3.70/1.75	YES
3.75/1.76	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.75/1.76	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.75/1.76	
3.75/1.76	
3.75/1.76	Left Termination of the query pattern
3.75/1.76	
3.75/1.76	max(a,g,a)
3.75/1.76	
3.75/1.76	w.r.t. the given Prolog program could successfully be proven:
3.75/1.76	
3.75/1.76	(0) Prolog
3.75/1.76	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.75/1.76	(2) PiTRS
3.75/1.76	(3) DependencyPairsProof [EQUIVALENT, 6 ms]
3.75/1.76	(4) PiDP
3.75/1.76	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.75/1.76	(6) AND
3.75/1.76	    (7) PiDP
3.75/1.76	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
3.75/1.76	        (9) PiDP
3.75/1.76	        (10) PiDPToQDPProof [SOUND, 0 ms]
3.75/1.76	        (11) QDP
3.75/1.76	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.75/1.76	        (13) YES
3.75/1.76	    (14) PiDP
3.75/1.76	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
3.75/1.76	        (16) PiDP
3.75/1.76	        (17) PiDPToQDPProof [SOUND, 0 ms]
3.75/1.76	        (18) QDP
3.75/1.76	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.75/1.76	        (20) YES
3.75/1.76	
3.75/1.76	
3.75/1.76	----------------------------------------
3.75/1.76	
3.75/1.76	(0)
3.75/1.76	Obligation:
3.75/1.76	Clauses:
3.75/1.76	
3.75/1.76	max(X, Y, X) :- less(Y, X).
3.75/1.76	max(X, Y, Y) :- less(X, s(Y)).
3.75/1.76	less(0, s(X1)).
3.75/1.76	less(s(X), s(Y)) :- less(X, Y).
3.75/1.76	
3.75/1.76	
3.75/1.76	Query: max(a,g,a)
3.75/1.76	----------------------------------------
3.75/1.76	
3.75/1.76	(1) PrologToPiTRSProof (SOUND)
3.75/1.76	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.75/1.76	
3.75/1.76	max_in_3: (f,b,f)
3.75/1.76	
3.75/1.76	less_in_2: (b,f) (f,b)
3.75/1.76	
3.75/1.76	Transforming Prolog into the following Term Rewriting System:
3.75/1.76	
3.75/1.76	Pi-finite rewrite system:
3.75/1.76	The TRS R consists of the following rules:
3.75/1.76	
3.75/1.76	   max_in_aga(X, Y, X) -> U1_aga(X, Y, less_in_ga(Y, X))
3.75/1.76	   less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1))
3.75/1.76	   less_in_ga(s(X), s(Y)) -> U3_ga(X, Y, less_in_ga(X, Y))
3.75/1.76	   U3_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
3.75/1.76	   U1_aga(X, Y, less_out_ga(Y, X)) -> max_out_aga(X, Y, X)
3.75/1.76	   max_in_aga(X, Y, Y) -> U2_aga(X, Y, less_in_ag(X, s(Y)))
3.75/1.76	   less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1))
3.75/1.76	   less_in_ag(s(X), s(Y)) -> U3_ag(X, Y, less_in_ag(X, Y))
3.75/1.76	   U3_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
3.75/1.76	   U2_aga(X, Y, less_out_ag(X, s(Y))) -> max_out_aga(X, Y, Y)
3.75/1.76	
3.75/1.76	The argument filtering Pi contains the following mapping:
3.75/1.76	max_in_aga(x1, x2, x3)  =  max_in_aga(x2)
3.75/1.76	
3.75/1.76	U1_aga(x1, x2, x3)  =  U1_aga(x3)
3.75/1.76	
3.75/1.76	less_in_ga(x1, x2)  =  less_in_ga(x1)
3.75/1.76	
3.75/1.76	0  =  0
3.75/1.76	
3.75/1.76	less_out_ga(x1, x2)  =  less_out_ga
3.75/1.76	
3.75/1.76	s(x1)  =  s(x1)
3.75/1.76	
3.75/1.76	U3_ga(x1, x2, x3)  =  U3_ga(x3)
3.75/1.76	
3.75/1.76	max_out_aga(x1, x2, x3)  =  max_out_aga
3.75/1.76	
3.75/1.76	U2_aga(x1, x2, x3)  =  U2_aga(x3)
3.75/1.76	
3.75/1.76	less_in_ag(x1, x2)  =  less_in_ag(x2)
3.75/1.76	
3.75/1.76	less_out_ag(x1, x2)  =  less_out_ag(x1)
3.75/1.76	
3.75/1.76	U3_ag(x1, x2, x3)  =  U3_ag(x3)
3.75/1.76	
3.75/1.76	
3.75/1.76	
3.75/1.76	
3.75/1.76	
3.75/1.76	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.75/1.76	
3.75/1.76	
3.75/1.76	
3.75/1.76	----------------------------------------
3.75/1.76	
3.75/1.76	(2)
3.75/1.76	Obligation:
3.75/1.76	Pi-finite rewrite system:
3.75/1.76	The TRS R consists of the following rules:
3.75/1.76	
3.75/1.76	   max_in_aga(X, Y, X) -> U1_aga(X, Y, less_in_ga(Y, X))
3.75/1.76	   less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1))
3.75/1.76	   less_in_ga(s(X), s(Y)) -> U3_ga(X, Y, less_in_ga(X, Y))
3.75/1.76	   U3_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
3.75/1.76	   U1_aga(X, Y, less_out_ga(Y, X)) -> max_out_aga(X, Y, X)
3.75/1.76	   max_in_aga(X, Y, Y) -> U2_aga(X, Y, less_in_ag(X, s(Y)))
3.75/1.76	   less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1))
3.75/1.76	   less_in_ag(s(X), s(Y)) -> U3_ag(X, Y, less_in_ag(X, Y))
3.75/1.76	   U3_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
3.75/1.76	   U2_aga(X, Y, less_out_ag(X, s(Y))) -> max_out_aga(X, Y, Y)
3.75/1.76	
3.75/1.76	The argument filtering Pi contains the following mapping:
3.75/1.76	max_in_aga(x1, x2, x3)  =  max_in_aga(x2)
3.75/1.76	
3.75/1.76	U1_aga(x1, x2, x3)  =  U1_aga(x3)
3.75/1.76	
3.75/1.76	less_in_ga(x1, x2)  =  less_in_ga(x1)
3.75/1.76	
3.75/1.76	0  =  0
3.75/1.76	
3.75/1.76	less_out_ga(x1, x2)  =  less_out_ga
3.75/1.76	
3.75/1.76	s(x1)  =  s(x1)
3.75/1.76	
3.75/1.76	U3_ga(x1, x2, x3)  =  U3_ga(x3)
3.75/1.76	
3.75/1.76	max_out_aga(x1, x2, x3)  =  max_out_aga
3.75/1.76	
3.75/1.76	U2_aga(x1, x2, x3)  =  U2_aga(x3)
3.75/1.76	
3.75/1.76	less_in_ag(x1, x2)  =  less_in_ag(x2)
3.75/1.76	
3.75/1.76	less_out_ag(x1, x2)  =  less_out_ag(x1)
3.75/1.76	
3.75/1.76	U3_ag(x1, x2, x3)  =  U3_ag(x3)
3.75/1.76	
3.75/1.76	
3.75/1.76	
3.75/1.76	----------------------------------------
3.75/1.76	
3.75/1.76	(3) DependencyPairsProof (EQUIVALENT)
3.75/1.76	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.75/1.76	Pi DP problem:
3.75/1.76	The TRS P consists of the following rules:
3.75/1.76	
3.75/1.76	   MAX_IN_AGA(X, Y, X) -> U1_AGA(X, Y, less_in_ga(Y, X))
3.75/1.76	   MAX_IN_AGA(X, Y, X) -> LESS_IN_GA(Y, X)
3.75/1.76	   LESS_IN_GA(s(X), s(Y)) -> U3_GA(X, Y, less_in_ga(X, Y))
3.75/1.76	   LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y)
3.75/1.76	   MAX_IN_AGA(X, Y, Y) -> U2_AGA(X, Y, less_in_ag(X, s(Y)))
3.75/1.76	   MAX_IN_AGA(X, Y, Y) -> LESS_IN_AG(X, s(Y))
3.75/1.76	   LESS_IN_AG(s(X), s(Y)) -> U3_AG(X, Y, less_in_ag(X, Y))
3.75/1.76	   LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y)
3.75/1.76	
3.75/1.76	The TRS R consists of the following rules:
3.75/1.76	
3.75/1.76	   max_in_aga(X, Y, X) -> U1_aga(X, Y, less_in_ga(Y, X))
3.75/1.76	   less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1))
3.75/1.76	   less_in_ga(s(X), s(Y)) -> U3_ga(X, Y, less_in_ga(X, Y))
3.75/1.76	   U3_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
3.75/1.76	   U1_aga(X, Y, less_out_ga(Y, X)) -> max_out_aga(X, Y, X)
3.75/1.76	   max_in_aga(X, Y, Y) -> U2_aga(X, Y, less_in_ag(X, s(Y)))
3.75/1.76	   less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1))
3.75/1.76	   less_in_ag(s(X), s(Y)) -> U3_ag(X, Y, less_in_ag(X, Y))
3.75/1.76	   U3_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
3.75/1.76	   U2_aga(X, Y, less_out_ag(X, s(Y))) -> max_out_aga(X, Y, Y)
3.75/1.76	
3.75/1.76	The argument filtering Pi contains the following mapping:
3.75/1.76	max_in_aga(x1, x2, x3)  =  max_in_aga(x2)
3.75/1.76	
3.75/1.76	U1_aga(x1, x2, x3)  =  U1_aga(x3)
3.75/1.76	
3.75/1.76	less_in_ga(x1, x2)  =  less_in_ga(x1)
3.75/1.76	
3.75/1.76	0  =  0
3.75/1.76	
3.75/1.76	less_out_ga(x1, x2)  =  less_out_ga
3.75/1.76	
3.75/1.76	s(x1)  =  s(x1)
3.75/1.76	
3.75/1.76	U3_ga(x1, x2, x3)  =  U3_ga(x3)
3.75/1.76	
3.75/1.76	max_out_aga(x1, x2, x3)  =  max_out_aga
3.75/1.76	
3.75/1.76	U2_aga(x1, x2, x3)  =  U2_aga(x3)
3.75/1.76	
3.75/1.76	less_in_ag(x1, x2)  =  less_in_ag(x2)
3.75/1.76	
3.75/1.76	less_out_ag(x1, x2)  =  less_out_ag(x1)
3.75/1.76	
3.75/1.76	U3_ag(x1, x2, x3)  =  U3_ag(x3)
3.75/1.76	
3.75/1.76	MAX_IN_AGA(x1, x2, x3)  =  MAX_IN_AGA(x2)
3.75/1.76	
3.75/1.76	U1_AGA(x1, x2, x3)  =  U1_AGA(x3)
3.75/1.76	
3.75/1.76	LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
3.75/1.76	
3.75/1.76	U3_GA(x1, x2, x3)  =  U3_GA(x3)
3.75/1.76	
3.75/1.76	U2_AGA(x1, x2, x3)  =  U2_AGA(x3)
3.75/1.76	
3.75/1.76	LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
3.75/1.76	
3.75/1.76	U3_AG(x1, x2, x3)  =  U3_AG(x3)
3.75/1.76	
3.75/1.76	
3.75/1.76	We have to consider all (P,R,Pi)-chains
3.75/1.76	----------------------------------------
3.75/1.76	
3.75/1.76	(4)
3.75/1.76	Obligation:
3.75/1.76	Pi DP problem:
3.75/1.76	The TRS P consists of the following rules:
3.75/1.76	
3.75/1.76	   MAX_IN_AGA(X, Y, X) -> U1_AGA(X, Y, less_in_ga(Y, X))
3.75/1.76	   MAX_IN_AGA(X, Y, X) -> LESS_IN_GA(Y, X)
3.75/1.76	   LESS_IN_GA(s(X), s(Y)) -> U3_GA(X, Y, less_in_ga(X, Y))
3.75/1.76	   LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y)
3.75/1.76	   MAX_IN_AGA(X, Y, Y) -> U2_AGA(X, Y, less_in_ag(X, s(Y)))
3.75/1.76	   MAX_IN_AGA(X, Y, Y) -> LESS_IN_AG(X, s(Y))
3.75/1.76	   LESS_IN_AG(s(X), s(Y)) -> U3_AG(X, Y, less_in_ag(X, Y))
3.75/1.76	   LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y)
3.75/1.76	
3.75/1.76	The TRS R consists of the following rules:
3.75/1.76	
3.75/1.76	   max_in_aga(X, Y, X) -> U1_aga(X, Y, less_in_ga(Y, X))
3.75/1.76	   less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1))
3.75/1.76	   less_in_ga(s(X), s(Y)) -> U3_ga(X, Y, less_in_ga(X, Y))
3.75/1.76	   U3_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
3.75/1.76	   U1_aga(X, Y, less_out_ga(Y, X)) -> max_out_aga(X, Y, X)
3.75/1.76	   max_in_aga(X, Y, Y) -> U2_aga(X, Y, less_in_ag(X, s(Y)))
3.75/1.76	   less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1))
3.75/1.76	   less_in_ag(s(X), s(Y)) -> U3_ag(X, Y, less_in_ag(X, Y))
3.75/1.76	   U3_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
3.75/1.76	   U2_aga(X, Y, less_out_ag(X, s(Y))) -> max_out_aga(X, Y, Y)
3.75/1.76	
3.75/1.76	The argument filtering Pi contains the following mapping:
3.75/1.76	max_in_aga(x1, x2, x3)  =  max_in_aga(x2)
3.75/1.76	
3.75/1.76	U1_aga(x1, x2, x3)  =  U1_aga(x3)
3.75/1.76	
3.75/1.76	less_in_ga(x1, x2)  =  less_in_ga(x1)
3.75/1.76	
3.75/1.76	0  =  0
3.75/1.76	
3.75/1.76	less_out_ga(x1, x2)  =  less_out_ga
3.75/1.76	
3.75/1.76	s(x1)  =  s(x1)
3.75/1.76	
3.75/1.76	U3_ga(x1, x2, x3)  =  U3_ga(x3)
3.75/1.76	
3.75/1.76	max_out_aga(x1, x2, x3)  =  max_out_aga
3.75/1.76	
3.75/1.76	U2_aga(x1, x2, x3)  =  U2_aga(x3)
3.75/1.76	
3.75/1.76	less_in_ag(x1, x2)  =  less_in_ag(x2)
3.75/1.76	
3.75/1.76	less_out_ag(x1, x2)  =  less_out_ag(x1)
3.75/1.76	
3.75/1.76	U3_ag(x1, x2, x3)  =  U3_ag(x3)
3.75/1.76	
3.75/1.76	MAX_IN_AGA(x1, x2, x3)  =  MAX_IN_AGA(x2)
3.75/1.76	
3.75/1.76	U1_AGA(x1, x2, x3)  =  U1_AGA(x3)
3.75/1.76	
3.75/1.76	LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
3.75/1.76	
3.75/1.76	U3_GA(x1, x2, x3)  =  U3_GA(x3)
3.75/1.76	
3.75/1.76	U2_AGA(x1, x2, x3)  =  U2_AGA(x3)
3.75/1.76	
3.75/1.76	LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
3.75/1.76	
3.75/1.76	U3_AG(x1, x2, x3)  =  U3_AG(x3)
3.75/1.76	
3.75/1.76	
3.75/1.76	We have to consider all (P,R,Pi)-chains
3.75/1.76	----------------------------------------
3.75/1.76	
3.75/1.76	(5) DependencyGraphProof (EQUIVALENT)
3.75/1.76	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.
3.75/1.76	----------------------------------------
3.75/1.76	
3.75/1.76	(6)
3.75/1.76	Complex Obligation (AND)
3.75/1.76	
3.75/1.76	----------------------------------------
3.75/1.76	
3.75/1.76	(7)
3.75/1.76	Obligation:
3.75/1.76	Pi DP problem:
3.75/1.76	The TRS P consists of the following rules:
3.75/1.76	
3.75/1.76	   LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y)
3.75/1.76	
3.75/1.76	The TRS R consists of the following rules:
3.75/1.76	
3.75/1.76	   max_in_aga(X, Y, X) -> U1_aga(X, Y, less_in_ga(Y, X))
3.75/1.76	   less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1))
3.75/1.76	   less_in_ga(s(X), s(Y)) -> U3_ga(X, Y, less_in_ga(X, Y))
3.75/1.76	   U3_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
3.75/1.76	   U1_aga(X, Y, less_out_ga(Y, X)) -> max_out_aga(X, Y, X)
3.75/1.76	   max_in_aga(X, Y, Y) -> U2_aga(X, Y, less_in_ag(X, s(Y)))
3.75/1.76	   less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1))
3.75/1.76	   less_in_ag(s(X), s(Y)) -> U3_ag(X, Y, less_in_ag(X, Y))
3.75/1.76	   U3_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
3.75/1.76	   U2_aga(X, Y, less_out_ag(X, s(Y))) -> max_out_aga(X, Y, Y)
3.75/1.76	
3.75/1.76	The argument filtering Pi contains the following mapping:
3.75/1.76	max_in_aga(x1, x2, x3)  =  max_in_aga(x2)
3.75/1.76	
3.75/1.76	U1_aga(x1, x2, x3)  =  U1_aga(x3)
3.75/1.76	
3.75/1.76	less_in_ga(x1, x2)  =  less_in_ga(x1)
3.75/1.76	
3.75/1.76	0  =  0
3.75/1.76	
3.75/1.76	less_out_ga(x1, x2)  =  less_out_ga
3.75/1.76	
3.75/1.76	s(x1)  =  s(x1)
3.75/1.76	
3.75/1.76	U3_ga(x1, x2, x3)  =  U3_ga(x3)
3.75/1.76	
3.75/1.76	max_out_aga(x1, x2, x3)  =  max_out_aga
3.75/1.76	
3.75/1.76	U2_aga(x1, x2, x3)  =  U2_aga(x3)
3.75/1.76	
3.75/1.76	less_in_ag(x1, x2)  =  less_in_ag(x2)
3.75/1.76	
3.75/1.76	less_out_ag(x1, x2)  =  less_out_ag(x1)
3.75/1.76	
3.75/1.76	U3_ag(x1, x2, x3)  =  U3_ag(x3)
3.75/1.76	
3.75/1.76	LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
3.75/1.76	
3.75/1.76	
3.75/1.76	We have to consider all (P,R,Pi)-chains
3.75/1.76	----------------------------------------
3.75/1.76	
3.75/1.76	(8) UsableRulesProof (EQUIVALENT)
3.75/1.76	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.75/1.76	----------------------------------------
3.75/1.76	
3.75/1.76	(9)
3.75/1.76	Obligation:
3.75/1.76	Pi DP problem:
3.75/1.76	The TRS P consists of the following rules:
3.75/1.76	
3.75/1.76	   LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y)
3.75/1.76	
3.75/1.76	R is empty.
3.75/1.76	The argument filtering Pi contains the following mapping:
3.75/1.76	s(x1)  =  s(x1)
3.75/1.76	
3.75/1.76	LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
3.75/1.76	
3.75/1.76	
3.75/1.76	We have to consider all (P,R,Pi)-chains
3.75/1.76	----------------------------------------
3.75/1.76	
3.75/1.76	(10) PiDPToQDPProof (SOUND)
3.75/1.76	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.75/1.76	----------------------------------------
3.75/1.76	
3.75/1.76	(11)
3.75/1.76	Obligation:
3.75/1.76	Q DP problem:
3.75/1.76	The TRS P consists of the following rules:
3.75/1.76	
3.75/1.76	   LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y)
3.75/1.76	
3.75/1.76	R is empty.
3.75/1.76	Q is empty.
3.75/1.76	We have to consider all (P,Q,R)-chains.
3.75/1.76	----------------------------------------
3.75/1.76	
3.75/1.76	(12) QDPSizeChangeProof (EQUIVALENT)
3.75/1.76	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. 
3.75/1.76	
3.75/1.76	From the DPs we obtained the following set of size-change graphs:
3.75/1.76	*LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y)
3.75/1.76	The graph contains the following edges 1 > 1
3.75/1.76	
3.75/1.76	
3.75/1.76	----------------------------------------
3.75/1.76	
3.75/1.76	(13)
3.75/1.76	YES
3.75/1.76	
3.75/1.76	----------------------------------------
3.75/1.76	
3.75/1.76	(14)
3.75/1.76	Obligation:
3.75/1.76	Pi DP problem:
3.75/1.76	The TRS P consists of the following rules:
3.75/1.76	
3.75/1.76	   LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y)
3.75/1.76	
3.75/1.76	The TRS R consists of the following rules:
3.75/1.76	
3.75/1.76	   max_in_aga(X, Y, X) -> U1_aga(X, Y, less_in_ga(Y, X))
3.75/1.76	   less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1))
3.75/1.76	   less_in_ga(s(X), s(Y)) -> U3_ga(X, Y, less_in_ga(X, Y))
3.75/1.76	   U3_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y))
3.75/1.76	   U1_aga(X, Y, less_out_ga(Y, X)) -> max_out_aga(X, Y, X)
3.75/1.76	   max_in_aga(X, Y, Y) -> U2_aga(X, Y, less_in_ag(X, s(Y)))
3.75/1.76	   less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1))
3.75/1.76	   less_in_ag(s(X), s(Y)) -> U3_ag(X, Y, less_in_ag(X, Y))
3.75/1.76	   U3_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
3.75/1.76	   U2_aga(X, Y, less_out_ag(X, s(Y))) -> max_out_aga(X, Y, Y)
3.75/1.76	
3.75/1.76	The argument filtering Pi contains the following mapping:
3.75/1.76	max_in_aga(x1, x2, x3)  =  max_in_aga(x2)
3.75/1.76	
3.75/1.76	U1_aga(x1, x2, x3)  =  U1_aga(x3)
3.75/1.76	
3.75/1.76	less_in_ga(x1, x2)  =  less_in_ga(x1)
3.75/1.76	
3.75/1.76	0  =  0
3.75/1.76	
3.75/1.76	less_out_ga(x1, x2)  =  less_out_ga
3.75/1.76	
3.75/1.76	s(x1)  =  s(x1)
3.75/1.76	
3.75/1.76	U3_ga(x1, x2, x3)  =  U3_ga(x3)
3.75/1.76	
3.75/1.76	max_out_aga(x1, x2, x3)  =  max_out_aga
3.75/1.76	
3.75/1.76	U2_aga(x1, x2, x3)  =  U2_aga(x3)
3.75/1.76	
3.75/1.76	less_in_ag(x1, x2)  =  less_in_ag(x2)
3.75/1.76	
3.75/1.76	less_out_ag(x1, x2)  =  less_out_ag(x1)
3.75/1.76	
3.75/1.76	U3_ag(x1, x2, x3)  =  U3_ag(x3)
3.75/1.76	
3.75/1.76	LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
3.75/1.76	
3.75/1.76	
3.75/1.76	We have to consider all (P,R,Pi)-chains
3.75/1.76	----------------------------------------
3.75/1.76	
3.75/1.76	(15) UsableRulesProof (EQUIVALENT)
3.75/1.76	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.75/1.76	----------------------------------------
3.75/1.76	
3.75/1.76	(16)
3.75/1.76	Obligation:
3.75/1.76	Pi DP problem:
3.75/1.76	The TRS P consists of the following rules:
3.75/1.76	
3.75/1.76	   LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y)
3.75/1.76	
3.75/1.76	R is empty.
3.75/1.76	The argument filtering Pi contains the following mapping:
3.75/1.76	s(x1)  =  s(x1)
3.75/1.76	
3.75/1.76	LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
3.75/1.76	
3.75/1.76	
3.75/1.76	We have to consider all (P,R,Pi)-chains
3.75/1.76	----------------------------------------
3.75/1.76	
3.75/1.76	(17) PiDPToQDPProof (SOUND)
3.75/1.76	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.75/1.76	----------------------------------------
3.75/1.76	
3.75/1.76	(18)
3.75/1.76	Obligation:
3.75/1.76	Q DP problem:
3.75/1.76	The TRS P consists of the following rules:
3.75/1.76	
3.75/1.76	   LESS_IN_GA(s(X)) -> LESS_IN_GA(X)
3.75/1.76	
3.75/1.76	R is empty.
3.75/1.76	Q is empty.
3.75/1.76	We have to consider all (P,Q,R)-chains.
3.75/1.76	----------------------------------------
3.75/1.76	
3.75/1.76	(19) QDPSizeChangeProof (EQUIVALENT)
3.75/1.76	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. 
3.75/1.76	
3.75/1.76	From the DPs we obtained the following set of size-change graphs:
3.75/1.76	*LESS_IN_GA(s(X)) -> LESS_IN_GA(X)
3.75/1.76	The graph contains the following edges 1 > 1
3.75/1.76	
3.75/1.76	
3.75/1.76	----------------------------------------
3.75/1.76	
3.75/1.76	(20)
3.75/1.76	YES
3.75/1.79	EOF