6.62/2.54	YES
7.00/2.58	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
7.00/2.58	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
7.00/2.58	
7.00/2.58	
7.00/2.58	Left Termination of the query pattern
7.00/2.58	
7.00/2.58	delete(a,g,a)
7.00/2.58	
7.00/2.58	w.r.t. the given Prolog program could successfully be proven:
7.00/2.58	
7.00/2.58	(0) Prolog
7.00/2.58	(1) PrologToPiTRSProof [SOUND, 0 ms]
7.00/2.58	(2) PiTRS
7.00/2.58	(3) DependencyPairsProof [EQUIVALENT, 12 ms]
7.00/2.58	(4) PiDP
7.00/2.58	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
7.00/2.58	(6) AND
7.00/2.58	    (7) PiDP
7.00/2.58	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
7.00/2.58	        (9) PiDP
7.00/2.58	        (10) PiDPToQDPProof [SOUND, 9 ms]
7.00/2.58	        (11) QDP
7.00/2.58	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
7.00/2.58	        (13) YES
7.00/2.58	    (14) PiDP
7.00/2.58	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
7.00/2.58	        (16) PiDP
7.00/2.58	        (17) PiDPToQDPProof [EQUIVALENT, 0 ms]
7.00/2.58	        (18) QDP
7.00/2.58	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
7.00/2.58	        (20) YES
7.00/2.58	    (21) PiDP
7.00/2.58	        (22) UsableRulesProof [EQUIVALENT, 0 ms]
7.00/2.58	        (23) PiDP
7.00/2.58	        (24) PiDPToQDPProof [SOUND, 0 ms]
7.00/2.58	        (25) QDP
7.00/2.58	        (26) QDPSizeChangeProof [EQUIVALENT, 0 ms]
7.00/2.58	        (27) YES
7.00/2.58	    (28) PiDP
7.00/2.58	        (29) UsableRulesProof [EQUIVALENT, 0 ms]
7.00/2.58	        (30) PiDP
7.00/2.58	        (31) PiDPToQDPProof [SOUND, 0 ms]
7.00/2.58	        (32) QDP
7.00/2.58	        (33) QDPSizeChangeProof [EQUIVALENT, 0 ms]
7.00/2.58	        (34) YES
7.00/2.58	    (35) PiDP
7.00/2.58	        (36) UsableRulesProof [EQUIVALENT, 0 ms]
7.00/2.58	        (37) PiDP
7.00/2.58	        (38) PiDPToQDPProof [SOUND, 0 ms]
7.00/2.58	        (39) QDP
7.00/2.58	        (40) QDPSizeChangeProof [EQUIVALENT, 0 ms]
7.00/2.58	        (41) YES
7.00/2.58	    (42) PiDP
7.00/2.58	        (43) UsableRulesProof [EQUIVALENT, 0 ms]
7.00/2.58	        (44) PiDP
7.00/2.58	        (45) PiDPToQDPProof [SOUND, 0 ms]
7.00/2.58	        (46) QDP
7.00/2.58	        (47) QDPSizeChangeProof [EQUIVALENT, 0 ms]
7.00/2.58	        (48) YES
7.00/2.58	
7.00/2.58	
7.00/2.58	----------------------------------------
7.00/2.58	
7.00/2.58	(0)
7.00/2.58	Obligation:
7.00/2.58	Clauses:
7.00/2.58	
7.00/2.58	delete(X, tree(X, void, Right), Right).
7.00/2.58	delete(X, tree(X, Left, void), Left).
7.00/2.58	delete(X, tree(X, Left, Right), tree(Y, Left, Right1)) :- delmin(Right, Y, Right1).
7.00/2.58	delete(X, tree(Y, Left, Right), tree(Y, Left1, Right)) :- ','(less(X, Y), delete(X, Left, Left1)).
7.00/2.58	delete(X, tree(Y, Left, Right), tree(Y, Left, Right1)) :- ','(less(Y, X), delete(X, Right, Right1)).
7.00/2.58	delmin(tree(Y, void, Right), Y, Right).
7.00/2.58	delmin(tree(X, Left, X1), Y, tree(X, Left1, X2)) :- delmin(Left, Y, Left1).
7.00/2.58	less(0, s(X3)).
7.00/2.58	less(s(X), s(Y)) :- less(X, Y).
7.00/2.58	
7.00/2.58	
7.00/2.58	Query: delete(a,g,a)
7.00/2.58	----------------------------------------
7.00/2.58	
7.00/2.58	(1) PrologToPiTRSProof (SOUND)
7.00/2.58	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
7.00/2.58	
7.00/2.58	delete_in_3: (f,b,f) (b,b,f)
7.00/2.58	
7.00/2.58	delmin_in_3: (b,f,f)
7.00/2.58	
7.00/2.58	less_in_2: (f,b) (b,b) (b,f)
7.00/2.58	
7.00/2.58	Transforming Prolog into the following Term Rewriting System:
7.00/2.58	
7.00/2.58	Pi-finite rewrite system:
7.00/2.58	The TRS R consists of the following rules:
7.00/2.58	
7.00/2.58	   delete_in_aga(X, tree(X, void, Right), Right) -> delete_out_aga(X, tree(X, void, Right), Right)
7.00/2.58	   delete_in_aga(X, tree(X, Left, void), Left) -> delete_out_aga(X, tree(X, Left, void), Left)
7.00/2.58	   delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.58	   delmin_in_gaa(tree(Y, void, Right), Y, Right) -> delmin_out_gaa(tree(Y, void, Right), Y, Right)
7.00/2.58	   delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
7.00/2.58	   U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) -> delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
7.00/2.58	   U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
7.00/2.58	   delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
7.00/2.58	   less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3))
7.00/2.58	   less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y))
7.00/2.58	   U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
7.00/2.58	   U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.58	   delete_in_gga(X, tree(X, void, Right), Right) -> delete_out_gga(X, tree(X, void, Right), Right)
7.00/2.58	   delete_in_gga(X, tree(X, Left, void), Left) -> delete_out_gga(X, tree(X, Left, void), Left)
7.00/2.58	   delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.58	   U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
7.00/2.58	   delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
7.00/2.58	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
7.00/2.58	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
7.00/2.58	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
7.00/2.58	   U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.58	   delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
7.00/2.58	   U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
7.00/2.58	   U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
7.00/2.58	   U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
7.00/2.58	   U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
7.00/2.58	   delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
7.00/2.58	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
7.00/2.58	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
7.00/2.58	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
7.00/2.58	   U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
7.00/2.58	   U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) -> delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
7.00/2.58	
7.00/2.58	The argument filtering Pi contains the following mapping:
7.00/2.58	delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
7.00/2.58	
7.00/2.58	tree(x1, x2, x3)  =  tree(x1, x2, x3)
7.00/2.58	
7.00/2.58	void  =  void
7.00/2.58	
7.00/2.58	delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
7.00/2.58	
7.00/2.58	U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
7.00/2.58	
7.00/2.58	delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
7.00/2.58	
7.00/2.58	delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
7.00/2.58	
7.00/2.58	U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
7.00/2.58	
7.00/2.58	U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
7.00/2.58	
7.00/2.58	less_in_ag(x1, x2)  =  less_in_ag(x2)
7.00/2.58	
7.00/2.58	s(x1)  =  s(x1)
7.00/2.58	
7.00/2.58	less_out_ag(x1, x2)  =  less_out_ag(x1)
7.00/2.58	
7.00/2.58	U7_ag(x1, x2, x3)  =  U7_ag(x3)
7.00/2.58	
7.00/2.58	U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
7.00/2.58	
7.00/2.58	delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
7.00/2.58	
7.00/2.58	delete_out_gga(x1, x2, x3)  =  delete_out_gga
7.00/2.58	
7.00/2.58	U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
7.00/2.58	
7.00/2.58	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
7.00/2.58	
7.00/2.58	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
7.00/2.58	
7.00/2.58	0  =  0
7.00/2.58	
7.00/2.58	less_out_gg(x1, x2)  =  less_out_gg
7.00/2.58	
7.00/2.58	U7_gg(x1, x2, x3)  =  U7_gg(x3)
7.00/2.58	
7.00/2.58	U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
7.00/2.58	
7.00/2.58	U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
7.00/2.58	
7.00/2.58	U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
7.00/2.58	
7.00/2.58	U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
7.00/2.58	
7.00/2.58	less_in_ga(x1, x2)  =  less_in_ga(x1)
7.00/2.58	
7.00/2.58	less_out_ga(x1, x2)  =  less_out_ga
7.00/2.58	
7.00/2.58	U7_ga(x1, x2, x3)  =  U7_ga(x3)
7.00/2.58	
7.00/2.58	U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
7.00/2.58	
7.00/2.58	
7.00/2.58	
7.00/2.58	
7.00/2.58	
7.00/2.58	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
7.00/2.58	
7.00/2.58	
7.00/2.58	
7.00/2.58	----------------------------------------
7.00/2.58	
7.00/2.58	(2)
7.00/2.58	Obligation:
7.00/2.58	Pi-finite rewrite system:
7.00/2.58	The TRS R consists of the following rules:
7.00/2.58	
7.00/2.58	   delete_in_aga(X, tree(X, void, Right), Right) -> delete_out_aga(X, tree(X, void, Right), Right)
7.00/2.58	   delete_in_aga(X, tree(X, Left, void), Left) -> delete_out_aga(X, tree(X, Left, void), Left)
7.00/2.58	   delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.58	   delmin_in_gaa(tree(Y, void, Right), Y, Right) -> delmin_out_gaa(tree(Y, void, Right), Y, Right)
7.00/2.58	   delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
7.00/2.58	   U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) -> delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
7.00/2.58	   U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
7.00/2.58	   delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
7.00/2.58	   less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3))
7.00/2.58	   less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y))
7.00/2.58	   U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
7.00/2.58	   U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.58	   delete_in_gga(X, tree(X, void, Right), Right) -> delete_out_gga(X, tree(X, void, Right), Right)
7.00/2.58	   delete_in_gga(X, tree(X, Left, void), Left) -> delete_out_gga(X, tree(X, Left, void), Left)
7.00/2.58	   delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.58	   U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
7.00/2.58	   delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
7.00/2.58	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
7.00/2.58	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
7.00/2.58	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
7.00/2.58	   U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.58	   delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
7.00/2.58	   U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
7.00/2.58	   U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
7.00/2.58	   U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
7.00/2.58	   U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
7.00/2.58	   delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
7.00/2.58	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
7.00/2.58	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
7.00/2.58	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
7.00/2.58	   U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
7.00/2.58	   U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) -> delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
7.00/2.58	
7.00/2.58	The argument filtering Pi contains the following mapping:
7.00/2.58	delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
7.00/2.58	
7.00/2.58	tree(x1, x2, x3)  =  tree(x1, x2, x3)
7.00/2.58	
7.00/2.58	void  =  void
7.00/2.58	
7.00/2.58	delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
7.00/2.58	
7.00/2.58	U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
7.00/2.58	
7.00/2.58	delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
7.00/2.58	
7.00/2.58	delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
7.00/2.58	
7.00/2.58	U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
7.00/2.58	
7.00/2.58	U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
7.00/2.58	
7.00/2.58	less_in_ag(x1, x2)  =  less_in_ag(x2)
7.00/2.58	
7.00/2.58	s(x1)  =  s(x1)
7.00/2.58	
7.00/2.58	less_out_ag(x1, x2)  =  less_out_ag(x1)
7.00/2.58	
7.00/2.58	U7_ag(x1, x2, x3)  =  U7_ag(x3)
7.00/2.58	
7.00/2.58	U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
7.00/2.58	
7.00/2.58	delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
7.00/2.58	
7.00/2.58	delete_out_gga(x1, x2, x3)  =  delete_out_gga
7.00/2.58	
7.00/2.58	U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
7.00/2.58	
7.00/2.58	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
7.00/2.58	
7.00/2.58	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
7.00/2.58	
7.00/2.58	0  =  0
7.00/2.58	
7.00/2.58	less_out_gg(x1, x2)  =  less_out_gg
7.00/2.58	
7.00/2.58	U7_gg(x1, x2, x3)  =  U7_gg(x3)
7.00/2.58	
7.00/2.58	U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
7.00/2.58	
7.00/2.58	U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
7.00/2.58	
7.00/2.58	U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
7.00/2.58	
7.00/2.58	U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
7.00/2.58	
7.00/2.58	less_in_ga(x1, x2)  =  less_in_ga(x1)
7.00/2.58	
7.00/2.58	less_out_ga(x1, x2)  =  less_out_ga
7.00/2.58	
7.00/2.58	U7_ga(x1, x2, x3)  =  U7_ga(x3)
7.00/2.58	
7.00/2.58	U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
7.00/2.58	
7.00/2.58	
7.00/2.58	
7.00/2.58	----------------------------------------
7.00/2.58	
7.00/2.58	(3) DependencyPairsProof (EQUIVALENT)
7.00/2.58	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
7.00/2.58	Pi DP problem:
7.00/2.58	The TRS P consists of the following rules:
7.00/2.58	
7.00/2.58	   DELETE_IN_AGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_AGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.58	   DELETE_IN_AGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> DELMIN_IN_GAA(Right, Y, Right1)
7.00/2.58	   DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_GAA(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
7.00/2.58	   DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_GAA(Left, Y, Left1)
7.00/2.58	   DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_AGA(X, Y, Left, Right, Left1, less_in_ag(X, Y))
7.00/2.58	   DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_AG(X, Y)
7.00/2.58	   LESS_IN_AG(s(X), s(Y)) -> U7_AG(X, Y, less_in_ag(X, Y))
7.00/2.58	   LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y)
7.00/2.58	   U2_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_AGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.58	   U2_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> DELETE_IN_GGA(X, Left, Left1)
7.00/2.58	   DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_GGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.58	   DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> DELMIN_IN_GAA(Right, Y, Right1)
7.00/2.58	   DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
7.00/2.58	   DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GG(X, Y)
7.00/2.58	   LESS_IN_GG(s(X), s(Y)) -> U7_GG(X, Y, less_in_gg(X, Y))
7.00/2.58	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
7.00/2.58	   U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_GGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.58	   U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> DELETE_IN_GGA(X, Left, Left1)
7.00/2.58	   DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
7.00/2.58	   DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_GG(Y, X)
7.00/2.58	   U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_GGA(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
7.00/2.58	   U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> DELETE_IN_GGA(X, Right, Right1)
7.00/2.58	   DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X))
7.00/2.58	   DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_GA(Y, X)
7.00/2.58	   LESS_IN_GA(s(X), s(Y)) -> U7_GA(X, Y, less_in_ga(X, Y))
7.00/2.58	   LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y)
7.00/2.58	   U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_AGA(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
7.00/2.58	   U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> DELETE_IN_AGA(X, Right, Right1)
7.00/2.58	
7.00/2.58	The TRS R consists of the following rules:
7.00/2.58	
7.00/2.58	   delete_in_aga(X, tree(X, void, Right), Right) -> delete_out_aga(X, tree(X, void, Right), Right)
7.00/2.58	   delete_in_aga(X, tree(X, Left, void), Left) -> delete_out_aga(X, tree(X, Left, void), Left)
7.00/2.58	   delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.58	   delmin_in_gaa(tree(Y, void, Right), Y, Right) -> delmin_out_gaa(tree(Y, void, Right), Y, Right)
7.00/2.58	   delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
7.00/2.58	   U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) -> delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
7.00/2.58	   U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
7.00/2.58	   delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
7.00/2.58	   less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3))
7.00/2.58	   less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y))
7.00/2.58	   U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
7.00/2.58	   U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.58	   delete_in_gga(X, tree(X, void, Right), Right) -> delete_out_gga(X, tree(X, void, Right), Right)
7.00/2.58	   delete_in_gga(X, tree(X, Left, void), Left) -> delete_out_gga(X, tree(X, Left, void), Left)
7.00/2.58	   delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.58	   U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
7.00/2.58	   delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
7.00/2.58	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
7.00/2.58	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
7.00/2.58	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
7.00/2.58	   U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.58	   delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
7.00/2.58	   U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
7.00/2.58	   U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
7.00/2.58	   U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
7.00/2.58	   U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
7.00/2.58	   delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
7.00/2.58	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
7.00/2.58	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
7.00/2.58	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
7.00/2.58	   U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
7.00/2.58	   U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) -> delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
7.00/2.58	
7.00/2.58	The argument filtering Pi contains the following mapping:
7.00/2.58	delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
7.00/2.58	
7.00/2.58	tree(x1, x2, x3)  =  tree(x1, x2, x3)
7.00/2.58	
7.00/2.58	void  =  void
7.00/2.58	
7.00/2.58	delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
7.00/2.58	
7.00/2.58	U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
7.00/2.58	
7.00/2.58	delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
7.00/2.58	
7.00/2.58	delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
7.00/2.58	
7.00/2.58	U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
7.00/2.58	
7.00/2.58	U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
7.00/2.58	
7.00/2.58	less_in_ag(x1, x2)  =  less_in_ag(x2)
7.00/2.58	
7.00/2.58	s(x1)  =  s(x1)
7.00/2.58	
7.00/2.58	less_out_ag(x1, x2)  =  less_out_ag(x1)
7.00/2.58	
7.00/2.58	U7_ag(x1, x2, x3)  =  U7_ag(x3)
7.00/2.58	
7.00/2.58	U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
7.00/2.58	
7.00/2.58	delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
7.00/2.58	
7.00/2.58	delete_out_gga(x1, x2, x3)  =  delete_out_gga
7.00/2.58	
7.00/2.58	U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
7.00/2.58	
7.00/2.58	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
7.00/2.58	
7.00/2.58	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
7.00/2.58	
7.00/2.58	0  =  0
7.00/2.58	
7.00/2.58	less_out_gg(x1, x2)  =  less_out_gg
7.00/2.58	
7.00/2.58	U7_gg(x1, x2, x3)  =  U7_gg(x3)
7.00/2.58	
7.00/2.58	U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
7.00/2.58	
7.00/2.58	U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
7.00/2.58	
7.00/2.58	U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
7.00/2.58	
7.00/2.58	U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
7.00/2.58	
7.00/2.58	less_in_ga(x1, x2)  =  less_in_ga(x1)
7.00/2.58	
7.00/2.58	less_out_ga(x1, x2)  =  less_out_ga
7.00/2.58	
7.00/2.58	U7_ga(x1, x2, x3)  =  U7_ga(x3)
7.00/2.58	
7.00/2.58	U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
7.00/2.58	
7.00/2.58	DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
7.00/2.58	
7.00/2.58	U1_AGA(x1, x2, x3, x4, x5, x6)  =  U1_AGA(x1, x6)
7.00/2.58	
7.00/2.58	DELMIN_IN_GAA(x1, x2, x3)  =  DELMIN_IN_GAA(x1)
7.00/2.58	
7.00/2.58	U6_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GAA(x7)
7.00/2.58	
7.00/2.58	U2_AGA(x1, x2, x3, x4, x5, x6)  =  U2_AGA(x3, x6)
7.00/2.58	
7.00/2.58	LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
7.00/2.58	
7.00/2.58	U7_AG(x1, x2, x3)  =  U7_AG(x3)
7.00/2.58	
7.00/2.58	U3_AGA(x1, x2, x3, x4, x5, x6)  =  U3_AGA(x1, x6)
7.00/2.58	
7.00/2.58	DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
7.00/2.58	
7.00/2.58	U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x6)
7.00/2.58	
7.00/2.58	U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x3, x6)
7.00/2.58	
7.00/2.58	LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
7.00/2.58	
7.00/2.58	U7_GG(x1, x2, x3)  =  U7_GG(x3)
7.00/2.58	
7.00/2.58	U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x6)
7.00/2.58	
7.00/2.58	U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x4, x6)
7.00/2.58	
7.00/2.58	U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x6)
7.00/2.58	
7.00/2.58	U4_AGA(x1, x2, x3, x4, x5, x6)  =  U4_AGA(x4, x6)
7.00/2.58	
7.00/2.58	LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
7.00/2.58	
7.00/2.58	U7_GA(x1, x2, x3)  =  U7_GA(x3)
7.00/2.58	
7.00/2.58	U5_AGA(x1, x2, x3, x4, x5, x6)  =  U5_AGA(x6)
7.00/2.58	
7.00/2.58	
7.00/2.58	We have to consider all (P,R,Pi)-chains
7.00/2.58	----------------------------------------
7.00/2.58	
7.00/2.58	(4)
7.00/2.58	Obligation:
7.00/2.58	Pi DP problem:
7.00/2.58	The TRS P consists of the following rules:
7.00/2.58	
7.00/2.58	   DELETE_IN_AGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_AGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.58	   DELETE_IN_AGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> DELMIN_IN_GAA(Right, Y, Right1)
7.00/2.58	   DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_GAA(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
7.00/2.58	   DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_GAA(Left, Y, Left1)
7.00/2.58	   DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_AGA(X, Y, Left, Right, Left1, less_in_ag(X, Y))
7.00/2.58	   DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_AG(X, Y)
7.00/2.59	   LESS_IN_AG(s(X), s(Y)) -> U7_AG(X, Y, less_in_ag(X, Y))
7.00/2.59	   LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y)
7.00/2.59	   U2_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_AGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.59	   U2_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> DELETE_IN_GGA(X, Left, Left1)
7.00/2.59	   DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_GGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.59	   DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> DELMIN_IN_GAA(Right, Y, Right1)
7.00/2.59	   DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
7.00/2.59	   DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GG(X, Y)
7.00/2.59	   LESS_IN_GG(s(X), s(Y)) -> U7_GG(X, Y, less_in_gg(X, Y))
7.00/2.59	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
7.00/2.59	   U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_GGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.59	   U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> DELETE_IN_GGA(X, Left, Left1)
7.00/2.59	   DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
7.00/2.59	   DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_GG(Y, X)
7.00/2.59	   U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_GGA(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
7.00/2.59	   U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> DELETE_IN_GGA(X, Right, Right1)
7.00/2.59	   DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X))
7.00/2.59	   DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_GA(Y, X)
7.00/2.59	   LESS_IN_GA(s(X), s(Y)) -> U7_GA(X, Y, less_in_ga(X, Y))
7.00/2.59	   LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y)
7.00/2.59	   U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_AGA(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
7.00/2.59	   U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> DELETE_IN_AGA(X, Right, Right1)
7.00/2.59	
7.00/2.59	The TRS R consists of the following rules:
7.00/2.59	
7.00/2.59	   delete_in_aga(X, tree(X, void, Right), Right) -> delete_out_aga(X, tree(X, void, Right), Right)
7.00/2.59	   delete_in_aga(X, tree(X, Left, void), Left) -> delete_out_aga(X, tree(X, Left, void), Left)
7.00/2.59	   delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.59	   delmin_in_gaa(tree(Y, void, Right), Y, Right) -> delmin_out_gaa(tree(Y, void, Right), Y, Right)
7.00/2.59	   delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
7.00/2.59	   U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) -> delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
7.00/2.59	   U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
7.00/2.59	   delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
7.00/2.59	   less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3))
7.00/2.59	   less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y))
7.00/2.59	   U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
7.00/2.59	   U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.59	   delete_in_gga(X, tree(X, void, Right), Right) -> delete_out_gga(X, tree(X, void, Right), Right)
7.00/2.59	   delete_in_gga(X, tree(X, Left, void), Left) -> delete_out_gga(X, tree(X, Left, void), Left)
7.00/2.59	   delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.59	   U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
7.00/2.59	   delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
7.00/2.59	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
7.00/2.59	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
7.00/2.59	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
7.00/2.59	   U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.59	   delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
7.00/2.59	   U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
7.00/2.59	   U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
7.00/2.59	   U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
7.00/2.59	   U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
7.00/2.59	   delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
7.00/2.59	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
7.00/2.59	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
7.00/2.59	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
7.00/2.59	   U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
7.00/2.59	   U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) -> delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
7.00/2.59	
7.00/2.59	The argument filtering Pi contains the following mapping:
7.00/2.59	delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
7.00/2.59	
7.00/2.59	tree(x1, x2, x3)  =  tree(x1, x2, x3)
7.00/2.59	
7.00/2.59	void  =  void
7.00/2.59	
7.00/2.59	delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
7.00/2.59	
7.00/2.59	U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
7.00/2.59	
7.00/2.59	delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
7.00/2.59	
7.00/2.59	delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
7.00/2.59	
7.00/2.59	U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
7.00/2.59	
7.00/2.59	U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
7.00/2.59	
7.00/2.59	less_in_ag(x1, x2)  =  less_in_ag(x2)
7.00/2.59	
7.00/2.59	s(x1)  =  s(x1)
7.00/2.59	
7.00/2.59	less_out_ag(x1, x2)  =  less_out_ag(x1)
7.00/2.59	
7.00/2.59	U7_ag(x1, x2, x3)  =  U7_ag(x3)
7.00/2.59	
7.00/2.59	U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
7.00/2.59	
7.00/2.59	delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
7.00/2.59	
7.00/2.59	delete_out_gga(x1, x2, x3)  =  delete_out_gga
7.00/2.59	
7.00/2.59	U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
7.00/2.59	
7.00/2.59	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
7.00/2.59	
7.00/2.59	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
7.00/2.59	
7.00/2.59	0  =  0
7.00/2.59	
7.00/2.59	less_out_gg(x1, x2)  =  less_out_gg
7.00/2.59	
7.00/2.59	U7_gg(x1, x2, x3)  =  U7_gg(x3)
7.00/2.59	
7.00/2.59	U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
7.00/2.59	
7.00/2.59	U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
7.00/2.59	
7.00/2.59	U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
7.00/2.59	
7.00/2.59	U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
7.00/2.59	
7.00/2.59	less_in_ga(x1, x2)  =  less_in_ga(x1)
7.00/2.59	
7.00/2.59	less_out_ga(x1, x2)  =  less_out_ga
7.00/2.59	
7.00/2.59	U7_ga(x1, x2, x3)  =  U7_ga(x3)
7.00/2.59	
7.00/2.59	U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
7.00/2.59	
7.00/2.59	DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
7.00/2.59	
7.00/2.59	U1_AGA(x1, x2, x3, x4, x5, x6)  =  U1_AGA(x1, x6)
7.00/2.59	
7.00/2.59	DELMIN_IN_GAA(x1, x2, x3)  =  DELMIN_IN_GAA(x1)
7.00/2.59	
7.00/2.59	U6_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GAA(x7)
7.00/2.59	
7.00/2.59	U2_AGA(x1, x2, x3, x4, x5, x6)  =  U2_AGA(x3, x6)
7.00/2.59	
7.00/2.59	LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
7.00/2.59	
7.00/2.59	U7_AG(x1, x2, x3)  =  U7_AG(x3)
7.00/2.59	
7.00/2.59	U3_AGA(x1, x2, x3, x4, x5, x6)  =  U3_AGA(x1, x6)
7.00/2.59	
7.00/2.59	DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
7.00/2.59	
7.00/2.59	U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x6)
7.00/2.59	
7.00/2.59	U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x3, x6)
7.00/2.59	
7.00/2.59	LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
7.00/2.59	
7.00/2.59	U7_GG(x1, x2, x3)  =  U7_GG(x3)
7.00/2.59	
7.00/2.59	U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x6)
7.00/2.59	
7.00/2.59	U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x4, x6)
7.00/2.59	
7.00/2.59	U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x6)
7.00/2.59	
7.00/2.59	U4_AGA(x1, x2, x3, x4, x5, x6)  =  U4_AGA(x4, x6)
7.00/2.59	
7.00/2.59	LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
7.00/2.59	
7.00/2.59	U7_GA(x1, x2, x3)  =  U7_GA(x3)
7.00/2.59	
7.00/2.59	U5_AGA(x1, x2, x3, x4, x5, x6)  =  U5_AGA(x6)
7.00/2.59	
7.00/2.59	
7.00/2.59	We have to consider all (P,R,Pi)-chains
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(5) DependencyGraphProof (EQUIVALENT)
7.00/2.59	The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 18 less nodes.
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(6)
7.00/2.59	Complex Obligation (AND)
7.00/2.59	
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(7)
7.00/2.59	Obligation:
7.00/2.59	Pi DP problem:
7.00/2.59	The TRS P consists of the following rules:
7.00/2.59	
7.00/2.59	   LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y)
7.00/2.59	
7.00/2.59	The TRS R consists of the following rules:
7.00/2.59	
7.00/2.59	   delete_in_aga(X, tree(X, void, Right), Right) -> delete_out_aga(X, tree(X, void, Right), Right)
7.00/2.59	   delete_in_aga(X, tree(X, Left, void), Left) -> delete_out_aga(X, tree(X, Left, void), Left)
7.00/2.59	   delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.59	   delmin_in_gaa(tree(Y, void, Right), Y, Right) -> delmin_out_gaa(tree(Y, void, Right), Y, Right)
7.00/2.59	   delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
7.00/2.59	   U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) -> delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
7.00/2.59	   U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
7.00/2.59	   delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
7.00/2.59	   less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3))
7.00/2.59	   less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y))
7.00/2.59	   U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
7.00/2.59	   U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.59	   delete_in_gga(X, tree(X, void, Right), Right) -> delete_out_gga(X, tree(X, void, Right), Right)
7.00/2.59	   delete_in_gga(X, tree(X, Left, void), Left) -> delete_out_gga(X, tree(X, Left, void), Left)
7.00/2.59	   delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.59	   U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
7.00/2.59	   delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
7.00/2.59	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
7.00/2.59	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
7.00/2.59	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
7.00/2.59	   U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.59	   delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
7.00/2.59	   U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
7.00/2.59	   U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
7.00/2.59	   U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
7.00/2.59	   U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
7.00/2.59	   delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
7.00/2.59	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
7.00/2.59	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
7.00/2.59	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
7.00/2.59	   U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
7.00/2.59	   U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) -> delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
7.00/2.59	
7.00/2.59	The argument filtering Pi contains the following mapping:
7.00/2.59	delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
7.00/2.59	
7.00/2.59	tree(x1, x2, x3)  =  tree(x1, x2, x3)
7.00/2.59	
7.00/2.59	void  =  void
7.00/2.59	
7.00/2.59	delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
7.00/2.59	
7.00/2.59	U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
7.00/2.59	
7.00/2.59	delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
7.00/2.59	
7.00/2.59	delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
7.00/2.59	
7.00/2.59	U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
7.00/2.59	
7.00/2.59	U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
7.00/2.59	
7.00/2.59	less_in_ag(x1, x2)  =  less_in_ag(x2)
7.00/2.59	
7.00/2.59	s(x1)  =  s(x1)
7.00/2.59	
7.00/2.59	less_out_ag(x1, x2)  =  less_out_ag(x1)
7.00/2.59	
7.00/2.59	U7_ag(x1, x2, x3)  =  U7_ag(x3)
7.00/2.59	
7.00/2.59	U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
7.00/2.59	
7.00/2.59	delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
7.00/2.59	
7.00/2.59	delete_out_gga(x1, x2, x3)  =  delete_out_gga
7.00/2.59	
7.00/2.59	U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
7.00/2.59	
7.00/2.59	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
7.00/2.59	
7.00/2.59	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
7.00/2.59	
7.00/2.59	0  =  0
7.00/2.59	
7.00/2.59	less_out_gg(x1, x2)  =  less_out_gg
7.00/2.59	
7.00/2.59	U7_gg(x1, x2, x3)  =  U7_gg(x3)
7.00/2.59	
7.00/2.59	U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
7.00/2.59	
7.00/2.59	U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
7.00/2.59	
7.00/2.59	U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
7.00/2.59	
7.00/2.59	U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
7.00/2.59	
7.00/2.59	less_in_ga(x1, x2)  =  less_in_ga(x1)
7.00/2.59	
7.00/2.59	less_out_ga(x1, x2)  =  less_out_ga
7.00/2.59	
7.00/2.59	U7_ga(x1, x2, x3)  =  U7_ga(x3)
7.00/2.59	
7.00/2.59	U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
7.00/2.59	
7.00/2.59	LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
7.00/2.59	
7.00/2.59	
7.00/2.59	We have to consider all (P,R,Pi)-chains
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(8) UsableRulesProof (EQUIVALENT)
7.00/2.59	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(9)
7.00/2.59	Obligation:
7.00/2.59	Pi DP problem:
7.00/2.59	The TRS P consists of the following rules:
7.00/2.59	
7.00/2.59	   LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y)
7.00/2.59	
7.00/2.59	R is empty.
7.00/2.59	The argument filtering Pi contains the following mapping:
7.00/2.59	s(x1)  =  s(x1)
7.00/2.59	
7.00/2.59	LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
7.00/2.59	
7.00/2.59	
7.00/2.59	We have to consider all (P,R,Pi)-chains
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(10) PiDPToQDPProof (SOUND)
7.00/2.59	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(11)
7.00/2.59	Obligation:
7.00/2.59	Q DP problem:
7.00/2.59	The TRS P consists of the following rules:
7.00/2.59	
7.00/2.59	   LESS_IN_GA(s(X)) -> LESS_IN_GA(X)
7.00/2.59	
7.00/2.59	R is empty.
7.00/2.59	Q is empty.
7.00/2.59	We have to consider all (P,Q,R)-chains.
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(12) QDPSizeChangeProof (EQUIVALENT)
7.00/2.59	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. 
7.00/2.59	
7.00/2.59	From the DPs we obtained the following set of size-change graphs:
7.00/2.59	*LESS_IN_GA(s(X)) -> LESS_IN_GA(X)
7.00/2.59	The graph contains the following edges 1 > 1
7.00/2.59	
7.00/2.59	
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(13)
7.00/2.59	YES
7.00/2.59	
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(14)
7.00/2.59	Obligation:
7.00/2.59	Pi DP problem:
7.00/2.59	The TRS P consists of the following rules:
7.00/2.59	
7.00/2.59	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
7.00/2.59	
7.00/2.59	The TRS R consists of the following rules:
7.00/2.59	
7.00/2.59	   delete_in_aga(X, tree(X, void, Right), Right) -> delete_out_aga(X, tree(X, void, Right), Right)
7.00/2.59	   delete_in_aga(X, tree(X, Left, void), Left) -> delete_out_aga(X, tree(X, Left, void), Left)
7.00/2.59	   delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.59	   delmin_in_gaa(tree(Y, void, Right), Y, Right) -> delmin_out_gaa(tree(Y, void, Right), Y, Right)
7.00/2.59	   delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
7.00/2.59	   U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) -> delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
7.00/2.59	   U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
7.00/2.59	   delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
7.00/2.59	   less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3))
7.00/2.59	   less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y))
7.00/2.59	   U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
7.00/2.59	   U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.59	   delete_in_gga(X, tree(X, void, Right), Right) -> delete_out_gga(X, tree(X, void, Right), Right)
7.00/2.59	   delete_in_gga(X, tree(X, Left, void), Left) -> delete_out_gga(X, tree(X, Left, void), Left)
7.00/2.59	   delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.59	   U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
7.00/2.59	   delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
7.00/2.59	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
7.00/2.59	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
7.00/2.59	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
7.00/2.59	   U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.59	   delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
7.00/2.59	   U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
7.00/2.59	   U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
7.00/2.59	   U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
7.00/2.59	   U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
7.00/2.59	   delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
7.00/2.59	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
7.00/2.59	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
7.00/2.59	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
7.00/2.59	   U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
7.00/2.59	   U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) -> delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
7.00/2.59	
7.00/2.59	The argument filtering Pi contains the following mapping:
7.00/2.59	delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
7.00/2.59	
7.00/2.59	tree(x1, x2, x3)  =  tree(x1, x2, x3)
7.00/2.59	
7.00/2.59	void  =  void
7.00/2.59	
7.00/2.59	delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
7.00/2.59	
7.00/2.59	U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
7.00/2.59	
7.00/2.59	delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
7.00/2.59	
7.00/2.59	delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
7.00/2.59	
7.00/2.59	U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
7.00/2.59	
7.00/2.59	U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
7.00/2.59	
7.00/2.59	less_in_ag(x1, x2)  =  less_in_ag(x2)
7.00/2.59	
7.00/2.59	s(x1)  =  s(x1)
7.00/2.59	
7.00/2.59	less_out_ag(x1, x2)  =  less_out_ag(x1)
7.00/2.59	
7.00/2.59	U7_ag(x1, x2, x3)  =  U7_ag(x3)
7.00/2.59	
7.00/2.59	U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
7.00/2.59	
7.00/2.59	delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
7.00/2.59	
7.00/2.59	delete_out_gga(x1, x2, x3)  =  delete_out_gga
7.00/2.59	
7.00/2.59	U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
7.00/2.59	
7.00/2.59	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
7.00/2.59	
7.00/2.59	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
7.00/2.59	
7.00/2.59	0  =  0
7.00/2.59	
7.00/2.59	less_out_gg(x1, x2)  =  less_out_gg
7.00/2.59	
7.00/2.59	U7_gg(x1, x2, x3)  =  U7_gg(x3)
7.00/2.59	
7.00/2.59	U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
7.00/2.59	
7.00/2.59	U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
7.00/2.59	
7.00/2.59	U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
7.00/2.59	
7.00/2.59	U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
7.00/2.59	
7.00/2.59	less_in_ga(x1, x2)  =  less_in_ga(x1)
7.00/2.59	
7.00/2.59	less_out_ga(x1, x2)  =  less_out_ga
7.00/2.59	
7.00/2.59	U7_ga(x1, x2, x3)  =  U7_ga(x3)
7.00/2.59	
7.00/2.59	U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
7.00/2.59	
7.00/2.59	LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
7.00/2.59	
7.00/2.59	
7.00/2.59	We have to consider all (P,R,Pi)-chains
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(15) UsableRulesProof (EQUIVALENT)
7.00/2.59	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(16)
7.00/2.59	Obligation:
7.00/2.59	Pi DP problem:
7.00/2.59	The TRS P consists of the following rules:
7.00/2.59	
7.00/2.59	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
7.00/2.59	
7.00/2.59	R is empty.
7.00/2.59	Pi is empty.
7.00/2.59	We have to consider all (P,R,Pi)-chains
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(17) PiDPToQDPProof (EQUIVALENT)
7.00/2.59	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(18)
7.00/2.59	Obligation:
7.00/2.59	Q DP problem:
7.00/2.59	The TRS P consists of the following rules:
7.00/2.59	
7.00/2.59	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
7.00/2.59	
7.00/2.59	R is empty.
7.00/2.59	Q is empty.
7.00/2.59	We have to consider all (P,Q,R)-chains.
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(19) QDPSizeChangeProof (EQUIVALENT)
7.00/2.59	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. 
7.00/2.59	
7.00/2.59	From the DPs we obtained the following set of size-change graphs:
7.00/2.59	*LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
7.00/2.59	The graph contains the following edges 1 > 1, 2 > 2
7.00/2.59	
7.00/2.59	
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(20)
7.00/2.59	YES
7.00/2.59	
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(21)
7.00/2.59	Obligation:
7.00/2.59	Pi DP problem:
7.00/2.59	The TRS P consists of the following rules:
7.00/2.59	
7.00/2.59	   LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y)
7.00/2.59	
7.00/2.59	The TRS R consists of the following rules:
7.00/2.59	
7.00/2.59	   delete_in_aga(X, tree(X, void, Right), Right) -> delete_out_aga(X, tree(X, void, Right), Right)
7.00/2.59	   delete_in_aga(X, tree(X, Left, void), Left) -> delete_out_aga(X, tree(X, Left, void), Left)
7.00/2.59	   delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.59	   delmin_in_gaa(tree(Y, void, Right), Y, Right) -> delmin_out_gaa(tree(Y, void, Right), Y, Right)
7.00/2.59	   delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
7.00/2.59	   U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) -> delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
7.00/2.59	   U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
7.00/2.59	   delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
7.00/2.59	   less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3))
7.00/2.59	   less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y))
7.00/2.59	   U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
7.00/2.59	   U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.59	   delete_in_gga(X, tree(X, void, Right), Right) -> delete_out_gga(X, tree(X, void, Right), Right)
7.00/2.59	   delete_in_gga(X, tree(X, Left, void), Left) -> delete_out_gga(X, tree(X, Left, void), Left)
7.00/2.59	   delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.59	   U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
7.00/2.59	   delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
7.00/2.59	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
7.00/2.59	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
7.00/2.59	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
7.00/2.59	   U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.59	   delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
7.00/2.59	   U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
7.00/2.59	   U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
7.00/2.59	   U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
7.00/2.59	   U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
7.00/2.59	   delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
7.00/2.59	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
7.00/2.59	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
7.00/2.59	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
7.00/2.59	   U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
7.00/2.59	   U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) -> delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
7.00/2.59	
7.00/2.59	The argument filtering Pi contains the following mapping:
7.00/2.59	delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
7.00/2.59	
7.00/2.59	tree(x1, x2, x3)  =  tree(x1, x2, x3)
7.00/2.59	
7.00/2.59	void  =  void
7.00/2.59	
7.00/2.59	delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
7.00/2.59	
7.00/2.59	U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
7.00/2.59	
7.00/2.59	delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
7.00/2.59	
7.00/2.59	delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
7.00/2.59	
7.00/2.59	U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
7.00/2.59	
7.00/2.59	U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
7.00/2.59	
7.00/2.59	less_in_ag(x1, x2)  =  less_in_ag(x2)
7.00/2.59	
7.00/2.59	s(x1)  =  s(x1)
7.00/2.59	
7.00/2.59	less_out_ag(x1, x2)  =  less_out_ag(x1)
7.00/2.59	
7.00/2.59	U7_ag(x1, x2, x3)  =  U7_ag(x3)
7.00/2.59	
7.00/2.59	U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
7.00/2.59	
7.00/2.59	delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
7.00/2.59	
7.00/2.59	delete_out_gga(x1, x2, x3)  =  delete_out_gga
7.00/2.59	
7.00/2.59	U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
7.00/2.59	
7.00/2.59	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
7.00/2.59	
7.00/2.59	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
7.00/2.59	
7.00/2.59	0  =  0
7.00/2.59	
7.00/2.59	less_out_gg(x1, x2)  =  less_out_gg
7.00/2.59	
7.00/2.59	U7_gg(x1, x2, x3)  =  U7_gg(x3)
7.00/2.59	
7.00/2.59	U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
7.00/2.59	
7.00/2.59	U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
7.00/2.59	
7.00/2.59	U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
7.00/2.59	
7.00/2.59	U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
7.00/2.59	
7.00/2.59	less_in_ga(x1, x2)  =  less_in_ga(x1)
7.00/2.59	
7.00/2.59	less_out_ga(x1, x2)  =  less_out_ga
7.00/2.59	
7.00/2.59	U7_ga(x1, x2, x3)  =  U7_ga(x3)
7.00/2.59	
7.00/2.59	U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
7.00/2.59	
7.00/2.59	LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
7.00/2.59	
7.00/2.59	
7.00/2.59	We have to consider all (P,R,Pi)-chains
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(22) UsableRulesProof (EQUIVALENT)
7.00/2.59	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(23)
7.00/2.59	Obligation:
7.00/2.59	Pi DP problem:
7.00/2.59	The TRS P consists of the following rules:
7.00/2.59	
7.00/2.59	   LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y)
7.00/2.59	
7.00/2.59	R is empty.
7.00/2.59	The argument filtering Pi contains the following mapping:
7.00/2.59	s(x1)  =  s(x1)
7.00/2.59	
7.00/2.59	LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
7.00/2.59	
7.00/2.59	
7.00/2.59	We have to consider all (P,R,Pi)-chains
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(24) PiDPToQDPProof (SOUND)
7.00/2.59	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(25)
7.00/2.59	Obligation:
7.00/2.59	Q DP problem:
7.00/2.59	The TRS P consists of the following rules:
7.00/2.59	
7.00/2.59	   LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y)
7.00/2.59	
7.00/2.59	R is empty.
7.00/2.59	Q is empty.
7.00/2.59	We have to consider all (P,Q,R)-chains.
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(26) QDPSizeChangeProof (EQUIVALENT)
7.00/2.59	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. 
7.00/2.59	
7.00/2.59	From the DPs we obtained the following set of size-change graphs:
7.00/2.59	*LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y)
7.00/2.59	The graph contains the following edges 1 > 1
7.00/2.59	
7.00/2.59	
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(27)
7.00/2.59	YES
7.00/2.59	
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(28)
7.00/2.59	Obligation:
7.00/2.59	Pi DP problem:
7.00/2.59	The TRS P consists of the following rules:
7.00/2.59	
7.00/2.59	   DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_GAA(Left, Y, Left1)
7.00/2.59	
7.00/2.59	The TRS R consists of the following rules:
7.00/2.59	
7.00/2.59	   delete_in_aga(X, tree(X, void, Right), Right) -> delete_out_aga(X, tree(X, void, Right), Right)
7.00/2.59	   delete_in_aga(X, tree(X, Left, void), Left) -> delete_out_aga(X, tree(X, Left, void), Left)
7.00/2.59	   delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.59	   delmin_in_gaa(tree(Y, void, Right), Y, Right) -> delmin_out_gaa(tree(Y, void, Right), Y, Right)
7.00/2.59	   delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
7.00/2.59	   U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) -> delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
7.00/2.59	   U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
7.00/2.59	   delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
7.00/2.59	   less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3))
7.00/2.59	   less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y))
7.00/2.59	   U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
7.00/2.59	   U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.59	   delete_in_gga(X, tree(X, void, Right), Right) -> delete_out_gga(X, tree(X, void, Right), Right)
7.00/2.59	   delete_in_gga(X, tree(X, Left, void), Left) -> delete_out_gga(X, tree(X, Left, void), Left)
7.00/2.59	   delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.59	   U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
7.00/2.59	   delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
7.00/2.59	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
7.00/2.59	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
7.00/2.59	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
7.00/2.59	   U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.59	   delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
7.00/2.59	   U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
7.00/2.59	   U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
7.00/2.59	   U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
7.00/2.59	   U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
7.00/2.59	   delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
7.00/2.59	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
7.00/2.59	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
7.00/2.59	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
7.00/2.59	   U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
7.00/2.59	   U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) -> delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
7.00/2.59	
7.00/2.59	The argument filtering Pi contains the following mapping:
7.00/2.59	delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
7.00/2.59	
7.00/2.59	tree(x1, x2, x3)  =  tree(x1, x2, x3)
7.00/2.59	
7.00/2.59	void  =  void
7.00/2.59	
7.00/2.59	delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
7.00/2.59	
7.00/2.59	U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
7.00/2.59	
7.00/2.59	delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
7.00/2.59	
7.00/2.59	delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
7.00/2.59	
7.00/2.59	U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
7.00/2.59	
7.00/2.59	U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
7.00/2.59	
7.00/2.59	less_in_ag(x1, x2)  =  less_in_ag(x2)
7.00/2.59	
7.00/2.59	s(x1)  =  s(x1)
7.00/2.59	
7.00/2.59	less_out_ag(x1, x2)  =  less_out_ag(x1)
7.00/2.59	
7.00/2.59	U7_ag(x1, x2, x3)  =  U7_ag(x3)
7.00/2.59	
7.00/2.59	U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
7.00/2.59	
7.00/2.59	delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
7.00/2.59	
7.00/2.59	delete_out_gga(x1, x2, x3)  =  delete_out_gga
7.00/2.59	
7.00/2.59	U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
7.00/2.59	
7.00/2.59	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
7.00/2.59	
7.00/2.59	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
7.00/2.59	
7.00/2.59	0  =  0
7.00/2.59	
7.00/2.59	less_out_gg(x1, x2)  =  less_out_gg
7.00/2.59	
7.00/2.59	U7_gg(x1, x2, x3)  =  U7_gg(x3)
7.00/2.59	
7.00/2.59	U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
7.00/2.59	
7.00/2.59	U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
7.00/2.59	
7.00/2.59	U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
7.00/2.59	
7.00/2.59	U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
7.00/2.59	
7.00/2.59	less_in_ga(x1, x2)  =  less_in_ga(x1)
7.00/2.59	
7.00/2.59	less_out_ga(x1, x2)  =  less_out_ga
7.00/2.59	
7.00/2.59	U7_ga(x1, x2, x3)  =  U7_ga(x3)
7.00/2.59	
7.00/2.59	U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
7.00/2.59	
7.00/2.59	DELMIN_IN_GAA(x1, x2, x3)  =  DELMIN_IN_GAA(x1)
7.00/2.59	
7.00/2.59	
7.00/2.59	We have to consider all (P,R,Pi)-chains
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(29) UsableRulesProof (EQUIVALENT)
7.00/2.59	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(30)
7.00/2.59	Obligation:
7.00/2.59	Pi DP problem:
7.00/2.59	The TRS P consists of the following rules:
7.00/2.59	
7.00/2.59	   DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_GAA(Left, Y, Left1)
7.00/2.59	
7.00/2.59	R is empty.
7.00/2.59	The argument filtering Pi contains the following mapping:
7.00/2.59	tree(x1, x2, x3)  =  tree(x1, x2, x3)
7.00/2.59	
7.00/2.59	DELMIN_IN_GAA(x1, x2, x3)  =  DELMIN_IN_GAA(x1)
7.00/2.59	
7.00/2.59	
7.00/2.59	We have to consider all (P,R,Pi)-chains
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(31) PiDPToQDPProof (SOUND)
7.00/2.59	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(32)
7.00/2.59	Obligation:
7.00/2.59	Q DP problem:
7.00/2.59	The TRS P consists of the following rules:
7.00/2.59	
7.00/2.59	   DELMIN_IN_GAA(tree(X, Left, X1)) -> DELMIN_IN_GAA(Left)
7.00/2.59	
7.00/2.59	R is empty.
7.00/2.59	Q is empty.
7.00/2.59	We have to consider all (P,Q,R)-chains.
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(33) QDPSizeChangeProof (EQUIVALENT)
7.00/2.59	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. 
7.00/2.59	
7.00/2.59	From the DPs we obtained the following set of size-change graphs:
7.00/2.59	*DELMIN_IN_GAA(tree(X, Left, X1)) -> DELMIN_IN_GAA(Left)
7.00/2.59	The graph contains the following edges 1 > 1
7.00/2.59	
7.00/2.59	
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(34)
7.00/2.59	YES
7.00/2.59	
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(35)
7.00/2.59	Obligation:
7.00/2.59	Pi DP problem:
7.00/2.59	The TRS P consists of the following rules:
7.00/2.59	
7.00/2.59	   DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
7.00/2.59	   U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> DELETE_IN_GGA(X, Left, Left1)
7.00/2.59	   DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
7.00/2.59	   U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> DELETE_IN_GGA(X, Right, Right1)
7.00/2.59	
7.00/2.59	The TRS R consists of the following rules:
7.00/2.59	
7.00/2.59	   delete_in_aga(X, tree(X, void, Right), Right) -> delete_out_aga(X, tree(X, void, Right), Right)
7.00/2.59	   delete_in_aga(X, tree(X, Left, void), Left) -> delete_out_aga(X, tree(X, Left, void), Left)
7.00/2.59	   delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.59	   delmin_in_gaa(tree(Y, void, Right), Y, Right) -> delmin_out_gaa(tree(Y, void, Right), Y, Right)
7.00/2.59	   delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
7.00/2.59	   U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) -> delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
7.00/2.59	   U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
7.00/2.59	   delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
7.00/2.59	   less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3))
7.00/2.59	   less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y))
7.00/2.59	   U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
7.00/2.59	   U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.59	   delete_in_gga(X, tree(X, void, Right), Right) -> delete_out_gga(X, tree(X, void, Right), Right)
7.00/2.59	   delete_in_gga(X, tree(X, Left, void), Left) -> delete_out_gga(X, tree(X, Left, void), Left)
7.00/2.59	   delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.59	   U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
7.00/2.59	   delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
7.00/2.59	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
7.00/2.59	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
7.00/2.59	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
7.00/2.59	   U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.59	   delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
7.00/2.59	   U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
7.00/2.59	   U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
7.00/2.59	   U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
7.00/2.59	   U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
7.00/2.59	   delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
7.00/2.59	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
7.00/2.59	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
7.00/2.59	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
7.00/2.59	   U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
7.00/2.59	   U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) -> delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
7.00/2.59	
7.00/2.59	The argument filtering Pi contains the following mapping:
7.00/2.59	delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
7.00/2.59	
7.00/2.59	tree(x1, x2, x3)  =  tree(x1, x2, x3)
7.00/2.59	
7.00/2.59	void  =  void
7.00/2.59	
7.00/2.59	delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
7.00/2.59	
7.00/2.59	U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
7.00/2.59	
7.00/2.59	delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
7.00/2.59	
7.00/2.59	delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
7.00/2.59	
7.00/2.59	U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
7.00/2.59	
7.00/2.59	U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
7.00/2.59	
7.00/2.59	less_in_ag(x1, x2)  =  less_in_ag(x2)
7.00/2.59	
7.00/2.59	s(x1)  =  s(x1)
7.00/2.59	
7.00/2.59	less_out_ag(x1, x2)  =  less_out_ag(x1)
7.00/2.59	
7.00/2.59	U7_ag(x1, x2, x3)  =  U7_ag(x3)
7.00/2.59	
7.00/2.59	U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
7.00/2.59	
7.00/2.59	delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
7.00/2.59	
7.00/2.59	delete_out_gga(x1, x2, x3)  =  delete_out_gga
7.00/2.59	
7.00/2.59	U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
7.00/2.59	
7.00/2.59	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
7.00/2.59	
7.00/2.59	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
7.00/2.59	
7.00/2.59	0  =  0
7.00/2.59	
7.00/2.59	less_out_gg(x1, x2)  =  less_out_gg
7.00/2.59	
7.00/2.59	U7_gg(x1, x2, x3)  =  U7_gg(x3)
7.00/2.59	
7.00/2.59	U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
7.00/2.59	
7.00/2.59	U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
7.00/2.59	
7.00/2.59	U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
7.00/2.59	
7.00/2.59	U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
7.00/2.59	
7.00/2.59	less_in_ga(x1, x2)  =  less_in_ga(x1)
7.00/2.59	
7.00/2.59	less_out_ga(x1, x2)  =  less_out_ga
7.00/2.59	
7.00/2.59	U7_ga(x1, x2, x3)  =  U7_ga(x3)
7.00/2.59	
7.00/2.59	U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
7.00/2.59	
7.00/2.59	DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
7.00/2.59	
7.00/2.59	U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x3, x6)
7.00/2.59	
7.00/2.59	U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x4, x6)
7.00/2.59	
7.00/2.59	
7.00/2.59	We have to consider all (P,R,Pi)-chains
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(36) UsableRulesProof (EQUIVALENT)
7.00/2.59	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(37)
7.00/2.59	Obligation:
7.00/2.59	Pi DP problem:
7.00/2.59	The TRS P consists of the following rules:
7.00/2.59	
7.00/2.59	   DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
7.00/2.59	   U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> DELETE_IN_GGA(X, Left, Left1)
7.00/2.59	   DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
7.00/2.59	   U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> DELETE_IN_GGA(X, Right, Right1)
7.00/2.59	
7.00/2.59	The TRS R consists of the following rules:
7.00/2.59	
7.00/2.59	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
7.00/2.59	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
7.00/2.59	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
7.00/2.59	
7.00/2.59	The argument filtering Pi contains the following mapping:
7.00/2.59	tree(x1, x2, x3)  =  tree(x1, x2, x3)
7.00/2.59	
7.00/2.59	s(x1)  =  s(x1)
7.00/2.59	
7.00/2.59	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
7.00/2.59	
7.00/2.59	0  =  0
7.00/2.59	
7.00/2.59	less_out_gg(x1, x2)  =  less_out_gg
7.00/2.59	
7.00/2.59	U7_gg(x1, x2, x3)  =  U7_gg(x3)
7.00/2.59	
7.00/2.59	DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
7.00/2.59	
7.00/2.59	U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x3, x6)
7.00/2.59	
7.00/2.59	U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x4, x6)
7.00/2.59	
7.00/2.59	
7.00/2.59	We have to consider all (P,R,Pi)-chains
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(38) PiDPToQDPProof (SOUND)
7.00/2.59	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(39)
7.00/2.59	Obligation:
7.00/2.59	Q DP problem:
7.00/2.59	The TRS P consists of the following rules:
7.00/2.59	
7.00/2.59	   DELETE_IN_GGA(X, tree(Y, Left, Right)) -> U2_GGA(X, Left, less_in_gg(X, Y))
7.00/2.59	   U2_GGA(X, Left, less_out_gg) -> DELETE_IN_GGA(X, Left)
7.00/2.59	   DELETE_IN_GGA(X, tree(Y, Left, Right)) -> U4_GGA(X, Right, less_in_gg(Y, X))
7.00/2.59	   U4_GGA(X, Right, less_out_gg) -> DELETE_IN_GGA(X, Right)
7.00/2.59	
7.00/2.59	The TRS R consists of the following rules:
7.00/2.59	
7.00/2.59	   less_in_gg(0, s(X3)) -> less_out_gg
7.00/2.59	   less_in_gg(s(X), s(Y)) -> U7_gg(less_in_gg(X, Y))
7.00/2.59	   U7_gg(less_out_gg) -> less_out_gg
7.00/2.59	
7.00/2.59	The set Q consists of the following terms:
7.00/2.59	
7.00/2.59	   less_in_gg(x0, x1)
7.00/2.59	   U7_gg(x0)
7.00/2.59	
7.00/2.59	We have to consider all (P,Q,R)-chains.
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(40) QDPSizeChangeProof (EQUIVALENT)
7.00/2.59	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. 
7.00/2.59	
7.00/2.59	From the DPs we obtained the following set of size-change graphs:
7.00/2.59	*U2_GGA(X, Left, less_out_gg) -> DELETE_IN_GGA(X, Left)
7.00/2.59	The graph contains the following edges 1 >= 1, 2 >= 2
7.00/2.59	
7.00/2.59	
7.00/2.59	*U4_GGA(X, Right, less_out_gg) -> DELETE_IN_GGA(X, Right)
7.00/2.59	The graph contains the following edges 1 >= 1, 2 >= 2
7.00/2.59	
7.00/2.59	
7.00/2.59	*DELETE_IN_GGA(X, tree(Y, Left, Right)) -> U2_GGA(X, Left, less_in_gg(X, Y))
7.00/2.59	The graph contains the following edges 1 >= 1, 2 > 2
7.00/2.59	
7.00/2.59	
7.00/2.59	*DELETE_IN_GGA(X, tree(Y, Left, Right)) -> U4_GGA(X, Right, less_in_gg(Y, X))
7.00/2.59	The graph contains the following edges 1 >= 1, 2 > 2
7.00/2.59	
7.00/2.59	
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(41)
7.00/2.59	YES
7.00/2.59	
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(42)
7.00/2.59	Obligation:
7.00/2.59	Pi DP problem:
7.00/2.59	The TRS P consists of the following rules:
7.00/2.59	
7.00/2.59	   DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X))
7.00/2.59	   U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> DELETE_IN_AGA(X, Right, Right1)
7.00/2.59	
7.00/2.59	The TRS R consists of the following rules:
7.00/2.59	
7.00/2.59	   delete_in_aga(X, tree(X, void, Right), Right) -> delete_out_aga(X, tree(X, void, Right), Right)
7.00/2.59	   delete_in_aga(X, tree(X, Left, void), Left) -> delete_out_aga(X, tree(X, Left, void), Left)
7.00/2.59	   delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.59	   delmin_in_gaa(tree(Y, void, Right), Y, Right) -> delmin_out_gaa(tree(Y, void, Right), Y, Right)
7.00/2.59	   delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
7.00/2.59	   U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) -> delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
7.00/2.59	   U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
7.00/2.59	   delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
7.00/2.59	   less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3))
7.00/2.59	   less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y))
7.00/2.59	   U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
7.00/2.59	   U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.59	   delete_in_gga(X, tree(X, void, Right), Right) -> delete_out_gga(X, tree(X, void, Right), Right)
7.00/2.59	   delete_in_gga(X, tree(X, Left, void), Left) -> delete_out_gga(X, tree(X, Left, void), Left)
7.00/2.59	   delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
7.00/2.59	   U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) -> delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
7.00/2.59	   delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
7.00/2.59	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
7.00/2.59	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
7.00/2.59	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
7.00/2.59	   U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
7.00/2.59	   delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
7.00/2.59	   U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
7.00/2.59	   U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
7.00/2.59	   U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
7.00/2.59	   U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) -> delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
7.00/2.59	   delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
7.00/2.59	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
7.00/2.59	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
7.00/2.59	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
7.00/2.59	   U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
7.00/2.59	   U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) -> delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
7.00/2.59	
7.00/2.59	The argument filtering Pi contains the following mapping:
7.00/2.59	delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
7.00/2.59	
7.00/2.59	tree(x1, x2, x3)  =  tree(x1, x2, x3)
7.00/2.59	
7.00/2.59	void  =  void
7.00/2.59	
7.00/2.59	delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
7.00/2.59	
7.00/2.59	U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
7.00/2.59	
7.00/2.59	delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
7.00/2.59	
7.00/2.59	delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
7.00/2.59	
7.00/2.59	U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
7.00/2.59	
7.00/2.59	U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
7.00/2.59	
7.00/2.59	less_in_ag(x1, x2)  =  less_in_ag(x2)
7.00/2.59	
7.00/2.59	s(x1)  =  s(x1)
7.00/2.59	
7.00/2.59	less_out_ag(x1, x2)  =  less_out_ag(x1)
7.00/2.59	
7.00/2.59	U7_ag(x1, x2, x3)  =  U7_ag(x3)
7.00/2.59	
7.00/2.59	U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
7.00/2.59	
7.00/2.59	delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
7.00/2.59	
7.00/2.59	delete_out_gga(x1, x2, x3)  =  delete_out_gga
7.00/2.59	
7.00/2.59	U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
7.00/2.59	
7.00/2.59	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
7.00/2.59	
7.00/2.59	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
7.00/2.59	
7.00/2.59	0  =  0
7.00/2.59	
7.00/2.59	less_out_gg(x1, x2)  =  less_out_gg
7.00/2.59	
7.00/2.59	U7_gg(x1, x2, x3)  =  U7_gg(x3)
7.00/2.59	
7.00/2.59	U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
7.00/2.59	
7.00/2.59	U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
7.00/2.59	
7.00/2.59	U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
7.00/2.59	
7.00/2.59	U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
7.00/2.59	
7.00/2.59	less_in_ga(x1, x2)  =  less_in_ga(x1)
7.00/2.59	
7.00/2.59	less_out_ga(x1, x2)  =  less_out_ga
7.00/2.59	
7.00/2.59	U7_ga(x1, x2, x3)  =  U7_ga(x3)
7.00/2.59	
7.00/2.59	U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
7.00/2.59	
7.00/2.59	DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
7.00/2.59	
7.00/2.59	U4_AGA(x1, x2, x3, x4, x5, x6)  =  U4_AGA(x4, x6)
7.00/2.59	
7.00/2.59	
7.00/2.59	We have to consider all (P,R,Pi)-chains
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(43) UsableRulesProof (EQUIVALENT)
7.00/2.59	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(44)
7.00/2.59	Obligation:
7.00/2.59	Pi DP problem:
7.00/2.59	The TRS P consists of the following rules:
7.00/2.59	
7.00/2.59	   DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X))
7.00/2.59	   U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> DELETE_IN_AGA(X, Right, Right1)
7.00/2.59	
7.00/2.59	The TRS R consists of the following rules:
7.00/2.59	
7.00/2.59	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
7.00/2.59	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
7.00/2.59	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
7.00/2.59	
7.00/2.59	The argument filtering Pi contains the following mapping:
7.00/2.59	tree(x1, x2, x3)  =  tree(x1, x2, x3)
7.00/2.59	
7.00/2.59	s(x1)  =  s(x1)
7.00/2.59	
7.00/2.59	0  =  0
7.00/2.59	
7.00/2.59	less_in_ga(x1, x2)  =  less_in_ga(x1)
7.00/2.59	
7.00/2.59	less_out_ga(x1, x2)  =  less_out_ga
7.00/2.59	
7.00/2.59	U7_ga(x1, x2, x3)  =  U7_ga(x3)
7.00/2.59	
7.00/2.59	DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
7.00/2.59	
7.00/2.59	U4_AGA(x1, x2, x3, x4, x5, x6)  =  U4_AGA(x4, x6)
7.00/2.59	
7.00/2.59	
7.00/2.59	We have to consider all (P,R,Pi)-chains
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(45) PiDPToQDPProof (SOUND)
7.00/2.59	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(46)
7.00/2.59	Obligation:
7.00/2.59	Q DP problem:
7.00/2.59	The TRS P consists of the following rules:
7.00/2.59	
7.00/2.59	   DELETE_IN_AGA(tree(Y, Left, Right)) -> U4_AGA(Right, less_in_ga(Y))
7.00/2.59	   U4_AGA(Right, less_out_ga) -> DELETE_IN_AGA(Right)
7.00/2.59	
7.00/2.59	The TRS R consists of the following rules:
7.00/2.59	
7.00/2.59	   less_in_ga(0) -> less_out_ga
7.00/2.59	   less_in_ga(s(X)) -> U7_ga(less_in_ga(X))
7.00/2.59	   U7_ga(less_out_ga) -> less_out_ga
7.00/2.59	
7.00/2.59	The set Q consists of the following terms:
7.00/2.59	
7.00/2.59	   less_in_ga(x0)
7.00/2.59	   U7_ga(x0)
7.00/2.59	
7.00/2.59	We have to consider all (P,Q,R)-chains.
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(47) QDPSizeChangeProof (EQUIVALENT)
7.00/2.59	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. 
7.00/2.59	
7.00/2.59	From the DPs we obtained the following set of size-change graphs:
7.00/2.59	*U4_AGA(Right, less_out_ga) -> DELETE_IN_AGA(Right)
7.00/2.59	The graph contains the following edges 1 >= 1
7.00/2.59	
7.00/2.59	
7.00/2.59	*DELETE_IN_AGA(tree(Y, Left, Right)) -> U4_AGA(Right, less_in_ga(Y))
7.00/2.59	The graph contains the following edges 1 > 1
7.00/2.59	
7.00/2.59	
7.00/2.59	----------------------------------------
7.00/2.59	
7.00/2.59	(48)
7.00/2.59	YES
7.23/2.65	EOF