4.13/1.83	YES
4.21/1.85	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
4.21/1.85	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.21/1.85	
4.21/1.85	
4.21/1.85	Left Termination of the query pattern
4.21/1.85	
4.21/1.85	div(g,g,a)
4.21/1.85	
4.21/1.85	w.r.t. the given Prolog program could successfully be proven:
4.21/1.85	
4.21/1.85	(0) Prolog
4.21/1.85	(1) PrologToPiTRSProof [SOUND, 0 ms]
4.21/1.85	(2) PiTRS
4.21/1.85	(3) DependencyPairsProof [EQUIVALENT, 2 ms]
4.21/1.85	(4) PiDP
4.21/1.85	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
4.21/1.85	(6) PiDP
4.21/1.85	(7) UsableRulesProof [EQUIVALENT, 0 ms]
4.21/1.85	(8) PiDP
4.21/1.85	(9) PiDPToQDPProof [SOUND, 0 ms]
4.21/1.85	(10) QDP
4.21/1.85	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
4.21/1.85	(12) YES
4.21/1.85	
4.21/1.85	
4.21/1.85	----------------------------------------
4.21/1.85	
4.21/1.85	(0)
4.21/1.85	Obligation:
4.21/1.85	Clauses:
4.21/1.85	
4.21/1.85	div(X, Y, Z) :- quot(X, Y, Y, Z).
4.21/1.85	quot(0, s(Y), s(Z), 0).
4.21/1.85	quot(s(X), s(Y), Z, U) :- quot(X, Y, Z, U).
4.21/1.85	quot(X, 0, s(Z), s(U)) :- quot(X, s(Z), s(Z), U).
4.21/1.85	
4.21/1.85	
4.21/1.85	Query: div(g,g,a)
4.21/1.85	----------------------------------------
4.21/1.85	
4.21/1.85	(1) PrologToPiTRSProof (SOUND)
4.21/1.85	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
4.21/1.85	
4.21/1.85	div_in_3: (b,b,f)
4.21/1.85	
4.21/1.85	quot_in_4: (b,b,b,f)
4.21/1.85	
4.21/1.85	Transforming Prolog into the following Term Rewriting System:
4.21/1.85	
4.21/1.85	Pi-finite rewrite system:
4.21/1.85	The TRS R consists of the following rules:
4.21/1.85	
4.21/1.85	   div_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, quot_in_ggga(X, Y, Y, Z))
4.21/1.85	   quot_in_ggga(0, s(Y), s(Z), 0) -> quot_out_ggga(0, s(Y), s(Z), 0)
4.21/1.85	   quot_in_ggga(s(X), s(Y), Z, U) -> U2_ggga(X, Y, Z, U, quot_in_ggga(X, Y, Z, U))
4.21/1.85	   quot_in_ggga(X, 0, s(Z), s(U)) -> U3_ggga(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U))
4.21/1.85	   U3_ggga(X, Z, U, quot_out_ggga(X, s(Z), s(Z), U)) -> quot_out_ggga(X, 0, s(Z), s(U))
4.21/1.85	   U2_ggga(X, Y, Z, U, quot_out_ggga(X, Y, Z, U)) -> quot_out_ggga(s(X), s(Y), Z, U)
4.21/1.85	   U1_gga(X, Y, Z, quot_out_ggga(X, Y, Y, Z)) -> div_out_gga(X, Y, Z)
4.21/1.85	
4.21/1.85	The argument filtering Pi contains the following mapping:
4.21/1.85	div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
4.21/1.85	
4.21/1.85	U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
4.21/1.85	
4.21/1.85	quot_in_ggga(x1, x2, x3, x4)  =  quot_in_ggga(x1, x2, x3)
4.21/1.85	
4.21/1.85	0  =  0
4.21/1.85	
4.21/1.85	s(x1)  =  s(x1)
4.21/1.85	
4.21/1.85	quot_out_ggga(x1, x2, x3, x4)  =  quot_out_ggga(x1, x2, x3, x4)
4.21/1.85	
4.21/1.85	U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)
4.21/1.85	
4.21/1.85	U3_ggga(x1, x2, x3, x4)  =  U3_ggga(x1, x2, x4)
4.21/1.85	
4.21/1.85	div_out_gga(x1, x2, x3)  =  div_out_gga(x1, x2, x3)
4.21/1.85	
4.21/1.85	
4.21/1.85	
4.21/1.85	
4.21/1.85	
4.21/1.85	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
4.21/1.85	
4.21/1.85	
4.21/1.85	
4.21/1.85	----------------------------------------
4.21/1.85	
4.21/1.85	(2)
4.21/1.85	Obligation:
4.21/1.85	Pi-finite rewrite system:
4.21/1.85	The TRS R consists of the following rules:
4.21/1.85	
4.21/1.85	   div_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, quot_in_ggga(X, Y, Y, Z))
4.21/1.85	   quot_in_ggga(0, s(Y), s(Z), 0) -> quot_out_ggga(0, s(Y), s(Z), 0)
4.21/1.85	   quot_in_ggga(s(X), s(Y), Z, U) -> U2_ggga(X, Y, Z, U, quot_in_ggga(X, Y, Z, U))
4.21/1.85	   quot_in_ggga(X, 0, s(Z), s(U)) -> U3_ggga(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U))
4.21/1.85	   U3_ggga(X, Z, U, quot_out_ggga(X, s(Z), s(Z), U)) -> quot_out_ggga(X, 0, s(Z), s(U))
4.21/1.85	   U2_ggga(X, Y, Z, U, quot_out_ggga(X, Y, Z, U)) -> quot_out_ggga(s(X), s(Y), Z, U)
4.21/1.85	   U1_gga(X, Y, Z, quot_out_ggga(X, Y, Y, Z)) -> div_out_gga(X, Y, Z)
4.21/1.85	
4.21/1.85	The argument filtering Pi contains the following mapping:
4.21/1.85	div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
4.21/1.85	
4.21/1.85	U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
4.21/1.85	
4.21/1.85	quot_in_ggga(x1, x2, x3, x4)  =  quot_in_ggga(x1, x2, x3)
4.21/1.85	
4.21/1.85	0  =  0
4.21/1.85	
4.21/1.85	s(x1)  =  s(x1)
4.21/1.85	
4.21/1.85	quot_out_ggga(x1, x2, x3, x4)  =  quot_out_ggga(x1, x2, x3, x4)
4.21/1.85	
4.21/1.85	U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)
4.21/1.85	
4.21/1.85	U3_ggga(x1, x2, x3, x4)  =  U3_ggga(x1, x2, x4)
4.21/1.85	
4.21/1.85	div_out_gga(x1, x2, x3)  =  div_out_gga(x1, x2, x3)
4.21/1.85	
4.21/1.85	
4.21/1.85	
4.21/1.85	----------------------------------------
4.21/1.85	
4.21/1.85	(3) DependencyPairsProof (EQUIVALENT)
4.21/1.85	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
4.21/1.85	Pi DP problem:
4.21/1.85	The TRS P consists of the following rules:
4.21/1.85	
4.21/1.85	   DIV_IN_GGA(X, Y, Z) -> U1_GGA(X, Y, Z, quot_in_ggga(X, Y, Y, Z))
4.21/1.85	   DIV_IN_GGA(X, Y, Z) -> QUOT_IN_GGGA(X, Y, Y, Z)
4.21/1.85	   QUOT_IN_GGGA(s(X), s(Y), Z, U) -> U2_GGGA(X, Y, Z, U, quot_in_ggga(X, Y, Z, U))
4.21/1.85	   QUOT_IN_GGGA(s(X), s(Y), Z, U) -> QUOT_IN_GGGA(X, Y, Z, U)
4.21/1.85	   QUOT_IN_GGGA(X, 0, s(Z), s(U)) -> U3_GGGA(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U))
4.21/1.85	   QUOT_IN_GGGA(X, 0, s(Z), s(U)) -> QUOT_IN_GGGA(X, s(Z), s(Z), U)
4.21/1.85	
4.21/1.85	The TRS R consists of the following rules:
4.21/1.85	
4.21/1.85	   div_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, quot_in_ggga(X, Y, Y, Z))
4.21/1.85	   quot_in_ggga(0, s(Y), s(Z), 0) -> quot_out_ggga(0, s(Y), s(Z), 0)
4.21/1.85	   quot_in_ggga(s(X), s(Y), Z, U) -> U2_ggga(X, Y, Z, U, quot_in_ggga(X, Y, Z, U))
4.21/1.85	   quot_in_ggga(X, 0, s(Z), s(U)) -> U3_ggga(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U))
4.21/1.85	   U3_ggga(X, Z, U, quot_out_ggga(X, s(Z), s(Z), U)) -> quot_out_ggga(X, 0, s(Z), s(U))
4.21/1.85	   U2_ggga(X, Y, Z, U, quot_out_ggga(X, Y, Z, U)) -> quot_out_ggga(s(X), s(Y), Z, U)
4.21/1.85	   U1_gga(X, Y, Z, quot_out_ggga(X, Y, Y, Z)) -> div_out_gga(X, Y, Z)
4.21/1.85	
4.21/1.85	The argument filtering Pi contains the following mapping:
4.21/1.85	div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
4.21/1.85	
4.21/1.85	U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
4.21/1.85	
4.21/1.85	quot_in_ggga(x1, x2, x3, x4)  =  quot_in_ggga(x1, x2, x3)
4.21/1.85	
4.21/1.85	0  =  0
4.21/1.85	
4.21/1.85	s(x1)  =  s(x1)
4.21/1.85	
4.21/1.85	quot_out_ggga(x1, x2, x3, x4)  =  quot_out_ggga(x1, x2, x3, x4)
4.21/1.85	
4.21/1.85	U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)
4.21/1.85	
4.21/1.85	U3_ggga(x1, x2, x3, x4)  =  U3_ggga(x1, x2, x4)
4.21/1.85	
4.21/1.85	div_out_gga(x1, x2, x3)  =  div_out_gga(x1, x2, x3)
4.21/1.85	
4.21/1.85	DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
4.21/1.85	
4.21/1.85	U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
4.21/1.85	
4.21/1.85	QUOT_IN_GGGA(x1, x2, x3, x4)  =  QUOT_IN_GGGA(x1, x2, x3)
4.21/1.85	
4.21/1.85	U2_GGGA(x1, x2, x3, x4, x5)  =  U2_GGGA(x1, x2, x3, x5)
4.21/1.85	
4.21/1.85	U3_GGGA(x1, x2, x3, x4)  =  U3_GGGA(x1, x2, x4)
4.21/1.85	
4.21/1.85	
4.21/1.85	We have to consider all (P,R,Pi)-chains
4.21/1.85	----------------------------------------
4.21/1.85	
4.21/1.85	(4)
4.21/1.85	Obligation:
4.21/1.85	Pi DP problem:
4.21/1.85	The TRS P consists of the following rules:
4.21/1.85	
4.21/1.85	   DIV_IN_GGA(X, Y, Z) -> U1_GGA(X, Y, Z, quot_in_ggga(X, Y, Y, Z))
4.21/1.85	   DIV_IN_GGA(X, Y, Z) -> QUOT_IN_GGGA(X, Y, Y, Z)
4.21/1.85	   QUOT_IN_GGGA(s(X), s(Y), Z, U) -> U2_GGGA(X, Y, Z, U, quot_in_ggga(X, Y, Z, U))
4.21/1.85	   QUOT_IN_GGGA(s(X), s(Y), Z, U) -> QUOT_IN_GGGA(X, Y, Z, U)
4.21/1.85	   QUOT_IN_GGGA(X, 0, s(Z), s(U)) -> U3_GGGA(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U))
4.21/1.85	   QUOT_IN_GGGA(X, 0, s(Z), s(U)) -> QUOT_IN_GGGA(X, s(Z), s(Z), U)
4.21/1.85	
4.21/1.85	The TRS R consists of the following rules:
4.21/1.85	
4.21/1.85	   div_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, quot_in_ggga(X, Y, Y, Z))
4.21/1.85	   quot_in_ggga(0, s(Y), s(Z), 0) -> quot_out_ggga(0, s(Y), s(Z), 0)
4.21/1.85	   quot_in_ggga(s(X), s(Y), Z, U) -> U2_ggga(X, Y, Z, U, quot_in_ggga(X, Y, Z, U))
4.21/1.85	   quot_in_ggga(X, 0, s(Z), s(U)) -> U3_ggga(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U))
4.21/1.85	   U3_ggga(X, Z, U, quot_out_ggga(X, s(Z), s(Z), U)) -> quot_out_ggga(X, 0, s(Z), s(U))
4.21/1.85	   U2_ggga(X, Y, Z, U, quot_out_ggga(X, Y, Z, U)) -> quot_out_ggga(s(X), s(Y), Z, U)
4.21/1.85	   U1_gga(X, Y, Z, quot_out_ggga(X, Y, Y, Z)) -> div_out_gga(X, Y, Z)
4.21/1.85	
4.21/1.85	The argument filtering Pi contains the following mapping:
4.21/1.85	div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
4.21/1.85	
4.21/1.85	U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
4.21/1.85	
4.21/1.85	quot_in_ggga(x1, x2, x3, x4)  =  quot_in_ggga(x1, x2, x3)
4.21/1.85	
4.21/1.85	0  =  0
4.21/1.85	
4.21/1.85	s(x1)  =  s(x1)
4.21/1.85	
4.21/1.85	quot_out_ggga(x1, x2, x3, x4)  =  quot_out_ggga(x1, x2, x3, x4)
4.21/1.85	
4.21/1.85	U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)
4.21/1.85	
4.21/1.85	U3_ggga(x1, x2, x3, x4)  =  U3_ggga(x1, x2, x4)
4.21/1.85	
4.21/1.85	div_out_gga(x1, x2, x3)  =  div_out_gga(x1, x2, x3)
4.21/1.85	
4.21/1.85	DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
4.21/1.85	
4.21/1.85	U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
4.21/1.85	
4.21/1.85	QUOT_IN_GGGA(x1, x2, x3, x4)  =  QUOT_IN_GGGA(x1, x2, x3)
4.21/1.85	
4.21/1.85	U2_GGGA(x1, x2, x3, x4, x5)  =  U2_GGGA(x1, x2, x3, x5)
4.21/1.85	
4.21/1.85	U3_GGGA(x1, x2, x3, x4)  =  U3_GGGA(x1, x2, x4)
4.21/1.85	
4.21/1.85	
4.21/1.85	We have to consider all (P,R,Pi)-chains
4.21/1.85	----------------------------------------
4.21/1.85	
4.21/1.85	(5) DependencyGraphProof (EQUIVALENT)
4.21/1.85	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 4 less nodes.
4.21/1.85	----------------------------------------
4.21/1.85	
4.21/1.85	(6)
4.21/1.85	Obligation:
4.21/1.85	Pi DP problem:
4.21/1.85	The TRS P consists of the following rules:
4.21/1.85	
4.21/1.85	   QUOT_IN_GGGA(X, 0, s(Z), s(U)) -> QUOT_IN_GGGA(X, s(Z), s(Z), U)
4.21/1.85	   QUOT_IN_GGGA(s(X), s(Y), Z, U) -> QUOT_IN_GGGA(X, Y, Z, U)
4.21/1.85	
4.21/1.85	The TRS R consists of the following rules:
4.21/1.85	
4.21/1.85	   div_in_gga(X, Y, Z) -> U1_gga(X, Y, Z, quot_in_ggga(X, Y, Y, Z))
4.21/1.85	   quot_in_ggga(0, s(Y), s(Z), 0) -> quot_out_ggga(0, s(Y), s(Z), 0)
4.21/1.85	   quot_in_ggga(s(X), s(Y), Z, U) -> U2_ggga(X, Y, Z, U, quot_in_ggga(X, Y, Z, U))
4.21/1.85	   quot_in_ggga(X, 0, s(Z), s(U)) -> U3_ggga(X, Z, U, quot_in_ggga(X, s(Z), s(Z), U))
4.21/1.85	   U3_ggga(X, Z, U, quot_out_ggga(X, s(Z), s(Z), U)) -> quot_out_ggga(X, 0, s(Z), s(U))
4.21/1.85	   U2_ggga(X, Y, Z, U, quot_out_ggga(X, Y, Z, U)) -> quot_out_ggga(s(X), s(Y), Z, U)
4.21/1.85	   U1_gga(X, Y, Z, quot_out_ggga(X, Y, Y, Z)) -> div_out_gga(X, Y, Z)
4.21/1.85	
4.21/1.85	The argument filtering Pi contains the following mapping:
4.21/1.85	div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
4.21/1.85	
4.21/1.85	U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
4.21/1.85	
4.21/1.85	quot_in_ggga(x1, x2, x3, x4)  =  quot_in_ggga(x1, x2, x3)
4.21/1.85	
4.21/1.85	0  =  0
4.21/1.85	
4.21/1.85	s(x1)  =  s(x1)
4.21/1.85	
4.21/1.85	quot_out_ggga(x1, x2, x3, x4)  =  quot_out_ggga(x1, x2, x3, x4)
4.21/1.85	
4.21/1.85	U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x1, x2, x3, x5)
4.21/1.85	
4.21/1.85	U3_ggga(x1, x2, x3, x4)  =  U3_ggga(x1, x2, x4)
4.21/1.85	
4.21/1.85	div_out_gga(x1, x2, x3)  =  div_out_gga(x1, x2, x3)
4.21/1.85	
4.21/1.85	QUOT_IN_GGGA(x1, x2, x3, x4)  =  QUOT_IN_GGGA(x1, x2, x3)
4.21/1.85	
4.21/1.85	
4.21/1.85	We have to consider all (P,R,Pi)-chains
4.21/1.85	----------------------------------------
4.21/1.85	
4.21/1.85	(7) UsableRulesProof (EQUIVALENT)
4.21/1.85	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
4.21/1.85	----------------------------------------
4.21/1.85	
4.21/1.85	(8)
4.21/1.85	Obligation:
4.21/1.85	Pi DP problem:
4.21/1.85	The TRS P consists of the following rules:
4.21/1.85	
4.21/1.85	   QUOT_IN_GGGA(X, 0, s(Z), s(U)) -> QUOT_IN_GGGA(X, s(Z), s(Z), U)
4.21/1.85	   QUOT_IN_GGGA(s(X), s(Y), Z, U) -> QUOT_IN_GGGA(X, Y, Z, U)
4.21/1.85	
4.21/1.85	R is empty.
4.21/1.85	The argument filtering Pi contains the following mapping:
4.21/1.85	0  =  0
4.21/1.85	
4.21/1.85	s(x1)  =  s(x1)
4.21/1.85	
4.21/1.85	QUOT_IN_GGGA(x1, x2, x3, x4)  =  QUOT_IN_GGGA(x1, x2, x3)
4.21/1.85	
4.21/1.85	
4.21/1.85	We have to consider all (P,R,Pi)-chains
4.21/1.85	----------------------------------------
4.21/1.85	
4.21/1.85	(9) PiDPToQDPProof (SOUND)
4.21/1.85	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
4.21/1.85	----------------------------------------
4.21/1.85	
4.21/1.85	(10)
4.21/1.85	Obligation:
4.21/1.85	Q DP problem:
4.21/1.85	The TRS P consists of the following rules:
4.21/1.85	
4.21/1.85	   QUOT_IN_GGGA(X, 0, s(Z)) -> QUOT_IN_GGGA(X, s(Z), s(Z))
4.21/1.85	   QUOT_IN_GGGA(s(X), s(Y), Z) -> QUOT_IN_GGGA(X, Y, Z)
4.21/1.85	
4.21/1.85	R is empty.
4.21/1.85	Q is empty.
4.21/1.85	We have to consider all (P,Q,R)-chains.
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4.21/1.85	
4.21/1.85	(11) QDPSizeChangeProof (EQUIVALENT)
4.21/1.85	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. 
4.21/1.85	
4.21/1.85	From the DPs we obtained the following set of size-change graphs:
4.21/1.85	*QUOT_IN_GGGA(s(X), s(Y), Z) -> QUOT_IN_GGGA(X, Y, Z)
4.21/1.85	The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
4.21/1.85	
4.21/1.85	
4.21/1.85	*QUOT_IN_GGGA(X, 0, s(Z)) -> QUOT_IN_GGGA(X, s(Z), s(Z))
4.21/1.85	The graph contains the following edges 1 >= 1, 3 >= 2, 3 >= 3
4.21/1.85	
4.21/1.85	
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4.21/1.85	
4.21/1.85	(12)
4.21/1.85	YES
4.21/1.88	EOF