3.89/1.80	YES
4.14/1.81	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
4.14/1.81	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.14/1.81	
4.14/1.81	
4.14/1.81	Left Termination of the query pattern
4.14/1.81	
4.14/1.81	times(g,g,a)
4.14/1.81	
4.14/1.81	w.r.t. the given Prolog program could successfully be proven:
4.14/1.81	
4.14/1.81	(0) Prolog
4.14/1.81	(1) PrologToPiTRSProof [SOUND, 0 ms]
4.14/1.81	(2) PiTRS
4.14/1.81	(3) DependencyPairsProof [EQUIVALENT, 15 ms]
4.14/1.81	(4) PiDP
4.14/1.81	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
4.14/1.81	(6) PiDP
4.14/1.81	(7) UsableRulesProof [EQUIVALENT, 0 ms]
4.14/1.81	(8) PiDP
4.14/1.81	(9) PiDPToQDPProof [SOUND, 8 ms]
4.14/1.81	(10) QDP
4.14/1.81	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
4.14/1.81	(12) YES
4.14/1.81	
4.14/1.81	
4.14/1.81	----------------------------------------
4.14/1.81	
4.14/1.81	(0)
4.14/1.81	Obligation:
4.14/1.81	Clauses:
4.14/1.81	
4.14/1.81	times(X, Y, Z) :- mult(X, Y, 0, Z).
4.14/1.81	mult(0, Y, 0, 0).
4.14/1.81	mult(s(U), Y, 0, Z) :- mult(U, Y, Y, Z).
4.14/1.81	mult(X, Y, s(W), s(Z)) :- mult(X, Y, W, Z).
4.14/1.81	
4.14/1.81	
4.14/1.81	Query: times(g,g,a)
4.14/1.81	----------------------------------------
4.14/1.81	
4.14/1.81	(1) PrologToPiTRSProof (SOUND)
4.14/1.81	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
4.14/1.81	
4.14/1.81	times_in_3: (b,b,f)
4.14/1.81	
4.14/1.81	mult_in_4: (b,b,b,f)
4.14/1.81	
4.14/1.81	Transforming Prolog into the following Term Rewriting System:
4.14/1.81	
4.14/1.81	Pi-finite rewrite system:
4.14/1.81	The TRS R consists of the following rules:
4.14/1.81	
4.14/1.81	   times_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
4.14/1.81	   mult_in_ggga(0, Y, 0, 0) -> mult_out_ggga(0, Y, 0, 0)
4.14/1.81	   mult_in_ggga(s(U), Y, 0, Z) -> U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
4.14/1.81	   mult_in_ggga(X, Y, s(W), s(Z)) -> U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
4.14/1.81	   U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) -> mult_out_ggga(X, Y, s(W), s(Z))
4.14/1.81	   U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) -> mult_out_ggga(s(U), Y, 0, Z)
4.14/1.81	   U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) -> times_out_gga(X, Y, Z)
4.14/1.81	
4.14/1.81	The argument filtering Pi contains the following mapping:
4.14/1.81	times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
4.14/1.81	
4.14/1.81	U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
4.14/1.81	
4.14/1.81	mult_in_ggga(x1, x2, x3, x4)  =  mult_in_ggga(x1, x2, x3)
4.14/1.81	
4.14/1.81	0  =  0
4.14/1.81	
4.14/1.81	mult_out_ggga(x1, x2, x3, x4)  =  mult_out_ggga(x4)
4.14/1.81	
4.14/1.81	s(x1)  =  s(x1)
4.14/1.81	
4.14/1.81	U2_ggga(x1, x2, x3, x4)  =  U2_ggga(x4)
4.14/1.81	
4.14/1.81	U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)
4.14/1.81	
4.14/1.81	times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
4.14/1.81	
4.14/1.81	
4.14/1.81	
4.14/1.81	
4.14/1.81	
4.14/1.81	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
4.14/1.81	
4.14/1.81	
4.14/1.81	
4.14/1.81	----------------------------------------
4.14/1.81	
4.14/1.81	(2)
4.14/1.81	Obligation:
4.14/1.81	Pi-finite rewrite system:
4.14/1.81	The TRS R consists of the following rules:
4.14/1.81	
4.14/1.81	   times_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
4.14/1.81	   mult_in_ggga(0, Y, 0, 0) -> mult_out_ggga(0, Y, 0, 0)
4.14/1.81	   mult_in_ggga(s(U), Y, 0, Z) -> U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
4.14/1.81	   mult_in_ggga(X, Y, s(W), s(Z)) -> U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
4.14/1.81	   U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) -> mult_out_ggga(X, Y, s(W), s(Z))
4.14/1.81	   U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) -> mult_out_ggga(s(U), Y, 0, Z)
4.14/1.81	   U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) -> times_out_gga(X, Y, Z)
4.14/1.81	
4.14/1.81	The argument filtering Pi contains the following mapping:
4.14/1.81	times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
4.14/1.81	
4.14/1.81	U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
4.14/1.81	
4.14/1.81	mult_in_ggga(x1, x2, x3, x4)  =  mult_in_ggga(x1, x2, x3)
4.14/1.81	
4.14/1.81	0  =  0
4.14/1.81	
4.14/1.81	mult_out_ggga(x1, x2, x3, x4)  =  mult_out_ggga(x4)
4.14/1.81	
4.14/1.81	s(x1)  =  s(x1)
4.14/1.81	
4.14/1.81	U2_ggga(x1, x2, x3, x4)  =  U2_ggga(x4)
4.14/1.81	
4.14/1.81	U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)
4.14/1.81	
4.14/1.81	times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
4.14/1.81	
4.14/1.81	
4.14/1.81	
4.14/1.81	----------------------------------------
4.14/1.81	
4.14/1.81	(3) DependencyPairsProof (EQUIVALENT)
4.14/1.81	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
4.14/1.81	Pi DP problem:
4.14/1.81	The TRS P consists of the following rules:
4.14/1.81	
4.14/1.81	   TIMES_IN_GGA(X, Y, Z) -> U1_GGA(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
4.14/1.81	   TIMES_IN_GGA(X, Y, Z) -> MULT_IN_GGGA(X, Y, 0, Z)
4.14/1.81	   MULT_IN_GGGA(s(U), Y, 0, Z) -> U2_GGGA(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
4.14/1.81	   MULT_IN_GGGA(s(U), Y, 0, Z) -> MULT_IN_GGGA(U, Y, Y, Z)
4.14/1.81	   MULT_IN_GGGA(X, Y, s(W), s(Z)) -> U3_GGGA(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
4.14/1.81	   MULT_IN_GGGA(X, Y, s(W), s(Z)) -> MULT_IN_GGGA(X, Y, W, Z)
4.14/1.81	
4.14/1.81	The TRS R consists of the following rules:
4.14/1.81	
4.14/1.81	   times_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
4.14/1.81	   mult_in_ggga(0, Y, 0, 0) -> mult_out_ggga(0, Y, 0, 0)
4.14/1.81	   mult_in_ggga(s(U), Y, 0, Z) -> U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
4.14/1.81	   mult_in_ggga(X, Y, s(W), s(Z)) -> U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
4.14/1.81	   U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) -> mult_out_ggga(X, Y, s(W), s(Z))
4.14/1.81	   U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) -> mult_out_ggga(s(U), Y, 0, Z)
4.14/1.81	   U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) -> times_out_gga(X, Y, Z)
4.14/1.81	
4.14/1.81	The argument filtering Pi contains the following mapping:
4.14/1.81	times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
4.14/1.81	
4.14/1.81	U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
4.14/1.81	
4.14/1.81	mult_in_ggga(x1, x2, x3, x4)  =  mult_in_ggga(x1, x2, x3)
4.14/1.81	
4.14/1.81	0  =  0
4.14/1.81	
4.14/1.81	mult_out_ggga(x1, x2, x3, x4)  =  mult_out_ggga(x4)
4.14/1.81	
4.14/1.81	s(x1)  =  s(x1)
4.14/1.81	
4.14/1.81	U2_ggga(x1, x2, x3, x4)  =  U2_ggga(x4)
4.14/1.81	
4.14/1.81	U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)
4.14/1.81	
4.14/1.81	times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
4.14/1.81	
4.14/1.81	TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)
4.14/1.81	
4.14/1.81	U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
4.14/1.81	
4.14/1.81	MULT_IN_GGGA(x1, x2, x3, x4)  =  MULT_IN_GGGA(x1, x2, x3)
4.14/1.81	
4.14/1.81	U2_GGGA(x1, x2, x3, x4)  =  U2_GGGA(x4)
4.14/1.81	
4.14/1.81	U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x5)
4.14/1.81	
4.14/1.81	
4.14/1.81	We have to consider all (P,R,Pi)-chains
4.14/1.81	----------------------------------------
4.14/1.81	
4.14/1.81	(4)
4.14/1.81	Obligation:
4.14/1.81	Pi DP problem:
4.14/1.81	The TRS P consists of the following rules:
4.14/1.81	
4.14/1.81	   TIMES_IN_GGA(X, Y, Z) -> U1_GGA(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
4.14/1.81	   TIMES_IN_GGA(X, Y, Z) -> MULT_IN_GGGA(X, Y, 0, Z)
4.14/1.81	   MULT_IN_GGGA(s(U), Y, 0, Z) -> U2_GGGA(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
4.14/1.81	   MULT_IN_GGGA(s(U), Y, 0, Z) -> MULT_IN_GGGA(U, Y, Y, Z)
4.14/1.81	   MULT_IN_GGGA(X, Y, s(W), s(Z)) -> U3_GGGA(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
4.14/1.81	   MULT_IN_GGGA(X, Y, s(W), s(Z)) -> MULT_IN_GGGA(X, Y, W, Z)
4.14/1.81	
4.14/1.81	The TRS R consists of the following rules:
4.14/1.81	
4.14/1.81	   times_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
4.14/1.81	   mult_in_ggga(0, Y, 0, 0) -> mult_out_ggga(0, Y, 0, 0)
4.14/1.81	   mult_in_ggga(s(U), Y, 0, Z) -> U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
4.14/1.81	   mult_in_ggga(X, Y, s(W), s(Z)) -> U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
4.14/1.81	   U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) -> mult_out_ggga(X, Y, s(W), s(Z))
4.14/1.81	   U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) -> mult_out_ggga(s(U), Y, 0, Z)
4.14/1.81	   U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) -> times_out_gga(X, Y, Z)
4.14/1.81	
4.14/1.81	The argument filtering Pi contains the following mapping:
4.14/1.81	times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
4.14/1.81	
4.14/1.81	U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
4.14/1.81	
4.14/1.81	mult_in_ggga(x1, x2, x3, x4)  =  mult_in_ggga(x1, x2, x3)
4.14/1.81	
4.14/1.81	0  =  0
4.14/1.81	
4.14/1.81	mult_out_ggga(x1, x2, x3, x4)  =  mult_out_ggga(x4)
4.14/1.81	
4.14/1.81	s(x1)  =  s(x1)
4.14/1.81	
4.14/1.81	U2_ggga(x1, x2, x3, x4)  =  U2_ggga(x4)
4.14/1.81	
4.14/1.81	U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)
4.14/1.81	
4.14/1.81	times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
4.14/1.81	
4.14/1.81	TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)
4.14/1.81	
4.14/1.81	U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
4.14/1.81	
4.14/1.81	MULT_IN_GGGA(x1, x2, x3, x4)  =  MULT_IN_GGGA(x1, x2, x3)
4.14/1.81	
4.14/1.81	U2_GGGA(x1, x2, x3, x4)  =  U2_GGGA(x4)
4.14/1.81	
4.14/1.81	U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x5)
4.14/1.81	
4.14/1.81	
4.14/1.81	We have to consider all (P,R,Pi)-chains
4.14/1.81	----------------------------------------
4.14/1.81	
4.14/1.81	(5) DependencyGraphProof (EQUIVALENT)
4.14/1.81	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 4 less nodes.
4.14/1.81	----------------------------------------
4.14/1.81	
4.14/1.81	(6)
4.14/1.81	Obligation:
4.14/1.81	Pi DP problem:
4.14/1.81	The TRS P consists of the following rules:
4.14/1.81	
4.14/1.81	   MULT_IN_GGGA(X, Y, s(W), s(Z)) -> MULT_IN_GGGA(X, Y, W, Z)
4.14/1.81	   MULT_IN_GGGA(s(U), Y, 0, Z) -> MULT_IN_GGGA(U, Y, Y, Z)
4.14/1.81	
4.14/1.81	The TRS R consists of the following rules:
4.14/1.81	
4.14/1.81	   times_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
4.14/1.81	   mult_in_ggga(0, Y, 0, 0) -> mult_out_ggga(0, Y, 0, 0)
4.14/1.81	   mult_in_ggga(s(U), Y, 0, Z) -> U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
4.14/1.81	   mult_in_ggga(X, Y, s(W), s(Z)) -> U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
4.14/1.81	   U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) -> mult_out_ggga(X, Y, s(W), s(Z))
4.14/1.81	   U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) -> mult_out_ggga(s(U), Y, 0, Z)
4.14/1.81	   U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) -> times_out_gga(X, Y, Z)
4.14/1.81	
4.14/1.81	The argument filtering Pi contains the following mapping:
4.14/1.81	times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
4.14/1.81	
4.14/1.81	U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
4.14/1.81	
4.14/1.81	mult_in_ggga(x1, x2, x3, x4)  =  mult_in_ggga(x1, x2, x3)
4.14/1.81	
4.14/1.81	0  =  0
4.14/1.81	
4.14/1.81	mult_out_ggga(x1, x2, x3, x4)  =  mult_out_ggga(x4)
4.14/1.81	
4.14/1.81	s(x1)  =  s(x1)
4.14/1.81	
4.14/1.81	U2_ggga(x1, x2, x3, x4)  =  U2_ggga(x4)
4.14/1.81	
4.14/1.81	U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)
4.14/1.81	
4.14/1.81	times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
4.14/1.81	
4.14/1.81	MULT_IN_GGGA(x1, x2, x3, x4)  =  MULT_IN_GGGA(x1, x2, x3)
4.14/1.81	
4.14/1.81	
4.14/1.81	We have to consider all (P,R,Pi)-chains
4.14/1.81	----------------------------------------
4.14/1.81	
4.14/1.81	(7) UsableRulesProof (EQUIVALENT)
4.14/1.81	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
4.14/1.81	----------------------------------------
4.14/1.81	
4.14/1.81	(8)
4.14/1.81	Obligation:
4.14/1.81	Pi DP problem:
4.14/1.81	The TRS P consists of the following rules:
4.14/1.81	
4.14/1.81	   MULT_IN_GGGA(X, Y, s(W), s(Z)) -> MULT_IN_GGGA(X, Y, W, Z)
4.14/1.81	   MULT_IN_GGGA(s(U), Y, 0, Z) -> MULT_IN_GGGA(U, Y, Y, Z)
4.14/1.81	
4.14/1.81	R is empty.
4.14/1.81	The argument filtering Pi contains the following mapping:
4.14/1.81	0  =  0
4.14/1.81	
4.14/1.81	s(x1)  =  s(x1)
4.14/1.81	
4.14/1.81	MULT_IN_GGGA(x1, x2, x3, x4)  =  MULT_IN_GGGA(x1, x2, x3)
4.14/1.81	
4.14/1.81	
4.14/1.81	We have to consider all (P,R,Pi)-chains
4.14/1.81	----------------------------------------
4.14/1.81	
4.14/1.81	(9) PiDPToQDPProof (SOUND)
4.14/1.81	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
4.14/1.81	----------------------------------------
4.14/1.81	
4.14/1.81	(10)
4.14/1.81	Obligation:
4.14/1.81	Q DP problem:
4.14/1.81	The TRS P consists of the following rules:
4.14/1.81	
4.14/1.81	   MULT_IN_GGGA(X, Y, s(W)) -> MULT_IN_GGGA(X, Y, W)
4.14/1.81	   MULT_IN_GGGA(s(U), Y, 0) -> MULT_IN_GGGA(U, Y, Y)
4.14/1.81	
4.14/1.81	R is empty.
4.14/1.81	Q is empty.
4.14/1.81	We have to consider all (P,Q,R)-chains.
4.14/1.81	----------------------------------------
4.14/1.81	
4.14/1.81	(11) QDPSizeChangeProof (EQUIVALENT)
4.14/1.81	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. 
4.14/1.81	
4.14/1.81	From the DPs we obtained the following set of size-change graphs:
4.14/1.81	*MULT_IN_GGGA(X, Y, s(W)) -> MULT_IN_GGGA(X, Y, W)
4.14/1.81	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
4.14/1.81	
4.14/1.81	
4.14/1.81	*MULT_IN_GGGA(s(U), Y, 0) -> MULT_IN_GGGA(U, Y, Y)
4.14/1.81	The graph contains the following edges 1 > 1, 2 >= 2, 2 >= 3
4.14/1.81	
4.14/1.81	
4.14/1.81	----------------------------------------
4.14/1.81	
4.14/1.81	(12)
4.14/1.81	YES
4.14/1.83	EOF