5.90/2.30	YES
5.92/2.33	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
5.92/2.33	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.92/2.33	
5.92/2.33	
5.92/2.33	Left Termination of the query pattern
5.92/2.33	
5.92/2.33	delete(a,a,g)
5.92/2.33	
5.92/2.33	w.r.t. the given Prolog program could successfully be proven:
5.92/2.33	
5.92/2.33	(0) Prolog
5.92/2.33	(1) PrologToPiTRSProof [SOUND, 0 ms]
5.92/2.33	(2) PiTRS
5.92/2.33	(3) DependencyPairsProof [EQUIVALENT, 22 ms]
5.92/2.33	(4) PiDP
5.92/2.33	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
5.92/2.33	(6) AND
5.92/2.33	    (7) PiDP
5.92/2.33	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
5.92/2.33	        (9) PiDP
5.92/2.33	        (10) PiDPToQDPProof [SOUND, 1 ms]
5.92/2.33	        (11) QDP
5.92/2.33	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
5.92/2.33	        (13) YES
5.92/2.33	    (14) PiDP
5.92/2.33	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
5.92/2.33	        (16) PiDP
5.92/2.33	        (17) PiDPToQDPProof [EQUIVALENT, 0 ms]
5.92/2.33	        (18) QDP
5.92/2.33	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
5.92/2.33	        (20) YES
5.92/2.33	    (21) PiDP
5.92/2.33	        (22) UsableRulesProof [EQUIVALENT, 0 ms]
5.92/2.33	        (23) PiDP
5.92/2.33	        (24) PiDPToQDPProof [SOUND, 0 ms]
5.92/2.33	        (25) QDP
5.92/2.33	        (26) QDPSizeChangeProof [EQUIVALENT, 0 ms]
5.92/2.33	        (27) YES
5.92/2.33	    (28) PiDP
5.92/2.33	        (29) UsableRulesProof [EQUIVALENT, 0 ms]
5.92/2.33	        (30) PiDP
5.92/2.33	        (31) PiDPToQDPProof [SOUND, 0 ms]
5.92/2.33	        (32) QDP
5.92/2.33	        (33) QDPSizeChangeProof [EQUIVALENT, 0 ms]
5.92/2.33	        (34) YES
5.92/2.33	    (35) PiDP
5.92/2.33	        (36) UsableRulesProof [EQUIVALENT, 0 ms]
5.92/2.33	        (37) PiDP
5.92/2.33	        (38) PiDPToQDPProof [SOUND, 0 ms]
5.92/2.33	        (39) QDP
5.92/2.33	        (40) QDPSizeChangeProof [EQUIVALENT, 0 ms]
5.92/2.33	        (41) YES
5.92/2.33	    (42) PiDP
5.92/2.33	        (43) UsableRulesProof [EQUIVALENT, 0 ms]
5.92/2.33	        (44) PiDP
5.92/2.33	        (45) PiDPToQDPProof [SOUND, 0 ms]
5.92/2.33	        (46) QDP
5.92/2.33	        (47) QDPSizeChangeProof [EQUIVALENT, 0 ms]
5.92/2.33	        (48) YES
5.92/2.33	
5.92/2.33	
5.92/2.33	----------------------------------------
5.92/2.33	
5.92/2.33	(0)
5.92/2.33	Obligation:
5.92/2.33	Clauses:
5.92/2.33	
5.92/2.33	delete(X, tree(X, void, Right), Right).
5.92/2.33	delete(X, tree(X, Left, void), Left).
5.92/2.33	delete(X, tree(X, Left, Right), tree(Y, Left, Right1)) :- delmin(Right, Y, Right1).
5.92/2.33	delete(X, tree(Y, Left, Right), tree(Y, Left1, Right)) :- ','(less(X, Y), delete(X, Left, Left1)).
5.92/2.33	delete(X, tree(Y, Left, Right), tree(Y, Left, Right1)) :- ','(less(Y, X), delete(X, Right, Right1)).
5.92/2.33	delmin(tree(Y, void, Right), Y, Right).
5.92/2.33	delmin(tree(X, Left, X1), Y, tree(X, Left1, X2)) :- delmin(Left, Y, Left1).
5.92/2.33	less(0, s(X3)).
5.92/2.33	less(s(X), s(Y)) :- less(X, Y).
5.92/2.33	
5.92/2.33	
5.92/2.33	Query: delete(a,a,g)
5.92/2.33	----------------------------------------
5.92/2.33	
5.92/2.33	(1) PrologToPiTRSProof (SOUND)
5.92/2.33	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
5.92/2.33	
5.92/2.33	delete_in_3: (f,f,b) (b,f,b)
5.92/2.33	
5.92/2.33	delmin_in_3: (f,b,b)
5.92/2.33	
5.92/2.33	less_in_2: (f,b) (b,b) (b,f)
5.92/2.33	
5.92/2.33	Transforming Prolog into the following Term Rewriting System:
5.92/2.33	
5.92/2.33	Pi-finite rewrite system:
5.92/2.33	The TRS R consists of the following rules:
5.92/2.33	
5.92/2.33	   delete_in_aag(X, tree(X, void, Right), Right) -> delete_out_aag(X, tree(X, void, Right), Right)
5.92/2.33	   delete_in_aag(X, tree(X, Left, void), Left) -> delete_out_aag(X, tree(X, Left, void), Left)
5.92/2.33	   delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.33	   delmin_in_agg(tree(Y, void, Right), Y, Right) -> delmin_out_agg(tree(Y, void, Right), Y, Right)
5.92/2.33	   delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
5.92/2.33	   U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) -> delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
5.92/2.33	   U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) -> delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
5.92/2.33	   delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
5.92/2.33	   less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3))
5.92/2.33	   less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y))
5.92/2.33	   U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
5.92/2.33	   U2_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.33	   delete_in_gag(X, tree(X, void, Right), Right) -> delete_out_gag(X, tree(X, void, Right), Right)
5.92/2.33	   delete_in_gag(X, tree(X, Left, void), Left) -> delete_out_gag(X, tree(X, Left, void), Left)
5.92/2.33	   delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.33	   U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) -> delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
5.92/2.33	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.92/2.33	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.92/2.33	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
5.92/2.33	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.92/2.33	   U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.33	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.92/2.33	   U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
5.92/2.33	   U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) -> delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
5.92/2.33	   U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) -> delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
5.92/2.33	   U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) -> delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
5.92/2.33	   delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
5.92/2.33	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
5.92/2.33	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
5.92/2.33	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
5.92/2.33	   U4_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
5.92/2.33	   U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) -> delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
5.92/2.33	
5.92/2.33	The argument filtering Pi contains the following mapping:
5.92/2.33	delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
5.92/2.33	
5.92/2.33	delete_out_aag(x1, x2, x3)  =  delete_out_aag
5.92/2.33	
5.92/2.33	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.92/2.33	
5.92/2.33	U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x6)
5.92/2.33	
5.92/2.33	delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
5.92/2.33	
5.92/2.33	delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
5.92/2.33	
5.92/2.33	U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
5.92/2.33	
5.92/2.33	U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x5, x6)
5.92/2.33	
5.92/2.33	less_in_ag(x1, x2)  =  less_in_ag(x2)
5.92/2.33	
5.92/2.33	s(x1)  =  s(x1)
5.92/2.33	
5.92/2.33	less_out_ag(x1, x2)  =  less_out_ag(x1)
5.92/2.33	
5.92/2.33	U7_ag(x1, x2, x3)  =  U7_ag(x3)
5.92/2.33	
5.92/2.33	U3_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x6)
5.92/2.33	
5.92/2.33	delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
5.92/2.33	
5.92/2.33	delete_out_gag(x1, x2, x3)  =  delete_out_gag
5.92/2.33	
5.92/2.33	U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
5.92/2.33	
5.92/2.33	U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
5.92/2.33	
5.92/2.33	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.92/2.33	
5.92/2.33	0  =  0
5.92/2.33	
5.92/2.33	less_out_gg(x1, x2)  =  less_out_gg
5.92/2.33	
5.92/2.33	U7_gg(x1, x2, x3)  =  U7_gg(x3)
5.92/2.33	
5.92/2.33	U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
5.92/2.33	
5.92/2.33	U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
5.92/2.33	
5.92/2.33	U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
5.92/2.33	
5.92/2.33	U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x5, x6)
5.92/2.33	
5.92/2.33	less_in_ga(x1, x2)  =  less_in_ga(x1)
5.92/2.33	
5.92/2.33	less_out_ga(x1, x2)  =  less_out_ga
5.92/2.33	
5.92/2.33	U7_ga(x1, x2, x3)  =  U7_ga(x3)
5.92/2.33	
5.92/2.33	U5_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x6)
5.92/2.33	
5.92/2.33	
5.92/2.33	
5.92/2.33	
5.92/2.33	
5.92/2.33	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
5.92/2.33	
5.92/2.33	
5.92/2.33	
5.92/2.33	----------------------------------------
5.92/2.33	
5.92/2.33	(2)
5.92/2.33	Obligation:
5.92/2.33	Pi-finite rewrite system:
5.92/2.33	The TRS R consists of the following rules:
5.92/2.33	
5.92/2.33	   delete_in_aag(X, tree(X, void, Right), Right) -> delete_out_aag(X, tree(X, void, Right), Right)
5.92/2.33	   delete_in_aag(X, tree(X, Left, void), Left) -> delete_out_aag(X, tree(X, Left, void), Left)
5.92/2.33	   delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.33	   delmin_in_agg(tree(Y, void, Right), Y, Right) -> delmin_out_agg(tree(Y, void, Right), Y, Right)
5.92/2.33	   delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
5.92/2.33	   U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) -> delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
5.92/2.33	   U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) -> delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
5.92/2.33	   delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
5.92/2.33	   less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3))
5.92/2.33	   less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y))
5.92/2.33	   U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
5.92/2.33	   U2_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.33	   delete_in_gag(X, tree(X, void, Right), Right) -> delete_out_gag(X, tree(X, void, Right), Right)
5.92/2.33	   delete_in_gag(X, tree(X, Left, void), Left) -> delete_out_gag(X, tree(X, Left, void), Left)
5.92/2.33	   delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.33	   U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) -> delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
5.92/2.33	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.92/2.33	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.92/2.33	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
5.92/2.33	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.92/2.33	   U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.33	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.92/2.33	   U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
5.92/2.33	   U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) -> delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
5.92/2.33	   U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) -> delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
5.92/2.33	   U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) -> delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
5.92/2.33	   delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
5.92/2.33	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
5.92/2.33	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
5.92/2.33	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
5.92/2.33	   U4_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
5.92/2.33	   U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) -> delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
5.92/2.33	
5.92/2.33	The argument filtering Pi contains the following mapping:
5.92/2.33	delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
5.92/2.33	
5.92/2.33	delete_out_aag(x1, x2, x3)  =  delete_out_aag
5.92/2.33	
5.92/2.33	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.92/2.33	
5.92/2.33	U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x6)
5.92/2.33	
5.92/2.33	delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
5.92/2.33	
5.92/2.33	delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
5.92/2.33	
5.92/2.33	U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
5.92/2.33	
5.92/2.33	U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x5, x6)
5.92/2.33	
5.92/2.33	less_in_ag(x1, x2)  =  less_in_ag(x2)
5.92/2.33	
5.92/2.33	s(x1)  =  s(x1)
5.92/2.33	
5.92/2.33	less_out_ag(x1, x2)  =  less_out_ag(x1)
5.92/2.33	
5.92/2.33	U7_ag(x1, x2, x3)  =  U7_ag(x3)
5.92/2.33	
5.92/2.33	U3_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x6)
5.92/2.33	
5.92/2.33	delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
5.92/2.33	
5.92/2.33	delete_out_gag(x1, x2, x3)  =  delete_out_gag
5.92/2.33	
5.92/2.33	U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
5.92/2.33	
5.92/2.33	U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
5.92/2.33	
5.92/2.33	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.92/2.33	
5.92/2.33	0  =  0
5.92/2.33	
5.92/2.33	less_out_gg(x1, x2)  =  less_out_gg
5.92/2.33	
5.92/2.33	U7_gg(x1, x2, x3)  =  U7_gg(x3)
5.92/2.33	
5.92/2.33	U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
5.92/2.33	
5.92/2.33	U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
5.92/2.33	
5.92/2.33	U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
5.92/2.33	
5.92/2.33	U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x5, x6)
5.92/2.33	
5.92/2.33	less_in_ga(x1, x2)  =  less_in_ga(x1)
5.92/2.33	
5.92/2.33	less_out_ga(x1, x2)  =  less_out_ga
5.92/2.33	
5.92/2.33	U7_ga(x1, x2, x3)  =  U7_ga(x3)
5.92/2.33	
5.92/2.33	U5_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x6)
5.92/2.33	
5.92/2.33	
5.92/2.33	
5.92/2.33	----------------------------------------
5.92/2.33	
5.92/2.33	(3) DependencyPairsProof (EQUIVALENT)
5.92/2.33	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
5.92/2.33	Pi DP problem:
5.92/2.33	The TRS P consists of the following rules:
5.92/2.33	
5.92/2.33	   DELETE_IN_AAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_AAG(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.33	   DELETE_IN_AAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> DELMIN_IN_AGG(Right, Y, Right1)
5.92/2.33	   DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_AGG(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
5.92/2.33	   DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AGG(Left, Y, Left1)
5.92/2.33	   DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_AAG(X, Y, Left, Right, Left1, less_in_ag(X, Y))
5.92/2.33	   DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_AG(X, Y)
5.92/2.33	   LESS_IN_AG(s(X), s(Y)) -> U7_AG(X, Y, less_in_ag(X, Y))
5.92/2.33	   LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y)
5.92/2.33	   U2_AAG(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_AAG(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.33	   U2_AAG(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> DELETE_IN_GAG(X, Left, Left1)
5.92/2.33	   DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_GAG(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.33	   DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> DELMIN_IN_AGG(Right, Y, Right1)
5.92/2.33	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.92/2.33	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GG(X, Y)
5.92/2.33	   LESS_IN_GG(s(X), s(Y)) -> U7_GG(X, Y, less_in_gg(X, Y))
5.92/2.33	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
5.92/2.33	   U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_GAG(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.33	   U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> DELETE_IN_GAG(X, Left, Left1)
5.92/2.33	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.92/2.33	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_GG(Y, X)
5.92/2.33	   U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_GAG(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
5.92/2.33	   U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> DELETE_IN_GAG(X, Right, Right1)
5.92/2.33	   DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_AAG(X, Y, Left, Right, Right1, less_in_ga(Y, X))
5.92/2.33	   DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_GA(Y, X)
5.92/2.33	   LESS_IN_GA(s(X), s(Y)) -> U7_GA(X, Y, less_in_ga(X, Y))
5.92/2.33	   LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y)
5.92/2.33	   U4_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_AAG(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
5.92/2.33	   U4_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> DELETE_IN_AAG(X, Right, Right1)
5.92/2.33	
5.92/2.33	The TRS R consists of the following rules:
5.92/2.33	
5.92/2.33	   delete_in_aag(X, tree(X, void, Right), Right) -> delete_out_aag(X, tree(X, void, Right), Right)
5.92/2.33	   delete_in_aag(X, tree(X, Left, void), Left) -> delete_out_aag(X, tree(X, Left, void), Left)
5.92/2.33	   delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.33	   delmin_in_agg(tree(Y, void, Right), Y, Right) -> delmin_out_agg(tree(Y, void, Right), Y, Right)
5.92/2.33	   delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
5.92/2.33	   U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) -> delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
5.92/2.33	   U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) -> delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
5.92/2.33	   delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
5.92/2.33	   less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3))
5.92/2.33	   less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y))
5.92/2.33	   U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
5.92/2.33	   U2_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.33	   delete_in_gag(X, tree(X, void, Right), Right) -> delete_out_gag(X, tree(X, void, Right), Right)
5.92/2.33	   delete_in_gag(X, tree(X, Left, void), Left) -> delete_out_gag(X, tree(X, Left, void), Left)
5.92/2.33	   delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.33	   U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) -> delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
5.92/2.33	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.92/2.33	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.92/2.33	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
5.92/2.33	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.92/2.33	   U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.33	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.92/2.33	   U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
5.92/2.33	   U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) -> delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
5.92/2.33	   U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) -> delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
5.92/2.33	   U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) -> delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
5.92/2.33	   delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
5.92/2.33	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
5.92/2.33	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
5.92/2.33	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
5.92/2.33	   U4_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
5.92/2.33	   U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) -> delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
5.92/2.33	
5.92/2.33	The argument filtering Pi contains the following mapping:
5.92/2.33	delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
5.92/2.33	
5.92/2.33	delete_out_aag(x1, x2, x3)  =  delete_out_aag
5.92/2.33	
5.92/2.33	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.92/2.33	
5.92/2.33	U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x6)
5.92/2.33	
5.92/2.33	delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
5.92/2.33	
5.92/2.33	delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
5.92/2.33	
5.92/2.33	U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
5.92/2.33	
5.92/2.33	U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x5, x6)
5.92/2.33	
5.92/2.33	less_in_ag(x1, x2)  =  less_in_ag(x2)
5.92/2.33	
5.92/2.33	s(x1)  =  s(x1)
5.92/2.33	
5.92/2.33	less_out_ag(x1, x2)  =  less_out_ag(x1)
5.92/2.33	
5.92/2.33	U7_ag(x1, x2, x3)  =  U7_ag(x3)
5.92/2.33	
5.92/2.33	U3_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x6)
5.92/2.33	
5.92/2.33	delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
5.92/2.33	
5.92/2.33	delete_out_gag(x1, x2, x3)  =  delete_out_gag
5.92/2.33	
5.92/2.33	U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
5.92/2.33	
5.92/2.33	U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
5.92/2.33	
5.92/2.33	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.92/2.33	
5.92/2.33	0  =  0
5.92/2.33	
5.92/2.33	less_out_gg(x1, x2)  =  less_out_gg
5.92/2.33	
5.92/2.33	U7_gg(x1, x2, x3)  =  U7_gg(x3)
5.92/2.33	
5.92/2.33	U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
5.92/2.33	
5.92/2.33	U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
5.92/2.33	
5.92/2.33	U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
5.92/2.33	
5.92/2.33	U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x5, x6)
5.92/2.33	
5.92/2.33	less_in_ga(x1, x2)  =  less_in_ga(x1)
5.92/2.33	
5.92/2.33	less_out_ga(x1, x2)  =  less_out_ga
5.92/2.33	
5.92/2.33	U7_ga(x1, x2, x3)  =  U7_ga(x3)
5.92/2.33	
5.92/2.33	U5_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x6)
5.92/2.33	
5.92/2.33	DELETE_IN_AAG(x1, x2, x3)  =  DELETE_IN_AAG(x3)
5.92/2.33	
5.92/2.33	U1_AAG(x1, x2, x3, x4, x5, x6)  =  U1_AAG(x6)
5.92/2.33	
5.92/2.33	DELMIN_IN_AGG(x1, x2, x3)  =  DELMIN_IN_AGG(x2, x3)
5.92/2.33	
5.92/2.33	U6_AGG(x1, x2, x3, x4, x5, x6, x7)  =  U6_AGG(x7)
5.92/2.33	
5.92/2.33	U2_AAG(x1, x2, x3, x4, x5, x6)  =  U2_AAG(x5, x6)
5.92/2.33	
5.92/2.33	LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
5.92/2.33	
5.92/2.33	U7_AG(x1, x2, x3)  =  U7_AG(x3)
5.92/2.33	
5.92/2.33	U3_AAG(x1, x2, x3, x4, x5, x6)  =  U3_AAG(x6)
5.92/2.33	
5.92/2.33	DELETE_IN_GAG(x1, x2, x3)  =  DELETE_IN_GAG(x1, x3)
5.92/2.33	
5.92/2.33	U1_GAG(x1, x2, x3, x4, x5, x6)  =  U1_GAG(x6)
5.92/2.33	
5.92/2.33	U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x1, x5, x6)
5.92/2.33	
5.92/2.33	LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
5.92/2.33	
5.92/2.33	U7_GG(x1, x2, x3)  =  U7_GG(x3)
5.92/2.33	
5.92/2.33	U3_GAG(x1, x2, x3, x4, x5, x6)  =  U3_GAG(x6)
5.92/2.33	
5.92/2.33	U4_GAG(x1, x2, x3, x4, x5, x6)  =  U4_GAG(x1, x5, x6)
5.92/2.33	
5.92/2.33	U5_GAG(x1, x2, x3, x4, x5, x6)  =  U5_GAG(x6)
5.92/2.33	
5.92/2.33	U4_AAG(x1, x2, x3, x4, x5, x6)  =  U4_AAG(x5, x6)
5.92/2.33	
5.92/2.33	LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
5.92/2.33	
5.92/2.33	U7_GA(x1, x2, x3)  =  U7_GA(x3)
5.92/2.33	
5.92/2.33	U5_AAG(x1, x2, x3, x4, x5, x6)  =  U5_AAG(x6)
5.92/2.33	
5.92/2.33	
5.92/2.33	We have to consider all (P,R,Pi)-chains
5.92/2.33	----------------------------------------
5.92/2.33	
5.92/2.33	(4)
5.92/2.33	Obligation:
5.92/2.33	Pi DP problem:
5.92/2.33	The TRS P consists of the following rules:
5.92/2.33	
5.92/2.33	   DELETE_IN_AAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_AAG(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.33	   DELETE_IN_AAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> DELMIN_IN_AGG(Right, Y, Right1)
5.92/2.33	   DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_AGG(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
5.92/2.33	   DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AGG(Left, Y, Left1)
5.92/2.33	   DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_AAG(X, Y, Left, Right, Left1, less_in_ag(X, Y))
5.92/2.33	   DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_AG(X, Y)
5.92/2.33	   LESS_IN_AG(s(X), s(Y)) -> U7_AG(X, Y, less_in_ag(X, Y))
5.92/2.33	   LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y)
5.92/2.33	   U2_AAG(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_AAG(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.33	   U2_AAG(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> DELETE_IN_GAG(X, Left, Left1)
5.92/2.33	   DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_GAG(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.33	   DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> DELMIN_IN_AGG(Right, Y, Right1)
5.92/2.33	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.92/2.33	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GG(X, Y)
5.92/2.33	   LESS_IN_GG(s(X), s(Y)) -> U7_GG(X, Y, less_in_gg(X, Y))
5.92/2.33	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
5.92/2.33	   U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_GAG(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.33	   U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> DELETE_IN_GAG(X, Left, Left1)
5.92/2.33	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.92/2.33	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_GG(Y, X)
5.92/2.33	   U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_GAG(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
5.92/2.33	   U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> DELETE_IN_GAG(X, Right, Right1)
5.92/2.33	   DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_AAG(X, Y, Left, Right, Right1, less_in_ga(Y, X))
5.92/2.33	   DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_GA(Y, X)
5.92/2.33	   LESS_IN_GA(s(X), s(Y)) -> U7_GA(X, Y, less_in_ga(X, Y))
5.92/2.33	   LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y)
5.92/2.33	   U4_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_AAG(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
5.92/2.33	   U4_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> DELETE_IN_AAG(X, Right, Right1)
5.92/2.33	
5.92/2.33	The TRS R consists of the following rules:
5.92/2.33	
5.92/2.33	   delete_in_aag(X, tree(X, void, Right), Right) -> delete_out_aag(X, tree(X, void, Right), Right)
5.92/2.33	   delete_in_aag(X, tree(X, Left, void), Left) -> delete_out_aag(X, tree(X, Left, void), Left)
5.92/2.33	   delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.33	   delmin_in_agg(tree(Y, void, Right), Y, Right) -> delmin_out_agg(tree(Y, void, Right), Y, Right)
5.92/2.33	   delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
5.92/2.33	   U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) -> delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
5.92/2.33	   U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) -> delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
5.92/2.33	   delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
5.92/2.33	   less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3))
5.92/2.33	   less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y))
5.92/2.33	   U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
5.92/2.33	   U2_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.33	   delete_in_gag(X, tree(X, void, Right), Right) -> delete_out_gag(X, tree(X, void, Right), Right)
5.92/2.33	   delete_in_gag(X, tree(X, Left, void), Left) -> delete_out_gag(X, tree(X, Left, void), Left)
5.92/2.33	   delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.33	   U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) -> delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
5.92/2.33	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.92/2.33	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.92/2.33	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
5.92/2.33	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.92/2.33	   U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.33	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.92/2.33	   U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
5.92/2.33	   U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) -> delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
5.92/2.33	   U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) -> delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
5.92/2.33	   U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) -> delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
5.92/2.33	   delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
5.92/2.33	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
5.92/2.33	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
5.92/2.33	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
5.92/2.33	   U4_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
5.92/2.33	   U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) -> delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
5.92/2.33	
5.92/2.33	The argument filtering Pi contains the following mapping:
5.92/2.33	delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
5.92/2.33	
5.92/2.33	delete_out_aag(x1, x2, x3)  =  delete_out_aag
5.92/2.33	
5.92/2.33	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.92/2.33	
5.92/2.33	U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x6)
5.92/2.33	
5.92/2.33	delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
5.92/2.33	
5.92/2.33	delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
5.92/2.33	
5.92/2.33	U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
5.92/2.33	
5.92/2.33	U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x5, x6)
5.92/2.33	
5.92/2.33	less_in_ag(x1, x2)  =  less_in_ag(x2)
5.92/2.33	
5.92/2.33	s(x1)  =  s(x1)
5.92/2.33	
5.92/2.33	less_out_ag(x1, x2)  =  less_out_ag(x1)
5.92/2.33	
5.92/2.33	U7_ag(x1, x2, x3)  =  U7_ag(x3)
5.92/2.33	
5.92/2.33	U3_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x6)
5.92/2.33	
5.92/2.33	delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
5.92/2.33	
5.92/2.33	delete_out_gag(x1, x2, x3)  =  delete_out_gag
5.92/2.33	
5.92/2.33	U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
5.92/2.33	
5.92/2.33	U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
5.92/2.33	
5.92/2.33	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.92/2.33	
5.92/2.33	0  =  0
5.92/2.33	
5.92/2.33	less_out_gg(x1, x2)  =  less_out_gg
5.92/2.33	
5.92/2.33	U7_gg(x1, x2, x3)  =  U7_gg(x3)
5.92/2.33	
5.92/2.33	U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
5.92/2.33	
5.92/2.33	U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
5.92/2.33	
5.92/2.33	U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
5.92/2.33	
5.92/2.33	U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x5, x6)
5.92/2.33	
5.92/2.33	less_in_ga(x1, x2)  =  less_in_ga(x1)
5.92/2.33	
5.92/2.33	less_out_ga(x1, x2)  =  less_out_ga
5.92/2.33	
5.92/2.33	U7_ga(x1, x2, x3)  =  U7_ga(x3)
5.92/2.33	
5.92/2.33	U5_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x6)
5.92/2.33	
5.92/2.33	DELETE_IN_AAG(x1, x2, x3)  =  DELETE_IN_AAG(x3)
5.92/2.33	
5.92/2.33	U1_AAG(x1, x2, x3, x4, x5, x6)  =  U1_AAG(x6)
5.92/2.33	
5.92/2.33	DELMIN_IN_AGG(x1, x2, x3)  =  DELMIN_IN_AGG(x2, x3)
5.92/2.33	
5.92/2.33	U6_AGG(x1, x2, x3, x4, x5, x6, x7)  =  U6_AGG(x7)
5.92/2.33	
5.92/2.33	U2_AAG(x1, x2, x3, x4, x5, x6)  =  U2_AAG(x5, x6)
5.92/2.33	
5.92/2.33	LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
5.92/2.33	
5.92/2.33	U7_AG(x1, x2, x3)  =  U7_AG(x3)
5.92/2.33	
5.92/2.33	U3_AAG(x1, x2, x3, x4, x5, x6)  =  U3_AAG(x6)
5.92/2.33	
5.92/2.33	DELETE_IN_GAG(x1, x2, x3)  =  DELETE_IN_GAG(x1, x3)
5.92/2.33	
5.92/2.33	U1_GAG(x1, x2, x3, x4, x5, x6)  =  U1_GAG(x6)
5.92/2.33	
5.92/2.33	U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x1, x5, x6)
5.92/2.33	
5.92/2.33	LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
5.92/2.33	
5.92/2.33	U7_GG(x1, x2, x3)  =  U7_GG(x3)
5.92/2.33	
5.92/2.33	U3_GAG(x1, x2, x3, x4, x5, x6)  =  U3_GAG(x6)
5.92/2.33	
5.92/2.33	U4_GAG(x1, x2, x3, x4, x5, x6)  =  U4_GAG(x1, x5, x6)
5.92/2.33	
5.92/2.33	U5_GAG(x1, x2, x3, x4, x5, x6)  =  U5_GAG(x6)
5.92/2.33	
5.92/2.33	U4_AAG(x1, x2, x3, x4, x5, x6)  =  U4_AAG(x5, x6)
5.92/2.33	
5.92/2.33	LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
5.92/2.33	
5.92/2.33	U7_GA(x1, x2, x3)  =  U7_GA(x3)
5.92/2.33	
5.92/2.33	U5_AAG(x1, x2, x3, x4, x5, x6)  =  U5_AAG(x6)
5.92/2.33	
5.92/2.33	
5.92/2.33	We have to consider all (P,R,Pi)-chains
5.92/2.33	----------------------------------------
5.92/2.33	
5.92/2.33	(5) DependencyGraphProof (EQUIVALENT)
5.92/2.33	The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 18 less nodes.
5.92/2.33	----------------------------------------
5.92/2.33	
5.92/2.33	(6)
5.92/2.33	Complex Obligation (AND)
5.92/2.33	
5.92/2.33	----------------------------------------
5.92/2.33	
5.92/2.33	(7)
5.92/2.33	Obligation:
5.92/2.33	Pi DP problem:
5.92/2.33	The TRS P consists of the following rules:
5.92/2.33	
5.92/2.33	   LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y)
5.92/2.33	
5.92/2.33	The TRS R consists of the following rules:
5.92/2.33	
5.92/2.33	   delete_in_aag(X, tree(X, void, Right), Right) -> delete_out_aag(X, tree(X, void, Right), Right)
5.92/2.33	   delete_in_aag(X, tree(X, Left, void), Left) -> delete_out_aag(X, tree(X, Left, void), Left)
5.92/2.33	   delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.33	   delmin_in_agg(tree(Y, void, Right), Y, Right) -> delmin_out_agg(tree(Y, void, Right), Y, Right)
5.92/2.33	   delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
5.92/2.33	   U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) -> delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
5.92/2.33	   U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) -> delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
5.92/2.33	   delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
5.92/2.33	   less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3))
5.92/2.33	   less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y))
5.92/2.33	   U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
5.92/2.33	   U2_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.33	   delete_in_gag(X, tree(X, void, Right), Right) -> delete_out_gag(X, tree(X, void, Right), Right)
5.92/2.33	   delete_in_gag(X, tree(X, Left, void), Left) -> delete_out_gag(X, tree(X, Left, void), Left)
5.92/2.33	   delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.33	   U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) -> delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
5.92/2.33	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.92/2.33	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.92/2.33	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
5.92/2.33	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.92/2.33	   U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.33	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.92/2.33	   U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
5.92/2.33	   U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) -> delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
5.92/2.33	   U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) -> delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
5.92/2.33	   U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) -> delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
5.92/2.33	   delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
5.92/2.33	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
5.92/2.33	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
5.92/2.33	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
5.92/2.33	   U4_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
5.92/2.33	   U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) -> delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
5.92/2.34	
5.92/2.34	The argument filtering Pi contains the following mapping:
5.92/2.34	delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
5.92/2.34	
5.92/2.34	delete_out_aag(x1, x2, x3)  =  delete_out_aag
5.92/2.34	
5.92/2.34	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.92/2.34	
5.92/2.34	U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x6)
5.92/2.34	
5.92/2.34	delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
5.92/2.34	
5.92/2.34	delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
5.92/2.34	
5.92/2.34	U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
5.92/2.34	
5.92/2.34	U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x5, x6)
5.92/2.34	
5.92/2.34	less_in_ag(x1, x2)  =  less_in_ag(x2)
5.92/2.34	
5.92/2.34	s(x1)  =  s(x1)
5.92/2.34	
5.92/2.34	less_out_ag(x1, x2)  =  less_out_ag(x1)
5.92/2.34	
5.92/2.34	U7_ag(x1, x2, x3)  =  U7_ag(x3)
5.92/2.34	
5.92/2.34	U3_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x6)
5.92/2.34	
5.92/2.34	delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
5.92/2.34	
5.92/2.34	delete_out_gag(x1, x2, x3)  =  delete_out_gag
5.92/2.34	
5.92/2.34	U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
5.92/2.34	
5.92/2.34	U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
5.92/2.34	
5.92/2.34	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.92/2.34	
5.92/2.34	0  =  0
5.92/2.34	
5.92/2.34	less_out_gg(x1, x2)  =  less_out_gg
5.92/2.34	
5.92/2.34	U7_gg(x1, x2, x3)  =  U7_gg(x3)
5.92/2.34	
5.92/2.34	U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
5.92/2.34	
5.92/2.34	U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
5.92/2.34	
5.92/2.34	U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
5.92/2.34	
5.92/2.34	U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x5, x6)
5.92/2.34	
5.92/2.34	less_in_ga(x1, x2)  =  less_in_ga(x1)
5.92/2.34	
5.92/2.34	less_out_ga(x1, x2)  =  less_out_ga
5.92/2.34	
5.92/2.34	U7_ga(x1, x2, x3)  =  U7_ga(x3)
5.92/2.34	
5.92/2.34	U5_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x6)
5.92/2.34	
5.92/2.34	LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
5.92/2.34	
5.92/2.34	
5.92/2.34	We have to consider all (P,R,Pi)-chains
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(8) UsableRulesProof (EQUIVALENT)
5.92/2.34	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(9)
5.92/2.34	Obligation:
5.92/2.34	Pi DP problem:
5.92/2.34	The TRS P consists of the following rules:
5.92/2.34	
5.92/2.34	   LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y)
5.92/2.34	
5.92/2.34	R is empty.
5.92/2.34	The argument filtering Pi contains the following mapping:
5.92/2.34	s(x1)  =  s(x1)
5.92/2.34	
5.92/2.34	LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
5.92/2.34	
5.92/2.34	
5.92/2.34	We have to consider all (P,R,Pi)-chains
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(10) PiDPToQDPProof (SOUND)
5.92/2.34	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(11)
5.92/2.34	Obligation:
5.92/2.34	Q DP problem:
5.92/2.34	The TRS P consists of the following rules:
5.92/2.34	
5.92/2.34	   LESS_IN_GA(s(X)) -> LESS_IN_GA(X)
5.92/2.34	
5.92/2.34	R is empty.
5.92/2.34	Q is empty.
5.92/2.34	We have to consider all (P,Q,R)-chains.
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(12) QDPSizeChangeProof (EQUIVALENT)
5.92/2.34	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. 
5.92/2.34	
5.92/2.34	From the DPs we obtained the following set of size-change graphs:
5.92/2.34	*LESS_IN_GA(s(X)) -> LESS_IN_GA(X)
5.92/2.34	The graph contains the following edges 1 > 1
5.92/2.34	
5.92/2.34	
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(13)
5.92/2.34	YES
5.92/2.34	
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(14)
5.92/2.34	Obligation:
5.92/2.34	Pi DP problem:
5.92/2.34	The TRS P consists of the following rules:
5.92/2.34	
5.92/2.34	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
5.92/2.34	
5.92/2.34	The TRS R consists of the following rules:
5.92/2.34	
5.92/2.34	   delete_in_aag(X, tree(X, void, Right), Right) -> delete_out_aag(X, tree(X, void, Right), Right)
5.92/2.34	   delete_in_aag(X, tree(X, Left, void), Left) -> delete_out_aag(X, tree(X, Left, void), Left)
5.92/2.34	   delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.34	   delmin_in_agg(tree(Y, void, Right), Y, Right) -> delmin_out_agg(tree(Y, void, Right), Y, Right)
5.92/2.34	   delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
5.92/2.34	   U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) -> delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
5.92/2.34	   U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) -> delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
5.92/2.34	   delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
5.92/2.34	   less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3))
5.92/2.34	   less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y))
5.92/2.34	   U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
5.92/2.34	   U2_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.34	   delete_in_gag(X, tree(X, void, Right), Right) -> delete_out_gag(X, tree(X, void, Right), Right)
5.92/2.34	   delete_in_gag(X, tree(X, Left, void), Left) -> delete_out_gag(X, tree(X, Left, void), Left)
5.92/2.34	   delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.34	   U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) -> delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
5.92/2.34	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.92/2.34	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.92/2.34	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
5.92/2.34	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.92/2.34	   U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.34	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.92/2.34	   U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
5.92/2.34	   U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) -> delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
5.92/2.34	   U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) -> delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
5.92/2.34	   U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) -> delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
5.92/2.34	   delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
5.92/2.34	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
5.92/2.34	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
5.92/2.34	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
5.92/2.34	   U4_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
5.92/2.34	   U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) -> delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
5.92/2.34	
5.92/2.34	The argument filtering Pi contains the following mapping:
5.92/2.34	delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
5.92/2.34	
5.92/2.34	delete_out_aag(x1, x2, x3)  =  delete_out_aag
5.92/2.34	
5.92/2.34	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.92/2.34	
5.92/2.34	U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x6)
5.92/2.34	
5.92/2.34	delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
5.92/2.34	
5.92/2.34	delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
5.92/2.34	
5.92/2.34	U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
5.92/2.34	
5.92/2.34	U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x5, x6)
5.92/2.34	
5.92/2.34	less_in_ag(x1, x2)  =  less_in_ag(x2)
5.92/2.34	
5.92/2.34	s(x1)  =  s(x1)
5.92/2.34	
5.92/2.34	less_out_ag(x1, x2)  =  less_out_ag(x1)
5.92/2.34	
5.92/2.34	U7_ag(x1, x2, x3)  =  U7_ag(x3)
5.92/2.34	
5.92/2.34	U3_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x6)
5.92/2.34	
5.92/2.34	delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
5.92/2.34	
5.92/2.34	delete_out_gag(x1, x2, x3)  =  delete_out_gag
5.92/2.34	
5.92/2.34	U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
5.92/2.34	
5.92/2.34	U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
5.92/2.34	
5.92/2.34	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.92/2.34	
5.92/2.34	0  =  0
5.92/2.34	
5.92/2.34	less_out_gg(x1, x2)  =  less_out_gg
5.92/2.34	
5.92/2.34	U7_gg(x1, x2, x3)  =  U7_gg(x3)
5.92/2.34	
5.92/2.34	U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
5.92/2.34	
5.92/2.34	U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
5.92/2.34	
5.92/2.34	U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
5.92/2.34	
5.92/2.34	U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x5, x6)
5.92/2.34	
5.92/2.34	less_in_ga(x1, x2)  =  less_in_ga(x1)
5.92/2.34	
5.92/2.34	less_out_ga(x1, x2)  =  less_out_ga
5.92/2.34	
5.92/2.34	U7_ga(x1, x2, x3)  =  U7_ga(x3)
5.92/2.34	
5.92/2.34	U5_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x6)
5.92/2.34	
5.92/2.34	LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
5.92/2.34	
5.92/2.34	
5.92/2.34	We have to consider all (P,R,Pi)-chains
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(15) UsableRulesProof (EQUIVALENT)
5.92/2.34	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(16)
5.92/2.34	Obligation:
5.92/2.34	Pi DP problem:
5.92/2.34	The TRS P consists of the following rules:
5.92/2.34	
5.92/2.34	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
5.92/2.34	
5.92/2.34	R is empty.
5.92/2.34	Pi is empty.
5.92/2.34	We have to consider all (P,R,Pi)-chains
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(17) PiDPToQDPProof (EQUIVALENT)
5.92/2.34	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(18)
5.92/2.34	Obligation:
5.92/2.34	Q DP problem:
5.92/2.34	The TRS P consists of the following rules:
5.92/2.34	
5.92/2.34	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
5.92/2.34	
5.92/2.34	R is empty.
5.92/2.34	Q is empty.
5.92/2.34	We have to consider all (P,Q,R)-chains.
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(19) QDPSizeChangeProof (EQUIVALENT)
5.92/2.34	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. 
5.92/2.34	
5.92/2.34	From the DPs we obtained the following set of size-change graphs:
5.92/2.34	*LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
5.92/2.34	The graph contains the following edges 1 > 1, 2 > 2
5.92/2.34	
5.92/2.34	
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(20)
5.92/2.34	YES
5.92/2.34	
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(21)
5.92/2.34	Obligation:
5.92/2.34	Pi DP problem:
5.92/2.34	The TRS P consists of the following rules:
5.92/2.34	
5.92/2.34	   LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y)
5.92/2.34	
5.92/2.34	The TRS R consists of the following rules:
5.92/2.34	
5.92/2.34	   delete_in_aag(X, tree(X, void, Right), Right) -> delete_out_aag(X, tree(X, void, Right), Right)
5.92/2.34	   delete_in_aag(X, tree(X, Left, void), Left) -> delete_out_aag(X, tree(X, Left, void), Left)
5.92/2.34	   delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.34	   delmin_in_agg(tree(Y, void, Right), Y, Right) -> delmin_out_agg(tree(Y, void, Right), Y, Right)
5.92/2.34	   delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
5.92/2.34	   U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) -> delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
5.92/2.34	   U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) -> delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
5.92/2.34	   delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
5.92/2.34	   less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3))
5.92/2.34	   less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y))
5.92/2.34	   U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
5.92/2.34	   U2_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.34	   delete_in_gag(X, tree(X, void, Right), Right) -> delete_out_gag(X, tree(X, void, Right), Right)
5.92/2.34	   delete_in_gag(X, tree(X, Left, void), Left) -> delete_out_gag(X, tree(X, Left, void), Left)
5.92/2.34	   delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.34	   U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) -> delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
5.92/2.34	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.92/2.34	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.92/2.34	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
5.92/2.34	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.92/2.34	   U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.34	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.92/2.34	   U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
5.92/2.34	   U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) -> delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
5.92/2.34	   U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) -> delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
5.92/2.34	   U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) -> delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
5.92/2.34	   delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
5.92/2.34	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
5.92/2.34	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
5.92/2.34	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
5.92/2.34	   U4_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
5.92/2.34	   U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) -> delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
5.92/2.34	
5.92/2.34	The argument filtering Pi contains the following mapping:
5.92/2.34	delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
5.92/2.34	
5.92/2.34	delete_out_aag(x1, x2, x3)  =  delete_out_aag
5.92/2.34	
5.92/2.34	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.92/2.34	
5.92/2.34	U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x6)
5.92/2.34	
5.92/2.34	delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
5.92/2.34	
5.92/2.34	delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
5.92/2.34	
5.92/2.34	U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
5.92/2.34	
5.92/2.34	U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x5, x6)
5.92/2.34	
5.92/2.34	less_in_ag(x1, x2)  =  less_in_ag(x2)
5.92/2.34	
5.92/2.34	s(x1)  =  s(x1)
5.92/2.34	
5.92/2.34	less_out_ag(x1, x2)  =  less_out_ag(x1)
5.92/2.34	
5.92/2.34	U7_ag(x1, x2, x3)  =  U7_ag(x3)
5.92/2.34	
5.92/2.34	U3_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x6)
5.92/2.34	
5.92/2.34	delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
5.92/2.34	
5.92/2.34	delete_out_gag(x1, x2, x3)  =  delete_out_gag
5.92/2.34	
5.92/2.34	U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
5.92/2.34	
5.92/2.34	U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
5.92/2.34	
5.92/2.34	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.92/2.34	
5.92/2.34	0  =  0
5.92/2.34	
5.92/2.34	less_out_gg(x1, x2)  =  less_out_gg
5.92/2.34	
5.92/2.34	U7_gg(x1, x2, x3)  =  U7_gg(x3)
5.92/2.34	
5.92/2.34	U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
5.92/2.34	
5.92/2.34	U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
5.92/2.34	
5.92/2.34	U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
5.92/2.34	
5.92/2.34	U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x5, x6)
5.92/2.34	
5.92/2.34	less_in_ga(x1, x2)  =  less_in_ga(x1)
5.92/2.34	
5.92/2.34	less_out_ga(x1, x2)  =  less_out_ga
5.92/2.34	
5.92/2.34	U7_ga(x1, x2, x3)  =  U7_ga(x3)
5.92/2.34	
5.92/2.34	U5_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x6)
5.92/2.34	
5.92/2.34	LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
5.92/2.34	
5.92/2.34	
5.92/2.34	We have to consider all (P,R,Pi)-chains
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(22) UsableRulesProof (EQUIVALENT)
5.92/2.34	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(23)
5.92/2.34	Obligation:
5.92/2.34	Pi DP problem:
5.92/2.34	The TRS P consists of the following rules:
5.92/2.34	
5.92/2.34	   LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y)
5.92/2.34	
5.92/2.34	R is empty.
5.92/2.34	The argument filtering Pi contains the following mapping:
5.92/2.34	s(x1)  =  s(x1)
5.92/2.34	
5.92/2.34	LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
5.92/2.34	
5.92/2.34	
5.92/2.34	We have to consider all (P,R,Pi)-chains
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(24) PiDPToQDPProof (SOUND)
5.92/2.34	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(25)
5.92/2.34	Obligation:
5.92/2.34	Q DP problem:
5.92/2.34	The TRS P consists of the following rules:
5.92/2.34	
5.92/2.34	   LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y)
5.92/2.34	
5.92/2.34	R is empty.
5.92/2.34	Q is empty.
5.92/2.34	We have to consider all (P,Q,R)-chains.
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(26) QDPSizeChangeProof (EQUIVALENT)
5.92/2.34	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. 
5.92/2.34	
5.92/2.34	From the DPs we obtained the following set of size-change graphs:
5.92/2.34	*LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y)
5.92/2.34	The graph contains the following edges 1 > 1
5.92/2.34	
5.92/2.34	
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(27)
5.92/2.34	YES
5.92/2.34	
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(28)
5.92/2.34	Obligation:
5.92/2.34	Pi DP problem:
5.92/2.34	The TRS P consists of the following rules:
5.92/2.34	
5.92/2.34	   DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AGG(Left, Y, Left1)
5.92/2.34	
5.92/2.34	The TRS R consists of the following rules:
5.92/2.34	
5.92/2.34	   delete_in_aag(X, tree(X, void, Right), Right) -> delete_out_aag(X, tree(X, void, Right), Right)
5.92/2.34	   delete_in_aag(X, tree(X, Left, void), Left) -> delete_out_aag(X, tree(X, Left, void), Left)
5.92/2.34	   delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.34	   delmin_in_agg(tree(Y, void, Right), Y, Right) -> delmin_out_agg(tree(Y, void, Right), Y, Right)
5.92/2.34	   delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
5.92/2.34	   U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) -> delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
5.92/2.34	   U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) -> delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
5.92/2.34	   delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
5.92/2.34	   less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3))
5.92/2.34	   less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y))
5.92/2.34	   U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
5.92/2.34	   U2_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.34	   delete_in_gag(X, tree(X, void, Right), Right) -> delete_out_gag(X, tree(X, void, Right), Right)
5.92/2.34	   delete_in_gag(X, tree(X, Left, void), Left) -> delete_out_gag(X, tree(X, Left, void), Left)
5.92/2.34	   delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.34	   U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) -> delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
5.92/2.34	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.92/2.34	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.92/2.34	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
5.92/2.34	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.92/2.34	   U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.34	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.92/2.34	   U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
5.92/2.34	   U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) -> delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
5.92/2.34	   U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) -> delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
5.92/2.34	   U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) -> delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
5.92/2.34	   delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
5.92/2.34	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
5.92/2.34	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
5.92/2.34	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
5.92/2.34	   U4_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
5.92/2.34	   U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) -> delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
5.92/2.34	
5.92/2.34	The argument filtering Pi contains the following mapping:
5.92/2.34	delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
5.92/2.34	
5.92/2.34	delete_out_aag(x1, x2, x3)  =  delete_out_aag
5.92/2.34	
5.92/2.34	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.92/2.34	
5.92/2.34	U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x6)
5.92/2.34	
5.92/2.34	delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
5.92/2.34	
5.92/2.34	delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
5.92/2.34	
5.92/2.34	U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
5.92/2.34	
5.92/2.34	U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x5, x6)
5.92/2.34	
5.92/2.34	less_in_ag(x1, x2)  =  less_in_ag(x2)
5.92/2.34	
5.92/2.34	s(x1)  =  s(x1)
5.92/2.34	
5.92/2.34	less_out_ag(x1, x2)  =  less_out_ag(x1)
5.92/2.34	
5.92/2.34	U7_ag(x1, x2, x3)  =  U7_ag(x3)
5.92/2.34	
5.92/2.34	U3_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x6)
5.92/2.34	
5.92/2.34	delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
5.92/2.34	
5.92/2.34	delete_out_gag(x1, x2, x3)  =  delete_out_gag
5.92/2.34	
5.92/2.34	U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
5.92/2.34	
5.92/2.34	U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
5.92/2.34	
5.92/2.34	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.92/2.34	
5.92/2.34	0  =  0
5.92/2.34	
5.92/2.34	less_out_gg(x1, x2)  =  less_out_gg
5.92/2.34	
5.92/2.34	U7_gg(x1, x2, x3)  =  U7_gg(x3)
5.92/2.34	
5.92/2.34	U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
5.92/2.34	
5.92/2.34	U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
5.92/2.34	
5.92/2.34	U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
5.92/2.34	
5.92/2.34	U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x5, x6)
5.92/2.34	
5.92/2.34	less_in_ga(x1, x2)  =  less_in_ga(x1)
5.92/2.34	
5.92/2.34	less_out_ga(x1, x2)  =  less_out_ga
5.92/2.34	
5.92/2.34	U7_ga(x1, x2, x3)  =  U7_ga(x3)
5.92/2.34	
5.92/2.34	U5_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x6)
5.92/2.34	
5.92/2.34	DELMIN_IN_AGG(x1, x2, x3)  =  DELMIN_IN_AGG(x2, x3)
5.92/2.34	
5.92/2.34	
5.92/2.34	We have to consider all (P,R,Pi)-chains
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(29) UsableRulesProof (EQUIVALENT)
5.92/2.34	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(30)
5.92/2.34	Obligation:
5.92/2.34	Pi DP problem:
5.92/2.34	The TRS P consists of the following rules:
5.92/2.34	
5.92/2.34	   DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AGG(Left, Y, Left1)
5.92/2.34	
5.92/2.34	R is empty.
5.92/2.34	The argument filtering Pi contains the following mapping:
5.92/2.34	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.92/2.34	
5.92/2.34	DELMIN_IN_AGG(x1, x2, x3)  =  DELMIN_IN_AGG(x2, x3)
5.92/2.34	
5.92/2.34	
5.92/2.34	We have to consider all (P,R,Pi)-chains
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(31) PiDPToQDPProof (SOUND)
5.92/2.34	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(32)
5.92/2.34	Obligation:
5.92/2.34	Q DP problem:
5.92/2.34	The TRS P consists of the following rules:
5.92/2.34	
5.92/2.34	   DELMIN_IN_AGG(Y, tree(X, Left1, X2)) -> DELMIN_IN_AGG(Y, Left1)
5.92/2.34	
5.92/2.34	R is empty.
5.92/2.34	Q is empty.
5.92/2.34	We have to consider all (P,Q,R)-chains.
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(33) QDPSizeChangeProof (EQUIVALENT)
5.92/2.34	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. 
5.92/2.34	
5.92/2.34	From the DPs we obtained the following set of size-change graphs:
5.92/2.34	*DELMIN_IN_AGG(Y, tree(X, Left1, X2)) -> DELMIN_IN_AGG(Y, Left1)
5.92/2.34	The graph contains the following edges 1 >= 1, 2 > 2
5.92/2.34	
5.92/2.34	
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(34)
5.92/2.34	YES
5.92/2.34	
5.92/2.34	----------------------------------------
5.92/2.34	
5.92/2.34	(35)
5.92/2.34	Obligation:
5.92/2.34	Pi DP problem:
5.92/2.34	The TRS P consists of the following rules:
5.92/2.34	
5.92/2.34	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.92/2.34	   U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> DELETE_IN_GAG(X, Left, Left1)
5.92/2.34	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.92/2.34	   U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> DELETE_IN_GAG(X, Right, Right1)
5.92/2.34	
5.92/2.34	The TRS R consists of the following rules:
5.92/2.34	
5.92/2.34	   delete_in_aag(X, tree(X, void, Right), Right) -> delete_out_aag(X, tree(X, void, Right), Right)
5.92/2.34	   delete_in_aag(X, tree(X, Left, void), Left) -> delete_out_aag(X, tree(X, Left, void), Left)
5.92/2.35	   delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.35	   delmin_in_agg(tree(Y, void, Right), Y, Right) -> delmin_out_agg(tree(Y, void, Right), Y, Right)
5.92/2.35	   delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
5.92/2.35	   U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) -> delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
5.92/2.35	   U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) -> delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
5.92/2.35	   delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
5.92/2.35	   less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3))
5.92/2.35	   less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y))
5.92/2.35	   U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
5.92/2.35	   U2_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.35	   delete_in_gag(X, tree(X, void, Right), Right) -> delete_out_gag(X, tree(X, void, Right), Right)
5.92/2.35	   delete_in_gag(X, tree(X, Left, void), Left) -> delete_out_gag(X, tree(X, Left, void), Left)
5.92/2.35	   delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.35	   U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) -> delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
5.92/2.35	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.92/2.35	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.92/2.35	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
5.92/2.35	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.92/2.35	   U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.35	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.92/2.35	   U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
5.92/2.35	   U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) -> delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
5.92/2.35	   U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) -> delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
5.92/2.35	   U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) -> delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
5.92/2.35	   delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
5.92/2.35	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
5.92/2.35	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
5.92/2.35	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
5.92/2.35	   U4_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
5.92/2.35	   U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) -> delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
5.92/2.35	
5.92/2.35	The argument filtering Pi contains the following mapping:
5.92/2.35	delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
5.92/2.35	
5.92/2.35	delete_out_aag(x1, x2, x3)  =  delete_out_aag
5.92/2.35	
5.92/2.35	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.92/2.35	
5.92/2.35	U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x6)
5.92/2.35	
5.92/2.35	delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
5.92/2.35	
5.92/2.35	delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
5.92/2.35	
5.92/2.35	U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
5.92/2.35	
5.92/2.35	U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x5, x6)
5.92/2.35	
5.92/2.35	less_in_ag(x1, x2)  =  less_in_ag(x2)
5.92/2.35	
5.92/2.35	s(x1)  =  s(x1)
5.92/2.35	
5.92/2.35	less_out_ag(x1, x2)  =  less_out_ag(x1)
5.92/2.35	
5.92/2.35	U7_ag(x1, x2, x3)  =  U7_ag(x3)
5.92/2.35	
5.92/2.35	U3_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x6)
5.92/2.35	
5.92/2.35	delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
5.92/2.35	
5.92/2.35	delete_out_gag(x1, x2, x3)  =  delete_out_gag
5.92/2.35	
5.92/2.35	U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
5.92/2.35	
5.92/2.35	U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
5.92/2.35	
5.92/2.35	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.92/2.35	
5.92/2.35	0  =  0
5.92/2.35	
5.92/2.35	less_out_gg(x1, x2)  =  less_out_gg
5.92/2.35	
5.92/2.35	U7_gg(x1, x2, x3)  =  U7_gg(x3)
5.92/2.35	
5.92/2.35	U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
5.92/2.35	
5.92/2.35	U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
5.92/2.35	
5.92/2.35	U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
5.92/2.35	
5.92/2.35	U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x5, x6)
5.92/2.35	
5.92/2.35	less_in_ga(x1, x2)  =  less_in_ga(x1)
5.92/2.35	
5.92/2.35	less_out_ga(x1, x2)  =  less_out_ga
5.92/2.35	
5.92/2.35	U7_ga(x1, x2, x3)  =  U7_ga(x3)
5.92/2.35	
5.92/2.35	U5_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x6)
5.92/2.35	
5.92/2.35	DELETE_IN_GAG(x1, x2, x3)  =  DELETE_IN_GAG(x1, x3)
5.92/2.35	
5.92/2.35	U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x1, x5, x6)
5.92/2.35	
5.92/2.35	U4_GAG(x1, x2, x3, x4, x5, x6)  =  U4_GAG(x1, x5, x6)
5.92/2.35	
5.92/2.35	
5.92/2.35	We have to consider all (P,R,Pi)-chains
5.92/2.35	----------------------------------------
5.92/2.35	
5.92/2.35	(36) UsableRulesProof (EQUIVALENT)
5.92/2.35	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.92/2.35	----------------------------------------
5.92/2.35	
5.92/2.35	(37)
5.92/2.35	Obligation:
5.92/2.35	Pi DP problem:
5.92/2.35	The TRS P consists of the following rules:
5.92/2.35	
5.92/2.35	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.92/2.35	   U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> DELETE_IN_GAG(X, Left, Left1)
5.92/2.35	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.92/2.35	   U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> DELETE_IN_GAG(X, Right, Right1)
5.92/2.35	
5.92/2.35	The TRS R consists of the following rules:
5.92/2.35	
5.92/2.35	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.92/2.35	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
5.92/2.35	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.92/2.35	
5.92/2.35	The argument filtering Pi contains the following mapping:
5.92/2.35	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.92/2.35	
5.92/2.35	s(x1)  =  s(x1)
5.92/2.35	
5.92/2.35	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.92/2.35	
5.92/2.35	0  =  0
5.92/2.35	
5.92/2.35	less_out_gg(x1, x2)  =  less_out_gg
5.92/2.35	
5.92/2.35	U7_gg(x1, x2, x3)  =  U7_gg(x3)
5.92/2.35	
5.92/2.35	DELETE_IN_GAG(x1, x2, x3)  =  DELETE_IN_GAG(x1, x3)
5.92/2.35	
5.92/2.35	U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x1, x5, x6)
5.92/2.35	
5.92/2.35	U4_GAG(x1, x2, x3, x4, x5, x6)  =  U4_GAG(x1, x5, x6)
5.92/2.35	
5.92/2.35	
5.92/2.35	We have to consider all (P,R,Pi)-chains
5.92/2.35	----------------------------------------
5.92/2.35	
5.92/2.35	(38) PiDPToQDPProof (SOUND)
5.92/2.35	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.92/2.35	----------------------------------------
5.92/2.35	
5.92/2.35	(39)
5.92/2.35	Obligation:
5.92/2.35	Q DP problem:
5.92/2.35	The TRS P consists of the following rules:
5.92/2.35	
5.92/2.35	   DELETE_IN_GAG(X, tree(Y, Left1, Right)) -> U2_GAG(X, Left1, less_in_gg(X, Y))
5.92/2.35	   U2_GAG(X, Left1, less_out_gg) -> DELETE_IN_GAG(X, Left1)
5.92/2.35	   DELETE_IN_GAG(X, tree(Y, Left, Right1)) -> U4_GAG(X, Right1, less_in_gg(Y, X))
5.92/2.35	   U4_GAG(X, Right1, less_out_gg) -> DELETE_IN_GAG(X, Right1)
5.92/2.35	
5.92/2.35	The TRS R consists of the following rules:
5.92/2.35	
5.92/2.35	   less_in_gg(0, s(X3)) -> less_out_gg
5.92/2.35	   less_in_gg(s(X), s(Y)) -> U7_gg(less_in_gg(X, Y))
5.92/2.35	   U7_gg(less_out_gg) -> less_out_gg
5.92/2.35	
5.92/2.35	The set Q consists of the following terms:
5.92/2.35	
5.92/2.35	   less_in_gg(x0, x1)
5.92/2.35	   U7_gg(x0)
5.92/2.35	
5.92/2.35	We have to consider all (P,Q,R)-chains.
5.92/2.35	----------------------------------------
5.92/2.35	
5.92/2.35	(40) QDPSizeChangeProof (EQUIVALENT)
5.92/2.35	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. 
5.92/2.35	
5.92/2.35	From the DPs we obtained the following set of size-change graphs:
5.92/2.35	*U2_GAG(X, Left1, less_out_gg) -> DELETE_IN_GAG(X, Left1)
5.92/2.35	The graph contains the following edges 1 >= 1, 2 >= 2
5.92/2.35	
5.92/2.35	
5.92/2.35	*U4_GAG(X, Right1, less_out_gg) -> DELETE_IN_GAG(X, Right1)
5.92/2.35	The graph contains the following edges 1 >= 1, 2 >= 2
5.92/2.35	
5.92/2.35	
5.92/2.35	*DELETE_IN_GAG(X, tree(Y, Left1, Right)) -> U2_GAG(X, Left1, less_in_gg(X, Y))
5.92/2.35	The graph contains the following edges 1 >= 1, 2 > 2
5.92/2.35	
5.92/2.35	
5.92/2.35	*DELETE_IN_GAG(X, tree(Y, Left, Right1)) -> U4_GAG(X, Right1, less_in_gg(Y, X))
5.92/2.35	The graph contains the following edges 1 >= 1, 2 > 2
5.92/2.35	
5.92/2.35	
5.92/2.35	----------------------------------------
5.92/2.35	
5.92/2.35	(41)
5.92/2.35	YES
5.92/2.35	
5.92/2.35	----------------------------------------
5.92/2.35	
5.92/2.35	(42)
5.92/2.35	Obligation:
5.92/2.35	Pi DP problem:
5.92/2.35	The TRS P consists of the following rules:
5.92/2.35	
5.92/2.35	   DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_AAG(X, Y, Left, Right, Right1, less_in_ga(Y, X))
5.92/2.35	   U4_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> DELETE_IN_AAG(X, Right, Right1)
5.92/2.35	
5.92/2.35	The TRS R consists of the following rules:
5.92/2.35	
5.92/2.35	   delete_in_aag(X, tree(X, void, Right), Right) -> delete_out_aag(X, tree(X, void, Right), Right)
5.92/2.35	   delete_in_aag(X, tree(X, Left, void), Left) -> delete_out_aag(X, tree(X, Left, void), Left)
5.92/2.35	   delete_in_aag(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_aag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.35	   delmin_in_agg(tree(Y, void, Right), Y, Right) -> delmin_out_agg(tree(Y, void, Right), Y, Right)
5.92/2.35	   delmin_in_agg(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> U6_agg(X, Left, X1, Y, Left1, X2, delmin_in_agg(Left, Y, Left1))
5.92/2.35	   U6_agg(X, Left, X1, Y, Left1, X2, delmin_out_agg(Left, Y, Left1)) -> delmin_out_agg(tree(X, Left, X1), Y, tree(X, Left1, X2))
5.92/2.35	   U1_aag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) -> delete_out_aag(X, tree(X, Left, Right), tree(Y, Left, Right1))
5.92/2.35	   delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_aag(X, Y, Left, Right, Left1, less_in_ag(X, Y))
5.92/2.35	   less_in_ag(0, s(X3)) -> less_out_ag(0, s(X3))
5.92/2.35	   less_in_ag(s(X), s(Y)) -> U7_ag(X, Y, less_in_ag(X, Y))
5.92/2.35	   U7_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
5.92/2.35	   U2_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U3_aag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.35	   delete_in_gag(X, tree(X, void, Right), Right) -> delete_out_gag(X, tree(X, void, Right), Right)
5.92/2.35	   delete_in_gag(X, tree(X, Left, void), Left) -> delete_out_gag(X, tree(X, Left, void), Left)
5.92/2.35	   delete_in_gag(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> U1_gag(X, Left, Right, Y, Right1, delmin_in_agg(Right, Y, Right1))
5.92/2.35	   U1_gag(X, Left, Right, Y, Right1, delmin_out_agg(Right, Y, Right1)) -> delete_out_gag(X, tree(X, Left, Right), tree(Y, Left, Right1))
5.92/2.35	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.92/2.35	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.92/2.35	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
5.92/2.35	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.92/2.35	   U2_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U3_gag(X, Y, Left, Right, Left1, delete_in_gag(X, Left, Left1))
5.92/2.35	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.92/2.35	   U4_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U5_gag(X, Y, Left, Right, Right1, delete_in_gag(X, Right, Right1))
5.92/2.35	   U5_gag(X, Y, Left, Right, Right1, delete_out_gag(X, Right, Right1)) -> delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
5.92/2.35	   U3_gag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) -> delete_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
5.92/2.35	   U3_aag(X, Y, Left, Right, Left1, delete_out_gag(X, Left, Left1)) -> delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
5.92/2.35	   delete_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_aag(X, Y, Left, Right, Right1, less_in_ga(Y, X))
5.92/2.35	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
5.92/2.35	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
5.92/2.35	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
5.92/2.35	   U4_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U5_aag(X, Y, Left, Right, Right1, delete_in_aag(X, Right, Right1))
5.92/2.35	   U5_aag(X, Y, Left, Right, Right1, delete_out_aag(X, Right, Right1)) -> delete_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
5.92/2.35	
5.92/2.35	The argument filtering Pi contains the following mapping:
5.92/2.35	delete_in_aag(x1, x2, x3)  =  delete_in_aag(x3)
5.92/2.35	
5.92/2.35	delete_out_aag(x1, x2, x3)  =  delete_out_aag
5.92/2.35	
5.92/2.35	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.92/2.35	
5.92/2.35	U1_aag(x1, x2, x3, x4, x5, x6)  =  U1_aag(x6)
5.92/2.35	
5.92/2.35	delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
5.92/2.35	
5.92/2.35	delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
5.92/2.35	
5.92/2.35	U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
5.92/2.35	
5.92/2.35	U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x5, x6)
5.92/2.35	
5.92/2.35	less_in_ag(x1, x2)  =  less_in_ag(x2)
5.92/2.35	
5.92/2.35	s(x1)  =  s(x1)
5.92/2.35	
5.92/2.35	less_out_ag(x1, x2)  =  less_out_ag(x1)
5.92/2.35	
5.92/2.35	U7_ag(x1, x2, x3)  =  U7_ag(x3)
5.92/2.35	
5.92/2.35	U3_aag(x1, x2, x3, x4, x5, x6)  =  U3_aag(x6)
5.92/2.35	
5.92/2.35	delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
5.92/2.35	
5.92/2.35	delete_out_gag(x1, x2, x3)  =  delete_out_gag
5.92/2.35	
5.92/2.35	U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
5.92/2.35	
5.92/2.35	U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
5.92/2.35	
5.92/2.35	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.92/2.35	
5.92/2.35	0  =  0
5.92/2.35	
5.92/2.35	less_out_gg(x1, x2)  =  less_out_gg
5.92/2.35	
5.92/2.35	U7_gg(x1, x2, x3)  =  U7_gg(x3)
5.92/2.35	
5.92/2.35	U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
5.92/2.35	
5.92/2.35	U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
5.92/2.35	
5.92/2.35	U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
5.92/2.35	
5.92/2.35	U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x5, x6)
5.92/2.35	
5.92/2.35	less_in_ga(x1, x2)  =  less_in_ga(x1)
5.92/2.35	
5.92/2.35	less_out_ga(x1, x2)  =  less_out_ga
5.92/2.35	
5.92/2.35	U7_ga(x1, x2, x3)  =  U7_ga(x3)
5.92/2.35	
5.92/2.35	U5_aag(x1, x2, x3, x4, x5, x6)  =  U5_aag(x6)
5.92/2.35	
5.92/2.35	DELETE_IN_AAG(x1, x2, x3)  =  DELETE_IN_AAG(x3)
5.92/2.35	
5.92/2.35	U4_AAG(x1, x2, x3, x4, x5, x6)  =  U4_AAG(x5, x6)
5.92/2.35	
5.92/2.35	
5.92/2.35	We have to consider all (P,R,Pi)-chains
5.92/2.35	----------------------------------------
5.92/2.35	
5.92/2.35	(43) UsableRulesProof (EQUIVALENT)
5.92/2.35	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.92/2.35	----------------------------------------
5.92/2.35	
5.92/2.35	(44)
5.92/2.35	Obligation:
5.92/2.35	Pi DP problem:
5.92/2.35	The TRS P consists of the following rules:
5.92/2.35	
5.92/2.35	   DELETE_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_AAG(X, Y, Left, Right, Right1, less_in_ga(Y, X))
5.92/2.35	   U4_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> DELETE_IN_AAG(X, Right, Right1)
5.92/2.35	
5.92/2.35	The TRS R consists of the following rules:
5.92/2.35	
5.92/2.35	   less_in_ga(0, s(X3)) -> less_out_ga(0, s(X3))
5.92/2.35	   less_in_ga(s(X), s(Y)) -> U7_ga(X, Y, less_in_ga(X, Y))
5.92/2.35	   U7_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
5.92/2.35	
5.92/2.35	The argument filtering Pi contains the following mapping:
5.92/2.35	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.92/2.35	
5.92/2.35	s(x1)  =  s(x1)
5.92/2.35	
5.92/2.35	0  =  0
5.92/2.35	
5.92/2.35	less_in_ga(x1, x2)  =  less_in_ga(x1)
5.92/2.35	
5.92/2.35	less_out_ga(x1, x2)  =  less_out_ga
5.92/2.35	
5.92/2.35	U7_ga(x1, x2, x3)  =  U7_ga(x3)
5.92/2.35	
5.92/2.35	DELETE_IN_AAG(x1, x2, x3)  =  DELETE_IN_AAG(x3)
5.92/2.35	
5.92/2.35	U4_AAG(x1, x2, x3, x4, x5, x6)  =  U4_AAG(x5, x6)
5.92/2.35	
5.92/2.35	
5.92/2.35	We have to consider all (P,R,Pi)-chains
5.92/2.35	----------------------------------------
5.92/2.35	
5.92/2.35	(45) PiDPToQDPProof (SOUND)
5.92/2.35	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.92/2.35	----------------------------------------
5.92/2.35	
5.92/2.35	(46)
5.92/2.35	Obligation:
5.92/2.35	Q DP problem:
5.92/2.35	The TRS P consists of the following rules:
5.92/2.35	
5.92/2.35	   DELETE_IN_AAG(tree(Y, Left, Right1)) -> U4_AAG(Right1, less_in_ga(Y))
5.92/2.35	   U4_AAG(Right1, less_out_ga) -> DELETE_IN_AAG(Right1)
5.92/2.35	
5.92/2.35	The TRS R consists of the following rules:
5.92/2.35	
5.92/2.35	   less_in_ga(0) -> less_out_ga
5.92/2.35	   less_in_ga(s(X)) -> U7_ga(less_in_ga(X))
5.92/2.35	   U7_ga(less_out_ga) -> less_out_ga
5.92/2.35	
5.92/2.35	The set Q consists of the following terms:
5.92/2.35	
5.92/2.35	   less_in_ga(x0)
5.92/2.35	   U7_ga(x0)
5.92/2.35	
5.92/2.35	We have to consider all (P,Q,R)-chains.
5.92/2.35	----------------------------------------
5.92/2.35	
5.92/2.35	(47) QDPSizeChangeProof (EQUIVALENT)
5.92/2.35	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. 
5.92/2.35	
5.92/2.35	From the DPs we obtained the following set of size-change graphs:
5.92/2.35	*U4_AAG(Right1, less_out_ga) -> DELETE_IN_AAG(Right1)
5.92/2.35	The graph contains the following edges 1 >= 1
5.92/2.35	
5.92/2.35	
5.92/2.35	*DELETE_IN_AAG(tree(Y, Left, Right1)) -> U4_AAG(Right1, less_in_ga(Y))
5.92/2.35	The graph contains the following edges 1 > 1
5.92/2.35	
5.92/2.35	
5.92/2.35	----------------------------------------
5.92/2.35	
5.92/2.35	(48)
5.92/2.35	YES
6.27/2.88	EOF