4.44/2.02	YES
4.44/2.03	proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl
4.44/2.03	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.44/2.03	
4.44/2.03	
4.44/2.03	Left Termination of the query pattern
4.44/2.03	
4.44/2.03	ackermann(g,g,a)
4.44/2.03	
4.44/2.03	w.r.t. the given Prolog program could successfully be proven:
4.44/2.03	
4.44/2.03	(0) Prolog
4.44/2.03	(1) PrologToPiTRSProof [SOUND, 0 ms]
4.44/2.03	(2) PiTRS
4.44/2.03	(3) DependencyPairsProof [EQUIVALENT, 3 ms]
4.44/2.03	(4) PiDP
4.44/2.03	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
4.44/2.03	(6) PiDP
4.44/2.03	(7) PiDPToQDPProof [SOUND, 13 ms]
4.44/2.03	(8) QDP
4.44/2.03	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
4.44/2.03	(10) YES
4.44/2.03	
4.44/2.03	
4.44/2.03	----------------------------------------
4.44/2.03	
4.44/2.03	(0)
4.44/2.03	Obligation:
4.44/2.03	Clauses:
4.44/2.03	
4.44/2.03	ackermann(0, N, s(N)).
4.44/2.03	ackermann(s(M), 0, Res) :- ackermann(M, s(0), Res).
4.44/2.03	ackermann(s(M), s(N), Res) :- ','(ackermann(s(M), N, Res1), ackermann(M, Res1, Res)).
4.44/2.03	
4.44/2.03	
4.44/2.03	Query: ackermann(g,g,a)
4.44/2.03	----------------------------------------
4.44/2.03	
4.44/2.03	(1) PrologToPiTRSProof (SOUND)
4.44/2.03	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
4.44/2.03	
4.44/2.03	ackermann_in_3: (b,b,f)
4.44/2.03	
4.44/2.03	Transforming Prolog into the following Term Rewriting System:
4.44/2.03	
4.44/2.03	Pi-finite rewrite system:
4.44/2.03	The TRS R consists of the following rules:
4.44/2.03	
4.44/2.03	   ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N))
4.44/2.03	   ackermann_in_gga(s(M), 0, Res) -> U1_gga(M, Res, ackermann_in_gga(M, s(0), Res))
4.44/2.03	   ackermann_in_gga(s(M), s(N), Res) -> U2_gga(M, N, Res, ackermann_in_gga(s(M), N, Res1))
4.44/2.03	   U2_gga(M, N, Res, ackermann_out_gga(s(M), N, Res1)) -> U3_gga(M, N, Res, ackermann_in_gga(M, Res1, Res))
4.44/2.03	   U3_gga(M, N, Res, ackermann_out_gga(M, Res1, Res)) -> ackermann_out_gga(s(M), s(N), Res)
4.44/2.03	   U1_gga(M, Res, ackermann_out_gga(M, s(0), Res)) -> ackermann_out_gga(s(M), 0, Res)
4.44/2.03	
4.44/2.03	The argument filtering Pi contains the following mapping:
4.44/2.03	ackermann_in_gga(x1, x2, x3)  =  ackermann_in_gga(x1, x2)
4.44/2.03	
4.44/2.03	0  =  0
4.44/2.03	
4.44/2.03	ackermann_out_gga(x1, x2, x3)  =  ackermann_out_gga(x3)
4.44/2.03	
4.44/2.03	s(x1)  =  s(x1)
4.44/2.03	
4.44/2.03	U1_gga(x1, x2, x3)  =  U1_gga(x3)
4.44/2.03	
4.44/2.03	U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
4.44/2.03	
4.44/2.03	U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
4.44/2.03	
4.44/2.03	
4.44/2.03	
4.44/2.03	
4.44/2.03	
4.44/2.03	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
4.44/2.03	
4.44/2.03	
4.44/2.03	
4.44/2.03	----------------------------------------
4.44/2.03	
4.44/2.03	(2)
4.44/2.03	Obligation:
4.44/2.03	Pi-finite rewrite system:
4.44/2.03	The TRS R consists of the following rules:
4.44/2.03	
4.44/2.03	   ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N))
4.44/2.03	   ackermann_in_gga(s(M), 0, Res) -> U1_gga(M, Res, ackermann_in_gga(M, s(0), Res))
4.44/2.03	   ackermann_in_gga(s(M), s(N), Res) -> U2_gga(M, N, Res, ackermann_in_gga(s(M), N, Res1))
4.44/2.03	   U2_gga(M, N, Res, ackermann_out_gga(s(M), N, Res1)) -> U3_gga(M, N, Res, ackermann_in_gga(M, Res1, Res))
4.44/2.03	   U3_gga(M, N, Res, ackermann_out_gga(M, Res1, Res)) -> ackermann_out_gga(s(M), s(N), Res)
4.44/2.03	   U1_gga(M, Res, ackermann_out_gga(M, s(0), Res)) -> ackermann_out_gga(s(M), 0, Res)
4.44/2.03	
4.44/2.03	The argument filtering Pi contains the following mapping:
4.44/2.03	ackermann_in_gga(x1, x2, x3)  =  ackermann_in_gga(x1, x2)
4.44/2.03	
4.44/2.03	0  =  0
4.44/2.03	
4.44/2.03	ackermann_out_gga(x1, x2, x3)  =  ackermann_out_gga(x3)
4.44/2.03	
4.44/2.03	s(x1)  =  s(x1)
4.44/2.03	
4.44/2.03	U1_gga(x1, x2, x3)  =  U1_gga(x3)
4.44/2.03	
4.44/2.03	U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
4.44/2.03	
4.44/2.03	U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
4.44/2.03	
4.44/2.03	
4.44/2.03	
4.44/2.03	----------------------------------------
4.44/2.03	
4.44/2.03	(3) DependencyPairsProof (EQUIVALENT)
4.44/2.03	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
4.44/2.03	Pi DP problem:
4.44/2.03	The TRS P consists of the following rules:
4.44/2.03	
4.44/2.03	   ACKERMANN_IN_GGA(s(M), 0, Res) -> U1_GGA(M, Res, ackermann_in_gga(M, s(0), Res))
4.44/2.03	   ACKERMANN_IN_GGA(s(M), 0, Res) -> ACKERMANN_IN_GGA(M, s(0), Res)
4.44/2.03	   ACKERMANN_IN_GGA(s(M), s(N), Res) -> U2_GGA(M, N, Res, ackermann_in_gga(s(M), N, Res1))
4.44/2.03	   ACKERMANN_IN_GGA(s(M), s(N), Res) -> ACKERMANN_IN_GGA(s(M), N, Res1)
4.44/2.03	   U2_GGA(M, N, Res, ackermann_out_gga(s(M), N, Res1)) -> U3_GGA(M, N, Res, ackermann_in_gga(M, Res1, Res))
4.44/2.03	   U2_GGA(M, N, Res, ackermann_out_gga(s(M), N, Res1)) -> ACKERMANN_IN_GGA(M, Res1, Res)
4.44/2.03	
4.44/2.03	The TRS R consists of the following rules:
4.44/2.03	
4.44/2.03	   ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N))
4.44/2.03	   ackermann_in_gga(s(M), 0, Res) -> U1_gga(M, Res, ackermann_in_gga(M, s(0), Res))
4.44/2.03	   ackermann_in_gga(s(M), s(N), Res) -> U2_gga(M, N, Res, ackermann_in_gga(s(M), N, Res1))
4.44/2.03	   U2_gga(M, N, Res, ackermann_out_gga(s(M), N, Res1)) -> U3_gga(M, N, Res, ackermann_in_gga(M, Res1, Res))
4.44/2.03	   U3_gga(M, N, Res, ackermann_out_gga(M, Res1, Res)) -> ackermann_out_gga(s(M), s(N), Res)
4.44/2.03	   U1_gga(M, Res, ackermann_out_gga(M, s(0), Res)) -> ackermann_out_gga(s(M), 0, Res)
4.44/2.03	
4.44/2.03	The argument filtering Pi contains the following mapping:
4.44/2.03	ackermann_in_gga(x1, x2, x3)  =  ackermann_in_gga(x1, x2)
4.44/2.03	
4.44/2.03	0  =  0
4.44/2.03	
4.44/2.03	ackermann_out_gga(x1, x2, x3)  =  ackermann_out_gga(x3)
4.44/2.03	
4.44/2.03	s(x1)  =  s(x1)
4.44/2.03	
4.44/2.03	U1_gga(x1, x2, x3)  =  U1_gga(x3)
4.44/2.03	
4.44/2.03	U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
4.44/2.03	
4.44/2.03	U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
4.44/2.03	
4.44/2.03	ACKERMANN_IN_GGA(x1, x2, x3)  =  ACKERMANN_IN_GGA(x1, x2)
4.44/2.03	
4.44/2.03	U1_GGA(x1, x2, x3)  =  U1_GGA(x3)
4.44/2.03	
4.44/2.03	U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x4)
4.44/2.03	
4.44/2.03	U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
4.44/2.03	
4.44/2.03	
4.44/2.03	We have to consider all (P,R,Pi)-chains
4.44/2.03	----------------------------------------
4.44/2.03	
4.44/2.03	(4)
4.44/2.03	Obligation:
4.44/2.03	Pi DP problem:
4.44/2.03	The TRS P consists of the following rules:
4.44/2.03	
4.44/2.03	   ACKERMANN_IN_GGA(s(M), 0, Res) -> U1_GGA(M, Res, ackermann_in_gga(M, s(0), Res))
4.44/2.03	   ACKERMANN_IN_GGA(s(M), 0, Res) -> ACKERMANN_IN_GGA(M, s(0), Res)
4.44/2.03	   ACKERMANN_IN_GGA(s(M), s(N), Res) -> U2_GGA(M, N, Res, ackermann_in_gga(s(M), N, Res1))
4.44/2.03	   ACKERMANN_IN_GGA(s(M), s(N), Res) -> ACKERMANN_IN_GGA(s(M), N, Res1)
4.44/2.03	   U2_GGA(M, N, Res, ackermann_out_gga(s(M), N, Res1)) -> U3_GGA(M, N, Res, ackermann_in_gga(M, Res1, Res))
4.44/2.03	   U2_GGA(M, N, Res, ackermann_out_gga(s(M), N, Res1)) -> ACKERMANN_IN_GGA(M, Res1, Res)
4.44/2.03	
4.44/2.03	The TRS R consists of the following rules:
4.44/2.03	
4.44/2.03	   ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N))
4.44/2.03	   ackermann_in_gga(s(M), 0, Res) -> U1_gga(M, Res, ackermann_in_gga(M, s(0), Res))
4.44/2.03	   ackermann_in_gga(s(M), s(N), Res) -> U2_gga(M, N, Res, ackermann_in_gga(s(M), N, Res1))
4.44/2.03	   U2_gga(M, N, Res, ackermann_out_gga(s(M), N, Res1)) -> U3_gga(M, N, Res, ackermann_in_gga(M, Res1, Res))
4.44/2.03	   U3_gga(M, N, Res, ackermann_out_gga(M, Res1, Res)) -> ackermann_out_gga(s(M), s(N), Res)
4.44/2.03	   U1_gga(M, Res, ackermann_out_gga(M, s(0), Res)) -> ackermann_out_gga(s(M), 0, Res)
4.44/2.03	
4.44/2.03	The argument filtering Pi contains the following mapping:
4.44/2.03	ackermann_in_gga(x1, x2, x3)  =  ackermann_in_gga(x1, x2)
4.44/2.03	
4.44/2.03	0  =  0
4.44/2.03	
4.44/2.03	ackermann_out_gga(x1, x2, x3)  =  ackermann_out_gga(x3)
4.44/2.03	
4.44/2.03	s(x1)  =  s(x1)
4.44/2.03	
4.44/2.03	U1_gga(x1, x2, x3)  =  U1_gga(x3)
4.44/2.03	
4.44/2.03	U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
4.44/2.03	
4.44/2.03	U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
4.44/2.03	
4.44/2.03	ACKERMANN_IN_GGA(x1, x2, x3)  =  ACKERMANN_IN_GGA(x1, x2)
4.44/2.03	
4.44/2.03	U1_GGA(x1, x2, x3)  =  U1_GGA(x3)
4.44/2.03	
4.44/2.03	U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x4)
4.44/2.03	
4.44/2.03	U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
4.44/2.03	
4.44/2.03	
4.44/2.03	We have to consider all (P,R,Pi)-chains
4.44/2.03	----------------------------------------
4.44/2.03	
4.44/2.03	(5) DependencyGraphProof (EQUIVALENT)
4.44/2.03	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.
4.44/2.03	----------------------------------------
4.44/2.03	
4.44/2.03	(6)
4.44/2.03	Obligation:
4.44/2.03	Pi DP problem:
4.44/2.03	The TRS P consists of the following rules:
4.44/2.03	
4.44/2.03	   ACKERMANN_IN_GGA(s(M), 0, Res) -> ACKERMANN_IN_GGA(M, s(0), Res)
4.44/2.03	   ACKERMANN_IN_GGA(s(M), s(N), Res) -> U2_GGA(M, N, Res, ackermann_in_gga(s(M), N, Res1))
4.44/2.03	   U2_GGA(M, N, Res, ackermann_out_gga(s(M), N, Res1)) -> ACKERMANN_IN_GGA(M, Res1, Res)
4.44/2.03	   ACKERMANN_IN_GGA(s(M), s(N), Res) -> ACKERMANN_IN_GGA(s(M), N, Res1)
4.44/2.03	
4.44/2.03	The TRS R consists of the following rules:
4.44/2.03	
4.44/2.03	   ackermann_in_gga(0, N, s(N)) -> ackermann_out_gga(0, N, s(N))
4.44/2.03	   ackermann_in_gga(s(M), 0, Res) -> U1_gga(M, Res, ackermann_in_gga(M, s(0), Res))
4.44/2.03	   ackermann_in_gga(s(M), s(N), Res) -> U2_gga(M, N, Res, ackermann_in_gga(s(M), N, Res1))
4.44/2.03	   U2_gga(M, N, Res, ackermann_out_gga(s(M), N, Res1)) -> U3_gga(M, N, Res, ackermann_in_gga(M, Res1, Res))
4.44/2.03	   U3_gga(M, N, Res, ackermann_out_gga(M, Res1, Res)) -> ackermann_out_gga(s(M), s(N), Res)
4.44/2.03	   U1_gga(M, Res, ackermann_out_gga(M, s(0), Res)) -> ackermann_out_gga(s(M), 0, Res)
4.44/2.03	
4.44/2.03	The argument filtering Pi contains the following mapping:
4.44/2.03	ackermann_in_gga(x1, x2, x3)  =  ackermann_in_gga(x1, x2)
4.44/2.03	
4.44/2.03	0  =  0
4.44/2.03	
4.44/2.03	ackermann_out_gga(x1, x2, x3)  =  ackermann_out_gga(x3)
4.44/2.03	
4.44/2.03	s(x1)  =  s(x1)
4.44/2.03	
4.44/2.03	U1_gga(x1, x2, x3)  =  U1_gga(x3)
4.44/2.03	
4.44/2.03	U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x4)
4.44/2.03	
4.44/2.03	U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
4.44/2.03	
4.44/2.03	ACKERMANN_IN_GGA(x1, x2, x3)  =  ACKERMANN_IN_GGA(x1, x2)
4.44/2.03	
4.44/2.03	U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x4)
4.44/2.04	
4.44/2.04	
4.44/2.04	We have to consider all (P,R,Pi)-chains
4.44/2.04	----------------------------------------
4.44/2.04	
4.44/2.04	(7) PiDPToQDPProof (SOUND)
4.44/2.04	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
4.44/2.04	----------------------------------------
4.44/2.04	
4.44/2.04	(8)
4.44/2.04	Obligation:
4.44/2.04	Q DP problem:
4.44/2.04	The TRS P consists of the following rules:
4.44/2.04	
4.44/2.04	   ACKERMANN_IN_GGA(s(M), 0) -> ACKERMANN_IN_GGA(M, s(0))
4.44/2.04	   ACKERMANN_IN_GGA(s(M), s(N)) -> U2_GGA(M, ackermann_in_gga(s(M), N))
4.44/2.04	   U2_GGA(M, ackermann_out_gga(Res1)) -> ACKERMANN_IN_GGA(M, Res1)
4.44/2.04	   ACKERMANN_IN_GGA(s(M), s(N)) -> ACKERMANN_IN_GGA(s(M), N)
4.44/2.04	
4.44/2.04	The TRS R consists of the following rules:
4.44/2.04	
4.44/2.04	   ackermann_in_gga(0, N) -> ackermann_out_gga(s(N))
4.44/2.04	   ackermann_in_gga(s(M), 0) -> U1_gga(ackermann_in_gga(M, s(0)))
4.44/2.04	   ackermann_in_gga(s(M), s(N)) -> U2_gga(M, ackermann_in_gga(s(M), N))
4.44/2.04	   U2_gga(M, ackermann_out_gga(Res1)) -> U3_gga(ackermann_in_gga(M, Res1))
4.44/2.04	   U3_gga(ackermann_out_gga(Res)) -> ackermann_out_gga(Res)
4.44/2.04	   U1_gga(ackermann_out_gga(Res)) -> ackermann_out_gga(Res)
4.44/2.04	
4.44/2.04	The set Q consists of the following terms:
4.44/2.04	
4.44/2.04	   ackermann_in_gga(x0, x1)
4.44/2.04	   U2_gga(x0, x1)
4.44/2.04	   U3_gga(x0)
4.44/2.04	   U1_gga(x0)
4.44/2.04	
4.44/2.04	We have to consider all (P,Q,R)-chains.
4.44/2.04	----------------------------------------
4.44/2.04	
4.44/2.04	(9) QDPSizeChangeProof (EQUIVALENT)
4.44/2.04	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. 
4.44/2.04	
4.44/2.04	From the DPs we obtained the following set of size-change graphs:
4.44/2.04	*ACKERMANN_IN_GGA(s(M), s(N)) -> ACKERMANN_IN_GGA(s(M), N)
4.44/2.04	The graph contains the following edges 1 >= 1, 2 > 2
4.44/2.04	
4.44/2.04	
4.44/2.04	*ACKERMANN_IN_GGA(s(M), s(N)) -> U2_GGA(M, ackermann_in_gga(s(M), N))
4.44/2.04	The graph contains the following edges 1 > 1
4.44/2.04	
4.44/2.04	
4.44/2.04	*U2_GGA(M, ackermann_out_gga(Res1)) -> ACKERMANN_IN_GGA(M, Res1)
4.44/2.04	The graph contains the following edges 1 >= 1, 2 > 2
4.44/2.04	
4.44/2.04	
4.44/2.04	*ACKERMANN_IN_GGA(s(M), 0) -> ACKERMANN_IN_GGA(M, s(0))
4.44/2.04	The graph contains the following edges 1 > 1
4.44/2.04	
4.44/2.04	
4.44/2.04	----------------------------------------
4.44/2.04	
4.44/2.04	(10)
4.44/2.04	YES
4.67/2.06	EOF