4.84/2.17	YES
5.25/2.19	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
5.25/2.19	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.25/2.19	
5.25/2.19	
5.25/2.19	Left Termination of the query pattern
5.25/2.19	
5.25/2.19	delete(g,a,g)
5.25/2.19	
5.25/2.19	w.r.t. the given Prolog program could successfully be proven:
5.25/2.19	
5.25/2.19	(0) Prolog
5.25/2.19	(1) PrologToPiTRSProof [SOUND, 0 ms]
5.25/2.19	(2) PiTRS
5.25/2.19	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
5.25/2.19	(4) PiDP
5.25/2.19	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
5.25/2.19	(6) AND
5.25/2.19	    (7) PiDP
5.25/2.19	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
5.25/2.19	        (9) PiDP
5.25/2.19	        (10) PiDPToQDPProof [EQUIVALENT, 1 ms]
5.25/2.19	        (11) QDP
5.25/2.19	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
5.25/2.19	        (13) YES
5.25/2.19	    (14) PiDP
5.25/2.19	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
5.25/2.19	        (16) PiDP
5.25/2.19	        (17) PiDPToQDPProof [SOUND, 0 ms]
5.25/2.19	        (18) QDP
5.25/2.19	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
5.25/2.19	        (20) YES
5.25/2.19	    (21) PiDP
5.25/2.19	        (22) UsableRulesProof [EQUIVALENT, 0 ms]
5.25/2.19	        (23) PiDP
5.25/2.19	        (24) PiDPToQDPProof [SOUND, 0 ms]
5.25/2.19	        (25) QDP
5.25/2.19	        (26) QDPSizeChangeProof [EQUIVALENT, 0 ms]
5.25/2.19	        (27) YES
5.25/2.19	
5.25/2.19	
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(0)
5.25/2.19	Obligation:
5.25/2.19	Clauses:
5.25/2.19	
5.25/2.19	delete(X, tree(X, void, Right), Right).
5.25/2.19	delete(X, tree(X, Left, void), Left).
5.25/2.19	delete(X, tree(X, Left, Right), tree(Y, Left, Right1)) :- delmin(Right, Y, Right1).
5.25/2.19	delete(X, tree(Y, Left, Right), tree(Y, Left1, Right)) :- ','(less(X, Y), delete(X, Left, Left1)).
5.25/2.19	delete(X, tree(Y, Left, Right), tree(Y, Left, Right1)) :- ','(less(Y, X), delete(X, Right, Right1)).
5.25/2.19	delmin(tree(Y, void, Right), Y, Right).
5.25/2.19	delmin(tree(X, Left, X1), Y, tree(X, Left1, X2)) :- delmin(Left, Y, Left1).
5.25/2.19	less(0, s(X3)).
5.25/2.19	less(s(X), s(Y)) :- less(X, Y).
5.25/2.19	
5.25/2.19	
5.25/2.19	Query: delete(g,a,g)
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(1) PrologToPiTRSProof (SOUND)
5.25/2.19	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
5.25/2.19	
5.25/2.19	delete_in_3: (b,f,b)
5.25/2.19	
5.25/2.19	delmin_in_3: (f,b,b)
5.25/2.19	
5.25/2.19	less_in_2: (b,b)
5.25/2.19	
5.25/2.19	Transforming Prolog into the following Term Rewriting System:
5.25/2.19	
5.25/2.19	Pi-finite rewrite system:
5.25/2.19	The TRS R consists of the following rules:
5.25/2.19	
5.25/2.19	   delete_in_gag(X, tree(X, void, Right), Right) -> delete_out_gag(X, tree(X, void, Right), Right)
5.25/2.19	   delete_in_gag(X, tree(X, Left, void), Left) -> delete_out_gag(X, tree(X, Left, void), Left)
5.25/2.19	   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))
5.25/2.19	   delmin_in_agg(tree(Y, void, Right), Y, Right) -> delmin_out_agg(tree(Y, void, Right), Y, Right)
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.25/2.19	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.25/2.19	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
5.25/2.19	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.25/2.19	   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))
5.25/2.19	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	
5.25/2.19	The argument filtering Pi contains the following mapping:
5.25/2.19	delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
5.25/2.19	
5.25/2.19	delete_out_gag(x1, x2, x3)  =  delete_out_gag
5.25/2.19	
5.25/2.19	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.25/2.19	
5.25/2.19	U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
5.25/2.19	
5.25/2.19	delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
5.25/2.19	
5.25/2.19	delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
5.25/2.19	
5.25/2.19	U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
5.25/2.19	
5.25/2.19	U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
5.25/2.19	
5.25/2.19	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.25/2.19	
5.25/2.19	0  =  0
5.25/2.19	
5.25/2.19	s(x1)  =  s(x1)
5.25/2.19	
5.25/2.19	less_out_gg(x1, x2)  =  less_out_gg
5.25/2.19	
5.25/2.19	U7_gg(x1, x2, x3)  =  U7_gg(x3)
5.25/2.19	
5.25/2.19	U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
5.25/2.19	
5.25/2.19	U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
5.25/2.19	
5.25/2.19	U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
5.25/2.19	
5.25/2.19	
5.25/2.19	
5.25/2.19	
5.25/2.19	
5.25/2.19	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
5.25/2.19	
5.25/2.19	
5.25/2.19	
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(2)
5.25/2.19	Obligation:
5.25/2.19	Pi-finite rewrite system:
5.25/2.19	The TRS R consists of the following rules:
5.25/2.19	
5.25/2.19	   delete_in_gag(X, tree(X, void, Right), Right) -> delete_out_gag(X, tree(X, void, Right), Right)
5.25/2.19	   delete_in_gag(X, tree(X, Left, void), Left) -> delete_out_gag(X, tree(X, Left, void), Left)
5.25/2.19	   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))
5.25/2.19	   delmin_in_agg(tree(Y, void, Right), Y, Right) -> delmin_out_agg(tree(Y, void, Right), Y, Right)
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.25/2.19	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.25/2.19	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
5.25/2.19	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.25/2.19	   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))
5.25/2.19	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	
5.25/2.19	The argument filtering Pi contains the following mapping:
5.25/2.19	delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
5.25/2.19	
5.25/2.19	delete_out_gag(x1, x2, x3)  =  delete_out_gag
5.25/2.19	
5.25/2.19	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.25/2.19	
5.25/2.19	U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
5.25/2.19	
5.25/2.19	delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
5.25/2.19	
5.25/2.19	delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
5.25/2.19	
5.25/2.19	U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
5.25/2.19	
5.25/2.19	U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
5.25/2.19	
5.25/2.19	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.25/2.19	
5.25/2.19	0  =  0
5.25/2.19	
5.25/2.19	s(x1)  =  s(x1)
5.25/2.19	
5.25/2.19	less_out_gg(x1, x2)  =  less_out_gg
5.25/2.19	
5.25/2.19	U7_gg(x1, x2, x3)  =  U7_gg(x3)
5.25/2.19	
5.25/2.19	U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
5.25/2.19	
5.25/2.19	U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
5.25/2.19	
5.25/2.19	U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
5.25/2.19	
5.25/2.19	
5.25/2.19	
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(3) DependencyPairsProof (EQUIVALENT)
5.25/2.19	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
5.25/2.19	Pi DP problem:
5.25/2.19	The TRS P consists of the following rules:
5.25/2.19	
5.25/2.19	   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))
5.25/2.19	   DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> DELMIN_IN_AGG(Right, Y, Right1)
5.25/2.19	   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))
5.25/2.19	   DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AGG(Left, Y, Left1)
5.25/2.19	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.25/2.19	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GG(X, Y)
5.25/2.19	   LESS_IN_GG(s(X), s(Y)) -> U7_GG(X, Y, less_in_gg(X, Y))
5.25/2.19	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
5.25/2.19	   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))
5.25/2.19	   U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> DELETE_IN_GAG(X, Left, Left1)
5.25/2.19	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.25/2.19	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_GG(Y, X)
5.25/2.19	   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))
5.25/2.19	   U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> DELETE_IN_GAG(X, Right, Right1)
5.25/2.19	
5.25/2.19	The TRS R consists of the following rules:
5.25/2.19	
5.25/2.19	   delete_in_gag(X, tree(X, void, Right), Right) -> delete_out_gag(X, tree(X, void, Right), Right)
5.25/2.19	   delete_in_gag(X, tree(X, Left, void), Left) -> delete_out_gag(X, tree(X, Left, void), Left)
5.25/2.19	   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))
5.25/2.19	   delmin_in_agg(tree(Y, void, Right), Y, Right) -> delmin_out_agg(tree(Y, void, Right), Y, Right)
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.25/2.19	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.25/2.19	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
5.25/2.19	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.25/2.19	   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))
5.25/2.19	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	
5.25/2.19	The argument filtering Pi contains the following mapping:
5.25/2.19	delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
5.25/2.19	
5.25/2.19	delete_out_gag(x1, x2, x3)  =  delete_out_gag
5.25/2.19	
5.25/2.19	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.25/2.19	
5.25/2.19	U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
5.25/2.19	
5.25/2.19	delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
5.25/2.19	
5.25/2.19	delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
5.25/2.19	
5.25/2.19	U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
5.25/2.19	
5.25/2.19	U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
5.25/2.19	
5.25/2.19	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.25/2.19	
5.25/2.19	0  =  0
5.25/2.19	
5.25/2.19	s(x1)  =  s(x1)
5.25/2.19	
5.25/2.19	less_out_gg(x1, x2)  =  less_out_gg
5.25/2.19	
5.25/2.19	U7_gg(x1, x2, x3)  =  U7_gg(x3)
5.25/2.19	
5.25/2.19	U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
5.25/2.19	
5.25/2.19	U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
5.25/2.19	
5.25/2.19	U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
5.25/2.19	
5.25/2.19	DELETE_IN_GAG(x1, x2, x3)  =  DELETE_IN_GAG(x1, x3)
5.25/2.19	
5.25/2.19	U1_GAG(x1, x2, x3, x4, x5, x6)  =  U1_GAG(x6)
5.25/2.19	
5.25/2.19	DELMIN_IN_AGG(x1, x2, x3)  =  DELMIN_IN_AGG(x2, x3)
5.25/2.19	
5.25/2.19	U6_AGG(x1, x2, x3, x4, x5, x6, x7)  =  U6_AGG(x7)
5.25/2.19	
5.25/2.19	U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x1, x5, x6)
5.25/2.19	
5.25/2.19	LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
5.25/2.19	
5.25/2.19	U7_GG(x1, x2, x3)  =  U7_GG(x3)
5.25/2.19	
5.25/2.19	U3_GAG(x1, x2, x3, x4, x5, x6)  =  U3_GAG(x6)
5.25/2.19	
5.25/2.19	U4_GAG(x1, x2, x3, x4, x5, x6)  =  U4_GAG(x1, x5, x6)
5.25/2.19	
5.25/2.19	U5_GAG(x1, x2, x3, x4, x5, x6)  =  U5_GAG(x6)
5.25/2.19	
5.25/2.19	
5.25/2.19	We have to consider all (P,R,Pi)-chains
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(4)
5.25/2.19	Obligation:
5.25/2.19	Pi DP problem:
5.25/2.19	The TRS P consists of the following rules:
5.25/2.19	
5.25/2.19	   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))
5.25/2.19	   DELETE_IN_GAG(X, tree(X, Left, Right), tree(Y, Left, Right1)) -> DELMIN_IN_AGG(Right, Y, Right1)
5.25/2.19	   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))
5.25/2.19	   DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AGG(Left, Y, Left1)
5.25/2.19	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.25/2.19	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GG(X, Y)
5.25/2.19	   LESS_IN_GG(s(X), s(Y)) -> U7_GG(X, Y, less_in_gg(X, Y))
5.25/2.19	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
5.25/2.19	   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))
5.25/2.19	   U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> DELETE_IN_GAG(X, Left, Left1)
5.25/2.19	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.25/2.19	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_GG(Y, X)
5.25/2.19	   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))
5.25/2.19	   U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> DELETE_IN_GAG(X, Right, Right1)
5.25/2.19	
5.25/2.19	The TRS R consists of the following rules:
5.25/2.19	
5.25/2.19	   delete_in_gag(X, tree(X, void, Right), Right) -> delete_out_gag(X, tree(X, void, Right), Right)
5.25/2.19	   delete_in_gag(X, tree(X, Left, void), Left) -> delete_out_gag(X, tree(X, Left, void), Left)
5.25/2.19	   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))
5.25/2.19	   delmin_in_agg(tree(Y, void, Right), Y, Right) -> delmin_out_agg(tree(Y, void, Right), Y, Right)
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.25/2.19	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.25/2.19	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
5.25/2.19	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.25/2.19	   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))
5.25/2.19	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	
5.25/2.19	The argument filtering Pi contains the following mapping:
5.25/2.19	delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
5.25/2.19	
5.25/2.19	delete_out_gag(x1, x2, x3)  =  delete_out_gag
5.25/2.19	
5.25/2.19	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.25/2.19	
5.25/2.19	U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
5.25/2.19	
5.25/2.19	delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
5.25/2.19	
5.25/2.19	delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
5.25/2.19	
5.25/2.19	U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
5.25/2.19	
5.25/2.19	U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
5.25/2.19	
5.25/2.19	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.25/2.19	
5.25/2.19	0  =  0
5.25/2.19	
5.25/2.19	s(x1)  =  s(x1)
5.25/2.19	
5.25/2.19	less_out_gg(x1, x2)  =  less_out_gg
5.25/2.19	
5.25/2.19	U7_gg(x1, x2, x3)  =  U7_gg(x3)
5.25/2.19	
5.25/2.19	U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
5.25/2.19	
5.25/2.19	U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
5.25/2.19	
5.25/2.19	U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
5.25/2.19	
5.25/2.19	DELETE_IN_GAG(x1, x2, x3)  =  DELETE_IN_GAG(x1, x3)
5.25/2.19	
5.25/2.19	U1_GAG(x1, x2, x3, x4, x5, x6)  =  U1_GAG(x6)
5.25/2.19	
5.25/2.19	DELMIN_IN_AGG(x1, x2, x3)  =  DELMIN_IN_AGG(x2, x3)
5.25/2.19	
5.25/2.19	U6_AGG(x1, x2, x3, x4, x5, x6, x7)  =  U6_AGG(x7)
5.25/2.19	
5.25/2.19	U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x1, x5, x6)
5.25/2.19	
5.25/2.19	LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
5.25/2.19	
5.25/2.19	U7_GG(x1, x2, x3)  =  U7_GG(x3)
5.25/2.19	
5.25/2.19	U3_GAG(x1, x2, x3, x4, x5, x6)  =  U3_GAG(x6)
5.25/2.19	
5.25/2.19	U4_GAG(x1, x2, x3, x4, x5, x6)  =  U4_GAG(x1, x5, x6)
5.25/2.19	
5.25/2.19	U5_GAG(x1, x2, x3, x4, x5, x6)  =  U5_GAG(x6)
5.25/2.19	
5.25/2.19	
5.25/2.19	We have to consider all (P,R,Pi)-chains
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(5) DependencyGraphProof (EQUIVALENT)
5.25/2.19	The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 8 less nodes.
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(6)
5.25/2.19	Complex Obligation (AND)
5.25/2.19	
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(7)
5.25/2.19	Obligation:
5.25/2.19	Pi DP problem:
5.25/2.19	The TRS P consists of the following rules:
5.25/2.19	
5.25/2.19	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
5.25/2.19	
5.25/2.19	The TRS R consists of the following rules:
5.25/2.19	
5.25/2.19	   delete_in_gag(X, tree(X, void, Right), Right) -> delete_out_gag(X, tree(X, void, Right), Right)
5.25/2.19	   delete_in_gag(X, tree(X, Left, void), Left) -> delete_out_gag(X, tree(X, Left, void), Left)
5.25/2.19	   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))
5.25/2.19	   delmin_in_agg(tree(Y, void, Right), Y, Right) -> delmin_out_agg(tree(Y, void, Right), Y, Right)
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.25/2.19	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.25/2.19	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
5.25/2.19	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.25/2.19	   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))
5.25/2.19	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	
5.25/2.19	The argument filtering Pi contains the following mapping:
5.25/2.19	delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
5.25/2.19	
5.25/2.19	delete_out_gag(x1, x2, x3)  =  delete_out_gag
5.25/2.19	
5.25/2.19	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.25/2.19	
5.25/2.19	U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
5.25/2.19	
5.25/2.19	delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
5.25/2.19	
5.25/2.19	delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
5.25/2.19	
5.25/2.19	U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
5.25/2.19	
5.25/2.19	U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
5.25/2.19	
5.25/2.19	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.25/2.19	
5.25/2.19	0  =  0
5.25/2.19	
5.25/2.19	s(x1)  =  s(x1)
5.25/2.19	
5.25/2.19	less_out_gg(x1, x2)  =  less_out_gg
5.25/2.19	
5.25/2.19	U7_gg(x1, x2, x3)  =  U7_gg(x3)
5.25/2.19	
5.25/2.19	U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
5.25/2.19	
5.25/2.19	U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
5.25/2.19	
5.25/2.19	U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
5.25/2.19	
5.25/2.19	LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
5.25/2.19	
5.25/2.19	
5.25/2.19	We have to consider all (P,R,Pi)-chains
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(8) UsableRulesProof (EQUIVALENT)
5.25/2.19	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(9)
5.25/2.19	Obligation:
5.25/2.19	Pi DP problem:
5.25/2.19	The TRS P consists of the following rules:
5.25/2.19	
5.25/2.19	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
5.25/2.19	
5.25/2.19	R is empty.
5.25/2.19	Pi is empty.
5.25/2.19	We have to consider all (P,R,Pi)-chains
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(10) PiDPToQDPProof (EQUIVALENT)
5.25/2.19	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(11)
5.25/2.19	Obligation:
5.25/2.19	Q DP problem:
5.25/2.19	The TRS P consists of the following rules:
5.25/2.19	
5.25/2.19	   LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
5.25/2.19	
5.25/2.19	R is empty.
5.25/2.19	Q is empty.
5.25/2.19	We have to consider all (P,Q,R)-chains.
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(12) QDPSizeChangeProof (EQUIVALENT)
5.25/2.19	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. 
5.25/2.19	
5.25/2.19	From the DPs we obtained the following set of size-change graphs:
5.25/2.19	*LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y)
5.25/2.19	The graph contains the following edges 1 > 1, 2 > 2
5.25/2.19	
5.25/2.19	
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(13)
5.25/2.19	YES
5.25/2.19	
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(14)
5.25/2.19	Obligation:
5.25/2.19	Pi DP problem:
5.25/2.19	The TRS P consists of the following rules:
5.25/2.19	
5.25/2.19	   DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AGG(Left, Y, Left1)
5.25/2.19	
5.25/2.19	The TRS R consists of the following rules:
5.25/2.19	
5.25/2.19	   delete_in_gag(X, tree(X, void, Right), Right) -> delete_out_gag(X, tree(X, void, Right), Right)
5.25/2.19	   delete_in_gag(X, tree(X, Left, void), Left) -> delete_out_gag(X, tree(X, Left, void), Left)
5.25/2.19	   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))
5.25/2.19	   delmin_in_agg(tree(Y, void, Right), Y, Right) -> delmin_out_agg(tree(Y, void, Right), Y, Right)
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.25/2.19	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.25/2.19	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
5.25/2.19	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.25/2.19	   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))
5.25/2.19	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	
5.25/2.19	The argument filtering Pi contains the following mapping:
5.25/2.19	delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
5.25/2.19	
5.25/2.19	delete_out_gag(x1, x2, x3)  =  delete_out_gag
5.25/2.19	
5.25/2.19	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.25/2.19	
5.25/2.19	U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
5.25/2.19	
5.25/2.19	delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
5.25/2.19	
5.25/2.19	delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
5.25/2.19	
5.25/2.19	U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
5.25/2.19	
5.25/2.19	U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
5.25/2.19	
5.25/2.19	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.25/2.19	
5.25/2.19	0  =  0
5.25/2.19	
5.25/2.19	s(x1)  =  s(x1)
5.25/2.19	
5.25/2.19	less_out_gg(x1, x2)  =  less_out_gg
5.25/2.19	
5.25/2.19	U7_gg(x1, x2, x3)  =  U7_gg(x3)
5.25/2.19	
5.25/2.19	U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
5.25/2.19	
5.25/2.19	U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
5.25/2.19	
5.25/2.19	U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
5.25/2.19	
5.25/2.19	DELMIN_IN_AGG(x1, x2, x3)  =  DELMIN_IN_AGG(x2, x3)
5.25/2.19	
5.25/2.19	
5.25/2.19	We have to consider all (P,R,Pi)-chains
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(15) UsableRulesProof (EQUIVALENT)
5.25/2.19	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(16)
5.25/2.19	Obligation:
5.25/2.19	Pi DP problem:
5.25/2.19	The TRS P consists of the following rules:
5.25/2.19	
5.25/2.19	   DELMIN_IN_AGG(tree(X, Left, X1), Y, tree(X, Left1, X2)) -> DELMIN_IN_AGG(Left, Y, Left1)
5.25/2.19	
5.25/2.19	R is empty.
5.25/2.19	The argument filtering Pi contains the following mapping:
5.25/2.19	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.25/2.19	
5.25/2.19	DELMIN_IN_AGG(x1, x2, x3)  =  DELMIN_IN_AGG(x2, x3)
5.25/2.19	
5.25/2.19	
5.25/2.19	We have to consider all (P,R,Pi)-chains
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(17) PiDPToQDPProof (SOUND)
5.25/2.19	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(18)
5.25/2.19	Obligation:
5.25/2.19	Q DP problem:
5.25/2.19	The TRS P consists of the following rules:
5.25/2.19	
5.25/2.19	   DELMIN_IN_AGG(Y, tree(X, Left1, X2)) -> DELMIN_IN_AGG(Y, Left1)
5.25/2.19	
5.25/2.19	R is empty.
5.25/2.19	Q is empty.
5.25/2.19	We have to consider all (P,Q,R)-chains.
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(19) QDPSizeChangeProof (EQUIVALENT)
5.25/2.19	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. 
5.25/2.19	
5.25/2.19	From the DPs we obtained the following set of size-change graphs:
5.25/2.19	*DELMIN_IN_AGG(Y, tree(X, Left1, X2)) -> DELMIN_IN_AGG(Y, Left1)
5.25/2.19	The graph contains the following edges 1 >= 1, 2 > 2
5.25/2.19	
5.25/2.19	
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(20)
5.25/2.19	YES
5.25/2.19	
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(21)
5.25/2.19	Obligation:
5.25/2.19	Pi DP problem:
5.25/2.19	The TRS P consists of the following rules:
5.25/2.19	
5.25/2.19	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.25/2.19	   U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> DELETE_IN_GAG(X, Left, Left1)
5.25/2.19	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.25/2.19	   U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> DELETE_IN_GAG(X, Right, Right1)
5.25/2.19	
5.25/2.19	The TRS R consists of the following rules:
5.25/2.19	
5.25/2.19	   delete_in_gag(X, tree(X, void, Right), Right) -> delete_out_gag(X, tree(X, void, Right), Right)
5.25/2.19	   delete_in_gag(X, tree(X, Left, void), Left) -> delete_out_gag(X, tree(X, Left, void), Left)
5.25/2.19	   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))
5.25/2.19	   delmin_in_agg(tree(Y, void, Right), Y, Right) -> delmin_out_agg(tree(Y, void, Right), Y, Right)
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_gag(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.25/2.19	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.25/2.19	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
5.25/2.19	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.25/2.19	   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))
5.25/2.19	   delete_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	   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))
5.25/2.19	
5.25/2.19	The argument filtering Pi contains the following mapping:
5.25/2.19	delete_in_gag(x1, x2, x3)  =  delete_in_gag(x1, x3)
5.25/2.19	
5.25/2.19	delete_out_gag(x1, x2, x3)  =  delete_out_gag
5.25/2.19	
5.25/2.19	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.25/2.19	
5.25/2.19	U1_gag(x1, x2, x3, x4, x5, x6)  =  U1_gag(x6)
5.25/2.19	
5.25/2.19	delmin_in_agg(x1, x2, x3)  =  delmin_in_agg(x2, x3)
5.25/2.19	
5.25/2.19	delmin_out_agg(x1, x2, x3)  =  delmin_out_agg
5.25/2.19	
5.25/2.19	U6_agg(x1, x2, x3, x4, x5, x6, x7)  =  U6_agg(x7)
5.25/2.19	
5.25/2.19	U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x5, x6)
5.25/2.19	
5.25/2.19	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.25/2.19	
5.25/2.19	0  =  0
5.25/2.19	
5.25/2.19	s(x1)  =  s(x1)
5.25/2.19	
5.25/2.19	less_out_gg(x1, x2)  =  less_out_gg
5.25/2.19	
5.25/2.19	U7_gg(x1, x2, x3)  =  U7_gg(x3)
5.25/2.19	
5.25/2.19	U3_gag(x1, x2, x3, x4, x5, x6)  =  U3_gag(x6)
5.25/2.19	
5.25/2.19	U4_gag(x1, x2, x3, x4, x5, x6)  =  U4_gag(x1, x5, x6)
5.25/2.19	
5.25/2.19	U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
5.25/2.19	
5.25/2.19	DELETE_IN_GAG(x1, x2, x3)  =  DELETE_IN_GAG(x1, x3)
5.25/2.19	
5.25/2.19	U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x1, x5, x6)
5.25/2.19	
5.25/2.19	U4_GAG(x1, x2, x3, x4, x5, x6)  =  U4_GAG(x1, x5, x6)
5.25/2.19	
5.25/2.19	
5.25/2.19	We have to consider all (P,R,Pi)-chains
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(22) UsableRulesProof (EQUIVALENT)
5.25/2.19	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(23)
5.25/2.19	Obligation:
5.25/2.19	Pi DP problem:
5.25/2.19	The TRS P consists of the following rules:
5.25/2.19	
5.25/2.19	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U2_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
5.25/2.19	   U2_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> DELETE_IN_GAG(X, Left, Left1)
5.25/2.19	   DELETE_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U4_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
5.25/2.19	   U4_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> DELETE_IN_GAG(X, Right, Right1)
5.25/2.19	
5.25/2.19	The TRS R consists of the following rules:
5.25/2.19	
5.25/2.19	   less_in_gg(0, s(X3)) -> less_out_gg(0, s(X3))
5.25/2.19	   less_in_gg(s(X), s(Y)) -> U7_gg(X, Y, less_in_gg(X, Y))
5.25/2.19	   U7_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y))
5.25/2.19	
5.25/2.19	The argument filtering Pi contains the following mapping:
5.25/2.19	tree(x1, x2, x3)  =  tree(x1, x2, x3)
5.25/2.19	
5.25/2.19	less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
5.25/2.19	
5.25/2.19	0  =  0
5.25/2.19	
5.25/2.19	s(x1)  =  s(x1)
5.25/2.19	
5.25/2.19	less_out_gg(x1, x2)  =  less_out_gg
5.25/2.19	
5.25/2.19	U7_gg(x1, x2, x3)  =  U7_gg(x3)
5.25/2.19	
5.25/2.19	DELETE_IN_GAG(x1, x2, x3)  =  DELETE_IN_GAG(x1, x3)
5.25/2.19	
5.25/2.19	U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x1, x5, x6)
5.25/2.19	
5.25/2.19	U4_GAG(x1, x2, x3, x4, x5, x6)  =  U4_GAG(x1, x5, x6)
5.25/2.19	
5.25/2.19	
5.25/2.19	We have to consider all (P,R,Pi)-chains
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(24) PiDPToQDPProof (SOUND)
5.25/2.19	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(25)
5.25/2.19	Obligation:
5.25/2.19	Q DP problem:
5.25/2.19	The TRS P consists of the following rules:
5.25/2.19	
5.25/2.19	   DELETE_IN_GAG(X, tree(Y, Left1, Right)) -> U2_GAG(X, Left1, less_in_gg(X, Y))
5.25/2.19	   U2_GAG(X, Left1, less_out_gg) -> DELETE_IN_GAG(X, Left1)
5.25/2.19	   DELETE_IN_GAG(X, tree(Y, Left, Right1)) -> U4_GAG(X, Right1, less_in_gg(Y, X))
5.25/2.19	   U4_GAG(X, Right1, less_out_gg) -> DELETE_IN_GAG(X, Right1)
5.25/2.19	
5.25/2.19	The TRS R consists of the following rules:
5.25/2.19	
5.25/2.19	   less_in_gg(0, s(X3)) -> less_out_gg
5.25/2.19	   less_in_gg(s(X), s(Y)) -> U7_gg(less_in_gg(X, Y))
5.25/2.19	   U7_gg(less_out_gg) -> less_out_gg
5.25/2.19	
5.25/2.19	The set Q consists of the following terms:
5.25/2.19	
5.25/2.19	   less_in_gg(x0, x1)
5.25/2.19	   U7_gg(x0)
5.25/2.19	
5.25/2.19	We have to consider all (P,Q,R)-chains.
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(26) QDPSizeChangeProof (EQUIVALENT)
5.25/2.19	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. 
5.25/2.19	
5.25/2.19	From the DPs we obtained the following set of size-change graphs:
5.25/2.19	*U2_GAG(X, Left1, less_out_gg) -> DELETE_IN_GAG(X, Left1)
5.25/2.19	The graph contains the following edges 1 >= 1, 2 >= 2
5.25/2.19	
5.25/2.19	
5.25/2.19	*U4_GAG(X, Right1, less_out_gg) -> DELETE_IN_GAG(X, Right1)
5.25/2.19	The graph contains the following edges 1 >= 1, 2 >= 2
5.25/2.19	
5.25/2.19	
5.25/2.19	*DELETE_IN_GAG(X, tree(Y, Left1, Right)) -> U2_GAG(X, Left1, less_in_gg(X, Y))
5.25/2.19	The graph contains the following edges 1 >= 1, 2 > 2
5.25/2.19	
5.25/2.19	
5.25/2.19	*DELETE_IN_GAG(X, tree(Y, Left, Right1)) -> U4_GAG(X, Right1, less_in_gg(Y, X))
5.25/2.19	The graph contains the following edges 1 >= 1, 2 > 2
5.25/2.19	
5.25/2.19	
5.25/2.19	----------------------------------------
5.25/2.19	
5.25/2.19	(27)
5.25/2.19	YES
5.29/2.23	EOF