3.62/1.76	YES
3.62/1.77	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.62/1.77	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.62/1.77	
3.62/1.77	
3.62/1.77	Left Termination of the query pattern
3.62/1.77	
3.62/1.77	sum(a,g,a)
3.62/1.77	
3.62/1.77	w.r.t. the given Prolog program could successfully be proven:
3.62/1.77	
3.62/1.77	(0) Prolog
3.62/1.77	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.62/1.77	(2) PiTRS
3.62/1.77	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.62/1.77	(4) PiDP
3.62/1.77	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.62/1.77	(6) PiDP
3.62/1.77	(7) UsableRulesProof [EQUIVALENT, 0 ms]
3.62/1.77	(8) PiDP
3.62/1.77	(9) PiDPToQDPProof [SOUND, 7 ms]
3.62/1.77	(10) QDP
3.62/1.77	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.62/1.77	(12) YES
3.62/1.77	
3.62/1.77	
3.62/1.77	----------------------------------------
3.62/1.77	
3.62/1.77	(0)
3.62/1.77	Obligation:
3.62/1.77	Clauses:
3.62/1.77	
3.62/1.77	sum(X, 0, X).
3.62/1.77	sum(X, s(Y), s(Z)) :- sum(X, Y, Z).
3.62/1.77	
3.62/1.77	
3.62/1.77	Query: sum(a,g,a)
3.62/1.77	----------------------------------------
3.62/1.77	
3.62/1.77	(1) PrologToPiTRSProof (SOUND)
3.62/1.77	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.62/1.77	
3.62/1.77	sum_in_3: (f,b,f)
3.62/1.77	
3.62/1.77	Transforming Prolog into the following Term Rewriting System:
3.62/1.77	
3.62/1.77	Pi-finite rewrite system:
3.62/1.77	The TRS R consists of the following rules:
3.62/1.77	
3.62/1.77	   sum_in_aga(X, 0, X) -> sum_out_aga(X, 0, X)
3.62/1.77	   sum_in_aga(X, s(Y), s(Z)) -> U1_aga(X, Y, Z, sum_in_aga(X, Y, Z))
3.62/1.77	   U1_aga(X, Y, Z, sum_out_aga(X, Y, Z)) -> sum_out_aga(X, s(Y), s(Z))
3.62/1.77	
3.62/1.77	The argument filtering Pi contains the following mapping:
3.62/1.77	sum_in_aga(x1, x2, x3)  =  sum_in_aga(x2)
3.62/1.77	
3.62/1.77	0  =  0
3.62/1.77	
3.62/1.77	sum_out_aga(x1, x2, x3)  =  sum_out_aga
3.62/1.77	
3.62/1.77	s(x1)  =  s(x1)
3.62/1.77	
3.62/1.77	U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
3.62/1.77	
3.62/1.77	
3.62/1.77	
3.62/1.77	
3.62/1.77	
3.62/1.77	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.62/1.77	
3.62/1.77	
3.62/1.77	
3.62/1.77	----------------------------------------
3.62/1.77	
3.62/1.77	(2)
3.62/1.77	Obligation:
3.62/1.77	Pi-finite rewrite system:
3.62/1.77	The TRS R consists of the following rules:
3.62/1.77	
3.62/1.77	   sum_in_aga(X, 0, X) -> sum_out_aga(X, 0, X)
3.62/1.77	   sum_in_aga(X, s(Y), s(Z)) -> U1_aga(X, Y, Z, sum_in_aga(X, Y, Z))
3.62/1.77	   U1_aga(X, Y, Z, sum_out_aga(X, Y, Z)) -> sum_out_aga(X, s(Y), s(Z))
3.62/1.77	
3.62/1.77	The argument filtering Pi contains the following mapping:
3.62/1.77	sum_in_aga(x1, x2, x3)  =  sum_in_aga(x2)
3.62/1.77	
3.62/1.77	0  =  0
3.62/1.77	
3.62/1.77	sum_out_aga(x1, x2, x3)  =  sum_out_aga
3.62/1.77	
3.62/1.77	s(x1)  =  s(x1)
3.62/1.77	
3.62/1.77	U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
3.62/1.77	
3.62/1.77	
3.62/1.77	
3.62/1.77	----------------------------------------
3.62/1.77	
3.62/1.77	(3) DependencyPairsProof (EQUIVALENT)
3.62/1.77	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.62/1.77	Pi DP problem:
3.62/1.77	The TRS P consists of the following rules:
3.62/1.77	
3.62/1.77	   SUM_IN_AGA(X, s(Y), s(Z)) -> U1_AGA(X, Y, Z, sum_in_aga(X, Y, Z))
3.62/1.77	   SUM_IN_AGA(X, s(Y), s(Z)) -> SUM_IN_AGA(X, Y, Z)
3.62/1.77	
3.62/1.77	The TRS R consists of the following rules:
3.62/1.77	
3.62/1.77	   sum_in_aga(X, 0, X) -> sum_out_aga(X, 0, X)
3.62/1.77	   sum_in_aga(X, s(Y), s(Z)) -> U1_aga(X, Y, Z, sum_in_aga(X, Y, Z))
3.62/1.77	   U1_aga(X, Y, Z, sum_out_aga(X, Y, Z)) -> sum_out_aga(X, s(Y), s(Z))
3.62/1.77	
3.62/1.77	The argument filtering Pi contains the following mapping:
3.62/1.77	sum_in_aga(x1, x2, x3)  =  sum_in_aga(x2)
3.62/1.77	
3.62/1.77	0  =  0
3.62/1.77	
3.62/1.77	sum_out_aga(x1, x2, x3)  =  sum_out_aga
3.62/1.77	
3.62/1.77	s(x1)  =  s(x1)
3.62/1.77	
3.62/1.77	U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
3.62/1.77	
3.62/1.77	SUM_IN_AGA(x1, x2, x3)  =  SUM_IN_AGA(x2)
3.62/1.77	
3.62/1.77	U1_AGA(x1, x2, x3, x4)  =  U1_AGA(x4)
3.62/1.77	
3.62/1.77	
3.62/1.77	We have to consider all (P,R,Pi)-chains
3.62/1.77	----------------------------------------
3.62/1.77	
3.62/1.77	(4)
3.62/1.77	Obligation:
3.62/1.77	Pi DP problem:
3.62/1.77	The TRS P consists of the following rules:
3.62/1.77	
3.62/1.77	   SUM_IN_AGA(X, s(Y), s(Z)) -> U1_AGA(X, Y, Z, sum_in_aga(X, Y, Z))
3.62/1.77	   SUM_IN_AGA(X, s(Y), s(Z)) -> SUM_IN_AGA(X, Y, Z)
3.62/1.77	
3.62/1.77	The TRS R consists of the following rules:
3.62/1.77	
3.62/1.77	   sum_in_aga(X, 0, X) -> sum_out_aga(X, 0, X)
3.62/1.77	   sum_in_aga(X, s(Y), s(Z)) -> U1_aga(X, Y, Z, sum_in_aga(X, Y, Z))
3.62/1.77	   U1_aga(X, Y, Z, sum_out_aga(X, Y, Z)) -> sum_out_aga(X, s(Y), s(Z))
3.62/1.77	
3.62/1.77	The argument filtering Pi contains the following mapping:
3.62/1.77	sum_in_aga(x1, x2, x3)  =  sum_in_aga(x2)
3.62/1.77	
3.62/1.77	0  =  0
3.62/1.77	
3.62/1.77	sum_out_aga(x1, x2, x3)  =  sum_out_aga
3.62/1.77	
3.62/1.77	s(x1)  =  s(x1)
3.62/1.77	
3.62/1.77	U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
3.62/1.77	
3.62/1.77	SUM_IN_AGA(x1, x2, x3)  =  SUM_IN_AGA(x2)
3.62/1.77	
3.62/1.77	U1_AGA(x1, x2, x3, x4)  =  U1_AGA(x4)
3.62/1.77	
3.62/1.77	
3.62/1.77	We have to consider all (P,R,Pi)-chains
3.62/1.77	----------------------------------------
3.62/1.77	
3.62/1.77	(5) DependencyGraphProof (EQUIVALENT)
3.62/1.77	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
3.62/1.77	----------------------------------------
3.62/1.77	
3.62/1.77	(6)
3.62/1.77	Obligation:
3.62/1.77	Pi DP problem:
3.62/1.77	The TRS P consists of the following rules:
3.62/1.77	
3.62/1.77	   SUM_IN_AGA(X, s(Y), s(Z)) -> SUM_IN_AGA(X, Y, Z)
3.62/1.77	
3.62/1.77	The TRS R consists of the following rules:
3.62/1.77	
3.62/1.77	   sum_in_aga(X, 0, X) -> sum_out_aga(X, 0, X)
3.62/1.77	   sum_in_aga(X, s(Y), s(Z)) -> U1_aga(X, Y, Z, sum_in_aga(X, Y, Z))
3.62/1.77	   U1_aga(X, Y, Z, sum_out_aga(X, Y, Z)) -> sum_out_aga(X, s(Y), s(Z))
3.62/1.77	
3.62/1.77	The argument filtering Pi contains the following mapping:
3.62/1.77	sum_in_aga(x1, x2, x3)  =  sum_in_aga(x2)
3.62/1.77	
3.62/1.77	0  =  0
3.62/1.77	
3.62/1.77	sum_out_aga(x1, x2, x3)  =  sum_out_aga
3.62/1.77	
3.62/1.77	s(x1)  =  s(x1)
3.62/1.77	
3.62/1.77	U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
3.62/1.77	
3.62/1.77	SUM_IN_AGA(x1, x2, x3)  =  SUM_IN_AGA(x2)
3.62/1.77	
3.62/1.77	
3.62/1.77	We have to consider all (P,R,Pi)-chains
3.62/1.77	----------------------------------------
3.62/1.77	
3.62/1.77	(7) UsableRulesProof (EQUIVALENT)
3.62/1.77	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.62/1.77	----------------------------------------
3.62/1.77	
3.62/1.77	(8)
3.62/1.77	Obligation:
3.62/1.77	Pi DP problem:
3.62/1.77	The TRS P consists of the following rules:
3.62/1.77	
3.62/1.77	   SUM_IN_AGA(X, s(Y), s(Z)) -> SUM_IN_AGA(X, Y, Z)
3.62/1.77	
3.62/1.77	R is empty.
3.62/1.77	The argument filtering Pi contains the following mapping:
3.62/1.77	s(x1)  =  s(x1)
3.62/1.77	
3.62/1.77	SUM_IN_AGA(x1, x2, x3)  =  SUM_IN_AGA(x2)
3.62/1.77	
3.62/1.77	
3.62/1.77	We have to consider all (P,R,Pi)-chains
3.62/1.77	----------------------------------------
3.62/1.77	
3.62/1.77	(9) PiDPToQDPProof (SOUND)
3.62/1.77	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.62/1.77	----------------------------------------
3.62/1.77	
3.62/1.77	(10)
3.62/1.77	Obligation:
3.62/1.77	Q DP problem:
3.62/1.77	The TRS P consists of the following rules:
3.62/1.77	
3.62/1.77	   SUM_IN_AGA(s(Y)) -> SUM_IN_AGA(Y)
3.62/1.77	
3.62/1.77	R is empty.
3.62/1.77	Q is empty.
3.62/1.77	We have to consider all (P,Q,R)-chains.
3.62/1.77	----------------------------------------
3.62/1.77	
3.62/1.77	(11) QDPSizeChangeProof (EQUIVALENT)
3.62/1.77	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. 
3.62/1.77	
3.62/1.77	From the DPs we obtained the following set of size-change graphs:
3.62/1.77	*SUM_IN_AGA(s(Y)) -> SUM_IN_AGA(Y)
3.62/1.77	The graph contains the following edges 1 > 1
3.62/1.77	
3.62/1.77	
3.62/1.77	----------------------------------------
3.62/1.77	
3.62/1.77	(12)
3.62/1.77	YES
3.62/1.80	EOF