3.59/1.69	YES
3.81/1.70	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.81/1.70	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.81/1.70	
3.81/1.70	
3.81/1.70	Left Termination of the query pattern
3.81/1.70	
3.81/1.70	reverse_concatenate(g,g,a)
3.81/1.70	
3.81/1.70	w.r.t. the given Prolog program could successfully be proven:
3.81/1.70	
3.81/1.70	(0) Prolog
3.81/1.70	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.81/1.70	(2) PiTRS
3.81/1.70	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.81/1.70	(4) PiDP
3.81/1.70	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.81/1.70	(6) PiDP
3.81/1.70	(7) UsableRulesProof [EQUIVALENT, 0 ms]
3.81/1.70	(8) PiDP
3.81/1.70	(9) PiDPToQDPProof [SOUND, 0 ms]
3.81/1.70	(10) QDP
3.81/1.70	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.81/1.70	(12) YES
3.81/1.70	
3.81/1.70	
3.81/1.70	----------------------------------------
3.81/1.70	
3.81/1.70	(0)
3.81/1.70	Obligation:
3.81/1.70	Clauses:
3.81/1.70	
3.81/1.70	concatenate([], L, L).
3.81/1.70	concatenate(.(X, L1), L2, .(X, L3)) :- concatenate(L1, L2, L3).
3.81/1.70	member(X, .(X, L)).
3.81/1.70	member(X, .(Y, L)) :- member(X, L).
3.81/1.70	reverse(L, L1) :- reverse_concatenate(L, [], L1).
3.81/1.70	reverse_concatenate([], L, L).
3.81/1.70	reverse_concatenate(.(X, L1), L2, L3) :- reverse_concatenate(L1, .(X, L2), L3).
3.81/1.70	
3.81/1.70	
3.81/1.70	Query: reverse_concatenate(g,g,a)
3.81/1.70	----------------------------------------
3.81/1.71	
3.81/1.71	(1) PrologToPiTRSProof (SOUND)
3.81/1.71	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.81/1.71	
3.81/1.71	reverse_concatenate_in_3: (b,b,f)
3.81/1.71	
3.81/1.71	Transforming Prolog into the following Term Rewriting System:
3.81/1.71	
3.81/1.71	Pi-finite rewrite system:
3.81/1.71	The TRS R consists of the following rules:
3.81/1.71	
3.81/1.71	   reverse_concatenate_in_gga([], L, L) -> reverse_concatenate_out_gga([], L, L)
3.81/1.71	   reverse_concatenate_in_gga(.(X, L1), L2, L3) -> U4_gga(X, L1, L2, L3, reverse_concatenate_in_gga(L1, .(X, L2), L3))
3.81/1.71	   U4_gga(X, L1, L2, L3, reverse_concatenate_out_gga(L1, .(X, L2), L3)) -> reverse_concatenate_out_gga(.(X, L1), L2, L3)
3.81/1.71	
3.81/1.71	The argument filtering Pi contains the following mapping:
3.81/1.71	reverse_concatenate_in_gga(x1, x2, x3)  =  reverse_concatenate_in_gga(x1, x2)
3.81/1.71	
3.81/1.71	[]  =  []
3.81/1.71	
3.81/1.71	reverse_concatenate_out_gga(x1, x2, x3)  =  reverse_concatenate_out_gga(x3)
3.81/1.71	
3.81/1.71	.(x1, x2)  =  .(x1, x2)
3.81/1.71	
3.81/1.71	U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
3.81/1.71	
3.81/1.71	
3.81/1.71	
3.81/1.71	
3.81/1.71	
3.81/1.71	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.81/1.71	
3.81/1.71	
3.81/1.71	
3.81/1.71	----------------------------------------
3.81/1.71	
3.81/1.71	(2)
3.81/1.71	Obligation:
3.81/1.71	Pi-finite rewrite system:
3.81/1.71	The TRS R consists of the following rules:
3.81/1.71	
3.81/1.71	   reverse_concatenate_in_gga([], L, L) -> reverse_concatenate_out_gga([], L, L)
3.81/1.71	   reverse_concatenate_in_gga(.(X, L1), L2, L3) -> U4_gga(X, L1, L2, L3, reverse_concatenate_in_gga(L1, .(X, L2), L3))
3.81/1.71	   U4_gga(X, L1, L2, L3, reverse_concatenate_out_gga(L1, .(X, L2), L3)) -> reverse_concatenate_out_gga(.(X, L1), L2, L3)
3.81/1.71	
3.81/1.71	The argument filtering Pi contains the following mapping:
3.81/1.71	reverse_concatenate_in_gga(x1, x2, x3)  =  reverse_concatenate_in_gga(x1, x2)
3.81/1.71	
3.81/1.71	[]  =  []
3.81/1.71	
3.81/1.71	reverse_concatenate_out_gga(x1, x2, x3)  =  reverse_concatenate_out_gga(x3)
3.81/1.71	
3.81/1.71	.(x1, x2)  =  .(x1, x2)
3.81/1.71	
3.81/1.71	U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
3.81/1.71	
3.81/1.71	
3.81/1.71	
3.81/1.71	----------------------------------------
3.81/1.71	
3.81/1.71	(3) DependencyPairsProof (EQUIVALENT)
3.81/1.71	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.81/1.71	Pi DP problem:
3.81/1.71	The TRS P consists of the following rules:
3.81/1.71	
3.81/1.71	   REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2, L3) -> U4_GGA(X, L1, L2, L3, reverse_concatenate_in_gga(L1, .(X, L2), L3))
3.81/1.71	   REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2, L3) -> REVERSE_CONCATENATE_IN_GGA(L1, .(X, L2), L3)
3.81/1.71	
3.81/1.71	The TRS R consists of the following rules:
3.81/1.71	
3.81/1.71	   reverse_concatenate_in_gga([], L, L) -> reverse_concatenate_out_gga([], L, L)
3.81/1.71	   reverse_concatenate_in_gga(.(X, L1), L2, L3) -> U4_gga(X, L1, L2, L3, reverse_concatenate_in_gga(L1, .(X, L2), L3))
3.81/1.71	   U4_gga(X, L1, L2, L3, reverse_concatenate_out_gga(L1, .(X, L2), L3)) -> reverse_concatenate_out_gga(.(X, L1), L2, L3)
3.81/1.71	
3.81/1.71	The argument filtering Pi contains the following mapping:
3.81/1.71	reverse_concatenate_in_gga(x1, x2, x3)  =  reverse_concatenate_in_gga(x1, x2)
3.81/1.71	
3.81/1.71	[]  =  []
3.81/1.71	
3.81/1.71	reverse_concatenate_out_gga(x1, x2, x3)  =  reverse_concatenate_out_gga(x3)
3.81/1.71	
3.81/1.71	.(x1, x2)  =  .(x1, x2)
3.81/1.71	
3.81/1.71	U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
3.81/1.71	
3.81/1.71	REVERSE_CONCATENATE_IN_GGA(x1, x2, x3)  =  REVERSE_CONCATENATE_IN_GGA(x1, x2)
3.81/1.71	
3.81/1.71	U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x5)
3.81/1.71	
3.81/1.71	
3.81/1.71	We have to consider all (P,R,Pi)-chains
3.81/1.71	----------------------------------------
3.81/1.71	
3.81/1.71	(4)
3.81/1.71	Obligation:
3.81/1.71	Pi DP problem:
3.81/1.71	The TRS P consists of the following rules:
3.81/1.71	
3.81/1.71	   REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2, L3) -> U4_GGA(X, L1, L2, L3, reverse_concatenate_in_gga(L1, .(X, L2), L3))
3.81/1.71	   REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2, L3) -> REVERSE_CONCATENATE_IN_GGA(L1, .(X, L2), L3)
3.81/1.71	
3.81/1.71	The TRS R consists of the following rules:
3.81/1.71	
3.81/1.71	   reverse_concatenate_in_gga([], L, L) -> reverse_concatenate_out_gga([], L, L)
3.81/1.71	   reverse_concatenate_in_gga(.(X, L1), L2, L3) -> U4_gga(X, L1, L2, L3, reverse_concatenate_in_gga(L1, .(X, L2), L3))
3.81/1.71	   U4_gga(X, L1, L2, L3, reverse_concatenate_out_gga(L1, .(X, L2), L3)) -> reverse_concatenate_out_gga(.(X, L1), L2, L3)
3.81/1.71	
3.81/1.71	The argument filtering Pi contains the following mapping:
3.81/1.71	reverse_concatenate_in_gga(x1, x2, x3)  =  reverse_concatenate_in_gga(x1, x2)
3.81/1.71	
3.81/1.71	[]  =  []
3.81/1.71	
3.81/1.71	reverse_concatenate_out_gga(x1, x2, x3)  =  reverse_concatenate_out_gga(x3)
3.81/1.71	
3.81/1.71	.(x1, x2)  =  .(x1, x2)
3.81/1.71	
3.81/1.71	U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
3.81/1.71	
3.81/1.71	REVERSE_CONCATENATE_IN_GGA(x1, x2, x3)  =  REVERSE_CONCATENATE_IN_GGA(x1, x2)
3.81/1.71	
3.81/1.71	U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x5)
3.81/1.71	
3.81/1.71	
3.81/1.71	We have to consider all (P,R,Pi)-chains
3.81/1.71	----------------------------------------
3.81/1.71	
3.81/1.71	(5) DependencyGraphProof (EQUIVALENT)
3.81/1.71	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
3.81/1.71	----------------------------------------
3.81/1.71	
3.81/1.71	(6)
3.81/1.71	Obligation:
3.81/1.71	Pi DP problem:
3.81/1.71	The TRS P consists of the following rules:
3.81/1.71	
3.81/1.71	   REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2, L3) -> REVERSE_CONCATENATE_IN_GGA(L1, .(X, L2), L3)
3.81/1.71	
3.81/1.71	The TRS R consists of the following rules:
3.81/1.71	
3.81/1.71	   reverse_concatenate_in_gga([], L, L) -> reverse_concatenate_out_gga([], L, L)
3.81/1.71	   reverse_concatenate_in_gga(.(X, L1), L2, L3) -> U4_gga(X, L1, L2, L3, reverse_concatenate_in_gga(L1, .(X, L2), L3))
3.81/1.71	   U4_gga(X, L1, L2, L3, reverse_concatenate_out_gga(L1, .(X, L2), L3)) -> reverse_concatenate_out_gga(.(X, L1), L2, L3)
3.81/1.71	
3.81/1.71	The argument filtering Pi contains the following mapping:
3.81/1.71	reverse_concatenate_in_gga(x1, x2, x3)  =  reverse_concatenate_in_gga(x1, x2)
3.81/1.71	
3.81/1.71	[]  =  []
3.81/1.71	
3.81/1.71	reverse_concatenate_out_gga(x1, x2, x3)  =  reverse_concatenate_out_gga(x3)
3.81/1.71	
3.81/1.71	.(x1, x2)  =  .(x1, x2)
3.81/1.71	
3.81/1.71	U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
3.81/1.71	
3.81/1.71	REVERSE_CONCATENATE_IN_GGA(x1, x2, x3)  =  REVERSE_CONCATENATE_IN_GGA(x1, x2)
3.81/1.71	
3.81/1.71	
3.81/1.71	We have to consider all (P,R,Pi)-chains
3.81/1.71	----------------------------------------
3.81/1.71	
3.81/1.71	(7) UsableRulesProof (EQUIVALENT)
3.81/1.71	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.81/1.71	----------------------------------------
3.81/1.71	
3.81/1.71	(8)
3.81/1.71	Obligation:
3.81/1.71	Pi DP problem:
3.81/1.71	The TRS P consists of the following rules:
3.81/1.71	
3.81/1.71	   REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2, L3) -> REVERSE_CONCATENATE_IN_GGA(L1, .(X, L2), L3)
3.81/1.71	
3.81/1.71	R is empty.
3.81/1.71	The argument filtering Pi contains the following mapping:
3.81/1.71	.(x1, x2)  =  .(x1, x2)
3.81/1.71	
3.81/1.71	REVERSE_CONCATENATE_IN_GGA(x1, x2, x3)  =  REVERSE_CONCATENATE_IN_GGA(x1, x2)
3.81/1.71	
3.81/1.71	
3.81/1.71	We have to consider all (P,R,Pi)-chains
3.81/1.71	----------------------------------------
3.81/1.71	
3.81/1.71	(9) PiDPToQDPProof (SOUND)
3.81/1.71	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.81/1.71	----------------------------------------
3.81/1.71	
3.81/1.71	(10)
3.81/1.71	Obligation:
3.81/1.71	Q DP problem:
3.81/1.71	The TRS P consists of the following rules:
3.81/1.71	
3.81/1.71	   REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2) -> REVERSE_CONCATENATE_IN_GGA(L1, .(X, L2))
3.81/1.71	
3.81/1.71	R is empty.
3.81/1.71	Q is empty.
3.81/1.71	We have to consider all (P,Q,R)-chains.
3.81/1.71	----------------------------------------
3.81/1.71	
3.81/1.71	(11) QDPSizeChangeProof (EQUIVALENT)
3.81/1.71	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. 
3.81/1.71	
3.81/1.71	From the DPs we obtained the following set of size-change graphs:
3.81/1.71	*REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2) -> REVERSE_CONCATENATE_IN_GGA(L1, .(X, L2))
3.81/1.71	The graph contains the following edges 1 > 1
3.81/1.71	
3.81/1.71	
3.81/1.71	----------------------------------------
3.81/1.71	
3.81/1.71	(12)
3.81/1.71	YES
4.01/1.81	EOF