Z)) -> U2_GAG(X, Y, Z, avg_in_gag(s(X), Y, Z))
3.67/1.81	   AVG_IN_GAG(X, s(s(s(Y))), s(Z)) -> AVG_IN_GAG(s(X), Y, Z)
3.67/1.81	
3.67/1.81	The TRS R consists of the following rules:
3.67/1.81	
3.67/1.81	   avg_in_gag(s(X), Y, Z) -> U1_gag(X, Y, Z, avg_in_gag(X, s(Y), Z))
3.67/1.81	   avg_in_gag(X, s(s(s(Y))), s(Z)) -> U2_gag(X, Y, Z, avg_in_gag(s(X), Y, Z))
3.67/1.81	   avg_in_gag(0, 0, 0) -> avg_out_gag(0, 0, 0)
3.67/1.81	   avg_in_gag(0, s(0), 0) -> avg_out_gag(0, s(0), 0)
3.67/1.81	   avg_in_gag(0, s(s(0)), s(0)) -> avg_out_gag(0, s(s(0)), s(0))
3.67/1.81	   U2_gag(X, Y, Z, avg_out_gag(s(X), Y, Z)) -> avg_out_gag(X, s(s(s(Y))), s(Z))
3.67/1.81	   U1_gag(X, Y, Z, avg_out_gag(X, s(Y), Z)) -> avg_out_gag(s(X), Y, Z)
3.67/1.81	
3.67/1.81	The argument filtering Pi contains the following mapping:
3.67/1.81	avg_in_gag(x1, x2, x3)  =  avg_in_gag(x1, x3)
3.67/1.81	
3.67/1.81	s(x1)  =  s(x1)
3.67/1.81	
3.67/1.81	U1_gag(x1, x2, x3, x4)  =  U1_gag(x4)
3.67/1.81	
3.67/1.81	U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
3.67/1.81	
3.67/1.81	0  =  0
3.67/1.81	
3.67/1.81	avg_out_gag(x1, x2, x3)  =  avg_out_gag(x2)
3.67/1.81	
3.67/1.81	AVG_IN_GAG(x1, x2, x3)  =  AVG_IN_GAG(x1, x3)
3.67/1.81	
3.67/1.81	U1_GAG(x1, x2, x3, x4)  =  U1_GAG(x4)
3.67/1.81	
3.67/1.81	U2_GAG(x1, x2, x3, x4)  =  U2_GAG(x4)
3.67/1.81	
3.67/1.81	
3.67/1.81	We have to consider all (P,R,Pi)-chains
3.67/1.81	----------------------------------------
3.67/1.81	
3.67/1.81	(5) DependencyGraphProof (EQUIVALENT)
3.67/1.81	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.
3.67/1.81	----------------------------------------
3.67/1.81	
3.67/1.81	(6)
3.67/1.81	Obligation:
3.67/1.81	Pi DP problem:
3.67/1.81	The TRS P consists of the following rules:
3.67/1.81	
3.67/1.81	   AVG_IN_GAG(X, s(s(s(Y))), s(Z)) -> AVG_IN_GAG(s(X), Y, Z)
3.67/1.81	   AVG_IN_GAG(s(X), Y, Z) -> AVG_IN_GAG(X, s(Y), Z)
3.67/1.81	
3.67/1.81	The TRS R consists of the following rules:
3.67/1.81	
3.67/1.81	   avg_in_gag(s(X), Y, Z) -> U1_gag(X, Y, Z, avg_in_gag(X, s(Y), Z))
3.67/1.81	   avg_in_gag(X, s(s(s(Y))), s(Z)) -> U2_gag(X, Y, Z, avg_in_gag(s(X), Y, Z))
3.67/1.81	   avg_in_gag(0, 0, 0) -> avg_out_gag(0, 0, 0)
3.67/1.81	   avg_in_gag(0, s(0), 0) -> avg_out_gag(0, s(0), 0)
3.67/1.81	   avg_in_gag(0, s(s(0)), s(0)) -> avg_out_gag(0, s(s(0)), s(0))
3.67/1.81	   U2_gag(X, Y, Z, avg_out_gag(s(X), Y, Z)) -> avg_out_gag(X, s(s(s(Y))), s(Z))
3.67/1.81	   U1_gag(X, Y, Z, avg_out_gag(X, s(Y), Z)) -> avg_out_gag(s(X), Y, Z)
3.67/1.81	
3.67/1.81	The argument filtering Pi contains the following mapping:
3.67/1.81	avg_in_gag(x1, x2, x3)  =  avg_in_gag(x1, x3)
3.67/1.81	
3.67/1.81	s(x1)  =  s(x1)
3.67/1.81	
3.67/1.81	U1_gag(x1, x2, x3, x4)  =  U1_gag(x4)
3.67/1.81	
3.67/1.81	U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
3.67/1.81	
3.67/1.81	0  =  0
3.67/1.81	
3.67/1.81	avg_out_gag(x1, x2, x3)  =  avg_out_gag(x2)
3.67/1.81	
3.67/1.81	AVG_IN_GAG(x1, x2, x3)  =  AVG_IN_GAG(x1, x3)
3.67/1.81	
3.67/1.81	
3.67/1.81	We have to consider all (P,R,Pi)-chains
3.67/1.81	----------------------------------------
3.67/1.81	
3.67/1.81	(7) UsableRulesProof (EQUIVALENT)
3.67/1.81	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.67/1.81	----------------------------------------
3.67/1.81	
3.67/1.81	(8)
3.67/1.81	Obligation:
3.67/1.81	Pi DP problem:
3.67/1.81	The TRS P consists of the following rules:
3.67/1.81	
3.67/1.81	   AVG_IN_GAG(X, s(s(s(Y))), s(Z)) -> AVG_IN_GAG(s(X), Y, Z)
3.67/1.81	   AVG_IN_GAG(s(X), Y, Z) -> AVG_IN_GAG(X, s(Y), Z)
3.67/1.81	
3.67/1.81	R is empty.
3.67/1.81	The argument filtering Pi contains the following mapping:
3.67/1.81	s(x1)  =  s(x1)
3.67/1.81	
3.67/1.81	AVG_IN_GAG(x1, x2, x3)  =  AVG_IN_GAG(x1, x3)
3.67/1.81	
3.67/1.81	
3.67/1.81	We have to consider all (P,R,Pi)-chains
3.67/1.81	----------------------------------------
3.67/1.81	
3.67/1.81	(9) PiDPToQDPProof (SOUND)
3.67/1.81	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.67/1.81	----------------------------------------
3.67/1.81	
3.67/1.81	(10)
3.67/1.81	Obligation:
3.67/1.81	Q DP problem:
3.67/1.81	The TRS P consists of the following rules:
3.67/1.81	
3.67/1.81	   AVG_IN_GAG(X, s(Z)) -> AVG_IN_GAG(s(X), Z)
3.67/1.81	   AVG_IN_GAG(s(X), Z) -> AVG_IN_GAG(X, Z)
3.67/1.81	
3.67/1.81	R is empty.
3.67/1.81	Q is empty.
3.67/1.81	We have to consider all (P,Q,R)-chains.
3.67/1.81	----------------------------------------
3.67/1.81	
3.67/1.81	(11) QDPSizeChangeProof (EQUIVALENT)
3.67/1.81	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. 
3.67/1.81	
3.67/1.81	From the DPs we obtained the following set of size-change graphs:
3.67/1.81	*AVG_IN_GAG(X, s(Z)) -> AVG_IN_GAG(s(X), Z)
3.67/1.81	The graph contains the following edges 2 > 2
3.67/1.81	
3.67/1.81	
3.67/1.81	*AVG_IN_GAG(s(X), Z) -> AVG_IN_GAG(X, Z)
3.67/1.81	The graph contains the following edges 1 > 1, 2 >= 2
3.67/1.81	
3.67/1.81	
3.67/1.81	----------------------------------------
3.67/1.81	
3.67/1.81	(12)
3.67/1.81	YES
3.85/1.85	EOF