3.95/1.95	YES
4.02/1.97	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
4.02/1.97	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.02/1.97	
4.02/1.97	
4.02/1.97	Left Termination of the query pattern
4.02/1.97	
4.02/1.97	sublist(a,g)
4.02/1.97	
4.02/1.97	w.r.t. the given Prolog program could successfully be proven:
4.02/1.97	
4.02/1.97	(0) Prolog
4.02/1.97	(1) PrologToPiTRSProof [SOUND, 0 ms]
4.02/1.97	(2) PiTRS
4.02/1.97	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
4.02/1.97	(4) PiDP
4.02/1.97	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
4.02/1.97	(6) PiDP
4.02/1.97	(7) UsableRulesProof [EQUIVALENT, 0 ms]
4.02/1.97	(8) PiDP
4.02/1.97	(9) PiDPToQDPProof [SOUND, 0 ms]
4.02/1.97	(10) QDP
4.02/1.97	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
4.02/1.97	(12) YES
4.02/1.97	
4.02/1.97	
4.02/1.97	----------------------------------------
4.02/1.97	
4.02/1.97	(0)
4.02/1.97	Obligation:
4.02/1.97	Clauses:
4.02/1.97	
4.02/1.97	append([], Ys, Ys).
4.02/1.97	append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs).
4.02/1.97	sublist(X, Y) :- ','(append(P, X1, Y), append(X2, X, P)).
4.02/1.97	
4.02/1.97	
4.02/1.97	Query: sublist(a,g)
4.02/1.97	----------------------------------------
4.02/1.97	
4.02/1.97	(1) PrologToPiTRSProof (SOUND)
4.02/1.97	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
4.02/1.97	
4.02/1.97	sublist_in_2: (f,b)
4.02/1.97	
4.02/1.97	append_in_3: (f,f,b)
4.02/1.97	
4.02/1.97	Transforming Prolog into the following Term Rewriting System:
4.02/1.97	
4.02/1.97	Pi-finite rewrite system:
4.02/1.97	The TRS R consists of the following rules:
4.02/1.97	
4.02/1.97	   sublist_in_ag(X, Y) -> U2_ag(X, Y, append_in_aag(P, X1, Y))
4.02/1.97	   append_in_aag([], Ys, Ys) -> append_out_aag([], Ys, Ys)
4.02/1.97	   append_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs))
4.02/1.97	   U1_aag(X, Xs, Ys, Zs, append_out_aag(Xs, Ys, Zs)) -> append_out_aag(.(X, Xs), Ys, .(X, Zs))
4.02/1.97	   U2_ag(X, Y, append_out_aag(P, X1, Y)) -> U3_ag(X, Y, append_in_aag(X2, X, P))
4.02/1.97	   U3_ag(X, Y, append_out_aag(X2, X, P)) -> sublist_out_ag(X, Y)
4.02/1.97	
4.02/1.97	The argument filtering Pi contains the following mapping:
4.02/1.97	sublist_in_ag(x1, x2)  =  sublist_in_ag(x2)
4.02/1.97	
4.02/1.97	U2_ag(x1, x2, x3)  =  U2_ag(x3)
4.02/1.97	
4.02/1.97	append_in_aag(x1, x2, x3)  =  append_in_aag(x3)
4.02/1.97	
4.02/1.97	append_out_aag(x1, x2, x3)  =  append_out_aag(x1, x2)
4.02/1.97	
4.02/1.97	.(x1, x2)  =  .(x1, x2)
4.02/1.97	
4.02/1.97	U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
4.02/1.97	
4.02/1.97	U3_ag(x1, x2, x3)  =  U3_ag(x3)
4.02/1.97	
4.02/1.97	sublist_out_ag(x1, x2)  =  sublist_out_ag(x1)
4.02/1.97	
4.02/1.97	
4.02/1.97	
4.02/1.97	
4.02/1.97	
4.02/1.97	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
4.02/1.97	
4.02/1.97	
4.02/1.97	
4.02/1.97	----------------------------------------
4.02/1.97	
4.02/1.97	(2)
4.02/1.97	Obligation:
4.02/1.97	Pi-finite rewrite system:
4.02/1.97	The TRS R consists of the following rules:
4.02/1.97	
4.02/1.97	   sublist_in_ag(X, Y) -> U2_ag(X, Y, append_in_aag(P, X1, Y))
4.02/1.97	   append_in_aag([], Ys, Ys) -> append_out_aag([], Ys, Ys)
4.02/1.97	   append_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs))
4.02/1.97	   U1_aag(X, Xs, Ys, Zs, append_out_aag(Xs, Ys, Zs)) -> append_out_aag(.(X, Xs), Ys, .(X, Zs))
4.02/1.97	   U2_ag(X, Y, append_out_aag(P, X1, Y)) -> U3_ag(X, Y, append_in_aag(X2, X, P))
4.02/1.97	   U3_ag(X, Y, append_out_aag(X2, X, P)) -> sublist_out_ag(X, Y)
4.02/1.97	
4.02/1.97	The argument filtering Pi contains the following mapping:
4.02/1.97	sublist_in_ag(x1, x2)  =  sublist_in_ag(x2)
4.02/1.97	
4.02/1.97	U2_ag(x1, x2, x3)  =  U2_ag(x3)
4.02/1.97	
4.02/1.97	append_in_aag(x1, x2, x3)  =  append_in_aag(x3)
4.02/1.97	
4.02/1.97	append_out_aag(x1, x2, x3)  =  append_out_aag(x1, x2)
4.02/1.97	
4.02/1.97	.(x1, x2)  =  .(x1, x2)
4.02/1.97	
4.02/1.97	U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
4.02/1.97	
4.02/1.97	U3_ag(x1, x2, x3)  =  U3_ag(x3)
4.02/1.97	
4.02/1.97	sublist_out_ag(x1, x2)  =  sublist_out_ag(x1)
4.02/1.97	
4.02/1.97	
4.02/1.97	
4.02/1.97	----------------------------------------
4.02/1.97	
4.02/1.97	(3) DependencyPairsProof (EQUIVALENT)
4.02/1.97	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
4.02/1.97	Pi DP problem:
4.02/1.97	The TRS P consists of the following rules:
4.02/1.97	
4.02/1.97	   SUBLIST_IN_AG(X, Y) -> U2_AG(X, Y, append_in_aag(P, X1, Y))
4.02/1.97	   SUBLIST_IN_AG(X, Y) -> APPEND_IN_AAG(P, X1, Y)
4.02/1.97	   APPEND_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U1_AAG(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs))
4.02/1.97	   APPEND_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAG(Xs, Ys, Zs)
4.02/1.97	   U2_AG(X, Y, append_out_aag(P, X1, Y)) -> U3_AG(X, Y, append_in_aag(X2, X, P))
4.02/1.97	   U2_AG(X, Y, append_out_aag(P, X1, Y)) -> APPEND_IN_AAG(X2, X, P)
4.02/1.97	
4.02/1.97	The TRS R consists of the following rules:
4.02/1.97	
4.02/1.97	   sublist_in_ag(X, Y) -> U2_ag(X, Y, append_in_aag(P, X1, Y))
4.02/1.97	   append_in_aag([], Ys, Ys) -> append_out_aag([], Ys, Ys)
4.02/1.97	   append_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs))
4.02/1.97	   U1_aag(X, Xs, Ys, Zs, append_out_aag(Xs, Ys, Zs)) -> append_out_aag(.(X, Xs), Ys, .(X, Zs))
4.02/1.97	   U2_ag(X, Y, append_out_aag(P, X1, Y)) -> U3_ag(X, Y, append_in_aag(X2, X, P))
4.02/1.97	   U3_ag(X, Y, append_out_aag(X2, X, P)) -> sublist_out_ag(X, Y)
4.02/1.97	
4.02/1.97	The argument filtering Pi contains the following mapping:
4.02/1.97	sublist_in_ag(x1, x2)  =  sublist_in_ag(x2)
4.02/1.97	
4.02/1.97	U2_ag(x1, x2, x3)  =  U2_ag(x3)
4.02/1.97	
4.02/1.97	append_in_aag(x1, x2, x3)  =  append_in_aag(x3)
4.02/1.97	
4.02/1.97	append_out_aag(x1, x2, x3)  =  append_out_aag(x1, x2)
4.02/1.97	
4.02/1.97	.(x1, x2)  =  .(x1, x2)
4.02/1.97	
4.02/1.97	U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
4.02/1.97	
4.02/1.97	U3_ag(x1, x2, x3)  =  U3_ag(x3)
4.02/1.97	
4.02/1.97	sublist_out_ag(x1, x2)  =  sublist_out_ag(x1)
4.02/1.97	
4.02/1.97	SUBLIST_IN_AG(x1, x2)  =  SUBLIST_IN_AG(x2)
4.02/1.97	
4.02/1.97	U2_AG(x1, x2, x3)  =  U2_AG(x3)
4.02/1.97	
4.02/1.97	APPEND_IN_AAG(x1, x2, x3)  =  APPEND_IN_AAG(x3)
4.02/1.97	
4.02/1.97	U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x1, x5)
4.02/1.97	
4.02/1.97	U3_AG(x1, x2, x3)  =  U3_AG(x3)
4.02/1.97	
4.02/1.97	
4.02/1.97	We have to consider all (P,R,Pi)-chains
4.02/1.97	----------------------------------------
4.02/1.97	
4.02/1.97	(4)
4.02/1.97	Obligation:
4.02/1.97	Pi DP problem:
4.02/1.97	The TRS P consists of the following rules:
4.02/1.97	
4.02/1.97	   SUBLIST_IN_AG(X, Y) -> U2_AG(X, Y, append_in_aag(P, X1, Y))
4.02/1.97	   SUBLIST_IN_AG(X, Y) -> APPEND_IN_AAG(P, X1, Y)
4.02/1.97	   APPEND_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U1_AAG(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs))
4.02/1.97	   APPEND_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAG(Xs, Ys, Zs)
4.02/1.97	   U2_AG(X, Y, append_out_aag(P, X1, Y)) -> U3_AG(X, Y, append_in_aag(X2, X, P))
4.02/1.97	   U2_AG(X, Y, append_out_aag(P, X1, Y)) -> APPEND_IN_AAG(X2, X, P)
4.02/1.97	
4.02/1.97	The TRS R consists of the following rules:
4.02/1.97	
4.02/1.97	   sublist_in_ag(X, Y) -> U2_ag(X, Y, append_in_aag(P, X1, Y))
4.02/1.97	   append_in_aag([], Ys, Ys) -> append_out_aag([], Ys, Ys)
4.02/1.97	   append_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs))
4.02/1.97	   U1_aag(X, Xs, Ys, Zs, append_out_aag(Xs, Ys, Zs)) -> append_out_aag(.(X, Xs), Ys, .(X, Zs))
4.02/1.97	   U2_ag(X, Y, append_out_aag(P, X1, Y)) -> U3_ag(X, Y, append_in_aag(X2, X, P))
4.02/1.97	   U3_ag(X, Y, append_out_aag(X2, X, P)) -> sublist_out_ag(X, Y)
4.02/1.97	
4.02/1.97	The argument filtering Pi contains the following mapping:
4.02/1.97	sublist_in_ag(x1, x2)  =  sublist_in_ag(x2)
4.02/1.97	
4.02/1.97	U2_ag(x1, x2, x3)  =  U2_ag(x3)
4.02/1.97	
4.02/1.97	append_in_aag(x1, x2, x3)  =  append_in_aag(x3)
4.02/1.97	
4.02/1.97	append_out_aag(x1, x2, x3)  =  append_out_aag(x1, x2)
4.02/1.97	
4.02/1.97	.(x1, x2)  =  .(x1, x2)
4.02/1.97	
4.02/1.97	U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
4.02/1.97	
4.02/1.97	U3_ag(x1, x2, x3)  =  U3_ag(x3)
4.02/1.97	
4.02/1.97	sublist_out_ag(x1, x2)  =  sublist_out_ag(x1)
4.02/1.97	
4.02/1.97	SUBLIST_IN_AG(x1, x2)  =  SUBLIST_IN_AG(x2)
4.02/1.97	
4.02/1.97	U2_AG(x1, x2, x3)  =  U2_AG(x3)
4.02/1.97	
4.02/1.97	APPEND_IN_AAG(x1, x2, x3)  =  APPEND_IN_AAG(x3)
4.02/1.97	
4.02/1.97	U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x1, x5)
4.02/1.97	
4.02/1.97	U3_AG(x1, x2, x3)  =  U3_AG(x3)
4.02/1.97	
4.02/1.97	
4.02/1.97	We have to consider all (P,R,Pi)-chains
4.02/1.97	----------------------------------------
4.02/1.97	
4.02/1.97	(5) DependencyGraphProof (EQUIVALENT)
4.02/1.97	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 5 less nodes.
4.02/1.97	----------------------------------------
4.02/1.97	
4.02/1.97	(6)
4.02/1.97	Obligation:
4.02/1.97	Pi DP problem:
4.02/1.97	The TRS P consists of the following rules:
4.02/1.97	
4.02/1.97	   APPEND_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAG(Xs, Ys, Zs)
4.02/1.97	
4.02/1.97	The TRS R consists of the following rules:
4.02/1.97	
4.02/1.97	   sublist_in_ag(X, Y) -> U2_ag(X, Y, append_in_aag(P, X1, Y))
4.02/1.97	   append_in_aag([], Ys, Ys) -> append_out_aag([], Ys, Ys)
4.02/1.97	   append_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs))
4.02/1.97	   U1_aag(X, Xs, Ys, Zs, append_out_aag(Xs, Ys, Zs)) -> append_out_aag(.(X, Xs), Ys, .(X, Zs))
4.02/1.97	   U2_ag(X, Y, append_out_aag(P, X1, Y)) -> U3_ag(X, Y, append_in_aag(X2, X, P))
4.02/1.97	   U3_ag(X, Y, append_out_aag(X2, X, P)) -> sublist_out_ag(X, Y)
4.02/1.97	
4.02/1.97	The argument filtering Pi contains the following mapping:
4.02/1.97	sublist_in_ag(x1, x2)  =  sublist_in_ag(x2)
4.02/1.97	
4.02/1.97	U2_ag(x1, x2, x3)  =  U2_ag(x3)
4.02/1.97	
4.02/1.97	append_in_aag(x1, x2, x3)  =  append_in_aag(x3)
4.02/1.97	
4.02/1.97	append_out_aag(x1, x2, x3)  =  append_out_aag(x1, x2)
4.02/1.97	
4.02/1.97	.(x1, x2)  =  .(x1, x2)
4.02/1.97	
4.02/1.97	U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
4.02/1.97	
4.02/1.97	U3_ag(x1, x2, x3)  =  U3_ag(x3)
4.02/1.97	
4.02/1.97	sublist_out_ag(x1, x2)  =  sublist_out_ag(x1)
4.02/1.97	
4.02/1.97	APPEND_IN_AAG(x1, x2, x3)  =  APPEND_IN_AAG(x3)
4.02/1.97	
4.02/1.97	
4.02/1.97	We have to consider all (P,R,Pi)-chains
4.02/1.97	----------------------------------------
4.02/1.97	
4.02/1.97	(7) UsableRulesProof (EQUIVALENT)
4.02/1.97	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
4.02/1.97	----------------------------------------
4.02/1.97	
4.02/1.97	(8)
4.02/1.97	Obligation:
4.02/1.97	Pi DP problem:
4.02/1.97	The TRS P consists of the following rules:
4.02/1.97	
4.02/1.97	   APPEND_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAG(Xs, Ys, Zs)
4.02/1.97	
4.02/1.97	R is empty.
4.02/1.97	The argument filtering Pi contains the following mapping:
4.02/1.97	.(x1, x2)  =  .(x1, x2)
4.02/1.97	
4.02/1.97	APPEND_IN_AAG(x1, x2, x3)  =  APPEND_IN_AAG(x3)
4.02/1.97	
4.02/1.97	
4.02/1.97	We have to consider all (P,R,Pi)-chains
4.02/1.97	----------------------------------------
4.02/1.97	
4.02/1.97	(9) PiDPToQDPProof (SOUND)
4.02/1.97	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
4.02/1.97	----------------------------------------
4.02/1.97	
4.02/1.97	(10)
4.02/1.97	Obligation:
4.02/1.97	Q DP problem:
4.02/1.97	The TRS P consists of the following rules:
4.02/1.97	
4.02/1.97	   APPEND_IN_AAG(.(X, Zs)) -> APPEND_IN_AAG(Zs)
4.02/1.97	
4.02/1.97	R is empty.
4.02/1.97	Q is empty.
4.02/1.97	We have to consider all (P,Q,R)-chains.
4.02/1.97	----------------------------------------
4.02/1.97	
4.02/1.97	(11) QDPSizeChangeProof (EQUIVALENT)
4.02/1.97	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. 
4.02/1.97	
4.02/1.97	From the DPs we obtained the following set of size-change graphs:
4.02/1.97	*APPEND_IN_AAG(.(X, Zs)) -> APPEND_IN_AAG(Zs)
4.02/1.97	The graph contains the following edges 1 > 1
4.02/1.97	
4.02/1.97	
4.02/1.97	----------------------------------------
4.02/1.97	
4.02/1.97	(12)
4.02/1.97	YES
4.02/2.00	EOF