4.01/1.85	YES
4.23/1.87	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
4.23/1.87	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.23/1.87	
4.23/1.87	
4.23/1.87	Left Termination of the query pattern
4.23/1.87	
4.23/1.87	f(g,g,a)
4.23/1.87	
4.23/1.87	w.r.t. the given Prolog program could successfully be proven:
4.23/1.87	
4.23/1.87	(0) Prolog
4.23/1.87	(1) PrologToPiTRSProof [SOUND, 0 ms]
4.23/1.87	(2) PiTRS
4.23/1.87	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
4.23/1.87	(4) PiDP
4.23/1.87	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
4.23/1.87	(6) AND
4.23/1.87	    (7) PiDP
4.23/1.87	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
4.23/1.87	        (9) PiDP
4.23/1.87	        (10) PiDPToQDPProof [SOUND, 17 ms]
4.23/1.87	        (11) QDP
4.23/1.87	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
4.23/1.87	        (13) YES
4.23/1.87	    (14) PiDP
4.23/1.87	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
4.23/1.87	        (16) PiDP
4.23/1.87	        (17) PiDPToQDPProof [SOUND, 0 ms]
4.23/1.87	        (18) QDP
4.23/1.87	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
4.23/1.87	        (20) YES
4.23/1.87	
4.23/1.87	
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(0)
4.23/1.87	Obligation:
4.23/1.87	Clauses:
4.23/1.87	
4.23/1.87	f(A, [], RES) :- g(A, [], RES).
4.23/1.87	f(.(A, As), .(B, Bs), RES) :- f(.(B, .(A, As)), Bs, RES).
4.23/1.87	g([], RES, RES).
4.23/1.87	g(.(C, Cs), D, RES) :- g(Cs, .(C, D), RES).
4.23/1.87	
4.23/1.87	
4.23/1.87	Query: f(g,g,a)
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(1) PrologToPiTRSProof (SOUND)
4.23/1.87	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
4.23/1.87	
4.23/1.87	f_in_3: (b,b,f)
4.23/1.87	
4.23/1.87	g_in_3: (b,b,f)
4.23/1.87	
4.23/1.87	Transforming Prolog into the following Term Rewriting System:
4.23/1.87	
4.23/1.87	Pi-finite rewrite system:
4.23/1.87	The TRS R consists of the following rules:
4.23/1.87	
4.23/1.87	   f_in_gga(A, [], RES) -> U1_gga(A, RES, g_in_gga(A, [], RES))
4.23/1.87	   g_in_gga([], RES, RES) -> g_out_gga([], RES, RES)
4.23/1.87	   g_in_gga(.(C, Cs), D, RES) -> U3_gga(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES))
4.23/1.87	   U3_gga(C, Cs, D, RES, g_out_gga(Cs, .(C, D), RES)) -> g_out_gga(.(C, Cs), D, RES)
4.23/1.87	   U1_gga(A, RES, g_out_gga(A, [], RES)) -> f_out_gga(A, [], RES)
4.23/1.87	   f_in_gga(.(A, As), .(B, Bs), RES) -> U2_gga(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES))
4.23/1.87	   U2_gga(A, As, B, Bs, RES, f_out_gga(.(B, .(A, As)), Bs, RES)) -> f_out_gga(.(A, As), .(B, Bs), RES)
4.23/1.87	
4.23/1.87	The argument filtering Pi contains the following mapping:
4.23/1.87	f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
4.23/1.87	
4.23/1.87	[]  =  []
4.23/1.87	
4.23/1.87	U1_gga(x1, x2, x3)  =  U1_gga(x3)
4.23/1.87	
4.23/1.87	g_in_gga(x1, x2, x3)  =  g_in_gga(x1, x2)
4.23/1.87	
4.23/1.87	g_out_gga(x1, x2, x3)  =  g_out_gga(x3)
4.23/1.87	
4.23/1.87	.(x1, x2)  =  .(x1, x2)
4.23/1.87	
4.23/1.87	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
4.23/1.87	
4.23/1.87	f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
4.23/1.87	
4.23/1.87	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x6)
4.23/1.87	
4.23/1.87	
4.23/1.87	
4.23/1.87	
4.23/1.87	
4.23/1.87	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
4.23/1.87	
4.23/1.87	
4.23/1.87	
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(2)
4.23/1.87	Obligation:
4.23/1.87	Pi-finite rewrite system:
4.23/1.87	The TRS R consists of the following rules:
4.23/1.87	
4.23/1.87	   f_in_gga(A, [], RES) -> U1_gga(A, RES, g_in_gga(A, [], RES))
4.23/1.87	   g_in_gga([], RES, RES) -> g_out_gga([], RES, RES)
4.23/1.87	   g_in_gga(.(C, Cs), D, RES) -> U3_gga(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES))
4.23/1.87	   U3_gga(C, Cs, D, RES, g_out_gga(Cs, .(C, D), RES)) -> g_out_gga(.(C, Cs), D, RES)
4.23/1.87	   U1_gga(A, RES, g_out_gga(A, [], RES)) -> f_out_gga(A, [], RES)
4.23/1.87	   f_in_gga(.(A, As), .(B, Bs), RES) -> U2_gga(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES))
4.23/1.87	   U2_gga(A, As, B, Bs, RES, f_out_gga(.(B, .(A, As)), Bs, RES)) -> f_out_gga(.(A, As), .(B, Bs), RES)
4.23/1.87	
4.23/1.87	The argument filtering Pi contains the following mapping:
4.23/1.87	f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
4.23/1.87	
4.23/1.87	[]  =  []
4.23/1.87	
4.23/1.87	U1_gga(x1, x2, x3)  =  U1_gga(x3)
4.23/1.87	
4.23/1.87	g_in_gga(x1, x2, x3)  =  g_in_gga(x1, x2)
4.23/1.87	
4.23/1.87	g_out_gga(x1, x2, x3)  =  g_out_gga(x3)
4.23/1.87	
4.23/1.87	.(x1, x2)  =  .(x1, x2)
4.23/1.87	
4.23/1.87	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
4.23/1.87	
4.23/1.87	f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
4.23/1.87	
4.23/1.87	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x6)
4.23/1.87	
4.23/1.87	
4.23/1.87	
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(3) DependencyPairsProof (EQUIVALENT)
4.23/1.87	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
4.23/1.87	Pi DP problem:
4.23/1.87	The TRS P consists of the following rules:
4.23/1.87	
4.23/1.87	   F_IN_GGA(A, [], RES) -> U1_GGA(A, RES, g_in_gga(A, [], RES))
4.23/1.87	   F_IN_GGA(A, [], RES) -> G_IN_GGA(A, [], RES)
4.23/1.87	   G_IN_GGA(.(C, Cs), D, RES) -> U3_GGA(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES))
4.23/1.87	   G_IN_GGA(.(C, Cs), D, RES) -> G_IN_GGA(Cs, .(C, D), RES)
4.23/1.87	   F_IN_GGA(.(A, As), .(B, Bs), RES) -> U2_GGA(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES))
4.23/1.87	   F_IN_GGA(.(A, As), .(B, Bs), RES) -> F_IN_GGA(.(B, .(A, As)), Bs, RES)
4.23/1.87	
4.23/1.87	The TRS R consists of the following rules:
4.23/1.87	
4.23/1.87	   f_in_gga(A, [], RES) -> U1_gga(A, RES, g_in_gga(A, [], RES))
4.23/1.87	   g_in_gga([], RES, RES) -> g_out_gga([], RES, RES)
4.23/1.87	   g_in_gga(.(C, Cs), D, RES) -> U3_gga(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES))
4.23/1.87	   U3_gga(C, Cs, D, RES, g_out_gga(Cs, .(C, D), RES)) -> g_out_gga(.(C, Cs), D, RES)
4.23/1.87	   U1_gga(A, RES, g_out_gga(A, [], RES)) -> f_out_gga(A, [], RES)
4.23/1.87	   f_in_gga(.(A, As), .(B, Bs), RES) -> U2_gga(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES))
4.23/1.87	   U2_gga(A, As, B, Bs, RES, f_out_gga(.(B, .(A, As)), Bs, RES)) -> f_out_gga(.(A, As), .(B, Bs), RES)
4.23/1.87	
4.23/1.87	The argument filtering Pi contains the following mapping:
4.23/1.87	f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
4.23/1.87	
4.23/1.87	[]  =  []
4.23/1.87	
4.23/1.87	U1_gga(x1, x2, x3)  =  U1_gga(x3)
4.23/1.87	
4.23/1.87	g_in_gga(x1, x2, x3)  =  g_in_gga(x1, x2)
4.23/1.87	
4.23/1.87	g_out_gga(x1, x2, x3)  =  g_out_gga(x3)
4.23/1.87	
4.23/1.87	.(x1, x2)  =  .(x1, x2)
4.23/1.87	
4.23/1.87	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
4.23/1.87	
4.23/1.87	f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
4.23/1.87	
4.23/1.87	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x6)
4.23/1.87	
4.23/1.87	F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)
4.23/1.87	
4.23/1.87	U1_GGA(x1, x2, x3)  =  U1_GGA(x3)
4.23/1.87	
4.23/1.87	G_IN_GGA(x1, x2, x3)  =  G_IN_GGA(x1, x2)
4.23/1.87	
4.23/1.87	U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x5)
4.23/1.87	
4.23/1.87	U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x6)
4.23/1.87	
4.23/1.87	
4.23/1.87	We have to consider all (P,R,Pi)-chains
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(4)
4.23/1.87	Obligation:
4.23/1.87	Pi DP problem:
4.23/1.87	The TRS P consists of the following rules:
4.23/1.87	
4.23/1.87	   F_IN_GGA(A, [], RES) -> U1_GGA(A, RES, g_in_gga(A, [], RES))
4.23/1.87	   F_IN_GGA(A, [], RES) -> G_IN_GGA(A, [], RES)
4.23/1.87	   G_IN_GGA(.(C, Cs), D, RES) -> U3_GGA(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES))
4.23/1.87	   G_IN_GGA(.(C, Cs), D, RES) -> G_IN_GGA(Cs, .(C, D), RES)
4.23/1.87	   F_IN_GGA(.(A, As), .(B, Bs), RES) -> U2_GGA(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES))
4.23/1.87	   F_IN_GGA(.(A, As), .(B, Bs), RES) -> F_IN_GGA(.(B, .(A, As)), Bs, RES)
4.23/1.87	
4.23/1.87	The TRS R consists of the following rules:
4.23/1.87	
4.23/1.87	   f_in_gga(A, [], RES) -> U1_gga(A, RES, g_in_gga(A, [], RES))
4.23/1.87	   g_in_gga([], RES, RES) -> g_out_gga([], RES, RES)
4.23/1.87	   g_in_gga(.(C, Cs), D, RES) -> U3_gga(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES))
4.23/1.87	   U3_gga(C, Cs, D, RES, g_out_gga(Cs, .(C, D), RES)) -> g_out_gga(.(C, Cs), D, RES)
4.23/1.87	   U1_gga(A, RES, g_out_gga(A, [], RES)) -> f_out_gga(A, [], RES)
4.23/1.87	   f_in_gga(.(A, As), .(B, Bs), RES) -> U2_gga(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES))
4.23/1.87	   U2_gga(A, As, B, Bs, RES, f_out_gga(.(B, .(A, As)), Bs, RES)) -> f_out_gga(.(A, As), .(B, Bs), RES)
4.23/1.87	
4.23/1.87	The argument filtering Pi contains the following mapping:
4.23/1.87	f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
4.23/1.87	
4.23/1.87	[]  =  []
4.23/1.87	
4.23/1.87	U1_gga(x1, x2, x3)  =  U1_gga(x3)
4.23/1.87	
4.23/1.87	g_in_gga(x1, x2, x3)  =  g_in_gga(x1, x2)
4.23/1.87	
4.23/1.87	g_out_gga(x1, x2, x3)  =  g_out_gga(x3)
4.23/1.87	
4.23/1.87	.(x1, x2)  =  .(x1, x2)
4.23/1.87	
4.23/1.87	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
4.23/1.87	
4.23/1.87	f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
4.23/1.87	
4.23/1.87	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x6)
4.23/1.87	
4.23/1.87	F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)
4.23/1.87	
4.23/1.87	U1_GGA(x1, x2, x3)  =  U1_GGA(x3)
4.23/1.87	
4.23/1.87	G_IN_GGA(x1, x2, x3)  =  G_IN_GGA(x1, x2)
4.23/1.87	
4.23/1.87	U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x5)
4.23/1.87	
4.23/1.87	U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x6)
4.23/1.87	
4.23/1.87	
4.23/1.87	We have to consider all (P,R,Pi)-chains
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(5) DependencyGraphProof (EQUIVALENT)
4.23/1.87	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes.
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(6)
4.23/1.87	Complex Obligation (AND)
4.23/1.87	
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(7)
4.23/1.87	Obligation:
4.23/1.87	Pi DP problem:
4.23/1.87	The TRS P consists of the following rules:
4.23/1.87	
4.23/1.87	   G_IN_GGA(.(C, Cs), D, RES) -> G_IN_GGA(Cs, .(C, D), RES)
4.23/1.87	
4.23/1.87	The TRS R consists of the following rules:
4.23/1.87	
4.23/1.87	   f_in_gga(A, [], RES) -> U1_gga(A, RES, g_in_gga(A, [], RES))
4.23/1.87	   g_in_gga([], RES, RES) -> g_out_gga([], RES, RES)
4.23/1.87	   g_in_gga(.(C, Cs), D, RES) -> U3_gga(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES))
4.23/1.87	   U3_gga(C, Cs, D, RES, g_out_gga(Cs, .(C, D), RES)) -> g_out_gga(.(C, Cs), D, RES)
4.23/1.87	   U1_gga(A, RES, g_out_gga(A, [], RES)) -> f_out_gga(A, [], RES)
4.23/1.87	   f_in_gga(.(A, As), .(B, Bs), RES) -> U2_gga(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES))
4.23/1.87	   U2_gga(A, As, B, Bs, RES, f_out_gga(.(B, .(A, As)), Bs, RES)) -> f_out_gga(.(A, As), .(B, Bs), RES)
4.23/1.87	
4.23/1.87	The argument filtering Pi contains the following mapping:
4.23/1.87	f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
4.23/1.87	
4.23/1.87	[]  =  []
4.23/1.87	
4.23/1.87	U1_gga(x1, x2, x3)  =  U1_gga(x3)
4.23/1.87	
4.23/1.87	g_in_gga(x1, x2, x3)  =  g_in_gga(x1, x2)
4.23/1.87	
4.23/1.87	g_out_gga(x1, x2, x3)  =  g_out_gga(x3)
4.23/1.87	
4.23/1.87	.(x1, x2)  =  .(x1, x2)
4.23/1.87	
4.23/1.87	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
4.23/1.87	
4.23/1.87	f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
4.23/1.87	
4.23/1.87	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x6)
4.23/1.87	
4.23/1.87	G_IN_GGA(x1, x2, x3)  =  G_IN_GGA(x1, x2)
4.23/1.87	
4.23/1.87	
4.23/1.87	We have to consider all (P,R,Pi)-chains
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(8) UsableRulesProof (EQUIVALENT)
4.23/1.87	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(9)
4.23/1.87	Obligation:
4.23/1.87	Pi DP problem:
4.23/1.87	The TRS P consists of the following rules:
4.23/1.87	
4.23/1.87	   G_IN_GGA(.(C, Cs), D, RES) -> G_IN_GGA(Cs, .(C, D), RES)
4.23/1.87	
4.23/1.87	R is empty.
4.23/1.87	The argument filtering Pi contains the following mapping:
4.23/1.87	.(x1, x2)  =  .(x1, x2)
4.23/1.87	
4.23/1.87	G_IN_GGA(x1, x2, x3)  =  G_IN_GGA(x1, x2)
4.23/1.87	
4.23/1.87	
4.23/1.87	We have to consider all (P,R,Pi)-chains
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(10) PiDPToQDPProof (SOUND)
4.23/1.87	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(11)
4.23/1.87	Obligation:
4.23/1.87	Q DP problem:
4.23/1.87	The TRS P consists of the following rules:
4.23/1.87	
4.23/1.87	   G_IN_GGA(.(C, Cs), D) -> G_IN_GGA(Cs, .(C, D))
4.23/1.87	
4.23/1.87	R is empty.
4.23/1.87	Q is empty.
4.23/1.87	We have to consider all (P,Q,R)-chains.
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(12) QDPSizeChangeProof (EQUIVALENT)
4.23/1.87	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. 
4.23/1.87	
4.23/1.87	From the DPs we obtained the following set of size-change graphs:
4.23/1.87	*G_IN_GGA(.(C, Cs), D) -> G_IN_GGA(Cs, .(C, D))
4.23/1.87	The graph contains the following edges 1 > 1
4.23/1.87	
4.23/1.87	
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(13)
4.23/1.87	YES
4.23/1.87	
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(14)
4.23/1.87	Obligation:
4.23/1.87	Pi DP problem:
4.23/1.87	The TRS P consists of the following rules:
4.23/1.87	
4.23/1.87	   F_IN_GGA(.(A, As), .(B, Bs), RES) -> F_IN_GGA(.(B, .(A, As)), Bs, RES)
4.23/1.87	
4.23/1.87	The TRS R consists of the following rules:
4.23/1.87	
4.23/1.87	   f_in_gga(A, [], RES) -> U1_gga(A, RES, g_in_gga(A, [], RES))
4.23/1.87	   g_in_gga([], RES, RES) -> g_out_gga([], RES, RES)
4.23/1.87	   g_in_gga(.(C, Cs), D, RES) -> U3_gga(C, Cs, D, RES, g_in_gga(Cs, .(C, D), RES))
4.23/1.87	   U3_gga(C, Cs, D, RES, g_out_gga(Cs, .(C, D), RES)) -> g_out_gga(.(C, Cs), D, RES)
4.23/1.87	   U1_gga(A, RES, g_out_gga(A, [], RES)) -> f_out_gga(A, [], RES)
4.23/1.87	   f_in_gga(.(A, As), .(B, Bs), RES) -> U2_gga(A, As, B, Bs, RES, f_in_gga(.(B, .(A, As)), Bs, RES))
4.23/1.87	   U2_gga(A, As, B, Bs, RES, f_out_gga(.(B, .(A, As)), Bs, RES)) -> f_out_gga(.(A, As), .(B, Bs), RES)
4.23/1.87	
4.23/1.87	The argument filtering Pi contains the following mapping:
4.23/1.87	f_in_gga(x1, x2, x3)  =  f_in_gga(x1, x2)
4.23/1.87	
4.23/1.87	[]  =  []
4.23/1.87	
4.23/1.87	U1_gga(x1, x2, x3)  =  U1_gga(x3)
4.23/1.87	
4.23/1.87	g_in_gga(x1, x2, x3)  =  g_in_gga(x1, x2)
4.23/1.87	
4.23/1.87	g_out_gga(x1, x2, x3)  =  g_out_gga(x3)
4.23/1.87	
4.23/1.87	.(x1, x2)  =  .(x1, x2)
4.23/1.87	
4.23/1.87	U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x5)
4.23/1.87	
4.23/1.87	f_out_gga(x1, x2, x3)  =  f_out_gga(x3)
4.23/1.87	
4.23/1.87	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x6)
4.23/1.87	
4.23/1.87	F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)
4.23/1.87	
4.23/1.87	
4.23/1.87	We have to consider all (P,R,Pi)-chains
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(15) UsableRulesProof (EQUIVALENT)
4.23/1.87	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(16)
4.23/1.87	Obligation:
4.23/1.87	Pi DP problem:
4.23/1.87	The TRS P consists of the following rules:
4.23/1.87	
4.23/1.87	   F_IN_GGA(.(A, As), .(B, Bs), RES) -> F_IN_GGA(.(B, .(A, As)), Bs, RES)
4.23/1.87	
4.23/1.87	R is empty.
4.23/1.87	The argument filtering Pi contains the following mapping:
4.23/1.87	.(x1, x2)  =  .(x1, x2)
4.23/1.87	
4.23/1.87	F_IN_GGA(x1, x2, x3)  =  F_IN_GGA(x1, x2)
4.23/1.87	
4.23/1.87	
4.23/1.87	We have to consider all (P,R,Pi)-chains
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(17) PiDPToQDPProof (SOUND)
4.23/1.87	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(18)
4.23/1.87	Obligation:
4.23/1.87	Q DP problem:
4.23/1.87	The TRS P consists of the following rules:
4.23/1.87	
4.23/1.87	   F_IN_GGA(.(A, As), .(B, Bs)) -> F_IN_GGA(.(B, .(A, As)), Bs)
4.23/1.87	
4.23/1.87	R is empty.
4.23/1.87	Q is empty.
4.23/1.87	We have to consider all (P,Q,R)-chains.
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(19) QDPSizeChangeProof (EQUIVALENT)
4.23/1.87	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. 
4.23/1.87	
4.23/1.87	From the DPs we obtained the following set of size-change graphs:
4.23/1.87	*F_IN_GGA(.(A, As), .(B, Bs)) -> F_IN_GGA(.(B, .(A, As)), Bs)
4.23/1.87	The graph contains the following edges 2 > 2
4.23/1.87	
4.23/1.87	
4.23/1.87	----------------------------------------
4.23/1.87	
4.23/1.87	(20)
4.23/1.87	YES
4.23/1.89	EOF