3.84/1.80	YES
3.84/1.82	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.84/1.82	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.84/1.82	
3.84/1.82	
3.84/1.82	Left Termination of the query pattern
3.84/1.82	
3.84/1.82	mult(g,g,a)
3.84/1.82	
3.84/1.82	w.r.t. the given Prolog program could successfully be proven:
3.84/1.82	
3.84/1.82	(0) Prolog
3.84/1.82	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.84/1.82	(2) PiTRS
3.84/1.82	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.84/1.82	(4) PiDP
3.84/1.82	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.84/1.82	(6) AND
3.84/1.82	    (7) PiDP
3.84/1.82	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
3.84/1.82	        (9) PiDP
3.84/1.82	        (10) PiDPToQDPProof [SOUND, 0 ms]
3.84/1.82	        (11) QDP
3.84/1.82	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.84/1.82	        (13) YES
3.84/1.82	    (14) PiDP
3.84/1.82	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
3.84/1.82	        (16) PiDP
3.84/1.82	        (17) PiDPToQDPProof [SOUND, 0 ms]
3.84/1.82	        (18) QDP
3.84/1.82	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.84/1.82	        (20) YES
3.84/1.82	
3.84/1.82	
3.84/1.82	----------------------------------------
3.84/1.82	
3.84/1.82	(0)
3.84/1.82	Obligation:
3.84/1.82	Clauses:
3.84/1.82	
3.84/1.82	mult(0, Y, 0).
3.84/1.82	mult(s(X), Y, Z) :- ','(mult(X, Y, Z1), add(Z1, Y, Z)).
3.84/1.82	add(0, Y, Y).
3.84/1.82	add(s(X), Y, s(Z)) :- add(X, Y, Z).
3.84/1.82	
3.84/1.82	
3.84/1.82	Query: mult(g,g,a)
3.84/1.82	----------------------------------------
3.84/1.82	
3.84/1.82	(1) PrologToPiTRSProof (SOUND)
3.84/1.82	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.84/1.82	
3.84/1.82	mult_in_3: (b,b,f)
3.84/1.82	
3.84/1.82	add_in_3: (b,b,f)
3.84/1.82	
3.84/1.82	Transforming Prolog into the following Term Rewriting System:
3.84/1.82	
3.84/1.82	Pi-finite rewrite system:
3.84/1.82	The TRS R consists of the following rules:
3.84/1.82	
3.84/1.82	   mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0)
3.84/1.82	   mult_in_gga(s(X), Y, Z) -> U1_gga(X, Y, Z, mult_in_gga(X, Y, Z1))
3.84/1.82	   U1_gga(X, Y, Z, mult_out_gga(X, Y, Z1)) -> U2_gga(X, Y, Z, add_in_gga(Z1, Y, Z))
3.84/1.82	   add_in_gga(0, Y, Y) -> add_out_gga(0, Y, Y)
3.84/1.82	   add_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, add_in_gga(X, Y, Z))
3.84/1.82	   U3_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z))
3.84/1.82	   U2_gga(X, Y, Z, add_out_gga(Z1, Y, Z)) -> mult_out_gga(s(X), Y, Z)
3.84/1.82	
3.84/1.82	The argument filtering Pi contains the following mapping:
3.84/1.82	mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
3.84/1.82	
3.84/1.82	0  =  0
3.84/1.82	
3.84/1.82	mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
3.84/1.82	
3.84/1.82	s(x1)  =  s(x1)
3.84/1.82	
3.84/1.82	U1_gga(x1, x2, x3, x4)  =  U1_gga(x2, x4)
3.84/1.82	
3.84/1.82	U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
3.84/1.82	
3.84/1.82	add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
3.84/1.82	
3.84/1.82	add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
3.84/1.82	
3.84/1.82	U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
3.84/1.82	
3.84/1.82	
3.84/1.82	
3.84/1.82	
3.84/1.82	
3.84/1.82	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.84/1.82	
3.84/1.82	
3.84/1.82	
3.84/1.82	----------------------------------------
3.84/1.82	
3.84/1.82	(2)
3.84/1.82	Obligation:
3.84/1.82	Pi-finite rewrite system:
3.84/1.82	The TRS R consists of the following rules:
3.84/1.82	
3.84/1.82	   mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0)
3.84/1.82	   mult_in_gga(s(X), Y, Z) -> U1_gga(X, Y, Z, mult_in_gga(X, Y, Z1))
3.84/1.82	   U1_gga(X, Y, Z, mult_out_gga(X, Y, Z1)) -> U2_gga(X, Y, Z, add_in_gga(Z1, Y, Z))
3.84/1.82	   add_in_gga(0, Y, Y) -> add_out_gga(0, Y, Y)
3.84/1.82	   add_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, add_in_gga(X, Y, Z))
3.84/1.82	   U3_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z))
3.84/1.82	   U2_gga(X, Y, Z, add_out_gga(Z1, Y, Z)) -> mult_out_gga(s(X), Y, Z)
3.84/1.82	
3.84/1.82	The argument filtering Pi contains the following mapping:
3.84/1.82	mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
3.84/1.82	
3.84/1.82	0  =  0
3.84/1.82	
3.84/1.82	mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
3.84/1.82	
3.84/1.82	s(x1)  =  s(x1)
3.84/1.82	
3.84/1.82	U1_gga(x1, x2, x3, x4)  =  U1_gga(x2, x4)
3.84/1.82	
3.84/1.82	U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
3.84/1.82	
3.84/1.82	add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
3.84/1.82	
3.84/1.82	add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
3.84/1.82	
3.84/1.82	U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
3.84/1.82	
3.84/1.82	
3.84/1.82	
3.84/1.82	----------------------------------------
3.84/1.82	
3.84/1.82	(3) DependencyPairsProof (EQUIVALENT)
3.84/1.82	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.84/1.82	Pi DP problem:
3.84/1.82	The TRS P consists of the following rules:
3.84/1.82	
3.84/1.82	   MULT_IN_GGA(s(X), Y, Z) -> U1_GGA(X, Y, Z, mult_in_gga(X, Y, Z1))
3.84/1.82	   MULT_IN_GGA(s(X), Y, Z) -> MULT_IN_GGA(X, Y, Z1)
3.84/1.82	   U1_GGA(X, Y, Z, mult_out_gga(X, Y, Z1)) -> U2_GGA(X, Y, Z, add_in_gga(Z1, Y, Z))
3.84/1.82	   U1_GGA(X, Y, Z, mult_out_gga(X, Y, Z1)) -> ADD_IN_GGA(Z1, Y, Z)
3.84/1.82	   ADD_IN_GGA(s(X), Y, s(Z)) -> U3_GGA(X, Y, Z, add_in_gga(X, Y, Z))
3.84/1.82	   ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z)
3.84/1.82	
3.84/1.82	The TRS R consists of the following rules:
3.84/1.82	
3.84/1.82	   mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0)
3.84/1.82	   mult_in_gga(s(X), Y, Z) -> U1_gga(X, Y, Z, mult_in_gga(X, Y, Z1))
3.84/1.82	   U1_gga(X, Y, Z, mult_out_gga(X, Y, Z1)) -> U2_gga(X, Y, Z, add_in_gga(Z1, Y, Z))
3.84/1.82	   add_in_gga(0, Y, Y) -> add_out_gga(0, Y, Y)
3.84/1.82	   add_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, add_in_gga(X, Y, Z))
3.84/1.82	   U3_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z))
3.84/1.82	   U2_gga(X, Y, Z, add_out_gga(Z1, Y, Z)) -> mult_out_gga(s(X), Y, Z)
3.84/1.82	
3.84/1.82	The argument filtering Pi contains the following mapping:
3.84/1.82	mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
3.84/1.82	
3.84/1.82	0  =  0
3.84/1.82	
3.84/1.82	mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
3.84/1.82	
3.84/1.82	s(x1)  =  s(x1)
3.84/1.82	
3.84/1.82	U1_gga(x1, x2, x3, x4)  =  U1_gga(x2, x4)
3.84/1.82	
3.84/1.82	U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
3.84/1.82	
3.84/1.82	add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
3.84/1.82	
3.84/1.82	add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
3.84/1.82	
3.84/1.82	U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
3.84/1.82	
3.84/1.82	MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)
3.84/1.82	
3.84/1.82	U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x2, x4)
3.84/1.82	
3.84/1.82	U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x4)
3.84/1.82	
3.84/1.82	ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
3.84/1.82	
3.84/1.82	U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
3.84/1.82	
3.84/1.82	
3.84/1.82	We have to consider all (P,R,Pi)-chains
3.84/1.82	----------------------------------------
3.84/1.82	
3.84/1.82	(4)
3.84/1.82	Obligation:
3.84/1.82	Pi DP problem:
3.84/1.82	The TRS P consists of the following rules:
3.84/1.82	
3.84/1.82	   MULT_IN_GGA(s(X), Y, Z) -> U1_GGA(X, Y, Z, mult_in_gga(X, Y, Z1))
3.84/1.82	   MULT_IN_GGA(s(X), Y, Z) -> MULT_IN_GGA(X, Y, Z1)
3.84/1.82	   U1_GGA(X, Y, Z, mult_out_gga(X, Y, Z1)) -> U2_GGA(X, Y, Z, add_in_gga(Z1, Y, Z))
3.84/1.82	   U1_GGA(X, Y, Z, mult_out_gga(X, Y, Z1)) -> ADD_IN_GGA(Z1, Y, Z)
3.84/1.82	   ADD_IN_GGA(s(X), Y, s(Z)) -> U3_GGA(X, Y, Z, add_in_gga(X, Y, Z))
3.84/1.82	   ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z)
3.84/1.82	
3.84/1.82	The TRS R consists of the following rules:
3.84/1.82	
3.84/1.82	   mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0)
3.84/1.82	   mult_in_gga(s(X), Y, Z) -> U1_gga(X, Y, Z, mult_in_gga(X, Y, Z1))
3.84/1.82	   U1_gga(X, Y, Z, mult_out_gga(X, Y, Z1)) -> U2_gga(X, Y, Z, add_in_gga(Z1, Y, Z))
3.84/1.82	   add_in_gga(0, Y, Y) -> add_out_gga(0, Y, Y)
3.84/1.82	   add_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, add_in_gga(X, Y, Z))
3.84/1.82	   U3_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z))
3.84/1.82	   U2_gga(X, Y, Z, add_out_gga(Z1, Y, Z)) -> mult_out_gga(s(X), Y, Z)
3.84/1.82	
3.84/1.82	The argument filtering Pi contains the following mapping:
3.84/1.82	mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
3.84/1.82	
3.84/1.82	0  =  0
3.84/1.82	
3.84/1.82	mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
3.84/1.82	
3.84/1.82	s(x1)  =  s(x1)
3.84/1.82	
3.84/1.82	U1_gga(x1, x2, x3, x4)  =  U1_gga(x2, x4)
3.84/1.82	
3.84/1.82	U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
3.84/1.82	
3.84/1.82	add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
3.84/1.82	
3.84/1.82	add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
3.84/1.82	
3.84/1.82	U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
3.84/1.82	
3.84/1.82	MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)
3.84/1.82	
3.84/1.82	U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x2, x4)
3.84/1.82	
3.84/1.82	U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x4)
3.84/1.82	
3.84/1.82	ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
3.84/1.82	
3.84/1.82	U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
3.84/1.82	
3.84/1.82	
3.84/1.82	We have to consider all (P,R,Pi)-chains
3.84/1.82	----------------------------------------
3.84/1.82	
3.84/1.82	(5) DependencyGraphProof (EQUIVALENT)
3.84/1.82	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes.
3.84/1.82	----------------------------------------
3.84/1.82	
3.84/1.82	(6)
3.84/1.82	Complex Obligation (AND)
3.84/1.82	
3.84/1.82	----------------------------------------
3.84/1.82	
3.84/1.82	(7)
3.84/1.82	Obligation:
3.84/1.82	Pi DP problem:
3.84/1.82	The TRS P consists of the following rules:
3.84/1.82	
3.84/1.82	   ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z)
3.84/1.82	
3.84/1.82	The TRS R consists of the following rules:
3.84/1.82	
3.84/1.82	   mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0)
3.84/1.82	   mult_in_gga(s(X), Y, Z) -> U1_gga(X, Y, Z, mult_in_gga(X, Y, Z1))
3.84/1.82	   U1_gga(X, Y, Z, mult_out_gga(X, Y, Z1)) -> U2_gga(X, Y, Z, add_in_gga(Z1, Y, Z))
3.84/1.82	   add_in_gga(0, Y, Y) -> add_out_gga(0, Y, Y)
3.84/1.82	   add_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, add_in_gga(X, Y, Z))
3.84/1.82	   U3_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z))
3.84/1.82	   U2_gga(X, Y, Z, add_out_gga(Z1, Y, Z)) -> mult_out_gga(s(X), Y, Z)
3.84/1.82	
3.84/1.82	The argument filtering Pi contains the following mapping:
3.84/1.82	mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
3.84/1.82	
3.84/1.82	0  =  0
3.84/1.82	
3.84/1.82	mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
3.84/1.82	
3.84/1.82	s(x1)  =  s(x1)
3.84/1.82	
3.84/1.82	U1_gga(x1, x2, x3, x4)  =  U1_gga(x2, x4)
3.84/1.82	
3.84/1.82	U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
3.84/1.82	
3.84/1.82	add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
3.84/1.82	
3.84/1.82	add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
3.84/1.82	
3.84/1.82	U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
3.84/1.82	
3.84/1.82	ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
3.84/1.82	
3.84/1.82	
3.84/1.82	We have to consider all (P,R,Pi)-chains
3.84/1.82	----------------------------------------
3.84/1.82	
3.84/1.82	(8) UsableRulesProof (EQUIVALENT)
3.84/1.82	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.84/1.82	----------------------------------------
3.84/1.82	
3.84/1.82	(9)
3.84/1.82	Obligation:
3.84/1.82	Pi DP problem:
3.84/1.82	The TRS P consists of the following rules:
3.84/1.82	
3.84/1.82	   ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z)
3.84/1.82	
3.84/1.82	R is empty.
3.84/1.82	The argument filtering Pi contains the following mapping:
3.84/1.82	s(x1)  =  s(x1)
3.84/1.82	
3.84/1.82	ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
3.84/1.82	
3.84/1.82	
3.84/1.82	We have to consider all (P,R,Pi)-chains
3.84/1.82	----------------------------------------
3.84/1.82	
3.84/1.82	(10) PiDPToQDPProof (SOUND)
3.84/1.82	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.84/1.82	----------------------------------------
3.84/1.82	
3.84/1.82	(11)
3.84/1.82	Obligation:
3.84/1.82	Q DP problem:
3.84/1.82	The TRS P consists of the following rules:
3.84/1.82	
3.84/1.82	   ADD_IN_GGA(s(X), Y) -> ADD_IN_GGA(X, Y)
3.84/1.82	
3.84/1.82	R is empty.
3.84/1.82	Q is empty.
3.84/1.82	We have to consider all (P,Q,R)-chains.
3.84/1.82	----------------------------------------
3.84/1.82	
3.84/1.82	(12) QDPSizeChangeProof (EQUIVALENT)
3.84/1.82	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. 
3.84/1.82	
3.84/1.82	From the DPs we obtained the following set of size-change graphs:
3.84/1.82	*ADD_IN_GGA(s(X), Y) -> ADD_IN_GGA(X, Y)
3.84/1.82	The graph contains the following edges 1 > 1, 2 >= 2
3.84/1.82	
3.84/1.82	
3.84/1.82	----------------------------------------
3.84/1.82	
3.84/1.82	(13)
3.84/1.82	YES
3.84/1.82	
3.84/1.82	----------------------------------------
3.84/1.82	
3.84/1.82	(14)
3.84/1.82	Obligation:
3.84/1.82	Pi DP problem:
3.84/1.82	The TRS P consists of the following rules:
3.84/1.82	
3.84/1.82	   MULT_IN_GGA(s(X), Y, Z) -> MULT_IN_GGA(X, Y, Z1)
3.84/1.82	
3.84/1.82	The TRS R consists of the following rules:
3.84/1.82	
3.84/1.82	   mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0)
3.84/1.82	   mult_in_gga(s(X), Y, Z) -> U1_gga(X, Y, Z, mult_in_gga(X, Y, Z1))
3.84/1.82	   U1_gga(X, Y, Z, mult_out_gga(X, Y, Z1)) -> U2_gga(X, Y, Z, add_in_gga(Z1, Y, Z))
3.84/1.82	   add_in_gga(0, Y, Y) -> add_out_gga(0, Y, Y)
3.84/1.82	   add_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, add_in_gga(X, Y, Z))
3.84/1.82	   U3_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z))
3.84/1.82	   U2_gga(X, Y, Z, add_out_gga(Z1, Y, Z)) -> mult_out_gga(s(X), Y, Z)
3.84/1.82	
3.84/1.82	The argument filtering Pi contains the following mapping:
3.84/1.82	mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)
3.84/1.82	
3.84/1.82	0  =  0
3.84/1.82	
3.84/1.82	mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)
3.84/1.82	
3.84/1.82	s(x1)  =  s(x1)
3.84/1.82	
3.84/1.82	U1_gga(x1, x2, x3, x4)  =  U1_gga(x2, x4)
3.84/1.82	
3.84/1.82	U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
3.84/1.82	
3.84/1.82	add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
3.84/1.82	
3.84/1.82	add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
3.84/1.82	
3.84/1.82	U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
3.84/1.82	
3.84/1.82	MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)
3.84/1.82	
3.84/1.82	
3.84/1.82	We have to consider all (P,R,Pi)-chains
3.84/1.82	----------------------------------------
3.84/1.82	
3.84/1.82	(15) UsableRulesProof (EQUIVALENT)
3.84/1.82	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.84/1.82	----------------------------------------
3.84/1.82	
3.84/1.82	(16)
3.84/1.82	Obligation:
3.84/1.82	Pi DP problem:
3.84/1.82	The TRS P consists of the following rules:
3.84/1.82	
3.84/1.82	   MULT_IN_GGA(s(X), Y, Z) -> MULT_IN_GGA(X, Y, Z1)
3.84/1.82	
3.84/1.82	R is empty.
3.84/1.82	The argument filtering Pi contains the following mapping:
3.84/1.82	s(x1)  =  s(x1)
3.84/1.82	
3.84/1.82	MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)
3.84/1.82	
3.84/1.82	
3.84/1.82	We have to consider all (P,R,Pi)-chains
3.84/1.82	----------------------------------------
3.84/1.82	
3.84/1.82	(17) PiDPToQDPProof (SOUND)
3.84/1.82	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.84/1.82	----------------------------------------
3.84/1.82	
3.84/1.82	(18)
3.84/1.82	Obligation:
3.84/1.82	Q DP problem:
3.84/1.82	The TRS P consists of the following rules:
3.84/1.82	
3.84/1.82	   MULT_IN_GGA(s(X), Y) -> MULT_IN_GGA(X, Y)
3.84/1.82	
3.84/1.82	R is empty.
3.84/1.82	Q is empty.
3.84/1.82	We have to consider all (P,Q,R)-chains.
3.84/1.82	----------------------------------------
3.84/1.82	
3.84/1.82	(19) QDPSizeChangeProof (EQUIVALENT)
3.84/1.82	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. 
3.84/1.82	
3.84/1.82	From the DPs we obtained the following set of size-change graphs:
3.84/1.82	*MULT_IN_GGA(s(X), Y) -> MULT_IN_GGA(X, Y)
3.84/1.82	The graph contains the following edges 1 > 1, 2 >= 2
3.84/1.82	
3.84/1.82	
3.84/1.82	----------------------------------------
3.84/1.82	
3.84/1.82	(20)
3.84/1.82	YES
3.84/1.85	EOF