6.72/2.59	MAYBE
6.72/2.63	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
6.72/2.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.72/2.63	
6.72/2.63	
6.72/2.63	Left Termination of the query pattern
6.72/2.63	
6.72/2.63	flat(a,g)
6.72/2.63	
6.72/2.63	w.r.t. the given Prolog program could not be shown:
6.72/2.63	
6.72/2.63	(0) Prolog
6.72/2.63	(1) PrologToPiTRSProof [SOUND, 0 ms]
6.72/2.63	(2) PiTRS
6.72/2.63	    (3) DependencyPairsProof [EQUIVALENT, 0 ms]
6.72/2.63	    (4) PiDP
6.72/2.63	    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
6.72/2.63	    (6) PiDP
6.72/2.63	    (7) UsableRulesProof [EQUIVALENT, 0 ms]
6.72/2.63	    (8) PiDP
6.72/2.63	    (9) PiDPToQDPProof [SOUND, 0 ms]
6.72/2.63	    (10) QDP
6.72/2.63	    (11) UsableRulesReductionPairsProof [EQUIVALENT, 7 ms]
6.72/2.63	    (12) QDP
6.72/2.63	    (13) DependencyGraphProof [EQUIVALENT, 0 ms]
6.72/2.63	    (14) QDP
6.72/2.63	    (15) UsableRulesProof [EQUIVALENT, 0 ms]
6.72/2.63	    (16) QDP
6.72/2.63	    (17) QReductionProof [EQUIVALENT, 0 ms]
6.72/2.63	    (18) QDP
6.72/2.63	    (19) NonTerminationLoopProof [COMPLETE, 0 ms]
6.72/2.63	    (20) NO
6.72/2.63	(21) PrologToPiTRSProof [SOUND, 0 ms]
6.72/2.63	(22) PiTRS
6.72/2.63	    (23) DependencyPairsProof [EQUIVALENT, 0 ms]
6.72/2.63	    (24) PiDP
6.72/2.63	    (25) DependencyGraphProof [EQUIVALENT, 0 ms]
6.72/2.63	    (26) PiDP
6.72/2.63	    (27) UsableRulesProof [EQUIVALENT, 0 ms]
6.72/2.63	    (28) PiDP
6.72/2.63	    (29) PiDPToQDPProof [SOUND, 0 ms]
6.72/2.63	    (30) QDP
6.72/2.63	    (31) UsableRulesReductionPairsProof [EQUIVALENT, 8 ms]
6.72/2.63	    (32) QDP
6.72/2.63	    (33) DependencyGraphProof [EQUIVALENT, 0 ms]
6.72/2.63	    (34) QDP
6.72/2.63	    (35) UsableRulesProof [EQUIVALENT, 0 ms]
6.72/2.63	    (36) QDP
6.72/2.63	    (37) QReductionProof [EQUIVALENT, 0 ms]
6.72/2.63	    (38) QDP
6.72/2.63	    (39) NonTerminationLoopProof [COMPLETE, 0 ms]
6.72/2.63	    (40) NO
6.72/2.63	(41) PrologToTRSTransformerProof [SOUND, 0 ms]
6.72/2.63	(42) QTRS
6.72/2.63	    (43) DependencyPairsProof [EQUIVALENT, 0 ms]
6.72/2.63	    (44) QDP
6.72/2.63	    (45) DependencyGraphProof [EQUIVALENT, 0 ms]
6.72/2.63	    (46) QDP
6.72/2.63	    (47) UsableRulesProof [EQUIVALENT, 0 ms]
6.72/2.63	    (48) QDP
6.72/2.63	    (49) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
6.72/2.63	    (50) QDP
6.72/2.63	    (51) NonTerminationLoopProof [COMPLETE, 0 ms]
6.72/2.63	    (52) NO
6.72/2.63	(53) PrologToIRSwTTransformerProof [SOUND, 0 ms]
6.72/2.63	(54) IRSwT
6.72/2.63	    (55) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
6.72/2.63	    (56) IRSwT
6.72/2.63	    (57) IntTRSCompressionProof [EQUIVALENT, 39 ms]
6.72/2.63	    (58) IRSwT
6.72/2.63	    (59) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
6.72/2.63	    (60) IRSwT
6.72/2.63	    (61) IRSwTTerminationDigraphProof [EQUIVALENT, 8 ms]
6.72/2.63	    (62) IRSwT
6.72/2.63	    (63) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 34 ms]
6.72/2.63	    (64) IRSwT
6.72/2.63	(65) PrologToDTProblemTransformerProof [SOUND, 0 ms]
6.72/2.63	(66) TRIPLES
6.72/2.63	    (67) TriplesToPiDPProof [SOUND, 0 ms]
6.72/2.63	    (68) PiDP
6.72/2.63	    (69) DependencyGraphProof [EQUIVALENT, 0 ms]
6.72/2.63	    (70) PiDP
6.72/2.63	    (71) PiDPToQDPProof [SOUND, 0 ms]
6.72/2.63	    (72) QDP
6.72/2.63	    (73) MRRProof [EQUIVALENT, 0 ms]
6.72/2.63	    (74) QDP
6.72/2.63	    (75) NonTerminationLoopProof [COMPLETE, 0 ms]
6.72/2.63	    (76) NO
6.72/2.63	
6.72/2.63	
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(0)
6.72/2.63	Obligation:
6.72/2.63	Clauses:
6.72/2.63	
6.72/2.63	right(tree(X, XS1, XS2), XS2).
6.72/2.63	flat(niltree, nil).
6.72/2.63	flat(tree(X, niltree, XS), cons(X, YS)) :- ','(right(tree(X, niltree, XS), ZS), flat(ZS, YS)).
6.72/2.63	flat(tree(X, tree(Y, YS1, YS2), XS), ZS) :- flat(tree(Y, YS1, tree(X, YS2, XS)), ZS).
6.72/2.63	
6.72/2.63	
6.72/2.63	Query: flat(a,g)
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(1) PrologToPiTRSProof (SOUND)
6.72/2.63	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.72/2.63	
6.72/2.63	flat_in_2: (f,b)
6.72/2.63	
6.72/2.63	Transforming Prolog into the following Term Rewriting System:
6.72/2.63	
6.72/2.63	Pi-finite rewrite system:
6.72/2.63	The TRS R consists of the following rules:
6.72/2.63	
6.72/2.63	   flat_in_ag(niltree, nil) -> flat_out_ag(niltree, nil)
6.72/2.63	   flat_in_ag(tree(X, niltree, XS), cons(X, YS)) -> U1_ag(X, XS, YS, right_in_aa(tree(X, niltree, XS), ZS))
6.72/2.63	   right_in_aa(tree(X, XS1, XS2), XS2) -> right_out_aa(tree(X, XS1, XS2), XS2)
6.72/2.63	   U1_ag(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> U2_ag(X, XS, YS, flat_in_ag(ZS, YS))
6.72/2.63	   flat_in_ag(tree(X, tree(Y, YS1, YS2), XS), ZS) -> U3_ag(X, Y, YS1, YS2, XS, ZS, flat_in_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS))
6.72/2.63	   U3_ag(X, Y, YS1, YS2, XS, ZS, flat_out_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS)) -> flat_out_ag(tree(X, tree(Y, YS1, YS2), XS), ZS)
6.72/2.63	   U2_ag(X, XS, YS, flat_out_ag(ZS, YS)) -> flat_out_ag(tree(X, niltree, XS), cons(X, YS))
6.72/2.63	
6.72/2.63	The argument filtering Pi contains the following mapping:
6.72/2.63	flat_in_ag(x1, x2)  =  flat_in_ag(x2)
6.72/2.63	
6.72/2.63	nil  =  nil
6.72/2.63	
6.72/2.63	flat_out_ag(x1, x2)  =  flat_out_ag
6.72/2.63	
6.72/2.63	cons(x1, x2)  =  cons(x1, x2)
6.72/2.63	
6.72/2.63	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
6.72/2.63	
6.72/2.63	right_in_aa(x1, x2)  =  right_in_aa
6.72/2.63	
6.72/2.63	right_out_aa(x1, x2)  =  right_out_aa
6.72/2.63	
6.72/2.63	tree(x1, x2, x3)  =  tree(x2, x3)
6.72/2.63	
6.72/2.63	U2_ag(x1, x2, x3, x4)  =  U2_ag(x4)
6.72/2.63	
6.72/2.63	U3_ag(x1, x2, x3, x4, x5, x6, x7)  =  U3_ag(x7)
6.72/2.63	
6.72/2.63	
6.72/2.63	
6.72/2.63	
6.72/2.63	
6.72/2.63	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.72/2.63	
6.72/2.63	
6.72/2.63	
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(2)
6.72/2.63	Obligation:
6.72/2.63	Pi-finite rewrite system:
6.72/2.63	The TRS R consists of the following rules:
6.72/2.63	
6.72/2.63	   flat_in_ag(niltree, nil) -> flat_out_ag(niltree, nil)
6.72/2.63	   flat_in_ag(tree(X, niltree, XS), cons(X, YS)) -> U1_ag(X, XS, YS, right_in_aa(tree(X, niltree, XS), ZS))
6.72/2.63	   right_in_aa(tree(X, XS1, XS2), XS2) -> right_out_aa(tree(X, XS1, XS2), XS2)
6.72/2.63	   U1_ag(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> U2_ag(X, XS, YS, flat_in_ag(ZS, YS))
6.72/2.63	   flat_in_ag(tree(X, tree(Y, YS1, YS2), XS), ZS) -> U3_ag(X, Y, YS1, YS2, XS, ZS, flat_in_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS))
6.72/2.63	   U3_ag(X, Y, YS1, YS2, XS, ZS, flat_out_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS)) -> flat_out_ag(tree(X, tree(Y, YS1, YS2), XS), ZS)
6.72/2.63	   U2_ag(X, XS, YS, flat_out_ag(ZS, YS)) -> flat_out_ag(tree(X, niltree, XS), cons(X, YS))
6.72/2.63	
6.72/2.63	The argument filtering Pi contains the following mapping:
6.72/2.63	flat_in_ag(x1, x2)  =  flat_in_ag(x2)
6.72/2.63	
6.72/2.63	nil  =  nil
6.72/2.63	
6.72/2.63	flat_out_ag(x1, x2)  =  flat_out_ag
6.72/2.63	
6.72/2.63	cons(x1, x2)  =  cons(x1, x2)
6.72/2.63	
6.72/2.63	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
6.72/2.63	
6.72/2.63	right_in_aa(x1, x2)  =  right_in_aa
6.72/2.63	
6.72/2.63	right_out_aa(x1, x2)  =  right_out_aa
6.72/2.63	
6.72/2.63	tree(x1, x2, x3)  =  tree(x2, x3)
6.72/2.63	
6.72/2.63	U2_ag(x1, x2, x3, x4)  =  U2_ag(x4)
6.72/2.63	
6.72/2.63	U3_ag(x1, x2, x3, x4, x5, x6, x7)  =  U3_ag(x7)
6.72/2.63	
6.72/2.63	
6.72/2.63	
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(3) DependencyPairsProof (EQUIVALENT)
6.72/2.63	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.72/2.63	Pi DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   FLAT_IN_AG(tree(X, niltree, XS), cons(X, YS)) -> U1_AG(X, XS, YS, right_in_aa(tree(X, niltree, XS), ZS))
6.72/2.63	   FLAT_IN_AG(tree(X, niltree, XS), cons(X, YS)) -> RIGHT_IN_AA(tree(X, niltree, XS), ZS)
6.72/2.63	   U1_AG(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> U2_AG(X, XS, YS, flat_in_ag(ZS, YS))
6.72/2.63	   U1_AG(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> FLAT_IN_AG(ZS, YS)
6.72/2.63	   FLAT_IN_AG(tree(X, tree(Y, YS1, YS2), XS), ZS) -> U3_AG(X, Y, YS1, YS2, XS, ZS, flat_in_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS))
6.72/2.63	   FLAT_IN_AG(tree(X, tree(Y, YS1, YS2), XS), ZS) -> FLAT_IN_AG(tree(Y, YS1, tree(X, YS2, XS)), ZS)
6.72/2.63	
6.72/2.63	The TRS R consists of the following rules:
6.72/2.63	
6.72/2.63	   flat_in_ag(niltree, nil) -> flat_out_ag(niltree, nil)
6.72/2.63	   flat_in_ag(tree(X, niltree, XS), cons(X, YS)) -> U1_ag(X, XS, YS, right_in_aa(tree(X, niltree, XS), ZS))
6.72/2.63	   right_in_aa(tree(X, XS1, XS2), XS2) -> right_out_aa(tree(X, XS1, XS2), XS2)
6.72/2.63	   U1_ag(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> U2_ag(X, XS, YS, flat_in_ag(ZS, YS))
6.72/2.63	   flat_in_ag(tree(X, tree(Y, YS1, YS2), XS), ZS) -> U3_ag(X, Y, YS1, YS2, XS, ZS, flat_in_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS))
6.72/2.63	   U3_ag(X, Y, YS1, YS2, XS, ZS, flat_out_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS)) -> flat_out_ag(tree(X, tree(Y, YS1, YS2), XS), ZS)
6.72/2.63	   U2_ag(X, XS, YS, flat_out_ag(ZS, YS)) -> flat_out_ag(tree(X, niltree, XS), cons(X, YS))
6.72/2.63	
6.72/2.63	The argument filtering Pi contains the following mapping:
6.72/2.63	flat_in_ag(x1, x2)  =  flat_in_ag(x2)
6.72/2.63	
6.72/2.63	nil  =  nil
6.72/2.63	
6.72/2.63	flat_out_ag(x1, x2)  =  flat_out_ag
6.72/2.63	
6.72/2.63	cons(x1, x2)  =  cons(x1, x2)
6.72/2.63	
6.72/2.63	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
6.72/2.63	
6.72/2.63	right_in_aa(x1, x2)  =  right_in_aa
6.72/2.63	
6.72/2.63	right_out_aa(x1, x2)  =  right_out_aa
6.72/2.63	
6.72/2.63	tree(x1, x2, x3)  =  tree(x2, x3)
6.72/2.63	
6.72/2.63	U2_ag(x1, x2, x3, x4)  =  U2_ag(x4)
6.72/2.63	
6.72/2.63	U3_ag(x1, x2, x3, x4, x5, x6, x7)  =  U3_ag(x7)
6.72/2.63	
6.72/2.63	FLAT_IN_AG(x1, x2)  =  FLAT_IN_AG(x2)
6.72/2.63	
6.72/2.63	U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
6.72/2.63	
6.72/2.63	RIGHT_IN_AA(x1, x2)  =  RIGHT_IN_AA
6.72/2.63	
6.72/2.63	U2_AG(x1, x2, x3, x4)  =  U2_AG(x4)
6.72/2.63	
6.72/2.63	U3_AG(x1, x2, x3, x4, x5, x6, x7)  =  U3_AG(x7)
6.72/2.63	
6.72/2.63	
6.72/2.63	We have to consider all (P,R,Pi)-chains
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(4)
6.72/2.63	Obligation:
6.72/2.63	Pi DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   FLAT_IN_AG(tree(X, niltree, XS), cons(X, YS)) -> U1_AG(X, XS, YS, right_in_aa(tree(X, niltree, XS), ZS))
6.72/2.63	   FLAT_IN_AG(tree(X, niltree, XS), cons(X, YS)) -> RIGHT_IN_AA(tree(X, niltree, XS), ZS)
6.72/2.63	   U1_AG(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> U2_AG(X, XS, YS, flat_in_ag(ZS, YS))
6.72/2.63	   U1_AG(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> FLAT_IN_AG(ZS, YS)
6.72/2.63	   FLAT_IN_AG(tree(X, tree(Y, YS1, YS2), XS), ZS) -> U3_AG(X, Y, YS1, YS2, XS, ZS, flat_in_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS))
6.72/2.63	   FLAT_IN_AG(tree(X, tree(Y, YS1, YS2), XS), ZS) -> FLAT_IN_AG(tree(Y, YS1, tree(X, YS2, XS)), ZS)
6.72/2.63	
6.72/2.63	The TRS R consists of the following rules:
6.72/2.63	
6.72/2.63	   flat_in_ag(niltree, nil) -> flat_out_ag(niltree, nil)
6.72/2.63	   flat_in_ag(tree(X, niltree, XS), cons(X, YS)) -> U1_ag(X, XS, YS, right_in_aa(tree(X, niltree, XS), ZS))
6.72/2.63	   right_in_aa(tree(X, XS1, XS2), XS2) -> right_out_aa(tree(X, XS1, XS2), XS2)
6.72/2.63	   U1_ag(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> U2_ag(X, XS, YS, flat_in_ag(ZS, YS))
6.72/2.63	   flat_in_ag(tree(X, tree(Y, YS1, YS2), XS), ZS) -> U3_ag(X, Y, YS1, YS2, XS, ZS, flat_in_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS))
6.72/2.63	   U3_ag(X, Y, YS1, YS2, XS, ZS, flat_out_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS)) -> flat_out_ag(tree(X, tree(Y, YS1, YS2), XS), ZS)
6.72/2.63	   U2_ag(X, XS, YS, flat_out_ag(ZS, YS)) -> flat_out_ag(tree(X, niltree, XS), cons(X, YS))
6.72/2.63	
6.72/2.63	The argument filtering Pi contains the following mapping:
6.72/2.63	flat_in_ag(x1, x2)  =  flat_in_ag(x2)
6.72/2.63	
6.72/2.63	nil  =  nil
6.72/2.63	
6.72/2.63	flat_out_ag(x1, x2)  =  flat_out_ag
6.72/2.63	
6.72/2.63	cons(x1, x2)  =  cons(x1, x2)
6.72/2.63	
6.72/2.63	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
6.72/2.63	
6.72/2.63	right_in_aa(x1, x2)  =  right_in_aa
6.72/2.63	
6.72/2.63	right_out_aa(x1, x2)  =  right_out_aa
6.72/2.63	
6.72/2.63	tree(x1, x2, x3)  =  tree(x2, x3)
6.72/2.63	
6.72/2.63	U2_ag(x1, x2, x3, x4)  =  U2_ag(x4)
6.72/2.63	
6.72/2.63	U3_ag(x1, x2, x3, x4, x5, x6, x7)  =  U3_ag(x7)
6.72/2.63	
6.72/2.63	FLAT_IN_AG(x1, x2)  =  FLAT_IN_AG(x2)
6.72/2.63	
6.72/2.63	U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
6.72/2.63	
6.72/2.63	RIGHT_IN_AA(x1, x2)  =  RIGHT_IN_AA
6.72/2.63	
6.72/2.63	U2_AG(x1, x2, x3, x4)  =  U2_AG(x4)
6.72/2.63	
6.72/2.63	U3_AG(x1, x2, x3, x4, x5, x6, x7)  =  U3_AG(x7)
6.72/2.63	
6.72/2.63	
6.72/2.63	We have to consider all (P,R,Pi)-chains
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(5) DependencyGraphProof (EQUIVALENT)
6.72/2.63	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(6)
6.72/2.63	Obligation:
6.72/2.63	Pi DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   U1_AG(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> FLAT_IN_AG(ZS, YS)
6.72/2.63	   FLAT_IN_AG(tree(X, niltree, XS), cons(X, YS)) -> U1_AG(X, XS, YS, right_in_aa(tree(X, niltree, XS), ZS))
6.72/2.63	   FLAT_IN_AG(tree(X, tree(Y, YS1, YS2), XS), ZS) -> FLAT_IN_AG(tree(Y, YS1, tree(X, YS2, XS)), ZS)
6.72/2.63	
6.72/2.63	The TRS R consists of the following rules:
6.72/2.63	
6.72/2.63	   flat_in_ag(niltree, nil) -> flat_out_ag(niltree, nil)
6.72/2.63	   flat_in_ag(tree(X, niltree, XS), cons(X, YS)) -> U1_ag(X, XS, YS, right_in_aa(tree(X, niltree, XS), ZS))
6.72/2.63	   right_in_aa(tree(X, XS1, XS2), XS2) -> right_out_aa(tree(X, XS1, XS2), XS2)
6.72/2.63	   U1_ag(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> U2_ag(X, XS, YS, flat_in_ag(ZS, YS))
6.72/2.63	   flat_in_ag(tree(X, tree(Y, YS1, YS2), XS), ZS) -> U3_ag(X, Y, YS1, YS2, XS, ZS, flat_in_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS))
6.72/2.63	   U3_ag(X, Y, YS1, YS2, XS, ZS, flat_out_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS)) -> flat_out_ag(tree(X, tree(Y, YS1, YS2), XS), ZS)
6.72/2.63	   U2_ag(X, XS, YS, flat_out_ag(ZS, YS)) -> flat_out_ag(tree(X, niltree, XS), cons(X, YS))
6.72/2.63	
6.72/2.63	The argument filtering Pi contains the following mapping:
6.72/2.63	flat_in_ag(x1, x2)  =  flat_in_ag(x2)
6.72/2.63	
6.72/2.63	nil  =  nil
6.72/2.63	
6.72/2.63	flat_out_ag(x1, x2)  =  flat_out_ag
6.72/2.63	
6.72/2.63	cons(x1, x2)  =  cons(x1, x2)
6.72/2.63	
6.72/2.63	U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)
6.72/2.63	
6.72/2.63	right_in_aa(x1, x2)  =  right_in_aa
6.72/2.63	
6.72/2.63	right_out_aa(x1, x2)  =  right_out_aa
6.72/2.63	
6.72/2.63	tree(x1, x2, x3)  =  tree(x2, x3)
6.72/2.63	
6.72/2.63	U2_ag(x1, x2, x3, x4)  =  U2_ag(x4)
6.72/2.63	
6.72/2.63	U3_ag(x1, x2, x3, x4, x5, x6, x7)  =  U3_ag(x7)
6.72/2.63	
6.72/2.63	FLAT_IN_AG(x1, x2)  =  FLAT_IN_AG(x2)
6.72/2.63	
6.72/2.63	U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
6.72/2.63	
6.72/2.63	
6.72/2.63	We have to consider all (P,R,Pi)-chains
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(7) UsableRulesProof (EQUIVALENT)
6.72/2.63	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(8)
6.72/2.63	Obligation:
6.72/2.63	Pi DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   U1_AG(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> FLAT_IN_AG(ZS, YS)
6.72/2.63	   FLAT_IN_AG(tree(X, niltree, XS), cons(X, YS)) -> U1_AG(X, XS, YS, right_in_aa(tree(X, niltree, XS), ZS))
6.72/2.63	   FLAT_IN_AG(tree(X, tree(Y, YS1, YS2), XS), ZS) -> FLAT_IN_AG(tree(Y, YS1, tree(X, YS2, XS)), ZS)
6.72/2.63	
6.72/2.63	The TRS R consists of the following rules:
6.72/2.63	
6.72/2.63	   right_in_aa(tree(X, XS1, XS2), XS2) -> right_out_aa(tree(X, XS1, XS2), XS2)
6.72/2.63	
6.72/2.63	The argument filtering Pi contains the following mapping:
6.72/2.63	cons(x1, x2)  =  cons(x1, x2)
6.72/2.63	
6.72/2.63	right_in_aa(x1, x2)  =  right_in_aa
6.72/2.63	
6.72/2.63	right_out_aa(x1, x2)  =  right_out_aa
6.72/2.63	
6.72/2.63	tree(x1, x2, x3)  =  tree(x2, x3)
6.72/2.63	
6.72/2.63	FLAT_IN_AG(x1, x2)  =  FLAT_IN_AG(x2)
6.72/2.63	
6.72/2.63	U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)
6.72/2.63	
6.72/2.63	
6.72/2.63	We have to consider all (P,R,Pi)-chains
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(9) PiDPToQDPProof (SOUND)
6.72/2.63	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(10)
6.72/2.63	Obligation:
6.72/2.63	Q DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   U1_AG(YS, right_out_aa) -> FLAT_IN_AG(YS)
6.72/2.63	   FLAT_IN_AG(cons(X, YS)) -> U1_AG(YS, right_in_aa)
6.72/2.63	   FLAT_IN_AG(ZS) -> FLAT_IN_AG(ZS)
6.72/2.63	
6.72/2.63	The TRS R consists of the following rules:
6.72/2.63	
6.72/2.63	   right_in_aa -> right_out_aa
6.72/2.63	
6.72/2.63	The set Q consists of the following terms:
6.72/2.63	
6.72/2.63	   right_in_aa
6.72/2.63	
6.72/2.63	We have to consider all (P,Q,R)-chains.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(11) UsableRulesReductionPairsProof (EQUIVALENT)
6.72/2.63	By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
6.72/2.63	
6.72/2.63	The following dependency pairs can be deleted:
6.72/2.63	
6.72/2.63	   FLAT_IN_AG(cons(X, YS)) -> U1_AG(YS, right_in_aa)
6.72/2.63	No rules are removed from R.
6.72/2.63	
6.72/2.63	Used ordering: POLO with Polynomial interpretation [POLO]:
6.72/2.63	
6.72/2.63	   POL(FLAT_IN_AG(x_1)) = x_1
6.72/2.63	   POL(U1_AG(x_1, x_2)) = x_1 + x_2
6.72/2.63	   POL(cons(x_1, x_2)) = x_1 + 2*x_2
6.72/2.63	   POL(right_in_aa) = 0
6.72/2.63	   POL(right_out_aa) = 0
6.72/2.63	
6.72/2.63	
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(12)
6.72/2.63	Obligation:
6.72/2.63	Q DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   U1_AG(YS, right_out_aa) -> FLAT_IN_AG(YS)
6.72/2.63	   FLAT_IN_AG(ZS) -> FLAT_IN_AG(ZS)
6.72/2.63	
6.72/2.63	The TRS R consists of the following rules:
6.72/2.63	
6.72/2.63	   right_in_aa -> right_out_aa
6.72/2.63	
6.72/2.63	The set Q consists of the following terms:
6.72/2.63	
6.72/2.63	   right_in_aa
6.72/2.63	
6.72/2.63	We have to consider all (P,Q,R)-chains.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(13) DependencyGraphProof (EQUIVALENT)
6.72/2.63	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(14)
6.72/2.63	Obligation:
6.72/2.63	Q DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   FLAT_IN_AG(ZS) -> FLAT_IN_AG(ZS)
6.72/2.63	
6.72/2.63	The TRS R consists of the following rules:
6.72/2.63	
6.72/2.63	   right_in_aa -> right_out_aa
6.72/2.63	
6.72/2.63	The set Q consists of the following terms:
6.72/2.63	
6.72/2.63	   right_in_aa
6.72/2.63	
6.72/2.63	We have to consider all (P,Q,R)-chains.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(15) UsableRulesProof (EQUIVALENT)
6.72/2.63	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.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(16)
6.72/2.63	Obligation:
6.72/2.63	Q DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   FLAT_IN_AG(ZS) -> FLAT_IN_AG(ZS)
6.72/2.63	
6.72/2.63	R is empty.
6.72/2.63	The set Q consists of the following terms:
6.72/2.63	
6.72/2.63	   right_in_aa
6.72/2.63	
6.72/2.63	We have to consider all (P,Q,R)-chains.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(17) QReductionProof (EQUIVALENT)
6.72/2.63	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
6.72/2.63	
6.72/2.63	   right_in_aa
6.72/2.63	
6.72/2.63	
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(18)
6.72/2.63	Obligation:
6.72/2.63	Q DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   FLAT_IN_AG(ZS) -> FLAT_IN_AG(ZS)
6.72/2.63	
6.72/2.63	R is empty.
6.72/2.63	Q is empty.
6.72/2.63	We have to consider all (P,Q,R)-chains.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(19) NonTerminationLoopProof (COMPLETE)
6.72/2.63	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
6.72/2.63	Found a loop by semiunifying a rule from P directly.
6.72/2.63	
6.72/2.63	s = FLAT_IN_AG(ZS) evaluates to  t =FLAT_IN_AG(ZS)
6.72/2.63	
6.72/2.63	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
6.72/2.63	* Matcher: [ ]
6.72/2.63	* Semiunifier: [ ]
6.72/2.63	
6.72/2.63	--------------------------------------------------------------------------------
6.72/2.63	Rewriting sequence
6.72/2.63	
6.72/2.63	The DP semiunifies directly so there is only one rewrite step from FLAT_IN_AG(ZS) to FLAT_IN_AG(ZS).
6.72/2.63	
6.72/2.63	
6.72/2.63	
6.72/2.63	
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(20)
6.72/2.63	NO
6.72/2.63	
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(21) PrologToPiTRSProof (SOUND)
6.72/2.63	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.72/2.63	
6.72/2.63	flat_in_2: (f,b)
6.72/2.63	
6.72/2.63	Transforming Prolog into the following Term Rewriting System:
6.72/2.63	
6.72/2.63	Pi-finite rewrite system:
6.72/2.63	The TRS R consists of the following rules:
6.72/2.63	
6.72/2.63	   flat_in_ag(niltree, nil) -> flat_out_ag(niltree, nil)
6.72/2.63	   flat_in_ag(tree(X, niltree, XS), cons(X, YS)) -> U1_ag(X, XS, YS, right_in_aa(tree(X, niltree, XS), ZS))
6.72/2.63	   right_in_aa(tree(X, XS1, XS2), XS2) -> right_out_aa(tree(X, XS1, XS2), XS2)
6.72/2.63	   U1_ag(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> U2_ag(X, XS, YS, flat_in_ag(ZS, YS))
6.72/2.63	   flat_in_ag(tree(X, tree(Y, YS1, YS2), XS), ZS) -> U3_ag(X, Y, YS1, YS2, XS, ZS, flat_in_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS))
6.72/2.63	   U3_ag(X, Y, YS1, YS2, XS, ZS, flat_out_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS)) -> flat_out_ag(tree(X, tree(Y, YS1, YS2), XS), ZS)
6.72/2.63	   U2_ag(X, XS, YS, flat_out_ag(ZS, YS)) -> flat_out_ag(tree(X, niltree, XS), cons(X, YS))
6.72/2.63	
6.72/2.63	The argument filtering Pi contains the following mapping:
6.72/2.63	flat_in_ag(x1, x2)  =  flat_in_ag(x2)
6.72/2.63	
6.72/2.63	nil  =  nil
6.72/2.63	
6.72/2.63	flat_out_ag(x1, x2)  =  flat_out_ag(x2)
6.72/2.63	
6.72/2.63	cons(x1, x2)  =  cons(x1, x2)
6.72/2.63	
6.72/2.63	U1_ag(x1, x2, x3, x4)  =  U1_ag(x1, x3, x4)
6.72/2.63	
6.72/2.63	right_in_aa(x1, x2)  =  right_in_aa
6.72/2.63	
6.72/2.63	right_out_aa(x1, x2)  =  right_out_aa
6.72/2.63	
6.72/2.63	tree(x1, x2, x3)  =  tree(x2, x3)
6.72/2.63	
6.72/2.63	U2_ag(x1, x2, x3, x4)  =  U2_ag(x1, x3, x4)
6.72/2.63	
6.72/2.63	U3_ag(x1, x2, x3, x4, x5, x6, x7)  =  U3_ag(x6, x7)
6.72/2.63	
6.72/2.63	
6.72/2.63	
6.72/2.63	
6.72/2.63	
6.72/2.63	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.72/2.63	
6.72/2.63	
6.72/2.63	
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(22)
6.72/2.63	Obligation:
6.72/2.63	Pi-finite rewrite system:
6.72/2.63	The TRS R consists of the following rules:
6.72/2.63	
6.72/2.63	   flat_in_ag(niltree, nil) -> flat_out_ag(niltree, nil)
6.72/2.63	   flat_in_ag(tree(X, niltree, XS), cons(X, YS)) -> U1_ag(X, XS, YS, right_in_aa(tree(X, niltree, XS), ZS))
6.72/2.63	   right_in_aa(tree(X, XS1, XS2), XS2) -> right_out_aa(tree(X, XS1, XS2), XS2)
6.72/2.63	   U1_ag(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> U2_ag(X, XS, YS, flat_in_ag(ZS, YS))
6.72/2.63	   flat_in_ag(tree(X, tree(Y, YS1, YS2), XS), ZS) -> U3_ag(X, Y, YS1, YS2, XS, ZS, flat_in_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS))
6.72/2.63	   U3_ag(X, Y, YS1, YS2, XS, ZS, flat_out_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS)) -> flat_out_ag(tree(X, tree(Y, YS1, YS2), XS), ZS)
6.72/2.63	   U2_ag(X, XS, YS, flat_out_ag(ZS, YS)) -> flat_out_ag(tree(X, niltree, XS), cons(X, YS))
6.72/2.63	
6.72/2.63	The argument filtering Pi contains the following mapping:
6.72/2.63	flat_in_ag(x1, x2)  =  flat_in_ag(x2)
6.72/2.63	
6.72/2.63	nil  =  nil
6.72/2.63	
6.72/2.63	flat_out_ag(x1, x2)  =  flat_out_ag(x2)
6.72/2.63	
6.72/2.63	cons(x1, x2)  =  cons(x1, x2)
6.72/2.63	
6.72/2.63	U1_ag(x1, x2, x3, x4)  =  U1_ag(x1, x3, x4)
6.72/2.63	
6.72/2.63	right_in_aa(x1, x2)  =  right_in_aa
6.72/2.63	
6.72/2.63	right_out_aa(x1, x2)  =  right_out_aa
6.72/2.63	
6.72/2.63	tree(x1, x2, x3)  =  tree(x2, x3)
6.72/2.63	
6.72/2.63	U2_ag(x1, x2, x3, x4)  =  U2_ag(x1, x3, x4)
6.72/2.63	
6.72/2.63	U3_ag(x1, x2, x3, x4, x5, x6, x7)  =  U3_ag(x6, x7)
6.72/2.63	
6.72/2.63	
6.72/2.63	
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(23) DependencyPairsProof (EQUIVALENT)
6.72/2.63	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.72/2.63	Pi DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   FLAT_IN_AG(tree(X, niltree, XS), cons(X, YS)) -> U1_AG(X, XS, YS, right_in_aa(tree(X, niltree, XS), ZS))
6.72/2.63	   FLAT_IN_AG(tree(X, niltree, XS), cons(X, YS)) -> RIGHT_IN_AA(tree(X, niltree, XS), ZS)
6.72/2.63	   U1_AG(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> U2_AG(X, XS, YS, flat_in_ag(ZS, YS))
6.72/2.63	   U1_AG(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> FLAT_IN_AG(ZS, YS)
6.72/2.63	   FLAT_IN_AG(tree(X, tree(Y, YS1, YS2), XS), ZS) -> U3_AG(X, Y, YS1, YS2, XS, ZS, flat_in_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS))
6.72/2.63	   FLAT_IN_AG(tree(X, tree(Y, YS1, YS2), XS), ZS) -> FLAT_IN_AG(tree(Y, YS1, tree(X, YS2, XS)), ZS)
6.72/2.63	
6.72/2.63	The TRS R consists of the following rules:
6.72/2.63	
6.72/2.63	   flat_in_ag(niltree, nil) -> flat_out_ag(niltree, nil)
6.72/2.63	   flat_in_ag(tree(X, niltree, XS), cons(X, YS)) -> U1_ag(X, XS, YS, right_in_aa(tree(X, niltree, XS), ZS))
6.72/2.63	   right_in_aa(tree(X, XS1, XS2), XS2) -> right_out_aa(tree(X, XS1, XS2), XS2)
6.72/2.63	   U1_ag(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> U2_ag(X, XS, YS, flat_in_ag(ZS, YS))
6.72/2.63	   flat_in_ag(tree(X, tree(Y, YS1, YS2), XS), ZS) -> U3_ag(X, Y, YS1, YS2, XS, ZS, flat_in_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS))
6.72/2.63	   U3_ag(X, Y, YS1, YS2, XS, ZS, flat_out_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS)) -> flat_out_ag(tree(X, tree(Y, YS1, YS2), XS), ZS)
6.72/2.63	   U2_ag(X, XS, YS, flat_out_ag(ZS, YS)) -> flat_out_ag(tree(X, niltree, XS), cons(X, YS))
6.72/2.63	
6.72/2.63	The argument filtering Pi contains the following mapping:
6.72/2.63	flat_in_ag(x1, x2)  =  flat_in_ag(x2)
6.72/2.63	
6.72/2.63	nil  =  nil
6.72/2.63	
6.72/2.63	flat_out_ag(x1, x2)  =  flat_out_ag(x2)
6.72/2.63	
6.72/2.63	cons(x1, x2)  =  cons(x1, x2)
6.72/2.63	
6.72/2.63	U1_ag(x1, x2, x3, x4)  =  U1_ag(x1, x3, x4)
6.72/2.63	
6.72/2.63	right_in_aa(x1, x2)  =  right_in_aa
6.72/2.63	
6.72/2.63	right_out_aa(x1, x2)  =  right_out_aa
6.72/2.63	
6.72/2.63	tree(x1, x2, x3)  =  tree(x2, x3)
6.72/2.63	
6.72/2.63	U2_ag(x1, x2, x3, x4)  =  U2_ag(x1, x3, x4)
6.72/2.63	
6.72/2.63	U3_ag(x1, x2, x3, x4, x5, x6, x7)  =  U3_ag(x6, x7)
6.72/2.63	
6.72/2.63	FLAT_IN_AG(x1, x2)  =  FLAT_IN_AG(x2)
6.72/2.63	
6.72/2.63	U1_AG(x1, x2, x3, x4)  =  U1_AG(x1, x3, x4)
6.72/2.63	
6.72/2.63	RIGHT_IN_AA(x1, x2)  =  RIGHT_IN_AA
6.72/2.63	
6.72/2.63	U2_AG(x1, x2, x3, x4)  =  U2_AG(x1, x3, x4)
6.72/2.63	
6.72/2.63	U3_AG(x1, x2, x3, x4, x5, x6, x7)  =  U3_AG(x6, x7)
6.72/2.63	
6.72/2.63	
6.72/2.63	We have to consider all (P,R,Pi)-chains
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(24)
6.72/2.63	Obligation:
6.72/2.63	Pi DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   FLAT_IN_AG(tree(X, niltree, XS), cons(X, YS)) -> U1_AG(X, XS, YS, right_in_aa(tree(X, niltree, XS), ZS))
6.72/2.63	   FLAT_IN_AG(tree(X, niltree, XS), cons(X, YS)) -> RIGHT_IN_AA(tree(X, niltree, XS), ZS)
6.72/2.63	   U1_AG(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> U2_AG(X, XS, YS, flat_in_ag(ZS, YS))
6.72/2.63	   U1_AG(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> FLAT_IN_AG(ZS, YS)
6.72/2.63	   FLAT_IN_AG(tree(X, tree(Y, YS1, YS2), XS), ZS) -> U3_AG(X, Y, YS1, YS2, XS, ZS, flat_in_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS))
6.72/2.63	   FLAT_IN_AG(tree(X, tree(Y, YS1, YS2), XS), ZS) -> FLAT_IN_AG(tree(Y, YS1, tree(X, YS2, XS)), ZS)
6.72/2.63	
6.72/2.63	The TRS R consists of the following rules:
6.72/2.63	
6.72/2.63	   flat_in_ag(niltree, nil) -> flat_out_ag(niltree, nil)
6.72/2.63	   flat_in_ag(tree(X, niltree, XS), cons(X, YS)) -> U1_ag(X, XS, YS, right_in_aa(tree(X, niltree, XS), ZS))
6.72/2.63	   right_in_aa(tree(X, XS1, XS2), XS2) -> right_out_aa(tree(X, XS1, XS2), XS2)
6.72/2.63	   U1_ag(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> U2_ag(X, XS, YS, flat_in_ag(ZS, YS))
6.72/2.63	   flat_in_ag(tree(X, tree(Y, YS1, YS2), XS), ZS) -> U3_ag(X, Y, YS1, YS2, XS, ZS, flat_in_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS))
6.72/2.63	   U3_ag(X, Y, YS1, YS2, XS, ZS, flat_out_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS)) -> flat_out_ag(tree(X, tree(Y, YS1, YS2), XS), ZS)
6.72/2.63	   U2_ag(X, XS, YS, flat_out_ag(ZS, YS)) -> flat_out_ag(tree(X, niltree, XS), cons(X, YS))
6.72/2.63	
6.72/2.63	The argument filtering Pi contains the following mapping:
6.72/2.63	flat_in_ag(x1, x2)  =  flat_in_ag(x2)
6.72/2.63	
6.72/2.63	nil  =  nil
6.72/2.63	
6.72/2.63	flat_out_ag(x1, x2)  =  flat_out_ag(x2)
6.72/2.63	
6.72/2.63	cons(x1, x2)  =  cons(x1, x2)
6.72/2.63	
6.72/2.63	U1_ag(x1, x2, x3, x4)  =  U1_ag(x1, x3, x4)
6.72/2.63	
6.72/2.63	right_in_aa(x1, x2)  =  right_in_aa
6.72/2.63	
6.72/2.63	right_out_aa(x1, x2)  =  right_out_aa
6.72/2.63	
6.72/2.63	tree(x1, x2, x3)  =  tree(x2, x3)
6.72/2.63	
6.72/2.63	U2_ag(x1, x2, x3, x4)  =  U2_ag(x1, x3, x4)
6.72/2.63	
6.72/2.63	U3_ag(x1, x2, x3, x4, x5, x6, x7)  =  U3_ag(x6, x7)
6.72/2.63	
6.72/2.63	FLAT_IN_AG(x1, x2)  =  FLAT_IN_AG(x2)
6.72/2.63	
6.72/2.63	U1_AG(x1, x2, x3, x4)  =  U1_AG(x1, x3, x4)
6.72/2.63	
6.72/2.63	RIGHT_IN_AA(x1, x2)  =  RIGHT_IN_AA
6.72/2.63	
6.72/2.63	U2_AG(x1, x2, x3, x4)  =  U2_AG(x1, x3, x4)
6.72/2.63	
6.72/2.63	U3_AG(x1, x2, x3, x4, x5, x6, x7)  =  U3_AG(x6, x7)
6.72/2.63	
6.72/2.63	
6.72/2.63	We have to consider all (P,R,Pi)-chains
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(25) DependencyGraphProof (EQUIVALENT)
6.72/2.63	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(26)
6.72/2.63	Obligation:
6.72/2.63	Pi DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   U1_AG(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> FLAT_IN_AG(ZS, YS)
6.72/2.63	   FLAT_IN_AG(tree(X, niltree, XS), cons(X, YS)) -> U1_AG(X, XS, YS, right_in_aa(tree(X, niltree, XS), ZS))
6.72/2.63	   FLAT_IN_AG(tree(X, tree(Y, YS1, YS2), XS), ZS) -> FLAT_IN_AG(tree(Y, YS1, tree(X, YS2, XS)), ZS)
6.72/2.63	
6.72/2.63	The TRS R consists of the following rules:
6.72/2.63	
6.72/2.63	   flat_in_ag(niltree, nil) -> flat_out_ag(niltree, nil)
6.72/2.63	   flat_in_ag(tree(X, niltree, XS), cons(X, YS)) -> U1_ag(X, XS, YS, right_in_aa(tree(X, niltree, XS), ZS))
6.72/2.63	   right_in_aa(tree(X, XS1, XS2), XS2) -> right_out_aa(tree(X, XS1, XS2), XS2)
6.72/2.63	   U1_ag(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> U2_ag(X, XS, YS, flat_in_ag(ZS, YS))
6.72/2.63	   flat_in_ag(tree(X, tree(Y, YS1, YS2), XS), ZS) -> U3_ag(X, Y, YS1, YS2, XS, ZS, flat_in_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS))
6.72/2.63	   U3_ag(X, Y, YS1, YS2, XS, ZS, flat_out_ag(tree(Y, YS1, tree(X, YS2, XS)), ZS)) -> flat_out_ag(tree(X, tree(Y, YS1, YS2), XS), ZS)
6.72/2.63	   U2_ag(X, XS, YS, flat_out_ag(ZS, YS)) -> flat_out_ag(tree(X, niltree, XS), cons(X, YS))
6.72/2.63	
6.72/2.63	The argument filtering Pi contains the following mapping:
6.72/2.63	flat_in_ag(x1, x2)  =  flat_in_ag(x2)
6.72/2.63	
6.72/2.63	nil  =  nil
6.72/2.63	
6.72/2.63	flat_out_ag(x1, x2)  =  flat_out_ag(x2)
6.72/2.63	
6.72/2.63	cons(x1, x2)  =  cons(x1, x2)
6.72/2.63	
6.72/2.63	U1_ag(x1, x2, x3, x4)  =  U1_ag(x1, x3, x4)
6.72/2.63	
6.72/2.63	right_in_aa(x1, x2)  =  right_in_aa
6.72/2.63	
6.72/2.63	right_out_aa(x1, x2)  =  right_out_aa
6.72/2.63	
6.72/2.63	tree(x1, x2, x3)  =  tree(x2, x3)
6.72/2.63	
6.72/2.63	U2_ag(x1, x2, x3, x4)  =  U2_ag(x1, x3, x4)
6.72/2.63	
6.72/2.63	U3_ag(x1, x2, x3, x4, x5, x6, x7)  =  U3_ag(x6, x7)
6.72/2.63	
6.72/2.63	FLAT_IN_AG(x1, x2)  =  FLAT_IN_AG(x2)
6.72/2.63	
6.72/2.63	U1_AG(x1, x2, x3, x4)  =  U1_AG(x1, x3, x4)
6.72/2.63	
6.72/2.63	
6.72/2.63	We have to consider all (P,R,Pi)-chains
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(27) UsableRulesProof (EQUIVALENT)
6.72/2.63	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(28)
6.72/2.63	Obligation:
6.72/2.63	Pi DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   U1_AG(X, XS, YS, right_out_aa(tree(X, niltree, XS), ZS)) -> FLAT_IN_AG(ZS, YS)
6.72/2.63	   FLAT_IN_AG(tree(X, niltree, XS), cons(X, YS)) -> U1_AG(X, XS, YS, right_in_aa(tree(X, niltree, XS), ZS))
6.72/2.63	   FLAT_IN_AG(tree(X, tree(Y, YS1, YS2), XS), ZS) -> FLAT_IN_AG(tree(Y, YS1, tree(X, YS2, XS)), ZS)
6.72/2.63	
6.72/2.63	The TRS R consists of the following rules:
6.72/2.63	
6.72/2.63	   right_in_aa(tree(X, XS1, XS2), XS2) -> right_out_aa(tree(X, XS1, XS2), XS2)
6.72/2.63	
6.72/2.63	The argument filtering Pi contains the following mapping:
6.72/2.63	cons(x1, x2)  =  cons(x1, x2)
6.72/2.63	
6.72/2.63	right_in_aa(x1, x2)  =  right_in_aa
6.72/2.63	
6.72/2.63	right_out_aa(x1, x2)  =  right_out_aa
6.72/2.63	
6.72/2.63	tree(x1, x2, x3)  =  tree(x2, x3)
6.72/2.63	
6.72/2.63	FLAT_IN_AG(x1, x2)  =  FLAT_IN_AG(x2)
6.72/2.63	
6.72/2.63	U1_AG(x1, x2, x3, x4)  =  U1_AG(x1, x3, x4)
6.72/2.63	
6.72/2.63	
6.72/2.63	We have to consider all (P,R,Pi)-chains
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(29) PiDPToQDPProof (SOUND)
6.72/2.63	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(30)
6.72/2.63	Obligation:
6.72/2.63	Q DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   U1_AG(X, YS, right_out_aa) -> FLAT_IN_AG(YS)
6.72/2.63	   FLAT_IN_AG(cons(X, YS)) -> U1_AG(X, YS, right_in_aa)
6.72/2.63	   FLAT_IN_AG(ZS) -> FLAT_IN_AG(ZS)
6.72/2.63	
6.72/2.63	The TRS R consists of the following rules:
6.72/2.63	
6.72/2.63	   right_in_aa -> right_out_aa
6.72/2.63	
6.72/2.63	The set Q consists of the following terms:
6.72/2.63	
6.72/2.63	   right_in_aa
6.72/2.63	
6.72/2.63	We have to consider all (P,Q,R)-chains.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(31) UsableRulesReductionPairsProof (EQUIVALENT)
6.72/2.63	By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
6.72/2.63	
6.72/2.63	The following dependency pairs can be deleted:
6.72/2.63	
6.72/2.63	   FLAT_IN_AG(cons(X, YS)) -> U1_AG(X, YS, right_in_aa)
6.72/2.63	No rules are removed from R.
6.72/2.63	
6.72/2.63	Used ordering: POLO with Polynomial interpretation [POLO]:
6.72/2.63	
6.72/2.63	   POL(FLAT_IN_AG(x_1)) = x_1
6.72/2.63	   POL(U1_AG(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3
6.72/2.63	   POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2
6.72/2.63	   POL(right_in_aa) = 0
6.72/2.63	   POL(right_out_aa) = 0
6.72/2.63	
6.72/2.63	
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(32)
6.72/2.63	Obligation:
6.72/2.63	Q DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   U1_AG(X, YS, right_out_aa) -> FLAT_IN_AG(YS)
6.72/2.63	   FLAT_IN_AG(ZS) -> FLAT_IN_AG(ZS)
6.72/2.63	
6.72/2.63	The TRS R consists of the following rules:
6.72/2.63	
6.72/2.63	   right_in_aa -> right_out_aa
6.72/2.63	
6.72/2.63	The set Q consists of the following terms:
6.72/2.63	
6.72/2.63	   right_in_aa
6.72/2.63	
6.72/2.63	We have to consider all (P,Q,R)-chains.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(33) DependencyGraphProof (EQUIVALENT)
6.72/2.63	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(34)
6.72/2.63	Obligation:
6.72/2.63	Q DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   FLAT_IN_AG(ZS) -> FLAT_IN_AG(ZS)
6.72/2.63	
6.72/2.63	The TRS R consists of the following rules:
6.72/2.63	
6.72/2.63	   right_in_aa -> right_out_aa
6.72/2.63	
6.72/2.63	The set Q consists of the following terms:
6.72/2.63	
6.72/2.63	   right_in_aa
6.72/2.63	
6.72/2.63	We have to consider all (P,Q,R)-chains.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(35) UsableRulesProof (EQUIVALENT)
6.72/2.63	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.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(36)
6.72/2.63	Obligation:
6.72/2.63	Q DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   FLAT_IN_AG(ZS) -> FLAT_IN_AG(ZS)
6.72/2.63	
6.72/2.63	R is empty.
6.72/2.63	The set Q consists of the following terms:
6.72/2.63	
6.72/2.63	   right_in_aa
6.72/2.63	
6.72/2.63	We have to consider all (P,Q,R)-chains.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(37) QReductionProof (EQUIVALENT)
6.72/2.63	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
6.72/2.63	
6.72/2.63	   right_in_aa
6.72/2.63	
6.72/2.63	
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(38)
6.72/2.63	Obligation:
6.72/2.63	Q DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   FLAT_IN_AG(ZS) -> FLAT_IN_AG(ZS)
6.72/2.63	
6.72/2.63	R is empty.
6.72/2.63	Q is empty.
6.72/2.63	We have to consider all (P,Q,R)-chains.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(39) NonTerminationLoopProof (COMPLETE)
6.72/2.63	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
6.72/2.63	Found a loop by semiunifying a rule from P directly.
6.72/2.63	
6.72/2.63	s = FLAT_IN_AG(ZS) evaluates to  t =FLAT_IN_AG(ZS)
6.72/2.63	
6.72/2.63	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
6.72/2.63	* Matcher: [ ]
6.72/2.63	* Semiunifier: [ ]
6.72/2.63	
6.72/2.63	--------------------------------------------------------------------------------
6.72/2.63	Rewriting sequence
6.72/2.63	
6.72/2.63	The DP semiunifies directly so there is only one rewrite step from FLAT_IN_AG(ZS) to FLAT_IN_AG(ZS).
6.72/2.63	
6.72/2.63	
6.72/2.63	
6.72/2.63	
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(40)
6.72/2.63	NO
6.72/2.63	
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(41) PrologToTRSTransformerProof (SOUND)
6.72/2.63	Transformed Prolog program to TRS.
6.72/2.63	
6.72/2.63	{
6.72/2.63	    "root": 96,
6.72/2.63	    "program": {
6.72/2.63	        "directives": [],
6.72/2.63	        "clauses": [
6.72/2.63	            [
6.72/2.63	                "(right (tree X XS1 XS2) XS2)",
6.72/2.63	                null
6.72/2.63	            ],
6.72/2.63	            [
6.72/2.63	                "(flat (niltree) (nil))",
6.72/2.63	                null
6.72/2.63	            ],
6.72/2.63	            [
6.72/2.63	                "(flat (tree X (niltree) XS) (cons X YS))",
6.72/2.63	                "(',' (right (tree X (niltree) XS) ZS) (flat ZS YS))"
6.72/2.63	            ],
6.72/2.63	            [
6.72/2.63	                "(flat (tree X (tree Y YS1 YS2) XS) ZS)",
6.72/2.63	                "(flat (tree Y YS1 (tree X YS2 XS)) ZS)"
6.72/2.63	            ]
6.72/2.63	        ]
6.72/2.63	    },
6.72/2.63	    "graph": {
6.72/2.63	        "nodes": {
6.72/2.63	            "190": {
6.72/2.63	                "goal": [{
6.72/2.63	                    "clause": -1,
6.72/2.63	                    "scope": -1,
6.72/2.63	                    "term": "(flat (tree T45 T46 (tree T47 T48 T49)) T44)"
6.72/2.63	                }],
6.72/2.63	                "kb": {
6.72/2.63	                    "nonunifying": [],
6.72/2.63	                    "intvars": {},
6.72/2.63	                    "arithmetic": {
6.72/2.63	                        "type": "PlainIntegerRelationState",
6.72/2.63	                        "relations": []
6.72/2.63	                    },
6.72/2.63	                    "ground": ["T44"],
6.72/2.63	                    "free": [],
6.72/2.63	                    "exprvars": []
6.72/2.63	                }
6.72/2.63	            },
6.72/2.63	            "191": {
6.72/2.63	                "goal": [],
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6.72/2.63	                        "relations": []
6.72/2.63	                    },
6.72/2.63	                    "ground": [],
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6.72/2.63	                }
6.72/2.63	            },
6.72/2.63	            "type": "Nodes",
6.72/2.63	            "161": {
6.72/2.63	                "goal": [{
6.72/2.63	                    "clause": 2,
6.72/2.63	                    "scope": 1,
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6.72/2.63	                }],
6.72/2.63	                "kb": {
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6.72/2.63	                        "relations": []
6.72/2.63	                    },
6.72/2.63	                    "ground": ["T2"],
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6.72/2.63	                }
6.72/2.63	            },
6.72/2.63	            "162": {
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6.72/2.63	                    "scope": 1,
6.72/2.63	                    "term": "(flat T1 T2)"
6.72/2.63	                }],
6.72/2.63	                "kb": {
6.72/2.63	                    "nonunifying": [],
6.72/2.63	                    "intvars": {},
6.72/2.63	                    "arithmetic": {
6.72/2.63	                        "type": "PlainIntegerRelationState",
6.72/2.63	                        "relations": []
6.72/2.63	                    },
6.72/2.63	                    "ground": ["T2"],
6.72/2.63	                    "free": [],
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6.72/2.63	                }
6.72/2.63	            },
6.72/2.63	            "163": {
6.72/2.63	                "goal": [{
6.72/2.63	                    "clause": -1,
6.72/2.63	                    "scope": -1,
6.72/2.63	                    "term": "(',' (right (tree T15 (niltree) T18) X18) (flat X18 T17))"
6.72/2.63	                }],
6.72/2.63	                "kb": {
6.72/2.63	                    "nonunifying": [],
6.72/2.63	                    "intvars": {},
6.72/2.63	                    "arithmetic": {
6.72/2.63	                        "type": "PlainIntegerRelationState",
6.72/2.63	                        "relations": []
6.72/2.63	                    },
6.72/2.63	                    "ground": [
6.72/2.63	                        "T15",
6.72/2.63	                        "T17"
6.72/2.63	                    ],
6.72/2.63	                    "free": ["X18"],
6.72/2.63	                    "exprvars": []
6.72/2.63	                }
6.72/2.63	            },
6.72/2.63	            "153": {
6.72/2.63	                "goal": [{
6.72/2.63	                    "clause": -1,
6.72/2.63	                    "scope": -1,
6.72/2.63	                    "term": "(true)"
6.72/2.63	                }],
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6.72/2.63	                        "relations": []
6.72/2.63	                    },
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6.72/2.63	                }
6.72/2.63	            },
6.72/2.63	            "164": {
6.72/2.63	                "goal": [],
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6.72/2.63	                    "intvars": {},
6.72/2.63	                    "arithmetic": {
6.72/2.63	                        "type": "PlainIntegerRelationState",
6.72/2.63	                        "relations": []
6.72/2.63	                    },
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6.72/2.63	                }
6.72/2.63	            },
6.72/2.63	            "175": {
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6.72/2.63	                    "clause": -1,
6.72/2.63	                    "scope": -1,
6.72/2.63	                    "term": "(flat T26 T17)"
6.72/2.63	                }],
6.72/2.63	                "kb": {
6.72/2.63	                    "nonunifying": [],
6.72/2.63	                    "intvars": {},
6.72/2.63	                    "arithmetic": {
6.72/2.63	                        "type": "PlainIntegerRelationState",
6.72/2.63	                        "relations": []
6.72/2.63	                    },
6.72/2.63	                    "ground": ["T17"],
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6.72/2.63	                }
6.72/2.63	            },
6.72/2.63	            "132": {
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6.72/2.63	                }],
6.72/2.63	                "kb": {
6.72/2.63	                    "nonunifying": [],
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6.72/2.63	                        "type": "PlainIntegerRelationState",
6.72/2.63	                        "relations": []
6.72/2.63	                    },
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6.72/2.63	                }
6.72/2.63	            },
6.72/2.63	            "154": {
6.72/2.63	                "goal": [],
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6.72/2.63	                    "nonunifying": [],
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6.72/2.63	                        "type": "PlainIntegerRelationState",
6.72/2.63	                        "relations": []
6.72/2.63	                    },
6.72/2.63	                    "ground": [],
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6.72/2.63	                }
6.72/2.63	            },
6.72/2.63	            "156": {
6.72/2.63	                "goal": [],
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6.72/2.63	                    "nonunifying": [],
6.72/2.63	                    "intvars": {},
6.72/2.63	                    "arithmetic": {
6.72/2.63	                        "type": "PlainIntegerRelationState",
6.72/2.63	                        "relations": []
6.72/2.63	                    },
6.72/2.63	                    "ground": [],
6.72/2.63	                    "free": [],
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6.72/2.63	                }
6.72/2.63	            },
6.72/2.63	            "167": {
6.72/2.63	                "goal": [{
6.72/2.63	                    "clause": 0,
6.72/2.63	                    "scope": 2,
6.72/2.63	                    "term": "(',' (right (tree T15 (niltree) T18) X18) (flat X18 T17))"
6.72/2.63	                }],
6.72/2.63	                "kb": {
6.72/2.63	                    "nonunifying": [],
6.72/2.63	                    "intvars": {},
6.72/2.63	                    "arithmetic": {
6.72/2.63	                        "type": "PlainIntegerRelationState",
6.72/2.63	                        "relations": []
6.72/2.63	                    },
6.72/2.63	                    "ground": [
6.72/2.63	                        "T15",
6.72/2.63	                        "T17"
6.72/2.63	                    ],
6.72/2.63	                    "free": ["X18"],
6.72/2.63	                    "exprvars": []
6.72/2.63	                }
6.72/2.63	            },
6.72/2.63	            "137": {
6.72/2.63	                "goal": [
6.72/2.63	                    {
6.72/2.63	                        "clause": 2,
6.72/2.63	                        "scope": 1,
6.72/2.63	                        "term": "(flat T1 T2)"
6.72/2.63	                    },
6.72/2.63	                    {
6.72/2.63	                        "clause": 3,
6.72/2.63	                        "scope": 1,
6.72/2.63	                        "term": "(flat T1 T2)"
6.72/2.63	                    }
6.72/2.63	                ],
6.72/2.63	                "kb": {
6.72/2.63	                    "nonunifying": [],
6.72/2.63	                    "intvars": {},
6.72/2.63	                    "arithmetic": {
6.72/2.63	                        "type": "PlainIntegerRelationState",
6.72/2.63	                        "relations": []
6.72/2.63	                    },
6.72/2.63	                    "ground": ["T2"],
6.72/2.63	                    "free": [],
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6.72/2.63	                }
6.72/2.63	            },
6.72/2.63	            "116": {
6.72/2.63	                "goal": [
6.72/2.63	                    {
6.72/2.63	                        "clause": 1,
6.72/2.63	                        "scope": 1,
6.72/2.63	                        "term": "(flat T1 T2)"
6.72/2.63	                    },
6.72/2.63	                    {
6.72/2.63	                        "clause": 2,
6.72/2.63	                        "scope": 1,
6.72/2.63	                        "term": "(flat T1 T2)"
6.72/2.63	                    },
6.72/2.63	                    {
6.72/2.63	                        "clause": 3,
6.72/2.63	                        "scope": 1,
6.72/2.63	                        "term": "(flat T1 T2)"
6.72/2.63	                    }
6.72/2.63	                ],
6.72/2.63	                "kb": {
6.72/2.63	                    "nonunifying": [],
6.72/2.63	                    "intvars": {},
6.72/2.63	                    "arithmetic": {
6.72/2.63	                        "type": "PlainIntegerRelationState",
6.72/2.63	                        "relations": []
6.72/2.63	                    },
6.72/2.63	                    "ground": ["T2"],
6.72/2.63	                    "free": [],
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6.72/2.63	                }
6.72/2.63	            },
6.72/2.63	            "96": {
6.72/2.63	                "goal": [{
6.72/2.63	                    "clause": -1,
6.72/2.63	                    "scope": -1,
6.72/2.63	                    "term": "(flat T1 T2)"
6.72/2.63	                }],
6.72/2.63	                "kb": {
6.72/2.63	                    "nonunifying": [],
6.72/2.63	                    "intvars": {},
6.72/2.63	                    "arithmetic": {
6.72/2.63	                        "type": "PlainIntegerRelationState",
6.72/2.63	                        "relations": []
6.72/2.63	                    },
6.72/2.63	                    "ground": ["T2"],
6.72/2.63	                    "free": [],
6.72/2.63	                    "exprvars": []
6.72/2.63	                }
6.72/2.63	            }
6.72/2.63	        },
6.72/2.63	        "edges": [
6.72/2.63	            {
6.72/2.63	                "from": 96,
6.72/2.63	                "to": 116,
6.72/2.63	                "label": "CASE"
6.72/2.63	            },
6.72/2.63	            {
6.72/2.63	                "from": 116,
6.72/2.63	                "to": 132,
6.72/2.63	                "label": "PARALLEL"
6.72/2.63	            },
6.72/2.63	            {
6.72/2.63	                "from": 116,
6.72/2.63	                "to": 137,
6.72/2.63	                "label": "PARALLEL"
6.72/2.63	            },
6.72/2.63	            {
6.72/2.63	                "from": 132,
6.72/2.63	                "to": 153,
6.72/2.63	                "label": "EVAL with clause\nflat(niltree, nil).\nand substitutionT1 -> niltree,\nT2 -> nil"
6.72/2.63	            },
6.72/2.63	            {
6.72/2.63	                "from": 132,
6.72/2.63	                "to": 154,
6.72/2.63	                "label": "EVAL-BACKTRACK"
6.72/2.63	            },
6.72/2.63	            {
6.72/2.63	                "from": 137,
6.72/2.63	                "to": 161,
6.72/2.63	                "label": "PARALLEL"
6.72/2.63	            },
6.72/2.63	            {
6.72/2.63	                "from": 137,
6.72/2.63	                "to": 162,
6.72/2.63	                "label": "PARALLEL"
6.72/2.63	            },
6.72/2.63	            {
6.72/2.63	                "from": 153,
6.72/2.63	                "to": 156,
6.72/2.63	                "label": "SUCCESS"
6.72/2.63	            },
6.72/2.63	            {
6.72/2.63	                "from": 161,
6.72/2.63	                "to": 163,
6.72/2.63	                "label": "EVAL with clause\nflat(tree(X15, niltree, X16), cons(X15, X17)) :- ','(right(tree(X15, niltree, X16), X18), flat(X18, X17)).\nand substitutionX15 -> T15,\nX16 -> T18,\nT1 -> tree(T15, niltree, T18),\nX17 -> T17,\nT2 -> cons(T15, T17),\nT16 -> T18"
6.72/2.63	            },
6.72/2.63	            {
6.72/2.63	                "from": 161,
6.72/2.63	                "to": 164,
6.72/2.63	                "label": "EVAL-BACKTRACK"
6.72/2.63	            },
6.72/2.63	            {
6.72/2.63	                "from": 162,
6.72/2.63	                "to": 190,
6.72/2.63	                "label": "EVAL with clause\nflat(tree(X40, tree(X41, X42, X43), X44), X45) :- flat(tree(X41, X42, tree(X40, X43, X44)), X45).\nand substitutionX40 -> T47,\nX41 -> T45,\nX42 -> T46,\nX43 -> T48,\nX44 -> T49,\nT1 -> tree(T47, tree(T45, T46, T48), T49),\nT2 -> T44,\nX45 -> T44,\nT40 -> T45,\nT41 -> T46,\nT39 -> T47,\nT42 -> T48,\nT43 -> T49"
6.72/2.63	            },
6.72/2.63	            {
6.72/2.63	                "from": 162,
6.72/2.63	                "to": 191,
6.72/2.63	                "label": "EVAL-BACKTRACK"
6.72/2.63	            },
6.72/2.63	            {
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6.72/2.63	                "to": 167,
6.72/2.63	                "label": "CASE"
6.72/2.63	            },
6.72/2.63	            {
6.72/2.63	                "from": 167,
6.72/2.63	                "to": 175,
6.72/2.63	                "label": "ONLY EVAL with clause\nright(tree(X25, X26, X27), X27).\nand substitutionT15 -> T24,\nX25 -> T24,\nX26 -> niltree,\nT18 -> T26,\nX27 -> T26,\nX18 -> T26,\nT25 -> T26"
6.72/2.63	            },
6.72/2.63	            {
6.72/2.63	                "from": 175,
6.72/2.63	                "to": 96,
6.72/2.63	                "label": "INSTANCE with matching:\nT1 -> T26\nT2 -> T17"
6.72/2.63	            },
6.72/2.63	            {
6.72/2.63	                "from": 190,
6.72/2.63	                "to": 96,
6.72/2.63	                "label": "INSTANCE with matching:\nT1 -> tree(T45, T46, tree(T47, T48, T49))\nT2 -> T44"
6.72/2.63	            }
6.72/2.63	        ],
6.72/2.63	        "type": "Graph"
6.72/2.63	    }
6.72/2.63	}
6.72/2.63	
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(42)
6.72/2.63	Obligation:
6.72/2.63	Q restricted rewrite system:
6.72/2.63	The TRS R consists of the following rules:
6.72/2.63	
6.72/2.63	   f96_in(nil) -> f96_out1
6.72/2.63	   f96_in(cons(T24, T17)) -> U1(f96_in(T17), cons(T24, T17))
6.72/2.63	   U1(f96_out1, cons(T24, T17)) -> f96_out1
6.72/2.63	   f96_in(T44) -> U2(f96_in(T44), T44)
6.72/2.63	   U2(f96_out1, T44) -> f96_out1
6.72/2.63	
6.72/2.63	Q is empty.
6.72/2.63	
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(43) DependencyPairsProof (EQUIVALENT)
6.72/2.63	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(44)
6.72/2.63	Obligation:
6.72/2.63	Q DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   F96_IN(cons(T24, T17)) -> U1^1(f96_in(T17), cons(T24, T17))
6.72/2.63	   F96_IN(cons(T24, T17)) -> F96_IN(T17)
6.72/2.63	   F96_IN(T44) -> U2^1(f96_in(T44), T44)
6.72/2.63	   F96_IN(T44) -> F96_IN(T44)
6.72/2.63	
6.72/2.63	The TRS R consists of the following rules:
6.72/2.63	
6.72/2.63	   f96_in(nil) -> f96_out1
6.72/2.63	   f96_in(cons(T24, T17)) -> U1(f96_in(T17), cons(T24, T17))
6.72/2.63	   U1(f96_out1, cons(T24, T17)) -> f96_out1
6.72/2.63	   f96_in(T44) -> U2(f96_in(T44), T44)
6.72/2.63	   U2(f96_out1, T44) -> f96_out1
6.72/2.63	
6.72/2.63	Q is empty.
6.72/2.63	We have to consider all minimal (P,Q,R)-chains.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(45) DependencyGraphProof (EQUIVALENT)
6.72/2.63	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(46)
6.72/2.63	Obligation:
6.72/2.63	Q DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   F96_IN(T44) -> F96_IN(T44)
6.72/2.63	   F96_IN(cons(T24, T17)) -> F96_IN(T17)
6.72/2.63	
6.72/2.63	The TRS R consists of the following rules:
6.72/2.63	
6.72/2.63	   f96_in(nil) -> f96_out1
6.72/2.63	   f96_in(cons(T24, T17)) -> U1(f96_in(T17), cons(T24, T17))
6.72/2.63	   U1(f96_out1, cons(T24, T17)) -> f96_out1
6.72/2.63	   f96_in(T44) -> U2(f96_in(T44), T44)
6.72/2.63	   U2(f96_out1, T44) -> f96_out1
6.72/2.63	
6.72/2.63	Q is empty.
6.72/2.63	We have to consider all minimal (P,Q,R)-chains.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(47) UsableRulesProof (EQUIVALENT)
6.72/2.63	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(48)
6.72/2.63	Obligation:
6.72/2.63	Q DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   F96_IN(T44) -> F96_IN(T44)
6.72/2.63	   F96_IN(cons(T24, T17)) -> F96_IN(T17)
6.72/2.63	
6.72/2.63	R is empty.
6.72/2.63	Q is empty.
6.72/2.63	We have to consider all minimal (P,Q,R)-chains.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(49) UsableRulesReductionPairsProof (EQUIVALENT)
6.72/2.63	By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
6.72/2.63	
6.72/2.63	The following dependency pairs can be deleted:
6.72/2.63	
6.72/2.63	   F96_IN(cons(T24, T17)) -> F96_IN(T17)
6.72/2.63	No rules are removed from R.
6.72/2.63	
6.72/2.63	Used ordering: POLO with Polynomial interpretation [POLO]:
6.72/2.63	
6.72/2.63	   POL(F96_IN(x_1)) = 2*x_1
6.72/2.63	   POL(cons(x_1, x_2)) = x_1 + x_2
6.72/2.63	
6.72/2.63	
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(50)
6.72/2.63	Obligation:
6.72/2.63	Q DP problem:
6.72/2.63	The TRS P consists of the following rules:
6.72/2.63	
6.72/2.63	   F96_IN(T44) -> F96_IN(T44)
6.72/2.63	
6.72/2.63	R is empty.
6.72/2.63	Q is empty.
6.72/2.63	We have to consider all minimal (P,Q,R)-chains.
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(51) NonTerminationLoopProof (COMPLETE)
6.72/2.63	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
6.72/2.63	Found a loop by semiunifying a rule from P directly.
6.72/2.63	
6.72/2.63	s = F96_IN(T44) evaluates to  t =F96_IN(T44)
6.72/2.63	
6.72/2.63	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
6.72/2.63	* Matcher: [ ]
6.72/2.63	* Semiunifier: [ ]
6.72/2.63	
6.72/2.63	--------------------------------------------------------------------------------
6.72/2.63	Rewriting sequence
6.72/2.63	
6.72/2.63	The DP semiunifies directly so there is only one rewrite step from F96_IN(T44) to F96_IN(T44).
6.72/2.63	
6.72/2.63	
6.72/2.63	
6.72/2.63	
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(52)
6.72/2.63	NO
6.72/2.63	
6.72/2.63	----------------------------------------
6.72/2.63	
6.72/2.63	(53) PrologToIRSwTTransformerProof (SOUND)
6.72/2.63	Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert
6.72/2.63	
6.72/2.63	{
6.72/2.63	    "root": 2,
6.72/2.63	    "program": {
6.72/2.63	        "directives": [],
6.72/2.63	        "clauses": [
6.72/2.63	            [
6.72/2.63	                "(right (tree X XS1 XS2) XS2)",
6.72/2.63	                null
6.72/2.63	            ],
6.72/2.63	            [
6.72/2.63	                "(flat (niltree) (nil))",
6.72/2.63	                null
6.72/2.63	            ],
6.72/2.63	            [
6.72/2.63	                "(flat (tree X (niltree) XS) (cons X YS))",
6.72/2.63	                "(',' (right (tree X (niltree) XS) ZS) (flat ZS YS))"
6.72/2.63	            ],
6.72/2.63	            [
6.72/2.63	                "(flat (tree X (tree Y YS1 YS2) XS) ZS)",
6.72/2.63	                "(flat (tree Y YS1 (tree X YS2 XS)) ZS)"
6.72/2.63	            ]
6.72/2.63	        ]
6.72/2.63	    },
6.72/2.63	    "graph": {
6.72/2.63	        "nodes": {
6.72/2.63	            "22": {
6.72/2.63	                "goal": [{
6.72/2.63	                    "clause": -1,
6.72/2.63	                    "scope": -1,
6.72/2.63	                    "term": "(true)"
6.72/2.63	                }],
6.72/2.63	                "kb": {
6.72/2.63	                    "nonunifying": [],
6.72/2.63	                    "intvars": {},
6.72/2.63	                    "arithmetic": {
6.72/2.63	                        "type": "PlainIntegerRelationState",
6.72/2.63	                        "relations": []
6.72/2.63	                    },
6.72/2.63	                    "ground": [],
6.72/2.63	                    "free": [],
6.72/2.63	                    "exprvars": []
6.72/2.63	                }
6.72/2.63	            },
6.72/2.63	            "12": {
6.72/2.63	                "goal": [{
6.72/2.63	                    "clause": 1,
6.72/2.63	                    "scope": 1,
6.72/2.63	                    "term": "(flat T1 T2)"
6.94/2.64	                }],
6.94/2.64	                "kb": {
6.94/2.64	                    "nonunifying": [],
6.94/2.64	                    "intvars": {},
6.94/2.64	                    "arithmetic": {
6.94/2.64	                        "type": "PlainIntegerRelationState",
6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": ["T2"],
6.94/2.64	                    "free": [],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            },
6.94/2.64	            "13": {
6.94/2.64	                "goal": [
6.94/2.64	                    {
6.94/2.64	                        "clause": 2,
6.94/2.64	                        "scope": 1,
6.94/2.64	                        "term": "(flat T1 T2)"
6.94/2.64	                    },
6.94/2.64	                    {
6.94/2.64	                        "clause": 3,
6.94/2.64	                        "scope": 1,
6.94/2.64	                        "term": "(flat T1 T2)"
6.94/2.64	                    }
6.94/2.64	                ],
6.94/2.64	                "kb": {
6.94/2.64	                    "nonunifying": [],
6.94/2.64	                    "intvars": {},
6.94/2.64	                    "arithmetic": {
6.94/2.64	                        "type": "PlainIntegerRelationState",
6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": ["T2"],
6.94/2.64	                    "free": [],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            },
6.94/2.64	            "180": {
6.94/2.64	                "goal": [{
6.94/2.64	                    "clause": 3,
6.94/2.64	                    "scope": 1,
6.94/2.64	                    "term": "(flat T1 T2)"
6.94/2.64	                }],
6.94/2.64	                "kb": {
6.94/2.64	                    "nonunifying": [],
6.94/2.64	                    "intvars": {},
6.94/2.64	                    "arithmetic": {
6.94/2.64	                        "type": "PlainIntegerRelationState",
6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": ["T2"],
6.94/2.64	                    "free": [],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            },
6.94/2.64	            "type": "Nodes",
6.94/2.64	            "183": {
6.94/2.64	                "goal": [{
6.94/2.64	                    "clause": -1,
6.94/2.64	                    "scope": -1,
6.94/2.64	                    "term": "(',' (right (tree T15 (niltree) T18) X18) (flat X18 T17))"
6.94/2.64	                }],
6.94/2.64	                "kb": {
6.94/2.64	                    "nonunifying": [],
6.94/2.64	                    "intvars": {},
6.94/2.64	                    "arithmetic": {
6.94/2.64	                        "type": "PlainIntegerRelationState",
6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": [
6.94/2.64	                        "T15",
6.94/2.64	                        "T17"
6.94/2.64	                    ],
6.94/2.64	                    "free": ["X18"],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            },
6.94/2.64	            "194": {
6.94/2.64	                "goal": [{
6.94/2.64	                    "clause": -1,
6.94/2.64	                    "scope": -1,
6.94/2.64	                    "term": "(flat (tree T45 T46 (tree T47 T48 T49)) T44)"
6.94/2.64	                }],
6.94/2.64	                "kb": {
6.94/2.64	                    "nonunifying": [],
6.94/2.64	                    "intvars": {},
6.94/2.64	                    "arithmetic": {
6.94/2.64	                        "type": "PlainIntegerRelationState",
6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": ["T44"],
6.94/2.64	                    "free": [],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            },
6.94/2.64	            "184": {
6.94/2.64	                "goal": [],
6.94/2.64	                "kb": {
6.94/2.64	                    "nonunifying": [],
6.94/2.64	                    "intvars": {},
6.94/2.64	                    "arithmetic": {
6.94/2.64	                        "type": "PlainIntegerRelationState",
6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": [],
6.94/2.64	                    "free": [],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            },
6.94/2.64	            "195": {
6.94/2.64	                "goal": [],
6.94/2.64	                "kb": {
6.94/2.64	                    "nonunifying": [],
6.94/2.64	                    "intvars": {},
6.94/2.64	                    "arithmetic": {
6.94/2.64	                        "type": "PlainIntegerRelationState",
6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": [],
6.94/2.64	                    "free": [],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            },
6.94/2.64	            "185": {
6.94/2.64	                "goal": [{
6.94/2.64	                    "clause": 0,
6.94/2.64	                    "scope": 2,
6.94/2.64	                    "term": "(',' (right (tree T15 (niltree) T18) X18) (flat X18 T17))"
6.94/2.64	                }],
6.94/2.64	                "kb": {
6.94/2.64	                    "nonunifying": [],
6.94/2.64	                    "intvars": {},
6.94/2.64	                    "arithmetic": {
6.94/2.64	                        "type": "PlainIntegerRelationState",
6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": [
6.94/2.64	                        "T15",
6.94/2.64	                        "T17"
6.94/2.64	                    ],
6.94/2.64	                    "free": ["X18"],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            },
6.94/2.64	            "187": {
6.94/2.64	                "goal": [{
6.94/2.64	                    "clause": -1,
6.94/2.64	                    "scope": -1,
6.94/2.64	                    "term": "(flat T26 T17)"
6.94/2.64	                }],
6.94/2.64	                "kb": {
6.94/2.64	                    "nonunifying": [],
6.94/2.64	                    "intvars": {},
6.94/2.64	                    "arithmetic": {
6.94/2.64	                        "type": "PlainIntegerRelationState",
6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": ["T17"],
6.94/2.64	                    "free": [],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            },
6.94/2.64	            "177": {
6.94/2.64	                "goal": [],
6.94/2.64	                "kb": {
6.94/2.64	                    "nonunifying": [],
6.94/2.64	                    "intvars": {},
6.94/2.64	                    "arithmetic": {
6.94/2.64	                        "type": "PlainIntegerRelationState",
6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": [],
6.94/2.64	                    "free": [],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            },
6.94/2.64	            "2": {
6.94/2.64	                "goal": [{
6.94/2.64	                    "clause": -1,
6.94/2.64	                    "scope": -1,
6.94/2.64	                    "term": "(flat T1 T2)"
6.94/2.64	                }],
6.94/2.64	                "kb": {
6.94/2.64	                    "nonunifying": [],
6.94/2.64	                    "intvars": {},
6.94/2.64	                    "arithmetic": {
6.94/2.64	                        "type": "PlainIntegerRelationState",
6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": ["T2"],
6.94/2.64	                    "free": [],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            },
6.94/2.64	            "178": {
6.94/2.64	                "goal": [],
6.94/2.64	                "kb": {
6.94/2.64	                    "nonunifying": [],
6.94/2.64	                    "intvars": {},
6.94/2.64	                    "arithmetic": {
6.94/2.64	                        "type": "PlainIntegerRelationState",
6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": [],
6.94/2.64	                    "free": [],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            },
6.94/2.64	            "179": {
6.94/2.64	                "goal": [{
6.94/2.64	                    "clause": 2,
6.94/2.64	                    "scope": 1,
6.94/2.64	                    "term": "(flat T1 T2)"
6.94/2.64	                }],
6.94/2.64	                "kb": {
6.94/2.64	                    "nonunifying": [],
6.94/2.64	                    "intvars": {},
6.94/2.64	                    "arithmetic": {
6.94/2.64	                        "type": "PlainIntegerRelationState",
6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": ["T2"],
6.94/2.64	                    "free": [],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            },
6.94/2.64	            "6": {
6.94/2.64	                "goal": [
6.94/2.64	                    {
6.94/2.64	                        "clause": 1,
6.94/2.64	                        "scope": 1,
6.94/2.64	                        "term": "(flat T1 T2)"
6.94/2.64	                    },
6.94/2.64	                    {
6.94/2.64	                        "clause": 2,
6.94/2.64	                        "scope": 1,
6.94/2.64	                        "term": "(flat T1 T2)"
6.94/2.64	                    },
6.94/2.64	                    {
6.94/2.64	                        "clause": 3,
6.94/2.64	                        "scope": 1,
6.94/2.64	                        "term": "(flat T1 T2)"
6.94/2.64	                    }
6.94/2.64	                ],
6.94/2.64	                "kb": {
6.94/2.64	                    "nonunifying": [],
6.94/2.64	                    "intvars": {},
6.94/2.64	                    "arithmetic": {
6.94/2.64	                        "type": "PlainIntegerRelationState",
6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": ["T2"],
6.94/2.64	                    "free": [],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            }
6.94/2.64	        },
6.94/2.64	        "edges": [
6.94/2.64	            {
6.94/2.64	                "from": 2,
6.94/2.64	                "to": 6,
6.94/2.64	                "label": "CASE"
6.94/2.64	            },
6.94/2.64	            {
6.94/2.64	                "from": 6,
6.94/2.64	                "to": 12,
6.94/2.64	                "label": "PARALLEL"
6.94/2.64	            },
6.94/2.64	            {
6.94/2.64	                "from": 6,
6.94/2.64	                "to": 13,
6.94/2.64	                "label": "PARALLEL"
6.94/2.64	            },
6.94/2.64	            {
6.94/2.64	                "from": 12,
6.94/2.64	                "to": 22,
6.94/2.64	                "label": "EVAL with clause\nflat(niltree, nil).\nand substitutionT1 -> niltree,\nT2 -> nil"
6.94/2.64	            },
6.94/2.64	            {
6.94/2.64	                "from": 12,
6.94/2.64	                "to": 177,
6.94/2.64	                "label": "EVAL-BACKTRACK"
6.94/2.64	            },
6.94/2.64	            {
6.94/2.64	                "from": 13,
6.94/2.64	                "to": 179,
6.94/2.64	                "label": "PARALLEL"
6.94/2.64	            },
6.94/2.64	            {
6.94/2.64	                "from": 13,
6.94/2.64	                "to": 180,
6.94/2.64	                "label": "PARALLEL"
6.94/2.64	            },
6.94/2.64	            {
6.94/2.64	                "from": 22,
6.94/2.64	                "to": 178,
6.94/2.64	                "label": "SUCCESS"
6.94/2.64	            },
6.94/2.64	            {
6.94/2.64	                "from": 179,
6.94/2.64	                "to": 183,
6.94/2.64	                "label": "EVAL with clause\nflat(tree(X15, niltree, X16), cons(X15, X17)) :- ','(right(tree(X15, niltree, X16), X18), flat(X18, X17)).\nand substitutionX15 -> T15,\nX16 -> T18,\nT1 -> tree(T15, niltree, T18),\nX17 -> T17,\nT2 -> cons(T15, T17),\nT16 -> T18"
6.94/2.64	            },
6.94/2.64	            {
6.94/2.64	                "from": 179,
6.94/2.64	                "to": 184,
6.94/2.64	                "label": "EVAL-BACKTRACK"
6.94/2.64	            },
6.94/2.64	            {
6.94/2.64	                "from": 180,
6.94/2.64	                "to": 194,
6.94/2.64	                "label": "EVAL with clause\nflat(tree(X40, tree(X41, X42, X43), X44), X45) :- flat(tree(X41, X42, tree(X40, X43, X44)), X45).\nand substitutionX40 -> T47,\nX41 -> T45,\nX42 -> T46,\nX43 -> T48,\nX44 -> T49,\nT1 -> tree(T47, tree(T45, T46, T48), T49),\nT2 -> T44,\nX45 -> T44,\nT40 -> T45,\nT41 -> T46,\nT39 -> T47,\nT42 -> T48,\nT43 -> T49"
6.94/2.64	            },
6.94/2.64	            {
6.94/2.64	                "from": 180,
6.94/2.64	                "to": 195,
6.94/2.64	                "label": "EVAL-BACKTRACK"
6.94/2.64	            },
6.94/2.64	            {
6.94/2.64	                "from": 183,
6.94/2.64	                "to": 185,
6.94/2.64	                "label": "CASE"
6.94/2.64	            },
6.94/2.64	            {
6.94/2.64	                "from": 185,
6.94/2.64	                "to": 187,
6.94/2.64	                "label": "ONLY EVAL with clause\nright(tree(X25, X26, X27), X27).\nand substitutionT15 -> T24,\nX25 -> T24,\nX26 -> niltree,\nT18 -> T26,\nX27 -> T26,\nX18 -> T26,\nT25 -> T26"
6.94/2.64	            },
6.94/2.64	            {
6.94/2.64	                "from": 187,
6.94/2.64	                "to": 2,
6.94/2.64	                "label": "INSTANCE with matching:\nT1 -> T26\nT2 -> T17"
6.94/2.64	            },
6.94/2.64	            {
6.94/2.64	                "from": 194,
6.94/2.64	                "to": 2,
6.94/2.64	                "label": "INSTANCE with matching:\nT1 -> tree(T45, T46, tree(T47, T48, T49))\nT2 -> T44"
6.94/2.64	            }
6.94/2.64	        ],
6.94/2.64	        "type": "Graph"
6.94/2.64	    }
6.94/2.64	}
6.94/2.64	
6.94/2.64	----------------------------------------
6.94/2.64	
6.94/2.64	(54)
6.94/2.64	Obligation:
6.94/2.64	Rules:
6.94/2.64	f180_in(T2) -> f195_in :|: TRUE
6.94/2.64	f180_in(T44) -> f194_in(T44) :|: TRUE
6.94/2.64	f194_out(x) -> f180_out(x) :|: TRUE
6.94/2.64	f195_out -> f180_out(x1) :|: TRUE
6.94/2.64	f2_out(x2) -> f194_out(x2) :|: TRUE
6.94/2.64	f194_in(x3) -> f2_in(x3) :|: TRUE
6.94/2.64	f185_out(T15, T17) -> f183_out(T15, T17) :|: TRUE
6.94/2.64	f183_in(x4, x5) -> f185_in(x4, x5) :|: TRUE
6.94/2.64	f13_out(x6) -> f6_out(x6) :|: TRUE
6.94/2.64	f12_out(x7) -> f6_out(x7) :|: TRUE
6.94/2.64	f6_in(x8) -> f13_in(x8) :|: TRUE
6.94/2.64	f6_in(x9) -> f12_in(x9) :|: TRUE
6.94/2.64	f13_in(x10) -> f179_in(x10) :|: TRUE
6.94/2.64	f179_out(x11) -> f13_out(x11) :|: TRUE
6.94/2.64	f13_in(x12) -> f180_in(x12) :|: TRUE
6.94/2.64	f180_out(x13) -> f13_out(x13) :|: TRUE
6.94/2.64	f179_in(cons(x14, x15)) -> f183_in(x14, x15) :|: TRUE
6.94/2.64	f184_out -> f179_out(x16) :|: TRUE
6.94/2.64	f183_out(x17, x18) -> f179_out(cons(x17, x18)) :|: TRUE
6.94/2.64	f179_in(x19) -> f184_in :|: TRUE
6.94/2.64	f2_in(x20) -> f6_in(x20) :|: TRUE
6.94/2.64	f6_out(x21) -> f2_out(x21) :|: TRUE
6.94/2.64	f185_in(x22, x23) -> f187_in(x23) :|: TRUE
6.94/2.64	f187_out(x24) -> f185_out(x25, x24) :|: TRUE
6.94/2.64	f2_out(x26) -> f187_out(x26) :|: TRUE
6.94/2.64	f187_in(x27) -> f2_in(x27) :|: TRUE
6.94/2.64	Start term: f2_in(T2)
6.94/2.64	
6.94/2.64	----------------------------------------
6.94/2.64	
6.94/2.64	(55) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
6.94/2.64	Constructed simple dependency graph.
6.94/2.64	
6.94/2.64	Simplified to the following IRSwTs:
6.94/2.64	
6.94/2.64	intTRSProblem:
6.94/2.64	f180_in(T44) -> f194_in(T44) :|: TRUE
6.94/2.64	f194_in(x3) -> f2_in(x3) :|: TRUE
6.94/2.64	f183_in(x4, x5) -> f185_in(x4, x5) :|: TRUE
6.94/2.64	f6_in(x8) -> f13_in(x8) :|: TRUE
6.94/2.64	f13_in(x10) -> f179_in(x10) :|: TRUE
6.94/2.64	f13_in(x12) -> f180_in(x12) :|: TRUE
6.94/2.64	f179_in(cons(x14, x15)) -> f183_in(x14, x15) :|: TRUE
6.94/2.64	f2_in(x20) -> f6_in(x20) :|: TRUE
6.94/2.64	f185_in(x22, x23) -> f187_in(x23) :|: TRUE
6.94/2.64	f187_in(x27) -> f2_in(x27) :|: TRUE
6.94/2.64	
6.94/2.64	
6.94/2.64	----------------------------------------
6.94/2.64	
6.94/2.64	(56)
6.94/2.64	Obligation:
6.94/2.64	Rules:
6.94/2.64	f180_in(T44) -> f194_in(T44) :|: TRUE
6.94/2.64	f194_in(x3) -> f2_in(x3) :|: TRUE
6.94/2.64	f183_in(x4, x5) -> f185_in(x4, x5) :|: TRUE
6.94/2.64	f6_in(x8) -> f13_in(x8) :|: TRUE
6.94/2.64	f13_in(x10) -> f179_in(x10) :|: TRUE
6.94/2.64	f13_in(x12) -> f180_in(x12) :|: TRUE
6.94/2.64	f179_in(cons(x14, x15)) -> f183_in(x14, x15) :|: TRUE
6.94/2.64	f2_in(x20) -> f6_in(x20) :|: TRUE
6.94/2.64	f185_in(x22, x23) -> f187_in(x23) :|: TRUE
6.94/2.64	f187_in(x27) -> f2_in(x27) :|: TRUE
6.94/2.64	
6.94/2.64	----------------------------------------
6.94/2.64	
6.94/2.64	(57) IntTRSCompressionProof (EQUIVALENT)
6.94/2.64	Compressed rules.
6.94/2.64	----------------------------------------
6.94/2.64	
6.94/2.64	(58)
6.94/2.64	Obligation:
6.94/2.64	Rules:
6.94/2.64	f6_in(x8:0) -> f6_in(x8:0) :|: TRUE
6.94/2.64	f6_in(cons(x14:0, x15:0)) -> f6_in(x15:0) :|: TRUE
6.94/2.64	
6.94/2.64	----------------------------------------
6.94/2.64	
6.94/2.64	(59) IRSFormatTransformerProof (EQUIVALENT)
6.94/2.64	Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
6.94/2.64	----------------------------------------
6.94/2.64	
6.94/2.64	(60)
6.94/2.64	Obligation:
6.94/2.64	Rules:
6.94/2.64	f6_in(x8:0) -> f6_in(x8:0) :|: TRUE
6.94/2.64	f6_in(cons(x14:0, x15:0)) -> f6_in(x15:0) :|: TRUE
6.94/2.64	
6.94/2.64	----------------------------------------
6.94/2.64	
6.94/2.64	(61) IRSwTTerminationDigraphProof (EQUIVALENT)
6.94/2.64	Constructed termination digraph!
6.94/2.64	Nodes:
6.94/2.64	(1) f6_in(x8:0) -> f6_in(x8:0) :|: TRUE
6.94/2.64	(2) f6_in(cons(x14:0, x15:0)) -> f6_in(x15:0) :|: TRUE
6.94/2.64	
6.94/2.64	Arcs:
6.94/2.64	(1) -> (1), (2)
6.94/2.64	(2) -> (1), (2)
6.94/2.64	
6.94/2.64	This digraph is fully evaluated!
6.94/2.64	----------------------------------------
6.94/2.64	
6.94/2.64	(62)
6.94/2.64	Obligation:
6.94/2.64	
6.94/2.64	Termination digraph:
6.94/2.64	Nodes:
6.94/2.64	(1) f6_in(x8:0) -> f6_in(x8:0) :|: TRUE
6.94/2.64	(2) f6_in(cons(x14:0, x15:0)) -> f6_in(x15:0) :|: TRUE
6.94/2.64	
6.94/2.64	Arcs:
6.94/2.64	(1) -> (1), (2)
6.94/2.64	(2) -> (1), (2)
6.94/2.64	
6.94/2.64	This digraph is fully evaluated!
6.94/2.64	
6.94/2.64	----------------------------------------
6.94/2.64	
6.94/2.64	(63) IntTRSUnneededArgumentFilterProof (EQUIVALENT)
6.94/2.64	Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements:
6.94/2.64	
6.94/2.64	   cons(x1, x2) -> cons(x2)
6.94/2.64	
6.94/2.64	----------------------------------------
6.94/2.64	
6.94/2.64	(64)
6.94/2.64	Obligation:
6.94/2.64	Rules:
6.94/2.64	f6_in(x8:0) -> f6_in(x8:0) :|: TRUE
6.94/2.64	f6_in(cons(x15:0)) -> f6_in(x15:0) :|: TRUE
6.94/2.64	
6.94/2.64	----------------------------------------
6.94/2.64	
6.94/2.64	(65) PrologToDTProblemTransformerProof (SOUND)
6.94/2.64	Built DT problem from termination graph DT10.
6.94/2.64	
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6.94/2.64	    "root": 1,
6.94/2.64	    "program": {
6.94/2.64	        "directives": [],
6.94/2.64	        "clauses": [
6.94/2.64	            [
6.94/2.64	                "(right (tree X XS1 XS2) XS2)",
6.94/2.64	                null
6.94/2.64	            ],
6.94/2.64	            [
6.94/2.64	                "(flat (niltree) (nil))",
6.94/2.64	                null
6.94/2.64	            ],
6.94/2.64	            [
6.94/2.64	                "(flat (tree X (niltree) XS) (cons X YS))",
6.94/2.64	                "(',' (right (tree X (niltree) XS) ZS) (flat ZS YS))"
6.94/2.64	            ],
6.94/2.64	            [
6.94/2.64	                "(flat (tree X (tree Y YS1 YS2) XS) ZS)",
6.94/2.64	                "(flat (tree Y YS1 (tree X YS2 XS)) ZS)"
6.94/2.64	            ]
6.94/2.64	        ]
6.94/2.64	    },
6.94/2.64	    "graph": {
6.94/2.64	        "nodes": {
6.94/2.64	            "192": {
6.94/2.64	                "goal": [{
6.94/2.64	                    "clause": 0,
6.94/2.64	                    "scope": 3,
6.94/2.64	                    "term": "(',' (right (tree T55 (niltree) T58) X43) (flat X43 T57))"
6.94/2.64	                }],
6.94/2.64	                "kb": {
6.94/2.64	                    "nonunifying": [],
6.94/2.64	                    "intvars": {},
6.94/2.64	                    "arithmetic": {
6.94/2.64	                        "type": "PlainIntegerRelationState",
6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": [
6.94/2.64	                        "T55",
6.94/2.64	                        "T57"
6.94/2.64	                    ],
6.94/2.64	                    "free": ["X43"],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            },
6.94/2.64	            "171": {
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6.94/2.64	                    "clause": -1,
6.94/2.64	                    "scope": -1,
6.94/2.64	                    "term": "(flat (tree T15 T16 (tree T17 T18 T19)) (nil))"
6.94/2.64	                }],
6.94/2.64	                "kb": {
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6.94/2.64	                    "intvars": {},
6.94/2.64	                    "arithmetic": {
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6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": [],
6.94/2.64	                    "free": [],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            },
6.94/2.64	            "193": {
6.94/2.64	                "goal": [
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6.94/2.64	                        "clause": -1,
6.94/2.64	                        "scope": 3,
6.94/2.64	                        "term": null
6.94/2.64	                    },
6.94/2.64	                    {
6.94/2.64	                        "clause": 3,
6.94/2.64	                        "scope": 1,
6.94/2.64	                        "term": "(flat T1 (cons T55 T57))"
6.94/2.64	                    }
6.94/2.64	                ],
6.94/2.64	                "kb": {
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6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": [
6.94/2.64	                        "T55",
6.94/2.64	                        "T57"
6.94/2.64	                    ],
6.94/2.64	                    "free": [],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            },
6.94/2.64	            "type": "Nodes",
6.94/2.64	            "172": {
6.94/2.64	                "goal": [],
6.94/2.64	                "kb": {
6.94/2.64	                    "nonunifying": [],
6.94/2.64	                    "intvars": {},
6.94/2.64	                    "arithmetic": {
6.94/2.64	                        "type": "PlainIntegerRelationState",
6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": [],
6.94/2.64	                    "free": [],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            },
6.94/2.64	            "173": {
6.94/2.64	                "goal": [
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6.94/2.64	                        "clause": 1,
6.94/2.64	                        "scope": 2,
6.94/2.64	                        "term": "(flat (tree T15 T16 (tree T17 T18 T19)) (nil))"
6.94/2.64	                    },
6.94/2.64	                    {
6.94/2.64	                        "clause": 2,
6.94/2.64	                        "scope": 2,
6.94/2.64	                        "term": "(flat (tree T15 T16 (tree T17 T18 T19)) (nil))"
6.94/2.64	                    },
6.94/2.64	                    {
6.94/2.64	                        "clause": 3,
6.94/2.64	                        "scope": 2,
6.94/2.64	                        "term": "(flat (tree T15 T16 (tree T17 T18 T19)) (nil))"
6.94/2.64	                    }
6.94/2.64	                ],
6.94/2.64	                "kb": {
6.94/2.64	                    "nonunifying": [],
6.94/2.64	                    "intvars": {},
6.94/2.64	                    "arithmetic": {
6.94/2.64	                        "type": "PlainIntegerRelationState",
6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": [],
6.94/2.64	                    "free": [],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            },
6.94/2.64	            "174": {
6.94/2.64	                "goal": [
6.94/2.64	                    {
6.94/2.64	                        "clause": 2,
6.94/2.64	                        "scope": 2,
6.94/2.64	                        "term": "(flat (tree T15 T16 (tree T17 T18 T19)) (nil))"
6.94/2.64	                    },
6.94/2.64	                    {
6.94/2.64	                        "clause": 3,
6.94/2.64	                        "scope": 2,
6.94/2.64	                        "term": "(flat (tree T15 T16 (tree T17 T18 T19)) (nil))"
6.94/2.64	                    }
6.94/2.64	                ],
6.94/2.64	                "kb": {
6.94/2.64	                    "nonunifying": [],
6.94/2.64	                    "intvars": {},
6.94/2.64	                    "arithmetic": {
6.94/2.64	                        "type": "PlainIntegerRelationState",
6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": [],
6.94/2.64	                    "free": [],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            },
6.94/2.64	            "196": {
6.94/2.64	                "goal": [{
6.94/2.64	                    "clause": -1,
6.94/2.64	                    "scope": -1,
6.94/2.64	                    "term": "(flat T71 T57)"
6.94/2.64	                }],
6.94/2.64	                "kb": {
6.94/2.64	                    "nonunifying": [],
6.94/2.64	                    "intvars": {},
6.94/2.64	                    "arithmetic": {
6.94/2.64	                        "type": "PlainIntegerRelationState",
6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": ["T57"],
6.94/2.64	                    "free": [],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            },
6.94/2.64	            "197": {
6.94/2.64	                "goal": [{
6.94/2.64	                    "clause": 3,
6.94/2.64	                    "scope": 1,
6.94/2.64	                    "term": "(flat T1 (cons T55 T57))"
6.94/2.64	                }],
6.94/2.64	                "kb": {
6.94/2.64	                    "nonunifying": [],
6.94/2.64	                    "intvars": {},
6.94/2.64	                    "arithmetic": {
6.94/2.64	                        "type": "PlainIntegerRelationState",
6.94/2.64	                        "relations": []
6.94/2.64	                    },
6.94/2.64	                    "ground": [
6.94/2.64	                        "T55",
6.94/2.64	                        "T57"
6.94/2.64	                    ],
6.94/2.64	                    "free": [],
6.94/2.64	                    "exprvars": []
6.94/2.64	                }
6.94/2.64	            },
6.94/2.64	            "230": {
6.94/2.64	                "goal": [{
6.94/2.64	                    "clause": -1,
6.94/2.64	                    "scope": -1,
6.94/2.65	                    "term": "(flat (tree T110 T111 (tree T112 T113 T114)) T109)"
6.94/2.65	                }],
6.94/2.65	                "kb": {
6.94/2.65	                    "nonunifying": [
6.94/2.65	                        [
6.94/2.65	                            "(flat T1 T109)",
6.94/2.65	                            "(flat (niltree) (nil))"
6.94/2.65	                        ],
6.94/2.65	                        [
6.94/2.65	                            "(flat T1 T109)",
6.94/2.65	                            "(flat (tree X40 (niltree) X41) (cons X40 X42))"
6.94/2.65	                        ]
6.94/2.65	                    ],
6.94/2.65	                    "intvars": {},
6.94/2.65	                    "arithmetic": {
6.94/2.65	                        "type": "PlainIntegerRelationState",
6.94/2.65	                        "relations": []
6.94/2.65	                    },
6.94/2.65	                    "ground": ["T109"],
6.94/2.65	                    "free": [
6.94/2.65	                        "X40",
6.94/2.65	                        "X41",
6.94/2.65	                        "X42"
6.94/2.65	                    ],
6.94/2.65	                    "exprvars": []
6.94/2.65	                }
6.94/2.65	            },
6.94/2.65	            "176": {
6.94/2.65	                "goal": [{
6.94/2.65	                    "clause": 3,
6.94/2.65	                    "scope": 2,
6.94/2.65	                    "term": "(flat (tree T15 T16 (tree T17 T18 T19)) (nil))"
6.94/2.65	                }],
6.94/2.65	                "kb": {
6.94/2.65	                    "nonunifying": [],
6.94/2.65	                    "intvars": {},
6.94/2.65	                    "arithmetic": {
6.94/2.65	                        "type": "PlainIntegerRelationState",
6.94/2.65	                        "relations": []
6.94/2.65	                    },
6.94/2.65	                    "ground": [],
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6.94/2.65	                    "exprvars": []
6.94/2.65	                }
6.94/2.65	            },
6.94/2.65	            "198": {
6.94/2.65	                "goal": [{
6.94/2.65	                    "clause": -1,
6.94/2.65	                    "scope": -1,
6.94/2.65	                    "term": "(flat (tree T93 T94 (tree T95 T96 T97)) (cons T91 T92))"
6.94/2.65	                }],
6.94/2.65	                "kb": {
6.94/2.65	                    "nonunifying": [],
6.94/2.65	                    "intvars": {},
6.94/2.65	                    "arithmetic": {
6.94/2.65	                        "type": "PlainIntegerRelationState",
6.94/2.65	                        "relations": []
6.94/2.65	                    },
6.94/2.65	                    "ground": [
6.94/2.65	                        "T91",
6.94/2.65	                        "T92"
6.94/2.65	                    ],
6.94/2.65	                    "free": [],
6.94/2.65	                    "exprvars": []
6.94/2.65	                }
6.94/2.65	            },
6.94/2.65	            "231": {
6.94/2.65	                "goal": [],
6.94/2.65	                "kb": {
6.94/2.65	                    "nonunifying": [],
6.94/2.65	                    "intvars": {},
6.94/2.65	                    "arithmetic": {
6.94/2.65	                        "type": "PlainIntegerRelationState",
6.94/2.65	                        "relations": []
6.94/2.65	                    },
6.94/2.65	                    "ground": [],
6.94/2.65	                    "free": [],
6.94/2.65	                    "exprvars": []
6.94/2.65	                }
6.94/2.65	            },
6.94/2.65	            "155": {
6.94/2.65	                "goal": [
6.94/2.65	                    {
6.94/2.65	                        "clause": 1,
6.94/2.65	                        "scope": 1,
6.94/2.65	                        "term": "(flat T1 T2)"
6.94/2.65	                    },
6.94/2.65	                    {
6.94/2.65	                        "clause": 2,
6.94/2.65	                        "scope": 1,
6.94/2.65	                        "term": "(flat T1 T2)"
6.94/2.65	                    },
6.94/2.65	                    {
6.94/2.65	                        "clause": 3,
6.94/2.65	                        "scope": 1,
6.94/2.65	                        "term": "(flat T1 T2)"
6.94/2.65	                    }
6.94/2.65	                ],
6.94/2.65	                "kb": {
6.94/2.65	                    "nonunifying": [],
6.94/2.65	                    "intvars": {},
6.94/2.65	                    "arithmetic": {
6.94/2.65	                        "type": "PlainIntegerRelationState",
6.94/2.65	                        "relations": []
6.94/2.65	                    },
6.94/2.65	                    "ground": ["T2"],
6.94/2.65	                    "free": [],
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6.94/2.65	                }
6.94/2.65	            },
6.94/2.65	            "199": {
6.94/2.65	                "goal": [],
6.94/2.65	                "kb": {
6.94/2.65	                    "nonunifying": [],
6.94/2.65	                    "intvars": {},
6.94/2.65	                    "arithmetic": {
6.94/2.65	                        "type": "PlainIntegerRelationState",
6.94/2.65	                        "relations": []
6.94/2.65	                    },
6.94/2.65	                    "ground": [],
6.94/2.65	                    "free": [],
6.94/2.65	                    "exprvars": []
6.94/2.65	                }
6.94/2.65	            },
6.94/2.65	            "232": {
6.94/2.65	                "goal": [
6.94/2.65	                    {
6.94/2.65	                        "clause": 1,
6.94/2.65	                        "scope": 4,
6.94/2.65	                        "term": "(flat (tree T110 T111 (tree T112 T113 T114)) T109)"
6.94/2.65	                    },
6.94/2.65	                    {
6.94/2.65	                        "clause": 2,
6.94/2.65	                        "scope": 4,
6.94/2.65	                        "term": "(flat (tree T110 T111 (tree T112 T113 T114)) T109)"
6.94/2.65	                    },
6.94/2.65	                    {
6.94/2.65	                        "clause": 3,
6.94/2.65	                        "scope": 4,
6.94/2.65	                        "term": "(flat (tree T110 T111 (tree T112 T113 T114)) T109)"
6.94/2.65	                    }
6.94/2.65	                ],
6.94/2.65	                "kb": {
6.94/2.65	                    "nonunifying": [
6.94/2.65	                        [
6.94/2.65	                            "(flat T1 T109)",
6.94/2.65	                            "(flat (niltree) (nil))"
6.94/2.65	                        ],
6.94/2.65	                        [
6.94/2.65	                            "(flat T1 T109)",
6.94/2.65	                            "(flat (tree X40 (niltree) X41) (cons X40 X42))"
6.94/2.65	                        ]
6.94/2.65	                    ],
6.94/2.65	                    "intvars": {},
6.94/2.65	                    "arithmetic": {
6.94/2.65	                        "type": "PlainIntegerRelationState",
6.94/2.65	                        "relations": []
6.94/2.65	                    },
6.94/2.65	                    "ground": ["T109"],
6.94/2.65	                    "free": [
6.94/2.65	                        "X40",
6.94/2.65	                        "X41",
6.94/2.65	                        "X42"
6.94/2.65	                    ],
6.94/2.65	                    "exprvars": []
6.94/2.65	                }
6.94/2.65	            },
6.94/2.65	            "233": {
6.94/2.65	                "goal": [
6.94/2.65	                    {
6.94/2.65	                        "clause": 2,
6.94/2.65	                        "scope": 4,
6.94/2.65	                        "term": "(flat (tree T110 T111 (tree T112 T113 T114)) T109)"
6.94/2.65	                    },
6.94/2.65	                    {
6.94/2.65	                        "clause": 3,
6.94/2.65	                        "scope": 4,
6.94/2.65	                        "term": "(flat (tree T110 T111 (tree T112 T113 T114)) T109)"
6.94/2.65	                    }
6.94/2.65	                ],
6.94/2.65	                "kb": {
6.94/2.65	                    "nonunifying": [
6.94/2.65	                        [
6.94/2.65	                            "(flat T1 T109)",
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6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 159,
6.94/2.65	                "to": 160,
6.94/2.65	                "label": "BACKTRACK\nfor clause: flat(tree(X, niltree, XS), cons(X, YS)) :- ','(right(tree(X, niltree, XS), ZS), flat(ZS, YS))because of non-unification"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 160,
6.94/2.65	                "to": 171,
6.94/2.65	                "label": "EVAL with clause\nflat(tree(X10, tree(X11, X12, X13), X14), X15) :- flat(tree(X11, X12, tree(X10, X13, X14)), X15).\nand substitutionX10 -> T17,\nX11 -> T15,\nX12 -> T16,\nX13 -> T18,\nX14 -> T19,\nT1 -> tree(T17, tree(T15, T16, T18), T19),\nX15 -> nil,\nT11 -> T15,\nT12 -> T16,\nT10 -> T17,\nT13 -> T18,\nT14 -> T19"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 160,
6.94/2.65	                "to": 172,
6.94/2.65	                "label": "EVAL-BACKTRACK"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 171,
6.94/2.65	                "to": 173,
6.94/2.65	                "label": "CASE"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 173,
6.94/2.65	                "to": 174,
6.94/2.65	                "label": "BACKTRACK\nfor clause: flat(niltree, nil)because of non-unification"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 174,
6.94/2.65	                "to": 176,
6.94/2.65	                "label": "BACKTRACK\nfor clause: flat(tree(X, niltree, XS), cons(X, YS)) :- ','(right(tree(X, niltree, XS), ZS), flat(ZS, YS))because of non-unification"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 176,
6.94/2.65	                "to": 181,
6.94/2.65	                "label": "EVAL with clause\nflat(tree(X31, tree(X32, X33, X34), X35), X36) :- flat(tree(X32, X33, tree(X31, X34, X35)), X36).\nand substitutionT15 -> T47,\nX31 -> T47,\nX32 -> T45,\nX33 -> T46,\nX34 -> T48,\nT16 -> tree(T45, T46, T48),\nT17 -> T49,\nT18 -> T50,\nT19 -> T51,\nX35 -> tree(T49, T50, T51),\nX36 -> nil,\nT39 -> T45,\nT40 -> T46,\nT38 -> T47,\nT41 -> T48,\nT42 -> T49,\nT43 -> T50,\nT44 -> T51"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 176,
6.94/2.65	                "to": 182,
6.94/2.65	                "label": "EVAL-BACKTRACK"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 181,
6.94/2.65	                "to": 1,
6.94/2.65	                "label": "INSTANCE with matching:\nT1 -> tree(T45, T46, tree(T47, T48, tree(T49, T50, T51)))\nT2 -> nil"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 186,
6.94/2.65	                "to": 189,
6.94/2.65	                "label": "CASE"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 188,
6.94/2.65	                "to": 230,
6.94/2.65	                "label": "EVAL with clause\nflat(tree(X83, tree(X84, X85, X86), X87), X88) :- flat(tree(X84, X85, tree(X83, X86, X87)), X88).\nand substitutionX83 -> T112,\nX84 -> T110,\nX85 -> T111,\nX86 -> T113,\nX87 -> T114,\nT1 -> tree(T112, tree(T110, T111, T113), T114),\nT2 -> T109,\nX88 -> T109,\nT105 -> T110,\nT106 -> T111,\nT104 -> T112,\nT107 -> T113,\nT108 -> T114"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 188,
6.94/2.65	                "to": 231,
6.94/2.65	                "label": "EVAL-BACKTRACK"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 189,
6.94/2.65	                "to": 192,
6.94/2.65	                "label": "PARALLEL"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 189,
6.94/2.65	                "to": 193,
6.94/2.65	                "label": "PARALLEL"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 192,
6.94/2.65	                "to": 196,
6.94/2.65	                "label": "ONLY EVAL with clause\nright(tree(X56, X57, X58), X58).\nand substitutionT55 -> T69,\nX56 -> T69,\nX57 -> niltree,\nT58 -> T71,\nX58 -> T71,\nX43 -> T71,\nT70 -> T71"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 193,
6.94/2.65	                "to": 197,
6.94/2.65	                "label": "FAILURE"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 196,
6.94/2.65	                "to": 1,
6.94/2.65	                "label": "INSTANCE with matching:\nT1 -> T71\nT2 -> T57"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 197,
6.94/2.65	                "to": 198,
6.94/2.65	                "label": "EVAL with clause\nflat(tree(X71, tree(X72, X73, X74), X75), X76) :- flat(tree(X72, X73, tree(X71, X74, X75)), X76).\nand substitutionX71 -> T95,\nX72 -> T93,\nX73 -> T94,\nX74 -> T96,\nX75 -> T97,\nT1 -> tree(T95, tree(T93, T94, T96), T97),\nT55 -> T91,\nT57 -> T92,\nX76 -> cons(T91, T92),\nT87 -> T93,\nT88 -> T94,\nT86 -> T95,\nT89 -> T96,\nT90 -> T97"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 197,
6.94/2.65	                "to": 199,
6.94/2.65	                "label": "EVAL-BACKTRACK"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 198,
6.94/2.65	                "to": 1,
6.94/2.65	                "label": "INSTANCE with matching:\nT1 -> tree(T93, T94, tree(T95, T96, T97))\nT2 -> cons(T91, T92)"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 230,
6.94/2.65	                "to": 232,
6.94/2.65	                "label": "CASE"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 232,
6.94/2.65	                "to": 233,
6.94/2.65	                "label": "BACKTRACK\nfor clause: flat(niltree, nil)because of non-unification"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 233,
6.94/2.65	                "to": 234,
6.94/2.65	                "label": "PARALLEL"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 233,
6.94/2.65	                "to": 235,
6.94/2.65	                "label": "PARALLEL"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 234,
6.94/2.65	                "to": 236,
6.94/2.65	                "label": "EVAL with clause\nflat(tree(X103, niltree, X104), cons(X103, X105)) :- ','(right(tree(X103, niltree, X104), X106), flat(X106, X105)).\nand substitutionT110 -> T135,\nX103 -> T135,\nT111 -> niltree,\nT112 -> T140,\nT113 -> T141,\nT114 -> T142,\nX104 -> tree(T140, T141, T142),\nX105 -> T139,\nT109 -> cons(T135, T139),\nT136 -> T140,\nT137 -> T141,\nT138 -> T142"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 234,
6.94/2.65	                "to": 237,
6.94/2.65	                "label": "EVAL-BACKTRACK"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 235,
6.94/2.65	                "to": 240,
6.94/2.65	                "label": "EVAL with clause\nflat(tree(X128, tree(X129, X130, X131), X132), X133) :- flat(tree(X129, X130, tree(X128, X131, X132)), X133).\nand substitutionT110 -> T187,\nX128 -> T187,\nX129 -> T185,\nX130 -> T186,\nX131 -> T188,\nT111 -> tree(T185, T186, T188),\nT112 -> T189,\nT113 -> T190,\nT114 -> T191,\nX132 -> tree(T189, T190, T191),\nT109 -> T184,\nX133 -> T184,\nT178 -> T185,\nT179 -> T186,\nT177 -> T187,\nT180 -> T188,\nT181 -> T189,\nT182 -> T190,\nT183 -> T191"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 235,
6.94/2.65	                "to": 241,
6.94/2.65	                "label": "EVAL-BACKTRACK"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 236,
6.94/2.65	                "to": 238,
6.94/2.65	                "label": "CASE"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 238,
6.94/2.65	                "to": 239,
6.94/2.65	                "label": "ONLY EVAL with clause\nright(tree(X113, X114, X115), X115).\nand substitutionT135 -> T154,\nX113 -> T154,\nX114 -> niltree,\nT140 -> T158,\nT141 -> T159,\nT142 -> T160,\nX115 -> tree(T158, T159, T160),\nX106 -> tree(T158, T159, T160),\nT155 -> T158,\nT156 -> T159,\nT157 -> T160"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 239,
6.94/2.65	                "to": 1,
6.94/2.65	                "label": "INSTANCE with matching:\nT1 -> tree(T158, T159, T160)\nT2 -> T139"
6.94/2.65	            },
6.94/2.65	            {
6.94/2.65	                "from": 240,
6.94/2.65	                "to": 1,
6.94/2.65	                "label": "INSTANCE with matching:\nT1 -> tree(T185, T186, tree(T187, T188, tree(T189, T190, T191)))\nT2 -> T184"
6.94/2.65	            }
6.94/2.65	        ],
6.94/2.65	        "type": "Graph"
6.94/2.65	    }
6.94/2.65	}
6.94/2.65	
6.94/2.65	----------------------------------------
6.94/2.65	
6.94/2.65	(66)
6.94/2.65	Obligation:
6.94/2.65	Triples:
6.94/2.65	
6.94/2.65	flatA(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7), nil) :- flatA(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))), nil).
6.94/2.65	flatA(tree(X1, niltree, X2), cons(X1, X3)) :- flatA(X2, X3).
6.94/2.65	flatA(tree(X1, tree(X2, X3, X4), X5), cons(X6, X7)) :- flatA(tree(X2, X3, tree(X1, X4, X5)), cons(X6, X7)).
6.94/2.65	flatA(tree(X1, tree(X2, niltree, X3), X4), cons(X2, X5)) :- flatA(tree(X1, X3, X4), X5).
6.94/2.65	flatA(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7), X8) :- flatA(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))), X8).
6.94/2.65	
6.94/2.65	Clauses:
6.94/2.65	
6.94/2.65	flatcA(niltree, nil).
6.94/2.65	flatcA(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7), nil) :- flatcA(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))), nil).
6.94/2.65	flatcA(tree(X1, niltree, X2), cons(X1, X3)) :- flatcA(X2, X3).
6.94/2.65	flatcA(tree(X1, tree(X2, X3, X4), X5), cons(X6, X7)) :- flatcA(tree(X2, X3, tree(X1, X4, X5)), cons(X6, X7)).
6.94/2.65	flatcA(tree(X1, tree(X2, niltree, X3), X4), cons(X2, X5)) :- flatcA(tree(X1, X3, X4), X5).
6.94/2.65	flatcA(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7), X8) :- flatcA(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))), X8).
6.94/2.65	
6.94/2.65	Afs:
6.94/2.65	
6.94/2.65	flatA(x1, x2)  =  flatA(x2)
6.94/2.65	
6.94/2.65	
6.94/2.65	----------------------------------------
6.94/2.65	
6.94/2.65	(67) TriplesToPiDPProof (SOUND)
6.94/2.65	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.94/2.65	
6.94/2.65	flatA_in_2: (f,b)
6.94/2.65	
6.94/2.65	Transforming TRIPLES into the following Term Rewriting System:
6.94/2.65	
6.94/2.65	Pi DP problem:
6.94/2.65	The TRS P consists of the following rules:
6.94/2.65	
6.94/2.65	   FLATA_IN_AG(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7), nil) -> U1_AG(X1, X2, X3, X4, X5, X6, X7, flatA_in_ag(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))), nil))
6.94/2.65	   FLATA_IN_AG(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7), nil) -> FLATA_IN_AG(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))), nil)
6.94/2.65	   FLATA_IN_AG(tree(X1, niltree, X2), cons(X1, X3)) -> U2_AG(X1, X2, X3, flatA_in_ag(X2, X3))
6.94/2.65	   FLATA_IN_AG(tree(X1, niltree, X2), cons(X1, X3)) -> FLATA_IN_AG(X2, X3)
6.94/2.65	   FLATA_IN_AG(tree(X1, tree(X2, X3, X4), X5), cons(X6, X7)) -> U3_AG(X1, X2, X3, X4, X5, X6, X7, flatA_in_ag(tree(X2, X3, tree(X1, X4, X5)), cons(X6, X7)))
6.94/2.65	   FLATA_IN_AG(tree(X1, tree(X2, X3, X4), X5), cons(X6, X7)) -> FLATA_IN_AG(tree(X2, X3, tree(X1, X4, X5)), cons(X6, X7))
6.94/2.65	   FLATA_IN_AG(tree(X1, tree(X2, niltree, X3), X4), cons(X2, X5)) -> U4_AG(X1, X2, X3, X4, X5, flatA_in_ag(tree(X1, X3, X4), X5))
6.94/2.65	   FLATA_IN_AG(tree(X1, tree(X2, niltree, X3), X4), cons(X2, X5)) -> FLATA_IN_AG(tree(X1, X3, X4), X5)
6.94/2.65	   FLATA_IN_AG(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7), X8) -> U5_AG(X1, X2, X3, X4, X5, X6, X7, X8, flatA_in_ag(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))), X8))
6.94/2.65	   FLATA_IN_AG(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7), X8) -> FLATA_IN_AG(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))), X8)
6.94/2.65	
6.94/2.65	R is empty.
6.94/2.65	The argument filtering Pi contains the following mapping:
6.94/2.65	flatA_in_ag(x1, x2)  =  flatA_in_ag(x2)
6.94/2.65	
6.94/2.65	nil  =  nil
6.94/2.65	
6.94/2.65	tree(x1, x2, x3)  =  tree(x2, x3)
6.94/2.65	
6.94/2.65	cons(x1, x2)  =  cons(x1, x2)
6.94/2.65	
6.94/2.65	niltree  =  niltree
6.94/2.65	
6.94/2.65	FLATA_IN_AG(x1, x2)  =  FLATA_IN_AG(x2)
6.94/2.65	
6.94/2.65	U1_AG(x1, x2, x3, x4, x5, x6, x7, x8)  =  U1_AG(x8)
6.94/2.65	
6.94/2.65	U2_AG(x1, x2, x3, x4)  =  U2_AG(x1, x3, x4)
6.94/2.65	
6.94/2.65	U3_AG(x1, x2, x3, x4, x5, x6, x7, x8)  =  U3_AG(x6, x7, x8)
6.94/2.65	
6.94/2.65	U4_AG(x1, x2, x3, x4, x5, x6)  =  U4_AG(x2, x5, x6)
6.94/2.65	
6.94/2.65	U5_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U5_AG(x8, x9)
6.94/2.65	
6.94/2.65	
6.94/2.65	We have to consider all (P,R,Pi)-chains
6.94/2.65	
6.94/2.65	
6.94/2.65	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
6.94/2.65	
6.94/2.65	
6.94/2.65	
6.94/2.65	----------------------------------------
6.94/2.65	
6.94/2.65	(68)
6.94/2.65	Obligation:
6.94/2.65	Pi DP problem:
6.94/2.65	The TRS P consists of the following rules:
6.94/2.65	
6.94/2.65	   FLATA_IN_AG(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7), nil) -> U1_AG(X1, X2, X3, X4, X5, X6, X7, flatA_in_ag(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))), nil))
6.94/2.65	   FLATA_IN_AG(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7), nil) -> FLATA_IN_AG(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))), nil)
6.94/2.65	   FLATA_IN_AG(tree(X1, niltree, X2), cons(X1, X3)) -> U2_AG(X1, X2, X3, flatA_in_ag(X2, X3))
6.94/2.65	   FLATA_IN_AG(tree(X1, niltree, X2), cons(X1, X3)) -> FLATA_IN_AG(X2, X3)
6.94/2.65	   FLATA_IN_AG(tree(X1, tree(X2, X3, X4), X5), cons(X6, X7)) -> U3_AG(X1, X2, X3, X4, X5, X6, X7, flatA_in_ag(tree(X2, X3, tree(X1, X4, X5)), cons(X6, X7)))
6.94/2.65	   FLATA_IN_AG(tree(X1, tree(X2, X3, X4), X5), cons(X6, X7)) -> FLATA_IN_AG(tree(X2, X3, tree(X1, X4, X5)), cons(X6, X7))
6.94/2.65	   FLATA_IN_AG(tree(X1, tree(X2, niltree, X3), X4), cons(X2, X5)) -> U4_AG(X1, X2, X3, X4, X5, flatA_in_ag(tree(X1, X3, X4), X5))
6.94/2.65	   FLATA_IN_AG(tree(X1, tree(X2, niltree, X3), X4), cons(X2, X5)) -> FLATA_IN_AG(tree(X1, X3, X4), X5)
6.94/2.65	   FLATA_IN_AG(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7), X8) -> U5_AG(X1, X2, X3, X4, X5, X6, X7, X8, flatA_in_ag(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))), X8))
6.94/2.65	   FLATA_IN_AG(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7), X8) -> FLATA_IN_AG(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))), X8)
6.94/2.65	
6.94/2.65	R is empty.
6.94/2.65	The argument filtering Pi contains the following mapping:
6.94/2.65	flatA_in_ag(x1, x2)  =  flatA_in_ag(x2)
6.94/2.65	
6.94/2.65	nil  =  nil
6.94/2.65	
6.94/2.65	tree(x1, x2, x3)  =  tree(x2, x3)
6.94/2.65	
6.94/2.65	cons(x1, x2)  =  cons(x1, x2)
6.94/2.65	
6.94/2.65	niltree  =  niltree
6.94/2.65	
6.94/2.65	FLATA_IN_AG(x1, x2)  =  FLATA_IN_AG(x2)
6.94/2.65	
6.94/2.65	U1_AG(x1, x2, x3, x4, x5, x6, x7, x8)  =  U1_AG(x8)
6.94/2.65	
6.94/2.65	U2_AG(x1, x2, x3, x4)  =  U2_AG(x1, x3, x4)
6.94/2.65	
6.94/2.65	U3_AG(x1, x2, x3, x4, x5, x6, x7, x8)  =  U3_AG(x6, x7, x8)
6.94/2.65	
6.94/2.65	U4_AG(x1, x2, x3, x4, x5, x6)  =  U4_AG(x2, x5, x6)
6.94/2.65	
6.94/2.65	U5_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U5_AG(x8, x9)
6.94/2.65	
6.94/2.65	
6.94/2.65	We have to consider all (P,R,Pi)-chains
6.94/2.65	----------------------------------------
6.94/2.65	
6.94/2.65	(69) DependencyGraphProof (EQUIVALENT)
6.94/2.65	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 5 less nodes.
6.94/2.65	----------------------------------------
6.94/2.65	
6.94/2.65	(70)
6.94/2.65	Obligation:
6.94/2.65	Pi DP problem:
6.94/2.65	The TRS P consists of the following rules:
6.94/2.65	
6.94/2.65	   FLATA_IN_AG(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7), X8) -> FLATA_IN_AG(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))), X8)
6.94/2.65	   FLATA_IN_AG(tree(X1, tree(X2, tree(X3, X4, X5), X6), X7), nil) -> FLATA_IN_AG(tree(X3, X4, tree(X2, X5, tree(X1, X6, X7))), nil)
6.94/2.65	   FLATA_IN_AG(tree(X1, niltree, X2), cons(X1, X3)) -> FLATA_IN_AG(X2, X3)
6.94/2.65	   FLATA_IN_AG(tree(X1, tree(X2, X3, X4), X5), cons(X6, X7)) -> FLATA_IN_AG(tree(X2, X3, tree(X1, X4, X5)), cons(X6, X7))
6.94/2.65	   FLATA_IN_AG(tree(X1, tree(X2, niltree, X3), X4), cons(X2, X5)) -> FLATA_IN_AG(tree(X1, X3, X4), X5)
6.94/2.65	
6.94/2.65	R is empty.
6.94/2.65	The argument filtering Pi contains the following mapping:
6.94/2.65	nil  =  nil
6.94/2.65	
6.94/2.65	tree(x1, x2, x3)  =  tree(x2, x3)
6.94/2.65	
6.94/2.65	cons(x1, x2)  =  cons(x1, x2)
6.94/2.65	
6.94/2.65	niltree  =  niltree
6.94/2.65	
6.94/2.65	FLATA_IN_AG(x1, x2)  =  FLATA_IN_AG(x2)
6.94/2.65	
6.94/2.65	
6.94/2.65	We have to consider all (P,R,Pi)-chains
6.94/2.65	----------------------------------------
6.94/2.65	
6.94/2.65	(71) PiDPToQDPProof (SOUND)
6.94/2.65	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.94/2.65	----------------------------------------
6.94/2.65	
6.94/2.65	(72)
6.94/2.65	Obligation:
6.94/2.65	Q DP problem:
6.94/2.65	The TRS P consists of the following rules:
6.94/2.65	
6.94/2.65	   FLATA_IN_AG(X8) -> FLATA_IN_AG(X8)
6.94/2.65	   FLATA_IN_AG(nil) -> FLATA_IN_AG(nil)
6.94/2.65	   FLATA_IN_AG(cons(X1, X3)) -> FLATA_IN_AG(X3)
6.94/2.65	   FLATA_IN_AG(cons(X6, X7)) -> FLATA_IN_AG(cons(X6, X7))
6.94/2.65	
6.94/2.65	R is empty.
6.94/2.65	Q is empty.
6.94/2.65	We have to consider all (P,Q,R)-chains.
6.94/2.65	----------------------------------------
6.94/2.65	
6.94/2.65	(73) MRRProof (EQUIVALENT)
6.94/2.65	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.
6.94/2.65	
6.94/2.65	Strictly oriented dependency pairs:
6.94/2.65	
6.94/2.65	   FLATA_IN_AG(cons(X1, X3)) -> FLATA_IN_AG(X3)
6.94/2.65	
6.94/2.65	
6.94/2.65	Used ordering: Knuth-Bendix order [KBO] with precedence:cons_2 > nil > FLATA_IN_AG_1
6.94/2.65	
6.94/2.65	and weight map:
6.94/2.65	
6.94/2.65	   nil=1
6.94/2.65	   FLATA_IN_AG_1=1
6.94/2.65	   cons_2=0
6.94/2.65	
6.94/2.65	The variable weight is 1
6.94/2.65	
6.94/2.65	----------------------------------------
6.94/2.65	
6.94/2.65	(74)
6.94/2.65	Obligation:
6.94/2.65	Q DP problem:
6.94/2.65	The TRS P consists of the following rules:
6.94/2.65	
6.94/2.65	   FLATA_IN_AG(X8) -> FLATA_IN_AG(X8)
6.94/2.65	   FLATA_IN_AG(nil) -> FLATA_IN_AG(nil)
6.94/2.65	   FLATA_IN_AG(cons(X6, X7)) -> FLATA_IN_AG(cons(X6, X7))
6.94/2.65	
6.94/2.65	R is empty.
6.94/2.65	Q is empty.
6.94/2.65	We have to consider all (P,Q,R)-chains.
6.94/2.65	----------------------------------------
6.94/2.65	
6.94/2.65	(75) NonTerminationLoopProof (COMPLETE)
6.94/2.65	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
6.94/2.65	Found a loop by semiunifying a rule from P directly.
6.94/2.65	
6.94/2.65	s = FLATA_IN_AG(X8) evaluates to  t =FLATA_IN_AG(X8)
6.94/2.65	
6.94/2.65	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
6.94/2.65	* Matcher: [ ]
6.94/2.65	* Semiunifier: [ ]
6.94/2.65	
6.94/2.65	--------------------------------------------------------------------------------
6.94/2.65	Rewriting sequence
6.94/2.65	
6.94/2.65	The DP semiunifies directly so there is only one rewrite step from FLATA_IN_AG(X8) to FLATA_IN_AG(X8).
6.94/2.65	
6.94/2.65	
6.94/2.65	
6.94/2.65	
6.94/2.65	----------------------------------------
6.94/2.65	
6.94/2.65	(76)
6.94/2.65	NO
7.05/2.79	EOF