4.22/1.96	YES
4.22/1.97	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
4.22/1.97	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.22/1.97	
4.22/1.97	
4.22/1.97	Left Termination of the query pattern
4.22/1.97	
4.22/1.97	sublist(g,g)
4.22/1.97	
4.22/1.97	w.r.t. the given Prolog program could successfully be proven:
4.22/1.97	
4.22/1.97	(0) Prolog
4.22/1.97	(1) PrologToPiTRSProof [SOUND, 0 ms]
4.22/1.97	(2) PiTRS
4.22/1.97	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
4.22/1.97	(4) PiDP
4.22/1.97	(5) DependencyGraphProof [EQUIVALENT, 3 ms]
4.22/1.97	(6) AND
4.22/1.97	    (7) PiDP
4.22/1.97	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
4.22/1.97	        (9) PiDP
4.22/1.97	        (10) PiDPToQDPProof [SOUND, 11 ms]
4.22/1.97	        (11) QDP
4.22/1.97	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
4.22/1.97	        (13) YES
4.22/1.97	    (14) PiDP
4.22/1.97	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
4.22/1.97	        (16) PiDP
4.22/1.97	        (17) PiDPToQDPProof [SOUND, 0 ms]
4.22/1.97	        (18) QDP
4.22/1.97	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
4.22/1.97	        (20) YES
4.22/1.97	
4.22/1.97	
4.22/1.97	----------------------------------------
4.22/1.97	
4.22/1.97	(0)
4.22/1.97	Obligation:
4.22/1.97	Clauses:
4.22/1.97	
4.22/1.97	append1([], Ys, Ys).
4.22/1.97	append1(.(X, Xs), Ys, .(X, Zs)) :- append1(Xs, Ys, Zs).
4.22/1.97	append2([], Ys, Ys).
4.22/1.97	append2(.(X, Xs), Ys, .(X, Zs)) :- append2(Xs, Ys, Zs).
4.22/1.97	sublist(X, Y) :- ','(append1(P, X1, Y), append2(X2, X, P)).
4.22/1.97	
4.22/1.97	
4.22/1.97	Query: sublist(g,g)
4.22/1.97	----------------------------------------
4.22/1.97	
4.22/1.97	(1) PrologToPiTRSProof (SOUND)
4.22/1.97	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
4.22/1.97	
4.22/1.97	sublist_in_2: (b,b)
4.22/1.97	
4.22/1.97	append1_in_3: (f,f,b)
4.22/1.97	
4.22/1.97	append2_in_3: (f,b,b)
4.22/1.97	
4.22/1.97	Transforming Prolog into the following Term Rewriting System:
4.22/1.97	
4.22/1.97	Pi-finite rewrite system:
4.22/1.97	The TRS R consists of the following rules:
4.22/1.97	
4.22/1.97	   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aag(P, X1, Y))
4.22/1.97	   append1_in_aag([], Ys, Ys) -> append1_out_aag([], Ys, Ys)
4.22/1.97	   append1_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, append1_in_aag(Xs, Ys, Zs))
4.22/1.97	   U1_aag(X, Xs, Ys, Zs, append1_out_aag(Xs, Ys, Zs)) -> append1_out_aag(.(X, Xs), Ys, .(X, Zs))
4.22/1.97	   U3_gg(X, Y, append1_out_aag(P, X1, Y)) -> U4_gg(X, Y, append2_in_agg(X2, X, P))
4.22/1.97	   append2_in_agg([], Ys, Ys) -> append2_out_agg([], Ys, Ys)
4.22/1.97	   append2_in_agg(.(X, Xs), Ys, .(X, Zs)) -> U2_agg(X, Xs, Ys, Zs, append2_in_agg(Xs, Ys, Zs))
4.22/1.97	   U2_agg(X, Xs, Ys, Zs, append2_out_agg(Xs, Ys, Zs)) -> append2_out_agg(.(X, Xs), Ys, .(X, Zs))
4.22/1.97	   U4_gg(X, Y, append2_out_agg(X2, X, P)) -> sublist_out_gg(X, Y)
4.22/1.97	
4.22/1.97	The argument filtering Pi contains the following mapping:
4.22/1.97	sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
4.22/1.97	
4.22/1.97	U3_gg(x1, x2, x3)  =  U3_gg(x1, x3)
4.22/1.97	
4.22/1.97	append1_in_aag(x1, x2, x3)  =  append1_in_aag(x3)
4.22/1.97	
4.22/1.97	append1_out_aag(x1, x2, x3)  =  append1_out_aag(x1, x2)
4.22/1.97	
4.22/1.97	.(x1, x2)  =  .(x1, x2)
4.22/1.97	
4.22/1.97	U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
4.22/1.97	
4.22/1.97	U4_gg(x1, x2, x3)  =  U4_gg(x3)
4.22/1.97	
4.22/1.97	append2_in_agg(x1, x2, x3)  =  append2_in_agg(x2, x3)
4.22/1.97	
4.22/1.97	append2_out_agg(x1, x2, x3)  =  append2_out_agg(x1)
4.22/1.97	
4.22/1.97	U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x1, x5)
4.22/1.97	
4.22/1.97	sublist_out_gg(x1, x2)  =  sublist_out_gg
4.22/1.97	
4.22/1.97	
4.22/1.97	
4.22/1.97	
4.22/1.97	
4.22/1.97	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
4.22/1.97	
4.22/1.97	
4.22/1.97	
4.22/1.97	----------------------------------------
4.22/1.97	
4.22/1.97	(2)
4.22/1.97	Obligation:
4.22/1.97	Pi-finite rewrite system:
4.22/1.97	The TRS R consists of the following rules:
4.22/1.97	
4.22/1.97	   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aag(P, X1, Y))
4.22/1.97	   append1_in_aag([], Ys, Ys) -> append1_out_aag([], Ys, Ys)
4.22/1.97	   append1_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, append1_in_aag(Xs, Ys, Zs))
4.22/1.97	   U1_aag(X, Xs, Ys, Zs, append1_out_aag(Xs, Ys, Zs)) -> append1_out_aag(.(X, Xs), Ys, .(X, Zs))
4.22/1.97	   U3_gg(X, Y, append1_out_aag(P, X1, Y)) -> U4_gg(X, Y, append2_in_agg(X2, X, P))
4.22/1.97	   append2_in_agg([], Ys, Ys) -> append2_out_agg([], Ys, Ys)
4.22/1.97	   append2_in_agg(.(X, Xs), Ys, .(X, Zs)) -> U2_agg(X, Xs, Ys, Zs, append2_in_agg(Xs, Ys, Zs))
4.22/1.97	   U2_agg(X, Xs, Ys, Zs, append2_out_agg(Xs, Ys, Zs)) -> append2_out_agg(.(X, Xs), Ys, .(X, Zs))
4.22/1.97	   U4_gg(X, Y, append2_out_agg(X2, X, P)) -> sublist_out_gg(X, Y)
4.22/1.97	
4.22/1.97	The argument filtering Pi contains the following mapping:
4.22/1.97	sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
4.22/1.97	
4.22/1.97	U3_gg(x1, x2, x3)  =  U3_gg(x1, x3)
4.22/1.97	
4.22/1.97	append1_in_aag(x1, x2, x3)  =  append1_in_aag(x3)
4.22/1.97	
4.22/1.97	append1_out_aag(x1, x2, x3)  =  append1_out_aag(x1, x2)
4.22/1.97	
4.22/1.97	.(x1, x2)  =  .(x1, x2)
4.22/1.97	
4.22/1.97	U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
4.22/1.97	
4.22/1.97	U4_gg(x1, x2, x3)  =  U4_gg(x3)
4.22/1.97	
4.22/1.97	append2_in_agg(x1, x2, x3)  =  append2_in_agg(x2, x3)
4.22/1.97	
4.22/1.97	append2_out_agg(x1, x2, x3)  =  append2_out_agg(x1)
4.22/1.97	
4.22/1.97	U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x1, x5)
4.22/1.97	
4.22/1.97	sublist_out_gg(x1, x2)  =  sublist_out_gg
4.22/1.97	
4.22/1.97	
4.22/1.97	
4.22/1.97	----------------------------------------
4.22/1.97	
4.22/1.97	(3) DependencyPairsProof (EQUIVALENT)
4.22/1.97	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
4.22/1.97	Pi DP problem:
4.22/1.97	The TRS P consists of the following rules:
4.22/1.97	
4.22/1.97	   SUBLIST_IN_GG(X, Y) -> U3_GG(X, Y, append1_in_aag(P, X1, Y))
4.22/1.97	   SUBLIST_IN_GG(X, Y) -> APPEND1_IN_AAG(P, X1, Y)
4.22/1.97	   APPEND1_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U1_AAG(X, Xs, Ys, Zs, append1_in_aag(Xs, Ys, Zs))
4.22/1.97	   APPEND1_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AAG(Xs, Ys, Zs)
4.22/1.97	   U3_GG(X, Y, append1_out_aag(P, X1, Y)) -> U4_GG(X, Y, append2_in_agg(X2, X, P))
4.22/1.97	   U3_GG(X, Y, append1_out_aag(P, X1, Y)) -> APPEND2_IN_AGG(X2, X, P)
4.22/1.97	   APPEND2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) -> U2_AGG(X, Xs, Ys, Zs, append2_in_agg(Xs, Ys, Zs))
4.22/1.97	   APPEND2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_AGG(Xs, Ys, Zs)
4.22/1.97	
4.22/1.97	The TRS R consists of the following rules:
4.22/1.97	
4.22/1.97	   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aag(P, X1, Y))
4.22/1.97	   append1_in_aag([], Ys, Ys) -> append1_out_aag([], Ys, Ys)
4.22/1.97	   append1_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, append1_in_aag(Xs, Ys, Zs))
4.22/1.97	   U1_aag(X, Xs, Ys, Zs, append1_out_aag(Xs, Ys, Zs)) -> append1_out_aag(.(X, Xs), Ys, .(X, Zs))
4.22/1.97	   U3_gg(X, Y, append1_out_aag(P, X1, Y)) -> U4_gg(X, Y, append2_in_agg(X2, X, P))
4.22/1.97	   append2_in_agg([], Ys, Ys) -> append2_out_agg([], Ys, Ys)
4.22/1.97	   append2_in_agg(.(X, Xs), Ys, .(X, Zs)) -> U2_agg(X, Xs, Ys, Zs, append2_in_agg(Xs, Ys, Zs))
4.22/1.97	   U2_agg(X, Xs, Ys, Zs, append2_out_agg(Xs, Ys, Zs)) -> append2_out_agg(.(X, Xs), Ys, .(X, Zs))
4.22/1.97	   U4_gg(X, Y, append2_out_agg(X2, X, P)) -> sublist_out_gg(X, Y)
4.22/1.97	
4.22/1.97	The argument filtering Pi contains the following mapping:
4.22/1.97	sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
4.22/1.97	
4.22/1.97	U3_gg(x1, x2, x3)  =  U3_gg(x1, x3)
4.22/1.97	
4.22/1.97	append1_in_aag(x1, x2, x3)  =  append1_in_aag(x3)
4.22/1.97	
4.22/1.97	append1_out_aag(x1, x2, x3)  =  append1_out_aag(x1, x2)
4.22/1.97	
4.22/1.97	.(x1, x2)  =  .(x1, x2)
4.22/1.97	
4.22/1.97	U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
4.22/1.97	
4.22/1.97	U4_gg(x1, x2, x3)  =  U4_gg(x3)
4.22/1.97	
4.22/1.97	append2_in_agg(x1, x2, x3)  =  append2_in_agg(x2, x3)
4.22/1.97	
4.22/1.97	append2_out_agg(x1, x2, x3)  =  append2_out_agg(x1)
4.22/1.97	
4.22/1.97	U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x1, x5)
4.22/1.97	
4.22/1.97	sublist_out_gg(x1, x2)  =  sublist_out_gg
4.22/1.97	
4.22/1.97	SUBLIST_IN_GG(x1, x2)  =  SUBLIST_IN_GG(x1, x2)
4.22/1.97	
4.22/1.97	U3_GG(x1, x2, x3)  =  U3_GG(x1, x3)
4.22/1.97	
4.22/1.97	APPEND1_IN_AAG(x1, x2, x3)  =  APPEND1_IN_AAG(x3)
4.22/1.97	
4.22/1.97	U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x1, x5)
4.22/1.97	
4.22/1.97	U4_GG(x1, x2, x3)  =  U4_GG(x3)
4.22/1.97	
4.22/1.97	APPEND2_IN_AGG(x1, x2, x3)  =  APPEND2_IN_AGG(x2, x3)
4.22/1.97	
4.22/1.97	U2_AGG(x1, x2, x3, x4, x5)  =  U2_AGG(x1, x5)
4.22/1.97	
4.22/1.97	
4.22/1.97	We have to consider all (P,R,Pi)-chains
4.22/1.97	----------------------------------------
4.22/1.97	
4.22/1.97	(4)
4.22/1.97	Obligation:
4.22/1.97	Pi DP problem:
4.22/1.97	The TRS P consists of the following rules:
4.22/1.97	
4.22/1.97	   SUBLIST_IN_GG(X, Y) -> U3_GG(X, Y, append1_in_aag(P, X1, Y))
4.22/1.97	   SUBLIST_IN_GG(X, Y) -> APPEND1_IN_AAG(P, X1, Y)
4.22/1.97	   APPEND1_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U1_AAG(X, Xs, Ys, Zs, append1_in_aag(Xs, Ys, Zs))
4.22/1.97	   APPEND1_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AAG(Xs, Ys, Zs)
4.22/1.97	   U3_GG(X, Y, append1_out_aag(P, X1, Y)) -> U4_GG(X, Y, append2_in_agg(X2, X, P))
4.22/1.97	   U3_GG(X, Y, append1_out_aag(P, X1, Y)) -> APPEND2_IN_AGG(X2, X, P)
4.22/1.97	   APPEND2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) -> U2_AGG(X, Xs, Ys, Zs, append2_in_agg(Xs, Ys, Zs))
4.22/1.97	   APPEND2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_AGG(Xs, Ys, Zs)
4.22/1.97	
4.22/1.97	The TRS R consists of the following rules:
4.22/1.97	
4.22/1.97	   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aag(P, X1, Y))
4.22/1.97	   append1_in_aag([], Ys, Ys) -> append1_out_aag([], Ys, Ys)
4.22/1.97	   append1_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, append1_in_aag(Xs, Ys, Zs))
4.22/1.97	   U1_aag(X, Xs, Ys, Zs, append1_out_aag(Xs, Ys, Zs)) -> append1_out_aag(.(X, Xs), Ys, .(X, Zs))
4.22/1.97	   U3_gg(X, Y, append1_out_aag(P, X1, Y)) -> U4_gg(X, Y, append2_in_agg(X2, X, P))
4.22/1.97	   append2_in_agg([], Ys, Ys) -> append2_out_agg([], Ys, Ys)
4.22/1.97	   append2_in_agg(.(X, Xs), Ys, .(X, Zs)) -> U2_agg(X, Xs, Ys, Zs, append2_in_agg(Xs, Ys, Zs))
4.22/1.97	   U2_agg(X, Xs, Ys, Zs, append2_out_agg(Xs, Ys, Zs)) -> append2_out_agg(.(X, Xs), Ys, .(X, Zs))
4.22/1.97	   U4_gg(X, Y, append2_out_agg(X2, X, P)) -> sublist_out_gg(X, Y)
4.22/1.97	
4.22/1.97	The argument filtering Pi contains the following mapping:
4.22/1.97	sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
4.22/1.97	
4.22/1.97	U3_gg(x1, x2, x3)  =  U3_gg(x1, x3)
4.22/1.97	
4.22/1.97	append1_in_aag(x1, x2, x3)  =  append1_in_aag(x3)
4.22/1.97	
4.22/1.97	append1_out_aag(x1, x2, x3)  =  append1_out_aag(x1, x2)
4.22/1.97	
4.22/1.97	.(x1, x2)  =  .(x1, x2)
4.22/1.97	
4.22/1.97	U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
4.22/1.97	
4.22/1.97	U4_gg(x1, x2, x3)  =  U4_gg(x3)
4.22/1.97	
4.22/1.97	append2_in_agg(x1, x2, x3)  =  append2_in_agg(x2, x3)
4.22/1.97	
4.22/1.97	append2_out_agg(x1, x2, x3)  =  append2_out_agg(x1)
4.22/1.97	
4.22/1.97	U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x1, x5)
4.22/1.97	
4.22/1.97	sublist_out_gg(x1, x2)  =  sublist_out_gg
4.22/1.97	
4.22/1.97	SUBLIST_IN_GG(x1, x2)  =  SUBLIST_IN_GG(x1, x2)
4.22/1.97	
4.22/1.97	U3_GG(x1, x2, x3)  =  U3_GG(x1, x3)
4.22/1.97	
4.22/1.97	APPEND1_IN_AAG(x1, x2, x3)  =  APPEND1_IN_AAG(x3)
4.22/1.97	
4.22/1.97	U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x1, x5)
4.22/1.97	
4.22/1.97	U4_GG(x1, x2, x3)  =  U4_GG(x3)
4.22/1.97	
4.22/1.97	APPEND2_IN_AGG(x1, x2, x3)  =  APPEND2_IN_AGG(x2, x3)
4.22/1.97	
4.22/1.97	U2_AGG(x1, x2, x3, x4, x5)  =  U2_AGG(x1, x5)
4.22/1.97	
4.22/1.97	
4.22/1.97	We have to consider all (P,R,Pi)-chains
4.22/1.97	----------------------------------------
4.22/1.97	
4.22/1.97	(5) DependencyGraphProof (EQUIVALENT)
4.22/1.97	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.
4.22/1.97	----------------------------------------
4.22/1.97	
4.22/1.97	(6)
4.22/1.97	Complex Obligation (AND)
4.22/1.97	
4.22/1.97	----------------------------------------
4.22/1.97	
4.22/1.97	(7)
4.22/1.97	Obligation:
4.22/1.97	Pi DP problem:
4.22/1.97	The TRS P consists of the following rules:
4.22/1.97	
4.22/1.97	   APPEND2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_AGG(Xs, Ys, Zs)
4.22/1.97	
4.22/1.97	The TRS R consists of the following rules:
4.22/1.97	
4.22/1.97	   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aag(P, X1, Y))
4.22/1.97	   append1_in_aag([], Ys, Ys) -> append1_out_aag([], Ys, Ys)
4.22/1.97	   append1_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, append1_in_aag(Xs, Ys, Zs))
4.22/1.97	   U1_aag(X, Xs, Ys, Zs, append1_out_aag(Xs, Ys, Zs)) -> append1_out_aag(.(X, Xs), Ys, .(X, Zs))
4.22/1.97	   U3_gg(X, Y, append1_out_aag(P, X1, Y)) -> U4_gg(X, Y, append2_in_agg(X2, X, P))
4.22/1.97	   append2_in_agg([], Ys, Ys) -> append2_out_agg([], Ys, Ys)
4.22/1.97	   append2_in_agg(.(X, Xs), Ys, .(X, Zs)) -> U2_agg(X, Xs, Ys, Zs, append2_in_agg(Xs, Ys, Zs))
4.22/1.97	   U2_agg(X, Xs, Ys, Zs, append2_out_agg(Xs, Ys, Zs)) -> append2_out_agg(.(X, Xs), Ys, .(X, Zs))
4.22/1.97	   U4_gg(X, Y, append2_out_agg(X2, X, P)) -> sublist_out_gg(X, Y)
4.22/1.97	
4.22/1.97	The argument filtering Pi contains the following mapping:
4.22/1.97	sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
4.22/1.97	
4.22/1.97	U3_gg(x1, x2, x3)  =  U3_gg(x1, x3)
4.22/1.97	
4.22/1.97	append1_in_aag(x1, x2, x3)  =  append1_in_aag(x3)
4.22/1.97	
4.22/1.97	append1_out_aag(x1, x2, x3)  =  append1_out_aag(x1, x2)
4.22/1.97	
4.22/1.97	.(x1, x2)  =  .(x1, x2)
4.22/1.97	
4.22/1.97	U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
4.22/1.97	
4.22/1.97	U4_gg(x1, x2, x3)  =  U4_gg(x3)
4.22/1.97	
4.22/1.97	append2_in_agg(x1, x2, x3)  =  append2_in_agg(x2, x3)
4.22/1.97	
4.22/1.97	append2_out_agg(x1, x2, x3)  =  append2_out_agg(x1)
4.22/1.97	
4.22/1.97	U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x1, x5)
4.22/1.97	
4.22/1.97	sublist_out_gg(x1, x2)  =  sublist_out_gg
4.22/1.97	
4.22/1.97	APPEND2_IN_AGG(x1, x2, x3)  =  APPEND2_IN_AGG(x2, x3)
4.22/1.97	
4.22/1.97	
4.22/1.97	We have to consider all (P,R,Pi)-chains
4.22/1.97	----------------------------------------
4.22/1.97	
4.22/1.97	(8) UsableRulesProof (EQUIVALENT)
4.22/1.97	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
4.22/1.97	----------------------------------------
4.22/1.97	
4.22/1.97	(9)
4.22/1.97	Obligation:
4.22/1.97	Pi DP problem:
4.22/1.97	The TRS P consists of the following rules:
4.22/1.97	
4.22/1.97	   APPEND2_IN_AGG(.(X, Xs), Ys, .(X, Zs)) -> APPEND2_IN_AGG(Xs, Ys, Zs)
4.22/1.97	
4.22/1.97	R is empty.
4.22/1.97	The argument filtering Pi contains the following mapping:
4.22/1.97	.(x1, x2)  =  .(x1, x2)
4.22/1.97	
4.22/1.97	APPEND2_IN_AGG(x1, x2, x3)  =  APPEND2_IN_AGG(x2, x3)
4.22/1.97	
4.22/1.97	
4.22/1.97	We have to consider all (P,R,Pi)-chains
4.22/1.97	----------------------------------------
4.22/1.97	
4.22/1.97	(10) PiDPToQDPProof (SOUND)
4.22/1.97	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
4.22/1.97	----------------------------------------
4.22/1.97	
4.22/1.97	(11)
4.22/1.97	Obligation:
4.22/1.97	Q DP problem:
4.22/1.97	The TRS P consists of the following rules:
4.22/1.97	
4.22/1.97	   APPEND2_IN_AGG(Ys, .(X, Zs)) -> APPEND2_IN_AGG(Ys, Zs)
4.22/1.97	
4.22/1.97	R is empty.
4.22/1.97	Q is empty.
4.22/1.97	We have to consider all (P,Q,R)-chains.
4.22/1.97	----------------------------------------
4.22/1.97	
4.22/1.97	(12) QDPSizeChangeProof (EQUIVALENT)
4.22/1.97	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. 
4.22/1.98	
4.22/1.98	From the DPs we obtained the following set of size-change graphs:
4.22/1.98	*APPEND2_IN_AGG(Ys, .(X, Zs)) -> APPEND2_IN_AGG(Ys, Zs)
4.22/1.98	The graph contains the following edges 1 >= 1, 2 > 2
4.22/1.98	
4.22/1.98	
4.22/1.98	----------------------------------------
4.22/1.98	
4.22/1.98	(13)
4.22/1.98	YES
4.22/1.98	
4.22/1.98	----------------------------------------
4.22/1.98	
4.22/1.98	(14)
4.22/1.98	Obligation:
4.22/1.98	Pi DP problem:
4.22/1.98	The TRS P consists of the following rules:
4.22/1.98	
4.22/1.98	   APPEND1_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AAG(Xs, Ys, Zs)
4.22/1.98	
4.22/1.98	The TRS R consists of the following rules:
4.22/1.98	
4.22/1.98	   sublist_in_gg(X, Y) -> U3_gg(X, Y, append1_in_aag(P, X1, Y))
4.22/1.98	   append1_in_aag([], Ys, Ys) -> append1_out_aag([], Ys, Ys)
4.22/1.98	   append1_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U1_aag(X, Xs, Ys, Zs, append1_in_aag(Xs, Ys, Zs))
4.22/1.98	   U1_aag(X, Xs, Ys, Zs, append1_out_aag(Xs, Ys, Zs)) -> append1_out_aag(.(X, Xs), Ys, .(X, Zs))
4.22/1.98	   U3_gg(X, Y, append1_out_aag(P, X1, Y)) -> U4_gg(X, Y, append2_in_agg(X2, X, P))
4.22/1.98	   append2_in_agg([], Ys, Ys) -> append2_out_agg([], Ys, Ys)
4.22/1.98	   append2_in_agg(.(X, Xs), Ys, .(X, Zs)) -> U2_agg(X, Xs, Ys, Zs, append2_in_agg(Xs, Ys, Zs))
4.22/1.98	   U2_agg(X, Xs, Ys, Zs, append2_out_agg(Xs, Ys, Zs)) -> append2_out_agg(.(X, Xs), Ys, .(X, Zs))
4.22/1.98	   U4_gg(X, Y, append2_out_agg(X2, X, P)) -> sublist_out_gg(X, Y)
4.22/1.98	
4.22/1.98	The argument filtering Pi contains the following mapping:
4.22/1.98	sublist_in_gg(x1, x2)  =  sublist_in_gg(x1, x2)
4.22/1.98	
4.22/1.98	U3_gg(x1, x2, x3)  =  U3_gg(x1, x3)
4.22/1.98	
4.22/1.98	append1_in_aag(x1, x2, x3)  =  append1_in_aag(x3)
4.22/1.98	
4.22/1.98	append1_out_aag(x1, x2, x3)  =  append1_out_aag(x1, x2)
4.22/1.98	
4.22/1.98	.(x1, x2)  =  .(x1, x2)
4.22/1.98	
4.22/1.98	U1_aag(x1, x2, x3, x4, x5)  =  U1_aag(x1, x5)
4.22/1.98	
4.22/1.98	U4_gg(x1, x2, x3)  =  U4_gg(x3)
4.22/1.98	
4.22/1.98	append2_in_agg(x1, x2, x3)  =  append2_in_agg(x2, x3)
4.22/1.98	
4.22/1.98	append2_out_agg(x1, x2, x3)  =  append2_out_agg(x1)
4.22/1.98	
4.22/1.98	U2_agg(x1, x2, x3, x4, x5)  =  U2_agg(x1, x5)
4.22/1.98	
4.22/1.98	sublist_out_gg(x1, x2)  =  sublist_out_gg
4.22/1.98	
4.22/1.98	APPEND1_IN_AAG(x1, x2, x3)  =  APPEND1_IN_AAG(x3)
4.22/1.98	
4.22/1.98	
4.22/1.98	We have to consider all (P,R,Pi)-chains
4.22/1.98	----------------------------------------
4.22/1.98	
4.22/1.98	(15) UsableRulesProof (EQUIVALENT)
4.22/1.98	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
4.22/1.98	----------------------------------------
4.22/1.98	
4.22/1.98	(16)
4.22/1.98	Obligation:
4.22/1.98	Pi DP problem:
4.22/1.98	The TRS P consists of the following rules:
4.22/1.98	
4.22/1.98	   APPEND1_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND1_IN_AAG(Xs, Ys, Zs)
4.22/1.98	
4.22/1.98	R is empty.
4.22/1.98	The argument filtering Pi contains the following mapping:
4.22/1.98	.(x1, x2)  =  .(x1, x2)
4.22/1.98	
4.22/1.98	APPEND1_IN_AAG(x1, x2, x3)  =  APPEND1_IN_AAG(x3)
4.22/1.98	
4.22/1.98	
4.22/1.98	We have to consider all (P,R,Pi)-chains
4.22/1.98	----------------------------------------
4.22/1.98	
4.22/1.98	(17) PiDPToQDPProof (SOUND)
4.22/1.98	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
4.22/1.98	----------------------------------------
4.22/1.98	
4.22/1.98	(18)
4.22/1.98	Obligation:
4.22/1.98	Q DP problem:
4.22/1.98	The TRS P consists of the following rules:
4.22/1.98	
4.22/1.98	   APPEND1_IN_AAG(.(X, Zs)) -> APPEND1_IN_AAG(Zs)
4.22/1.98	
4.22/1.98	R is empty.
4.22/1.98	Q is empty.
4.22/1.98	We have to consider all (P,Q,R)-chains.
4.22/1.98	----------------------------------------
4.22/1.98	
4.22/1.98	(19) QDPSizeChangeProof (EQUIVALENT)
4.22/1.98	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. 
4.22/1.98	
4.22/1.98	From the DPs we obtained the following set of size-change graphs:
4.22/1.98	*APPEND1_IN_AAG(.(X, Zs)) -> APPEND1_IN_AAG(Zs)
4.22/1.98	The graph contains the following edges 1 > 1
4.22/1.98	
4.22/1.98	
4.22/1.98	----------------------------------------
4.22/1.98	
4.22/1.98	(20)
4.22/1.98	YES
4.44/2.07	EOF