5.50/2.23	YES
5.50/2.25	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
5.50/2.25	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.50/2.25	
5.50/2.25	
5.50/2.25	Left Termination of the query pattern
5.50/2.25	
5.50/2.25	search_tree(g)
5.50/2.25	
5.50/2.25	w.r.t. the given Prolog program could successfully be proven:
5.50/2.25	
5.50/2.25	(0) Prolog
5.50/2.25	(1) PrologToPiTRSProof [SOUND, 0 ms]
5.50/2.25	(2) PiTRS
5.50/2.25	(3) DependencyPairsProof [EQUIVALENT, 3 ms]
5.50/2.25	(4) PiDP
5.50/2.25	(5) DependencyGraphProof [EQUIVALENT, 2 ms]
5.50/2.25	(6) AND
5.50/2.25	    (7) PiDP
5.50/2.25	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
5.50/2.25	        (9) PiDP
5.50/2.25	        (10) PiDPToQDPProof [EQUIVALENT, 0 ms]
5.50/2.25	        (11) QDP
5.50/2.25	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
5.50/2.25	        (13) YES
5.50/2.25	    (14) PiDP
5.50/2.25	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
5.50/2.25	        (16) PiDP
5.50/2.25	        (17) PiDPToQDPProof [SOUND, 9 ms]
5.50/2.25	        (18) QDP
5.50/2.25	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
5.50/2.25	        (20) YES
5.50/2.25	
5.50/2.25	
5.50/2.25	----------------------------------------
5.50/2.25	
5.50/2.25	(0)
5.50/2.25	Obligation:
5.50/2.25	Clauses:
5.50/2.25	
5.50/2.25	search_tree(void).
5.50/2.25	search_tree(T) :- search_tree(T, X1, X2).
5.50/2.25	search_tree(tree(X, void, void), X, X).
5.50/2.25	search_tree(tree(X, void, Right), X, Max) :- ','(search_tree(Right, Min, Max), less(X, Min)).
5.50/2.25	search_tree(tree(X, Left, void), Min, X) :- ','(search_tree(Left, Min, Max), less(Max, X)).
5.50/2.25	search_tree(tree(X, Left, Right), Min1, Max2) :- ','(search_tree(Left, Min1, Max1), ','(less(Max1, X), ','(search_tree(Right, Min2, Max2), less(X, Min2)))).
5.50/2.25	less(0, s(X3)).
5.50/2.25	less(s(X), s(Y)) :- less(X, Y).
5.50/2.25	
5.50/2.25	
5.50/2.25	Query: search_tree(g)
5.50/2.25	----------------------------------------
5.50/2.25	
5.50/2.25	(1) PrologToPiTRSProof (SOUND)
5.50/2.25	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
5.50/2.25	
5.50/2.25	search_tree_in_1: (b)
5.50/2.25	
5.50/2.25	search_tree_in_3: (b,f,f)
5.50/2.25	
5.50/2.25	less_in_2: (b,b)
5.50/2.25	
5.50/2.25	Transforming Prolog into the following Term Rewriting System:
5.50/2.25	
5.50/2.25	Pi-finite rewrite system:
5.50/2.25	The TRS R consists of the following rules:
5.50/2.25	
5.50/2.25	   search_tree_in_g(void) -> search_tree_out_g(void)
5.50/2.25	   search_tree_in_g(T) -> U1_g(T, search_tree_in_gaa(T, X1, X2))
5.50/2.25	   search_tree_in_gaa(tree(X, void, void), X, X) -> search_tree_out_gaa(tree(X, void, void), X, X)
5.50/2.25	   search_tree_in_gaa(tree(X, void, Right), X, Max) -> U2_gaa(X, Right, Max, search_tree_in_gaa(Right, Min, Max))
5.50/2.25	   search_tree_in_gaa(tree(X, Left, void), Min, X) -> U4_gaa(X, Left, Min, search_tree_in_gaa(Left, Min, Max))
5.50/2.25	   search_tree_in_gaa(tree(X, Left, Right), Min1, Max2) -> U6_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1))
5.50/2.25	   U6_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_gaa(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X))
5.50/2.25	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.50/2.25	   less_in_gg(s(X), s(Y)) -> U10_gg(X, Y, less_in_gg(X, Y))
5.50/2.25	   U10_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.50/2.25	   U7_gaa(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> U8_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Right, Min2, Max2))
5.50/2.25	   U8_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> U9_gaa(X, Left, Right, Min1, Max2, less_in_gg(X, Min2))
5.50/2.25	   U9_gaa(X, Left, Right, Min1, Max2, less_out_gg(X, Min2)) -> search_tree_out_gaa(tree(X, Left, Right), Min1, Max2)
5.50/2.25	   U4_gaa(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> U5_gaa(X, Left, Min, less_in_gg(Max, X))
5.50/2.25	   U5_gaa(X, Left, Min, less_out_gg(Max, X)) -> search_tree_out_gaa(tree(X, Left, void), Min, X)
5.50/2.25	   U2_gaa(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> U3_gaa(X, Right, Max, less_in_gg(X, Min))
5.50/2.25	   U3_gaa(X, Right, Max, less_out_gg(X, Min)) -> search_tree_out_gaa(tree(X, void, Right), X, Max)
5.50/2.25	   U1_g(T, search_tree_out_gaa(T, X1, X2)) -> search_tree_out_g(T)
5.50/2.25	
5.50/2.25	The argument filtering Pi contains the following mapping:
5.50/2.25	search_tree_in_g(x1)  =  search_tree_in_g(x1)
5.50/2.25	
5.50/2.25	void  =  void
5.50/2.25	
5.50/2.25	search_tree_out_g(x1)  =  search_tree_out_g
5.50/2.25	
5.50/2.25	U1_g(x1, x2)  =  U1_g(x2)
5.50/2.25	
5.50/2.25	search_tree_in_gaa(x1, x2, x3)  =  search_tree_in_gaa(x1)
5.50/2.25	
5.50/2.25	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.50/2.25	
5.50/2.25	search_tree_out_gaa(x1, x2, x3)  =  search_tree_out_gaa(x2, x3)
5.50/2.25	
5.50/2.25	U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x1, x4)
5.50/2.25	
5.50/2.25	U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x1, x4)
5.50/2.25	
5.50/2.25	U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x1, x3, x6)
5.50/2.25	
5.50/2.25	U7_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U7_gaa(x1, x3, x4, x7)
5.50/2.25	
5.50/2.25	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.50/2.25	
5.50/2.25	0  =  0
5.50/2.25	
5.50/2.25	s(x1)  =  s(x1)
5.50/2.25	
5.50/2.25	less_out_gg(x1, x2)  =  less_out_gg
5.50/2.25	
5.50/2.25	U10_gg(x1, x2, x3)  =  U10_gg(x3)
5.50/2.25	
5.50/2.25	U8_gaa(x1, x2, x3, x4, x5, x6)  =  U8_gaa(x1, x4, x6)
5.50/2.25	
5.50/2.25	U9_gaa(x1, x2, x3, x4, x5, x6)  =  U9_gaa(x4, x5, x6)
5.50/2.25	
5.50/2.25	U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x1, x3, x4)
5.50/2.25	
5.50/2.25	U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x1, x3, x4)
5.50/2.25	
5.50/2.25	
5.50/2.25	
5.50/2.25	
5.50/2.25	
5.50/2.25	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
5.50/2.25	
5.50/2.25	
5.50/2.25	
5.50/2.25	----------------------------------------
5.50/2.25	
5.50/2.25	(2)
5.50/2.25	Obligation:
5.50/2.25	Pi-finite rewrite system:
5.50/2.25	The TRS R consists of the following rules:
5.50/2.25	
5.50/2.25	   search_tree_in_g(void) -> search_tree_out_g(void)
5.50/2.25	   search_tree_in_g(T) -> U1_g(T, search_tree_in_gaa(T, X1, X2))
5.50/2.25	   search_tree_in_gaa(tree(X, void, void), X, X) -> search_tree_out_gaa(tree(X, void, void), X, X)
5.50/2.25	   search_tree_in_gaa(tree(X, void, Right), X, Max) -> U2_gaa(X, Right, Max, search_tree_in_gaa(Right, Min, Max))
5.50/2.25	   search_tree_in_gaa(tree(X, Left, void), Min, X) -> U4_gaa(X, Left, Min, search_tree_in_gaa(Left, Min, Max))
5.50/2.25	   search_tree_in_gaa(tree(X, Left, Right), Min1, Max2) -> U6_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1))
5.50/2.25	   U6_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_gaa(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X))
5.50/2.25	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.50/2.25	   less_in_gg(s(X), s(Y)) -> U10_gg(X, Y, less_in_gg(X, Y))
5.50/2.25	   U10_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.50/2.25	   U7_gaa(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> U8_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Right, Min2, Max2))
5.50/2.25	   U8_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> U9_gaa(X, Left, Right, Min1, Max2, less_in_gg(X, Min2))
5.50/2.25	   U9_gaa(X, Left, Right, Min1, Max2, less_out_gg(X, Min2)) -> search_tree_out_gaa(tree(X, Left, Right), Min1, Max2)
5.50/2.25	   U4_gaa(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> U5_gaa(X, Left, Min, less_in_gg(Max, X))
5.50/2.25	   U5_gaa(X, Left, Min, less_out_gg(Max, X)) -> search_tree_out_gaa(tree(X, Left, void), Min, X)
5.50/2.25	   U2_gaa(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> U3_gaa(X, Right, Max, less_in_gg(X, Min))
5.50/2.25	   U3_gaa(X, Right, Max, less_out_gg(X, Min)) -> search_tree_out_gaa(tree(X, void, Right), X, Max)
5.50/2.25	   U1_g(T, search_tree_out_gaa(T, X1, X2)) -> search_tree_out_g(T)
5.50/2.25	
5.50/2.25	The argument filtering Pi contains the following mapping:
5.50/2.25	search_tree_in_g(x1)  =  search_tree_in_g(x1)
5.50/2.25	
5.50/2.25	void  =  void
5.50/2.25	
5.50/2.25	search_tree_out_g(x1)  =  search_tree_out_g
5.50/2.25	
5.50/2.25	U1_g(x1, x2)  =  U1_g(x2)
5.50/2.25	
5.50/2.25	search_tree_in_gaa(x1, x2, x3)  =  search_tree_in_gaa(x1)
5.50/2.25	
5.50/2.25	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.50/2.25	
5.50/2.25	search_tree_out_gaa(x1, x2, x3)  =  search_tree_out_gaa(x2, x3)
5.50/2.25	
5.50/2.25	U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x1, x4)
5.50/2.25	
5.50/2.25	U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x1, x4)
5.50/2.25	
5.50/2.25	U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x1, x3, x6)
5.50/2.25	
5.50/2.25	U7_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U7_gaa(x1, x3, x4, x7)
5.50/2.25	
5.50/2.25	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.50/2.25	
5.50/2.25	0  =  0
5.50/2.25	
5.50/2.25	s(x1)  =  s(x1)
5.50/2.25	
5.50/2.25	less_out_gg(x1, x2)  =  less_out_gg
5.50/2.25	
5.50/2.25	U10_gg(x1, x2, x3)  =  U10_gg(x3)
5.50/2.25	
5.50/2.25	U8_gaa(x1, x2, x3, x4, x5, x6)  =  U8_gaa(x1, x4, x6)
5.50/2.25	
5.50/2.25	U9_gaa(x1, x2, x3, x4, x5, x6)  =  U9_gaa(x4, x5, x6)
5.50/2.25	
5.50/2.25	U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x1, x3, x4)
5.50/2.25	
5.50/2.25	U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x1, x3, x4)
5.50/2.25	
5.50/2.25	
5.50/2.25	
5.50/2.25	----------------------------------------
5.50/2.25	
5.50/2.25	(3) DependencyPairsProof (EQUIVALENT)
5.50/2.25	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
5.50/2.25	Pi DP problem:
5.50/2.25	The TRS P consists of the following rules:
5.50/2.25	
5.50/2.25	   SEARCH_TREE_IN_G(T) -> U1_G(T, search_tree_in_gaa(T, X1, X2))
5.50/2.25	   SEARCH_TREE_IN_G(T) -> SEARCH_TREE_IN_GAA(T, X1, X2)
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, void, Right), X, Max) -> U2_GAA(X, Right, Max, search_tree_in_gaa(Right, Min, Max))
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, void, Right), X, Max) -> SEARCH_TREE_IN_GAA(Right, Min, Max)
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, Left, void), Min, X) -> U4_GAA(X, Left, Min, search_tree_in_gaa(Left, Min, Max))
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, Left, void), Min, X) -> SEARCH_TREE_IN_GAA(Left, Min, Max)
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, Left, Right), Min1, Max2) -> U6_GAA(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1))
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, Left, Right), Min1, Max2) -> SEARCH_TREE_IN_GAA(Left, Min1, Max1)
5.50/2.25	   U6_GAA(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_GAA(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X))
5.50/2.25	   U6_GAA(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> LESS_IN_GG(Max1, X)
5.50/2.25	   LESS_IN_GG(s(X), s(Y)) -> U10_GG(X, Y, less_in_gg(X, Y))
5.50/2.25	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
5.50/2.25	   U7_GAA(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> U8_GAA(X, Left, Right, Min1, Max2, search_tree_in_gaa(Right, Min2, Max2))
5.50/2.25	   U7_GAA(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> SEARCH_TREE_IN_GAA(Right, Min2, Max2)
5.50/2.25	   U8_GAA(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> U9_GAA(X, Left, Right, Min1, Max2, less_in_gg(X, Min2))
5.50/2.25	   U8_GAA(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> LESS_IN_GG(X, Min2)
5.50/2.25	   U4_GAA(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> U5_GAA(X, Left, Min, less_in_gg(Max, X))
5.50/2.25	   U4_GAA(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> LESS_IN_GG(Max, X)
5.50/2.25	   U2_GAA(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> U3_GAA(X, Right, Max, less_in_gg(X, Min))
5.50/2.25	   U2_GAA(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> LESS_IN_GG(X, Min)
5.50/2.25	
5.50/2.25	The TRS R consists of the following rules:
5.50/2.25	
5.50/2.25	   search_tree_in_g(void) -> search_tree_out_g(void)
5.50/2.25	   search_tree_in_g(T) -> U1_g(T, search_tree_in_gaa(T, X1, X2))
5.50/2.25	   search_tree_in_gaa(tree(X, void, void), X, X) -> search_tree_out_gaa(tree(X, void, void), X, X)
5.50/2.25	   search_tree_in_gaa(tree(X, void, Right), X, Max) -> U2_gaa(X, Right, Max, search_tree_in_gaa(Right, Min, Max))
5.50/2.25	   search_tree_in_gaa(tree(X, Left, void), Min, X) -> U4_gaa(X, Left, Min, search_tree_in_gaa(Left, Min, Max))
5.50/2.25	   search_tree_in_gaa(tree(X, Left, Right), Min1, Max2) -> U6_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1))
5.50/2.25	   U6_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_gaa(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X))
5.50/2.25	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.50/2.25	   less_in_gg(s(X), s(Y)) -> U10_gg(X, Y, less_in_gg(X, Y))
5.50/2.25	   U10_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.50/2.25	   U7_gaa(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> U8_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Right, Min2, Max2))
5.50/2.25	   U8_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> U9_gaa(X, Left, Right, Min1, Max2, less_in_gg(X, Min2))
5.50/2.25	   U9_gaa(X, Left, Right, Min1, Max2, less_out_gg(X, Min2)) -> search_tree_out_gaa(tree(X, Left, Right), Min1, Max2)
5.50/2.25	   U4_gaa(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> U5_gaa(X, Left, Min, less_in_gg(Max, X))
5.50/2.25	   U5_gaa(X, Left, Min, less_out_gg(Max, X)) -> search_tree_out_gaa(tree(X, Left, void), Min, X)
5.50/2.25	   U2_gaa(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> U3_gaa(X, Right, Max, less_in_gg(X, Min))
5.50/2.25	   U3_gaa(X, Right, Max, less_out_gg(X, Min)) -> search_tree_out_gaa(tree(X, void, Right), X, Max)
5.50/2.25	   U1_g(T, search_tree_out_gaa(T, X1, X2)) -> search_tree_out_g(T)
5.50/2.25	
5.50/2.25	The argument filtering Pi contains the following mapping:
5.50/2.25	search_tree_in_g(x1)  =  search_tree_in_g(x1)
5.50/2.25	
5.50/2.25	void  =  void
5.50/2.25	
5.50/2.25	search_tree_out_g(x1)  =  search_tree_out_g
5.50/2.25	
5.50/2.25	U1_g(x1, x2)  =  U1_g(x2)
5.50/2.25	
5.50/2.25	search_tree_in_gaa(x1, x2, x3)  =  search_tree_in_gaa(x1)
5.50/2.25	
5.50/2.25	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.50/2.25	
5.50/2.25	search_tree_out_gaa(x1, x2, x3)  =  search_tree_out_gaa(x2, x3)
5.50/2.25	
5.50/2.25	U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x1, x4)
5.50/2.25	
5.50/2.25	U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x1, x4)
5.50/2.25	
5.50/2.25	U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x1, x3, x6)
5.50/2.25	
5.50/2.25	U7_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U7_gaa(x1, x3, x4, x7)
5.50/2.25	
5.50/2.25	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.50/2.25	
5.50/2.25	0  =  0
5.50/2.25	
5.50/2.25	s(x1)  =  s(x1)
5.50/2.25	
5.50/2.25	less_out_gg(x1, x2)  =  less_out_gg
5.50/2.25	
5.50/2.25	U10_gg(x1, x2, x3)  =  U10_gg(x3)
5.50/2.25	
5.50/2.25	U8_gaa(x1, x2, x3, x4, x5, x6)  =  U8_gaa(x1, x4, x6)
5.50/2.25	
5.50/2.25	U9_gaa(x1, x2, x3, x4, x5, x6)  =  U9_gaa(x4, x5, x6)
5.50/2.25	
5.50/2.25	U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x1, x3, x4)
5.50/2.25	
5.50/2.25	U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x1, x3, x4)
5.50/2.25	
5.50/2.25	SEARCH_TREE_IN_G(x1)  =  SEARCH_TREE_IN_G(x1)
5.50/2.25	
5.50/2.25	U1_G(x1, x2)  =  U1_G(x2)
5.50/2.25	
5.50/2.25	SEARCH_TREE_IN_GAA(x1, x2, x3)  =  SEARCH_TREE_IN_GAA(x1)
5.50/2.25	
5.50/2.25	U2_GAA(x1, x2, x3, x4)  =  U2_GAA(x1, x4)
5.50/2.25	
5.50/2.25	U4_GAA(x1, x2, x3, x4)  =  U4_GAA(x1, x4)
5.50/2.25	
5.50/2.25	U6_GAA(x1, x2, x3, x4, x5, x6)  =  U6_GAA(x1, x3, x6)
5.50/2.25	
5.50/2.25	U7_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U7_GAA(x1, x3, x4, x7)
5.50/2.25	
5.50/2.25	LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
5.50/2.25	
5.50/2.25	U10_GG(x1, x2, x3)  =  U10_GG(x3)
5.50/2.25	
5.50/2.25	U8_GAA(x1, x2, x3, x4, x5, x6)  =  U8_GAA(x1, x4, x6)
5.50/2.25	
5.50/2.25	U9_GAA(x1, x2, x3, x4, x5, x6)  =  U9_GAA(x4, x5, x6)
5.50/2.25	
5.50/2.25	U5_GAA(x1, x2, x3, x4)  =  U5_GAA(x1, x3, x4)
5.50/2.25	
5.50/2.25	U3_GAA(x1, x2, x3, x4)  =  U3_GAA(x1, x3, x4)
5.50/2.25	
5.50/2.25	
5.50/2.25	We have to consider all (P,R,Pi)-chains
5.50/2.25	----------------------------------------
5.50/2.25	
5.50/2.25	(4)
5.50/2.25	Obligation:
5.50/2.25	Pi DP problem:
5.50/2.25	The TRS P consists of the following rules:
5.50/2.25	
5.50/2.25	   SEARCH_TREE_IN_G(T) -> U1_G(T, search_tree_in_gaa(T, X1, X2))
5.50/2.25	   SEARCH_TREE_IN_G(T) -> SEARCH_TREE_IN_GAA(T, X1, X2)
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, void, Right), X, Max) -> U2_GAA(X, Right, Max, search_tree_in_gaa(Right, Min, Max))
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, void, Right), X, Max) -> SEARCH_TREE_IN_GAA(Right, Min, Max)
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, Left, void), Min, X) -> U4_GAA(X, Left, Min, search_tree_in_gaa(Left, Min, Max))
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, Left, void), Min, X) -> SEARCH_TREE_IN_GAA(Left, Min, Max)
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, Left, Right), Min1, Max2) -> U6_GAA(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1))
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, Left, Right), Min1, Max2) -> SEARCH_TREE_IN_GAA(Left, Min1, Max1)
5.50/2.25	   U6_GAA(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_GAA(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X))
5.50/2.25	   U6_GAA(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> LESS_IN_GG(Max1, X)
5.50/2.25	   LESS_IN_GG(s(X), s(Y)) -> U10_GG(X, Y, less_in_gg(X, Y))
5.50/2.25	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
5.50/2.25	   U7_GAA(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> U8_GAA(X, Left, Right, Min1, Max2, search_tree_in_gaa(Right, Min2, Max2))
5.50/2.25	   U7_GAA(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> SEARCH_TREE_IN_GAA(Right, Min2, Max2)
5.50/2.25	   U8_GAA(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> U9_GAA(X, Left, Right, Min1, Max2, less_in_gg(X, Min2))
5.50/2.25	   U8_GAA(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> LESS_IN_GG(X, Min2)
5.50/2.25	   U4_GAA(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> U5_GAA(X, Left, Min, less_in_gg(Max, X))
5.50/2.25	   U4_GAA(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> LESS_IN_GG(Max, X)
5.50/2.25	   U2_GAA(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> U3_GAA(X, Right, Max, less_in_gg(X, Min))
5.50/2.25	   U2_GAA(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> LESS_IN_GG(X, Min)
5.50/2.25	
5.50/2.25	The TRS R consists of the following rules:
5.50/2.25	
5.50/2.25	   search_tree_in_g(void) -> search_tree_out_g(void)
5.50/2.25	   search_tree_in_g(T) -> U1_g(T, search_tree_in_gaa(T, X1, X2))
5.50/2.25	   search_tree_in_gaa(tree(X, void, void), X, X) -> search_tree_out_gaa(tree(X, void, void), X, X)
5.50/2.25	   search_tree_in_gaa(tree(X, void, Right), X, Max) -> U2_gaa(X, Right, Max, search_tree_in_gaa(Right, Min, Max))
5.50/2.25	   search_tree_in_gaa(tree(X, Left, void), Min, X) -> U4_gaa(X, Left, Min, search_tree_in_gaa(Left, Min, Max))
5.50/2.25	   search_tree_in_gaa(tree(X, Left, Right), Min1, Max2) -> U6_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1))
5.50/2.25	   U6_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_gaa(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X))
5.50/2.25	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.50/2.25	   less_in_gg(s(X), s(Y)) -> U10_gg(X, Y, less_in_gg(X, Y))
5.50/2.25	   U10_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.50/2.25	   U7_gaa(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> U8_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Right, Min2, Max2))
5.50/2.25	   U8_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> U9_gaa(X, Left, Right, Min1, Max2, less_in_gg(X, Min2))
5.50/2.25	   U9_gaa(X, Left, Right, Min1, Max2, less_out_gg(X, Min2)) -> search_tree_out_gaa(tree(X, Left, Right), Min1, Max2)
5.50/2.25	   U4_gaa(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> U5_gaa(X, Left, Min, less_in_gg(Max, X))
5.50/2.25	   U5_gaa(X, Left, Min, less_out_gg(Max, X)) -> search_tree_out_gaa(tree(X, Left, void), Min, X)
5.50/2.25	   U2_gaa(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> U3_gaa(X, Right, Max, less_in_gg(X, Min))
5.50/2.25	   U3_gaa(X, Right, Max, less_out_gg(X, Min)) -> search_tree_out_gaa(tree(X, void, Right), X, Max)
5.50/2.25	   U1_g(T, search_tree_out_gaa(T, X1, X2)) -> search_tree_out_g(T)
5.50/2.25	
5.50/2.25	The argument filtering Pi contains the following mapping:
5.50/2.25	search_tree_in_g(x1)  =  search_tree_in_g(x1)
5.50/2.25	
5.50/2.25	void  =  void
5.50/2.25	
5.50/2.25	search_tree_out_g(x1)  =  search_tree_out_g
5.50/2.25	
5.50/2.25	U1_g(x1, x2)  =  U1_g(x2)
5.50/2.25	
5.50/2.25	search_tree_in_gaa(x1, x2, x3)  =  search_tree_in_gaa(x1)
5.50/2.25	
5.50/2.25	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.50/2.25	
5.50/2.25	search_tree_out_gaa(x1, x2, x3)  =  search_tree_out_gaa(x2, x3)
5.50/2.25	
5.50/2.25	U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x1, x4)
5.50/2.25	
5.50/2.25	U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x1, x4)
5.50/2.25	
5.50/2.25	U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x1, x3, x6)
5.50/2.25	
5.50/2.25	U7_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U7_gaa(x1, x3, x4, x7)
5.50/2.25	
5.50/2.25	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.50/2.25	
5.50/2.25	0  =  0
5.50/2.25	
5.50/2.25	s(x1)  =  s(x1)
5.50/2.25	
5.50/2.25	less_out_gg(x1, x2)  =  less_out_gg
5.50/2.25	
5.50/2.25	U10_gg(x1, x2, x3)  =  U10_gg(x3)
5.50/2.25	
5.50/2.25	U8_gaa(x1, x2, x3, x4, x5, x6)  =  U8_gaa(x1, x4, x6)
5.50/2.25	
5.50/2.25	U9_gaa(x1, x2, x3, x4, x5, x6)  =  U9_gaa(x4, x5, x6)
5.50/2.25	
5.50/2.25	U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x1, x3, x4)
5.50/2.25	
5.50/2.25	U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x1, x3, x4)
5.50/2.25	
5.50/2.25	SEARCH_TREE_IN_G(x1)  =  SEARCH_TREE_IN_G(x1)
5.50/2.25	
5.50/2.25	U1_G(x1, x2)  =  U1_G(x2)
5.50/2.25	
5.50/2.25	SEARCH_TREE_IN_GAA(x1, x2, x3)  =  SEARCH_TREE_IN_GAA(x1)
5.50/2.25	
5.50/2.25	U2_GAA(x1, x2, x3, x4)  =  U2_GAA(x1, x4)
5.50/2.25	
5.50/2.25	U4_GAA(x1, x2, x3, x4)  =  U4_GAA(x1, x4)
5.50/2.25	
5.50/2.25	U6_GAA(x1, x2, x3, x4, x5, x6)  =  U6_GAA(x1, x3, x6)
5.50/2.25	
5.50/2.25	U7_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U7_GAA(x1, x3, x4, x7)
5.50/2.25	
5.50/2.25	LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
5.50/2.25	
5.50/2.25	U10_GG(x1, x2, x3)  =  U10_GG(x3)
5.50/2.25	
5.50/2.25	U8_GAA(x1, x2, x3, x4, x5, x6)  =  U8_GAA(x1, x4, x6)
5.50/2.25	
5.50/2.25	U9_GAA(x1, x2, x3, x4, x5, x6)  =  U9_GAA(x4, x5, x6)
5.50/2.25	
5.50/2.25	U5_GAA(x1, x2, x3, x4)  =  U5_GAA(x1, x3, x4)
5.50/2.25	
5.50/2.25	U3_GAA(x1, x2, x3, x4)  =  U3_GAA(x1, x3, x4)
5.50/2.25	
5.50/2.25	
5.50/2.25	We have to consider all (P,R,Pi)-chains
5.50/2.25	----------------------------------------
5.50/2.25	
5.50/2.25	(5) DependencyGraphProof (EQUIVALENT)
5.50/2.25	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 13 less nodes.
5.50/2.25	----------------------------------------
5.50/2.25	
5.50/2.25	(6)
5.50/2.25	Complex Obligation (AND)
5.50/2.25	
5.50/2.25	----------------------------------------
5.50/2.25	
5.50/2.25	(7)
5.50/2.25	Obligation:
5.50/2.25	Pi DP problem:
5.50/2.25	The TRS P consists of the following rules:
5.50/2.25	
5.50/2.25	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
5.50/2.25	
5.50/2.25	The TRS R consists of the following rules:
5.50/2.25	
5.50/2.25	   search_tree_in_g(void) -> search_tree_out_g(void)
5.50/2.25	   search_tree_in_g(T) -> U1_g(T, search_tree_in_gaa(T, X1, X2))
5.50/2.25	   search_tree_in_gaa(tree(X, void, void), X, X) -> search_tree_out_gaa(tree(X, void, void), X, X)
5.50/2.25	   search_tree_in_gaa(tree(X, void, Right), X, Max) -> U2_gaa(X, Right, Max, search_tree_in_gaa(Right, Min, Max))
5.50/2.25	   search_tree_in_gaa(tree(X, Left, void), Min, X) -> U4_gaa(X, Left, Min, search_tree_in_gaa(Left, Min, Max))
5.50/2.25	   search_tree_in_gaa(tree(X, Left, Right), Min1, Max2) -> U6_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1))
5.50/2.25	   U6_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_gaa(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X))
5.50/2.25	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.50/2.25	   less_in_gg(s(X), s(Y)) -> U10_gg(X, Y, less_in_gg(X, Y))
5.50/2.25	   U10_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.50/2.25	   U7_gaa(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> U8_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Right, Min2, Max2))
5.50/2.25	   U8_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> U9_gaa(X, Left, Right, Min1, Max2, less_in_gg(X, Min2))
5.50/2.25	   U9_gaa(X, Left, Right, Min1, Max2, less_out_gg(X, Min2)) -> search_tree_out_gaa(tree(X, Left, Right), Min1, Max2)
5.50/2.25	   U4_gaa(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> U5_gaa(X, Left, Min, less_in_gg(Max, X))
5.50/2.25	   U5_gaa(X, Left, Min, less_out_gg(Max, X)) -> search_tree_out_gaa(tree(X, Left, void), Min, X)
5.50/2.25	   U2_gaa(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> U3_gaa(X, Right, Max, less_in_gg(X, Min))
5.50/2.25	   U3_gaa(X, Right, Max, less_out_gg(X, Min)) -> search_tree_out_gaa(tree(X, void, Right), X, Max)
5.50/2.25	   U1_g(T, search_tree_out_gaa(T, X1, X2)) -> search_tree_out_g(T)
5.50/2.25	
5.50/2.25	The argument filtering Pi contains the following mapping:
5.50/2.25	search_tree_in_g(x1)  =  search_tree_in_g(x1)
5.50/2.25	
5.50/2.25	void  =  void
5.50/2.25	
5.50/2.25	search_tree_out_g(x1)  =  search_tree_out_g
5.50/2.25	
5.50/2.25	U1_g(x1, x2)  =  U1_g(x2)
5.50/2.25	
5.50/2.25	search_tree_in_gaa(x1, x2, x3)  =  search_tree_in_gaa(x1)
5.50/2.25	
5.50/2.25	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.50/2.25	
5.50/2.25	search_tree_out_gaa(x1, x2, x3)  =  search_tree_out_gaa(x2, x3)
5.50/2.25	
5.50/2.25	U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x1, x4)
5.50/2.25	
5.50/2.25	U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x1, x4)
5.50/2.25	
5.50/2.25	U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x1, x3, x6)
5.50/2.25	
5.50/2.25	U7_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U7_gaa(x1, x3, x4, x7)
5.50/2.25	
5.50/2.25	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.50/2.25	
5.50/2.25	0  =  0
5.50/2.25	
5.50/2.25	s(x1)  =  s(x1)
5.50/2.25	
5.50/2.25	less_out_gg(x1, x2)  =  less_out_gg
5.50/2.25	
5.50/2.25	U10_gg(x1, x2, x3)  =  U10_gg(x3)
5.50/2.25	
5.50/2.25	U8_gaa(x1, x2, x3, x4, x5, x6)  =  U8_gaa(x1, x4, x6)
5.50/2.25	
5.50/2.25	U9_gaa(x1, x2, x3, x4, x5, x6)  =  U9_gaa(x4, x5, x6)
5.50/2.25	
5.50/2.25	U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x1, x3, x4)
5.50/2.25	
5.50/2.25	U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x1, x3, x4)
5.50/2.25	
5.50/2.25	LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
5.50/2.25	
5.50/2.25	
5.50/2.25	We have to consider all (P,R,Pi)-chains
5.50/2.25	----------------------------------------
5.50/2.25	
5.50/2.25	(8) UsableRulesProof (EQUIVALENT)
5.50/2.25	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.50/2.25	----------------------------------------
5.50/2.25	
5.50/2.25	(9)
5.50/2.25	Obligation:
5.50/2.25	Pi DP problem:
5.50/2.25	The TRS P consists of the following rules:
5.50/2.25	
5.50/2.25	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
5.50/2.25	
5.50/2.25	R is empty.
5.50/2.25	Pi is empty.
5.50/2.25	We have to consider all (P,R,Pi)-chains
5.50/2.25	----------------------------------------
5.50/2.25	
5.50/2.25	(10) PiDPToQDPProof (EQUIVALENT)
5.50/2.25	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.50/2.25	----------------------------------------
5.50/2.25	
5.50/2.25	(11)
5.50/2.25	Obligation:
5.50/2.25	Q DP problem:
5.50/2.25	The TRS P consists of the following rules:
5.50/2.25	
5.50/2.25	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
5.50/2.25	
5.50/2.25	R is empty.
5.50/2.25	Q is empty.
5.50/2.25	We have to consider all (P,Q,R)-chains.
5.50/2.25	----------------------------------------
5.50/2.25	
5.50/2.25	(12) QDPSizeChangeProof (EQUIVALENT)
5.50/2.25	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. 
5.50/2.25	
5.50/2.25	From the DPs we obtained the following set of size-change graphs:
5.50/2.25	*LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
5.50/2.25	The graph contains the following edges 1 > 1, 2 > 2
5.50/2.25	
5.50/2.25	
5.50/2.25	----------------------------------------
5.50/2.25	
5.50/2.25	(13)
5.50/2.25	YES
5.50/2.25	
5.50/2.25	----------------------------------------
5.50/2.25	
5.50/2.25	(14)
5.50/2.25	Obligation:
5.50/2.25	Pi DP problem:
5.50/2.25	The TRS P consists of the following rules:
5.50/2.25	
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, Left, void), Min, X) -> SEARCH_TREE_IN_GAA(Left, Min, Max)
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, void, Right), X, Max) -> SEARCH_TREE_IN_GAA(Right, Min, Max)
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, Left, Right), Min1, Max2) -> U6_GAA(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1))
5.50/2.25	   U6_GAA(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_GAA(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X))
5.50/2.25	   U7_GAA(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> SEARCH_TREE_IN_GAA(Right, Min2, Max2)
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, Left, Right), Min1, Max2) -> SEARCH_TREE_IN_GAA(Left, Min1, Max1)
5.50/2.25	
5.50/2.25	The TRS R consists of the following rules:
5.50/2.25	
5.50/2.25	   search_tree_in_g(void) -> search_tree_out_g(void)
5.50/2.25	   search_tree_in_g(T) -> U1_g(T, search_tree_in_gaa(T, X1, X2))
5.50/2.25	   search_tree_in_gaa(tree(X, void, void), X, X) -> search_tree_out_gaa(tree(X, void, void), X, X)
5.50/2.25	   search_tree_in_gaa(tree(X, void, Right), X, Max) -> U2_gaa(X, Right, Max, search_tree_in_gaa(Right, Min, Max))
5.50/2.25	   search_tree_in_gaa(tree(X, Left, void), Min, X) -> U4_gaa(X, Left, Min, search_tree_in_gaa(Left, Min, Max))
5.50/2.25	   search_tree_in_gaa(tree(X, Left, Right), Min1, Max2) -> U6_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1))
5.50/2.25	   U6_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_gaa(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X))
5.50/2.25	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.50/2.25	   less_in_gg(s(X), s(Y)) -> U10_gg(X, Y, less_in_gg(X, Y))
5.50/2.25	   U10_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.50/2.25	   U7_gaa(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> U8_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Right, Min2, Max2))
5.50/2.25	   U8_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> U9_gaa(X, Left, Right, Min1, Max2, less_in_gg(X, Min2))
5.50/2.25	   U9_gaa(X, Left, Right, Min1, Max2, less_out_gg(X, Min2)) -> search_tree_out_gaa(tree(X, Left, Right), Min1, Max2)
5.50/2.25	   U4_gaa(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> U5_gaa(X, Left, Min, less_in_gg(Max, X))
5.50/2.25	   U5_gaa(X, Left, Min, less_out_gg(Max, X)) -> search_tree_out_gaa(tree(X, Left, void), Min, X)
5.50/2.25	   U2_gaa(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> U3_gaa(X, Right, Max, less_in_gg(X, Min))
5.50/2.25	   U3_gaa(X, Right, Max, less_out_gg(X, Min)) -> search_tree_out_gaa(tree(X, void, Right), X, Max)
5.50/2.25	   U1_g(T, search_tree_out_gaa(T, X1, X2)) -> search_tree_out_g(T)
5.50/2.25	
5.50/2.25	The argument filtering Pi contains the following mapping:
5.50/2.25	search_tree_in_g(x1)  =  search_tree_in_g(x1)
5.50/2.25	
5.50/2.25	void  =  void
5.50/2.25	
5.50/2.25	search_tree_out_g(x1)  =  search_tree_out_g
5.50/2.25	
5.50/2.25	U1_g(x1, x2)  =  U1_g(x2)
5.50/2.25	
5.50/2.25	search_tree_in_gaa(x1, x2, x3)  =  search_tree_in_gaa(x1)
5.50/2.25	
5.50/2.25	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.50/2.25	
5.50/2.25	search_tree_out_gaa(x1, x2, x3)  =  search_tree_out_gaa(x2, x3)
5.50/2.25	
5.50/2.25	U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x1, x4)
5.50/2.25	
5.50/2.25	U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x1, x4)
5.50/2.25	
5.50/2.25	U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x1, x3, x6)
5.50/2.25	
5.50/2.25	U7_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U7_gaa(x1, x3, x4, x7)
5.50/2.25	
5.50/2.25	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.50/2.25	
5.50/2.25	0  =  0
5.50/2.25	
5.50/2.25	s(x1)  =  s(x1)
5.50/2.25	
5.50/2.25	less_out_gg(x1, x2)  =  less_out_gg
5.50/2.25	
5.50/2.25	U10_gg(x1, x2, x3)  =  U10_gg(x3)
5.50/2.25	
5.50/2.25	U8_gaa(x1, x2, x3, x4, x5, x6)  =  U8_gaa(x1, x4, x6)
5.50/2.25	
5.50/2.25	U9_gaa(x1, x2, x3, x4, x5, x6)  =  U9_gaa(x4, x5, x6)
5.50/2.25	
5.50/2.25	U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x1, x3, x4)
5.50/2.25	
5.50/2.25	U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x1, x3, x4)
5.50/2.25	
5.50/2.25	SEARCH_TREE_IN_GAA(x1, x2, x3)  =  SEARCH_TREE_IN_GAA(x1)
5.50/2.25	
5.50/2.25	U6_GAA(x1, x2, x3, x4, x5, x6)  =  U6_GAA(x1, x3, x6)
5.50/2.25	
5.50/2.25	U7_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U7_GAA(x1, x3, x4, x7)
5.50/2.25	
5.50/2.25	
5.50/2.25	We have to consider all (P,R,Pi)-chains
5.50/2.25	----------------------------------------
5.50/2.25	
5.50/2.25	(15) UsableRulesProof (EQUIVALENT)
5.50/2.25	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.50/2.25	----------------------------------------
5.50/2.25	
5.50/2.25	(16)
5.50/2.25	Obligation:
5.50/2.25	Pi DP problem:
5.50/2.25	The TRS P consists of the following rules:
5.50/2.25	
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, Left, void), Min, X) -> SEARCH_TREE_IN_GAA(Left, Min, Max)
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, void, Right), X, Max) -> SEARCH_TREE_IN_GAA(Right, Min, Max)
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, Left, Right), Min1, Max2) -> U6_GAA(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1))
5.50/2.25	   U6_GAA(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_GAA(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X))
5.50/2.25	   U7_GAA(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> SEARCH_TREE_IN_GAA(Right, Min2, Max2)
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, Left, Right), Min1, Max2) -> SEARCH_TREE_IN_GAA(Left, Min1, Max1)
5.50/2.25	
5.50/2.25	The TRS R consists of the following rules:
5.50/2.25	
5.50/2.25	   search_tree_in_gaa(tree(X, void, void), X, X) -> search_tree_out_gaa(tree(X, void, void), X, X)
5.50/2.25	   search_tree_in_gaa(tree(X, void, Right), X, Max) -> U2_gaa(X, Right, Max, search_tree_in_gaa(Right, Min, Max))
5.50/2.25	   search_tree_in_gaa(tree(X, Left, void), Min, X) -> U4_gaa(X, Left, Min, search_tree_in_gaa(Left, Min, Max))
5.50/2.25	   search_tree_in_gaa(tree(X, Left, Right), Min1, Max2) -> U6_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Left, Min1, Max1))
5.50/2.25	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.50/2.25	   less_in_gg(s(X), s(Y)) -> U10_gg(X, Y, less_in_gg(X, Y))
5.50/2.25	   U2_gaa(X, Right, Max, search_tree_out_gaa(Right, Min, Max)) -> U3_gaa(X, Right, Max, less_in_gg(X, Min))
5.50/2.25	   U4_gaa(X, Left, Min, search_tree_out_gaa(Left, Min, Max)) -> U5_gaa(X, Left, Min, less_in_gg(Max, X))
5.50/2.25	   U6_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Left, Min1, Max1)) -> U7_gaa(X, Left, Right, Min1, Max2, Max1, less_in_gg(Max1, X))
5.50/2.25	   U10_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.50/2.25	   U3_gaa(X, Right, Max, less_out_gg(X, Min)) -> search_tree_out_gaa(tree(X, void, Right), X, Max)
5.50/2.25	   U5_gaa(X, Left, Min, less_out_gg(Max, X)) -> search_tree_out_gaa(tree(X, Left, void), Min, X)
5.50/2.25	   U7_gaa(X, Left, Right, Min1, Max2, Max1, less_out_gg(Max1, X)) -> U8_gaa(X, Left, Right, Min1, Max2, search_tree_in_gaa(Right, Min2, Max2))
5.50/2.25	   U8_gaa(X, Left, Right, Min1, Max2, search_tree_out_gaa(Right, Min2, Max2)) -> U9_gaa(X, Left, Right, Min1, Max2, less_in_gg(X, Min2))
5.50/2.25	   U9_gaa(X, Left, Right, Min1, Max2, less_out_gg(X, Min2)) -> search_tree_out_gaa(tree(X, Left, Right), Min1, Max2)
5.50/2.25	
5.50/2.25	The argument filtering Pi contains the following mapping:
5.50/2.25	void  =  void
5.50/2.25	
5.50/2.25	search_tree_in_gaa(x1, x2, x3)  =  search_tree_in_gaa(x1)
5.50/2.25	
5.50/2.25	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.50/2.25	
5.50/2.25	search_tree_out_gaa(x1, x2, x3)  =  search_tree_out_gaa(x2, x3)
5.50/2.25	
5.50/2.25	U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x1, x4)
5.50/2.25	
5.50/2.25	U4_gaa(x1, x2, x3, x4)  =  U4_gaa(x1, x4)
5.50/2.25	
5.50/2.25	U6_gaa(x1, x2, x3, x4, x5, x6)  =  U6_gaa(x1, x3, x6)
5.50/2.25	
5.50/2.25	U7_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U7_gaa(x1, x3, x4, x7)
5.50/2.25	
5.50/2.25	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.50/2.25	
5.50/2.25	0  =  0
5.50/2.25	
5.50/2.25	s(x1)  =  s(x1)
5.50/2.25	
5.50/2.25	less_out_gg(x1, x2)  =  less_out_gg
5.50/2.25	
5.50/2.25	U10_gg(x1, x2, x3)  =  U10_gg(x3)
5.50/2.25	
5.50/2.25	U8_gaa(x1, x2, x3, x4, x5, x6)  =  U8_gaa(x1, x4, x6)
5.50/2.25	
5.50/2.25	U9_gaa(x1, x2, x3, x4, x5, x6)  =  U9_gaa(x4, x5, x6)
5.50/2.25	
5.50/2.25	U5_gaa(x1, x2, x3, x4)  =  U5_gaa(x1, x3, x4)
5.50/2.25	
5.50/2.25	U3_gaa(x1, x2, x3, x4)  =  U3_gaa(x1, x3, x4)
5.50/2.25	
5.50/2.25	SEARCH_TREE_IN_GAA(x1, x2, x3)  =  SEARCH_TREE_IN_GAA(x1)
5.50/2.25	
5.50/2.25	U6_GAA(x1, x2, x3, x4, x5, x6)  =  U6_GAA(x1, x3, x6)
5.50/2.25	
5.50/2.25	U7_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U7_GAA(x1, x3, x4, x7)
5.50/2.25	
5.50/2.25	
5.50/2.25	We have to consider all (P,R,Pi)-chains
5.50/2.25	----------------------------------------
5.50/2.25	
5.50/2.25	(17) PiDPToQDPProof (SOUND)
5.50/2.25	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.50/2.25	----------------------------------------
5.50/2.25	
5.50/2.25	(18)
5.50/2.25	Obligation:
5.50/2.25	Q DP problem:
5.50/2.25	The TRS P consists of the following rules:
5.50/2.25	
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, Left, void)) -> SEARCH_TREE_IN_GAA(Left)
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, void, Right)) -> SEARCH_TREE_IN_GAA(Right)
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, Left, Right)) -> U6_GAA(X, Right, search_tree_in_gaa(Left))
5.50/2.25	   U6_GAA(X, Right, search_tree_out_gaa(Min1, Max1)) -> U7_GAA(X, Right, Min1, less_in_gg(Max1, X))
5.50/2.25	   U7_GAA(X, Right, Min1, less_out_gg) -> SEARCH_TREE_IN_GAA(Right)
5.50/2.25	   SEARCH_TREE_IN_GAA(tree(X, Left, Right)) -> SEARCH_TREE_IN_GAA(Left)
5.50/2.25	
5.50/2.25	The TRS R consists of the following rules:
5.50/2.25	
5.50/2.25	   search_tree_in_gaa(tree(X, void, void)) -> search_tree_out_gaa(X, X)
5.50/2.25	   search_tree_in_gaa(tree(X, void, Right)) -> U2_gaa(X, search_tree_in_gaa(Right))
5.50/2.25	   search_tree_in_gaa(tree(X, Left, void)) -> U4_gaa(X, search_tree_in_gaa(Left))
5.50/2.25	   search_tree_in_gaa(tree(X, Left, Right)) -> U6_gaa(X, Right, search_tree_in_gaa(Left))
5.50/2.25	   less_in_gg(0, s(X3)) -> less_out_gg
5.50/2.25	   less_in_gg(s(X), s(Y)) -> U10_gg(less_in_gg(X, Y))
5.50/2.25	   U2_gaa(X, search_tree_out_gaa(Min, Max)) -> U3_gaa(X, Max, less_in_gg(X, Min))
5.50/2.25	   U4_gaa(X, search_tree_out_gaa(Min, Max)) -> U5_gaa(X, Min, less_in_gg(Max, X))
5.50/2.25	   U6_gaa(X, Right, search_tree_out_gaa(Min1, Max1)) -> U7_gaa(X, Right, Min1, less_in_gg(Max1, X))
5.50/2.25	   U10_gg(less_out_gg) -> less_out_gg
5.50/2.25	   U3_gaa(X, Max, less_out_gg) -> search_tree_out_gaa(X, Max)
5.50/2.25	   U5_gaa(X, Min, less_out_gg) -> search_tree_out_gaa(Min, X)
5.50/2.25	   U7_gaa(X, Right, Min1, less_out_gg) -> U8_gaa(X, Min1, search_tree_in_gaa(Right))
5.50/2.25	   U8_gaa(X, Min1, search_tree_out_gaa(Min2, Max2)) -> U9_gaa(Min1, Max2, less_in_gg(X, Min2))
5.50/2.25	   U9_gaa(Min1, Max2, less_out_gg) -> search_tree_out_gaa(Min1, Max2)
5.50/2.25	
5.50/2.25	The set Q consists of the following terms:
5.50/2.25	
5.50/2.25	   search_tree_in_gaa(x0)
5.50/2.25	   less_in_gg(x0, x1)
5.50/2.25	   U2_gaa(x0, x1)
5.50/2.25	   U4_gaa(x0, x1)
5.50/2.25	   U6_gaa(x0, x1, x2)
5.50/2.25	   U10_gg(x0)
5.50/2.25	   U3_gaa(x0, x1, x2)
5.50/2.25	   U5_gaa(x0, x1, x2)
5.50/2.25	   U7_gaa(x0, x1, x2, x3)
5.50/2.25	   U8_gaa(x0, x1, x2)
5.50/2.25	   U9_gaa(x0, x1, x2)
5.50/2.25	
5.50/2.25	We have to consider all (P,Q,R)-chains.
5.50/2.25	----------------------------------------
5.50/2.25	
5.50/2.25	(19) QDPSizeChangeProof (EQUIVALENT)
5.50/2.25	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. 
5.50/2.25	
5.50/2.25	From the DPs we obtained the following set of size-change graphs:
5.50/2.25	*SEARCH_TREE_IN_GAA(tree(X, Left, Right)) -> U6_GAA(X, Right, search_tree_in_gaa(Left))
5.50/2.25	The graph contains the following edges 1 > 1, 1 > 2
5.50/2.25	
5.50/2.25	
5.50/2.25	*U7_GAA(X, Right, Min1, less_out_gg) -> SEARCH_TREE_IN_GAA(Right)
5.50/2.25	The graph contains the following edges 2 >= 1
5.50/2.25	
5.50/2.25	
5.50/2.25	*U6_GAA(X, Right, search_tree_out_gaa(Min1, Max1)) -> U7_GAA(X, Right, Min1, less_in_gg(Max1, X))
5.50/2.25	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
5.50/2.25	
5.50/2.25	
5.50/2.25	*SEARCH_TREE_IN_GAA(tree(X, Left, void)) -> SEARCH_TREE_IN_GAA(Left)
5.50/2.25	The graph contains the following edges 1 > 1
5.50/2.25	
5.50/2.25	
5.50/2.25	*SEARCH_TREE_IN_GAA(tree(X, void, Right)) -> SEARCH_TREE_IN_GAA(Right)
5.50/2.25	The graph contains the following edges 1 > 1
5.50/2.25	
5.50/2.25	
5.50/2.25	*SEARCH_TREE_IN_GAA(tree(X, Left, Right)) -> SEARCH_TREE_IN_GAA(Left)
5.50/2.25	The graph contains the following edges 1 > 1
5.50/2.25	
5.50/2.25	
5.50/2.25	----------------------------------------
5.50/2.25	
5.50/2.25	(20)
5.50/2.25	YES
5.50/2.28	EOF