6.00/2.48	YES
6.00/2.50	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
6.00/2.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.00/2.50	
6.00/2.50	
6.00/2.50	Left Termination of the query pattern
6.00/2.50	
6.00/2.50	div(g,g,a)
6.00/2.50	
6.00/2.50	w.r.t. the given Prolog program could successfully be proven:
6.00/2.50	
6.00/2.50	(0) Prolog
6.00/2.50	(1) PrologToPiTRSProof [SOUND, 0 ms]
6.00/2.50	(2) PiTRS
6.00/2.50	(3) DependencyPairsProof [EQUIVALENT, 17 ms]
6.00/2.50	(4) PiDP
6.00/2.50	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
6.00/2.50	(6) AND
6.00/2.50	    (7) PiDP
6.00/2.50	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
6.00/2.50	        (9) PiDP
6.00/2.50	        (10) PiDPToQDPProof [SOUND, 0 ms]
6.00/2.50	        (11) QDP
6.00/2.50	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.00/2.50	        (13) YES
6.00/2.50	    (14) PiDP
6.00/2.50	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
6.00/2.50	        (16) PiDP
6.00/2.50	        (17) PiDPToQDPProof [EQUIVALENT, 0 ms]
6.00/2.50	        (18) QDP
6.00/2.50	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.00/2.50	        (20) YES
6.00/2.50	    (21) PiDP
6.00/2.50	        (22) UsableRulesProof [EQUIVALENT, 0 ms]
6.00/2.50	        (23) PiDP
6.00/2.50	        (24) PiDPToQDPProof [SOUND, 0 ms]
6.00/2.50	        (25) QDP
6.00/2.50	        (26) QDPOrderProof [EQUIVALENT, 25 ms]
6.00/2.50	        (27) QDP
6.00/2.50	        (28) PisEmptyProof [EQUIVALENT, 0 ms]
6.00/2.50	        (29) YES
6.00/2.50	
6.00/2.50	
6.00/2.50	----------------------------------------
6.00/2.50	
6.00/2.50	(0)
6.00/2.50	Obligation:
6.00/2.50	Clauses:
6.00/2.50	
6.00/2.50	div(X, s(Y), Z) :- div_s(X, Y, Z).
6.00/2.50	div_s(0, Y, 0).
6.00/2.50	div_s(s(X), Y, 0) :- lss(X, Y).
6.00/2.50	div_s(s(X), Y, s(Z)) :- ','(sub(X, Y, R), div_s(R, Y, Z)).
6.00/2.50	lss(s(X), s(Y)) :- lss(X, Y).
6.00/2.50	lss(0, s(Y)).
6.00/2.50	sub(s(X), s(Y), Z) :- sub(X, Y, Z).
6.00/2.50	sub(X, 0, X).
6.00/2.50	
6.00/2.50	
6.00/2.50	Query: div(g,g,a)
6.00/2.50	----------------------------------------
6.00/2.50	
6.00/2.50	(1) PrologToPiTRSProof (SOUND)
6.00/2.50	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.00/2.50	
6.00/2.50	div_in_3: (b,b,f)
6.00/2.50	
6.00/2.50	div_s_in_3: (b,b,f)
6.00/2.50	
6.00/2.50	lss_in_2: (b,b)
6.00/2.50	
6.00/2.50	sub_in_3: (b,b,f)
6.00/2.50	
6.00/2.50	Transforming Prolog into the following Term Rewriting System:
6.00/2.50	
6.00/2.50	Pi-finite rewrite system:
6.00/2.50	The TRS R consists of the following rules:
6.00/2.50	
6.00/2.50	   div_in_gga(X, s(Y), Z) -> U1_gga(X, Y, Z, div_s_in_gga(X, Y, Z))
6.00/2.50	   div_s_in_gga(0, Y, 0) -> div_s_out_gga(0, Y, 0)
6.00/2.50	   div_s_in_gga(s(X), Y, 0) -> U2_gga(X, Y, lss_in_gg(X, Y))
6.00/2.50	   lss_in_gg(s(X), s(Y)) -> U5_gg(X, Y, lss_in_gg(X, Y))
6.00/2.50	   lss_in_gg(0, s(Y)) -> lss_out_gg(0, s(Y))
6.00/2.50	   U5_gg(X, Y, lss_out_gg(X, Y)) -> lss_out_gg(s(X), s(Y))
6.00/2.50	   U2_gga(X, Y, lss_out_gg(X, Y)) -> div_s_out_gga(s(X), Y, 0)
6.00/2.50	   div_s_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, sub_in_gga(X, Y, R))
6.00/2.50	   sub_in_gga(s(X), s(Y), Z) -> U6_gga(X, Y, Z, sub_in_gga(X, Y, Z))
6.00/2.50	   sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X)
6.00/2.50	   U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z)
6.00/2.50	   U3_gga(X, Y, Z, sub_out_gga(X, Y, R)) -> U4_gga(X, Y, Z, div_s_in_gga(R, Y, Z))
6.00/2.50	   U4_gga(X, Y, Z, div_s_out_gga(R, Y, Z)) -> div_s_out_gga(s(X), Y, s(Z))
6.00/2.50	   U1_gga(X, Y, Z, div_s_out_gga(X, Y, Z)) -> div_out_gga(X, s(Y), Z)
6.00/2.50	
6.00/2.50	The argument filtering Pi contains the following mapping:
6.00/2.50	div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
6.00/2.50	
6.00/2.50	s(x1)  =  s(x1)
6.00/2.50	
6.00/2.50	U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
6.00/2.50	
6.00/2.50	div_s_in_gga(x1, x2, x3)  =  div_s_in_gga(x1, x2)
6.00/2.50	
6.00/2.50	0  =  0
6.00/2.50	
6.00/2.50	div_s_out_gga(x1, x2, x3)  =  div_s_out_gga(x3)
6.00/2.50	
6.00/2.50	U2_gga(x1, x2, x3)  =  U2_gga(x3)
6.00/2.50	
6.00/2.50	lss_in_gg(x1, x2)  =  lss_in_gg(x1, x2)
6.00/2.50	
6.00/2.50	U5_gg(x1, x2, x3)  =  U5_gg(x3)
6.00/2.50	
6.00/2.50	lss_out_gg(x1, x2)  =  lss_out_gg
6.00/2.50	
6.00/2.50	U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
6.00/2.50	
6.00/2.50	sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
6.00/2.50	
6.00/2.50	U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
6.00/2.50	
6.00/2.50	sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
6.00/2.50	
6.00/2.50	U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
6.00/2.50	
6.00/2.50	div_out_gga(x1, x2, x3)  =  div_out_gga(x3)
6.00/2.50	
6.00/2.50	
6.00/2.50	
6.00/2.50	
6.00/2.50	
6.00/2.50	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.00/2.50	
6.00/2.50	
6.00/2.50	
6.00/2.50	----------------------------------------
6.00/2.50	
6.00/2.50	(2)
6.00/2.50	Obligation:
6.00/2.50	Pi-finite rewrite system:
6.00/2.50	The TRS R consists of the following rules:
6.00/2.50	
6.00/2.50	   div_in_gga(X, s(Y), Z) -> U1_gga(X, Y, Z, div_s_in_gga(X, Y, Z))
6.00/2.50	   div_s_in_gga(0, Y, 0) -> div_s_out_gga(0, Y, 0)
6.00/2.50	   div_s_in_gga(s(X), Y, 0) -> U2_gga(X, Y, lss_in_gg(X, Y))
6.00/2.50	   lss_in_gg(s(X), s(Y)) -> U5_gg(X, Y, lss_in_gg(X, Y))
6.00/2.50	   lss_in_gg(0, s(Y)) -> lss_out_gg(0, s(Y))
6.00/2.50	   U5_gg(X, Y, lss_out_gg(X, Y)) -> lss_out_gg(s(X), s(Y))
6.00/2.50	   U2_gga(X, Y, lss_out_gg(X, Y)) -> div_s_out_gga(s(X), Y, 0)
6.00/2.50	   div_s_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, sub_in_gga(X, Y, R))
6.00/2.51	   sub_in_gga(s(X), s(Y), Z) -> U6_gga(X, Y, Z, sub_in_gga(X, Y, Z))
6.00/2.51	   sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X)
6.00/2.51	   U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z)
6.00/2.51	   U3_gga(X, Y, Z, sub_out_gga(X, Y, R)) -> U4_gga(X, Y, Z, div_s_in_gga(R, Y, Z))
6.00/2.51	   U4_gga(X, Y, Z, div_s_out_gga(R, Y, Z)) -> div_s_out_gga(s(X), Y, s(Z))
6.00/2.51	   U1_gga(X, Y, Z, div_s_out_gga(X, Y, Z)) -> div_out_gga(X, s(Y), Z)
6.00/2.51	
6.00/2.51	The argument filtering Pi contains the following mapping:
6.00/2.51	div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
6.00/2.51	
6.00/2.51	s(x1)  =  s(x1)
6.00/2.51	
6.00/2.51	U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
6.00/2.51	
6.00/2.51	div_s_in_gga(x1, x2, x3)  =  div_s_in_gga(x1, x2)
6.00/2.51	
6.00/2.51	0  =  0
6.00/2.51	
6.00/2.51	div_s_out_gga(x1, x2, x3)  =  div_s_out_gga(x3)
6.00/2.51	
6.00/2.51	U2_gga(x1, x2, x3)  =  U2_gga(x3)
6.00/2.51	
6.00/2.51	lss_in_gg(x1, x2)  =  lss_in_gg(x1, x2)
6.00/2.51	
6.00/2.51	U5_gg(x1, x2, x3)  =  U5_gg(x3)
6.00/2.51	
6.00/2.51	lss_out_gg(x1, x2)  =  lss_out_gg
6.00/2.51	
6.00/2.51	U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
6.00/2.51	
6.00/2.51	sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
6.00/2.51	
6.00/2.51	U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
6.00/2.51	
6.00/2.51	sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
6.00/2.51	
6.00/2.51	U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
6.00/2.51	
6.00/2.51	div_out_gga(x1, x2, x3)  =  div_out_gga(x3)
6.00/2.51	
6.00/2.51	
6.00/2.51	
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(3) DependencyPairsProof (EQUIVALENT)
6.00/2.51	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.00/2.51	Pi DP problem:
6.00/2.51	The TRS P consists of the following rules:
6.00/2.51	
6.00/2.51	   DIV_IN_GGA(X, s(Y), Z) -> U1_GGA(X, Y, Z, div_s_in_gga(X, Y, Z))
6.00/2.51	   DIV_IN_GGA(X, s(Y), Z) -> DIV_S_IN_GGA(X, Y, Z)
6.00/2.51	   DIV_S_IN_GGA(s(X), Y, 0) -> U2_GGA(X, Y, lss_in_gg(X, Y))
6.00/2.51	   DIV_S_IN_GGA(s(X), Y, 0) -> LSS_IN_GG(X, Y)
6.00/2.51	   LSS_IN_GG(s(X), s(Y)) -> U5_GG(X, Y, lss_in_gg(X, Y))
6.00/2.51	   LSS_IN_GG(s(X), s(Y)) -> LSS_IN_GG(X, Y)
6.00/2.51	   DIV_S_IN_GGA(s(X), Y, s(Z)) -> U3_GGA(X, Y, Z, sub_in_gga(X, Y, R))
6.00/2.51	   DIV_S_IN_GGA(s(X), Y, s(Z)) -> SUB_IN_GGA(X, Y, R)
6.00/2.51	   SUB_IN_GGA(s(X), s(Y), Z) -> U6_GGA(X, Y, Z, sub_in_gga(X, Y, Z))
6.00/2.51	   SUB_IN_GGA(s(X), s(Y), Z) -> SUB_IN_GGA(X, Y, Z)
6.00/2.51	   U3_GGA(X, Y, Z, sub_out_gga(X, Y, R)) -> U4_GGA(X, Y, Z, div_s_in_gga(R, Y, Z))
6.00/2.51	   U3_GGA(X, Y, Z, sub_out_gga(X, Y, R)) -> DIV_S_IN_GGA(R, Y, Z)
6.00/2.51	
6.00/2.51	The TRS R consists of the following rules:
6.00/2.51	
6.00/2.51	   div_in_gga(X, s(Y), Z) -> U1_gga(X, Y, Z, div_s_in_gga(X, Y, Z))
6.00/2.51	   div_s_in_gga(0, Y, 0) -> div_s_out_gga(0, Y, 0)
6.00/2.51	   div_s_in_gga(s(X), Y, 0) -> U2_gga(X, Y, lss_in_gg(X, Y))
6.00/2.51	   lss_in_gg(s(X), s(Y)) -> U5_gg(X, Y, lss_in_gg(X, Y))
6.00/2.51	   lss_in_gg(0, s(Y)) -> lss_out_gg(0, s(Y))
6.00/2.51	   U5_gg(X, Y, lss_out_gg(X, Y)) -> lss_out_gg(s(X), s(Y))
6.00/2.51	   U2_gga(X, Y, lss_out_gg(X, Y)) -> div_s_out_gga(s(X), Y, 0)
6.00/2.51	   div_s_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, sub_in_gga(X, Y, R))
6.00/2.51	   sub_in_gga(s(X), s(Y), Z) -> U6_gga(X, Y, Z, sub_in_gga(X, Y, Z))
6.00/2.51	   sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X)
6.00/2.51	   U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z)
6.00/2.51	   U3_gga(X, Y, Z, sub_out_gga(X, Y, R)) -> U4_gga(X, Y, Z, div_s_in_gga(R, Y, Z))
6.00/2.51	   U4_gga(X, Y, Z, div_s_out_gga(R, Y, Z)) -> div_s_out_gga(s(X), Y, s(Z))
6.00/2.51	   U1_gga(X, Y, Z, div_s_out_gga(X, Y, Z)) -> div_out_gga(X, s(Y), Z)
6.00/2.51	
6.00/2.51	The argument filtering Pi contains the following mapping:
6.00/2.51	div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
6.00/2.51	
6.00/2.51	s(x1)  =  s(x1)
6.00/2.51	
6.00/2.51	U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
6.00/2.51	
6.00/2.51	div_s_in_gga(x1, x2, x3)  =  div_s_in_gga(x1, x2)
6.00/2.51	
6.00/2.51	0  =  0
6.00/2.51	
6.00/2.51	div_s_out_gga(x1, x2, x3)  =  div_s_out_gga(x3)
6.00/2.51	
6.00/2.51	U2_gga(x1, x2, x3)  =  U2_gga(x3)
6.00/2.51	
6.00/2.51	lss_in_gg(x1, x2)  =  lss_in_gg(x1, x2)
6.00/2.51	
6.00/2.51	U5_gg(x1, x2, x3)  =  U5_gg(x3)
6.00/2.51	
6.00/2.51	lss_out_gg(x1, x2)  =  lss_out_gg
6.00/2.51	
6.00/2.51	U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
6.00/2.51	
6.00/2.51	sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
6.00/2.51	
6.00/2.51	U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
6.00/2.51	
6.00/2.51	sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
6.00/2.51	
6.00/2.51	U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
6.00/2.51	
6.00/2.51	div_out_gga(x1, x2, x3)  =  div_out_gga(x3)
6.00/2.51	
6.00/2.51	DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
6.00/2.51	
6.00/2.51	U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
6.00/2.51	
6.00/2.51	DIV_S_IN_GGA(x1, x2, x3)  =  DIV_S_IN_GGA(x1, x2)
6.00/2.51	
6.00/2.51	U2_GGA(x1, x2, x3)  =  U2_GGA(x3)
6.00/2.51	
6.00/2.51	LSS_IN_GG(x1, x2)  =  LSS_IN_GG(x1, x2)
6.00/2.51	
6.00/2.51	U5_GG(x1, x2, x3)  =  U5_GG(x3)
6.00/2.51	
6.00/2.51	U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x2, x4)
6.00/2.51	
6.00/2.51	SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)
6.00/2.51	
6.00/2.51	U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x4)
6.00/2.51	
6.00/2.51	U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x4)
6.00/2.51	
6.00/2.51	
6.00/2.51	We have to consider all (P,R,Pi)-chains
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(4)
6.00/2.51	Obligation:
6.00/2.51	Pi DP problem:
6.00/2.51	The TRS P consists of the following rules:
6.00/2.51	
6.00/2.51	   DIV_IN_GGA(X, s(Y), Z) -> U1_GGA(X, Y, Z, div_s_in_gga(X, Y, Z))
6.00/2.51	   DIV_IN_GGA(X, s(Y), Z) -> DIV_S_IN_GGA(X, Y, Z)
6.00/2.51	   DIV_S_IN_GGA(s(X), Y, 0) -> U2_GGA(X, Y, lss_in_gg(X, Y))
6.00/2.51	   DIV_S_IN_GGA(s(X), Y, 0) -> LSS_IN_GG(X, Y)
6.00/2.51	   LSS_IN_GG(s(X), s(Y)) -> U5_GG(X, Y, lss_in_gg(X, Y))
6.00/2.51	   LSS_IN_GG(s(X), s(Y)) -> LSS_IN_GG(X, Y)
6.00/2.51	   DIV_S_IN_GGA(s(X), Y, s(Z)) -> U3_GGA(X, Y, Z, sub_in_gga(X, Y, R))
6.00/2.51	   DIV_S_IN_GGA(s(X), Y, s(Z)) -> SUB_IN_GGA(X, Y, R)
6.00/2.51	   SUB_IN_GGA(s(X), s(Y), Z) -> U6_GGA(X, Y, Z, sub_in_gga(X, Y, Z))
6.00/2.51	   SUB_IN_GGA(s(X), s(Y), Z) -> SUB_IN_GGA(X, Y, Z)
6.00/2.51	   U3_GGA(X, Y, Z, sub_out_gga(X, Y, R)) -> U4_GGA(X, Y, Z, div_s_in_gga(R, Y, Z))
6.00/2.51	   U3_GGA(X, Y, Z, sub_out_gga(X, Y, R)) -> DIV_S_IN_GGA(R, Y, Z)
6.00/2.51	
6.00/2.51	The TRS R consists of the following rules:
6.00/2.51	
6.00/2.51	   div_in_gga(X, s(Y), Z) -> U1_gga(X, Y, Z, div_s_in_gga(X, Y, Z))
6.00/2.51	   div_s_in_gga(0, Y, 0) -> div_s_out_gga(0, Y, 0)
6.00/2.51	   div_s_in_gga(s(X), Y, 0) -> U2_gga(X, Y, lss_in_gg(X, Y))
6.00/2.51	   lss_in_gg(s(X), s(Y)) -> U5_gg(X, Y, lss_in_gg(X, Y))
6.00/2.51	   lss_in_gg(0, s(Y)) -> lss_out_gg(0, s(Y))
6.00/2.51	   U5_gg(X, Y, lss_out_gg(X, Y)) -> lss_out_gg(s(X), s(Y))
6.00/2.51	   U2_gga(X, Y, lss_out_gg(X, Y)) -> div_s_out_gga(s(X), Y, 0)
6.00/2.51	   div_s_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, sub_in_gga(X, Y, R))
6.00/2.51	   sub_in_gga(s(X), s(Y), Z) -> U6_gga(X, Y, Z, sub_in_gga(X, Y, Z))
6.00/2.51	   sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X)
6.00/2.51	   U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z)
6.00/2.51	   U3_gga(X, Y, Z, sub_out_gga(X, Y, R)) -> U4_gga(X, Y, Z, div_s_in_gga(R, Y, Z))
6.00/2.51	   U4_gga(X, Y, Z, div_s_out_gga(R, Y, Z)) -> div_s_out_gga(s(X), Y, s(Z))
6.00/2.51	   U1_gga(X, Y, Z, div_s_out_gga(X, Y, Z)) -> div_out_gga(X, s(Y), Z)
6.00/2.51	
6.00/2.51	The argument filtering Pi contains the following mapping:
6.00/2.51	div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
6.00/2.51	
6.00/2.51	s(x1)  =  s(x1)
6.00/2.51	
6.00/2.51	U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
6.00/2.51	
6.00/2.51	div_s_in_gga(x1, x2, x3)  =  div_s_in_gga(x1, x2)
6.00/2.51	
6.00/2.51	0  =  0
6.00/2.51	
6.00/2.51	div_s_out_gga(x1, x2, x3)  =  div_s_out_gga(x3)
6.00/2.51	
6.00/2.51	U2_gga(x1, x2, x3)  =  U2_gga(x3)
6.00/2.51	
6.00/2.51	lss_in_gg(x1, x2)  =  lss_in_gg(x1, x2)
6.00/2.51	
6.00/2.51	U5_gg(x1, x2, x3)  =  U5_gg(x3)
6.00/2.51	
6.00/2.51	lss_out_gg(x1, x2)  =  lss_out_gg
6.00/2.51	
6.00/2.51	U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
6.00/2.51	
6.00/2.51	sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
6.00/2.51	
6.00/2.51	U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
6.00/2.51	
6.00/2.51	sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
6.00/2.51	
6.00/2.51	U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
6.00/2.51	
6.00/2.51	div_out_gga(x1, x2, x3)  =  div_out_gga(x3)
6.00/2.51	
6.00/2.51	DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
6.00/2.51	
6.00/2.51	U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
6.00/2.51	
6.00/2.51	DIV_S_IN_GGA(x1, x2, x3)  =  DIV_S_IN_GGA(x1, x2)
6.00/2.51	
6.00/2.51	U2_GGA(x1, x2, x3)  =  U2_GGA(x3)
6.00/2.51	
6.00/2.51	LSS_IN_GG(x1, x2)  =  LSS_IN_GG(x1, x2)
6.00/2.51	
6.00/2.51	U5_GG(x1, x2, x3)  =  U5_GG(x3)
6.00/2.51	
6.00/2.51	U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x2, x4)
6.00/2.51	
6.00/2.51	SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)
6.00/2.51	
6.00/2.51	U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x4)
6.00/2.51	
6.00/2.51	U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x4)
6.00/2.51	
6.00/2.51	
6.00/2.51	We have to consider all (P,R,Pi)-chains
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(5) DependencyGraphProof (EQUIVALENT)
6.00/2.51	The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 8 less nodes.
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(6)
6.00/2.51	Complex Obligation (AND)
6.00/2.51	
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(7)
6.00/2.51	Obligation:
6.00/2.51	Pi DP problem:
6.00/2.51	The TRS P consists of the following rules:
6.00/2.51	
6.00/2.51	   SUB_IN_GGA(s(X), s(Y), Z) -> SUB_IN_GGA(X, Y, Z)
6.00/2.51	
6.00/2.51	The TRS R consists of the following rules:
6.00/2.51	
6.00/2.51	   div_in_gga(X, s(Y), Z) -> U1_gga(X, Y, Z, div_s_in_gga(X, Y, Z))
6.00/2.51	   div_s_in_gga(0, Y, 0) -> div_s_out_gga(0, Y, 0)
6.00/2.51	   div_s_in_gga(s(X), Y, 0) -> U2_gga(X, Y, lss_in_gg(X, Y))
6.00/2.51	   lss_in_gg(s(X), s(Y)) -> U5_gg(X, Y, lss_in_gg(X, Y))
6.00/2.51	   lss_in_gg(0, s(Y)) -> lss_out_gg(0, s(Y))
6.00/2.51	   U5_gg(X, Y, lss_out_gg(X, Y)) -> lss_out_gg(s(X), s(Y))
6.00/2.51	   U2_gga(X, Y, lss_out_gg(X, Y)) -> div_s_out_gga(s(X), Y, 0)
6.00/2.51	   div_s_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, sub_in_gga(X, Y, R))
6.00/2.51	   sub_in_gga(s(X), s(Y), Z) -> U6_gga(X, Y, Z, sub_in_gga(X, Y, Z))
6.00/2.51	   sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X)
6.00/2.51	   U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z)
6.00/2.51	   U3_gga(X, Y, Z, sub_out_gga(X, Y, R)) -> U4_gga(X, Y, Z, div_s_in_gga(R, Y, Z))
6.00/2.51	   U4_gga(X, Y, Z, div_s_out_gga(R, Y, Z)) -> div_s_out_gga(s(X), Y, s(Z))
6.00/2.51	   U1_gga(X, Y, Z, div_s_out_gga(X, Y, Z)) -> div_out_gga(X, s(Y), Z)
6.00/2.51	
6.00/2.51	The argument filtering Pi contains the following mapping:
6.00/2.51	div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
6.00/2.51	
6.00/2.51	s(x1)  =  s(x1)
6.00/2.51	
6.00/2.51	U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
6.00/2.51	
6.00/2.51	div_s_in_gga(x1, x2, x3)  =  div_s_in_gga(x1, x2)
6.00/2.51	
6.00/2.51	0  =  0
6.00/2.51	
6.00/2.51	div_s_out_gga(x1, x2, x3)  =  div_s_out_gga(x3)
6.00/2.51	
6.00/2.51	U2_gga(x1, x2, x3)  =  U2_gga(x3)
6.00/2.51	
6.00/2.51	lss_in_gg(x1, x2)  =  lss_in_gg(x1, x2)
6.00/2.51	
6.00/2.51	U5_gg(x1, x2, x3)  =  U5_gg(x3)
6.00/2.51	
6.00/2.51	lss_out_gg(x1, x2)  =  lss_out_gg
6.00/2.51	
6.00/2.51	U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
6.00/2.51	
6.00/2.51	sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
6.00/2.51	
6.00/2.51	U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
6.00/2.51	
6.00/2.51	sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
6.00/2.51	
6.00/2.51	U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
6.00/2.51	
6.00/2.51	div_out_gga(x1, x2, x3)  =  div_out_gga(x3)
6.00/2.51	
6.00/2.51	SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)
6.00/2.51	
6.00/2.51	
6.00/2.51	We have to consider all (P,R,Pi)-chains
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(8) UsableRulesProof (EQUIVALENT)
6.00/2.51	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(9)
6.00/2.51	Obligation:
6.00/2.51	Pi DP problem:
6.00/2.51	The TRS P consists of the following rules:
6.00/2.51	
6.00/2.51	   SUB_IN_GGA(s(X), s(Y), Z) -> SUB_IN_GGA(X, Y, Z)
6.00/2.51	
6.00/2.51	R is empty.
6.00/2.51	The argument filtering Pi contains the following mapping:
6.00/2.51	s(x1)  =  s(x1)
6.00/2.51	
6.00/2.51	SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)
6.00/2.51	
6.00/2.51	
6.00/2.51	We have to consider all (P,R,Pi)-chains
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(10) PiDPToQDPProof (SOUND)
6.00/2.51	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(11)
6.00/2.51	Obligation:
6.00/2.51	Q DP problem:
6.00/2.51	The TRS P consists of the following rules:
6.00/2.51	
6.00/2.51	   SUB_IN_GGA(s(X), s(Y)) -> SUB_IN_GGA(X, Y)
6.00/2.51	
6.00/2.51	R is empty.
6.00/2.51	Q is empty.
6.00/2.51	We have to consider all (P,Q,R)-chains.
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(12) QDPSizeChangeProof (EQUIVALENT)
6.00/2.51	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. 
6.00/2.51	
6.00/2.51	From the DPs we obtained the following set of size-change graphs:
6.00/2.51	*SUB_IN_GGA(s(X), s(Y)) -> SUB_IN_GGA(X, Y)
6.00/2.51	The graph contains the following edges 1 > 1, 2 > 2
6.00/2.51	
6.00/2.51	
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(13)
6.00/2.51	YES
6.00/2.51	
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(14)
6.00/2.51	Obligation:
6.00/2.51	Pi DP problem:
6.00/2.51	The TRS P consists of the following rules:
6.00/2.51	
6.00/2.51	   LSS_IN_GG(s(X), s(Y)) -> LSS_IN_GG(X, Y)
6.00/2.51	
6.00/2.51	The TRS R consists of the following rules:
6.00/2.51	
6.00/2.51	   div_in_gga(X, s(Y), Z) -> U1_gga(X, Y, Z, div_s_in_gga(X, Y, Z))
6.00/2.51	   div_s_in_gga(0, Y, 0) -> div_s_out_gga(0, Y, 0)
6.00/2.51	   div_s_in_gga(s(X), Y, 0) -> U2_gga(X, Y, lss_in_gg(X, Y))
6.00/2.51	   lss_in_gg(s(X), s(Y)) -> U5_gg(X, Y, lss_in_gg(X, Y))
6.00/2.51	   lss_in_gg(0, s(Y)) -> lss_out_gg(0, s(Y))
6.00/2.51	   U5_gg(X, Y, lss_out_gg(X, Y)) -> lss_out_gg(s(X), s(Y))
6.00/2.51	   U2_gga(X, Y, lss_out_gg(X, Y)) -> div_s_out_gga(s(X), Y, 0)
6.00/2.51	   div_s_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, sub_in_gga(X, Y, R))
6.00/2.51	   sub_in_gga(s(X), s(Y), Z) -> U6_gga(X, Y, Z, sub_in_gga(X, Y, Z))
6.00/2.51	   sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X)
6.00/2.51	   U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z)
6.00/2.51	   U3_gga(X, Y, Z, sub_out_gga(X, Y, R)) -> U4_gga(X, Y, Z, div_s_in_gga(R, Y, Z))
6.00/2.51	   U4_gga(X, Y, Z, div_s_out_gga(R, Y, Z)) -> div_s_out_gga(s(X), Y, s(Z))
6.00/2.51	   U1_gga(X, Y, Z, div_s_out_gga(X, Y, Z)) -> div_out_gga(X, s(Y), Z)
6.00/2.51	
6.00/2.51	The argument filtering Pi contains the following mapping:
6.00/2.51	div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
6.00/2.51	
6.00/2.51	s(x1)  =  s(x1)
6.00/2.51	
6.00/2.51	U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
6.00/2.51	
6.00/2.51	div_s_in_gga(x1, x2, x3)  =  div_s_in_gga(x1, x2)
6.00/2.51	
6.00/2.51	0  =  0
6.00/2.51	
6.00/2.51	div_s_out_gga(x1, x2, x3)  =  div_s_out_gga(x3)
6.00/2.51	
6.00/2.51	U2_gga(x1, x2, x3)  =  U2_gga(x3)
6.00/2.51	
6.00/2.51	lss_in_gg(x1, x2)  =  lss_in_gg(x1, x2)
6.00/2.51	
6.00/2.51	U5_gg(x1, x2, x3)  =  U5_gg(x3)
6.00/2.51	
6.00/2.51	lss_out_gg(x1, x2)  =  lss_out_gg
6.00/2.51	
6.00/2.51	U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
6.00/2.51	
6.00/2.51	sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
6.00/2.51	
6.00/2.51	U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
6.00/2.51	
6.00/2.51	sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
6.00/2.51	
6.00/2.51	U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
6.00/2.51	
6.00/2.51	div_out_gga(x1, x2, x3)  =  div_out_gga(x3)
6.00/2.51	
6.00/2.51	LSS_IN_GG(x1, x2)  =  LSS_IN_GG(x1, x2)
6.00/2.51	
6.00/2.51	
6.00/2.51	We have to consider all (P,R,Pi)-chains
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(15) UsableRulesProof (EQUIVALENT)
6.00/2.51	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(16)
6.00/2.51	Obligation:
6.00/2.51	Pi DP problem:
6.00/2.51	The TRS P consists of the following rules:
6.00/2.51	
6.00/2.51	   LSS_IN_GG(s(X), s(Y)) -> LSS_IN_GG(X, Y)
6.00/2.51	
6.00/2.51	R is empty.
6.00/2.51	Pi is empty.
6.00/2.51	We have to consider all (P,R,Pi)-chains
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(17) PiDPToQDPProof (EQUIVALENT)
6.00/2.51	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(18)
6.00/2.51	Obligation:
6.00/2.51	Q DP problem:
6.00/2.51	The TRS P consists of the following rules:
6.00/2.51	
6.00/2.51	   LSS_IN_GG(s(X), s(Y)) -> LSS_IN_GG(X, Y)
6.00/2.51	
6.00/2.51	R is empty.
6.00/2.51	Q is empty.
6.00/2.51	We have to consider all (P,Q,R)-chains.
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(19) QDPSizeChangeProof (EQUIVALENT)
6.00/2.51	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. 
6.00/2.51	
6.00/2.51	From the DPs we obtained the following set of size-change graphs:
6.00/2.51	*LSS_IN_GG(s(X), s(Y)) -> LSS_IN_GG(X, Y)
6.00/2.51	The graph contains the following edges 1 > 1, 2 > 2
6.00/2.51	
6.00/2.51	
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(20)
6.00/2.51	YES
6.00/2.51	
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(21)
6.00/2.51	Obligation:
6.00/2.51	Pi DP problem:
6.00/2.51	The TRS P consists of the following rules:
6.00/2.51	
6.00/2.51	   DIV_S_IN_GGA(s(X), Y, s(Z)) -> U3_GGA(X, Y, Z, sub_in_gga(X, Y, R))
6.00/2.51	   U3_GGA(X, Y, Z, sub_out_gga(X, Y, R)) -> DIV_S_IN_GGA(R, Y, Z)
6.00/2.51	
6.00/2.51	The TRS R consists of the following rules:
6.00/2.51	
6.00/2.51	   div_in_gga(X, s(Y), Z) -> U1_gga(X, Y, Z, div_s_in_gga(X, Y, Z))
6.00/2.51	   div_s_in_gga(0, Y, 0) -> div_s_out_gga(0, Y, 0)
6.00/2.51	   div_s_in_gga(s(X), Y, 0) -> U2_gga(X, Y, lss_in_gg(X, Y))
6.00/2.51	   lss_in_gg(s(X), s(Y)) -> U5_gg(X, Y, lss_in_gg(X, Y))
6.00/2.51	   lss_in_gg(0, s(Y)) -> lss_out_gg(0, s(Y))
6.00/2.51	   U5_gg(X, Y, lss_out_gg(X, Y)) -> lss_out_gg(s(X), s(Y))
6.00/2.51	   U2_gga(X, Y, lss_out_gg(X, Y)) -> div_s_out_gga(s(X), Y, 0)
6.00/2.51	   div_s_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, sub_in_gga(X, Y, R))
6.00/2.51	   sub_in_gga(s(X), s(Y), Z) -> U6_gga(X, Y, Z, sub_in_gga(X, Y, Z))
6.00/2.51	   sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X)
6.00/2.51	   U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z)
6.00/2.51	   U3_gga(X, Y, Z, sub_out_gga(X, Y, R)) -> U4_gga(X, Y, Z, div_s_in_gga(R, Y, Z))
6.00/2.51	   U4_gga(X, Y, Z, div_s_out_gga(R, Y, Z)) -> div_s_out_gga(s(X), Y, s(Z))
6.00/2.51	   U1_gga(X, Y, Z, div_s_out_gga(X, Y, Z)) -> div_out_gga(X, s(Y), Z)
6.00/2.51	
6.00/2.51	The argument filtering Pi contains the following mapping:
6.00/2.51	div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
6.00/2.51	
6.00/2.51	s(x1)  =  s(x1)
6.00/2.51	
6.00/2.51	U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
6.00/2.51	
6.00/2.51	div_s_in_gga(x1, x2, x3)  =  div_s_in_gga(x1, x2)
6.00/2.51	
6.00/2.51	0  =  0
6.00/2.51	
6.00/2.51	div_s_out_gga(x1, x2, x3)  =  div_s_out_gga(x3)
6.00/2.51	
6.00/2.51	U2_gga(x1, x2, x3)  =  U2_gga(x3)
6.00/2.51	
6.00/2.51	lss_in_gg(x1, x2)  =  lss_in_gg(x1, x2)
6.00/2.51	
6.00/2.51	U5_gg(x1, x2, x3)  =  U5_gg(x3)
6.00/2.51	
6.00/2.51	lss_out_gg(x1, x2)  =  lss_out_gg
6.00/2.51	
6.00/2.51	U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
6.00/2.51	
6.00/2.51	sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
6.00/2.51	
6.00/2.51	U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
6.00/2.51	
6.00/2.51	sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
6.00/2.51	
6.00/2.51	U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
6.00/2.51	
6.00/2.51	div_out_gga(x1, x2, x3)  =  div_out_gga(x3)
6.00/2.51	
6.00/2.51	DIV_S_IN_GGA(x1, x2, x3)  =  DIV_S_IN_GGA(x1, x2)
6.00/2.51	
6.00/2.51	U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x2, x4)
6.00/2.51	
6.00/2.51	
6.00/2.51	We have to consider all (P,R,Pi)-chains
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(22) UsableRulesProof (EQUIVALENT)
6.00/2.51	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(23)
6.00/2.51	Obligation:
6.00/2.51	Pi DP problem:
6.00/2.51	The TRS P consists of the following rules:
6.00/2.51	
6.00/2.51	   DIV_S_IN_GGA(s(X), Y, s(Z)) -> U3_GGA(X, Y, Z, sub_in_gga(X, Y, R))
6.00/2.51	   U3_GGA(X, Y, Z, sub_out_gga(X, Y, R)) -> DIV_S_IN_GGA(R, Y, Z)
6.00/2.51	
6.00/2.51	The TRS R consists of the following rules:
6.00/2.51	
6.00/2.51	   sub_in_gga(s(X), s(Y), Z) -> U6_gga(X, Y, Z, sub_in_gga(X, Y, Z))
6.00/2.51	   sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X)
6.00/2.51	   U6_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z)
6.00/2.51	
6.00/2.51	The argument filtering Pi contains the following mapping:
6.00/2.51	s(x1)  =  s(x1)
6.00/2.51	
6.00/2.51	0  =  0
6.00/2.51	
6.00/2.51	sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)
6.00/2.51	
6.00/2.51	U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
6.00/2.51	
6.00/2.51	sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)
6.00/2.51	
6.00/2.51	DIV_S_IN_GGA(x1, x2, x3)  =  DIV_S_IN_GGA(x1, x2)
6.00/2.51	
6.00/2.51	U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x2, x4)
6.00/2.51	
6.00/2.51	
6.00/2.51	We have to consider all (P,R,Pi)-chains
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(24) PiDPToQDPProof (SOUND)
6.00/2.51	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(25)
6.00/2.51	Obligation:
6.00/2.51	Q DP problem:
6.00/2.51	The TRS P consists of the following rules:
6.00/2.51	
6.00/2.51	   DIV_S_IN_GGA(s(X), Y) -> U3_GGA(Y, sub_in_gga(X, Y))
6.00/2.51	   U3_GGA(Y, sub_out_gga(R)) -> DIV_S_IN_GGA(R, Y)
6.00/2.51	
6.00/2.51	The TRS R consists of the following rules:
6.00/2.51	
6.00/2.51	   sub_in_gga(s(X), s(Y)) -> U6_gga(sub_in_gga(X, Y))
6.00/2.51	   sub_in_gga(X, 0) -> sub_out_gga(X)
6.00/2.51	   U6_gga(sub_out_gga(Z)) -> sub_out_gga(Z)
6.00/2.51	
6.00/2.51	The set Q consists of the following terms:
6.00/2.51	
6.00/2.51	   sub_in_gga(x0, x1)
6.00/2.51	   U6_gga(x0)
6.00/2.51	
6.00/2.51	We have to consider all (P,Q,R)-chains.
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(26) QDPOrderProof (EQUIVALENT)
6.00/2.51	We use the reduction pair processor [LPAR04,JAR06].
6.00/2.51	
6.00/2.51	
6.00/2.51	The following pairs can be oriented strictly and are deleted.
6.00/2.51	
6.00/2.51	   DIV_S_IN_GGA(s(X), Y) -> U3_GGA(Y, sub_in_gga(X, Y))
6.00/2.51	   U3_GGA(Y, sub_out_gga(R)) -> DIV_S_IN_GGA(R, Y)
6.00/2.51	The remaining pairs can at least be oriented weakly.
6.00/2.51	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
6.00/2.51	
6.00/2.51	POL( U3_GGA_2(x_1, x_2) ) = 2x_1 + x_2 + 2
6.00/2.51	POL( sub_in_gga_2(x_1, x_2) ) = 2x_1 + 1
6.00/2.51	POL( s_1(x_1) ) = 2x_1 + 2
6.00/2.51	POL( U6_gga_1(x_1) ) = 2x_1
6.00/2.51	POL( 0 ) = 2
6.00/2.51	POL( sub_out_gga_1(x_1) ) = 2x_1 + 1
6.00/2.51	POL( DIV_S_IN_GGA_2(x_1, x_2) ) = 2x_1 + 2x_2
6.00/2.51	
6.00/2.51	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
6.00/2.51	
6.00/2.51	   sub_in_gga(s(X), s(Y)) -> U6_gga(sub_in_gga(X, Y))
6.00/2.51	   sub_in_gga(X, 0) -> sub_out_gga(X)
6.00/2.51	   U6_gga(sub_out_gga(Z)) -> sub_out_gga(Z)
6.00/2.51	
6.00/2.51	
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(27)
6.00/2.51	Obligation:
6.00/2.51	Q DP problem:
6.00/2.51	P is empty.
6.00/2.51	The TRS R consists of the following rules:
6.00/2.51	
6.00/2.51	   sub_in_gga(s(X), s(Y)) -> U6_gga(sub_in_gga(X, Y))
6.00/2.51	   sub_in_gga(X, 0) -> sub_out_gga(X)
6.00/2.51	   U6_gga(sub_out_gga(Z)) -> sub_out_gga(Z)
6.00/2.51	
6.00/2.51	The set Q consists of the following terms:
6.00/2.51	
6.00/2.51	   sub_in_gga(x0, x1)
6.00/2.51	   U6_gga(x0)
6.00/2.51	
6.00/2.51	We have to consider all (P,Q,R)-chains.
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(28) PisEmptyProof (EQUIVALENT)
6.00/2.51	The TRS P is empty. Hence, there is no (P,Q,R) chain.
6.00/2.51	----------------------------------------
6.00/2.51	
6.00/2.51	(29)
6.00/2.51	YES
6.31/2.54	EOF