3.97/1.85	YES
3.97/1.87	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.97/1.87	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.97/1.87	
3.97/1.87	
3.97/1.87	Left Termination of the query pattern
3.97/1.87	
3.97/1.87	goal(g)
3.97/1.87	
3.97/1.87	w.r.t. the given Prolog program could successfully be proven:
3.97/1.87	
3.97/1.87	(0) Prolog
3.97/1.87	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.97/1.87	(2) PiTRS
3.97/1.87	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.97/1.87	(4) PiDP
3.97/1.87	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.97/1.87	(6) AND
3.97/1.87	    (7) PiDP
3.97/1.87	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
3.97/1.87	        (9) PiDP
3.97/1.87	        (10) PiDPToQDPProof [SOUND, 0 ms]
3.97/1.87	        (11) QDP
3.97/1.87	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.97/1.87	        (13) YES
3.97/1.87	    (14) PiDP
3.97/1.87	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
3.97/1.87	        (16) PiDP
3.97/1.87	        (17) PiDPToQDPProof [SOUND, 0 ms]
3.97/1.87	        (18) QDP
3.97/1.87	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.97/1.87	        (20) YES
3.97/1.87	
3.97/1.87	
3.97/1.87	----------------------------------------
3.97/1.87	
3.97/1.87	(0)
3.97/1.87	Obligation:
3.97/1.87	Clauses:
3.97/1.87	
3.97/1.87	list([]).
3.97/1.87	list(.(X, XS)) :- list(XS).
3.97/1.87	s2l(s(X), .(Y, Xs)) :- s2l(X, Xs).
3.97/1.87	s2l(0, []).
3.97/1.87	goal(X) :- ','(s2l(X, XS), list(XS)).
3.97/1.87	
3.97/1.87	
3.97/1.87	Query: goal(g)
3.97/1.87	----------------------------------------
3.97/1.87	
3.97/1.87	(1) PrologToPiTRSProof (SOUND)
3.97/1.87	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.97/1.87	
3.97/1.87	goal_in_1: (b)
3.97/1.87	
3.97/1.87	s2l_in_2: (b,f)
3.97/1.87	
3.97/1.87	list_in_1: (b)
3.97/1.87	
3.97/1.87	Transforming Prolog into the following Term Rewriting System:
3.97/1.87	
3.97/1.87	Pi-finite rewrite system:
3.97/1.87	The TRS R consists of the following rules:
3.97/1.87	
3.97/1.87	   goal_in_g(X) -> U3_g(X, s2l_in_ga(X, XS))
3.97/1.87	   s2l_in_ga(s(X), .(Y, Xs)) -> U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
3.97/1.87	   s2l_in_ga(0, []) -> s2l_out_ga(0, [])
3.97/1.87	   U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs))
3.97/1.87	   U3_g(X, s2l_out_ga(X, XS)) -> U4_g(X, list_in_g(XS))
3.97/1.87	   list_in_g([]) -> list_out_g([])
3.97/1.87	   list_in_g(.(X, XS)) -> U1_g(X, XS, list_in_g(XS))
3.97/1.87	   U1_g(X, XS, list_out_g(XS)) -> list_out_g(.(X, XS))
3.97/1.87	   U4_g(X, list_out_g(XS)) -> goal_out_g(X)
3.97/1.87	
3.97/1.87	The argument filtering Pi contains the following mapping:
3.97/1.87	goal_in_g(x1)  =  goal_in_g(x1)
3.97/1.87	
3.97/1.87	U3_g(x1, x2)  =  U3_g(x1, x2)
3.97/1.87	
3.97/1.87	s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
3.97/1.87	
3.97/1.87	s(x1)  =  s(x1)
3.97/1.87	
3.97/1.87	U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
3.97/1.87	
3.97/1.87	0  =  0
3.97/1.87	
3.97/1.87	s2l_out_ga(x1, x2)  =  s2l_out_ga(x1, x2)
3.97/1.87	
3.97/1.87	.(x1, x2)  =  .(x2)
3.97/1.87	
3.97/1.87	U4_g(x1, x2)  =  U4_g(x1, x2)
3.97/1.87	
3.97/1.87	list_in_g(x1)  =  list_in_g(x1)
3.97/1.87	
3.97/1.87	[]  =  []
3.97/1.87	
3.97/1.87	list_out_g(x1)  =  list_out_g(x1)
3.97/1.87	
3.97/1.87	U1_g(x1, x2, x3)  =  U1_g(x2, x3)
3.97/1.87	
3.97/1.87	goal_out_g(x1)  =  goal_out_g(x1)
3.97/1.87	
3.97/1.87	
3.97/1.87	
3.97/1.87	
3.97/1.87	
3.97/1.87	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.97/1.87	
3.97/1.87	
3.97/1.87	
3.97/1.87	----------------------------------------
3.97/1.87	
3.97/1.87	(2)
3.97/1.87	Obligation:
3.97/1.87	Pi-finite rewrite system:
3.97/1.87	The TRS R consists of the following rules:
3.97/1.87	
3.97/1.87	   goal_in_g(X) -> U3_g(X, s2l_in_ga(X, XS))
3.97/1.87	   s2l_in_ga(s(X), .(Y, Xs)) -> U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
3.97/1.87	   s2l_in_ga(0, []) -> s2l_out_ga(0, [])
3.97/1.87	   U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs))
3.97/1.87	   U3_g(X, s2l_out_ga(X, XS)) -> U4_g(X, list_in_g(XS))
3.97/1.87	   list_in_g([]) -> list_out_g([])
3.97/1.87	   list_in_g(.(X, XS)) -> U1_g(X, XS, list_in_g(XS))
3.97/1.87	   U1_g(X, XS, list_out_g(XS)) -> list_out_g(.(X, XS))
3.97/1.87	   U4_g(X, list_out_g(XS)) -> goal_out_g(X)
3.97/1.87	
3.97/1.87	The argument filtering Pi contains the following mapping:
3.97/1.87	goal_in_g(x1)  =  goal_in_g(x1)
3.97/1.87	
3.97/1.87	U3_g(x1, x2)  =  U3_g(x1, x2)
3.97/1.87	
3.97/1.87	s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
3.97/1.87	
3.97/1.87	s(x1)  =  s(x1)
3.97/1.87	
3.97/1.87	U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
3.97/1.87	
3.97/1.87	0  =  0
3.97/1.87	
3.97/1.87	s2l_out_ga(x1, x2)  =  s2l_out_ga(x1, x2)
3.97/1.87	
3.97/1.87	.(x1, x2)  =  .(x2)
3.97/1.87	
3.97/1.87	U4_g(x1, x2)  =  U4_g(x1, x2)
3.97/1.87	
3.97/1.87	list_in_g(x1)  =  list_in_g(x1)
3.97/1.87	
3.97/1.87	[]  =  []
3.97/1.87	
3.97/1.87	list_out_g(x1)  =  list_out_g(x1)
3.97/1.87	
3.97/1.87	U1_g(x1, x2, x3)  =  U1_g(x2, x3)
3.97/1.87	
3.97/1.87	goal_out_g(x1)  =  goal_out_g(x1)
3.97/1.87	
3.97/1.87	
3.97/1.87	
3.97/1.87	----------------------------------------
3.97/1.87	
3.97/1.87	(3) DependencyPairsProof (EQUIVALENT)
3.97/1.87	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.97/1.87	Pi DP problem:
3.97/1.87	The TRS P consists of the following rules:
3.97/1.87	
3.97/1.87	   GOAL_IN_G(X) -> U3_G(X, s2l_in_ga(X, XS))
3.97/1.87	   GOAL_IN_G(X) -> S2L_IN_GA(X, XS)
3.97/1.87	   S2L_IN_GA(s(X), .(Y, Xs)) -> U2_GA(X, Y, Xs, s2l_in_ga(X, Xs))
3.97/1.87	   S2L_IN_GA(s(X), .(Y, Xs)) -> S2L_IN_GA(X, Xs)
3.97/1.87	   U3_G(X, s2l_out_ga(X, XS)) -> U4_G(X, list_in_g(XS))
3.97/1.87	   U3_G(X, s2l_out_ga(X, XS)) -> LIST_IN_G(XS)
3.97/1.87	   LIST_IN_G(.(X, XS)) -> U1_G(X, XS, list_in_g(XS))
3.97/1.87	   LIST_IN_G(.(X, XS)) -> LIST_IN_G(XS)
3.97/1.87	
3.97/1.87	The TRS R consists of the following rules:
3.97/1.87	
3.97/1.87	   goal_in_g(X) -> U3_g(X, s2l_in_ga(X, XS))
3.97/1.87	   s2l_in_ga(s(X), .(Y, Xs)) -> U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
3.97/1.87	   s2l_in_ga(0, []) -> s2l_out_ga(0, [])
3.97/1.87	   U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs))
3.97/1.87	   U3_g(X, s2l_out_ga(X, XS)) -> U4_g(X, list_in_g(XS))
3.97/1.87	   list_in_g([]) -> list_out_g([])
3.97/1.87	   list_in_g(.(X, XS)) -> U1_g(X, XS, list_in_g(XS))
3.97/1.87	   U1_g(X, XS, list_out_g(XS)) -> list_out_g(.(X, XS))
3.97/1.87	   U4_g(X, list_out_g(XS)) -> goal_out_g(X)
3.97/1.87	
3.97/1.87	The argument filtering Pi contains the following mapping:
3.97/1.87	goal_in_g(x1)  =  goal_in_g(x1)
3.97/1.87	
3.97/1.87	U3_g(x1, x2)  =  U3_g(x1, x2)
3.97/1.87	
3.97/1.87	s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
3.97/1.87	
3.97/1.87	s(x1)  =  s(x1)
3.97/1.87	
3.97/1.87	U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
3.97/1.87	
3.97/1.87	0  =  0
3.97/1.87	
3.97/1.87	s2l_out_ga(x1, x2)  =  s2l_out_ga(x1, x2)
3.97/1.87	
3.97/1.87	.(x1, x2)  =  .(x2)
3.97/1.87	
3.97/1.87	U4_g(x1, x2)  =  U4_g(x1, x2)
3.97/1.87	
3.97/1.87	list_in_g(x1)  =  list_in_g(x1)
3.97/1.87	
3.97/1.87	[]  =  []
3.97/1.87	
3.97/1.87	list_out_g(x1)  =  list_out_g(x1)
3.97/1.87	
3.97/1.87	U1_g(x1, x2, x3)  =  U1_g(x2, x3)
3.97/1.87	
3.97/1.87	goal_out_g(x1)  =  goal_out_g(x1)
3.97/1.87	
3.97/1.87	GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
3.97/1.87	
3.97/1.87	U3_G(x1, x2)  =  U3_G(x1, x2)
3.97/1.87	
3.97/1.87	S2L_IN_GA(x1, x2)  =  S2L_IN_GA(x1)
3.97/1.87	
3.97/1.87	U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
3.97/1.87	
3.97/1.87	U4_G(x1, x2)  =  U4_G(x1, x2)
3.97/1.87	
3.97/1.87	LIST_IN_G(x1)  =  LIST_IN_G(x1)
3.97/1.87	
3.97/1.87	U1_G(x1, x2, x3)  =  U1_G(x2, x3)
3.97/1.87	
3.97/1.87	
3.97/1.87	We have to consider all (P,R,Pi)-chains
3.97/1.87	----------------------------------------
3.97/1.87	
3.97/1.87	(4)
3.97/1.87	Obligation:
3.97/1.87	Pi DP problem:
3.97/1.87	The TRS P consists of the following rules:
3.97/1.87	
3.97/1.87	   GOAL_IN_G(X) -> U3_G(X, s2l_in_ga(X, XS))
3.97/1.87	   GOAL_IN_G(X) -> S2L_IN_GA(X, XS)
3.97/1.87	   S2L_IN_GA(s(X), .(Y, Xs)) -> U2_GA(X, Y, Xs, s2l_in_ga(X, Xs))
3.97/1.87	   S2L_IN_GA(s(X), .(Y, Xs)) -> S2L_IN_GA(X, Xs)
3.97/1.87	   U3_G(X, s2l_out_ga(X, XS)) -> U4_G(X, list_in_g(XS))
3.97/1.87	   U3_G(X, s2l_out_ga(X, XS)) -> LIST_IN_G(XS)
3.97/1.87	   LIST_IN_G(.(X, XS)) -> U1_G(X, XS, list_in_g(XS))
3.97/1.87	   LIST_IN_G(.(X, XS)) -> LIST_IN_G(XS)
3.97/1.87	
3.97/1.87	The TRS R consists of the following rules:
3.97/1.87	
3.97/1.87	   goal_in_g(X) -> U3_g(X, s2l_in_ga(X, XS))
3.97/1.87	   s2l_in_ga(s(X), .(Y, Xs)) -> U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
3.97/1.87	   s2l_in_ga(0, []) -> s2l_out_ga(0, [])
3.97/1.87	   U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs))
3.97/1.87	   U3_g(X, s2l_out_ga(X, XS)) -> U4_g(X, list_in_g(XS))
3.97/1.87	   list_in_g([]) -> list_out_g([])
3.97/1.87	   list_in_g(.(X, XS)) -> U1_g(X, XS, list_in_g(XS))
3.97/1.87	   U1_g(X, XS, list_out_g(XS)) -> list_out_g(.(X, XS))
3.97/1.87	   U4_g(X, list_out_g(XS)) -> goal_out_g(X)
3.97/1.87	
3.97/1.87	The argument filtering Pi contains the following mapping:
3.97/1.87	goal_in_g(x1)  =  goal_in_g(x1)
3.97/1.87	
3.97/1.87	U3_g(x1, x2)  =  U3_g(x1, x2)
3.97/1.87	
3.97/1.87	s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
3.97/1.87	
3.97/1.87	s(x1)  =  s(x1)
3.97/1.87	
3.97/1.87	U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
3.97/1.87	
3.97/1.87	0  =  0
3.97/1.87	
3.97/1.87	s2l_out_ga(x1, x2)  =  s2l_out_ga(x1, x2)
3.97/1.87	
3.97/1.87	.(x1, x2)  =  .(x2)
3.97/1.87	
3.97/1.87	U4_g(x1, x2)  =  U4_g(x1, x2)
3.97/1.87	
3.97/1.87	list_in_g(x1)  =  list_in_g(x1)
3.97/1.87	
3.97/1.87	[]  =  []
3.97/1.87	
3.97/1.87	list_out_g(x1)  =  list_out_g(x1)
3.97/1.87	
3.97/1.87	U1_g(x1, x2, x3)  =  U1_g(x2, x3)
3.97/1.87	
3.97/1.87	goal_out_g(x1)  =  goal_out_g(x1)
3.97/1.87	
3.97/1.87	GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
3.97/1.87	
3.97/1.87	U3_G(x1, x2)  =  U3_G(x1, x2)
3.97/1.87	
3.97/1.87	S2L_IN_GA(x1, x2)  =  S2L_IN_GA(x1)
3.97/1.87	
3.97/1.87	U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
3.97/1.87	
3.97/1.87	U4_G(x1, x2)  =  U4_G(x1, x2)
3.97/1.87	
3.97/1.87	LIST_IN_G(x1)  =  LIST_IN_G(x1)
3.97/1.87	
3.97/1.87	U1_G(x1, x2, x3)  =  U1_G(x2, x3)
3.97/1.87	
3.97/1.87	
3.97/1.87	We have to consider all (P,R,Pi)-chains
3.97/1.87	----------------------------------------
3.97/1.87	
3.97/1.87	(5) DependencyGraphProof (EQUIVALENT)
3.97/1.87	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.
3.97/1.87	----------------------------------------
3.97/1.87	
3.97/1.87	(6)
3.97/1.87	Complex Obligation (AND)
3.97/1.87	
3.97/1.87	----------------------------------------
3.97/1.87	
3.97/1.87	(7)
3.97/1.87	Obligation:
3.97/1.87	Pi DP problem:
3.97/1.87	The TRS P consists of the following rules:
3.97/1.87	
3.97/1.87	   LIST_IN_G(.(X, XS)) -> LIST_IN_G(XS)
3.97/1.87	
3.97/1.87	The TRS R consists of the following rules:
3.97/1.87	
3.97/1.87	   goal_in_g(X) -> U3_g(X, s2l_in_ga(X, XS))
3.97/1.87	   s2l_in_ga(s(X), .(Y, Xs)) -> U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
3.97/1.87	   s2l_in_ga(0, []) -> s2l_out_ga(0, [])
3.97/1.87	   U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs))
3.97/1.87	   U3_g(X, s2l_out_ga(X, XS)) -> U4_g(X, list_in_g(XS))
3.97/1.87	   list_in_g([]) -> list_out_g([])
3.97/1.87	   list_in_g(.(X, XS)) -> U1_g(X, XS, list_in_g(XS))
3.97/1.87	   U1_g(X, XS, list_out_g(XS)) -> list_out_g(.(X, XS))
3.97/1.87	   U4_g(X, list_out_g(XS)) -> goal_out_g(X)
3.97/1.87	
3.97/1.87	The argument filtering Pi contains the following mapping:
3.97/1.87	goal_in_g(x1)  =  goal_in_g(x1)
3.97/1.87	
3.97/1.87	U3_g(x1, x2)  =  U3_g(x1, x2)
3.97/1.87	
3.97/1.87	s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
3.97/1.87	
3.97/1.87	s(x1)  =  s(x1)
3.97/1.87	
3.97/1.87	U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
3.97/1.87	
3.97/1.87	0  =  0
3.97/1.87	
3.97/1.87	s2l_out_ga(x1, x2)  =  s2l_out_ga(x1, x2)
3.97/1.87	
3.97/1.87	.(x1, x2)  =  .(x2)
3.97/1.87	
3.97/1.87	U4_g(x1, x2)  =  U4_g(x1, x2)
3.97/1.87	
3.97/1.87	list_in_g(x1)  =  list_in_g(x1)
3.97/1.87	
3.97/1.87	[]  =  []
3.97/1.87	
3.97/1.87	list_out_g(x1)  =  list_out_g(x1)
3.97/1.87	
3.97/1.87	U1_g(x1, x2, x3)  =  U1_g(x2, x3)
3.97/1.87	
3.97/1.87	goal_out_g(x1)  =  goal_out_g(x1)
3.97/1.87	
3.97/1.87	LIST_IN_G(x1)  =  LIST_IN_G(x1)
3.97/1.87	
3.97/1.87	
3.97/1.87	We have to consider all (P,R,Pi)-chains
3.97/1.87	----------------------------------------
3.97/1.87	
3.97/1.87	(8) UsableRulesProof (EQUIVALENT)
3.97/1.87	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.97/1.87	----------------------------------------
3.97/1.87	
3.97/1.87	(9)
3.97/1.87	Obligation:
3.97/1.87	Pi DP problem:
3.97/1.87	The TRS P consists of the following rules:
3.97/1.87	
3.97/1.87	   LIST_IN_G(.(X, XS)) -> LIST_IN_G(XS)
3.97/1.87	
3.97/1.87	R is empty.
3.97/1.87	The argument filtering Pi contains the following mapping:
3.97/1.87	.(x1, x2)  =  .(x2)
3.97/1.87	
3.97/1.87	LIST_IN_G(x1)  =  LIST_IN_G(x1)
3.97/1.87	
3.97/1.87	
3.97/1.87	We have to consider all (P,R,Pi)-chains
3.97/1.87	----------------------------------------
3.97/1.87	
3.97/1.87	(10) PiDPToQDPProof (SOUND)
3.97/1.87	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.97/1.87	----------------------------------------
3.97/1.87	
3.97/1.87	(11)
3.97/1.87	Obligation:
3.97/1.87	Q DP problem:
3.97/1.87	The TRS P consists of the following rules:
3.97/1.87	
3.97/1.87	   LIST_IN_G(.(XS)) -> LIST_IN_G(XS)
3.97/1.87	
3.97/1.87	R is empty.
3.97/1.87	Q is empty.
3.97/1.87	We have to consider all (P,Q,R)-chains.
3.97/1.87	----------------------------------------
3.97/1.87	
3.97/1.87	(12) QDPSizeChangeProof (EQUIVALENT)
3.97/1.87	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. 
3.97/1.87	
3.97/1.87	From the DPs we obtained the following set of size-change graphs:
3.97/1.87	*LIST_IN_G(.(XS)) -> LIST_IN_G(XS)
3.97/1.87	The graph contains the following edges 1 > 1
3.97/1.87	
3.97/1.87	
3.97/1.87	----------------------------------------
3.97/1.87	
3.97/1.87	(13)
3.97/1.87	YES
3.97/1.87	
3.97/1.87	----------------------------------------
3.97/1.87	
3.97/1.87	(14)
3.97/1.87	Obligation:
3.97/1.87	Pi DP problem:
3.97/1.87	The TRS P consists of the following rules:
3.97/1.87	
3.97/1.87	   S2L_IN_GA(s(X), .(Y, Xs)) -> S2L_IN_GA(X, Xs)
3.97/1.87	
3.97/1.87	The TRS R consists of the following rules:
3.97/1.87	
3.97/1.87	   goal_in_g(X) -> U3_g(X, s2l_in_ga(X, XS))
3.97/1.87	   s2l_in_ga(s(X), .(Y, Xs)) -> U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
3.97/1.87	   s2l_in_ga(0, []) -> s2l_out_ga(0, [])
3.97/1.87	   U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs))
3.97/1.87	   U3_g(X, s2l_out_ga(X, XS)) -> U4_g(X, list_in_g(XS))
3.97/1.87	   list_in_g([]) -> list_out_g([])
3.97/1.87	   list_in_g(.(X, XS)) -> U1_g(X, XS, list_in_g(XS))
3.97/1.87	   U1_g(X, XS, list_out_g(XS)) -> list_out_g(.(X, XS))
3.97/1.87	   U4_g(X, list_out_g(XS)) -> goal_out_g(X)
3.97/1.87	
3.97/1.87	The argument filtering Pi contains the following mapping:
3.97/1.87	goal_in_g(x1)  =  goal_in_g(x1)
3.97/1.87	
3.97/1.87	U3_g(x1, x2)  =  U3_g(x1, x2)
3.97/1.87	
3.97/1.87	s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
3.97/1.87	
3.97/1.87	s(x1)  =  s(x1)
3.97/1.87	
3.97/1.87	U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
3.97/1.87	
3.97/1.87	0  =  0
3.97/1.87	
3.97/1.87	s2l_out_ga(x1, x2)  =  s2l_out_ga(x1, x2)
3.97/1.87	
3.97/1.87	.(x1, x2)  =  .(x2)
3.97/1.87	
3.97/1.87	U4_g(x1, x2)  =  U4_g(x1, x2)
3.97/1.87	
3.97/1.87	list_in_g(x1)  =  list_in_g(x1)
3.97/1.87	
3.97/1.87	[]  =  []
3.97/1.87	
3.97/1.87	list_out_g(x1)  =  list_out_g(x1)
3.97/1.87	
3.97/1.87	U1_g(x1, x2, x3)  =  U1_g(x2, x3)
3.97/1.87	
3.97/1.87	goal_out_g(x1)  =  goal_out_g(x1)
3.97/1.87	
3.97/1.87	S2L_IN_GA(x1, x2)  =  S2L_IN_GA(x1)
3.97/1.87	
3.97/1.87	
3.97/1.87	We have to consider all (P,R,Pi)-chains
3.97/1.87	----------------------------------------
3.97/1.87	
3.97/1.87	(15) UsableRulesProof (EQUIVALENT)
3.97/1.87	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.97/1.87	----------------------------------------
3.97/1.87	
3.97/1.87	(16)
3.97/1.87	Obligation:
3.97/1.87	Pi DP problem:
3.97/1.87	The TRS P consists of the following rules:
3.97/1.87	
3.97/1.87	   S2L_IN_GA(s(X), .(Y, Xs)) -> S2L_IN_GA(X, Xs)
3.97/1.87	
3.97/1.87	R is empty.
3.97/1.87	The argument filtering Pi contains the following mapping:
3.97/1.87	s(x1)  =  s(x1)
3.97/1.87	
3.97/1.87	.(x1, x2)  =  .(x2)
3.97/1.87	
3.97/1.87	S2L_IN_GA(x1, x2)  =  S2L_IN_GA(x1)
3.97/1.87	
3.97/1.87	
3.97/1.87	We have to consider all (P,R,Pi)-chains
3.97/1.87	----------------------------------------
3.97/1.87	
3.97/1.87	(17) PiDPToQDPProof (SOUND)
3.97/1.87	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.97/1.87	----------------------------------------
3.97/1.87	
3.97/1.87	(18)
3.97/1.87	Obligation:
3.97/1.87	Q DP problem:
3.97/1.87	The TRS P consists of the following rules:
3.97/1.87	
3.97/1.87	   S2L_IN_GA(s(X)) -> S2L_IN_GA(X)
3.97/1.87	
3.97/1.87	R is empty.
3.97/1.87	Q is empty.
3.97/1.87	We have to consider all (P,Q,R)-chains.
3.97/1.87	----------------------------------------
3.97/1.87	
3.97/1.87	(19) QDPSizeChangeProof (EQUIVALENT)
3.97/1.87	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. 
3.97/1.87	
3.97/1.87	From the DPs we obtained the following set of size-change graphs:
3.97/1.87	*S2L_IN_GA(s(X)) -> S2L_IN_GA(X)
3.97/1.87	The graph contains the following edges 1 > 1
3.97/1.87	
3.97/1.87	
3.97/1.87	----------------------------------------
3.97/1.87	
3.97/1.87	(20)
3.97/1.87	YES
3.97/1.89	EOF