3.46/1.69	YES
3.46/1.69	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.46/1.69	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.46/1.69	
3.46/1.69	
3.46/1.69	Left Termination of the query pattern
3.46/1.69	
3.46/1.69	minimum(g,a)
3.46/1.69	
3.46/1.69	w.r.t. the given Prolog program could successfully be proven:
3.46/1.69	
3.46/1.69	(0) Prolog
3.46/1.69	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.46/1.69	(2) PiTRS
3.46/1.69	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.46/1.69	(4) PiDP
3.46/1.69	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.46/1.69	(6) PiDP
3.46/1.69	(7) UsableRulesProof [EQUIVALENT, 0 ms]
3.46/1.69	(8) PiDP
3.46/1.69	(9) PiDPToQDPProof [SOUND, 0 ms]
3.46/1.69	(10) QDP
3.46/1.69	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.46/1.69	(12) YES
3.46/1.69	
3.46/1.69	
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(0)
3.46/1.69	Obligation:
3.46/1.69	Clauses:
3.46/1.69	
3.46/1.69	minimum(tree(X, void, X1), X).
3.46/1.69	minimum(tree(X2, Left, X3), X) :- minimum(Left, X).
3.46/1.69	
3.46/1.69	
3.46/1.69	Query: minimum(g,a)
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(1) PrologToPiTRSProof (SOUND)
3.46/1.69	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.46/1.69	
3.46/1.69	minimum_in_2: (b,f)
3.46/1.69	
3.46/1.69	Transforming Prolog into the following Term Rewriting System:
3.46/1.69	
3.46/1.69	Pi-finite rewrite system:
3.46/1.69	The TRS R consists of the following rules:
3.46/1.69	
3.46/1.69	   minimum_in_ga(tree(X, void, X1), X) -> minimum_out_ga(tree(X, void, X1), X)
3.46/1.69	   minimum_in_ga(tree(X2, Left, X3), X) -> U1_ga(X2, Left, X3, X, minimum_in_ga(Left, X))
3.46/1.69	   U1_ga(X2, Left, X3, X, minimum_out_ga(Left, X)) -> minimum_out_ga(tree(X2, Left, X3), X)
3.46/1.69	
3.46/1.69	The argument filtering Pi contains the following mapping:
3.46/1.69	minimum_in_ga(x1, x2)  =  minimum_in_ga(x1)
3.46/1.69	
3.46/1.69	tree(x1, x2, x3)  =  tree(x1, x2, x3)
3.46/1.69	
3.46/1.69	void  =  void
3.46/1.69	
3.46/1.69	minimum_out_ga(x1, x2)  =  minimum_out_ga(x2)
3.46/1.69	
3.46/1.69	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
3.46/1.69	
3.46/1.69	
3.46/1.69	
3.46/1.69	
3.46/1.69	
3.46/1.69	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.46/1.69	
3.46/1.69	
3.46/1.69	
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(2)
3.46/1.69	Obligation:
3.46/1.69	Pi-finite rewrite system:
3.46/1.69	The TRS R consists of the following rules:
3.46/1.69	
3.46/1.69	   minimum_in_ga(tree(X, void, X1), X) -> minimum_out_ga(tree(X, void, X1), X)
3.46/1.69	   minimum_in_ga(tree(X2, Left, X3), X) -> U1_ga(X2, Left, X3, X, minimum_in_ga(Left, X))
3.46/1.69	   U1_ga(X2, Left, X3, X, minimum_out_ga(Left, X)) -> minimum_out_ga(tree(X2, Left, X3), X)
3.46/1.69	
3.46/1.69	The argument filtering Pi contains the following mapping:
3.46/1.69	minimum_in_ga(x1, x2)  =  minimum_in_ga(x1)
3.46/1.69	
3.46/1.69	tree(x1, x2, x3)  =  tree(x1, x2, x3)
3.46/1.69	
3.46/1.69	void  =  void
3.46/1.69	
3.46/1.69	minimum_out_ga(x1, x2)  =  minimum_out_ga(x2)
3.46/1.69	
3.46/1.69	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
3.46/1.69	
3.46/1.69	
3.46/1.69	
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(3) DependencyPairsProof (EQUIVALENT)
3.46/1.69	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.46/1.69	Pi DP problem:
3.46/1.69	The TRS P consists of the following rules:
3.46/1.69	
3.46/1.69	   MINIMUM_IN_GA(tree(X2, Left, X3), X) -> U1_GA(X2, Left, X3, X, minimum_in_ga(Left, X))
3.46/1.69	   MINIMUM_IN_GA(tree(X2, Left, X3), X) -> MINIMUM_IN_GA(Left, X)
3.46/1.69	
3.46/1.69	The TRS R consists of the following rules:
3.46/1.69	
3.46/1.69	   minimum_in_ga(tree(X, void, X1), X) -> minimum_out_ga(tree(X, void, X1), X)
3.46/1.69	   minimum_in_ga(tree(X2, Left, X3), X) -> U1_ga(X2, Left, X3, X, minimum_in_ga(Left, X))
3.46/1.69	   U1_ga(X2, Left, X3, X, minimum_out_ga(Left, X)) -> minimum_out_ga(tree(X2, Left, X3), X)
3.46/1.69	
3.46/1.69	The argument filtering Pi contains the following mapping:
3.46/1.69	minimum_in_ga(x1, x2)  =  minimum_in_ga(x1)
3.46/1.69	
3.46/1.69	tree(x1, x2, x3)  =  tree(x1, x2, x3)
3.46/1.69	
3.46/1.69	void  =  void
3.46/1.69	
3.46/1.69	minimum_out_ga(x1, x2)  =  minimum_out_ga(x2)
3.46/1.69	
3.46/1.69	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
3.46/1.69	
3.46/1.69	MINIMUM_IN_GA(x1, x2)  =  MINIMUM_IN_GA(x1)
3.46/1.69	
3.46/1.69	U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
3.46/1.69	
3.46/1.69	
3.46/1.69	We have to consider all (P,R,Pi)-chains
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(4)
3.46/1.69	Obligation:
3.46/1.69	Pi DP problem:
3.46/1.69	The TRS P consists of the following rules:
3.46/1.69	
3.46/1.69	   MINIMUM_IN_GA(tree(X2, Left, X3), X) -> U1_GA(X2, Left, X3, X, minimum_in_ga(Left, X))
3.46/1.69	   MINIMUM_IN_GA(tree(X2, Left, X3), X) -> MINIMUM_IN_GA(Left, X)
3.46/1.69	
3.46/1.69	The TRS R consists of the following rules:
3.46/1.69	
3.46/1.69	   minimum_in_ga(tree(X, void, X1), X) -> minimum_out_ga(tree(X, void, X1), X)
3.46/1.69	   minimum_in_ga(tree(X2, Left, X3), X) -> U1_ga(X2, Left, X3, X, minimum_in_ga(Left, X))
3.46/1.69	   U1_ga(X2, Left, X3, X, minimum_out_ga(Left, X)) -> minimum_out_ga(tree(X2, Left, X3), X)
3.46/1.69	
3.46/1.69	The argument filtering Pi contains the following mapping:
3.46/1.69	minimum_in_ga(x1, x2)  =  minimum_in_ga(x1)
3.46/1.69	
3.46/1.69	tree(x1, x2, x3)  =  tree(x1, x2, x3)
3.46/1.69	
3.46/1.69	void  =  void
3.46/1.69	
3.46/1.69	minimum_out_ga(x1, x2)  =  minimum_out_ga(x2)
3.46/1.69	
3.46/1.69	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
3.46/1.69	
3.46/1.69	MINIMUM_IN_GA(x1, x2)  =  MINIMUM_IN_GA(x1)
3.46/1.69	
3.46/1.69	U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
3.46/1.69	
3.46/1.69	
3.46/1.69	We have to consider all (P,R,Pi)-chains
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(5) DependencyGraphProof (EQUIVALENT)
3.46/1.69	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(6)
3.46/1.69	Obligation:
3.46/1.69	Pi DP problem:
3.46/1.69	The TRS P consists of the following rules:
3.46/1.69	
3.46/1.69	   MINIMUM_IN_GA(tree(X2, Left, X3), X) -> MINIMUM_IN_GA(Left, X)
3.46/1.69	
3.46/1.69	The TRS R consists of the following rules:
3.46/1.69	
3.46/1.69	   minimum_in_ga(tree(X, void, X1), X) -> minimum_out_ga(tree(X, void, X1), X)
3.46/1.69	   minimum_in_ga(tree(X2, Left, X3), X) -> U1_ga(X2, Left, X3, X, minimum_in_ga(Left, X))
3.46/1.69	   U1_ga(X2, Left, X3, X, minimum_out_ga(Left, X)) -> minimum_out_ga(tree(X2, Left, X3), X)
3.46/1.69	
3.46/1.69	The argument filtering Pi contains the following mapping:
3.46/1.69	minimum_in_ga(x1, x2)  =  minimum_in_ga(x1)
3.46/1.69	
3.46/1.69	tree(x1, x2, x3)  =  tree(x1, x2, x3)
3.46/1.69	
3.46/1.69	void  =  void
3.46/1.69	
3.46/1.69	minimum_out_ga(x1, x2)  =  minimum_out_ga(x2)
3.46/1.69	
3.46/1.69	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
3.46/1.69	
3.46/1.69	MINIMUM_IN_GA(x1, x2)  =  MINIMUM_IN_GA(x1)
3.46/1.69	
3.46/1.69	
3.46/1.69	We have to consider all (P,R,Pi)-chains
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(7) UsableRulesProof (EQUIVALENT)
3.46/1.69	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(8)
3.46/1.69	Obligation:
3.46/1.69	Pi DP problem:
3.46/1.69	The TRS P consists of the following rules:
3.46/1.69	
3.46/1.69	   MINIMUM_IN_GA(tree(X2, Left, X3), X) -> MINIMUM_IN_GA(Left, X)
3.46/1.69	
3.46/1.69	R is empty.
3.46/1.69	The argument filtering Pi contains the following mapping:
3.46/1.69	tree(x1, x2, x3)  =  tree(x1, x2, x3)
3.46/1.69	
3.46/1.69	MINIMUM_IN_GA(x1, x2)  =  MINIMUM_IN_GA(x1)
3.46/1.69	
3.46/1.69	
3.46/1.69	We have to consider all (P,R,Pi)-chains
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(9) PiDPToQDPProof (SOUND)
3.46/1.69	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.46/1.69	----------------------------------------
3.46/1.69	
3.46/1.69	(10)
3.46/1.69	Obligation:
3.69/1.69	Q DP problem:
3.69/1.69	The TRS P consists of the following rules:
3.69/1.69	
3.69/1.69	   MINIMUM_IN_GA(tree(X2, Left, X3)) -> MINIMUM_IN_GA(Left)
3.69/1.69	
3.69/1.69	R is empty.
3.69/1.69	Q is empty.
3.69/1.69	We have to consider all (P,Q,R)-chains.
3.69/1.69	----------------------------------------
3.69/1.69	
3.69/1.69	(11) QDPSizeChangeProof (EQUIVALENT)
3.69/1.69	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. 
3.69/1.69	
3.69/1.69	From the DPs we obtained the following set of size-change graphs:
3.69/1.69	*MINIMUM_IN_GA(tree(X2, Left, X3)) -> MINIMUM_IN_GA(Left)
3.69/1.69	The graph contains the following edges 1 > 1
3.69/1.69	
3.69/1.69	
3.69/1.69	----------------------------------------
3.69/1.69	
3.69/1.69	(12)
3.69/1.69	YES
3.69/1.73	EOF