3.71/1.73	YES
3.71/1.75	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.71/1.75	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.71/1.75	
3.71/1.75	
3.71/1.75	Left Termination of the query pattern
3.71/1.75	
3.71/1.75	preorder(g,a)
3.71/1.75	
3.71/1.75	w.r.t. the given Prolog program could successfully be proven:
3.71/1.75	
3.71/1.75	(0) Prolog
3.71/1.75	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.71/1.75	(2) PiTRS
3.71/1.75	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.71/1.75	(4) PiDP
3.71/1.75	(5) DependencyGraphProof [EQUIVALENT, 1 ms]
3.71/1.75	(6) PiDP
3.71/1.75	(7) UsableRulesProof [EQUIVALENT, 0 ms]
3.71/1.75	(8) PiDP
3.71/1.75	(9) PiDPToQDPProof [SOUND, 0 ms]
3.71/1.75	(10) QDP
3.71/1.75	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.71/1.75	(12) YES
3.71/1.75	
3.71/1.75	
3.71/1.75	----------------------------------------
3.71/1.75	
3.71/1.75	(0)
3.71/1.75	Obligation:
3.71/1.75	Clauses:
3.71/1.75	
3.71/1.75	preorder(T, Xs) :- preorder_dl(T, -(Xs, [])).
3.71/1.75	preorder_dl(nil, -(X, X)).
3.71/1.75	preorder_dl(tree(L, X, R), -(.(X, Xs), Zs)) :- ','(preorder_dl(L, -(Xs, Ys)), preorder_dl(R, -(Ys, Zs))).
3.71/1.75	
3.71/1.75	
3.71/1.75	Query: preorder(g,a)
3.71/1.75	----------------------------------------
3.71/1.75	
3.71/1.75	(1) PrologToPiTRSProof (SOUND)
3.71/1.75	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.71/1.75	
3.71/1.75	preorder_in_2: (b,f)
3.71/1.75	
3.71/1.75	preorder_dl_in_2: (b,f)
3.71/1.75	
3.71/1.75	Transforming Prolog into the following Term Rewriting System:
3.71/1.75	
3.71/1.75	Pi-finite rewrite system:
3.71/1.75	The TRS R consists of the following rules:
3.71/1.75	
3.71/1.75	   preorder_in_ga(T, Xs) -> U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
3.71/1.75	   preorder_dl_in_ga(nil, -(X, X)) -> preorder_dl_out_ga(nil, -(X, X))
3.71/1.75	   preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) -> U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
3.71/1.75	   U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
3.71/1.75	   U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) -> preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))
3.71/1.75	   U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) -> preorder_out_ga(T, Xs)
3.71/1.75	
3.71/1.75	The argument filtering Pi contains the following mapping:
3.71/1.75	preorder_in_ga(x1, x2)  =  preorder_in_ga(x1)
3.71/1.75	
3.71/1.75	U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
3.71/1.75	
3.71/1.75	preorder_dl_in_ga(x1, x2)  =  preorder_dl_in_ga(x1)
3.71/1.75	
3.71/1.75	nil  =  nil
3.71/1.75	
3.71/1.75	preorder_dl_out_ga(x1, x2)  =  preorder_dl_out_ga(x1)
3.71/1.75	
3.71/1.75	tree(x1, x2, x3)  =  tree(x1, x2, x3)
3.71/1.75	
3.71/1.75	U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x1, x2, x3, x6)
3.71/1.75	
3.71/1.75	U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x1, x2, x3, x6)
3.71/1.75	
3.71/1.75	preorder_out_ga(x1, x2)  =  preorder_out_ga(x1)
3.71/1.75	
3.71/1.75	
3.71/1.75	
3.71/1.75	
3.71/1.75	
3.71/1.75	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.71/1.75	
3.71/1.75	
3.71/1.75	
3.71/1.75	----------------------------------------
3.71/1.75	
3.71/1.75	(2)
3.71/1.75	Obligation:
3.71/1.75	Pi-finite rewrite system:
3.71/1.75	The TRS R consists of the following rules:
3.71/1.75	
3.71/1.75	   preorder_in_ga(T, Xs) -> U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
3.71/1.75	   preorder_dl_in_ga(nil, -(X, X)) -> preorder_dl_out_ga(nil, -(X, X))
3.71/1.75	   preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) -> U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
3.71/1.75	   U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
3.71/1.75	   U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) -> preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))
3.71/1.75	   U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) -> preorder_out_ga(T, Xs)
3.71/1.75	
3.71/1.75	The argument filtering Pi contains the following mapping:
3.71/1.75	preorder_in_ga(x1, x2)  =  preorder_in_ga(x1)
3.71/1.75	
3.71/1.75	U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
3.71/1.75	
3.71/1.75	preorder_dl_in_ga(x1, x2)  =  preorder_dl_in_ga(x1)
3.71/1.75	
3.71/1.75	nil  =  nil
3.71/1.75	
3.71/1.75	preorder_dl_out_ga(x1, x2)  =  preorder_dl_out_ga(x1)
3.71/1.75	
3.71/1.75	tree(x1, x2, x3)  =  tree(x1, x2, x3)
3.71/1.75	
3.71/1.75	U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x1, x2, x3, x6)
3.71/1.75	
3.71/1.75	U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x1, x2, x3, x6)
3.71/1.75	
3.71/1.75	preorder_out_ga(x1, x2)  =  preorder_out_ga(x1)
3.71/1.75	
3.71/1.75	
3.71/1.75	
3.71/1.75	----------------------------------------
3.71/1.75	
3.71/1.75	(3) DependencyPairsProof (EQUIVALENT)
3.71/1.75	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.71/1.75	Pi DP problem:
3.71/1.75	The TRS P consists of the following rules:
3.71/1.75	
3.71/1.75	   PREORDER_IN_GA(T, Xs) -> U1_GA(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
3.71/1.75	   PREORDER_IN_GA(T, Xs) -> PREORDER_DL_IN_GA(T, -(Xs, []))
3.71/1.75	   PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) -> U2_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
3.71/1.75	   PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) -> PREORDER_DL_IN_GA(L, -(Xs, Ys))
3.71/1.75	   U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> U3_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
3.71/1.75	   U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> PREORDER_DL_IN_GA(R, -(Ys, Zs))
3.71/1.75	
3.71/1.75	The TRS R consists of the following rules:
3.71/1.75	
3.71/1.75	   preorder_in_ga(T, Xs) -> U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
3.71/1.75	   preorder_dl_in_ga(nil, -(X, X)) -> preorder_dl_out_ga(nil, -(X, X))
3.71/1.75	   preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) -> U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
3.71/1.75	   U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
3.71/1.75	   U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) -> preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))
3.71/1.75	   U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) -> preorder_out_ga(T, Xs)
3.71/1.75	
3.71/1.75	The argument filtering Pi contains the following mapping:
3.71/1.75	preorder_in_ga(x1, x2)  =  preorder_in_ga(x1)
3.71/1.75	
3.71/1.75	U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
3.71/1.75	
3.71/1.75	preorder_dl_in_ga(x1, x2)  =  preorder_dl_in_ga(x1)
3.71/1.75	
3.71/1.75	nil  =  nil
3.71/1.75	
3.71/1.75	preorder_dl_out_ga(x1, x2)  =  preorder_dl_out_ga(x1)
3.71/1.75	
3.71/1.75	tree(x1, x2, x3)  =  tree(x1, x2, x3)
3.71/1.75	
3.71/1.75	U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x1, x2, x3, x6)
3.71/1.75	
3.71/1.75	U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x1, x2, x3, x6)
3.71/1.75	
3.71/1.75	preorder_out_ga(x1, x2)  =  preorder_out_ga(x1)
3.71/1.75	
3.71/1.75	PREORDER_IN_GA(x1, x2)  =  PREORDER_IN_GA(x1)
3.71/1.75	
3.71/1.75	U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
3.71/1.75	
3.71/1.75	PREORDER_DL_IN_GA(x1, x2)  =  PREORDER_DL_IN_GA(x1)
3.71/1.75	
3.71/1.75	U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x1, x2, x3, x6)
3.71/1.75	
3.71/1.75	U3_GA(x1, x2, x3, x4, x5, x6)  =  U3_GA(x1, x2, x3, x6)
3.71/1.75	
3.71/1.75	
3.71/1.75	We have to consider all (P,R,Pi)-chains
3.71/1.75	----------------------------------------
3.71/1.75	
3.71/1.75	(4)
3.71/1.75	Obligation:
3.71/1.75	Pi DP problem:
3.71/1.75	The TRS P consists of the following rules:
3.71/1.75	
3.71/1.75	   PREORDER_IN_GA(T, Xs) -> U1_GA(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
3.71/1.75	   PREORDER_IN_GA(T, Xs) -> PREORDER_DL_IN_GA(T, -(Xs, []))
3.71/1.75	   PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) -> U2_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
3.71/1.75	   PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) -> PREORDER_DL_IN_GA(L, -(Xs, Ys))
3.71/1.75	   U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> U3_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
3.71/1.75	   U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> PREORDER_DL_IN_GA(R, -(Ys, Zs))
3.71/1.75	
3.71/1.75	The TRS R consists of the following rules:
3.71/1.75	
3.71/1.75	   preorder_in_ga(T, Xs) -> U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
3.71/1.75	   preorder_dl_in_ga(nil, -(X, X)) -> preorder_dl_out_ga(nil, -(X, X))
3.71/1.75	   preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) -> U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
3.71/1.75	   U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
3.71/1.75	   U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) -> preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))
3.71/1.75	   U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) -> preorder_out_ga(T, Xs)
3.71/1.75	
3.71/1.75	The argument filtering Pi contains the following mapping:
3.71/1.75	preorder_in_ga(x1, x2)  =  preorder_in_ga(x1)
3.71/1.75	
3.71/1.75	U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
3.71/1.75	
3.71/1.75	preorder_dl_in_ga(x1, x2)  =  preorder_dl_in_ga(x1)
3.71/1.75	
3.71/1.75	nil  =  nil
3.71/1.75	
3.71/1.75	preorder_dl_out_ga(x1, x2)  =  preorder_dl_out_ga(x1)
3.71/1.75	
3.71/1.75	tree(x1, x2, x3)  =  tree(x1, x2, x3)
3.71/1.75	
3.71/1.75	U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x1, x2, x3, x6)
3.71/1.75	
3.71/1.75	U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x1, x2, x3, x6)
3.71/1.75	
3.71/1.75	preorder_out_ga(x1, x2)  =  preorder_out_ga(x1)
3.71/1.75	
3.71/1.75	PREORDER_IN_GA(x1, x2)  =  PREORDER_IN_GA(x1)
3.71/1.75	
3.71/1.75	U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
3.71/1.75	
3.71/1.75	PREORDER_DL_IN_GA(x1, x2)  =  PREORDER_DL_IN_GA(x1)
3.71/1.75	
3.71/1.75	U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x1, x2, x3, x6)
3.71/1.75	
3.71/1.75	U3_GA(x1, x2, x3, x4, x5, x6)  =  U3_GA(x1, x2, x3, x6)
3.71/1.75	
3.71/1.75	
3.71/1.75	We have to consider all (P,R,Pi)-chains
3.71/1.75	----------------------------------------
3.71/1.75	
3.71/1.75	(5) DependencyGraphProof (EQUIVALENT)
3.71/1.75	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
3.71/1.75	----------------------------------------
3.71/1.75	
3.71/1.75	(6)
3.71/1.75	Obligation:
3.71/1.75	Pi DP problem:
3.71/1.75	The TRS P consists of the following rules:
3.71/1.75	
3.71/1.75	   U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> PREORDER_DL_IN_GA(R, -(Ys, Zs))
3.71/1.75	   PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) -> U2_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
3.71/1.75	   PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) -> PREORDER_DL_IN_GA(L, -(Xs, Ys))
3.71/1.75	
3.71/1.75	The TRS R consists of the following rules:
3.71/1.75	
3.71/1.75	   preorder_in_ga(T, Xs) -> U1_ga(T, Xs, preorder_dl_in_ga(T, -(Xs, [])))
3.71/1.75	   preorder_dl_in_ga(nil, -(X, X)) -> preorder_dl_out_ga(nil, -(X, X))
3.71/1.75	   preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) -> U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
3.71/1.75	   U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
3.71/1.75	   U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) -> preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))
3.71/1.75	   U1_ga(T, Xs, preorder_dl_out_ga(T, -(Xs, []))) -> preorder_out_ga(T, Xs)
3.71/1.75	
3.71/1.75	The argument filtering Pi contains the following mapping:
3.71/1.75	preorder_in_ga(x1, x2)  =  preorder_in_ga(x1)
3.71/1.75	
3.71/1.75	U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
3.71/1.75	
3.71/1.75	preorder_dl_in_ga(x1, x2)  =  preorder_dl_in_ga(x1)
3.71/1.75	
3.71/1.75	nil  =  nil
3.71/1.75	
3.71/1.75	preorder_dl_out_ga(x1, x2)  =  preorder_dl_out_ga(x1)
3.71/1.75	
3.71/1.75	tree(x1, x2, x3)  =  tree(x1, x2, x3)
3.71/1.75	
3.71/1.75	U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x1, x2, x3, x6)
3.71/1.75	
3.71/1.75	U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x1, x2, x3, x6)
3.71/1.75	
3.71/1.75	preorder_out_ga(x1, x2)  =  preorder_out_ga(x1)
3.71/1.75	
3.71/1.75	PREORDER_DL_IN_GA(x1, x2)  =  PREORDER_DL_IN_GA(x1)
3.71/1.75	
3.71/1.75	U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x1, x2, x3, x6)
3.71/1.75	
3.71/1.75	
3.71/1.75	We have to consider all (P,R,Pi)-chains
3.71/1.75	----------------------------------------
3.71/1.75	
3.71/1.75	(7) UsableRulesProof (EQUIVALENT)
3.71/1.75	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.71/1.75	----------------------------------------
3.71/1.75	
3.71/1.75	(8)
3.71/1.75	Obligation:
3.71/1.75	Pi DP problem:
3.71/1.75	The TRS P consists of the following rules:
3.71/1.75	
3.71/1.75	   U2_GA(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> PREORDER_DL_IN_GA(R, -(Ys, Zs))
3.71/1.75	   PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) -> U2_GA(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
3.71/1.75	   PREORDER_DL_IN_GA(tree(L, X, R), -(.(X, Xs), Zs)) -> PREORDER_DL_IN_GA(L, -(Xs, Ys))
3.71/1.75	
3.71/1.75	The TRS R consists of the following rules:
3.71/1.75	
3.71/1.75	   preorder_dl_in_ga(nil, -(X, X)) -> preorder_dl_out_ga(nil, -(X, X))
3.71/1.75	   preorder_dl_in_ga(tree(L, X, R), -(.(X, Xs), Zs)) -> U2_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(L, -(Xs, Ys)))
3.71/1.75	   U2_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(L, -(Xs, Ys))) -> U3_ga(L, X, R, Xs, Zs, preorder_dl_in_ga(R, -(Ys, Zs)))
3.71/1.75	   U3_ga(L, X, R, Xs, Zs, preorder_dl_out_ga(R, -(Ys, Zs))) -> preorder_dl_out_ga(tree(L, X, R), -(.(X, Xs), Zs))
3.71/1.75	
3.71/1.75	The argument filtering Pi contains the following mapping:
3.71/1.75	preorder_dl_in_ga(x1, x2)  =  preorder_dl_in_ga(x1)
3.71/1.75	
3.71/1.75	nil  =  nil
3.71/1.75	
3.71/1.75	preorder_dl_out_ga(x1, x2)  =  preorder_dl_out_ga(x1)
3.71/1.75	
3.71/1.75	tree(x1, x2, x3)  =  tree(x1, x2, x3)
3.71/1.75	
3.71/1.75	U2_ga(x1, x2, x3, x4, x5, x6)  =  U2_ga(x1, x2, x3, x6)
3.71/1.75	
3.71/1.75	U3_ga(x1, x2, x3, x4, x5, x6)  =  U3_ga(x1, x2, x3, x6)
3.71/1.75	
3.71/1.75	PREORDER_DL_IN_GA(x1, x2)  =  PREORDER_DL_IN_GA(x1)
3.71/1.75	
3.71/1.75	U2_GA(x1, x2, x3, x4, x5, x6)  =  U2_GA(x1, x2, x3, x6)
3.71/1.75	
3.71/1.75	
3.71/1.75	We have to consider all (P,R,Pi)-chains
3.71/1.75	----------------------------------------
3.71/1.75	
3.71/1.75	(9) PiDPToQDPProof (SOUND)
3.71/1.75	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.71/1.75	----------------------------------------
3.71/1.75	
3.71/1.75	(10)
3.71/1.75	Obligation:
3.71/1.75	Q DP problem:
3.71/1.75	The TRS P consists of the following rules:
3.71/1.75	
3.71/1.75	   U2_GA(L, X, R, preorder_dl_out_ga(L)) -> PREORDER_DL_IN_GA(R)
3.71/1.75	   PREORDER_DL_IN_GA(tree(L, X, R)) -> U2_GA(L, X, R, preorder_dl_in_ga(L))
3.71/1.75	   PREORDER_DL_IN_GA(tree(L, X, R)) -> PREORDER_DL_IN_GA(L)
3.71/1.75	
3.71/1.75	The TRS R consists of the following rules:
3.71/1.75	
3.71/1.75	   preorder_dl_in_ga(nil) -> preorder_dl_out_ga(nil)
3.71/1.75	   preorder_dl_in_ga(tree(L, X, R)) -> U2_ga(L, X, R, preorder_dl_in_ga(L))
3.71/1.75	   U2_ga(L, X, R, preorder_dl_out_ga(L)) -> U3_ga(L, X, R, preorder_dl_in_ga(R))
3.71/1.75	   U3_ga(L, X, R, preorder_dl_out_ga(R)) -> preorder_dl_out_ga(tree(L, X, R))
3.71/1.75	
3.71/1.75	The set Q consists of the following terms:
3.71/1.75	
3.71/1.75	   preorder_dl_in_ga(x0)
3.71/1.75	   U2_ga(x0, x1, x2, x3)
3.71/1.75	   U3_ga(x0, x1, x2, x3)
3.71/1.75	
3.71/1.75	We have to consider all (P,Q,R)-chains.
3.71/1.75	----------------------------------------
3.71/1.75	
3.71/1.75	(11) QDPSizeChangeProof (EQUIVALENT)
3.71/1.75	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
3.71/1.75	
3.71/1.75	From the DPs we obtained the following set of size-change graphs:
3.71/1.75	*PREORDER_DL_IN_GA(tree(L, X, R)) -> U2_GA(L, X, R, preorder_dl_in_ga(L))
3.71/1.75	The graph contains the following edges 1 > 1, 1 > 2, 1 > 3
3.71/1.75	
3.71/1.75	
3.71/1.75	*PREORDER_DL_IN_GA(tree(L, X, R)) -> PREORDER_DL_IN_GA(L)
3.71/1.75	The graph contains the following edges 1 > 1
3.71/1.75	
3.71/1.75	
3.71/1.75	*U2_GA(L, X, R, preorder_dl_out_ga(L)) -> PREORDER_DL_IN_GA(R)
3.71/1.75	The graph contains the following edges 3 >= 1
3.71/1.75	
3.71/1.75	
3.71/1.75	----------------------------------------
3.71/1.75	
3.71/1.75	(12)
3.71/1.75	YES
3.71/1.78	EOF