3.63/1.72	YES
3.63/1.73	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.63/1.73	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.63/1.73	
3.63/1.73	
3.63/1.73	Left Termination of the query pattern
3.63/1.73	
3.63/1.73	ack(g,g,a)
3.63/1.73	
3.63/1.73	w.r.t. the given Prolog program could successfully be proven:
3.63/1.73	
3.63/1.73	(0) Prolog
3.63/1.73	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.63/1.73	(2) PiTRS
3.63/1.73	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.63/1.73	(4) PiDP
3.63/1.73	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.63/1.73	(6) PiDP
3.63/1.73	(7) PiDPToQDPProof [SOUND, 2 ms]
3.63/1.73	(8) QDP
3.63/1.73	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.63/1.73	(10) YES
3.63/1.73	
3.63/1.73	
3.63/1.73	----------------------------------------
3.63/1.73	
3.63/1.73	(0)
3.63/1.73	Obligation:
3.63/1.73	Clauses:
3.63/1.73	
3.63/1.73	ack(0, N, s(N)).
3.63/1.73	ack(s(M), 0, A) :- ack(M, s(0), A).
3.63/1.73	ack(s(M), s(N), A) :- ','(ack(s(M), N, A1), ack(M, A1, A)).
3.63/1.73	
3.63/1.73	
3.63/1.73	Query: ack(g,g,a)
3.63/1.73	----------------------------------------
3.63/1.73	
3.63/1.73	(1) PrologToPiTRSProof (SOUND)
3.63/1.73	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.63/1.73	
3.63/1.73	ack_in_3: (b,b,f)
3.63/1.73	
3.63/1.73	Transforming Prolog into the following Term Rewriting System:
3.63/1.73	
3.63/1.73	Pi-finite rewrite system:
3.63/1.73	The TRS R consists of the following rules:
3.63/1.73	
3.63/1.73	   ack_in_gga(0, N, s(N)) -> ack_out_gga(0, N, s(N))
3.63/1.73	   ack_in_gga(s(M), 0, A) -> U1_gga(M, A, ack_in_gga(M, s(0), A))
3.63/1.73	   ack_in_gga(s(M), s(N), A) -> U2_gga(M, N, A, ack_in_gga(s(M), N, A1))
3.63/1.73	   U2_gga(M, N, A, ack_out_gga(s(M), N, A1)) -> U3_gga(M, N, A, ack_in_gga(M, A1, A))
3.63/1.73	   U3_gga(M, N, A, ack_out_gga(M, A1, A)) -> ack_out_gga(s(M), s(N), A)
3.63/1.73	   U1_gga(M, A, ack_out_gga(M, s(0), A)) -> ack_out_gga(s(M), 0, A)
3.63/1.73	
3.63/1.73	The argument filtering Pi contains the following mapping:
3.63/1.73	ack_in_gga(x1, x2, x3)  =  ack_in_gga(x1, x2)
3.63/1.73	
3.63/1.73	0  =  0
3.63/1.73	
3.63/1.73	ack_out_gga(x1, x2, x3)  =  ack_out_gga(x1, x2, x3)
3.63/1.73	
3.63/1.73	s(x1)  =  s(x1)
3.63/1.73	
3.63/1.73	U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
3.63/1.73	
3.63/1.73	U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
3.63/1.73	
3.63/1.73	U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
3.63/1.73	
3.63/1.73	
3.63/1.73	
3.63/1.73	
3.63/1.73	
3.63/1.73	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.63/1.73	
3.63/1.73	
3.63/1.73	
3.63/1.73	----------------------------------------
3.63/1.73	
3.63/1.73	(2)
3.63/1.73	Obligation:
3.63/1.73	Pi-finite rewrite system:
3.63/1.73	The TRS R consists of the following rules:
3.63/1.73	
3.63/1.73	   ack_in_gga(0, N, s(N)) -> ack_out_gga(0, N, s(N))
3.63/1.73	   ack_in_gga(s(M), 0, A) -> U1_gga(M, A, ack_in_gga(M, s(0), A))
3.63/1.73	   ack_in_gga(s(M), s(N), A) -> U2_gga(M, N, A, ack_in_gga(s(M), N, A1))
3.63/1.73	   U2_gga(M, N, A, ack_out_gga(s(M), N, A1)) -> U3_gga(M, N, A, ack_in_gga(M, A1, A))
3.63/1.73	   U3_gga(M, N, A, ack_out_gga(M, A1, A)) -> ack_out_gga(s(M), s(N), A)
3.63/1.73	   U1_gga(M, A, ack_out_gga(M, s(0), A)) -> ack_out_gga(s(M), 0, A)
3.63/1.73	
3.63/1.73	The argument filtering Pi contains the following mapping:
3.63/1.73	ack_in_gga(x1, x2, x3)  =  ack_in_gga(x1, x2)
3.63/1.73	
3.63/1.73	0  =  0
3.63/1.73	
3.63/1.73	ack_out_gga(x1, x2, x3)  =  ack_out_gga(x1, x2, x3)
3.63/1.73	
3.63/1.73	s(x1)  =  s(x1)
3.63/1.73	
3.63/1.73	U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
3.63/1.73	
3.63/1.73	U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
3.63/1.73	
3.63/1.73	U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
3.63/1.73	
3.63/1.73	
3.63/1.73	
3.63/1.73	----------------------------------------
3.63/1.73	
3.63/1.73	(3) DependencyPairsProof (EQUIVALENT)
3.63/1.73	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.63/1.73	Pi DP problem:
3.63/1.73	The TRS P consists of the following rules:
3.63/1.73	
3.63/1.73	   ACK_IN_GGA(s(M), 0, A) -> U1_GGA(M, A, ack_in_gga(M, s(0), A))
3.63/1.73	   ACK_IN_GGA(s(M), 0, A) -> ACK_IN_GGA(M, s(0), A)
3.63/1.73	   ACK_IN_GGA(s(M), s(N), A) -> U2_GGA(M, N, A, ack_in_gga(s(M), N, A1))
3.63/1.73	   ACK_IN_GGA(s(M), s(N), A) -> ACK_IN_GGA(s(M), N, A1)
3.63/1.73	   U2_GGA(M, N, A, ack_out_gga(s(M), N, A1)) -> U3_GGA(M, N, A, ack_in_gga(M, A1, A))
3.63/1.73	   U2_GGA(M, N, A, ack_out_gga(s(M), N, A1)) -> ACK_IN_GGA(M, A1, A)
3.63/1.73	
3.63/1.73	The TRS R consists of the following rules:
3.63/1.73	
3.63/1.73	   ack_in_gga(0, N, s(N)) -> ack_out_gga(0, N, s(N))
3.63/1.73	   ack_in_gga(s(M), 0, A) -> U1_gga(M, A, ack_in_gga(M, s(0), A))
3.63/1.73	   ack_in_gga(s(M), s(N), A) -> U2_gga(M, N, A, ack_in_gga(s(M), N, A1))
3.63/1.73	   U2_gga(M, N, A, ack_out_gga(s(M), N, A1)) -> U3_gga(M, N, A, ack_in_gga(M, A1, A))
3.63/1.73	   U3_gga(M, N, A, ack_out_gga(M, A1, A)) -> ack_out_gga(s(M), s(N), A)
3.63/1.73	   U1_gga(M, A, ack_out_gga(M, s(0), A)) -> ack_out_gga(s(M), 0, A)
3.63/1.73	
3.63/1.73	The argument filtering Pi contains the following mapping:
3.63/1.73	ack_in_gga(x1, x2, x3)  =  ack_in_gga(x1, x2)
3.63/1.73	
3.63/1.73	0  =  0
3.63/1.73	
3.63/1.73	ack_out_gga(x1, x2, x3)  =  ack_out_gga(x1, x2, x3)
3.63/1.73	
3.63/1.73	s(x1)  =  s(x1)
3.63/1.73	
3.63/1.73	U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
3.63/1.73	
3.63/1.73	U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
3.63/1.73	
3.63/1.73	U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
3.63/1.73	
3.63/1.73	ACK_IN_GGA(x1, x2, x3)  =  ACK_IN_GGA(x1, x2)
3.63/1.73	
3.63/1.73	U1_GGA(x1, x2, x3)  =  U1_GGA(x1, x3)
3.63/1.73	
3.63/1.73	U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
3.63/1.73	
3.63/1.73	U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
3.63/1.73	
3.63/1.73	
3.63/1.73	We have to consider all (P,R,Pi)-chains
3.63/1.73	----------------------------------------
3.63/1.73	
3.63/1.73	(4)
3.63/1.73	Obligation:
3.63/1.73	Pi DP problem:
3.63/1.73	The TRS P consists of the following rules:
3.63/1.73	
3.63/1.73	   ACK_IN_GGA(s(M), 0, A) -> U1_GGA(M, A, ack_in_gga(M, s(0), A))
3.63/1.73	   ACK_IN_GGA(s(M), 0, A) -> ACK_IN_GGA(M, s(0), A)
3.63/1.73	   ACK_IN_GGA(s(M), s(N), A) -> U2_GGA(M, N, A, ack_in_gga(s(M), N, A1))
3.63/1.73	   ACK_IN_GGA(s(M), s(N), A) -> ACK_IN_GGA(s(M), N, A1)
3.63/1.73	   U2_GGA(M, N, A, ack_out_gga(s(M), N, A1)) -> U3_GGA(M, N, A, ack_in_gga(M, A1, A))
3.63/1.73	   U2_GGA(M, N, A, ack_out_gga(s(M), N, A1)) -> ACK_IN_GGA(M, A1, A)
3.63/1.73	
3.63/1.73	The TRS R consists of the following rules:
3.63/1.73	
3.63/1.73	   ack_in_gga(0, N, s(N)) -> ack_out_gga(0, N, s(N))
3.63/1.73	   ack_in_gga(s(M), 0, A) -> U1_gga(M, A, ack_in_gga(M, s(0), A))
3.63/1.73	   ack_in_gga(s(M), s(N), A) -> U2_gga(M, N, A, ack_in_gga(s(M), N, A1))
3.63/1.73	   U2_gga(M, N, A, ack_out_gga(s(M), N, A1)) -> U3_gga(M, N, A, ack_in_gga(M, A1, A))
3.63/1.73	   U3_gga(M, N, A, ack_out_gga(M, A1, A)) -> ack_out_gga(s(M), s(N), A)
3.63/1.73	   U1_gga(M, A, ack_out_gga(M, s(0), A)) -> ack_out_gga(s(M), 0, A)
3.63/1.73	
3.63/1.73	The argument filtering Pi contains the following mapping:
3.63/1.73	ack_in_gga(x1, x2, x3)  =  ack_in_gga(x1, x2)
3.63/1.73	
3.63/1.73	0  =  0
3.63/1.73	
3.63/1.73	ack_out_gga(x1, x2, x3)  =  ack_out_gga(x1, x2, x3)
3.63/1.73	
3.63/1.73	s(x1)  =  s(x1)
3.63/1.73	
3.63/1.73	U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
3.63/1.73	
3.63/1.73	U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
3.63/1.73	
3.63/1.73	U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
3.63/1.73	
3.63/1.73	ACK_IN_GGA(x1, x2, x3)  =  ACK_IN_GGA(x1, x2)
3.63/1.73	
3.63/1.73	U1_GGA(x1, x2, x3)  =  U1_GGA(x1, x3)
3.63/1.73	
3.63/1.73	U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
3.63/1.73	
3.63/1.73	U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
3.63/1.73	
3.63/1.73	
3.63/1.73	We have to consider all (P,R,Pi)-chains
3.63/1.73	----------------------------------------
3.63/1.73	
3.63/1.73	(5) DependencyGraphProof (EQUIVALENT)
3.63/1.73	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.
3.63/1.73	----------------------------------------
3.63/1.73	
3.63/1.73	(6)
3.63/1.73	Obligation:
3.63/1.73	Pi DP problem:
3.63/1.73	The TRS P consists of the following rules:
3.63/1.73	
3.63/1.73	   ACK_IN_GGA(s(M), 0, A) -> ACK_IN_GGA(M, s(0), A)
3.63/1.73	   ACK_IN_GGA(s(M), s(N), A) -> U2_GGA(M, N, A, ack_in_gga(s(M), N, A1))
3.63/1.73	   U2_GGA(M, N, A, ack_out_gga(s(M), N, A1)) -> ACK_IN_GGA(M, A1, A)
3.63/1.73	   ACK_IN_GGA(s(M), s(N), A) -> ACK_IN_GGA(s(M), N, A1)
3.63/1.73	
3.63/1.73	The TRS R consists of the following rules:
3.63/1.73	
3.63/1.73	   ack_in_gga(0, N, s(N)) -> ack_out_gga(0, N, s(N))
3.63/1.73	   ack_in_gga(s(M), 0, A) -> U1_gga(M, A, ack_in_gga(M, s(0), A))
3.63/1.73	   ack_in_gga(s(M), s(N), A) -> U2_gga(M, N, A, ack_in_gga(s(M), N, A1))
3.63/1.73	   U2_gga(M, N, A, ack_out_gga(s(M), N, A1)) -> U3_gga(M, N, A, ack_in_gga(M, A1, A))
3.63/1.73	   U3_gga(M, N, A, ack_out_gga(M, A1, A)) -> ack_out_gga(s(M), s(N), A)
3.63/1.73	   U1_gga(M, A, ack_out_gga(M, s(0), A)) -> ack_out_gga(s(M), 0, A)
3.63/1.73	
3.63/1.73	The argument filtering Pi contains the following mapping:
3.63/1.73	ack_in_gga(x1, x2, x3)  =  ack_in_gga(x1, x2)
3.63/1.73	
3.63/1.73	0  =  0
3.63/1.73	
3.63/1.73	ack_out_gga(x1, x2, x3)  =  ack_out_gga(x1, x2, x3)
3.63/1.73	
3.63/1.73	s(x1)  =  s(x1)
3.63/1.73	
3.63/1.73	U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
3.63/1.73	
3.63/1.73	U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
3.63/1.73	
3.63/1.73	U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
3.63/1.73	
3.63/1.73	ACK_IN_GGA(x1, x2, x3)  =  ACK_IN_GGA(x1, x2)
3.63/1.73	
3.63/1.73	U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
3.63/1.73	
3.63/1.73	
3.63/1.73	We have to consider all (P,R,Pi)-chains
3.63/1.73	----------------------------------------
3.63/1.73	
3.63/1.73	(7) PiDPToQDPProof (SOUND)
3.63/1.73	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.63/1.73	----------------------------------------
3.63/1.73	
3.63/1.73	(8)
3.63/1.73	Obligation:
3.63/1.73	Q DP problem:
3.63/1.73	The TRS P consists of the following rules:
3.63/1.73	
3.63/1.73	   ACK_IN_GGA(s(M), 0) -> ACK_IN_GGA(M, s(0))
3.63/1.73	   ACK_IN_GGA(s(M), s(N)) -> U2_GGA(M, N, ack_in_gga(s(M), N))
3.63/1.73	   U2_GGA(M, N, ack_out_gga(s(M), N, A1)) -> ACK_IN_GGA(M, A1)
3.63/1.73	   ACK_IN_GGA(s(M), s(N)) -> ACK_IN_GGA(s(M), N)
3.63/1.73	
3.63/1.73	The TRS R consists of the following rules:
3.63/1.73	
3.63/1.73	   ack_in_gga(0, N) -> ack_out_gga(0, N, s(N))
3.63/1.73	   ack_in_gga(s(M), 0) -> U1_gga(M, ack_in_gga(M, s(0)))
3.63/1.73	   ack_in_gga(s(M), s(N)) -> U2_gga(M, N, ack_in_gga(s(M), N))
3.63/1.73	   U2_gga(M, N, ack_out_gga(s(M), N, A1)) -> U3_gga(M, N, ack_in_gga(M, A1))
3.63/1.73	   U3_gga(M, N, ack_out_gga(M, A1, A)) -> ack_out_gga(s(M), s(N), A)
3.63/1.73	   U1_gga(M, ack_out_gga(M, s(0), A)) -> ack_out_gga(s(M), 0, A)
3.63/1.73	
3.63/1.73	The set Q consists of the following terms:
3.63/1.73	
3.63/1.73	   ack_in_gga(x0, x1)
3.63/1.73	   U2_gga(x0, x1, x2)
3.63/1.73	   U3_gga(x0, x1, x2)
3.63/1.73	   U1_gga(x0, x1)
3.63/1.73	
3.63/1.73	We have to consider all (P,Q,R)-chains.
3.63/1.73	----------------------------------------
3.63/1.73	
3.63/1.73	(9) QDPSizeChangeProof (EQUIVALENT)
3.63/1.73	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. 
3.63/1.73	
3.63/1.73	From the DPs we obtained the following set of size-change graphs:
3.63/1.73	*ACK_IN_GGA(s(M), s(N)) -> ACK_IN_GGA(s(M), N)
3.63/1.73	The graph contains the following edges 1 >= 1, 2 > 2
3.63/1.73	
3.63/1.73	
3.63/1.73	*ACK_IN_GGA(s(M), s(N)) -> U2_GGA(M, N, ack_in_gga(s(M), N))
3.63/1.73	The graph contains the following edges 1 > 1, 2 > 2
3.63/1.73	
3.63/1.73	
3.63/1.73	*U2_GGA(M, N, ack_out_gga(s(M), N, A1)) -> ACK_IN_GGA(M, A1)
3.63/1.73	The graph contains the following edges 1 >= 1, 3 > 1, 3 > 2
3.63/1.73	
3.63/1.73	
3.63/1.73	*ACK_IN_GGA(s(M), 0) -> ACK_IN_GGA(M, s(0))
3.63/1.73	The graph contains the following edges 1 > 1
3.63/1.73	
3.63/1.73	
3.63/1.73	----------------------------------------
3.63/1.73	
3.63/1.73	(10)
3.63/1.73	YES
3.63/1.76	EOF