3.96/1.91	YES
3.96/1.93	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.96/1.93	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.96/1.93	
3.96/1.93	
3.96/1.93	Left Termination of the query pattern
3.96/1.93	
3.96/1.93	fib(g,a)
3.96/1.93	
3.96/1.93	w.r.t. the given Prolog program could successfully be proven:
3.96/1.93	
3.96/1.93	(0) Prolog
3.96/1.93	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.96/1.93	(2) PiTRS
3.96/1.93	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.96/1.93	(4) PiDP
3.96/1.93	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.96/1.93	(6) AND
3.96/1.93	    (7) PiDP
3.96/1.93	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
3.96/1.93	        (9) PiDP
3.96/1.93	        (10) PiDPToQDPProof [SOUND, 0 ms]
3.96/1.93	        (11) QDP
3.96/1.93	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.96/1.93	        (13) YES
3.96/1.93	    (14) PiDP
3.96/1.93	        (15) PiDPToQDPProof [SOUND, 0 ms]
3.96/1.93	        (16) QDP
3.96/1.93	        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.96/1.93	        (18) YES
3.96/1.93	
3.96/1.93	
3.96/1.93	----------------------------------------
3.96/1.93	
3.96/1.93	(0)
3.96/1.93	Obligation:
3.96/1.93	Clauses:
3.96/1.93	
3.96/1.93	add(0, 0, 0).
3.96/1.93	add(s(X), Y, s(N)) :- add(X, Y, N).
3.96/1.93	add(X, s(Y), s(N)) :- add(X, Y, N).
3.96/1.93	fib(0, 0).
3.96/1.93	fib(s(0), s(0)).
3.96/1.93	fib(s(s(X)), N) :- ','(fib(s(X), N1), ','(fib(X, N2), add(N1, N2, N))).
3.96/1.93	
3.96/1.93	
3.96/1.93	Query: fib(g,a)
3.96/1.93	----------------------------------------
3.96/1.93	
3.96/1.93	(1) PrologToPiTRSProof (SOUND)
3.96/1.93	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.96/1.93	
3.96/1.93	fib_in_2: (b,f)
3.96/1.93	
3.96/1.93	add_in_3: (b,b,f)
3.96/1.93	
3.96/1.93	Transforming Prolog into the following Term Rewriting System:
3.96/1.93	
3.96/1.93	Pi-finite rewrite system:
3.96/1.93	The TRS R consists of the following rules:
3.96/1.93	
3.96/1.93	   fib_in_ga(0, 0) -> fib_out_ga(0, 0)
3.96/1.93	   fib_in_ga(s(0), s(0)) -> fib_out_ga(s(0), s(0))
3.96/1.93	   fib_in_ga(s(s(X)), N) -> U3_ga(X, N, fib_in_ga(s(X), N1))
3.96/1.93	   U3_ga(X, N, fib_out_ga(s(X), N1)) -> U4_ga(X, N, N1, fib_in_ga(X, N2))
3.96/1.93	   U4_ga(X, N, N1, fib_out_ga(X, N2)) -> U5_ga(X, N, add_in_gga(N1, N2, N))
3.96/1.93	   add_in_gga(0, 0, 0) -> add_out_gga(0, 0, 0)
3.96/1.93	   add_in_gga(s(X), Y, s(N)) -> U1_gga(X, Y, N, add_in_gga(X, Y, N))
3.96/1.93	   add_in_gga(X, s(Y), s(N)) -> U2_gga(X, Y, N, add_in_gga(X, Y, N))
3.96/1.93	   U2_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(X, s(Y), s(N))
3.96/1.93	   U1_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(s(X), Y, s(N))
3.96/1.93	   U5_ga(X, N, add_out_gga(N1, N2, N)) -> fib_out_ga(s(s(X)), N)
3.96/1.93	
3.96/1.93	The argument filtering Pi contains the following mapping:
3.96/1.93	fib_in_ga(x1, x2)  =  fib_in_ga(x1)
3.96/1.93	
3.96/1.93	0  =  0
3.96/1.93	
3.96/1.93	fib_out_ga(x1, x2)  =  fib_out_ga(x1, x2)
3.96/1.93	
3.96/1.93	s(x1)  =  s(x1)
3.96/1.93	
3.96/1.93	U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
3.96/1.93	
3.96/1.93	U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x3, x4)
3.96/1.93	
3.96/1.93	U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
3.96/1.93	
3.96/1.93	add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
3.96/1.93	
3.96/1.93	add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
3.96/1.93	
3.96/1.93	U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
3.96/1.93	
3.96/1.93	U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
3.96/1.93	
3.96/1.93	
3.96/1.93	
3.96/1.93	
3.96/1.93	
3.96/1.93	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.96/1.93	
3.96/1.93	
3.96/1.93	
3.96/1.93	----------------------------------------
3.96/1.93	
3.96/1.93	(2)
3.96/1.93	Obligation:
3.96/1.93	Pi-finite rewrite system:
3.96/1.93	The TRS R consists of the following rules:
3.96/1.93	
3.96/1.93	   fib_in_ga(0, 0) -> fib_out_ga(0, 0)
3.96/1.93	   fib_in_ga(s(0), s(0)) -> fib_out_ga(s(0), s(0))
3.96/1.93	   fib_in_ga(s(s(X)), N) -> U3_ga(X, N, fib_in_ga(s(X), N1))
3.96/1.93	   U3_ga(X, N, fib_out_ga(s(X), N1)) -> U4_ga(X, N, N1, fib_in_ga(X, N2))
3.96/1.93	   U4_ga(X, N, N1, fib_out_ga(X, N2)) -> U5_ga(X, N, add_in_gga(N1, N2, N))
3.96/1.93	   add_in_gga(0, 0, 0) -> add_out_gga(0, 0, 0)
3.96/1.93	   add_in_gga(s(X), Y, s(N)) -> U1_gga(X, Y, N, add_in_gga(X, Y, N))
3.96/1.93	   add_in_gga(X, s(Y), s(N)) -> U2_gga(X, Y, N, add_in_gga(X, Y, N))
3.96/1.93	   U2_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(X, s(Y), s(N))
3.96/1.93	   U1_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(s(X), Y, s(N))
3.96/1.93	   U5_ga(X, N, add_out_gga(N1, N2, N)) -> fib_out_ga(s(s(X)), N)
3.96/1.93	
3.96/1.93	The argument filtering Pi contains the following mapping:
3.96/1.93	fib_in_ga(x1, x2)  =  fib_in_ga(x1)
3.96/1.93	
3.96/1.93	0  =  0
3.96/1.93	
3.96/1.93	fib_out_ga(x1, x2)  =  fib_out_ga(x1, x2)
3.96/1.93	
3.96/1.93	s(x1)  =  s(x1)
3.96/1.93	
3.96/1.93	U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
3.96/1.93	
3.96/1.93	U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x3, x4)
3.96/1.93	
3.96/1.93	U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
3.96/1.93	
3.96/1.93	add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
3.96/1.93	
3.96/1.93	add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
3.96/1.93	
3.96/1.93	U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
3.96/1.93	
3.96/1.93	U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
3.96/1.93	
3.96/1.93	
3.96/1.93	
3.96/1.93	----------------------------------------
3.96/1.93	
3.96/1.93	(3) DependencyPairsProof (EQUIVALENT)
3.96/1.93	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.96/1.93	Pi DP problem:
3.96/1.93	The TRS P consists of the following rules:
3.96/1.93	
3.96/1.93	   FIB_IN_GA(s(s(X)), N) -> U3_GA(X, N, fib_in_ga(s(X), N1))
3.96/1.93	   FIB_IN_GA(s(s(X)), N) -> FIB_IN_GA(s(X), N1)
3.96/1.93	   U3_GA(X, N, fib_out_ga(s(X), N1)) -> U4_GA(X, N, N1, fib_in_ga(X, N2))
3.96/1.93	   U3_GA(X, N, fib_out_ga(s(X), N1)) -> FIB_IN_GA(X, N2)
3.96/1.93	   U4_GA(X, N, N1, fib_out_ga(X, N2)) -> U5_GA(X, N, add_in_gga(N1, N2, N))
3.96/1.93	   U4_GA(X, N, N1, fib_out_ga(X, N2)) -> ADD_IN_GGA(N1, N2, N)
3.96/1.93	   ADD_IN_GGA(s(X), Y, s(N)) -> U1_GGA(X, Y, N, add_in_gga(X, Y, N))
3.96/1.93	   ADD_IN_GGA(s(X), Y, s(N)) -> ADD_IN_GGA(X, Y, N)
3.96/1.93	   ADD_IN_GGA(X, s(Y), s(N)) -> U2_GGA(X, Y, N, add_in_gga(X, Y, N))
3.96/1.93	   ADD_IN_GGA(X, s(Y), s(N)) -> ADD_IN_GGA(X, Y, N)
3.96/1.93	
3.96/1.93	The TRS R consists of the following rules:
3.96/1.93	
3.96/1.93	   fib_in_ga(0, 0) -> fib_out_ga(0, 0)
3.96/1.93	   fib_in_ga(s(0), s(0)) -> fib_out_ga(s(0), s(0))
3.96/1.93	   fib_in_ga(s(s(X)), N) -> U3_ga(X, N, fib_in_ga(s(X), N1))
3.96/1.93	   U3_ga(X, N, fib_out_ga(s(X), N1)) -> U4_ga(X, N, N1, fib_in_ga(X, N2))
3.96/1.93	   U4_ga(X, N, N1, fib_out_ga(X, N2)) -> U5_ga(X, N, add_in_gga(N1, N2, N))
3.96/1.93	   add_in_gga(0, 0, 0) -> add_out_gga(0, 0, 0)
3.96/1.93	   add_in_gga(s(X), Y, s(N)) -> U1_gga(X, Y, N, add_in_gga(X, Y, N))
3.96/1.93	   add_in_gga(X, s(Y), s(N)) -> U2_gga(X, Y, N, add_in_gga(X, Y, N))
3.96/1.93	   U2_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(X, s(Y), s(N))
3.96/1.93	   U1_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(s(X), Y, s(N))
3.96/1.93	   U5_ga(X, N, add_out_gga(N1, N2, N)) -> fib_out_ga(s(s(X)), N)
3.96/1.93	
3.96/1.93	The argument filtering Pi contains the following mapping:
3.96/1.93	fib_in_ga(x1, x2)  =  fib_in_ga(x1)
3.96/1.93	
3.96/1.93	0  =  0
3.96/1.93	
3.96/1.93	fib_out_ga(x1, x2)  =  fib_out_ga(x1, x2)
3.96/1.93	
3.96/1.93	s(x1)  =  s(x1)
3.96/1.93	
3.96/1.93	U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
3.96/1.93	
3.96/1.93	U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x3, x4)
3.96/1.93	
3.96/1.93	U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
3.96/1.93	
3.96/1.93	add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
3.96/1.93	
3.96/1.93	add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
3.96/1.93	
3.96/1.93	U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
3.96/1.93	
3.96/1.93	U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
3.96/1.93	
3.96/1.93	FIB_IN_GA(x1, x2)  =  FIB_IN_GA(x1)
3.96/1.93	
3.96/1.93	U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
3.96/1.93	
3.96/1.93	U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x3, x4)
3.96/1.93	
3.96/1.93	U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
3.96/1.93	
3.96/1.93	ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
3.96/1.93	
3.96/1.93	U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
3.96/1.93	
3.96/1.93	U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
3.96/1.93	
3.96/1.93	
3.96/1.93	We have to consider all (P,R,Pi)-chains
3.96/1.93	----------------------------------------
3.96/1.93	
3.96/1.93	(4)
3.96/1.93	Obligation:
3.96/1.93	Pi DP problem:
3.96/1.93	The TRS P consists of the following rules:
3.96/1.93	
3.96/1.93	   FIB_IN_GA(s(s(X)), N) -> U3_GA(X, N, fib_in_ga(s(X), N1))
3.96/1.93	   FIB_IN_GA(s(s(X)), N) -> FIB_IN_GA(s(X), N1)
3.96/1.93	   U3_GA(X, N, fib_out_ga(s(X), N1)) -> U4_GA(X, N, N1, fib_in_ga(X, N2))
3.96/1.93	   U3_GA(X, N, fib_out_ga(s(X), N1)) -> FIB_IN_GA(X, N2)
3.96/1.93	   U4_GA(X, N, N1, fib_out_ga(X, N2)) -> U5_GA(X, N, add_in_gga(N1, N2, N))
3.96/1.93	   U4_GA(X, N, N1, fib_out_ga(X, N2)) -> ADD_IN_GGA(N1, N2, N)
3.96/1.93	   ADD_IN_GGA(s(X), Y, s(N)) -> U1_GGA(X, Y, N, add_in_gga(X, Y, N))
3.96/1.93	   ADD_IN_GGA(s(X), Y, s(N)) -> ADD_IN_GGA(X, Y, N)
3.96/1.93	   ADD_IN_GGA(X, s(Y), s(N)) -> U2_GGA(X, Y, N, add_in_gga(X, Y, N))
3.96/1.93	   ADD_IN_GGA(X, s(Y), s(N)) -> ADD_IN_GGA(X, Y, N)
3.96/1.93	
3.96/1.93	The TRS R consists of the following rules:
3.96/1.93	
3.96/1.93	   fib_in_ga(0, 0) -> fib_out_ga(0, 0)
3.96/1.93	   fib_in_ga(s(0), s(0)) -> fib_out_ga(s(0), s(0))
3.96/1.93	   fib_in_ga(s(s(X)), N) -> U3_ga(X, N, fib_in_ga(s(X), N1))
3.96/1.93	   U3_ga(X, N, fib_out_ga(s(X), N1)) -> U4_ga(X, N, N1, fib_in_ga(X, N2))
3.96/1.93	   U4_ga(X, N, N1, fib_out_ga(X, N2)) -> U5_ga(X, N, add_in_gga(N1, N2, N))
3.96/1.93	   add_in_gga(0, 0, 0) -> add_out_gga(0, 0, 0)
3.96/1.93	   add_in_gga(s(X), Y, s(N)) -> U1_gga(X, Y, N, add_in_gga(X, Y, N))
3.96/1.93	   add_in_gga(X, s(Y), s(N)) -> U2_gga(X, Y, N, add_in_gga(X, Y, N))
3.96/1.93	   U2_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(X, s(Y), s(N))
3.96/1.93	   U1_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(s(X), Y, s(N))
3.96/1.93	   U5_ga(X, N, add_out_gga(N1, N2, N)) -> fib_out_ga(s(s(X)), N)
3.96/1.93	
3.96/1.93	The argument filtering Pi contains the following mapping:
3.96/1.93	fib_in_ga(x1, x2)  =  fib_in_ga(x1)
3.96/1.93	
3.96/1.93	0  =  0
3.96/1.93	
3.96/1.93	fib_out_ga(x1, x2)  =  fib_out_ga(x1, x2)
3.96/1.93	
3.96/1.93	s(x1)  =  s(x1)
3.96/1.93	
3.96/1.93	U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
3.96/1.93	
3.96/1.93	U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x3, x4)
3.96/1.93	
3.96/1.93	U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
3.96/1.93	
3.96/1.93	add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
3.96/1.93	
3.96/1.93	add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
3.96/1.93	
3.96/1.93	U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
3.96/1.93	
3.96/1.93	U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
3.96/1.93	
3.96/1.93	FIB_IN_GA(x1, x2)  =  FIB_IN_GA(x1)
3.96/1.93	
3.96/1.93	U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
3.96/1.93	
3.96/1.93	U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x3, x4)
3.96/1.93	
3.96/1.93	U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
3.96/1.93	
3.96/1.93	ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
3.96/1.93	
3.96/1.93	U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
3.96/1.93	
3.96/1.93	U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
3.96/1.93	
3.96/1.93	
3.96/1.93	We have to consider all (P,R,Pi)-chains
3.96/1.93	----------------------------------------
3.96/1.93	
3.96/1.93	(5) DependencyGraphProof (EQUIVALENT)
3.96/1.93	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.
3.96/1.93	----------------------------------------
3.96/1.93	
3.96/1.93	(6)
3.96/1.93	Complex Obligation (AND)
3.96/1.93	
3.96/1.93	----------------------------------------
3.96/1.93	
3.96/1.93	(7)
3.96/1.93	Obligation:
3.96/1.93	Pi DP problem:
3.96/1.93	The TRS P consists of the following rules:
3.96/1.93	
3.96/1.93	   ADD_IN_GGA(X, s(Y), s(N)) -> ADD_IN_GGA(X, Y, N)
3.96/1.93	   ADD_IN_GGA(s(X), Y, s(N)) -> ADD_IN_GGA(X, Y, N)
3.96/1.93	
3.96/1.93	The TRS R consists of the following rules:
3.96/1.93	
3.96/1.93	   fib_in_ga(0, 0) -> fib_out_ga(0, 0)
3.96/1.93	   fib_in_ga(s(0), s(0)) -> fib_out_ga(s(0), s(0))
3.96/1.93	   fib_in_ga(s(s(X)), N) -> U3_ga(X, N, fib_in_ga(s(X), N1))
3.96/1.93	   U3_ga(X, N, fib_out_ga(s(X), N1)) -> U4_ga(X, N, N1, fib_in_ga(X, N2))
3.96/1.93	   U4_ga(X, N, N1, fib_out_ga(X, N2)) -> U5_ga(X, N, add_in_gga(N1, N2, N))
3.96/1.93	   add_in_gga(0, 0, 0) -> add_out_gga(0, 0, 0)
3.96/1.93	   add_in_gga(s(X), Y, s(N)) -> U1_gga(X, Y, N, add_in_gga(X, Y, N))
3.96/1.93	   add_in_gga(X, s(Y), s(N)) -> U2_gga(X, Y, N, add_in_gga(X, Y, N))
3.96/1.93	   U2_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(X, s(Y), s(N))
3.96/1.93	   U1_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(s(X), Y, s(N))
3.96/1.93	   U5_ga(X, N, add_out_gga(N1, N2, N)) -> fib_out_ga(s(s(X)), N)
3.96/1.93	
3.96/1.93	The argument filtering Pi contains the following mapping:
3.96/1.93	fib_in_ga(x1, x2)  =  fib_in_ga(x1)
3.96/1.93	
3.96/1.93	0  =  0
3.96/1.93	
3.96/1.93	fib_out_ga(x1, x2)  =  fib_out_ga(x1, x2)
3.96/1.93	
3.96/1.93	s(x1)  =  s(x1)
3.96/1.93	
3.96/1.93	U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
3.96/1.93	
3.96/1.93	U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x3, x4)
3.96/1.93	
3.96/1.93	U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
3.96/1.93	
3.96/1.93	add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
3.96/1.93	
3.96/1.93	add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
3.96/1.93	
3.96/1.93	U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
3.96/1.93	
3.96/1.93	U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
3.96/1.93	
3.96/1.93	ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
3.96/1.93	
3.96/1.93	
3.96/1.93	We have to consider all (P,R,Pi)-chains
3.96/1.93	----------------------------------------
3.96/1.93	
3.96/1.93	(8) UsableRulesProof (EQUIVALENT)
3.96/1.93	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.96/1.93	----------------------------------------
3.96/1.93	
3.96/1.93	(9)
3.96/1.93	Obligation:
3.96/1.93	Pi DP problem:
3.96/1.93	The TRS P consists of the following rules:
3.96/1.93	
3.96/1.93	   ADD_IN_GGA(X, s(Y), s(N)) -> ADD_IN_GGA(X, Y, N)
3.96/1.93	   ADD_IN_GGA(s(X), Y, s(N)) -> ADD_IN_GGA(X, Y, N)
3.96/1.93	
3.96/1.93	R is empty.
3.96/1.93	The argument filtering Pi contains the following mapping:
3.96/1.93	s(x1)  =  s(x1)
3.96/1.93	
3.96/1.93	ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
3.96/1.93	
3.96/1.93	
3.96/1.93	We have to consider all (P,R,Pi)-chains
3.96/1.93	----------------------------------------
3.96/1.93	
3.96/1.93	(10) PiDPToQDPProof (SOUND)
3.96/1.93	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.96/1.93	----------------------------------------
3.96/1.93	
3.96/1.93	(11)
3.96/1.93	Obligation:
3.96/1.93	Q DP problem:
3.96/1.93	The TRS P consists of the following rules:
3.96/1.93	
3.96/1.93	   ADD_IN_GGA(X, s(Y)) -> ADD_IN_GGA(X, Y)
3.96/1.93	   ADD_IN_GGA(s(X), Y) -> ADD_IN_GGA(X, Y)
3.96/1.93	
3.96/1.93	R is empty.
3.96/1.93	Q is empty.
3.96/1.93	We have to consider all (P,Q,R)-chains.
3.96/1.93	----------------------------------------
3.96/1.93	
3.96/1.93	(12) QDPSizeChangeProof (EQUIVALENT)
3.96/1.93	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. 
3.96/1.93	
3.96/1.93	From the DPs we obtained the following set of size-change graphs:
3.96/1.93	*ADD_IN_GGA(X, s(Y)) -> ADD_IN_GGA(X, Y)
3.96/1.93	The graph contains the following edges 1 >= 1, 2 > 2
3.96/1.93	
3.96/1.93	
3.96/1.93	*ADD_IN_GGA(s(X), Y) -> ADD_IN_GGA(X, Y)
3.96/1.93	The graph contains the following edges 1 > 1, 2 >= 2
3.96/1.93	
3.96/1.93	
3.96/1.93	----------------------------------------
3.96/1.93	
3.96/1.93	(13)
3.96/1.93	YES
3.96/1.93	
3.96/1.93	----------------------------------------
3.96/1.93	
3.96/1.93	(14)
3.96/1.93	Obligation:
3.96/1.93	Pi DP problem:
3.96/1.93	The TRS P consists of the following rules:
3.96/1.93	
3.96/1.93	   U3_GA(X, N, fib_out_ga(s(X), N1)) -> FIB_IN_GA(X, N2)
3.96/1.93	   FIB_IN_GA(s(s(X)), N) -> U3_GA(X, N, fib_in_ga(s(X), N1))
3.96/1.93	   FIB_IN_GA(s(s(X)), N) -> FIB_IN_GA(s(X), N1)
3.96/1.93	
3.96/1.93	The TRS R consists of the following rules:
3.96/1.93	
3.96/1.93	   fib_in_ga(0, 0) -> fib_out_ga(0, 0)
3.96/1.93	   fib_in_ga(s(0), s(0)) -> fib_out_ga(s(0), s(0))
3.96/1.93	   fib_in_ga(s(s(X)), N) -> U3_ga(X, N, fib_in_ga(s(X), N1))
3.96/1.93	   U3_ga(X, N, fib_out_ga(s(X), N1)) -> U4_ga(X, N, N1, fib_in_ga(X, N2))
3.96/1.93	   U4_ga(X, N, N1, fib_out_ga(X, N2)) -> U5_ga(X, N, add_in_gga(N1, N2, N))
3.96/1.93	   add_in_gga(0, 0, 0) -> add_out_gga(0, 0, 0)
3.96/1.93	   add_in_gga(s(X), Y, s(N)) -> U1_gga(X, Y, N, add_in_gga(X, Y, N))
3.96/1.93	   add_in_gga(X, s(Y), s(N)) -> U2_gga(X, Y, N, add_in_gga(X, Y, N))
3.96/1.93	   U2_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(X, s(Y), s(N))
3.96/1.93	   U1_gga(X, Y, N, add_out_gga(X, Y, N)) -> add_out_gga(s(X), Y, s(N))
3.96/1.93	   U5_ga(X, N, add_out_gga(N1, N2, N)) -> fib_out_ga(s(s(X)), N)
3.96/1.93	
3.96/1.93	The argument filtering Pi contains the following mapping:
3.96/1.93	fib_in_ga(x1, x2)  =  fib_in_ga(x1)
3.96/1.93	
3.96/1.93	0  =  0
3.96/1.93	
3.96/1.93	fib_out_ga(x1, x2)  =  fib_out_ga(x1, x2)
3.96/1.93	
3.96/1.93	s(x1)  =  s(x1)
3.96/1.93	
3.96/1.93	U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
3.96/1.93	
3.96/1.93	U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x3, x4)
3.96/1.93	
3.96/1.93	U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
3.96/1.93	
3.96/1.93	add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
3.96/1.93	
3.96/1.93	add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
3.96/1.93	
3.96/1.93	U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
3.96/1.93	
3.96/1.93	U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
3.96/1.93	
3.96/1.93	FIB_IN_GA(x1, x2)  =  FIB_IN_GA(x1)
3.96/1.93	
3.96/1.93	U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
3.96/1.93	
3.96/1.93	
3.96/1.93	We have to consider all (P,R,Pi)-chains
3.96/1.93	----------------------------------------
3.96/1.93	
3.96/1.93	(15) PiDPToQDPProof (SOUND)
3.96/1.93	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.96/1.93	----------------------------------------
3.96/1.93	
3.96/1.93	(16)
3.96/1.93	Obligation:
3.96/1.93	Q DP problem:
3.96/1.93	The TRS P consists of the following rules:
3.96/1.93	
3.96/1.93	   U3_GA(X, fib_out_ga(s(X), N1)) -> FIB_IN_GA(X)
3.96/1.93	   FIB_IN_GA(s(s(X))) -> U3_GA(X, fib_in_ga(s(X)))
3.96/1.93	   FIB_IN_GA(s(s(X))) -> FIB_IN_GA(s(X))
3.96/1.93	
3.96/1.93	The TRS R consists of the following rules:
3.96/1.93	
3.96/1.93	   fib_in_ga(0) -> fib_out_ga(0, 0)
3.96/1.93	   fib_in_ga(s(0)) -> fib_out_ga(s(0), s(0))
3.96/1.93	   fib_in_ga(s(s(X))) -> U3_ga(X, fib_in_ga(s(X)))
3.96/1.93	   U3_ga(X, fib_out_ga(s(X), N1)) -> U4_ga(X, N1, fib_in_ga(X))
3.96/1.93	   U4_ga(X, N1, fib_out_ga(X, N2)) -> U5_ga(X, add_in_gga(N1, N2))
3.96/1.93	   add_in_gga(0, 0) -> add_out_gga(0, 0, 0)
3.96/1.93	   add_in_gga(s(X), Y) -> U1_gga(X, Y, add_in_gga(X, Y))
3.96/1.93	   add_in_gga(X, s(Y)) -> U2_gga(X, Y, add_in_gga(X, Y))
3.96/1.93	   U2_gga(X, Y, add_out_gga(X, Y, N)) -> add_out_gga(X, s(Y), s(N))
3.96/1.93	   U1_gga(X, Y, add_out_gga(X, Y, N)) -> add_out_gga(s(X), Y, s(N))
3.96/1.93	   U5_ga(X, add_out_gga(N1, N2, N)) -> fib_out_ga(s(s(X)), N)
3.96/1.93	
3.96/1.93	The set Q consists of the following terms:
3.96/1.93	
3.96/1.93	   fib_in_ga(x0)
3.96/1.93	   U3_ga(x0, x1)
3.96/1.93	   U4_ga(x0, x1, x2)
3.96/1.93	   add_in_gga(x0, x1)
3.96/1.93	   U2_gga(x0, x1, x2)
3.96/1.93	   U1_gga(x0, x1, x2)
3.96/1.93	   U5_ga(x0, x1)
3.96/1.93	
3.96/1.93	We have to consider all (P,Q,R)-chains.
3.96/1.93	----------------------------------------
3.96/1.93	
3.96/1.93	(17) QDPSizeChangeProof (EQUIVALENT)
3.96/1.93	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. 
3.96/1.93	
3.96/1.93	From the DPs we obtained the following set of size-change graphs:
3.96/1.93	*FIB_IN_GA(s(s(X))) -> U3_GA(X, fib_in_ga(s(X)))
3.96/1.93	The graph contains the following edges 1 > 1
3.96/1.93	
3.96/1.93	
3.96/1.93	*FIB_IN_GA(s(s(X))) -> FIB_IN_GA(s(X))
3.96/1.93	The graph contains the following edges 1 > 1
3.96/1.93	
3.96/1.93	
3.96/1.93	*U3_GA(X, fib_out_ga(s(X), N1)) -> FIB_IN_GA(X)
3.96/1.93	The graph contains the following edges 1 >= 1, 2 > 1
3.96/1.93	
3.96/1.93	
3.96/1.93	----------------------------------------
3.96/1.93	
3.96/1.93	(18)
3.96/1.93	YES
4.22/2.00	EOF