3.91/1.87	YES
3.91/1.88	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.91/1.88	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.91/1.88	
3.91/1.88	
3.91/1.88	Left Termination of the query pattern
3.91/1.88	
3.91/1.88	rev(g,a)
3.91/1.88	
3.91/1.88	w.r.t. the given Prolog program could successfully be proven:
3.91/1.88	
3.91/1.88	(0) Prolog
3.91/1.88	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.91/1.88	(2) PiTRS
3.91/1.88	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.91/1.88	(4) PiDP
3.91/1.88	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.91/1.88	(6) PiDP
3.91/1.88	(7) UsableRulesProof [EQUIVALENT, 0 ms]
3.91/1.88	(8) PiDP
3.91/1.88	(9) PiDPToQDPProof [SOUND, 10 ms]
3.91/1.88	(10) QDP
3.91/1.88	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.91/1.88	(12) YES
3.91/1.88	
3.91/1.88	
3.91/1.88	----------------------------------------
3.91/1.88	
3.91/1.88	(0)
3.91/1.88	Obligation:
3.91/1.88	Clauses:
3.91/1.88	
3.91/1.88	rev(LS, RES) :- r1(LS, [], RES).
3.91/1.88	r1([], RES, RES).
3.91/1.88	r1(.(X, Xs), Accm, RES) :- r1(Xs, .(X, Accm), RES).
3.91/1.88	
3.91/1.88	
3.91/1.88	Query: rev(g,a)
3.91/1.88	----------------------------------------
3.91/1.88	
3.91/1.88	(1) PrologToPiTRSProof (SOUND)
3.91/1.88	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.91/1.88	
3.91/1.88	rev_in_2: (b,f)
3.91/1.88	
3.91/1.88	r1_in_3: (b,b,f)
3.91/1.88	
3.91/1.88	Transforming Prolog into the following Term Rewriting System:
3.91/1.88	
3.91/1.88	Pi-finite rewrite system:
3.91/1.88	The TRS R consists of the following rules:
3.91/1.88	
3.91/1.88	   rev_in_ga(LS, RES) -> U1_ga(LS, RES, r1_in_gga(LS, [], RES))
3.91/1.88	   r1_in_gga([], RES, RES) -> r1_out_gga([], RES, RES)
3.91/1.88	   r1_in_gga(.(X, Xs), Accm, RES) -> U2_gga(X, Xs, Accm, RES, r1_in_gga(Xs, .(X, Accm), RES))
3.91/1.88	   U2_gga(X, Xs, Accm, RES, r1_out_gga(Xs, .(X, Accm), RES)) -> r1_out_gga(.(X, Xs), Accm, RES)
3.91/1.88	   U1_ga(LS, RES, r1_out_gga(LS, [], RES)) -> rev_out_ga(LS, RES)
3.91/1.88	
3.91/1.88	The argument filtering Pi contains the following mapping:
3.91/1.88	rev_in_ga(x1, x2)  =  rev_in_ga(x1)
3.91/1.88	
3.91/1.88	U1_ga(x1, x2, x3)  =  U1_ga(x3)
3.91/1.88	
3.91/1.88	r1_in_gga(x1, x2, x3)  =  r1_in_gga(x1, x2)
3.91/1.88	
3.91/1.88	[]  =  []
3.91/1.88	
3.91/1.88	r1_out_gga(x1, x2, x3)  =  r1_out_gga(x3)
3.91/1.88	
3.91/1.88	.(x1, x2)  =  .(x1, x2)
3.91/1.88	
3.91/1.88	U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
3.91/1.88	
3.91/1.88	rev_out_ga(x1, x2)  =  rev_out_ga(x2)
3.91/1.88	
3.91/1.88	
3.91/1.88	
3.91/1.88	
3.91/1.88	
3.91/1.88	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.91/1.88	
3.91/1.88	
3.91/1.88	
3.91/1.88	----------------------------------------
3.91/1.88	
3.91/1.88	(2)
3.91/1.88	Obligation:
3.91/1.88	Pi-finite rewrite system:
3.91/1.88	The TRS R consists of the following rules:
3.91/1.88	
3.91/1.88	   rev_in_ga(LS, RES) -> U1_ga(LS, RES, r1_in_gga(LS, [], RES))
3.91/1.88	   r1_in_gga([], RES, RES) -> r1_out_gga([], RES, RES)
3.91/1.88	   r1_in_gga(.(X, Xs), Accm, RES) -> U2_gga(X, Xs, Accm, RES, r1_in_gga(Xs, .(X, Accm), RES))
3.91/1.88	   U2_gga(X, Xs, Accm, RES, r1_out_gga(Xs, .(X, Accm), RES)) -> r1_out_gga(.(X, Xs), Accm, RES)
3.91/1.88	   U1_ga(LS, RES, r1_out_gga(LS, [], RES)) -> rev_out_ga(LS, RES)
3.91/1.88	
3.91/1.88	The argument filtering Pi contains the following mapping:
3.91/1.88	rev_in_ga(x1, x2)  =  rev_in_ga(x1)
3.91/1.88	
3.91/1.88	U1_ga(x1, x2, x3)  =  U1_ga(x3)
3.91/1.88	
3.91/1.88	r1_in_gga(x1, x2, x3)  =  r1_in_gga(x1, x2)
3.91/1.88	
3.91/1.88	[]  =  []
3.91/1.88	
3.91/1.88	r1_out_gga(x1, x2, x3)  =  r1_out_gga(x3)
3.91/1.88	
3.91/1.88	.(x1, x2)  =  .(x1, x2)
3.91/1.88	
3.91/1.88	U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
3.91/1.88	
3.91/1.88	rev_out_ga(x1, x2)  =  rev_out_ga(x2)
3.91/1.88	
3.91/1.88	
3.91/1.88	
3.91/1.88	----------------------------------------
3.91/1.88	
3.91/1.88	(3) DependencyPairsProof (EQUIVALENT)
3.91/1.88	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.91/1.88	Pi DP problem:
3.91/1.88	The TRS P consists of the following rules:
3.91/1.88	
3.91/1.88	   REV_IN_GA(LS, RES) -> U1_GA(LS, RES, r1_in_gga(LS, [], RES))
3.91/1.88	   REV_IN_GA(LS, RES) -> R1_IN_GGA(LS, [], RES)
3.91/1.88	   R1_IN_GGA(.(X, Xs), Accm, RES) -> U2_GGA(X, Xs, Accm, RES, r1_in_gga(Xs, .(X, Accm), RES))
3.91/1.88	   R1_IN_GGA(.(X, Xs), Accm, RES) -> R1_IN_GGA(Xs, .(X, Accm), RES)
3.91/1.88	
3.91/1.88	The TRS R consists of the following rules:
3.91/1.88	
3.91/1.88	   rev_in_ga(LS, RES) -> U1_ga(LS, RES, r1_in_gga(LS, [], RES))
3.91/1.88	   r1_in_gga([], RES, RES) -> r1_out_gga([], RES, RES)
3.91/1.88	   r1_in_gga(.(X, Xs), Accm, RES) -> U2_gga(X, Xs, Accm, RES, r1_in_gga(Xs, .(X, Accm), RES))
3.91/1.88	   U2_gga(X, Xs, Accm, RES, r1_out_gga(Xs, .(X, Accm), RES)) -> r1_out_gga(.(X, Xs), Accm, RES)
3.91/1.88	   U1_ga(LS, RES, r1_out_gga(LS, [], RES)) -> rev_out_ga(LS, RES)
3.91/1.88	
3.91/1.88	The argument filtering Pi contains the following mapping:
3.91/1.88	rev_in_ga(x1, x2)  =  rev_in_ga(x1)
3.91/1.88	
3.91/1.88	U1_ga(x1, x2, x3)  =  U1_ga(x3)
3.91/1.88	
3.91/1.88	r1_in_gga(x1, x2, x3)  =  r1_in_gga(x1, x2)
3.91/1.88	
3.91/1.88	[]  =  []
3.91/1.88	
3.91/1.88	r1_out_gga(x1, x2, x3)  =  r1_out_gga(x3)
3.91/1.88	
3.91/1.88	.(x1, x2)  =  .(x1, x2)
3.91/1.88	
3.91/1.88	U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
3.91/1.88	
3.91/1.88	rev_out_ga(x1, x2)  =  rev_out_ga(x2)
3.91/1.88	
3.91/1.88	REV_IN_GA(x1, x2)  =  REV_IN_GA(x1)
3.91/1.88	
3.91/1.88	U1_GA(x1, x2, x3)  =  U1_GA(x3)
3.91/1.88	
3.91/1.88	R1_IN_GGA(x1, x2, x3)  =  R1_IN_GGA(x1, x2)
3.91/1.88	
3.91/1.88	U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x5)
3.91/1.88	
3.91/1.88	
3.91/1.88	We have to consider all (P,R,Pi)-chains
3.91/1.88	----------------------------------------
3.91/1.88	
3.91/1.88	(4)
3.91/1.88	Obligation:
3.91/1.88	Pi DP problem:
3.91/1.88	The TRS P consists of the following rules:
3.91/1.88	
3.91/1.88	   REV_IN_GA(LS, RES) -> U1_GA(LS, RES, r1_in_gga(LS, [], RES))
3.91/1.88	   REV_IN_GA(LS, RES) -> R1_IN_GGA(LS, [], RES)
3.91/1.88	   R1_IN_GGA(.(X, Xs), Accm, RES) -> U2_GGA(X, Xs, Accm, RES, r1_in_gga(Xs, .(X, Accm), RES))
3.91/1.88	   R1_IN_GGA(.(X, Xs), Accm, RES) -> R1_IN_GGA(Xs, .(X, Accm), RES)
3.91/1.88	
3.91/1.88	The TRS R consists of the following rules:
3.91/1.88	
3.91/1.88	   rev_in_ga(LS, RES) -> U1_ga(LS, RES, r1_in_gga(LS, [], RES))
3.91/1.88	   r1_in_gga([], RES, RES) -> r1_out_gga([], RES, RES)
3.91/1.88	   r1_in_gga(.(X, Xs), Accm, RES) -> U2_gga(X, Xs, Accm, RES, r1_in_gga(Xs, .(X, Accm), RES))
3.91/1.88	   U2_gga(X, Xs, Accm, RES, r1_out_gga(Xs, .(X, Accm), RES)) -> r1_out_gga(.(X, Xs), Accm, RES)
3.91/1.88	   U1_ga(LS, RES, r1_out_gga(LS, [], RES)) -> rev_out_ga(LS, RES)
3.91/1.88	
3.91/1.88	The argument filtering Pi contains the following mapping:
3.91/1.88	rev_in_ga(x1, x2)  =  rev_in_ga(x1)
3.91/1.88	
3.91/1.88	U1_ga(x1, x2, x3)  =  U1_ga(x3)
3.91/1.88	
3.91/1.88	r1_in_gga(x1, x2, x3)  =  r1_in_gga(x1, x2)
3.91/1.88	
3.91/1.88	[]  =  []
3.91/1.88	
3.91/1.88	r1_out_gga(x1, x2, x3)  =  r1_out_gga(x3)
3.91/1.88	
3.91/1.88	.(x1, x2)  =  .(x1, x2)
3.91/1.88	
3.91/1.88	U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
3.91/1.88	
3.91/1.88	rev_out_ga(x1, x2)  =  rev_out_ga(x2)
3.91/1.88	
3.91/1.88	REV_IN_GA(x1, x2)  =  REV_IN_GA(x1)
3.91/1.88	
3.91/1.88	U1_GA(x1, x2, x3)  =  U1_GA(x3)
3.91/1.88	
3.91/1.88	R1_IN_GGA(x1, x2, x3)  =  R1_IN_GGA(x1, x2)
3.91/1.88	
3.91/1.88	U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x5)
3.91/1.88	
3.91/1.88	
3.91/1.88	We have to consider all (P,R,Pi)-chains
3.91/1.88	----------------------------------------
3.91/1.88	
3.91/1.88	(5) DependencyGraphProof (EQUIVALENT)
3.91/1.88	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
3.91/1.88	----------------------------------------
3.91/1.88	
3.91/1.88	(6)
3.91/1.88	Obligation:
3.91/1.88	Pi DP problem:
3.91/1.88	The TRS P consists of the following rules:
3.91/1.88	
3.91/1.88	   R1_IN_GGA(.(X, Xs), Accm, RES) -> R1_IN_GGA(Xs, .(X, Accm), RES)
3.91/1.88	
3.91/1.88	The TRS R consists of the following rules:
3.91/1.88	
3.91/1.88	   rev_in_ga(LS, RES) -> U1_ga(LS, RES, r1_in_gga(LS, [], RES))
3.91/1.88	   r1_in_gga([], RES, RES) -> r1_out_gga([], RES, RES)
3.91/1.88	   r1_in_gga(.(X, Xs), Accm, RES) -> U2_gga(X, Xs, Accm, RES, r1_in_gga(Xs, .(X, Accm), RES))
3.91/1.88	   U2_gga(X, Xs, Accm, RES, r1_out_gga(Xs, .(X, Accm), RES)) -> r1_out_gga(.(X, Xs), Accm, RES)
3.91/1.88	   U1_ga(LS, RES, r1_out_gga(LS, [], RES)) -> rev_out_ga(LS, RES)
3.91/1.88	
3.91/1.88	The argument filtering Pi contains the following mapping:
3.91/1.88	rev_in_ga(x1, x2)  =  rev_in_ga(x1)
3.91/1.88	
3.91/1.88	U1_ga(x1, x2, x3)  =  U1_ga(x3)
3.91/1.88	
3.91/1.88	r1_in_gga(x1, x2, x3)  =  r1_in_gga(x1, x2)
3.91/1.88	
3.91/1.88	[]  =  []
3.91/1.88	
3.91/1.88	r1_out_gga(x1, x2, x3)  =  r1_out_gga(x3)
3.91/1.88	
3.91/1.88	.(x1, x2)  =  .(x1, x2)
3.91/1.88	
3.91/1.88	U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
3.91/1.88	
3.91/1.88	rev_out_ga(x1, x2)  =  rev_out_ga(x2)
3.91/1.88	
3.91/1.88	R1_IN_GGA(x1, x2, x3)  =  R1_IN_GGA(x1, x2)
3.91/1.88	
3.91/1.88	
3.91/1.88	We have to consider all (P,R,Pi)-chains
3.91/1.88	----------------------------------------
3.91/1.88	
3.91/1.88	(7) UsableRulesProof (EQUIVALENT)
3.91/1.88	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.91/1.88	----------------------------------------
3.91/1.88	
3.91/1.88	(8)
3.91/1.88	Obligation:
3.91/1.88	Pi DP problem:
3.91/1.88	The TRS P consists of the following rules:
3.91/1.88	
3.91/1.88	   R1_IN_GGA(.(X, Xs), Accm, RES) -> R1_IN_GGA(Xs, .(X, Accm), RES)
3.91/1.88	
3.91/1.88	R is empty.
3.91/1.88	The argument filtering Pi contains the following mapping:
3.91/1.88	.(x1, x2)  =  .(x1, x2)
3.91/1.88	
3.91/1.88	R1_IN_GGA(x1, x2, x3)  =  R1_IN_GGA(x1, x2)
3.91/1.88	
3.91/1.88	
3.91/1.88	We have to consider all (P,R,Pi)-chains
3.91/1.88	----------------------------------------
3.91/1.88	
3.91/1.88	(9) PiDPToQDPProof (SOUND)
3.91/1.88	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.91/1.88	----------------------------------------
3.91/1.88	
3.91/1.88	(10)
3.91/1.88	Obligation:
3.91/1.88	Q DP problem:
3.91/1.88	The TRS P consists of the following rules:
3.91/1.88	
3.91/1.88	   R1_IN_GGA(.(X, Xs), Accm) -> R1_IN_GGA(Xs, .(X, Accm))
3.91/1.88	
3.91/1.88	R is empty.
3.91/1.88	Q is empty.
3.91/1.88	We have to consider all (P,Q,R)-chains.
3.91/1.88	----------------------------------------
3.91/1.88	
3.91/1.88	(11) QDPSizeChangeProof (EQUIVALENT)
3.91/1.88	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. 
3.91/1.88	
3.91/1.88	From the DPs we obtained the following set of size-change graphs:
3.91/1.88	*R1_IN_GGA(.(X, Xs), Accm) -> R1_IN_GGA(Xs, .(X, Accm))
3.91/1.88	The graph contains the following edges 1 > 1
3.91/1.88	
3.91/1.88	
3.91/1.88	----------------------------------------
3.91/1.88	
3.91/1.88	(12)
3.91/1.88	YES
4.06/1.92	EOF