4.93/2.02	YES
4.93/2.05	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
4.93/2.05	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.93/2.05	
4.93/2.05	
4.93/2.05	Left Termination of the query pattern
4.93/2.05	
4.93/2.05	normal(g,a)
4.93/2.05	
4.93/2.05	w.r.t. the given Prolog program could successfully be proven:
4.93/2.05	
4.93/2.05	(0) Prolog
4.93/2.05	(1) PrologToPiTRSProof [SOUND, 0 ms]
4.93/2.05	(2) PiTRS
4.93/2.05	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
4.93/2.05	(4) PiDP
4.93/2.05	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
4.93/2.05	(6) AND
4.93/2.05	    (7) PiDP
4.93/2.05	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
4.93/2.05	        (9) PiDP
4.93/2.05	        (10) PiDPToQDPProof [SOUND, 0 ms]
4.93/2.05	        (11) QDP
4.93/2.05	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
4.93/2.05	        (13) YES
4.93/2.05	    (14) PiDP
4.93/2.05	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
4.93/2.05	        (16) PiDP
4.93/2.05	        (17) PiDPToQDPProof [SOUND, 0 ms]
4.93/2.05	        (18) QDP
4.93/2.05	        (19) MRRProof [EQUIVALENT, 13 ms]
4.93/2.05	        (20) QDP
4.93/2.05	        (21) PisEmptyProof [EQUIVALENT, 0 ms]
4.93/2.05	        (22) YES
4.93/2.05	
4.93/2.05	
4.93/2.05	----------------------------------------
4.93/2.05	
4.93/2.05	(0)
4.93/2.05	Obligation:
4.93/2.05	Clauses:
4.93/2.05	
4.93/2.05	normal(F, N) :- ','(rewrite(F, F1), normal(F1, N)).
4.93/2.05	normal(F, F).
4.93/2.05	rewrite(op(op(A, B), C), op(A, op(B, C))).
4.93/2.05	rewrite(op(A, op(B, C)), op(A, L)) :- rewrite(op(B, C), L).
4.93/2.05	
4.93/2.05	
4.93/2.05	Query: normal(g,a)
4.93/2.05	----------------------------------------
4.93/2.05	
4.93/2.05	(1) PrologToPiTRSProof (SOUND)
4.93/2.05	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
4.93/2.05	
4.93/2.05	normal_in_2: (b,f)
4.93/2.05	
4.93/2.05	rewrite_in_2: (b,f)
4.93/2.05	
4.93/2.05	Transforming Prolog into the following Term Rewriting System:
4.93/2.05	
4.93/2.05	Pi-finite rewrite system:
4.93/2.05	The TRS R consists of the following rules:
4.93/2.05	
4.93/2.05	   normal_in_ga(F, N) -> U1_ga(F, N, rewrite_in_ga(F, F1))
4.93/2.05	   rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) -> rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
4.93/2.05	   rewrite_in_ga(op(A, op(B, C)), op(A, L)) -> U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L))
4.93/2.05	   U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) -> rewrite_out_ga(op(A, op(B, C)), op(A, L))
4.93/2.05	   U1_ga(F, N, rewrite_out_ga(F, F1)) -> U2_ga(F, N, normal_in_ga(F1, N))
4.93/2.05	   normal_in_ga(F, F) -> normal_out_ga(F, F)
4.93/2.05	   U2_ga(F, N, normal_out_ga(F1, N)) -> normal_out_ga(F, N)
4.93/2.05	
4.93/2.05	The argument filtering Pi contains the following mapping:
4.93/2.05	normal_in_ga(x1, x2)  =  normal_in_ga(x1)
4.93/2.05	
4.93/2.05	U1_ga(x1, x2, x3)  =  U1_ga(x3)
4.93/2.05	
4.93/2.05	rewrite_in_ga(x1, x2)  =  rewrite_in_ga(x1)
4.93/2.05	
4.93/2.05	op(x1, x2)  =  op(x1, x2)
4.93/2.05	
4.93/2.05	rewrite_out_ga(x1, x2)  =  rewrite_out_ga(x2)
4.93/2.05	
4.93/2.05	U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
4.93/2.05	
4.93/2.05	U2_ga(x1, x2, x3)  =  U2_ga(x3)
4.93/2.05	
4.93/2.05	normal_out_ga(x1, x2)  =  normal_out_ga(x2)
4.93/2.05	
4.93/2.05	
4.93/2.05	
4.93/2.05	
4.93/2.05	
4.93/2.05	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
4.93/2.05	
4.93/2.05	
4.93/2.05	
4.93/2.05	----------------------------------------
4.93/2.05	
4.93/2.05	(2)
4.93/2.05	Obligation:
4.93/2.05	Pi-finite rewrite system:
4.93/2.05	The TRS R consists of the following rules:
4.93/2.05	
4.93/2.05	   normal_in_ga(F, N) -> U1_ga(F, N, rewrite_in_ga(F, F1))
4.93/2.05	   rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) -> rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
4.93/2.05	   rewrite_in_ga(op(A, op(B, C)), op(A, L)) -> U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L))
4.93/2.05	   U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) -> rewrite_out_ga(op(A, op(B, C)), op(A, L))
4.93/2.05	   U1_ga(F, N, rewrite_out_ga(F, F1)) -> U2_ga(F, N, normal_in_ga(F1, N))
4.93/2.05	   normal_in_ga(F, F) -> normal_out_ga(F, F)
4.93/2.05	   U2_ga(F, N, normal_out_ga(F1, N)) -> normal_out_ga(F, N)
4.93/2.05	
4.93/2.05	The argument filtering Pi contains the following mapping:
4.93/2.05	normal_in_ga(x1, x2)  =  normal_in_ga(x1)
4.93/2.05	
4.93/2.05	U1_ga(x1, x2, x3)  =  U1_ga(x3)
4.93/2.05	
4.93/2.05	rewrite_in_ga(x1, x2)  =  rewrite_in_ga(x1)
4.93/2.05	
4.93/2.05	op(x1, x2)  =  op(x1, x2)
4.93/2.05	
4.93/2.05	rewrite_out_ga(x1, x2)  =  rewrite_out_ga(x2)
4.93/2.05	
4.93/2.05	U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
4.93/2.05	
4.93/2.05	U2_ga(x1, x2, x3)  =  U2_ga(x3)
4.93/2.05	
4.93/2.05	normal_out_ga(x1, x2)  =  normal_out_ga(x2)
4.93/2.05	
4.93/2.05	
4.93/2.05	
4.93/2.05	----------------------------------------
4.93/2.05	
4.93/2.05	(3) DependencyPairsProof (EQUIVALENT)
4.93/2.05	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
4.93/2.05	Pi DP problem:
4.93/2.05	The TRS P consists of the following rules:
4.93/2.05	
4.93/2.05	   NORMAL_IN_GA(F, N) -> U1_GA(F, N, rewrite_in_ga(F, F1))
4.93/2.05	   NORMAL_IN_GA(F, N) -> REWRITE_IN_GA(F, F1)
4.93/2.05	   REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) -> U3_GA(A, B, C, L, rewrite_in_ga(op(B, C), L))
4.93/2.05	   REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) -> REWRITE_IN_GA(op(B, C), L)
4.93/2.05	   U1_GA(F, N, rewrite_out_ga(F, F1)) -> U2_GA(F, N, normal_in_ga(F1, N))
4.93/2.05	   U1_GA(F, N, rewrite_out_ga(F, F1)) -> NORMAL_IN_GA(F1, N)
4.93/2.05	
4.93/2.05	The TRS R consists of the following rules:
4.93/2.05	
4.93/2.05	   normal_in_ga(F, N) -> U1_ga(F, N, rewrite_in_ga(F, F1))
4.93/2.05	   rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) -> rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
4.93/2.05	   rewrite_in_ga(op(A, op(B, C)), op(A, L)) -> U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L))
4.93/2.05	   U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) -> rewrite_out_ga(op(A, op(B, C)), op(A, L))
4.93/2.05	   U1_ga(F, N, rewrite_out_ga(F, F1)) -> U2_ga(F, N, normal_in_ga(F1, N))
4.93/2.05	   normal_in_ga(F, F) -> normal_out_ga(F, F)
4.93/2.05	   U2_ga(F, N, normal_out_ga(F1, N)) -> normal_out_ga(F, N)
4.93/2.05	
4.93/2.05	The argument filtering Pi contains the following mapping:
4.93/2.05	normal_in_ga(x1, x2)  =  normal_in_ga(x1)
4.93/2.05	
4.93/2.05	U1_ga(x1, x2, x3)  =  U1_ga(x3)
4.93/2.05	
4.93/2.05	rewrite_in_ga(x1, x2)  =  rewrite_in_ga(x1)
4.93/2.05	
4.93/2.05	op(x1, x2)  =  op(x1, x2)
4.93/2.05	
4.93/2.05	rewrite_out_ga(x1, x2)  =  rewrite_out_ga(x2)
4.93/2.05	
4.93/2.05	U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
4.93/2.05	
4.93/2.05	U2_ga(x1, x2, x3)  =  U2_ga(x3)
4.93/2.05	
4.93/2.05	normal_out_ga(x1, x2)  =  normal_out_ga(x2)
4.93/2.05	
4.93/2.05	NORMAL_IN_GA(x1, x2)  =  NORMAL_IN_GA(x1)
4.93/2.05	
4.93/2.05	U1_GA(x1, x2, x3)  =  U1_GA(x3)
4.93/2.05	
4.93/2.05	REWRITE_IN_GA(x1, x2)  =  REWRITE_IN_GA(x1)
4.93/2.05	
4.93/2.05	U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x5)
4.93/2.05	
4.93/2.05	U2_GA(x1, x2, x3)  =  U2_GA(x3)
4.93/2.05	
4.93/2.05	
4.93/2.05	We have to consider all (P,R,Pi)-chains
4.93/2.05	----------------------------------------
4.93/2.05	
4.93/2.05	(4)
4.93/2.05	Obligation:
4.93/2.05	Pi DP problem:
4.93/2.05	The TRS P consists of the following rules:
4.93/2.05	
4.93/2.05	   NORMAL_IN_GA(F, N) -> U1_GA(F, N, rewrite_in_ga(F, F1))
4.93/2.05	   NORMAL_IN_GA(F, N) -> REWRITE_IN_GA(F, F1)
4.93/2.05	   REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) -> U3_GA(A, B, C, L, rewrite_in_ga(op(B, C), L))
4.93/2.05	   REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) -> REWRITE_IN_GA(op(B, C), L)
4.93/2.05	   U1_GA(F, N, rewrite_out_ga(F, F1)) -> U2_GA(F, N, normal_in_ga(F1, N))
4.93/2.05	   U1_GA(F, N, rewrite_out_ga(F, F1)) -> NORMAL_IN_GA(F1, N)
4.93/2.05	
4.93/2.05	The TRS R consists of the following rules:
4.93/2.05	
4.93/2.05	   normal_in_ga(F, N) -> U1_ga(F, N, rewrite_in_ga(F, F1))
4.93/2.05	   rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) -> rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
4.93/2.05	   rewrite_in_ga(op(A, op(B, C)), op(A, L)) -> U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L))
4.93/2.05	   U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) -> rewrite_out_ga(op(A, op(B, C)), op(A, L))
4.93/2.05	   U1_ga(F, N, rewrite_out_ga(F, F1)) -> U2_ga(F, N, normal_in_ga(F1, N))
4.93/2.05	   normal_in_ga(F, F) -> normal_out_ga(F, F)
4.93/2.05	   U2_ga(F, N, normal_out_ga(F1, N)) -> normal_out_ga(F, N)
4.93/2.05	
4.93/2.05	The argument filtering Pi contains the following mapping:
4.93/2.05	normal_in_ga(x1, x2)  =  normal_in_ga(x1)
4.93/2.05	
4.93/2.05	U1_ga(x1, x2, x3)  =  U1_ga(x3)
4.93/2.05	
4.93/2.05	rewrite_in_ga(x1, x2)  =  rewrite_in_ga(x1)
4.93/2.05	
4.93/2.05	op(x1, x2)  =  op(x1, x2)
4.93/2.05	
4.93/2.05	rewrite_out_ga(x1, x2)  =  rewrite_out_ga(x2)
4.93/2.05	
4.93/2.05	U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
4.93/2.05	
4.93/2.05	U2_ga(x1, x2, x3)  =  U2_ga(x3)
4.93/2.05	
4.93/2.05	normal_out_ga(x1, x2)  =  normal_out_ga(x2)
4.93/2.05	
4.93/2.05	NORMAL_IN_GA(x1, x2)  =  NORMAL_IN_GA(x1)
4.93/2.05	
4.93/2.05	U1_GA(x1, x2, x3)  =  U1_GA(x3)
4.93/2.05	
4.93/2.05	REWRITE_IN_GA(x1, x2)  =  REWRITE_IN_GA(x1)
4.93/2.05	
4.93/2.05	U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x5)
4.93/2.05	
4.93/2.05	U2_GA(x1, x2, x3)  =  U2_GA(x3)
4.93/2.05	
4.93/2.05	
4.93/2.05	We have to consider all (P,R,Pi)-chains
4.93/2.05	----------------------------------------
4.93/2.05	
4.93/2.05	(5) DependencyGraphProof (EQUIVALENT)
4.93/2.05	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes.
4.93/2.05	----------------------------------------
4.93/2.05	
4.93/2.05	(6)
4.93/2.05	Complex Obligation (AND)
4.93/2.05	
4.93/2.05	----------------------------------------
4.93/2.05	
4.93/2.05	(7)
4.93/2.05	Obligation:
4.93/2.05	Pi DP problem:
4.93/2.05	The TRS P consists of the following rules:
4.93/2.05	
4.93/2.05	   REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) -> REWRITE_IN_GA(op(B, C), L)
4.93/2.05	
4.93/2.05	The TRS R consists of the following rules:
4.93/2.05	
4.93/2.05	   normal_in_ga(F, N) -> U1_ga(F, N, rewrite_in_ga(F, F1))
4.93/2.05	   rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) -> rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
4.93/2.05	   rewrite_in_ga(op(A, op(B, C)), op(A, L)) -> U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L))
4.93/2.05	   U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) -> rewrite_out_ga(op(A, op(B, C)), op(A, L))
4.93/2.05	   U1_ga(F, N, rewrite_out_ga(F, F1)) -> U2_ga(F, N, normal_in_ga(F1, N))
4.93/2.05	   normal_in_ga(F, F) -> normal_out_ga(F, F)
4.93/2.05	   U2_ga(F, N, normal_out_ga(F1, N)) -> normal_out_ga(F, N)
4.93/2.05	
4.93/2.05	The argument filtering Pi contains the following mapping:
4.93/2.05	normal_in_ga(x1, x2)  =  normal_in_ga(x1)
4.93/2.06	
4.93/2.06	U1_ga(x1, x2, x3)  =  U1_ga(x3)
4.93/2.06	
4.93/2.06	rewrite_in_ga(x1, x2)  =  rewrite_in_ga(x1)
4.93/2.06	
4.93/2.06	op(x1, x2)  =  op(x1, x2)
4.93/2.06	
4.93/2.06	rewrite_out_ga(x1, x2)  =  rewrite_out_ga(x2)
4.93/2.06	
4.93/2.06	U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
4.93/2.06	
4.93/2.06	U2_ga(x1, x2, x3)  =  U2_ga(x3)
4.93/2.06	
4.93/2.06	normal_out_ga(x1, x2)  =  normal_out_ga(x2)
4.93/2.06	
4.93/2.06	REWRITE_IN_GA(x1, x2)  =  REWRITE_IN_GA(x1)
4.93/2.06	
4.93/2.06	
4.93/2.06	We have to consider all (P,R,Pi)-chains
4.93/2.06	----------------------------------------
4.93/2.06	
4.93/2.06	(8) UsableRulesProof (EQUIVALENT)
4.93/2.06	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
4.93/2.06	----------------------------------------
4.93/2.06	
4.93/2.06	(9)
4.93/2.06	Obligation:
4.93/2.06	Pi DP problem:
4.93/2.06	The TRS P consists of the following rules:
4.93/2.06	
4.93/2.06	   REWRITE_IN_GA(op(A, op(B, C)), op(A, L)) -> REWRITE_IN_GA(op(B, C), L)
4.93/2.06	
4.93/2.06	R is empty.
4.93/2.06	The argument filtering Pi contains the following mapping:
4.93/2.06	op(x1, x2)  =  op(x1, x2)
4.93/2.06	
4.93/2.06	REWRITE_IN_GA(x1, x2)  =  REWRITE_IN_GA(x1)
4.93/2.06	
4.93/2.06	
4.93/2.06	We have to consider all (P,R,Pi)-chains
4.93/2.06	----------------------------------------
4.93/2.06	
4.93/2.06	(10) PiDPToQDPProof (SOUND)
4.93/2.06	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
4.93/2.06	----------------------------------------
4.93/2.06	
4.93/2.06	(11)
4.93/2.06	Obligation:
4.93/2.06	Q DP problem:
4.93/2.06	The TRS P consists of the following rules:
4.93/2.06	
4.93/2.06	   REWRITE_IN_GA(op(A, op(B, C))) -> REWRITE_IN_GA(op(B, C))
4.93/2.06	
4.93/2.06	R is empty.
4.93/2.06	Q is empty.
4.93/2.06	We have to consider all (P,Q,R)-chains.
4.93/2.06	----------------------------------------
4.93/2.06	
4.93/2.06	(12) QDPSizeChangeProof (EQUIVALENT)
4.93/2.06	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. 
4.93/2.06	
4.93/2.06	From the DPs we obtained the following set of size-change graphs:
4.93/2.06	*REWRITE_IN_GA(op(A, op(B, C))) -> REWRITE_IN_GA(op(B, C))
4.93/2.06	The graph contains the following edges 1 > 1
4.93/2.06	
4.93/2.06	
4.93/2.06	----------------------------------------
4.93/2.06	
4.93/2.06	(13)
4.93/2.06	YES
4.93/2.06	
4.93/2.06	----------------------------------------
4.93/2.06	
4.93/2.06	(14)
4.93/2.06	Obligation:
4.93/2.06	Pi DP problem:
4.93/2.06	The TRS P consists of the following rules:
4.93/2.06	
4.93/2.06	   U1_GA(F, N, rewrite_out_ga(F, F1)) -> NORMAL_IN_GA(F1, N)
4.93/2.06	   NORMAL_IN_GA(F, N) -> U1_GA(F, N, rewrite_in_ga(F, F1))
4.93/2.06	
4.93/2.06	The TRS R consists of the following rules:
4.93/2.06	
4.93/2.06	   normal_in_ga(F, N) -> U1_ga(F, N, rewrite_in_ga(F, F1))
4.93/2.06	   rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) -> rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
4.93/2.06	   rewrite_in_ga(op(A, op(B, C)), op(A, L)) -> U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L))
4.93/2.06	   U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) -> rewrite_out_ga(op(A, op(B, C)), op(A, L))
4.93/2.06	   U1_ga(F, N, rewrite_out_ga(F, F1)) -> U2_ga(F, N, normal_in_ga(F1, N))
4.93/2.06	   normal_in_ga(F, F) -> normal_out_ga(F, F)
4.93/2.06	   U2_ga(F, N, normal_out_ga(F1, N)) -> normal_out_ga(F, N)
4.93/2.06	
4.93/2.06	The argument filtering Pi contains the following mapping:
4.93/2.06	normal_in_ga(x1, x2)  =  normal_in_ga(x1)
4.93/2.06	
4.93/2.06	U1_ga(x1, x2, x3)  =  U1_ga(x3)
4.93/2.06	
4.93/2.06	rewrite_in_ga(x1, x2)  =  rewrite_in_ga(x1)
4.93/2.06	
4.93/2.06	op(x1, x2)  =  op(x1, x2)
4.93/2.06	
4.93/2.06	rewrite_out_ga(x1, x2)  =  rewrite_out_ga(x2)
4.93/2.06	
4.93/2.06	U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
4.93/2.06	
4.93/2.06	U2_ga(x1, x2, x3)  =  U2_ga(x3)
4.93/2.06	
4.93/2.06	normal_out_ga(x1, x2)  =  normal_out_ga(x2)
4.93/2.06	
4.93/2.06	NORMAL_IN_GA(x1, x2)  =  NORMAL_IN_GA(x1)
4.93/2.06	
4.93/2.06	U1_GA(x1, x2, x3)  =  U1_GA(x3)
4.93/2.06	
4.93/2.06	
4.93/2.06	We have to consider all (P,R,Pi)-chains
4.93/2.06	----------------------------------------
4.93/2.06	
4.93/2.06	(15) UsableRulesProof (EQUIVALENT)
4.93/2.06	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
4.93/2.06	----------------------------------------
4.93/2.06	
4.93/2.06	(16)
4.93/2.06	Obligation:
4.93/2.06	Pi DP problem:
4.93/2.06	The TRS P consists of the following rules:
4.93/2.06	
4.93/2.06	   U1_GA(F, N, rewrite_out_ga(F, F1)) -> NORMAL_IN_GA(F1, N)
4.93/2.06	   NORMAL_IN_GA(F, N) -> U1_GA(F, N, rewrite_in_ga(F, F1))
4.93/2.06	
4.93/2.06	The TRS R consists of the following rules:
4.93/2.06	
4.93/2.06	   rewrite_in_ga(op(op(A, B), C), op(A, op(B, C))) -> rewrite_out_ga(op(op(A, B), C), op(A, op(B, C)))
4.93/2.06	   rewrite_in_ga(op(A, op(B, C)), op(A, L)) -> U3_ga(A, B, C, L, rewrite_in_ga(op(B, C), L))
4.93/2.06	   U3_ga(A, B, C, L, rewrite_out_ga(op(B, C), L)) -> rewrite_out_ga(op(A, op(B, C)), op(A, L))
4.93/2.06	
4.93/2.06	The argument filtering Pi contains the following mapping:
4.93/2.06	rewrite_in_ga(x1, x2)  =  rewrite_in_ga(x1)
4.93/2.06	
4.93/2.06	op(x1, x2)  =  op(x1, x2)
4.93/2.06	
4.93/2.06	rewrite_out_ga(x1, x2)  =  rewrite_out_ga(x2)
4.93/2.06	
4.93/2.06	U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x5)
4.93/2.06	
4.93/2.06	NORMAL_IN_GA(x1, x2)  =  NORMAL_IN_GA(x1)
4.93/2.06	
4.93/2.06	U1_GA(x1, x2, x3)  =  U1_GA(x3)
4.93/2.06	
4.93/2.06	
4.93/2.06	We have to consider all (P,R,Pi)-chains
4.93/2.06	----------------------------------------
4.93/2.06	
4.93/2.06	(17) PiDPToQDPProof (SOUND)
4.93/2.06	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
4.93/2.06	----------------------------------------
4.93/2.06	
4.93/2.06	(18)
4.93/2.06	Obligation:
4.93/2.06	Q DP problem:
4.93/2.06	The TRS P consists of the following rules:
4.93/2.06	
4.93/2.06	   U1_GA(rewrite_out_ga(F1)) -> NORMAL_IN_GA(F1)
4.93/2.06	   NORMAL_IN_GA(F) -> U1_GA(rewrite_in_ga(F))
4.93/2.06	
4.93/2.06	The TRS R consists of the following rules:
4.93/2.06	
4.93/2.06	   rewrite_in_ga(op(op(A, B), C)) -> rewrite_out_ga(op(A, op(B, C)))
4.93/2.06	   rewrite_in_ga(op(A, op(B, C))) -> U3_ga(A, rewrite_in_ga(op(B, C)))
4.93/2.06	   U3_ga(A, rewrite_out_ga(L)) -> rewrite_out_ga(op(A, L))
4.93/2.06	
4.93/2.06	The set Q consists of the following terms:
4.93/2.06	
4.93/2.06	   rewrite_in_ga(x0)
4.93/2.06	   U3_ga(x0, x1)
4.93/2.06	
4.93/2.06	We have to consider all (P,Q,R)-chains.
4.93/2.06	----------------------------------------
4.93/2.06	
4.93/2.06	(19) MRRProof (EQUIVALENT)
4.93/2.06	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
4.93/2.06	
4.93/2.06	Strictly oriented dependency pairs:
4.93/2.06	
4.93/2.06	   U1_GA(rewrite_out_ga(F1)) -> NORMAL_IN_GA(F1)
4.93/2.06	   NORMAL_IN_GA(F) -> U1_GA(rewrite_in_ga(F))
4.93/2.06	
4.93/2.06	
4.93/2.06	Used ordering: Polynomial interpretation [POLO]:
4.93/2.06	
4.93/2.06	   POL(NORMAL_IN_GA(x_1)) = 1 + x_1
4.93/2.06	   POL(U1_GA(x_1)) = x_1
4.93/2.06	   POL(U3_ga(x_1, x_2)) = 2 + 2*x_1 + x_2
4.93/2.06	   POL(op(x_1, x_2)) = 2 + 2*x_1 + x_2
4.93/2.06	   POL(rewrite_in_ga(x_1)) = x_1
4.93/2.06	   POL(rewrite_out_ga(x_1)) = 2 + x_1
4.93/2.06	
4.93/2.06	
4.93/2.06	----------------------------------------
4.93/2.06	
4.93/2.06	(20)
4.93/2.06	Obligation:
4.93/2.06	Q DP problem:
4.93/2.06	P is empty.
4.93/2.06	The TRS R consists of the following rules:
4.93/2.06	
4.93/2.06	   rewrite_in_ga(op(op(A, B), C)) -> rewrite_out_ga(op(A, op(B, C)))
4.93/2.06	   rewrite_in_ga(op(A, op(B, C))) -> U3_ga(A, rewrite_in_ga(op(B, C)))
4.93/2.06	   U3_ga(A, rewrite_out_ga(L)) -> rewrite_out_ga(op(A, L))
4.93/2.06	
4.93/2.06	The set Q consists of the following terms:
4.93/2.06	
4.93/2.06	   rewrite_in_ga(x0)
4.93/2.06	   U3_ga(x0, x1)
4.93/2.06	
4.93/2.06	We have to consider all (P,Q,R)-chains.
4.93/2.06	----------------------------------------
4.93/2.06	
4.93/2.06	(21) PisEmptyProof (EQUIVALENT)
4.93/2.06	The TRS P is empty. Hence, there is no (P,Q,R) chain.
4.93/2.06	----------------------------------------
4.93/2.06	
4.93/2.06	(22)
4.93/2.06	YES
4.93/2.09	EOF