3.63/1.77	YES
3.94/1.78	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.94/1.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.94/1.78	
3.94/1.78	
3.94/1.78	Left Termination of the query pattern
3.94/1.78	
3.94/1.78	rev(g,a)
3.94/1.78	
3.94/1.78	w.r.t. the given Prolog program could successfully be proven:
3.94/1.78	
3.94/1.78	(0) Prolog
3.94/1.78	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.94/1.78	(2) PiTRS
3.94/1.78	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.94/1.78	(4) PiDP
3.94/1.78	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.94/1.78	(6) AND
3.94/1.78	    (7) PiDP
3.94/1.78	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
3.94/1.78	        (9) PiDP
3.94/1.78	        (10) PiDPToQDPProof [SOUND, 0 ms]
3.94/1.78	        (11) QDP
3.94/1.78	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.94/1.78	        (13) YES
3.94/1.78	    (14) PiDP
3.94/1.78	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
3.94/1.78	        (16) PiDP
3.94/1.78	        (17) PiDPToQDPProof [SOUND, 0 ms]
3.94/1.78	        (18) QDP
3.94/1.78	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.94/1.78	        (20) YES
3.94/1.78	
3.94/1.78	
3.94/1.78	----------------------------------------
3.94/1.78	
3.94/1.78	(0)
3.94/1.78	Obligation:
3.94/1.78	Clauses:
3.94/1.78	
3.94/1.78	rev([], []).
3.94/1.78	rev(.(X, Xs), Ys) :- ','(rev(Xs, Zs), app(Zs, .(X, []), Ys)).
3.94/1.78	app([], X, X).
3.94/1.78	app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
3.94/1.78	
3.94/1.78	
3.94/1.78	Query: rev(g,a)
3.94/1.78	----------------------------------------
3.94/1.78	
3.94/1.78	(1) PrologToPiTRSProof (SOUND)
3.94/1.78	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.94/1.78	
3.94/1.78	rev_in_2: (b,f)
3.94/1.78	
3.94/1.78	app_in_3: (b,b,f)
3.94/1.78	
3.94/1.78	Transforming Prolog into the following Term Rewriting System:
3.94/1.78	
3.94/1.78	Pi-finite rewrite system:
3.94/1.78	The TRS R consists of the following rules:
3.94/1.78	
3.94/1.78	   rev_in_ga([], []) -> rev_out_ga([], [])
3.94/1.78	   rev_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, rev_in_ga(Xs, Zs))
3.94/1.78	   U1_ga(X, Xs, Ys, rev_out_ga(Xs, Zs)) -> U2_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
3.94/1.78	   app_in_gga([], X, X) -> app_out_gga([], X, X)
3.94/1.78	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
3.94/1.78	   U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
3.94/1.78	   U2_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> rev_out_ga(.(X, Xs), Ys)
3.94/1.78	
3.94/1.78	The argument filtering Pi contains the following mapping:
3.94/1.78	rev_in_ga(x1, x2)  =  rev_in_ga(x1)
3.94/1.78	
3.94/1.78	[]  =  []
3.94/1.78	
3.94/1.78	rev_out_ga(x1, x2)  =  rev_out_ga(x2)
3.94/1.78	
3.94/1.78	.(x1, x2)  =  .(x1, x2)
3.94/1.78	
3.94/1.78	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
3.94/1.78	
3.94/1.78	U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
3.94/1.78	
3.94/1.78	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
3.94/1.78	
3.94/1.78	app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
3.94/1.78	
3.94/1.78	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)
3.94/1.78	
3.94/1.78	
3.94/1.78	
3.94/1.78	
3.94/1.78	
3.94/1.78	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.94/1.78	
3.94/1.78	
3.94/1.78	
3.94/1.78	----------------------------------------
3.94/1.78	
3.94/1.78	(2)
3.94/1.78	Obligation:
3.94/1.78	Pi-finite rewrite system:
3.94/1.78	The TRS R consists of the following rules:
3.94/1.78	
3.94/1.78	   rev_in_ga([], []) -> rev_out_ga([], [])
3.94/1.78	   rev_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, rev_in_ga(Xs, Zs))
3.94/1.78	   U1_ga(X, Xs, Ys, rev_out_ga(Xs, Zs)) -> U2_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
3.94/1.78	   app_in_gga([], X, X) -> app_out_gga([], X, X)
3.94/1.78	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
3.94/1.78	   U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
3.94/1.78	   U2_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> rev_out_ga(.(X, Xs), Ys)
3.94/1.78	
3.94/1.78	The argument filtering Pi contains the following mapping:
3.94/1.78	rev_in_ga(x1, x2)  =  rev_in_ga(x1)
3.94/1.78	
3.94/1.78	[]  =  []
3.94/1.78	
3.94/1.78	rev_out_ga(x1, x2)  =  rev_out_ga(x2)
3.94/1.78	
3.94/1.78	.(x1, x2)  =  .(x1, x2)
3.94/1.78	
3.94/1.78	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
3.94/1.78	
3.94/1.78	U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
3.94/1.78	
3.94/1.78	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
3.94/1.78	
3.94/1.78	app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
3.94/1.78	
3.94/1.78	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)
3.94/1.78	
3.94/1.78	
3.94/1.78	
3.94/1.78	----------------------------------------
3.94/1.78	
3.94/1.78	(3) DependencyPairsProof (EQUIVALENT)
3.94/1.78	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.94/1.78	Pi DP problem:
3.94/1.78	The TRS P consists of the following rules:
3.94/1.78	
3.94/1.78	   REV_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, rev_in_ga(Xs, Zs))
3.94/1.78	   REV_IN_GA(.(X, Xs), Ys) -> REV_IN_GA(Xs, Zs)
3.94/1.78	   U1_GA(X, Xs, Ys, rev_out_ga(Xs, Zs)) -> U2_GA(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
3.94/1.78	   U1_GA(X, Xs, Ys, rev_out_ga(Xs, Zs)) -> APP_IN_GGA(Zs, .(X, []), Ys)
3.94/1.78	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U3_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
3.94/1.78	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
3.94/1.78	
3.94/1.78	The TRS R consists of the following rules:
3.94/1.78	
3.94/1.78	   rev_in_ga([], []) -> rev_out_ga([], [])
3.94/1.78	   rev_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, rev_in_ga(Xs, Zs))
3.94/1.78	   U1_ga(X, Xs, Ys, rev_out_ga(Xs, Zs)) -> U2_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
3.94/1.78	   app_in_gga([], X, X) -> app_out_gga([], X, X)
3.94/1.78	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
3.94/1.78	   U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
3.94/1.78	   U2_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> rev_out_ga(.(X, Xs), Ys)
3.94/1.78	
3.94/1.78	The argument filtering Pi contains the following mapping:
3.94/1.78	rev_in_ga(x1, x2)  =  rev_in_ga(x1)
3.94/1.78	
3.94/1.78	[]  =  []
3.94/1.78	
3.94/1.78	rev_out_ga(x1, x2)  =  rev_out_ga(x2)
3.94/1.78	
3.94/1.78	.(x1, x2)  =  .(x1, x2)
3.94/1.78	
3.94/1.78	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
3.94/1.78	
3.94/1.78	U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
3.94/1.78	
3.94/1.78	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
3.94/1.78	
3.94/1.78	app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
3.94/1.78	
3.94/1.78	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)
3.94/1.78	
3.94/1.78	REV_IN_GA(x1, x2)  =  REV_IN_GA(x1)
3.94/1.78	
3.94/1.78	U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
3.94/1.78	
3.94/1.78	U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
3.94/1.78	
3.94/1.78	APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
3.94/1.78	
3.94/1.78	U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x5)
3.94/1.78	
3.94/1.78	
3.94/1.78	We have to consider all (P,R,Pi)-chains
3.94/1.78	----------------------------------------
3.94/1.78	
3.94/1.78	(4)
3.94/1.78	Obligation:
3.94/1.78	Pi DP problem:
3.94/1.78	The TRS P consists of the following rules:
3.94/1.78	
3.94/1.78	   REV_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, rev_in_ga(Xs, Zs))
3.94/1.78	   REV_IN_GA(.(X, Xs), Ys) -> REV_IN_GA(Xs, Zs)
3.94/1.78	   U1_GA(X, Xs, Ys, rev_out_ga(Xs, Zs)) -> U2_GA(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
3.94/1.78	   U1_GA(X, Xs, Ys, rev_out_ga(Xs, Zs)) -> APP_IN_GGA(Zs, .(X, []), Ys)
3.94/1.78	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U3_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
3.94/1.78	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
3.94/1.78	
3.94/1.78	The TRS R consists of the following rules:
3.94/1.78	
3.94/1.78	   rev_in_ga([], []) -> rev_out_ga([], [])
3.94/1.78	   rev_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, rev_in_ga(Xs, Zs))
3.94/1.78	   U1_ga(X, Xs, Ys, rev_out_ga(Xs, Zs)) -> U2_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
3.94/1.78	   app_in_gga([], X, X) -> app_out_gga([], X, X)
3.94/1.78	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
3.94/1.78	   U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
3.94/1.78	   U2_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> rev_out_ga(.(X, Xs), Ys)
3.94/1.78	
3.94/1.78	The argument filtering Pi contains the following mapping:
3.94/1.78	rev_in_ga(x1, x2)  =  rev_in_ga(x1)
3.94/1.78	
3.94/1.78	[]  =  []
3.94/1.78	
3.94/1.78	rev_out_ga(x1, x2)  =  rev_out_ga(x2)
3.94/1.78	
3.94/1.78	.(x1, x2)  =  .(x1, x2)
3.94/1.78	
3.94/1.78	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
3.94/1.78	
3.94/1.78	U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
3.94/1.78	
3.94/1.78	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
3.94/1.78	
3.94/1.78	app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
3.94/1.78	
3.94/1.78	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)
3.94/1.78	
3.94/1.78	REV_IN_GA(x1, x2)  =  REV_IN_GA(x1)
3.94/1.78	
3.94/1.78	U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
3.94/1.78	
3.94/1.78	U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
3.94/1.78	
3.94/1.78	APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
3.94/1.78	
3.94/1.78	U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x5)
3.94/1.78	
3.94/1.78	
3.94/1.78	We have to consider all (P,R,Pi)-chains
3.94/1.78	----------------------------------------
3.94/1.78	
3.94/1.78	(5) DependencyGraphProof (EQUIVALENT)
3.94/1.78	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes.
3.94/1.78	----------------------------------------
3.94/1.78	
3.94/1.78	(6)
3.94/1.78	Complex Obligation (AND)
3.94/1.78	
3.94/1.78	----------------------------------------
3.94/1.78	
3.94/1.78	(7)
3.94/1.78	Obligation:
3.94/1.78	Pi DP problem:
3.94/1.78	The TRS P consists of the following rules:
3.94/1.78	
3.94/1.78	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
3.94/1.78	
3.94/1.78	The TRS R consists of the following rules:
3.94/1.78	
3.94/1.78	   rev_in_ga([], []) -> rev_out_ga([], [])
3.94/1.78	   rev_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, rev_in_ga(Xs, Zs))
3.94/1.78	   U1_ga(X, Xs, Ys, rev_out_ga(Xs, Zs)) -> U2_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
3.94/1.78	   app_in_gga([], X, X) -> app_out_gga([], X, X)
3.94/1.78	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
3.94/1.78	   U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
3.94/1.78	   U2_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> rev_out_ga(.(X, Xs), Ys)
3.94/1.78	
3.94/1.78	The argument filtering Pi contains the following mapping:
3.94/1.78	rev_in_ga(x1, x2)  =  rev_in_ga(x1)
3.94/1.78	
3.94/1.78	[]  =  []
3.94/1.78	
3.94/1.78	rev_out_ga(x1, x2)  =  rev_out_ga(x2)
3.94/1.78	
3.94/1.78	.(x1, x2)  =  .(x1, x2)
3.94/1.78	
3.94/1.78	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
3.94/1.78	
3.94/1.78	U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
3.94/1.78	
3.94/1.78	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
3.94/1.78	
3.94/1.78	app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
3.94/1.78	
3.94/1.78	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)
3.94/1.78	
3.94/1.78	APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
3.94/1.78	
3.94/1.78	
3.94/1.78	We have to consider all (P,R,Pi)-chains
3.94/1.78	----------------------------------------
3.94/1.78	
3.94/1.78	(8) UsableRulesProof (EQUIVALENT)
3.94/1.78	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.94/1.78	----------------------------------------
3.94/1.78	
3.94/1.78	(9)
3.94/1.78	Obligation:
3.94/1.78	Pi DP problem:
3.94/1.78	The TRS P consists of the following rules:
3.94/1.78	
3.94/1.78	   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
3.94/1.78	
3.94/1.78	R is empty.
3.94/1.78	The argument filtering Pi contains the following mapping:
3.94/1.78	.(x1, x2)  =  .(x1, x2)
3.94/1.78	
3.94/1.78	APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
3.94/1.78	
3.94/1.78	
3.94/1.78	We have to consider all (P,R,Pi)-chains
3.94/1.78	----------------------------------------
3.94/1.78	
3.94/1.78	(10) PiDPToQDPProof (SOUND)
3.94/1.78	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.94/1.78	----------------------------------------
3.94/1.78	
3.94/1.78	(11)
3.94/1.78	Obligation:
3.94/1.78	Q DP problem:
3.94/1.78	The TRS P consists of the following rules:
3.94/1.78	
3.94/1.78	   APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys)
3.94/1.78	
3.94/1.78	R is empty.
3.94/1.78	Q is empty.
3.94/1.78	We have to consider all (P,Q,R)-chains.
3.94/1.78	----------------------------------------
3.94/1.78	
3.94/1.78	(12) QDPSizeChangeProof (EQUIVALENT)
3.94/1.78	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. 
3.94/1.78	
3.94/1.78	From the DPs we obtained the following set of size-change graphs:
3.94/1.78	*APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys)
3.94/1.78	The graph contains the following edges 1 > 1, 2 >= 2
3.94/1.78	
3.94/1.78	
3.94/1.78	----------------------------------------
3.94/1.78	
3.94/1.78	(13)
3.94/1.78	YES
3.94/1.78	
3.94/1.78	----------------------------------------
3.94/1.78	
3.94/1.78	(14)
3.94/1.78	Obligation:
3.94/1.78	Pi DP problem:
3.94/1.78	The TRS P consists of the following rules:
3.94/1.78	
3.94/1.78	   REV_IN_GA(.(X, Xs), Ys) -> REV_IN_GA(Xs, Zs)
3.94/1.78	
3.94/1.78	The TRS R consists of the following rules:
3.94/1.78	
3.94/1.78	   rev_in_ga([], []) -> rev_out_ga([], [])
3.94/1.78	   rev_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, rev_in_ga(Xs, Zs))
3.94/1.78	   U1_ga(X, Xs, Ys, rev_out_ga(Xs, Zs)) -> U2_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
3.94/1.78	   app_in_gga([], X, X) -> app_out_gga([], X, X)
3.94/1.78	   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
3.94/1.78	   U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
3.94/1.78	   U2_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) -> rev_out_ga(.(X, Xs), Ys)
3.94/1.78	
3.94/1.78	The argument filtering Pi contains the following mapping:
3.94/1.78	rev_in_ga(x1, x2)  =  rev_in_ga(x1)
3.94/1.78	
3.94/1.78	[]  =  []
3.94/1.78	
3.94/1.78	rev_out_ga(x1, x2)  =  rev_out_ga(x2)
3.94/1.78	
3.94/1.78	.(x1, x2)  =  .(x1, x2)
3.94/1.78	
3.94/1.78	U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
3.94/1.78	
3.94/1.78	U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
3.94/1.78	
3.94/1.78	app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
3.94/1.78	
3.94/1.78	app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
3.94/1.78	
3.94/1.78	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)
3.94/1.78	
3.94/1.78	REV_IN_GA(x1, x2)  =  REV_IN_GA(x1)
3.94/1.78	
3.94/1.78	
3.94/1.78	We have to consider all (P,R,Pi)-chains
3.94/1.78	----------------------------------------
3.94/1.78	
3.94/1.78	(15) UsableRulesProof (EQUIVALENT)
3.94/1.78	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.94/1.78	----------------------------------------
3.94/1.78	
3.94/1.78	(16)
3.94/1.78	Obligation:
3.94/1.78	Pi DP problem:
3.94/1.78	The TRS P consists of the following rules:
3.94/1.78	
3.94/1.78	   REV_IN_GA(.(X, Xs), Ys) -> REV_IN_GA(Xs, Zs)
3.94/1.78	
3.94/1.78	R is empty.
3.94/1.78	The argument filtering Pi contains the following mapping:
3.94/1.78	.(x1, x2)  =  .(x1, x2)
3.94/1.78	
3.94/1.78	REV_IN_GA(x1, x2)  =  REV_IN_GA(x1)
3.94/1.78	
3.94/1.78	
3.94/1.78	We have to consider all (P,R,Pi)-chains
3.94/1.78	----------------------------------------
3.94/1.78	
3.94/1.78	(17) PiDPToQDPProof (SOUND)
3.94/1.78	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.94/1.78	----------------------------------------
3.94/1.78	
3.94/1.78	(18)
3.94/1.78	Obligation:
3.94/1.78	Q DP problem:
3.94/1.78	The TRS P consists of the following rules:
3.94/1.78	
3.94/1.78	   REV_IN_GA(.(X, Xs)) -> REV_IN_GA(Xs)
3.94/1.78	
3.94/1.78	R is empty.
3.94/1.78	Q is empty.
3.94/1.78	We have to consider all (P,Q,R)-chains.
3.94/1.78	----------------------------------------
3.94/1.78	
3.94/1.78	(19) QDPSizeChangeProof (EQUIVALENT)
3.94/1.78	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. 
3.94/1.78	
3.94/1.78	From the DPs we obtained the following set of size-change graphs:
3.94/1.78	*REV_IN_GA(.(X, Xs)) -> REV_IN_GA(Xs)
3.94/1.78	The graph contains the following edges 1 > 1
3.94/1.78	
3.94/1.78	
3.94/1.78	----------------------------------------
3.94/1.78	
3.94/1.78	(20)
3.94/1.78	YES
3.99/1.81	EOF