3.97/1.71	YES
3.97/1.73	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.97/1.73	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.97/1.73	
3.97/1.73	
3.97/1.73	Left Termination of the query pattern
3.97/1.73	
3.97/1.73	max(g,a,a)
3.97/1.73	
3.97/1.73	w.r.t. the given Prolog program could successfully be proven:
3.97/1.73	
3.97/1.73	(0) Prolog
3.97/1.73	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.97/1.73	(2) PiTRS
3.97/1.73	(3) DependencyPairsProof [EQUIVALENT, 11 ms]
3.97/1.73	(4) PiDP
3.97/1.73	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.97/1.73	(6) AND
3.97/1.73	    (7) PiDP
3.97/1.73	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
3.97/1.73	        (9) PiDP
3.97/1.73	        (10) PiDPToQDPProof [SOUND, 0 ms]
3.97/1.73	        (11) QDP
3.97/1.73	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.97/1.73	        (13) YES
3.97/1.73	    (14) PiDP
3.97/1.73	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
3.97/1.73	        (16) PiDP
3.97/1.73	        (17) PiDPToQDPProof [SOUND, 0 ms]
3.97/1.73	        (18) QDP
3.97/1.73	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.97/1.73	        (20) YES
3.97/1.73	
3.97/1.73	
3.97/1.73	----------------------------------------
3.97/1.73	
3.97/1.73	(0)
3.97/1.73	Obligation:
3.97/1.73	Clauses:
3.97/1.73	
3.97/1.73	max(X, Y, X) :- less(Y, X).
3.97/1.73	max(X, Y, Y) :- less(X, s(Y)).
3.97/1.73	less(0, s(X1)).
3.97/1.73	less(s(X), s(Y)) :- less(X, Y).
3.97/1.73	
3.97/1.73	
3.97/1.73	Query: max(g,a,a)
3.97/1.73	----------------------------------------
3.97/1.73	
3.97/1.73	(1) PrologToPiTRSProof (SOUND)
3.97/1.73	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.97/1.73	
3.97/1.73	max_in_3: (b,f,f)
3.97/1.73	
3.97/1.73	less_in_2: (f,b) (b,f)
3.97/1.73	
3.97/1.73	Transforming Prolog into the following Term Rewriting System:
3.97/1.73	
3.97/1.73	Pi-finite rewrite system:
3.97/1.73	The TRS R consists of the following rules:
3.97/1.73	
3.97/1.73	   max_in_gaa(X, Y, X) -> U1_gaa(X, Y, less_in_ag(Y, X))
3.97/1.73	   less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1))
3.97/1.73	   less_in_ag(s(X), s(Y)) -> U3_ag(X, Y, less_in_ag(X, Y))
3.97/1.73	   U3_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
3.97/1.73	   U1_gaa(X, Y, less_out_ag(Y, X)) -> max_out_gaa(X, Y, X)
3.97/1.73	   max_in_gaa(X, Y, Y) -> U2_gaa(X, Y, less_in_ga(X, s(Y)))
3.97/1.73	   less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1))
3.97/1.73	   less_in_ga(s(X), s(Y)) -> U3_ga(X, Y, less_in_ga(X, Y))
3.97/1.73	   U3_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
3.97/1.73	   U2_gaa(X, Y, less_out_ga(X, s(Y))) -> max_out_gaa(X, Y, Y)
3.97/1.73	
3.97/1.73	The argument filtering Pi contains the following mapping:
3.97/1.73	max_in_gaa(x1, x2, x3)  =  max_in_gaa(x1)
3.97/1.73	
3.97/1.73	U1_gaa(x1, x2, x3)  =  U1_gaa(x3)
3.97/1.73	
3.97/1.73	less_in_ag(x1, x2)  =  less_in_ag(x2)
3.97/1.73	
3.97/1.73	s(x1)  =  s(x1)
3.97/1.73	
3.97/1.73	less_out_ag(x1, x2)  =  less_out_ag(x1)
3.97/1.73	
3.97/1.73	U3_ag(x1, x2, x3)  =  U3_ag(x3)
3.97/1.73	
3.97/1.73	max_out_gaa(x1, x2, x3)  =  max_out_gaa
3.97/1.73	
3.97/1.73	U2_gaa(x1, x2, x3)  =  U2_gaa(x3)
3.97/1.73	
3.97/1.73	less_in_ga(x1, x2)  =  less_in_ga(x1)
3.97/1.73	
3.97/1.73	0  =  0
3.97/1.73	
3.97/1.73	less_out_ga(x1, x2)  =  less_out_ga
3.97/1.73	
3.97/1.73	U3_ga(x1, x2, x3)  =  U3_ga(x3)
3.97/1.73	
3.97/1.73	
3.97/1.73	
3.97/1.73	
3.97/1.73	
3.97/1.73	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.97/1.73	
3.97/1.73	
3.97/1.73	
3.97/1.73	----------------------------------------
3.97/1.73	
3.97/1.73	(2)
3.97/1.73	Obligation:
3.97/1.73	Pi-finite rewrite system:
3.97/1.73	The TRS R consists of the following rules:
3.97/1.73	
3.97/1.73	   max_in_gaa(X, Y, X) -> U1_gaa(X, Y, less_in_ag(Y, X))
3.97/1.73	   less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1))
3.97/1.73	   less_in_ag(s(X), s(Y)) -> U3_ag(X, Y, less_in_ag(X, Y))
3.97/1.73	   U3_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
3.97/1.73	   U1_gaa(X, Y, less_out_ag(Y, X)) -> max_out_gaa(X, Y, X)
3.97/1.73	   max_in_gaa(X, Y, Y) -> U2_gaa(X, Y, less_in_ga(X, s(Y)))
3.97/1.73	   less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1))
3.97/1.73	   less_in_ga(s(X), s(Y)) -> U3_ga(X, Y, less_in_ga(X, Y))
3.97/1.73	   U3_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
3.97/1.73	   U2_gaa(X, Y, less_out_ga(X, s(Y))) -> max_out_gaa(X, Y, Y)
3.97/1.73	
3.97/1.73	The argument filtering Pi contains the following mapping:
3.97/1.73	max_in_gaa(x1, x2, x3)  =  max_in_gaa(x1)
3.97/1.73	
3.97/1.73	U1_gaa(x1, x2, x3)  =  U1_gaa(x3)
3.97/1.73	
3.97/1.73	less_in_ag(x1, x2)  =  less_in_ag(x2)
3.97/1.73	
3.97/1.73	s(x1)  =  s(x1)
3.97/1.73	
3.97/1.73	less_out_ag(x1, x2)  =  less_out_ag(x1)
3.97/1.73	
3.97/1.73	U3_ag(x1, x2, x3)  =  U3_ag(x3)
3.97/1.73	
3.97/1.73	max_out_gaa(x1, x2, x3)  =  max_out_gaa
3.97/1.73	
3.97/1.73	U2_gaa(x1, x2, x3)  =  U2_gaa(x3)
3.97/1.73	
3.97/1.73	less_in_ga(x1, x2)  =  less_in_ga(x1)
3.97/1.73	
3.97/1.73	0  =  0
3.97/1.73	
3.97/1.73	less_out_ga(x1, x2)  =  less_out_ga
3.97/1.73	
3.97/1.73	U3_ga(x1, x2, x3)  =  U3_ga(x3)
3.97/1.73	
3.97/1.73	
3.97/1.73	
3.97/1.73	----------------------------------------
3.97/1.73	
3.97/1.73	(3) DependencyPairsProof (EQUIVALENT)
3.97/1.73	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.97/1.73	Pi DP problem:
3.97/1.73	The TRS P consists of the following rules:
3.97/1.73	
3.97/1.73	   MAX_IN_GAA(X, Y, X) -> U1_GAA(X, Y, less_in_ag(Y, X))
3.97/1.73	   MAX_IN_GAA(X, Y, X) -> LESS_IN_AG(Y, X)
3.97/1.73	   LESS_IN_AG(s(X), s(Y)) -> U3_AG(X, Y, less_in_ag(X, Y))
3.97/1.73	   LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y)
3.97/1.73	   MAX_IN_GAA(X, Y, Y) -> U2_GAA(X, Y, less_in_ga(X, s(Y)))
3.97/1.73	   MAX_IN_GAA(X, Y, Y) -> LESS_IN_GA(X, s(Y))
3.97/1.73	   LESS_IN_GA(s(X), s(Y)) -> U3_GA(X, Y, less_in_ga(X, Y))
3.97/1.73	   LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y)
3.97/1.73	
3.97/1.73	The TRS R consists of the following rules:
3.97/1.73	
3.97/1.73	   max_in_gaa(X, Y, X) -> U1_gaa(X, Y, less_in_ag(Y, X))
3.97/1.73	   less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1))
3.97/1.73	   less_in_ag(s(X), s(Y)) -> U3_ag(X, Y, less_in_ag(X, Y))
3.97/1.73	   U3_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
3.97/1.73	   U1_gaa(X, Y, less_out_ag(Y, X)) -> max_out_gaa(X, Y, X)
3.97/1.73	   max_in_gaa(X, Y, Y) -> U2_gaa(X, Y, less_in_ga(X, s(Y)))
3.97/1.73	   less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1))
3.97/1.73	   less_in_ga(s(X), s(Y)) -> U3_ga(X, Y, less_in_ga(X, Y))
3.97/1.73	   U3_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
3.97/1.73	   U2_gaa(X, Y, less_out_ga(X, s(Y))) -> max_out_gaa(X, Y, Y)
3.97/1.73	
3.97/1.73	The argument filtering Pi contains the following mapping:
3.97/1.73	max_in_gaa(x1, x2, x3)  =  max_in_gaa(x1)
3.97/1.73	
3.97/1.73	U1_gaa(x1, x2, x3)  =  U1_gaa(x3)
3.97/1.73	
3.97/1.73	less_in_ag(x1, x2)  =  less_in_ag(x2)
3.97/1.73	
3.97/1.73	s(x1)  =  s(x1)
3.97/1.73	
3.97/1.73	less_out_ag(x1, x2)  =  less_out_ag(x1)
3.97/1.73	
3.97/1.73	U3_ag(x1, x2, x3)  =  U3_ag(x3)
3.97/1.73	
3.97/1.73	max_out_gaa(x1, x2, x3)  =  max_out_gaa
3.97/1.73	
3.97/1.73	U2_gaa(x1, x2, x3)  =  U2_gaa(x3)
3.97/1.73	
3.97/1.73	less_in_ga(x1, x2)  =  less_in_ga(x1)
3.97/1.73	
3.97/1.73	0  =  0
3.97/1.73	
3.97/1.73	less_out_ga(x1, x2)  =  less_out_ga
3.97/1.73	
3.97/1.73	U3_ga(x1, x2, x3)  =  U3_ga(x3)
3.97/1.73	
3.97/1.73	MAX_IN_GAA(x1, x2, x3)  =  MAX_IN_GAA(x1)
3.97/1.73	
3.97/1.73	U1_GAA(x1, x2, x3)  =  U1_GAA(x3)
3.97/1.73	
3.97/1.73	LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
3.97/1.73	
3.97/1.73	U3_AG(x1, x2, x3)  =  U3_AG(x3)
3.97/1.73	
3.97/1.73	U2_GAA(x1, x2, x3)  =  U2_GAA(x3)
3.97/1.73	
3.97/1.73	LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
3.97/1.73	
3.97/1.73	U3_GA(x1, x2, x3)  =  U3_GA(x3)
3.97/1.73	
3.97/1.73	
3.97/1.73	We have to consider all (P,R,Pi)-chains
3.97/1.73	----------------------------------------
3.97/1.73	
3.97/1.73	(4)
3.97/1.73	Obligation:
3.97/1.73	Pi DP problem:
3.97/1.73	The TRS P consists of the following rules:
3.97/1.73	
3.97/1.73	   MAX_IN_GAA(X, Y, X) -> U1_GAA(X, Y, less_in_ag(Y, X))
3.97/1.73	   MAX_IN_GAA(X, Y, X) -> LESS_IN_AG(Y, X)
3.97/1.73	   LESS_IN_AG(s(X), s(Y)) -> U3_AG(X, Y, less_in_ag(X, Y))
3.97/1.73	   LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y)
3.97/1.73	   MAX_IN_GAA(X, Y, Y) -> U2_GAA(X, Y, less_in_ga(X, s(Y)))
3.97/1.73	   MAX_IN_GAA(X, Y, Y) -> LESS_IN_GA(X, s(Y))
3.97/1.73	   LESS_IN_GA(s(X), s(Y)) -> U3_GA(X, Y, less_in_ga(X, Y))
3.97/1.73	   LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y)
3.97/1.73	
3.97/1.73	The TRS R consists of the following rules:
3.97/1.73	
3.97/1.73	   max_in_gaa(X, Y, X) -> U1_gaa(X, Y, less_in_ag(Y, X))
3.97/1.73	   less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1))
3.97/1.73	   less_in_ag(s(X), s(Y)) -> U3_ag(X, Y, less_in_ag(X, Y))
3.97/1.73	   U3_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
3.97/1.73	   U1_gaa(X, Y, less_out_ag(Y, X)) -> max_out_gaa(X, Y, X)
3.97/1.73	   max_in_gaa(X, Y, Y) -> U2_gaa(X, Y, less_in_ga(X, s(Y)))
3.97/1.73	   less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1))
3.97/1.73	   less_in_ga(s(X), s(Y)) -> U3_ga(X, Y, less_in_ga(X, Y))
3.97/1.73	   U3_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
3.97/1.73	   U2_gaa(X, Y, less_out_ga(X, s(Y))) -> max_out_gaa(X, Y, Y)
3.97/1.73	
3.97/1.73	The argument filtering Pi contains the following mapping:
3.97/1.73	max_in_gaa(x1, x2, x3)  =  max_in_gaa(x1)
3.97/1.73	
3.97/1.73	U1_gaa(x1, x2, x3)  =  U1_gaa(x3)
3.97/1.73	
3.97/1.73	less_in_ag(x1, x2)  =  less_in_ag(x2)
3.97/1.73	
3.97/1.73	s(x1)  =  s(x1)
3.97/1.73	
3.97/1.73	less_out_ag(x1, x2)  =  less_out_ag(x1)
3.97/1.73	
3.97/1.73	U3_ag(x1, x2, x3)  =  U3_ag(x3)
3.97/1.73	
3.97/1.73	max_out_gaa(x1, x2, x3)  =  max_out_gaa
3.97/1.73	
3.97/1.73	U2_gaa(x1, x2, x3)  =  U2_gaa(x3)
3.97/1.73	
3.97/1.73	less_in_ga(x1, x2)  =  less_in_ga(x1)
3.97/1.73	
3.97/1.73	0  =  0
3.97/1.73	
3.97/1.73	less_out_ga(x1, x2)  =  less_out_ga
3.97/1.73	
3.97/1.73	U3_ga(x1, x2, x3)  =  U3_ga(x3)
3.97/1.73	
3.97/1.73	MAX_IN_GAA(x1, x2, x3)  =  MAX_IN_GAA(x1)
3.97/1.73	
3.97/1.73	U1_GAA(x1, x2, x3)  =  U1_GAA(x3)
3.97/1.73	
3.97/1.73	LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
3.97/1.73	
3.97/1.73	U3_AG(x1, x2, x3)  =  U3_AG(x3)
3.97/1.73	
3.97/1.73	U2_GAA(x1, x2, x3)  =  U2_GAA(x3)
3.97/1.73	
3.97/1.73	LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
3.97/1.73	
3.97/1.73	U3_GA(x1, x2, x3)  =  U3_GA(x3)
3.97/1.73	
3.97/1.73	
3.97/1.73	We have to consider all (P,R,Pi)-chains
3.97/1.73	----------------------------------------
3.97/1.73	
3.97/1.73	(5) DependencyGraphProof (EQUIVALENT)
3.97/1.73	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.
3.97/1.73	----------------------------------------
3.97/1.73	
3.97/1.73	(6)
3.97/1.73	Complex Obligation (AND)
3.97/1.73	
3.97/1.73	----------------------------------------
3.97/1.73	
3.97/1.73	(7)
3.97/1.73	Obligation:
3.97/1.73	Pi DP problem:
3.97/1.73	The TRS P consists of the following rules:
3.97/1.73	
3.97/1.73	   LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y)
3.97/1.73	
3.97/1.73	The TRS R consists of the following rules:
3.97/1.73	
3.97/1.73	   max_in_gaa(X, Y, X) -> U1_gaa(X, Y, less_in_ag(Y, X))
3.97/1.73	   less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1))
3.97/1.73	   less_in_ag(s(X), s(Y)) -> U3_ag(X, Y, less_in_ag(X, Y))
3.97/1.73	   U3_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
3.97/1.73	   U1_gaa(X, Y, less_out_ag(Y, X)) -> max_out_gaa(X, Y, X)
3.97/1.73	   max_in_gaa(X, Y, Y) -> U2_gaa(X, Y, less_in_ga(X, s(Y)))
3.97/1.73	   less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1))
3.97/1.73	   less_in_ga(s(X), s(Y)) -> U3_ga(X, Y, less_in_ga(X, Y))
3.97/1.73	   U3_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
3.97/1.73	   U2_gaa(X, Y, less_out_ga(X, s(Y))) -> max_out_gaa(X, Y, Y)
3.97/1.73	
3.97/1.73	The argument filtering Pi contains the following mapping:
3.97/1.73	max_in_gaa(x1, x2, x3)  =  max_in_gaa(x1)
3.97/1.73	
3.97/1.73	U1_gaa(x1, x2, x3)  =  U1_gaa(x3)
3.97/1.73	
3.97/1.73	less_in_ag(x1, x2)  =  less_in_ag(x2)
3.97/1.73	
3.97/1.73	s(x1)  =  s(x1)
3.97/1.73	
3.97/1.73	less_out_ag(x1, x2)  =  less_out_ag(x1)
3.97/1.73	
3.97/1.73	U3_ag(x1, x2, x3)  =  U3_ag(x3)
3.97/1.73	
3.97/1.73	max_out_gaa(x1, x2, x3)  =  max_out_gaa
3.97/1.73	
3.97/1.73	U2_gaa(x1, x2, x3)  =  U2_gaa(x3)
3.97/1.73	
3.97/1.73	less_in_ga(x1, x2)  =  less_in_ga(x1)
3.97/1.73	
3.97/1.73	0  =  0
3.97/1.73	
3.97/1.73	less_out_ga(x1, x2)  =  less_out_ga
3.97/1.73	
3.97/1.73	U3_ga(x1, x2, x3)  =  U3_ga(x3)
3.97/1.73	
3.97/1.73	LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
3.97/1.73	
3.97/1.73	
3.97/1.73	We have to consider all (P,R,Pi)-chains
3.97/1.73	----------------------------------------
3.97/1.73	
3.97/1.73	(8) UsableRulesProof (EQUIVALENT)
3.97/1.73	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.97/1.73	----------------------------------------
3.97/1.73	
3.97/1.73	(9)
3.97/1.73	Obligation:
3.97/1.73	Pi DP problem:
3.97/1.73	The TRS P consists of the following rules:
3.97/1.73	
3.97/1.73	   LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y)
3.97/1.73	
3.97/1.73	R is empty.
3.97/1.73	The argument filtering Pi contains the following mapping:
3.97/1.73	s(x1)  =  s(x1)
3.97/1.73	
3.97/1.73	LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
3.97/1.73	
3.97/1.73	
3.97/1.73	We have to consider all (P,R,Pi)-chains
3.97/1.73	----------------------------------------
3.97/1.73	
3.97/1.73	(10) PiDPToQDPProof (SOUND)
3.97/1.73	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.97/1.73	----------------------------------------
3.97/1.73	
3.97/1.73	(11)
3.97/1.73	Obligation:
3.97/1.73	Q DP problem:
3.97/1.73	The TRS P consists of the following rules:
3.97/1.73	
3.97/1.73	   LESS_IN_GA(s(X)) -> LESS_IN_GA(X)
3.97/1.73	
3.97/1.73	R is empty.
3.97/1.73	Q is empty.
3.97/1.73	We have to consider all (P,Q,R)-chains.
3.97/1.73	----------------------------------------
3.97/1.73	
3.97/1.73	(12) QDPSizeChangeProof (EQUIVALENT)
3.97/1.73	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
3.97/1.73	
3.97/1.73	From the DPs we obtained the following set of size-change graphs:
3.97/1.73	*LESS_IN_GA(s(X)) -> LESS_IN_GA(X)
3.97/1.73	The graph contains the following edges 1 > 1
3.97/1.73	
3.97/1.73	
3.97/1.73	----------------------------------------
3.97/1.73	
3.97/1.73	(13)
3.97/1.73	YES
3.97/1.73	
3.97/1.73	----------------------------------------
3.97/1.73	
3.97/1.73	(14)
3.97/1.73	Obligation:
3.97/1.73	Pi DP problem:
3.97/1.73	The TRS P consists of the following rules:
3.97/1.73	
3.97/1.73	   LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y)
3.97/1.73	
3.97/1.73	The TRS R consists of the following rules:
3.97/1.73	
3.97/1.73	   max_in_gaa(X, Y, X) -> U1_gaa(X, Y, less_in_ag(Y, X))
3.97/1.73	   less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1))
3.97/1.73	   less_in_ag(s(X), s(Y)) -> U3_ag(X, Y, less_in_ag(X, Y))
3.97/1.73	   U3_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
3.97/1.73	   U1_gaa(X, Y, less_out_ag(Y, X)) -> max_out_gaa(X, Y, X)
3.97/1.73	   max_in_gaa(X, Y, Y) -> U2_gaa(X, Y, less_in_ga(X, s(Y)))
3.97/1.73	   less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1))
3.97/1.73	   less_in_ga(s(X), s(Y)) -> U3_ga(X, Y, less_in_ga(X, Y))
3.97/1.73	   U3_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
3.97/1.73	   U2_gaa(X, Y, less_out_ga(X, s(Y))) -> max_out_gaa(X, Y, Y)
3.97/1.73	
3.97/1.73	The argument filtering Pi contains the following mapping:
3.97/1.73	max_in_gaa(x1, x2, x3)  =  max_in_gaa(x1)
3.97/1.73	
3.97/1.73	U1_gaa(x1, x2, x3)  =  U1_gaa(x3)
3.97/1.73	
3.97/1.73	less_in_ag(x1, x2)  =  less_in_ag(x2)
3.97/1.73	
3.97/1.73	s(x1)  =  s(x1)
3.97/1.73	
3.97/1.73	less_out_ag(x1, x2)  =  less_out_ag(x1)
3.97/1.73	
3.97/1.73	U3_ag(x1, x2, x3)  =  U3_ag(x3)
3.97/1.73	
3.97/1.73	max_out_gaa(x1, x2, x3)  =  max_out_gaa
3.97/1.73	
3.97/1.73	U2_gaa(x1, x2, x3)  =  U2_gaa(x3)
3.97/1.73	
3.97/1.73	less_in_ga(x1, x2)  =  less_in_ga(x1)
3.97/1.73	
3.97/1.73	0  =  0
3.97/1.73	
3.97/1.73	less_out_ga(x1, x2)  =  less_out_ga
3.97/1.73	
3.97/1.73	U3_ga(x1, x2, x3)  =  U3_ga(x3)
3.97/1.73	
3.97/1.73	LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
3.97/1.73	
3.97/1.73	
3.97/1.73	We have to consider all (P,R,Pi)-chains
3.97/1.73	----------------------------------------
3.97/1.73	
3.97/1.73	(15) UsableRulesProof (EQUIVALENT)
3.97/1.73	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.97/1.73	----------------------------------------
3.97/1.73	
3.97/1.73	(16)
3.97/1.73	Obligation:
3.97/1.73	Pi DP problem:
3.97/1.73	The TRS P consists of the following rules:
3.97/1.73	
3.97/1.73	   LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y)
3.97/1.73	
3.97/1.73	R is empty.
3.97/1.73	The argument filtering Pi contains the following mapping:
3.97/1.73	s(x1)  =  s(x1)
3.97/1.73	
3.97/1.73	LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
3.97/1.73	
3.97/1.73	
3.97/1.73	We have to consider all (P,R,Pi)-chains
3.97/1.73	----------------------------------------
3.97/1.73	
3.97/1.73	(17) PiDPToQDPProof (SOUND)
3.97/1.73	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.97/1.73	----------------------------------------
3.97/1.73	
3.97/1.73	(18)
3.97/1.73	Obligation:
3.97/1.73	Q DP problem:
3.97/1.73	The TRS P consists of the following rules:
3.97/1.73	
3.97/1.73	   LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y)
3.97/1.73	
3.97/1.73	R is empty.
3.97/1.73	Q is empty.
3.97/1.73	We have to consider all (P,Q,R)-chains.
3.97/1.73	----------------------------------------
3.97/1.73	
3.97/1.73	(19) QDPSizeChangeProof (EQUIVALENT)
3.97/1.73	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
3.97/1.73	
3.97/1.73	From the DPs we obtained the following set of size-change graphs:
3.97/1.73	*LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y)
3.97/1.73	The graph contains the following edges 1 > 1
3.97/1.73	
3.97/1.73	
3.97/1.73	----------------------------------------
3.97/1.73	
3.97/1.73	(20)
3.97/1.73	YES
3.97/1.78	EOF