5.42/2.68	MAYBE
5.71/2.70	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
5.71/2.70	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.71/2.70	
5.71/2.70	
5.71/2.70	Left Termination of the query pattern
5.71/2.70	
5.71/2.70	minimum(a,g)
5.71/2.70	
5.71/2.70	w.r.t. the given Prolog program could not be shown:
5.71/2.70	
5.71/2.70	(0) Prolog
5.71/2.70	(1) PrologToPiTRSProof [SOUND, 0 ms]
5.71/2.70	(2) PiTRS
5.71/2.70	    (3) DependencyPairsProof [EQUIVALENT, 0 ms]
5.71/2.70	    (4) PiDP
5.71/2.70	    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
5.71/2.70	    (6) PiDP
5.71/2.70	    (7) UsableRulesProof [EQUIVALENT, 0 ms]
5.71/2.70	    (8) PiDP
5.71/2.70	    (9) PiDPToQDPProof [SOUND, 0 ms]
5.71/2.70	    (10) QDP
5.71/2.70	(11) PrologToPiTRSProof [SOUND, 0 ms]
5.71/2.70	(12) PiTRS
5.71/2.70	    (13) DependencyPairsProof [EQUIVALENT, 0 ms]
5.71/2.70	    (14) PiDP
5.71/2.70	    (15) DependencyGraphProof [EQUIVALENT, 0 ms]
5.71/2.70	    (16) PiDP
5.71/2.70	    (17) UsableRulesProof [EQUIVALENT, 0 ms]
5.71/2.70	    (18) PiDP
5.71/2.70	    (19) PiDPToQDPProof [SOUND, 0 ms]
5.71/2.70	    (20) QDP
5.71/2.70	(21) PrologToTRSTransformerProof [SOUND, 0 ms]
5.71/2.70	(22) QTRS
5.71/2.70	    (23) DependencyPairsProof [EQUIVALENT, 0 ms]
5.71/2.70	    (24) QDP
5.71/2.70	    (25) DependencyGraphProof [EQUIVALENT, 0 ms]
5.71/2.70	    (26) QDP
5.71/2.70	    (27) MNOCProof [EQUIVALENT, 0 ms]
5.71/2.70	    (28) QDP
5.71/2.70	    (29) UsableRulesProof [EQUIVALENT, 0 ms]
5.71/2.70	    (30) QDP
5.71/2.70	    (31) QReductionProof [EQUIVALENT, 0 ms]
5.71/2.70	    (32) QDP
5.71/2.70	(33) PrologToIRSwTTransformerProof [SOUND, 0 ms]
5.71/2.70	(34) IRSwT
5.71/2.70	    (35) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
5.71/2.70	    (36) IRSwT
5.71/2.70	    (37) IntTRSCompressionProof [EQUIVALENT, 18 ms]
5.71/2.70	    (38) IRSwT
5.71/2.70	    (39) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
5.71/2.70	    (40) IRSwT
5.71/2.70	    (41) IRSwTTerminationDigraphProof [EQUIVALENT, 6 ms]
5.71/2.70	    (42) IRSwT
5.71/2.70	    (43) FilterProof [EQUIVALENT, 0 ms]
5.71/2.70	    (44) IntTRS
5.71/2.70	    (45) IntTRSPeriodicNontermProof [COMPLETE, 5 ms]
5.71/2.70	    (46) NO
5.71/2.70	(47) PrologToDTProblemTransformerProof [SOUND, 0 ms]
5.71/2.70	(48) TRIPLES
5.71/2.70	    (49) TriplesToPiDPProof [SOUND, 0 ms]
5.71/2.70	    (50) PiDP
5.71/2.70	    (51) DependencyGraphProof [EQUIVALENT, 0 ms]
5.71/2.70	    (52) PiDP
5.71/2.70	    (53) PiDPToQDPProof [SOUND, 0 ms]
5.71/2.70	    (54) QDP
5.71/2.70	
5.71/2.70	
5.71/2.70	----------------------------------------
5.71/2.70	
5.71/2.70	(0)
5.71/2.70	Obligation:
5.71/2.70	Clauses:
5.71/2.70	
5.71/2.70	minimum(tree(X, void, X1), X).
5.71/2.70	minimum(tree(X2, Left, X3), X) :- minimum(Left, X).
5.71/2.70	
5.71/2.70	
5.71/2.70	Query: minimum(a,g)
5.71/2.70	----------------------------------------
5.71/2.70	
5.71/2.70	(1) PrologToPiTRSProof (SOUND)
5.71/2.70	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
5.71/2.70	
5.71/2.70	minimum_in_2: (f,b)
5.71/2.70	
5.71/2.70	Transforming Prolog into the following Term Rewriting System:
5.71/2.70	
5.71/2.70	Pi-finite rewrite system:
5.71/2.70	The TRS R consists of the following rules:
5.71/2.70	
5.71/2.70	   minimum_in_ag(tree(X, void, X1), X) -> minimum_out_ag(tree(X, void, X1), X)
5.71/2.70	   minimum_in_ag(tree(X2, Left, X3), X) -> U1_ag(X2, Left, X3, X, minimum_in_ag(Left, X))
5.71/2.70	   U1_ag(X2, Left, X3, X, minimum_out_ag(Left, X)) -> minimum_out_ag(tree(X2, Left, X3), X)
5.71/2.70	
5.71/2.70	The argument filtering Pi contains the following mapping:
5.71/2.70	minimum_in_ag(x1, x2)  =  minimum_in_ag(x2)
5.71/2.70	
5.71/2.70	minimum_out_ag(x1, x2)  =  minimum_out_ag(x1)
5.71/2.70	
5.71/2.70	tree(x1, x2, x3)  =  tree(x2)
5.71/2.70	
5.71/2.70	U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x5)
5.71/2.70	
5.71/2.70	
5.71/2.70	
5.71/2.70	
5.71/2.70	
5.71/2.70	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
5.71/2.70	
5.71/2.70	
5.71/2.70	
5.71/2.70	----------------------------------------
5.71/2.70	
5.71/2.70	(2)
5.71/2.70	Obligation:
5.71/2.70	Pi-finite rewrite system:
5.71/2.70	The TRS R consists of the following rules:
5.71/2.70	
5.71/2.70	   minimum_in_ag(tree(X, void, X1), X) -> minimum_out_ag(tree(X, void, X1), X)
5.71/2.70	   minimum_in_ag(tree(X2, Left, X3), X) -> U1_ag(X2, Left, X3, X, minimum_in_ag(Left, X))
5.71/2.70	   U1_ag(X2, Left, X3, X, minimum_out_ag(Left, X)) -> minimum_out_ag(tree(X2, Left, X3), X)
5.71/2.70	
5.71/2.70	The argument filtering Pi contains the following mapping:
5.71/2.70	minimum_in_ag(x1, x2)  =  minimum_in_ag(x2)
5.71/2.70	
5.71/2.70	minimum_out_ag(x1, x2)  =  minimum_out_ag(x1)
5.71/2.70	
5.71/2.70	tree(x1, x2, x3)  =  tree(x2)
5.71/2.70	
5.71/2.70	U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x5)
5.71/2.70	
5.71/2.70	
5.71/2.70	
5.71/2.70	----------------------------------------
5.71/2.70	
5.71/2.70	(3) DependencyPairsProof (EQUIVALENT)
5.71/2.70	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
5.71/2.70	Pi DP problem:
5.71/2.70	The TRS P consists of the following rules:
5.71/2.70	
5.71/2.70	   MINIMUM_IN_AG(tree(X2, Left, X3), X) -> U1_AG(X2, Left, X3, X, minimum_in_ag(Left, X))
5.71/2.70	   MINIMUM_IN_AG(tree(X2, Left, X3), X) -> MINIMUM_IN_AG(Left, X)
5.71/2.70	
5.71/2.70	The TRS R consists of the following rules:
5.71/2.70	
5.71/2.70	   minimum_in_ag(tree(X, void, X1), X) -> minimum_out_ag(tree(X, void, X1), X)
5.71/2.70	   minimum_in_ag(tree(X2, Left, X3), X) -> U1_ag(X2, Left, X3, X, minimum_in_ag(Left, X))
5.71/2.70	   U1_ag(X2, Left, X3, X, minimum_out_ag(Left, X)) -> minimum_out_ag(tree(X2, Left, X3), X)
5.71/2.70	
5.71/2.70	The argument filtering Pi contains the following mapping:
5.71/2.70	minimum_in_ag(x1, x2)  =  minimum_in_ag(x2)
5.71/2.70	
5.71/2.70	minimum_out_ag(x1, x2)  =  minimum_out_ag(x1)
5.71/2.70	
5.71/2.70	tree(x1, x2, x3)  =  tree(x2)
5.71/2.70	
5.71/2.70	U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x5)
5.71/2.70	
5.71/2.70	MINIMUM_IN_AG(x1, x2)  =  MINIMUM_IN_AG(x2)
5.71/2.70	
5.71/2.70	U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x5)
5.71/2.70	
5.71/2.70	
5.71/2.70	We have to consider all (P,R,Pi)-chains
5.71/2.70	----------------------------------------
5.71/2.70	
5.71/2.70	(4)
5.71/2.70	Obligation:
5.71/2.70	Pi DP problem:
5.71/2.70	The TRS P consists of the following rules:
5.71/2.70	
5.71/2.70	   MINIMUM_IN_AG(tree(X2, Left, X3), X) -> U1_AG(X2, Left, X3, X, minimum_in_ag(Left, X))
5.71/2.70	   MINIMUM_IN_AG(tree(X2, Left, X3), X) -> MINIMUM_IN_AG(Left, X)
5.71/2.70	
5.71/2.70	The TRS R consists of the following rules:
5.71/2.70	
5.71/2.70	   minimum_in_ag(tree(X, void, X1), X) -> minimum_out_ag(tree(X, void, X1), X)
5.71/2.70	   minimum_in_ag(tree(X2, Left, X3), X) -> U1_ag(X2, Left, X3, X, minimum_in_ag(Left, X))
5.71/2.70	   U1_ag(X2, Left, X3, X, minimum_out_ag(Left, X)) -> minimum_out_ag(tree(X2, Left, X3), X)
5.71/2.70	
5.71/2.70	The argument filtering Pi contains the following mapping:
5.71/2.70	minimum_in_ag(x1, x2)  =  minimum_in_ag(x2)
5.71/2.70	
5.71/2.70	minimum_out_ag(x1, x2)  =  minimum_out_ag(x1)
5.71/2.70	
5.71/2.70	tree(x1, x2, x3)  =  tree(x2)
5.71/2.70	
5.71/2.70	U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x5)
5.71/2.70	
5.71/2.70	MINIMUM_IN_AG(x1, x2)  =  MINIMUM_IN_AG(x2)
5.71/2.70	
5.71/2.70	U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x5)
5.71/2.70	
5.71/2.70	
5.71/2.70	We have to consider all (P,R,Pi)-chains
5.71/2.70	----------------------------------------
5.71/2.70	
5.71/2.70	(5) DependencyGraphProof (EQUIVALENT)
5.71/2.70	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
5.71/2.70	----------------------------------------
5.71/2.70	
5.71/2.70	(6)
5.71/2.70	Obligation:
5.71/2.70	Pi DP problem:
5.71/2.70	The TRS P consists of the following rules:
5.71/2.70	
5.71/2.70	   MINIMUM_IN_AG(tree(X2, Left, X3), X) -> MINIMUM_IN_AG(Left, X)
5.71/2.70	
5.71/2.70	The TRS R consists of the following rules:
5.71/2.70	
5.71/2.70	   minimum_in_ag(tree(X, void, X1), X) -> minimum_out_ag(tree(X, void, X1), X)
5.71/2.70	   minimum_in_ag(tree(X2, Left, X3), X) -> U1_ag(X2, Left, X3, X, minimum_in_ag(Left, X))
5.71/2.70	   U1_ag(X2, Left, X3, X, minimum_out_ag(Left, X)) -> minimum_out_ag(tree(X2, Left, X3), X)
5.71/2.70	
5.71/2.70	The argument filtering Pi contains the following mapping:
5.71/2.70	minimum_in_ag(x1, x2)  =  minimum_in_ag(x2)
5.71/2.70	
5.71/2.70	minimum_out_ag(x1, x2)  =  minimum_out_ag(x1)
5.71/2.70	
5.71/2.70	tree(x1, x2, x3)  =  tree(x2)
5.71/2.70	
5.71/2.70	U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x5)
5.71/2.70	
5.71/2.70	MINIMUM_IN_AG(x1, x2)  =  MINIMUM_IN_AG(x2)
5.71/2.70	
5.71/2.70	
5.71/2.70	We have to consider all (P,R,Pi)-chains
5.71/2.70	----------------------------------------
5.71/2.70	
5.71/2.70	(7) UsableRulesProof (EQUIVALENT)
5.71/2.70	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.71/2.70	----------------------------------------
5.71/2.70	
5.71/2.70	(8)
5.71/2.70	Obligation:
5.71/2.70	Pi DP problem:
5.71/2.70	The TRS P consists of the following rules:
5.71/2.70	
5.71/2.70	   MINIMUM_IN_AG(tree(X2, Left, X3), X) -> MINIMUM_IN_AG(Left, X)
5.71/2.70	
5.71/2.70	R is empty.
5.71/2.70	The argument filtering Pi contains the following mapping:
5.71/2.70	tree(x1, x2, x3)  =  tree(x2)
5.71/2.70	
5.71/2.70	MINIMUM_IN_AG(x1, x2)  =  MINIMUM_IN_AG(x2)
5.71/2.70	
5.71/2.70	
5.71/2.70	We have to consider all (P,R,Pi)-chains
5.71/2.70	----------------------------------------
5.71/2.70	
5.71/2.70	(9) PiDPToQDPProof (SOUND)
5.71/2.70	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.71/2.70	----------------------------------------
5.71/2.70	
5.71/2.70	(10)
5.71/2.70	Obligation:
5.71/2.70	Q DP problem:
5.71/2.70	The TRS P consists of the following rules:
5.71/2.70	
5.71/2.70	   MINIMUM_IN_AG(X) -> MINIMUM_IN_AG(X)
5.71/2.70	
5.71/2.70	R is empty.
5.71/2.70	Q is empty.
5.71/2.70	We have to consider all (P,Q,R)-chains.
5.71/2.70	----------------------------------------
5.71/2.70	
5.71/2.70	(11) PrologToPiTRSProof (SOUND)
5.71/2.70	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
5.71/2.70	
5.71/2.70	minimum_in_2: (f,b)
5.71/2.70	
5.71/2.70	Transforming Prolog into the following Term Rewriting System:
5.71/2.70	
5.71/2.70	Pi-finite rewrite system:
5.71/2.70	The TRS R consists of the following rules:
5.71/2.70	
5.71/2.70	   minimum_in_ag(tree(X, void, X1), X) -> minimum_out_ag(tree(X, void, X1), X)
5.71/2.70	   minimum_in_ag(tree(X2, Left, X3), X) -> U1_ag(X2, Left, X3, X, minimum_in_ag(Left, X))
5.71/2.70	   U1_ag(X2, Left, X3, X, minimum_out_ag(Left, X)) -> minimum_out_ag(tree(X2, Left, X3), X)
5.71/2.70	
5.71/2.70	The argument filtering Pi contains the following mapping:
5.71/2.70	minimum_in_ag(x1, x2)  =  minimum_in_ag(x2)
5.71/2.70	
5.71/2.70	minimum_out_ag(x1, x2)  =  minimum_out_ag(x1, x2)
5.71/2.70	
5.71/2.70	tree(x1, x2, x3)  =  tree(x2)
5.71/2.70	
5.71/2.70	U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
5.71/2.70	
5.71/2.70	
5.71/2.70	
5.71/2.70	
5.71/2.70	
5.71/2.70	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
5.71/2.70	
5.71/2.70	
5.71/2.70	
5.71/2.70	----------------------------------------
5.71/2.70	
5.71/2.70	(12)
5.71/2.70	Obligation:
5.71/2.70	Pi-finite rewrite system:
5.71/2.70	The TRS R consists of the following rules:
5.71/2.70	
5.71/2.70	   minimum_in_ag(tree(X, void, X1), X) -> minimum_out_ag(tree(X, void, X1), X)
5.71/2.70	   minimum_in_ag(tree(X2, Left, X3), X) -> U1_ag(X2, Left, X3, X, minimum_in_ag(Left, X))
5.71/2.70	   U1_ag(X2, Left, X3, X, minimum_out_ag(Left, X)) -> minimum_out_ag(tree(X2, Left, X3), X)
5.71/2.70	
5.71/2.70	The argument filtering Pi contains the following mapping:
5.71/2.70	minimum_in_ag(x1, x2)  =  minimum_in_ag(x2)
5.71/2.70	
5.71/2.70	minimum_out_ag(x1, x2)  =  minimum_out_ag(x1, x2)
5.71/2.70	
5.71/2.70	tree(x1, x2, x3)  =  tree(x2)
5.71/2.70	
5.71/2.70	U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
5.71/2.70	
5.71/2.70	
5.71/2.70	
5.71/2.70	----------------------------------------
5.71/2.70	
5.71/2.70	(13) DependencyPairsProof (EQUIVALENT)
5.71/2.70	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
5.71/2.70	Pi DP problem:
5.71/2.70	The TRS P consists of the following rules:
5.71/2.70	
5.71/2.70	   MINIMUM_IN_AG(tree(X2, Left, X3), X) -> U1_AG(X2, Left, X3, X, minimum_in_ag(Left, X))
5.71/2.70	   MINIMUM_IN_AG(tree(X2, Left, X3), X) -> MINIMUM_IN_AG(Left, X)
5.71/2.70	
5.71/2.70	The TRS R consists of the following rules:
5.71/2.70	
5.71/2.70	   minimum_in_ag(tree(X, void, X1), X) -> minimum_out_ag(tree(X, void, X1), X)
5.71/2.70	   minimum_in_ag(tree(X2, Left, X3), X) -> U1_ag(X2, Left, X3, X, minimum_in_ag(Left, X))
5.71/2.70	   U1_ag(X2, Left, X3, X, minimum_out_ag(Left, X)) -> minimum_out_ag(tree(X2, Left, X3), X)
5.71/2.70	
5.71/2.70	The argument filtering Pi contains the following mapping:
5.71/2.70	minimum_in_ag(x1, x2)  =  minimum_in_ag(x2)
5.71/2.70	
5.71/2.70	minimum_out_ag(x1, x2)  =  minimum_out_ag(x1, x2)
5.71/2.70	
5.71/2.70	tree(x1, x2, x3)  =  tree(x2)
5.71/2.70	
5.71/2.70	U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
5.71/2.70	
5.71/2.70	MINIMUM_IN_AG(x1, x2)  =  MINIMUM_IN_AG(x2)
5.71/2.70	
5.71/2.70	U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x4, x5)
5.71/2.70	
5.71/2.70	
5.71/2.70	We have to consider all (P,R,Pi)-chains
5.71/2.70	----------------------------------------
5.71/2.70	
5.71/2.70	(14)
5.71/2.70	Obligation:
5.71/2.70	Pi DP problem:
5.71/2.70	The TRS P consists of the following rules:
5.71/2.70	
5.71/2.70	   MINIMUM_IN_AG(tree(X2, Left, X3), X) -> U1_AG(X2, Left, X3, X, minimum_in_ag(Left, X))
5.71/2.70	   MINIMUM_IN_AG(tree(X2, Left, X3), X) -> MINIMUM_IN_AG(Left, X)
5.71/2.70	
5.71/2.70	The TRS R consists of the following rules:
5.71/2.70	
5.71/2.70	   minimum_in_ag(tree(X, void, X1), X) -> minimum_out_ag(tree(X, void, X1), X)
5.71/2.70	   minimum_in_ag(tree(X2, Left, X3), X) -> U1_ag(X2, Left, X3, X, minimum_in_ag(Left, X))
5.71/2.70	   U1_ag(X2, Left, X3, X, minimum_out_ag(Left, X)) -> minimum_out_ag(tree(X2, Left, X3), X)
5.71/2.70	
5.71/2.70	The argument filtering Pi contains the following mapping:
5.71/2.70	minimum_in_ag(x1, x2)  =  minimum_in_ag(x2)
5.71/2.70	
5.71/2.70	minimum_out_ag(x1, x2)  =  minimum_out_ag(x1, x2)
5.71/2.70	
5.71/2.70	tree(x1, x2, x3)  =  tree(x2)
5.71/2.70	
5.71/2.70	U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
5.71/2.70	
5.71/2.70	MINIMUM_IN_AG(x1, x2)  =  MINIMUM_IN_AG(x2)
5.71/2.70	
5.71/2.70	U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x4, x5)
5.71/2.70	
5.71/2.70	
5.71/2.70	We have to consider all (P,R,Pi)-chains
5.71/2.70	----------------------------------------
5.71/2.70	
5.71/2.70	(15) DependencyGraphProof (EQUIVALENT)
5.71/2.70	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
5.71/2.70	----------------------------------------
5.71/2.70	
5.71/2.70	(16)
5.71/2.70	Obligation:
5.71/2.71	Pi DP problem:
5.71/2.71	The TRS P consists of the following rules:
5.71/2.71	
5.71/2.71	   MINIMUM_IN_AG(tree(X2, Left, X3), X) -> MINIMUM_IN_AG(Left, X)
5.71/2.71	
5.71/2.71	The TRS R consists of the following rules:
5.71/2.71	
5.71/2.71	   minimum_in_ag(tree(X, void, X1), X) -> minimum_out_ag(tree(X, void, X1), X)
5.71/2.71	   minimum_in_ag(tree(X2, Left, X3), X) -> U1_ag(X2, Left, X3, X, minimum_in_ag(Left, X))
5.71/2.71	   U1_ag(X2, Left, X3, X, minimum_out_ag(Left, X)) -> minimum_out_ag(tree(X2, Left, X3), X)
5.71/2.71	
5.71/2.71	The argument filtering Pi contains the following mapping:
5.71/2.71	minimum_in_ag(x1, x2)  =  minimum_in_ag(x2)
5.71/2.71	
5.71/2.71	minimum_out_ag(x1, x2)  =  minimum_out_ag(x1, x2)
5.71/2.71	
5.71/2.71	tree(x1, x2, x3)  =  tree(x2)
5.71/2.71	
5.71/2.71	U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x4, x5)
5.71/2.71	
5.71/2.71	MINIMUM_IN_AG(x1, x2)  =  MINIMUM_IN_AG(x2)
5.71/2.71	
5.71/2.71	
5.71/2.71	We have to consider all (P,R,Pi)-chains
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(17) UsableRulesProof (EQUIVALENT)
5.71/2.71	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(18)
5.71/2.71	Obligation:
5.71/2.71	Pi DP problem:
5.71/2.71	The TRS P consists of the following rules:
5.71/2.71	
5.71/2.71	   MINIMUM_IN_AG(tree(X2, Left, X3), X) -> MINIMUM_IN_AG(Left, X)
5.71/2.71	
5.71/2.71	R is empty.
5.71/2.71	The argument filtering Pi contains the following mapping:
5.71/2.71	tree(x1, x2, x3)  =  tree(x2)
5.71/2.71	
5.71/2.71	MINIMUM_IN_AG(x1, x2)  =  MINIMUM_IN_AG(x2)
5.71/2.71	
5.71/2.71	
5.71/2.71	We have to consider all (P,R,Pi)-chains
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(19) PiDPToQDPProof (SOUND)
5.71/2.71	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(20)
5.71/2.71	Obligation:
5.71/2.71	Q DP problem:
5.71/2.71	The TRS P consists of the following rules:
5.71/2.71	
5.71/2.71	   MINIMUM_IN_AG(X) -> MINIMUM_IN_AG(X)
5.71/2.71	
5.71/2.71	R is empty.
5.71/2.71	Q is empty.
5.71/2.71	We have to consider all (P,Q,R)-chains.
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(21) PrologToTRSTransformerProof (SOUND)
5.71/2.71	Transformed Prolog program to TRS.
5.71/2.71	
5.71/2.71	{
5.71/2.71	    "root": 1,
5.71/2.71	    "program": {
5.71/2.71	        "directives": [],
5.71/2.71	        "clauses": [
5.71/2.71	            [
5.71/2.71	                "(minimum (tree X (void) X1) X)",
5.71/2.71	                null
5.71/2.71	            ],
5.71/2.71	            [
5.71/2.71	                "(minimum (tree X2 Left X3) X)",
5.71/2.71	                "(minimum Left X)"
5.71/2.71	            ]
5.71/2.71	        ]
5.71/2.71	    },
5.71/2.71	    "graph": {
5.71/2.71	        "nodes": {
5.71/2.71	            "1": {
5.71/2.71	                "goal": [{
5.71/2.71	                    "clause": -1,
5.71/2.71	                    "scope": -1,
5.71/2.71	                    "term": "(minimum T1 T2)"
5.71/2.71	                }],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": ["T2"],
5.71/2.71	                    "free": [],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "122": {
5.71/2.71	                "goal": [],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": [],
5.71/2.71	                    "free": [],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "102": {
5.71/2.71	                "goal": [{
5.71/2.71	                    "clause": 0,
5.71/2.71	                    "scope": 1,
5.71/2.71	                    "term": "(minimum T1 T2)"
5.71/2.71	                }],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": ["T2"],
5.71/2.71	                    "free": [],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "103": {
5.71/2.71	                "goal": [{
5.71/2.71	                    "clause": 1,
5.71/2.71	                    "scope": 1,
5.71/2.71	                    "term": "(minimum T1 T2)"
5.71/2.71	                }],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": ["T2"],
5.71/2.71	                    "free": [],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "104": {
5.71/2.71	                "goal": [{
5.71/2.71	                    "clause": -1,
5.71/2.71	                    "scope": -1,
5.71/2.71	                    "term": "(true)"
5.71/2.71	                }],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": [],
5.71/2.71	                    "free": [],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "105": {
5.71/2.71	                "goal": [],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": [],
5.71/2.71	                    "free": [],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "106": {
5.71/2.71	                "goal": [],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": [],
5.71/2.71	                    "free": [],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "107": {
5.71/2.71	                "goal": [{
5.71/2.71	                    "clause": -1,
5.71/2.71	                    "scope": -1,
5.71/2.71	                    "term": "(minimum T25 T24)"
5.71/2.71	                }],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": ["T24"],
5.71/2.71	                    "free": [],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "type": "Nodes",
5.71/2.71	            "62": {
5.71/2.71	                "goal": [
5.71/2.71	                    {
5.71/2.71	                        "clause": 0,
5.71/2.71	                        "scope": 1,
5.71/2.71	                        "term": "(minimum T1 T2)"
5.71/2.71	                    },
5.71/2.71	                    {
5.71/2.71	                        "clause": 1,
5.71/2.71	                        "scope": 1,
5.71/2.71	                        "term": "(minimum T1 T2)"
5.71/2.71	                    }
5.71/2.71	                ],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": ["T2"],
5.71/2.71	                    "free": [],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            }
5.71/2.71	        },
5.71/2.71	        "edges": [
5.71/2.71	            {
5.71/2.71	                "from": 1,
5.71/2.71	                "to": 62,
5.71/2.71	                "label": "CASE"
5.71/2.71	            },
5.71/2.71	            {
5.71/2.71	                "from": 62,
5.71/2.71	                "to": 102,
5.71/2.71	                "label": "PARALLEL"
5.71/2.71	            },
5.71/2.71	            {
5.71/2.71	                "from": 62,
5.71/2.71	                "to": 103,
5.71/2.71	                "label": "PARALLEL"
5.71/2.71	            },
5.71/2.71	            {
5.71/2.71	                "from": 102,
5.71/2.71	                "to": 104,
5.71/2.71	                "label": "EVAL with clause\nminimum(tree(X12, void, X13), X12).\nand substitutionX12 -> T11,\nX13 -> T12,\nT1 -> tree(T11, void, T12),\nT2 -> T11"
5.71/2.71	            },
5.71/2.71	            {
5.71/2.71	                "from": 102,
5.71/2.71	                "to": 105,
5.71/2.71	                "label": "EVAL-BACKTRACK"
5.71/2.71	            },
5.71/2.71	            {
5.71/2.71	                "from": 103,
5.71/2.71	                "to": 107,
5.71/2.71	                "label": "EVAL with clause\nminimum(tree(X22, X23, X24), X25) :- minimum(X23, X25).\nand substitutionX22 -> T21,\nX23 -> T25,\nX24 -> T23,\nT1 -> tree(T21, T25, T23),\nT2 -> T24,\nX25 -> T24,\nT22 -> T25"
5.71/2.71	            },
5.71/2.71	            {
5.71/2.71	                "from": 103,
5.71/2.71	                "to": 122,
5.71/2.71	                "label": "EVAL-BACKTRACK"
5.71/2.71	            },
5.71/2.71	            {
5.71/2.71	                "from": 104,
5.71/2.71	                "to": 106,
5.71/2.71	                "label": "SUCCESS"
5.71/2.71	            },
5.71/2.71	            {
5.71/2.71	                "from": 107,
5.71/2.71	                "to": 1,
5.71/2.71	                "label": "INSTANCE with matching:\nT1 -> T25\nT2 -> T24"
5.71/2.71	            }
5.71/2.71	        ],
5.71/2.71	        "type": "Graph"
5.71/2.71	    }
5.71/2.71	}
5.71/2.71	
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(22)
5.71/2.71	Obligation:
5.71/2.71	Q restricted rewrite system:
5.71/2.71	The TRS R consists of the following rules:
5.71/2.71	
5.71/2.71	   f1_in(T11) -> f1_out1
5.71/2.71	   f1_in(T24) -> U1(f1_in(T24), T24)
5.71/2.71	   U1(f1_out1, T24) -> f1_out1
5.71/2.71	
5.71/2.71	Q is empty.
5.71/2.71	
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(23) DependencyPairsProof (EQUIVALENT)
5.71/2.71	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(24)
5.71/2.71	Obligation:
5.71/2.71	Q DP problem:
5.71/2.71	The TRS P consists of the following rules:
5.71/2.71	
5.71/2.71	   F1_IN(T24) -> U1^1(f1_in(T24), T24)
5.71/2.71	   F1_IN(T24) -> F1_IN(T24)
5.71/2.71	
5.71/2.71	The TRS R consists of the following rules:
5.71/2.71	
5.71/2.71	   f1_in(T11) -> f1_out1
5.71/2.71	   f1_in(T24) -> U1(f1_in(T24), T24)
5.71/2.71	   U1(f1_out1, T24) -> f1_out1
5.71/2.71	
5.71/2.71	Q is empty.
5.71/2.71	We have to consider all minimal (P,Q,R)-chains.
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(25) DependencyGraphProof (EQUIVALENT)
5.71/2.71	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(26)
5.71/2.71	Obligation:
5.71/2.71	Q DP problem:
5.71/2.71	The TRS P consists of the following rules:
5.71/2.71	
5.71/2.71	   F1_IN(T24) -> F1_IN(T24)
5.71/2.71	
5.71/2.71	The TRS R consists of the following rules:
5.71/2.71	
5.71/2.71	   f1_in(T11) -> f1_out1
5.71/2.71	   f1_in(T24) -> U1(f1_in(T24), T24)
5.71/2.71	   U1(f1_out1, T24) -> f1_out1
5.71/2.71	
5.71/2.71	Q is empty.
5.71/2.71	We have to consider all minimal (P,Q,R)-chains.
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(27) MNOCProof (EQUIVALENT)
5.71/2.71	We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(28)
5.71/2.71	Obligation:
5.71/2.71	Q DP problem:
5.71/2.71	The TRS P consists of the following rules:
5.71/2.71	
5.71/2.71	   F1_IN(T24) -> F1_IN(T24)
5.71/2.71	
5.71/2.71	The TRS R consists of the following rules:
5.71/2.71	
5.71/2.71	   f1_in(T11) -> f1_out1
5.71/2.71	   f1_in(T24) -> U1(f1_in(T24), T24)
5.71/2.71	   U1(f1_out1, T24) -> f1_out1
5.71/2.71	
5.71/2.71	The set Q consists of the following terms:
5.71/2.71	
5.71/2.71	   f1_in(x0)
5.71/2.71	   U1(f1_out1, x0)
5.71/2.71	
5.71/2.71	We have to consider all minimal (P,Q,R)-chains.
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(29) UsableRulesProof (EQUIVALENT)
5.71/2.71	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(30)
5.71/2.71	Obligation:
5.71/2.71	Q DP problem:
5.71/2.71	The TRS P consists of the following rules:
5.71/2.71	
5.71/2.71	   F1_IN(T24) -> F1_IN(T24)
5.71/2.71	
5.71/2.71	R is empty.
5.71/2.71	The set Q consists of the following terms:
5.71/2.71	
5.71/2.71	   f1_in(x0)
5.71/2.71	   U1(f1_out1, x0)
5.71/2.71	
5.71/2.71	We have to consider all minimal (P,Q,R)-chains.
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(31) QReductionProof (EQUIVALENT)
5.71/2.71	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
5.71/2.71	
5.71/2.71	   f1_in(x0)
5.71/2.71	   U1(f1_out1, x0)
5.71/2.71	
5.71/2.71	
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(32)
5.71/2.71	Obligation:
5.71/2.71	Q DP problem:
5.71/2.71	The TRS P consists of the following rules:
5.71/2.71	
5.71/2.71	   F1_IN(T24) -> F1_IN(T24)
5.71/2.71	
5.71/2.71	R is empty.
5.71/2.71	Q is empty.
5.71/2.71	We have to consider all minimal (P,Q,R)-chains.
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(33) PrologToIRSwTTransformerProof (SOUND)
5.71/2.71	Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert
5.71/2.71	
5.71/2.71	{
5.71/2.71	    "root": 2,
5.71/2.71	    "program": {
5.71/2.71	        "directives": [],
5.71/2.71	        "clauses": [
5.71/2.71	            [
5.71/2.71	                "(minimum (tree X (void) X1) X)",
5.71/2.71	                null
5.71/2.71	            ],
5.71/2.71	            [
5.71/2.71	                "(minimum (tree X2 Left X3) X)",
5.71/2.71	                "(minimum Left X)"
5.71/2.71	            ]
5.71/2.71	        ]
5.71/2.71	    },
5.71/2.71	    "graph": {
5.71/2.71	        "nodes": {
5.71/2.71	            "88": {
5.71/2.71	                "goal": [{
5.71/2.71	                    "clause": 0,
5.71/2.71	                    "scope": 1,
5.71/2.71	                    "term": "(minimum T1 T2)"
5.71/2.71	                }],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": ["T2"],
5.71/2.71	                    "free": [],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "99": {
5.71/2.71	                "goal": [],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": [],
5.71/2.71	                    "free": [],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "89": {
5.71/2.71	                "goal": [{
5.71/2.71	                    "clause": 1,
5.71/2.71	                    "scope": 1,
5.71/2.71	                    "term": "(minimum T1 T2)"
5.71/2.71	                }],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": ["T2"],
5.71/2.71	                    "free": [],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "2": {
5.71/2.71	                "goal": [{
5.71/2.71	                    "clause": -1,
5.71/2.71	                    "scope": -1,
5.71/2.71	                    "term": "(minimum T1 T2)"
5.71/2.71	                }],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": ["T2"],
5.71/2.71	                    "free": [],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "13": {
5.71/2.71	                "goal": [
5.71/2.71	                    {
5.71/2.71	                        "clause": 0,
5.71/2.71	                        "scope": 1,
5.71/2.71	                        "term": "(minimum T1 T2)"
5.71/2.71	                    },
5.71/2.71	                    {
5.71/2.71	                        "clause": 1,
5.71/2.71	                        "scope": 1,
5.71/2.71	                        "term": "(minimum T1 T2)"
5.71/2.71	                    }
5.71/2.71	                ],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": ["T2"],
5.71/2.71	                    "free": [],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "92": {
5.71/2.71	                "goal": [{
5.71/2.71	                    "clause": -1,
5.71/2.71	                    "scope": -1,
5.71/2.71	                    "term": "(true)"
5.71/2.71	                }],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": [],
5.71/2.71	                    "free": [],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "94": {
5.71/2.71	                "goal": [],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": [],
5.71/2.71	                    "free": [],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "type": "Nodes",
5.71/2.71	            "95": {
5.71/2.71	                "goal": [],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": [],
5.71/2.71	                    "free": [],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "98": {
5.71/2.71	                "goal": [{
5.71/2.71	                    "clause": -1,
5.71/2.71	                    "scope": -1,
5.71/2.71	                    "term": "(minimum T25 T24)"
5.71/2.71	                }],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": ["T24"],
5.71/2.71	                    "free": [],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            }
5.71/2.71	        },
5.71/2.71	        "edges": [
5.71/2.71	            {
5.71/2.71	                "from": 2,
5.71/2.71	                "to": 13,
5.71/2.71	                "label": "CASE"
5.71/2.71	            },
5.71/2.71	            {
5.71/2.71	                "from": 13,
5.71/2.71	                "to": 88,
5.71/2.71	                "label": "PARALLEL"
5.71/2.71	            },
5.71/2.71	            {
5.71/2.71	                "from": 13,
5.71/2.71	                "to": 89,
5.71/2.71	                "label": "PARALLEL"
5.71/2.71	            },
5.71/2.71	            {
5.71/2.71	                "from": 88,
5.71/2.71	                "to": 92,
5.71/2.71	                "label": "EVAL with clause\nminimum(tree(X12, void, X13), X12).\nand substitutionX12 -> T11,\nX13 -> T12,\nT1 -> tree(T11, void, T12),\nT2 -> T11"
5.71/2.71	            },
5.71/2.71	            {
5.71/2.71	                "from": 88,
5.71/2.71	                "to": 94,
5.71/2.71	                "label": "EVAL-BACKTRACK"
5.71/2.71	            },
5.71/2.71	            {
5.71/2.71	                "from": 89,
5.71/2.71	                "to": 98,
5.71/2.71	                "label": "EVAL with clause\nminimum(tree(X22, X23, X24), X25) :- minimum(X23, X25).\nand substitutionX22 -> T21,\nX23 -> T25,\nX24 -> T23,\nT1 -> tree(T21, T25, T23),\nT2 -> T24,\nX25 -> T24,\nT22 -> T25"
5.71/2.71	            },
5.71/2.71	            {
5.71/2.71	                "from": 89,
5.71/2.71	                "to": 99,
5.71/2.71	                "label": "EVAL-BACKTRACK"
5.71/2.71	            },
5.71/2.71	            {
5.71/2.71	                "from": 92,
5.71/2.71	                "to": 95,
5.71/2.71	                "label": "SUCCESS"
5.71/2.71	            },
5.71/2.71	            {
5.71/2.71	                "from": 98,
5.71/2.71	                "to": 2,
5.71/2.71	                "label": "INSTANCE with matching:\nT1 -> T25\nT2 -> T24"
5.71/2.71	            }
5.71/2.71	        ],
5.71/2.71	        "type": "Graph"
5.71/2.71	    }
5.71/2.71	}
5.71/2.71	
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(34)
5.71/2.71	Obligation:
5.71/2.71	Rules:
5.71/2.71	f99_out -> f89_out(T2) :|: TRUE
5.71/2.71	f89_in(x) -> f99_in :|: TRUE
5.71/2.71	f89_in(T24) -> f98_in(T24) :|: TRUE
5.71/2.71	f98_out(x1) -> f89_out(x1) :|: TRUE
5.71/2.71	f88_out(x2) -> f13_out(x2) :|: TRUE
5.71/2.71	f89_out(x3) -> f13_out(x3) :|: TRUE
5.71/2.71	f13_in(x4) -> f88_in(x4) :|: TRUE
5.71/2.71	f13_in(x5) -> f89_in(x5) :|: TRUE
5.71/2.71	f98_in(x6) -> f2_in(x6) :|: TRUE
5.71/2.71	f2_out(x7) -> f98_out(x7) :|: TRUE
5.71/2.71	f2_in(x8) -> f13_in(x8) :|: TRUE
5.71/2.71	f13_out(x9) -> f2_out(x9) :|: TRUE
5.71/2.71	Start term: f2_in(T2)
5.71/2.71	
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(35) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
5.71/2.71	Constructed simple dependency graph.
5.71/2.71	
5.71/2.71	Simplified to the following IRSwTs:
5.71/2.71	
5.71/2.71	intTRSProblem:
5.71/2.71	f89_in(T24) -> f98_in(T24) :|: TRUE
5.71/2.71	f13_in(x5) -> f89_in(x5) :|: TRUE
5.71/2.71	f98_in(x6) -> f2_in(x6) :|: TRUE
5.71/2.71	f2_in(x8) -> f13_in(x8) :|: TRUE
5.71/2.71	
5.71/2.71	
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(36)
5.71/2.71	Obligation:
5.71/2.71	Rules:
5.71/2.71	f89_in(T24) -> f98_in(T24) :|: TRUE
5.71/2.71	f13_in(x5) -> f89_in(x5) :|: TRUE
5.71/2.71	f98_in(x6) -> f2_in(x6) :|: TRUE
5.71/2.71	f2_in(x8) -> f13_in(x8) :|: TRUE
5.71/2.71	
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(37) IntTRSCompressionProof (EQUIVALENT)
5.71/2.71	Compressed rules.
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(38)
5.71/2.71	Obligation:
5.71/2.71	Rules:
5.71/2.71	f13_in(x5:0) -> f13_in(x5:0) :|: TRUE
5.71/2.71	
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(39) IRSFormatTransformerProof (EQUIVALENT)
5.71/2.71	Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(40)
5.71/2.71	Obligation:
5.71/2.71	Rules:
5.71/2.71	f13_in(x5:0) -> f13_in(x5:0) :|: TRUE
5.71/2.71	
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(41) IRSwTTerminationDigraphProof (EQUIVALENT)
5.71/2.71	Constructed termination digraph!
5.71/2.71	Nodes:
5.71/2.71	(1) f13_in(x5:0) -> f13_in(x5:0) :|: TRUE
5.71/2.71	
5.71/2.71	Arcs:
5.71/2.71	(1) -> (1)
5.71/2.71	
5.71/2.71	This digraph is fully evaluated!
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(42)
5.71/2.71	Obligation:
5.71/2.71	
5.71/2.71	Termination digraph:
5.71/2.71	Nodes:
5.71/2.71	(1) f13_in(x5:0) -> f13_in(x5:0) :|: TRUE
5.71/2.71	
5.71/2.71	Arcs:
5.71/2.71	(1) -> (1)
5.71/2.71	
5.71/2.71	This digraph is fully evaluated!
5.71/2.71	
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(43) FilterProof (EQUIVALENT)
5.71/2.71	Used the following sort dictionary for filtering: 
5.71/2.71	f13_in(VARIABLE)
5.71/2.71	Replaced non-predefined constructor symbols by 0.
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(44)
5.71/2.71	Obligation:
5.71/2.71	Rules:
5.71/2.71	f13_in(x5:0) -> f13_in(x5:0) :|: TRUE
5.71/2.71	
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(45) IntTRSPeriodicNontermProof (COMPLETE)
5.71/2.71	Normalized system to the following form:
5.71/2.71	f(pc, x5:0) -> f(1, x5:0) :|: pc = 1 && TRUE
5.71/2.71	Witness term starting non-terminating reduction: f(1, -8)
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(46)
5.71/2.71	NO
5.71/2.71	
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(47) PrologToDTProblemTransformerProof (SOUND)
5.71/2.71	Built DT problem from termination graph DT10.
5.71/2.71	
5.71/2.71	{
5.71/2.71	    "root": 3,
5.71/2.71	    "program": {
5.71/2.71	        "directives": [],
5.71/2.71	        "clauses": [
5.71/2.71	            [
5.71/2.71	                "(minimum (tree X (void) X1) X)",
5.71/2.71	                null
5.71/2.71	            ],
5.71/2.71	            [
5.71/2.71	                "(minimum (tree X2 Left X3) X)",
5.71/2.71	                "(minimum Left X)"
5.71/2.71	            ]
5.71/2.71	        ]
5.71/2.71	    },
5.71/2.71	    "graph": {
5.71/2.71	        "nodes": {
5.71/2.71	            "67": {
5.71/2.71	                "goal": [
5.71/2.71	                    {
5.71/2.71	                        "clause": -1,
5.71/2.71	                        "scope": -1,
5.71/2.71	                        "term": "(true)"
5.71/2.71	                    },
5.71/2.71	                    {
5.71/2.71	                        "clause": 1,
5.71/2.71	                        "scope": 1,
5.71/2.71	                        "term": "(minimum T1 T5)"
5.71/2.71	                    }
5.71/2.71	                ],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": ["T5"],
5.71/2.71	                    "free": [],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "68": {
5.71/2.71	                "goal": [{
5.71/2.71	                    "clause": 1,
5.71/2.71	                    "scope": 1,
5.71/2.71	                    "term": "(minimum T1 T2)"
5.71/2.71	                }],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [[
5.71/2.71	                        "(minimum T1 T2)",
5.71/2.71	                        "(minimum (tree X6 (void) X7) X6)"
5.71/2.71	                    ]],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": ["T2"],
5.71/2.71	                    "free": [
5.71/2.71	                        "X6",
5.71/2.71	                        "X7"
5.71/2.71	                    ],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "69": {
5.71/2.71	                "goal": [{
5.71/2.71	                    "clause": 1,
5.71/2.71	                    "scope": 1,
5.71/2.71	                    "term": "(minimum T1 T5)"
5.71/2.71	                }],
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5.71/2.71	                    "nonunifying": [],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": ["T5"],
5.71/2.71	                    "free": [],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "type": "Nodes",
5.71/2.71	            "110": {
5.71/2.71	                "goal": [{
5.71/2.71	                    "clause": -1,
5.71/2.71	                    "scope": -1,
5.71/2.71	                    "term": "(minimum T51 T50)"
5.71/2.71	                }],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [[
5.71/2.71	                        "(minimum T1 T50)",
5.71/2.71	                        "(minimum (tree X6 (void) X7) X6)"
5.71/2.71	                    ]],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": ["T50"],
5.71/2.71	                    "free": [
5.71/2.71	                        "X6",
5.71/2.71	                        "X7"
5.71/2.71	                    ],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "111": {
5.71/2.71	                "goal": [],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": [],
5.71/2.71	                    "free": [],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "112": {
5.71/2.71	                "goal": [
5.71/2.71	                    {
5.71/2.71	                        "clause": 0,
5.71/2.71	                        "scope": 3,
5.71/2.71	                        "term": "(minimum T51 T50)"
5.71/2.71	                    },
5.71/2.71	                    {
5.71/2.71	                        "clause": 1,
5.71/2.71	                        "scope": 3,
5.71/2.71	                        "term": "(minimum T51 T50)"
5.71/2.71	                    }
5.71/2.71	                ],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [[
5.71/2.71	                        "(minimum T1 T50)",
5.71/2.71	                        "(minimum (tree X6 (void) X7) X6)"
5.71/2.71	                    ]],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": ["T50"],
5.71/2.71	                    "free": [
5.71/2.71	                        "X6",
5.71/2.71	                        "X7"
5.71/2.71	                    ],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "113": {
5.71/2.71	                "goal": [{
5.71/2.71	                    "clause": 0,
5.71/2.71	                    "scope": 3,
5.71/2.71	                    "term": "(minimum T51 T50)"
5.71/2.71	                }],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [[
5.71/2.71	                        "(minimum T1 T50)",
5.71/2.71	                        "(minimum (tree X6 (void) X7) X6)"
5.71/2.71	                    ]],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": ["T50"],
5.71/2.71	                    "free": [
5.71/2.71	                        "X6",
5.71/2.71	                        "X7"
5.71/2.71	                    ],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "114": {
5.71/2.71	                "goal": [{
5.71/2.71	                    "clause": 1,
5.71/2.71	                    "scope": 3,
5.71/2.71	                    "term": "(minimum T51 T50)"
5.71/2.71	                }],
5.71/2.71	                "kb": {
5.71/2.71	                    "nonunifying": [[
5.71/2.71	                        "(minimum T1 T50)",
5.71/2.71	                        "(minimum (tree X6 (void) X7) X6)"
5.71/2.71	                    ]],
5.71/2.71	                    "intvars": {},
5.71/2.71	                    "arithmetic": {
5.71/2.71	                        "type": "PlainIntegerRelationState",
5.71/2.71	                        "relations": []
5.71/2.71	                    },
5.71/2.71	                    "ground": ["T50"],
5.71/2.71	                    "free": [
5.71/2.71	                        "X6",
5.71/2.71	                        "X7"
5.71/2.71	                    ],
5.71/2.71	                    "exprvars": []
5.71/2.71	                }
5.71/2.71	            },
5.71/2.71	            "115": {
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5.71/2.71	                        "(minimum T1 T73)",
5.71/2.71	                        "(minimum (tree X6 (void) X7) X6)"
5.71/2.71	                    ]],
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5.71/2.71	                        "X6",
5.71/2.71	                        "X7"
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5.71/2.71	    }
5.71/2.71	}
5.71/2.71	
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(48)
5.71/2.71	Obligation:
5.71/2.71	Triples:
5.71/2.71	
5.71/2.71	minimumA(tree(X1, tree(X2, X3, X4), X5), X6) :- minimumA(X3, X6).
5.71/2.71	minimumA(tree(X1, tree(X2, X3, X4), X5), X6) :- minimumA(X3, X6).
5.71/2.71	
5.71/2.71	Clauses:
5.71/2.71	
5.71/2.71	minimumcA(tree(X1, void, X2), X1).
5.71/2.71	minimumcA(tree(X1, tree(X2, void, X3), X4), X2).
5.71/2.71	minimumcA(tree(X1, tree(X2, X3, X4), X5), X6) :- minimumcA(X3, X6).
5.71/2.71	minimumcA(tree(X1, tree(X2, void, X3), X4), X2).
5.71/2.71	minimumcA(tree(X1, tree(X2, X3, X4), X5), X6) :- minimumcA(X3, X6).
5.71/2.71	
5.71/2.71	Afs:
5.71/2.71	
5.71/2.71	minimumA(x1, x2)  =  minimumA(x2)
5.71/2.71	
5.71/2.71	
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(49) TriplesToPiDPProof (SOUND)
5.71/2.71	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
5.71/2.71	
5.71/2.71	minimumA_in_2: (f,b)
5.71/2.71	
5.71/2.71	Transforming TRIPLES into the following Term Rewriting System:
5.71/2.71	
5.71/2.71	Pi DP problem:
5.71/2.71	The TRS P consists of the following rules:
5.71/2.71	
5.71/2.71	   MINIMUMA_IN_AG(tree(X1, tree(X2, X3, X4), X5), X6) -> U1_AG(X1, X2, X3, X4, X5, X6, minimumA_in_ag(X3, X6))
5.71/2.71	   MINIMUMA_IN_AG(tree(X1, tree(X2, X3, X4), X5), X6) -> MINIMUMA_IN_AG(X3, X6)
5.71/2.71	
5.71/2.71	R is empty.
5.71/2.71	The argument filtering Pi contains the following mapping:
5.71/2.71	minimumA_in_ag(x1, x2)  =  minimumA_in_ag(x2)
5.71/2.71	
5.71/2.71	tree(x1, x2, x3)  =  tree(x2)
5.71/2.71	
5.71/2.71	MINIMUMA_IN_AG(x1, x2)  =  MINIMUMA_IN_AG(x2)
5.71/2.71	
5.71/2.71	U1_AG(x1, x2, x3, x4, x5, x6, x7)  =  U1_AG(x6, x7)
5.71/2.71	
5.71/2.71	
5.71/2.71	We have to consider all (P,R,Pi)-chains
5.71/2.71	
5.71/2.71	
5.71/2.71	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
5.71/2.71	
5.71/2.71	
5.71/2.71	
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(50)
5.71/2.71	Obligation:
5.71/2.71	Pi DP problem:
5.71/2.71	The TRS P consists of the following rules:
5.71/2.71	
5.71/2.71	   MINIMUMA_IN_AG(tree(X1, tree(X2, X3, X4), X5), X6) -> U1_AG(X1, X2, X3, X4, X5, X6, minimumA_in_ag(X3, X6))
5.71/2.71	   MINIMUMA_IN_AG(tree(X1, tree(X2, X3, X4), X5), X6) -> MINIMUMA_IN_AG(X3, X6)
5.71/2.71	
5.71/2.71	R is empty.
5.71/2.71	The argument filtering Pi contains the following mapping:
5.71/2.71	minimumA_in_ag(x1, x2)  =  minimumA_in_ag(x2)
5.71/2.71	
5.71/2.71	tree(x1, x2, x3)  =  tree(x2)
5.71/2.71	
5.71/2.71	MINIMUMA_IN_AG(x1, x2)  =  MINIMUMA_IN_AG(x2)
5.71/2.71	
5.71/2.71	U1_AG(x1, x2, x3, x4, x5, x6, x7)  =  U1_AG(x6, x7)
5.71/2.71	
5.71/2.71	
5.71/2.71	We have to consider all (P,R,Pi)-chains
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(51) DependencyGraphProof (EQUIVALENT)
5.71/2.71	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(52)
5.71/2.71	Obligation:
5.71/2.71	Pi DP problem:
5.71/2.71	The TRS P consists of the following rules:
5.71/2.71	
5.71/2.71	   MINIMUMA_IN_AG(tree(X1, tree(X2, X3, X4), X5), X6) -> MINIMUMA_IN_AG(X3, X6)
5.71/2.71	
5.71/2.71	R is empty.
5.71/2.71	The argument filtering Pi contains the following mapping:
5.71/2.71	tree(x1, x2, x3)  =  tree(x2)
5.71/2.71	
5.71/2.71	MINIMUMA_IN_AG(x1, x2)  =  MINIMUMA_IN_AG(x2)
5.71/2.71	
5.71/2.71	
5.71/2.71	We have to consider all (P,R,Pi)-chains
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(53) PiDPToQDPProof (SOUND)
5.71/2.71	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.71/2.71	----------------------------------------
5.71/2.71	
5.71/2.71	(54)
5.71/2.71	Obligation:
5.71/2.71	Q DP problem:
5.71/2.71	The TRS P consists of the following rules:
5.71/2.71	
5.71/2.71	   MINIMUMA_IN_AG(X6) -> MINIMUMA_IN_AG(X6)
5.71/2.71	
5.71/2.71	R is empty.
5.71/2.71	Q is empty.
5.71/2.71	We have to consider all (P,Q,R)-chains.
5.71/2.74	EOF