10.22/3.54	YES
10.61/3.58	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
10.61/3.58	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.61/3.58	
10.61/3.58	
10.61/3.58	Left Termination of the query pattern
10.61/3.58	
10.61/3.58	d(g,g,a)
10.61/3.58	
10.61/3.58	w.r.t. the given Prolog program could successfully be proven:
10.61/3.58	
10.61/3.58	(0) Prolog
10.61/3.58	(1) PrologToPiTRSProof [SOUND, 0 ms]
10.61/3.58	(2) PiTRS
10.61/3.58	(3) DependencyPairsProof [EQUIVALENT, 5 ms]
10.61/3.58	(4) PiDP
10.61/3.58	(5) DependencyGraphProof [EQUIVALENT, 1 ms]
10.61/3.58	(6) AND
10.61/3.58	    (7) PiDP
10.61/3.58	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
10.61/3.58	        (9) PiDP
10.61/3.58	        (10) PiDPToQDPProof [EQUIVALENT, 0 ms]
10.61/3.58	        (11) QDP
10.61/3.58	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.61/3.58	        (13) YES
10.61/3.58	    (14) PiDP
10.61/3.58	        (15) PiDPToQDPProof [SOUND, 0 ms]
10.61/3.58	        (16) QDP
10.61/3.58	        (17) QDPQMonotonicMRRProof [EQUIVALENT, 122 ms]
10.61/3.58	        (18) QDP
10.61/3.58	        (19) QDPQMonotonicMRRProof [EQUIVALENT, 44 ms]
10.61/3.58	        (20) QDP
10.61/3.58	        (21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.61/3.58	        (22) YES
10.61/3.58	
10.61/3.58	
10.61/3.58	----------------------------------------
10.61/3.58	
10.61/3.58	(0)
10.61/3.58	Obligation:
10.61/3.58	Clauses:
10.61/3.58	
10.61/3.58	d(X, X, 1).
10.61/3.58	d(T, X1, 0) :- isnumber(T).
10.61/3.58	d(times(U, V), X, +(times(B, U), times(A, V))) :- ','(d(U, X, A), d(V, X, B)).
10.61/3.58	d(div(U, V), X, W) :- d(times(U, power(V, p(0))), X, W).
10.61/3.58	d(power(U, V), X, times(V, times(W, power(U, p(V))))) :- ','(isnumber(V), d(U, X, W)).
10.61/3.58	isnumber(0).
10.61/3.58	isnumber(s(X)) :- isnumber(X).
10.61/3.58	isnumber(p(X)) :- isnumber(X).
10.61/3.58	
10.61/3.58	
10.61/3.58	Query: d(g,g,a)
10.61/3.58	----------------------------------------
10.61/3.58	
10.61/3.58	(1) PrologToPiTRSProof (SOUND)
10.61/3.58	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
10.61/3.58	
10.61/3.58	d_in_3: (b,b,f)
10.61/3.58	
10.61/3.58	isnumber_in_1: (b)
10.61/3.58	
10.61/3.58	Transforming Prolog into the following Term Rewriting System:
10.61/3.58	
10.61/3.58	Pi-finite rewrite system:
10.61/3.58	The TRS R consists of the following rules:
10.61/3.58	
10.61/3.58	   d_in_gga(X, X, 1) -> d_out_gga(X, X, 1)
10.61/3.58	   d_in_gga(T, X1, 0) -> U1_gga(T, X1, isnumber_in_g(T))
10.61/3.58	   isnumber_in_g(0) -> isnumber_out_g(0)
10.61/3.58	   isnumber_in_g(s(X)) -> U7_g(X, isnumber_in_g(X))
10.61/3.58	   isnumber_in_g(p(X)) -> U8_g(X, isnumber_in_g(X))
10.61/3.58	   U8_g(X, isnumber_out_g(X)) -> isnumber_out_g(p(X))
10.61/3.58	   U7_g(X, isnumber_out_g(X)) -> isnumber_out_g(s(X))
10.61/3.58	   U1_gga(T, X1, isnumber_out_g(T)) -> d_out_gga(T, X1, 0)
10.61/3.58	   d_in_gga(times(U, V), X, +(times(B, U), times(A, V))) -> U2_gga(U, V, X, B, A, d_in_gga(U, X, A))
10.61/3.58	   d_in_gga(div(U, V), X, W) -> U4_gga(U, V, X, W, d_in_gga(times(U, power(V, p(0))), X, W))
10.61/3.58	   d_in_gga(power(U, V), X, times(V, times(W, power(U, p(V))))) -> U5_gga(U, V, X, W, isnumber_in_g(V))
10.61/3.58	   U5_gga(U, V, X, W, isnumber_out_g(V)) -> U6_gga(U, V, X, W, d_in_gga(U, X, W))
10.61/3.58	   U6_gga(U, V, X, W, d_out_gga(U, X, W)) -> d_out_gga(power(U, V), X, times(V, times(W, power(U, p(V)))))
10.61/3.58	   U4_gga(U, V, X, W, d_out_gga(times(U, power(V, p(0))), X, W)) -> d_out_gga(div(U, V), X, W)
10.61/3.58	   U2_gga(U, V, X, B, A, d_out_gga(U, X, A)) -> U3_gga(U, V, X, B, A, d_in_gga(V, X, B))
10.61/3.58	   U3_gga(U, V, X, B, A, d_out_gga(V, X, B)) -> d_out_gga(times(U, V), X, +(times(B, U), times(A, V)))
10.61/3.58	
10.61/3.58	The argument filtering Pi contains the following mapping:
10.61/3.58	d_in_gga(x1, x2, x3)  =  d_in_gga(x1, x2)
10.61/3.58	
10.61/3.58	d_out_gga(x1, x2, x3)  =  d_out_gga(x1, x2, x3)
10.61/3.58	
10.61/3.58	U1_gga(x1, x2, x3)  =  U1_gga(x1, x2, x3)
10.61/3.58	
10.61/3.58	isnumber_in_g(x1)  =  isnumber_in_g(x1)
10.61/3.58	
10.61/3.58	0  =  0
10.61/3.58	
10.61/3.58	isnumber_out_g(x1)  =  isnumber_out_g(x1)
10.61/3.58	
10.61/3.58	s(x1)  =  s(x1)
10.61/3.58	
10.61/3.58	U7_g(x1, x2)  =  U7_g(x1, x2)
10.61/3.58	
10.61/3.58	p(x1)  =  p(x1)
10.61/3.58	
10.61/3.58	U8_g(x1, x2)  =  U8_g(x1, x2)
10.61/3.58	
10.61/3.58	times(x1, x2)  =  times(x1, x2)
10.61/3.58	
10.61/3.58	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x6)
10.61/3.58	
10.61/3.58	div(x1, x2)  =  div(x1, x2)
10.61/3.58	
10.61/3.58	U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
10.61/3.58	
10.61/3.58	power(x1, x2)  =  power(x1, x2)
10.61/3.58	
10.61/3.58	U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
10.61/3.58	
10.61/3.58	U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
10.61/3.58	
10.61/3.58	U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x5, x6)
10.61/3.58	
10.61/3.58	+(x1, x2)  =  +(x1, x2)
10.61/3.58	
10.61/3.58	
10.61/3.58	
10.61/3.58	
10.61/3.58	
10.61/3.58	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
10.61/3.58	
10.61/3.58	
10.61/3.58	
10.61/3.58	----------------------------------------
10.61/3.58	
10.61/3.58	(2)
10.61/3.58	Obligation:
10.61/3.58	Pi-finite rewrite system:
10.61/3.58	The TRS R consists of the following rules:
10.61/3.58	
10.61/3.58	   d_in_gga(X, X, 1) -> d_out_gga(X, X, 1)
10.61/3.58	   d_in_gga(T, X1, 0) -> U1_gga(T, X1, isnumber_in_g(T))
10.61/3.58	   isnumber_in_g(0) -> isnumber_out_g(0)
10.61/3.58	   isnumber_in_g(s(X)) -> U7_g(X, isnumber_in_g(X))
10.61/3.58	   isnumber_in_g(p(X)) -> U8_g(X, isnumber_in_g(X))
10.61/3.58	   U8_g(X, isnumber_out_g(X)) -> isnumber_out_g(p(X))
10.61/3.58	   U7_g(X, isnumber_out_g(X)) -> isnumber_out_g(s(X))
10.61/3.58	   U1_gga(T, X1, isnumber_out_g(T)) -> d_out_gga(T, X1, 0)
10.61/3.58	   d_in_gga(times(U, V), X, +(times(B, U), times(A, V))) -> U2_gga(U, V, X, B, A, d_in_gga(U, X, A))
10.61/3.58	   d_in_gga(div(U, V), X, W) -> U4_gga(U, V, X, W, d_in_gga(times(U, power(V, p(0))), X, W))
10.61/3.58	   d_in_gga(power(U, V), X, times(V, times(W, power(U, p(V))))) -> U5_gga(U, V, X, W, isnumber_in_g(V))
10.61/3.58	   U5_gga(U, V, X, W, isnumber_out_g(V)) -> U6_gga(U, V, X, W, d_in_gga(U, X, W))
10.61/3.58	   U6_gga(U, V, X, W, d_out_gga(U, X, W)) -> d_out_gga(power(U, V), X, times(V, times(W, power(U, p(V)))))
10.61/3.58	   U4_gga(U, V, X, W, d_out_gga(times(U, power(V, p(0))), X, W)) -> d_out_gga(div(U, V), X, W)
10.61/3.58	   U2_gga(U, V, X, B, A, d_out_gga(U, X, A)) -> U3_gga(U, V, X, B, A, d_in_gga(V, X, B))
10.61/3.58	   U3_gga(U, V, X, B, A, d_out_gga(V, X, B)) -> d_out_gga(times(U, V), X, +(times(B, U), times(A, V)))
10.61/3.58	
10.61/3.58	The argument filtering Pi contains the following mapping:
10.61/3.58	d_in_gga(x1, x2, x3)  =  d_in_gga(x1, x2)
10.61/3.58	
10.61/3.58	d_out_gga(x1, x2, x3)  =  d_out_gga(x1, x2, x3)
10.61/3.58	
10.61/3.58	U1_gga(x1, x2, x3)  =  U1_gga(x1, x2, x3)
10.61/3.58	
10.61/3.58	isnumber_in_g(x1)  =  isnumber_in_g(x1)
10.61/3.58	
10.61/3.58	0  =  0
10.61/3.58	
10.61/3.58	isnumber_out_g(x1)  =  isnumber_out_g(x1)
10.61/3.58	
10.61/3.58	s(x1)  =  s(x1)
10.61/3.58	
10.61/3.58	U7_g(x1, x2)  =  U7_g(x1, x2)
10.61/3.58	
10.61/3.58	p(x1)  =  p(x1)
10.61/3.58	
10.61/3.58	U8_g(x1, x2)  =  U8_g(x1, x2)
10.61/3.58	
10.61/3.58	times(x1, x2)  =  times(x1, x2)
10.61/3.58	
10.61/3.58	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x6)
10.61/3.58	
10.61/3.58	div(x1, x2)  =  div(x1, x2)
10.61/3.58	
10.61/3.58	U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
10.61/3.58	
10.61/3.58	power(x1, x2)  =  power(x1, x2)
10.61/3.58	
10.61/3.58	U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
10.61/3.58	
10.61/3.58	U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
10.61/3.58	
10.61/3.58	U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x5, x6)
10.61/3.58	
10.61/3.58	+(x1, x2)  =  +(x1, x2)
10.61/3.58	
10.61/3.58	
10.61/3.58	
10.61/3.58	----------------------------------------
10.61/3.58	
10.61/3.58	(3) DependencyPairsProof (EQUIVALENT)
10.61/3.58	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
10.61/3.58	Pi DP problem:
10.61/3.58	The TRS P consists of the following rules:
10.61/3.58	
10.61/3.58	   D_IN_GGA(T, X1, 0) -> U1_GGA(T, X1, isnumber_in_g(T))
10.61/3.58	   D_IN_GGA(T, X1, 0) -> ISNUMBER_IN_G(T)
10.61/3.58	   ISNUMBER_IN_G(s(X)) -> U7_G(X, isnumber_in_g(X))
10.61/3.58	   ISNUMBER_IN_G(s(X)) -> ISNUMBER_IN_G(X)
10.61/3.58	   ISNUMBER_IN_G(p(X)) -> U8_G(X, isnumber_in_g(X))
10.61/3.58	   ISNUMBER_IN_G(p(X)) -> ISNUMBER_IN_G(X)
10.61/3.58	   D_IN_GGA(times(U, V), X, +(times(B, U), times(A, V))) -> U2_GGA(U, V, X, B, A, d_in_gga(U, X, A))
10.61/3.58	   D_IN_GGA(times(U, V), X, +(times(B, U), times(A, V))) -> D_IN_GGA(U, X, A)
10.61/3.58	   D_IN_GGA(div(U, V), X, W) -> U4_GGA(U, V, X, W, d_in_gga(times(U, power(V, p(0))), X, W))
10.61/3.58	   D_IN_GGA(div(U, V), X, W) -> D_IN_GGA(times(U, power(V, p(0))), X, W)
10.61/3.58	   D_IN_GGA(power(U, V), X, times(V, times(W, power(U, p(V))))) -> U5_GGA(U, V, X, W, isnumber_in_g(V))
10.61/3.58	   D_IN_GGA(power(U, V), X, times(V, times(W, power(U, p(V))))) -> ISNUMBER_IN_G(V)
10.61/3.58	   U5_GGA(U, V, X, W, isnumber_out_g(V)) -> U6_GGA(U, V, X, W, d_in_gga(U, X, W))
10.61/3.58	   U5_GGA(U, V, X, W, isnumber_out_g(V)) -> D_IN_GGA(U, X, W)
10.61/3.58	   U2_GGA(U, V, X, B, A, d_out_gga(U, X, A)) -> U3_GGA(U, V, X, B, A, d_in_gga(V, X, B))
10.61/3.58	   U2_GGA(U, V, X, B, A, d_out_gga(U, X, A)) -> D_IN_GGA(V, X, B)
10.61/3.58	
10.61/3.58	The TRS R consists of the following rules:
10.61/3.58	
10.61/3.58	   d_in_gga(X, X, 1) -> d_out_gga(X, X, 1)
10.61/3.58	   d_in_gga(T, X1, 0) -> U1_gga(T, X1, isnumber_in_g(T))
10.61/3.58	   isnumber_in_g(0) -> isnumber_out_g(0)
10.61/3.58	   isnumber_in_g(s(X)) -> U7_g(X, isnumber_in_g(X))
10.61/3.58	   isnumber_in_g(p(X)) -> U8_g(X, isnumber_in_g(X))
10.61/3.58	   U8_g(X, isnumber_out_g(X)) -> isnumber_out_g(p(X))
10.61/3.58	   U7_g(X, isnumber_out_g(X)) -> isnumber_out_g(s(X))
10.61/3.58	   U1_gga(T, X1, isnumber_out_g(T)) -> d_out_gga(T, X1, 0)
10.61/3.58	   d_in_gga(times(U, V), X, +(times(B, U), times(A, V))) -> U2_gga(U, V, X, B, A, d_in_gga(U, X, A))
10.61/3.58	   d_in_gga(div(U, V), X, W) -> U4_gga(U, V, X, W, d_in_gga(times(U, power(V, p(0))), X, W))
10.61/3.58	   d_in_gga(power(U, V), X, times(V, times(W, power(U, p(V))))) -> U5_gga(U, V, X, W, isnumber_in_g(V))
10.61/3.58	   U5_gga(U, V, X, W, isnumber_out_g(V)) -> U6_gga(U, V, X, W, d_in_gga(U, X, W))
10.61/3.58	   U6_gga(U, V, X, W, d_out_gga(U, X, W)) -> d_out_gga(power(U, V), X, times(V, times(W, power(U, p(V)))))
10.61/3.58	   U4_gga(U, V, X, W, d_out_gga(times(U, power(V, p(0))), X, W)) -> d_out_gga(div(U, V), X, W)
10.61/3.59	   U2_gga(U, V, X, B, A, d_out_gga(U, X, A)) -> U3_gga(U, V, X, B, A, d_in_gga(V, X, B))
10.61/3.59	   U3_gga(U, V, X, B, A, d_out_gga(V, X, B)) -> d_out_gga(times(U, V), X, +(times(B, U), times(A, V)))
10.61/3.59	
10.61/3.59	The argument filtering Pi contains the following mapping:
10.61/3.59	d_in_gga(x1, x2, x3)  =  d_in_gga(x1, x2)
10.61/3.59	
10.61/3.59	d_out_gga(x1, x2, x3)  =  d_out_gga(x1, x2, x3)
10.61/3.59	
10.61/3.59	U1_gga(x1, x2, x3)  =  U1_gga(x1, x2, x3)
10.61/3.59	
10.61/3.59	isnumber_in_g(x1)  =  isnumber_in_g(x1)
10.61/3.59	
10.61/3.59	0  =  0
10.61/3.59	
10.61/3.59	isnumber_out_g(x1)  =  isnumber_out_g(x1)
10.61/3.59	
10.61/3.59	s(x1)  =  s(x1)
10.61/3.59	
10.61/3.59	U7_g(x1, x2)  =  U7_g(x1, x2)
10.61/3.59	
10.61/3.59	p(x1)  =  p(x1)
10.61/3.59	
10.61/3.59	U8_g(x1, x2)  =  U8_g(x1, x2)
10.61/3.59	
10.61/3.59	times(x1, x2)  =  times(x1, x2)
10.61/3.59	
10.61/3.59	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x6)
10.61/3.59	
10.61/3.59	div(x1, x2)  =  div(x1, x2)
10.61/3.59	
10.61/3.59	U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
10.61/3.59	
10.61/3.59	power(x1, x2)  =  power(x1, x2)
10.61/3.59	
10.61/3.59	U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
10.61/3.59	
10.61/3.59	U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
10.61/3.59	
10.61/3.59	U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x5, x6)
10.61/3.59	
10.61/3.59	+(x1, x2)  =  +(x1, x2)
10.61/3.59	
10.61/3.59	D_IN_GGA(x1, x2, x3)  =  D_IN_GGA(x1, x2)
10.61/3.59	
10.61/3.59	U1_GGA(x1, x2, x3)  =  U1_GGA(x1, x2, x3)
10.61/3.59	
10.61/3.59	ISNUMBER_IN_G(x1)  =  ISNUMBER_IN_G(x1)
10.61/3.59	
10.61/3.59	U7_G(x1, x2)  =  U7_G(x1, x2)
10.61/3.59	
10.61/3.59	U8_G(x1, x2)  =  U8_G(x1, x2)
10.61/3.59	
10.61/3.59	U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, x6)
10.61/3.59	
10.61/3.59	U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
10.61/3.59	
10.61/3.59	U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)
10.61/3.59	
10.61/3.59	U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x3, x5)
10.61/3.59	
10.61/3.59	U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x1, x2, x3, x5, x6)
10.61/3.59	
10.61/3.59	
10.61/3.59	We have to consider all (P,R,Pi)-chains
10.61/3.59	----------------------------------------
10.61/3.59	
10.61/3.59	(4)
10.61/3.59	Obligation:
10.61/3.59	Pi DP problem:
10.61/3.59	The TRS P consists of the following rules:
10.61/3.59	
10.61/3.59	   D_IN_GGA(T, X1, 0) -> U1_GGA(T, X1, isnumber_in_g(T))
10.61/3.59	   D_IN_GGA(T, X1, 0) -> ISNUMBER_IN_G(T)
10.61/3.59	   ISNUMBER_IN_G(s(X)) -> U7_G(X, isnumber_in_g(X))
10.61/3.59	   ISNUMBER_IN_G(s(X)) -> ISNUMBER_IN_G(X)
10.61/3.59	   ISNUMBER_IN_G(p(X)) -> U8_G(X, isnumber_in_g(X))
10.61/3.59	   ISNUMBER_IN_G(p(X)) -> ISNUMBER_IN_G(X)
10.61/3.59	   D_IN_GGA(times(U, V), X, +(times(B, U), times(A, V))) -> U2_GGA(U, V, X, B, A, d_in_gga(U, X, A))
10.61/3.59	   D_IN_GGA(times(U, V), X, +(times(B, U), times(A, V))) -> D_IN_GGA(U, X, A)
10.61/3.59	   D_IN_GGA(div(U, V), X, W) -> U4_GGA(U, V, X, W, d_in_gga(times(U, power(V, p(0))), X, W))
10.61/3.59	   D_IN_GGA(div(U, V), X, W) -> D_IN_GGA(times(U, power(V, p(0))), X, W)
10.61/3.59	   D_IN_GGA(power(U, V), X, times(V, times(W, power(U, p(V))))) -> U5_GGA(U, V, X, W, isnumber_in_g(V))
10.61/3.59	   D_IN_GGA(power(U, V), X, times(V, times(W, power(U, p(V))))) -> ISNUMBER_IN_G(V)
10.61/3.59	   U5_GGA(U, V, X, W, isnumber_out_g(V)) -> U6_GGA(U, V, X, W, d_in_gga(U, X, W))
10.61/3.59	   U5_GGA(U, V, X, W, isnumber_out_g(V)) -> D_IN_GGA(U, X, W)
10.61/3.59	   U2_GGA(U, V, X, B, A, d_out_gga(U, X, A)) -> U3_GGA(U, V, X, B, A, d_in_gga(V, X, B))
10.61/3.59	   U2_GGA(U, V, X, B, A, d_out_gga(U, X, A)) -> D_IN_GGA(V, X, B)
10.61/3.59	
10.61/3.59	The TRS R consists of the following rules:
10.61/3.59	
10.61/3.59	   d_in_gga(X, X, 1) -> d_out_gga(X, X, 1)
10.61/3.59	   d_in_gga(T, X1, 0) -> U1_gga(T, X1, isnumber_in_g(T))
10.61/3.59	   isnumber_in_g(0) -> isnumber_out_g(0)
10.61/3.59	   isnumber_in_g(s(X)) -> U7_g(X, isnumber_in_g(X))
10.61/3.59	   isnumber_in_g(p(X)) -> U8_g(X, isnumber_in_g(X))
10.61/3.59	   U8_g(X, isnumber_out_g(X)) -> isnumber_out_g(p(X))
10.61/3.59	   U7_g(X, isnumber_out_g(X)) -> isnumber_out_g(s(X))
10.61/3.59	   U1_gga(T, X1, isnumber_out_g(T)) -> d_out_gga(T, X1, 0)
10.61/3.59	   d_in_gga(times(U, V), X, +(times(B, U), times(A, V))) -> U2_gga(U, V, X, B, A, d_in_gga(U, X, A))
10.61/3.59	   d_in_gga(div(U, V), X, W) -> U4_gga(U, V, X, W, d_in_gga(times(U, power(V, p(0))), X, W))
10.61/3.59	   d_in_gga(power(U, V), X, times(V, times(W, power(U, p(V))))) -> U5_gga(U, V, X, W, isnumber_in_g(V))
10.61/3.59	   U5_gga(U, V, X, W, isnumber_out_g(V)) -> U6_gga(U, V, X, W, d_in_gga(U, X, W))
10.61/3.59	   U6_gga(U, V, X, W, d_out_gga(U, X, W)) -> d_out_gga(power(U, V), X, times(V, times(W, power(U, p(V)))))
10.61/3.59	   U4_gga(U, V, X, W, d_out_gga(times(U, power(V, p(0))), X, W)) -> d_out_gga(div(U, V), X, W)
10.61/3.59	   U2_gga(U, V, X, B, A, d_out_gga(U, X, A)) -> U3_gga(U, V, X, B, A, d_in_gga(V, X, B))
10.61/3.59	   U3_gga(U, V, X, B, A, d_out_gga(V, X, B)) -> d_out_gga(times(U, V), X, +(times(B, U), times(A, V)))
10.61/3.59	
10.61/3.59	The argument filtering Pi contains the following mapping:
10.61/3.59	d_in_gga(x1, x2, x3)  =  d_in_gga(x1, x2)
10.61/3.59	
10.61/3.59	d_out_gga(x1, x2, x3)  =  d_out_gga(x1, x2, x3)
10.61/3.59	
10.61/3.59	U1_gga(x1, x2, x3)  =  U1_gga(x1, x2, x3)
10.61/3.59	
10.61/3.59	isnumber_in_g(x1)  =  isnumber_in_g(x1)
10.61/3.59	
10.61/3.59	0  =  0
10.61/3.59	
10.61/3.59	isnumber_out_g(x1)  =  isnumber_out_g(x1)
10.61/3.59	
10.61/3.59	s(x1)  =  s(x1)
10.61/3.59	
10.61/3.59	U7_g(x1, x2)  =  U7_g(x1, x2)
10.61/3.59	
10.61/3.59	p(x1)  =  p(x1)
10.61/3.59	
10.61/3.59	U8_g(x1, x2)  =  U8_g(x1, x2)
10.61/3.59	
10.61/3.59	times(x1, x2)  =  times(x1, x2)
10.61/3.59	
10.61/3.59	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x6)
10.61/3.59	
10.61/3.59	div(x1, x2)  =  div(x1, x2)
10.61/3.59	
10.61/3.59	U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
10.61/3.59	
10.61/3.59	power(x1, x2)  =  power(x1, x2)
10.61/3.59	
10.61/3.59	U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
10.61/3.59	
10.61/3.59	U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
10.61/3.59	
10.61/3.59	U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x5, x6)
10.61/3.59	
10.61/3.59	+(x1, x2)  =  +(x1, x2)
10.61/3.59	
10.61/3.59	D_IN_GGA(x1, x2, x3)  =  D_IN_GGA(x1, x2)
10.61/3.59	
10.61/3.59	U1_GGA(x1, x2, x3)  =  U1_GGA(x1, x2, x3)
10.61/3.59	
10.61/3.59	ISNUMBER_IN_G(x1)  =  ISNUMBER_IN_G(x1)
10.61/3.59	
10.61/3.59	U7_G(x1, x2)  =  U7_G(x1, x2)
10.61/3.59	
10.61/3.59	U8_G(x1, x2)  =  U8_G(x1, x2)
10.61/3.59	
10.61/3.59	U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, x6)
10.61/3.59	
10.61/3.59	U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
10.61/3.59	
10.61/3.59	U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)
10.61/3.59	
10.61/3.59	U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x3, x5)
10.61/3.59	
10.61/3.59	U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x1, x2, x3, x5, x6)
10.61/3.59	
10.61/3.59	
10.61/3.59	We have to consider all (P,R,Pi)-chains
10.61/3.59	----------------------------------------
10.61/3.59	
10.61/3.59	(5) DependencyGraphProof (EQUIVALENT)
10.61/3.59	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 8 less nodes.
10.61/3.59	----------------------------------------
10.61/3.59	
10.61/3.59	(6)
10.61/3.59	Complex Obligation (AND)
10.61/3.59	
10.61/3.59	----------------------------------------
10.61/3.59	
10.61/3.59	(7)
10.61/3.59	Obligation:
10.61/3.59	Pi DP problem:
10.61/3.59	The TRS P consists of the following rules:
10.61/3.59	
10.61/3.59	   ISNUMBER_IN_G(p(X)) -> ISNUMBER_IN_G(X)
10.61/3.59	   ISNUMBER_IN_G(s(X)) -> ISNUMBER_IN_G(X)
10.61/3.59	
10.61/3.59	The TRS R consists of the following rules:
10.61/3.59	
10.61/3.59	   d_in_gga(X, X, 1) -> d_out_gga(X, X, 1)
10.61/3.59	   d_in_gga(T, X1, 0) -> U1_gga(T, X1, isnumber_in_g(T))
10.61/3.59	   isnumber_in_g(0) -> isnumber_out_g(0)
10.61/3.59	   isnumber_in_g(s(X)) -> U7_g(X, isnumber_in_g(X))
10.61/3.59	   isnumber_in_g(p(X)) -> U8_g(X, isnumber_in_g(X))
10.61/3.59	   U8_g(X, isnumber_out_g(X)) -> isnumber_out_g(p(X))
10.61/3.59	   U7_g(X, isnumber_out_g(X)) -> isnumber_out_g(s(X))
10.61/3.59	   U1_gga(T, X1, isnumber_out_g(T)) -> d_out_gga(T, X1, 0)
10.61/3.59	   d_in_gga(times(U, V), X, +(times(B, U), times(A, V))) -> U2_gga(U, V, X, B, A, d_in_gga(U, X, A))
10.61/3.59	   d_in_gga(div(U, V), X, W) -> U4_gga(U, V, X, W, d_in_gga(times(U, power(V, p(0))), X, W))
10.61/3.59	   d_in_gga(power(U, V), X, times(V, times(W, power(U, p(V))))) -> U5_gga(U, V, X, W, isnumber_in_g(V))
10.61/3.59	   U5_gga(U, V, X, W, isnumber_out_g(V)) -> U6_gga(U, V, X, W, d_in_gga(U, X, W))
10.61/3.59	   U6_gga(U, V, X, W, d_out_gga(U, X, W)) -> d_out_gga(power(U, V), X, times(V, times(W, power(U, p(V)))))
10.61/3.59	   U4_gga(U, V, X, W, d_out_gga(times(U, power(V, p(0))), X, W)) -> d_out_gga(div(U, V), X, W)
10.61/3.59	   U2_gga(U, V, X, B, A, d_out_gga(U, X, A)) -> U3_gga(U, V, X, B, A, d_in_gga(V, X, B))
10.61/3.59	   U3_gga(U, V, X, B, A, d_out_gga(V, X, B)) -> d_out_gga(times(U, V), X, +(times(B, U), times(A, V)))
10.61/3.59	
10.61/3.59	The argument filtering Pi contains the following mapping:
10.61/3.59	d_in_gga(x1, x2, x3)  =  d_in_gga(x1, x2)
10.61/3.59	
10.61/3.59	d_out_gga(x1, x2, x3)  =  d_out_gga(x1, x2, x3)
10.61/3.59	
10.61/3.59	U1_gga(x1, x2, x3)  =  U1_gga(x1, x2, x3)
10.61/3.59	
10.61/3.59	isnumber_in_g(x1)  =  isnumber_in_g(x1)
10.61/3.59	
10.61/3.59	0  =  0
10.61/3.59	
10.61/3.59	isnumber_out_g(x1)  =  isnumber_out_g(x1)
10.61/3.59	
10.61/3.59	s(x1)  =  s(x1)
10.61/3.59	
10.61/3.59	U7_g(x1, x2)  =  U7_g(x1, x2)
10.61/3.59	
10.61/3.59	p(x1)  =  p(x1)
10.61/3.59	
10.61/3.59	U8_g(x1, x2)  =  U8_g(x1, x2)
10.61/3.59	
10.61/3.59	times(x1, x2)  =  times(x1, x2)
10.61/3.59	
10.61/3.59	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x6)
10.61/3.59	
10.61/3.59	div(x1, x2)  =  div(x1, x2)
10.61/3.59	
10.61/3.59	U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
10.61/3.59	
10.61/3.59	power(x1, x2)  =  power(x1, x2)
10.61/3.59	
10.61/3.59	U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
10.61/3.59	
10.61/3.59	U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
10.61/3.59	
10.61/3.59	U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x5, x6)
10.61/3.59	
10.61/3.59	+(x1, x2)  =  +(x1, x2)
10.61/3.59	
10.61/3.59	ISNUMBER_IN_G(x1)  =  ISNUMBER_IN_G(x1)
10.61/3.59	
10.61/3.59	
10.61/3.59	We have to consider all (P,R,Pi)-chains
10.61/3.59	----------------------------------------
10.61/3.59	
10.61/3.59	(8) UsableRulesProof (EQUIVALENT)
10.61/3.59	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
10.61/3.59	----------------------------------------
10.61/3.59	
10.61/3.59	(9)
10.61/3.59	Obligation:
10.61/3.59	Pi DP problem:
10.61/3.59	The TRS P consists of the following rules:
10.61/3.59	
10.61/3.59	   ISNUMBER_IN_G(p(X)) -> ISNUMBER_IN_G(X)
10.61/3.59	   ISNUMBER_IN_G(s(X)) -> ISNUMBER_IN_G(X)
10.61/3.59	
10.61/3.59	R is empty.
10.61/3.59	Pi is empty.
10.61/3.59	We have to consider all (P,R,Pi)-chains
10.61/3.59	----------------------------------------
10.61/3.59	
10.61/3.59	(10) PiDPToQDPProof (EQUIVALENT)
10.61/3.59	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
10.61/3.59	----------------------------------------
10.61/3.59	
10.61/3.59	(11)
10.61/3.59	Obligation:
10.61/3.59	Q DP problem:
10.61/3.59	The TRS P consists of the following rules:
10.61/3.59	
10.61/3.59	   ISNUMBER_IN_G(p(X)) -> ISNUMBER_IN_G(X)
10.61/3.59	   ISNUMBER_IN_G(s(X)) -> ISNUMBER_IN_G(X)
10.61/3.59	
10.61/3.59	R is empty.
10.61/3.59	Q is empty.
10.61/3.59	We have to consider all (P,Q,R)-chains.
10.61/3.59	----------------------------------------
10.61/3.59	
10.61/3.59	(12) QDPSizeChangeProof (EQUIVALENT)
10.61/3.59	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. 
10.61/3.59	
10.61/3.59	From the DPs we obtained the following set of size-change graphs:
10.61/3.59	*ISNUMBER_IN_G(p(X)) -> ISNUMBER_IN_G(X)
10.61/3.59	The graph contains the following edges 1 > 1
10.61/3.59	
10.61/3.59	
10.61/3.59	*ISNUMBER_IN_G(s(X)) -> ISNUMBER_IN_G(X)
10.61/3.59	The graph contains the following edges 1 > 1
10.61/3.59	
10.61/3.59	
10.61/3.59	----------------------------------------
10.61/3.59	
10.61/3.59	(13)
10.61/3.59	YES
10.61/3.59	
10.61/3.59	----------------------------------------
10.61/3.59	
10.61/3.59	(14)
10.61/3.59	Obligation:
10.61/3.59	Pi DP problem:
10.61/3.59	The TRS P consists of the following rules:
10.61/3.59	
10.61/3.59	   D_IN_GGA(times(U, V), X, +(times(B, U), times(A, V))) -> U2_GGA(U, V, X, B, A, d_in_gga(U, X, A))
10.61/3.59	   U2_GGA(U, V, X, B, A, d_out_gga(U, X, A)) -> D_IN_GGA(V, X, B)
10.61/3.59	   D_IN_GGA(times(U, V), X, +(times(B, U), times(A, V))) -> D_IN_GGA(U, X, A)
10.61/3.59	   D_IN_GGA(div(U, V), X, W) -> D_IN_GGA(times(U, power(V, p(0))), X, W)
10.61/3.59	   D_IN_GGA(power(U, V), X, times(V, times(W, power(U, p(V))))) -> U5_GGA(U, V, X, W, isnumber_in_g(V))
10.61/3.59	   U5_GGA(U, V, X, W, isnumber_out_g(V)) -> D_IN_GGA(U, X, W)
10.61/3.59	
10.61/3.59	The TRS R consists of the following rules:
10.61/3.59	
10.61/3.59	   d_in_gga(X, X, 1) -> d_out_gga(X, X, 1)
10.61/3.59	   d_in_gga(T, X1, 0) -> U1_gga(T, X1, isnumber_in_g(T))
10.61/3.59	   isnumber_in_g(0) -> isnumber_out_g(0)
10.61/3.59	   isnumber_in_g(s(X)) -> U7_g(X, isnumber_in_g(X))
10.61/3.59	   isnumber_in_g(p(X)) -> U8_g(X, isnumber_in_g(X))
10.61/3.59	   U8_g(X, isnumber_out_g(X)) -> isnumber_out_g(p(X))
10.61/3.59	   U7_g(X, isnumber_out_g(X)) -> isnumber_out_g(s(X))
10.61/3.59	   U1_gga(T, X1, isnumber_out_g(T)) -> d_out_gga(T, X1, 0)
10.61/3.59	   d_in_gga(times(U, V), X, +(times(B, U), times(A, V))) -> U2_gga(U, V, X, B, A, d_in_gga(U, X, A))
10.61/3.59	   d_in_gga(div(U, V), X, W) -> U4_gga(U, V, X, W, d_in_gga(times(U, power(V, p(0))), X, W))
10.61/3.59	   d_in_gga(power(U, V), X, times(V, times(W, power(U, p(V))))) -> U5_gga(U, V, X, W, isnumber_in_g(V))
10.61/3.59	   U5_gga(U, V, X, W, isnumber_out_g(V)) -> U6_gga(U, V, X, W, d_in_gga(U, X, W))
10.61/3.59	   U6_gga(U, V, X, W, d_out_gga(U, X, W)) -> d_out_gga(power(U, V), X, times(V, times(W, power(U, p(V)))))
10.61/3.59	   U4_gga(U, V, X, W, d_out_gga(times(U, power(V, p(0))), X, W)) -> d_out_gga(div(U, V), X, W)
10.61/3.59	   U2_gga(U, V, X, B, A, d_out_gga(U, X, A)) -> U3_gga(U, V, X, B, A, d_in_gga(V, X, B))
10.61/3.59	   U3_gga(U, V, X, B, A, d_out_gga(V, X, B)) -> d_out_gga(times(U, V), X, +(times(B, U), times(A, V)))
10.61/3.59	
10.61/3.59	The argument filtering Pi contains the following mapping:
10.61/3.59	d_in_gga(x1, x2, x3)  =  d_in_gga(x1, x2)
10.61/3.59	
10.61/3.59	d_out_gga(x1, x2, x3)  =  d_out_gga(x1, x2, x3)
10.61/3.59	
10.61/3.59	U1_gga(x1, x2, x3)  =  U1_gga(x1, x2, x3)
10.61/3.59	
10.61/3.59	isnumber_in_g(x1)  =  isnumber_in_g(x1)
10.61/3.59	
10.61/3.59	0  =  0
10.61/3.59	
10.61/3.59	isnumber_out_g(x1)  =  isnumber_out_g(x1)
10.61/3.59	
10.61/3.59	s(x1)  =  s(x1)
10.61/3.59	
10.61/3.59	U7_g(x1, x2)  =  U7_g(x1, x2)
10.61/3.59	
10.61/3.59	p(x1)  =  p(x1)
10.61/3.59	
10.61/3.59	U8_g(x1, x2)  =  U8_g(x1, x2)
10.61/3.59	
10.61/3.59	times(x1, x2)  =  times(x1, x2)
10.61/3.59	
10.61/3.59	U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x6)
10.61/3.59	
10.61/3.59	div(x1, x2)  =  div(x1, x2)
10.61/3.59	
10.61/3.59	U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
10.61/3.59	
10.61/3.59	power(x1, x2)  =  power(x1, x2)
10.61/3.59	
10.61/3.59	U5_gga(x1, x2, x3, x4, x5)  =  U5_gga(x1, x2, x3, x5)
10.61/3.59	
10.61/3.59	U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x3, x5)
10.61/3.59	
10.61/3.59	U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x5, x6)
10.61/3.59	
10.61/3.59	+(x1, x2)  =  +(x1, x2)
10.61/3.59	
10.61/3.59	D_IN_GGA(x1, x2, x3)  =  D_IN_GGA(x1, x2)
10.61/3.59	
10.61/3.59	U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, x6)
10.61/3.59	
10.61/3.59	U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)
10.61/3.59	
10.61/3.59	
10.61/3.59	We have to consider all (P,R,Pi)-chains
10.61/3.59	----------------------------------------
10.61/3.59	
10.61/3.59	(15) PiDPToQDPProof (SOUND)
10.61/3.59	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
10.61/3.59	----------------------------------------
10.61/3.59	
10.61/3.59	(16)
10.61/3.59	Obligation:
10.61/3.59	Q DP problem:
10.61/3.59	The TRS P consists of the following rules:
10.61/3.59	
10.61/3.59	   D_IN_GGA(times(U, V), X) -> U2_GGA(U, V, X, d_in_gga(U, X))
10.61/3.59	   U2_GGA(U, V, X, d_out_gga(U, X, A)) -> D_IN_GGA(V, X)
10.61/3.59	   D_IN_GGA(times(U, V), X) -> D_IN_GGA(U, X)
10.61/3.59	   D_IN_GGA(div(U, V), X) -> D_IN_GGA(times(U, power(V, p(0))), X)
10.61/3.59	   D_IN_GGA(power(U, V), X) -> U5_GGA(U, V, X, isnumber_in_g(V))
10.61/3.59	   U5_GGA(U, V, X, isnumber_out_g(V)) -> D_IN_GGA(U, X)
10.61/3.59	
10.61/3.59	The TRS R consists of the following rules:
10.61/3.59	
10.61/3.59	   d_in_gga(X, X) -> d_out_gga(X, X, 1)
10.61/3.59	   d_in_gga(T, X1) -> U1_gga(T, X1, isnumber_in_g(T))
10.61/3.59	   isnumber_in_g(0) -> isnumber_out_g(0)
10.61/3.59	   isnumber_in_g(s(X)) -> U7_g(X, isnumber_in_g(X))
10.61/3.59	   isnumber_in_g(p(X)) -> U8_g(X, isnumber_in_g(X))
10.61/3.59	   U8_g(X, isnumber_out_g(X)) -> isnumber_out_g(p(X))
10.61/3.59	   U7_g(X, isnumber_out_g(X)) -> isnumber_out_g(s(X))
10.61/3.59	   U1_gga(T, X1, isnumber_out_g(T)) -> d_out_gga(T, X1, 0)
10.61/3.59	   d_in_gga(times(U, V), X) -> U2_gga(U, V, X, d_in_gga(U, X))
10.61/3.59	   d_in_gga(div(U, V), X) -> U4_gga(U, V, X, d_in_gga(times(U, power(V, p(0))), X))
10.61/3.59	   d_in_gga(power(U, V), X) -> U5_gga(U, V, X, isnumber_in_g(V))
10.61/3.59	   U5_gga(U, V, X, isnumber_out_g(V)) -> U6_gga(U, V, X, d_in_gga(U, X))
10.61/3.59	   U6_gga(U, V, X, d_out_gga(U, X, W)) -> d_out_gga(power(U, V), X, times(V, times(W, power(U, p(V)))))
10.61/3.59	   U4_gga(U, V, X, d_out_gga(times(U, power(V, p(0))), X, W)) -> d_out_gga(div(U, V), X, W)
10.61/3.59	   U2_gga(U, V, X, d_out_gga(U, X, A)) -> U3_gga(U, V, X, A, d_in_gga(V, X))
10.61/3.59	   U3_gga(U, V, X, A, d_out_gga(V, X, B)) -> d_out_gga(times(U, V), X, +(times(B, U), times(A, V)))
10.61/3.59	
10.61/3.59	The set Q consists of the following terms:
10.61/3.59	
10.61/3.59	   d_in_gga(x0, x1)
10.61/3.59	   isnumber_in_g(x0)
10.61/3.59	   U8_g(x0, x1)
10.61/3.59	   U7_g(x0, x1)
10.61/3.59	   U1_gga(x0, x1, x2)
10.61/3.59	   U5_gga(x0, x1, x2, x3)
10.61/3.59	   U6_gga(x0, x1, x2, x3)
10.61/3.59	   U4_gga(x0, x1, x2, x3)
10.61/3.59	   U2_gga(x0, x1, x2, x3)
10.61/3.59	   U3_gga(x0, x1, x2, x3, x4)
10.61/3.59	
10.61/3.59	We have to consider all (P,Q,R)-chains.
10.61/3.59	----------------------------------------
10.61/3.59	
10.61/3.59	(17) QDPQMonotonicMRRProof (EQUIVALENT)
10.61/3.59	By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.
10.61/3.59	
10.61/3.59	Strictly oriented dependency pairs:
10.61/3.59	
10.61/3.59	   D_IN_GGA(power(U, V), X) -> U5_GGA(U, V, X, isnumber_in_g(V))
10.61/3.59	   U5_GGA(U, V, X, isnumber_out_g(V)) -> D_IN_GGA(U, X)
10.61/3.59	
10.61/3.59	
10.61/3.59	Used ordering: Polynomial interpretation [POLO]:
10.61/3.59	
10.61/3.59	   POL(+(x_1, x_2)) = 2
10.61/3.59	   POL(0) = 0
10.61/3.59	   POL(1) = 2
10.61/3.59	   POL(D_IN_GGA(x_1, x_2)) = 2*x_1
10.61/3.59	   POL(U1_gga(x_1, x_2, x_3)) = 1
10.61/3.59	   POL(U2_GGA(x_1, x_2, x_3, x_4)) = 2*x_2
10.61/3.59	   POL(U2_gga(x_1, x_2, x_3, x_4)) = 1
10.61/3.59	   POL(U3_gga(x_1, x_2, x_3, x_4, x_5)) = 1
10.61/3.59	   POL(U4_gga(x_1, x_2, x_3, x_4)) = 2
10.61/3.59	   POL(U5_GGA(x_1, x_2, x_3, x_4)) = 2 + 2*x_1
10.61/3.59	   POL(U5_gga(x_1, x_2, x_3, x_4)) = 0
10.61/3.59	   POL(U6_gga(x_1, x_2, x_3, x_4)) = 0
10.61/3.59	   POL(U7_g(x_1, x_2)) = 1
10.61/3.59	   POL(U8_g(x_1, x_2)) = 2
10.61/3.59	   POL(d_in_gga(x_1, x_2)) = 2
10.61/3.59	   POL(d_out_gga(x_1, x_2, x_3)) = 0
10.61/3.59	   POL(div(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
10.61/3.59	   POL(isnumber_in_g(x_1)) = 2 + x_1
10.61/3.59	   POL(isnumber_out_g(x_1)) = 0
10.61/3.59	   POL(p(x_1)) = 0
10.61/3.59	   POL(power(x_1, x_2)) = 2 + 2*x_1 + x_2
10.61/3.59	   POL(s(x_1)) = 2*x_1
10.61/3.59	   POL(times(x_1, x_2)) = 2*x_1 + x_2
10.61/3.59	
10.61/3.59	
10.61/3.59	----------------------------------------
10.61/3.59	
10.61/3.59	(18)
10.61/3.59	Obligation:
10.61/3.59	Q DP problem:
10.61/3.59	The TRS P consists of the following rules:
10.61/3.59	
10.61/3.59	   D_IN_GGA(times(U, V), X) -> U2_GGA(U, V, X, d_in_gga(U, X))
10.61/3.59	   U2_GGA(U, V, X, d_out_gga(U, X, A)) -> D_IN_GGA(V, X)
10.61/3.59	   D_IN_GGA(times(U, V), X) -> D_IN_GGA(U, X)
10.61/3.59	   D_IN_GGA(div(U, V), X) -> D_IN_GGA(times(U, power(V, p(0))), X)
10.61/3.59	
10.61/3.59	The TRS R consists of the following rules:
10.61/3.59	
10.61/3.59	   d_in_gga(X, X) -> d_out_gga(X, X, 1)
10.61/3.59	   d_in_gga(T, X1) -> U1_gga(T, X1, isnumber_in_g(T))
10.61/3.59	   isnumber_in_g(0) -> isnumber_out_g(0)
10.61/3.59	   isnumber_in_g(s(X)) -> U7_g(X, isnumber_in_g(X))
10.61/3.59	   isnumber_in_g(p(X)) -> U8_g(X, isnumber_in_g(X))
10.61/3.59	   U8_g(X, isnumber_out_g(X)) -> isnumber_out_g(p(X))
10.61/3.59	   U7_g(X, isnumber_out_g(X)) -> isnumber_out_g(s(X))
10.61/3.59	   U1_gga(T, X1, isnumber_out_g(T)) -> d_out_gga(T, X1, 0)
10.61/3.59	   d_in_gga(times(U, V), X) -> U2_gga(U, V, X, d_in_gga(U, X))
10.61/3.59	   d_in_gga(div(U, V), X) -> U4_gga(U, V, X, d_in_gga(times(U, power(V, p(0))), X))
10.61/3.59	   d_in_gga(power(U, V), X) -> U5_gga(U, V, X, isnumber_in_g(V))
10.61/3.59	   U5_gga(U, V, X, isnumber_out_g(V)) -> U6_gga(U, V, X, d_in_gga(U, X))
10.61/3.59	   U6_gga(U, V, X, d_out_gga(U, X, W)) -> d_out_gga(power(U, V), X, times(V, times(W, power(U, p(V)))))
10.61/3.59	   U4_gga(U, V, X, d_out_gga(times(U, power(V, p(0))), X, W)) -> d_out_gga(div(U, V), X, W)
10.61/3.59	   U2_gga(U, V, X, d_out_gga(U, X, A)) -> U3_gga(U, V, X, A, d_in_gga(V, X))
10.61/3.59	   U3_gga(U, V, X, A, d_out_gga(V, X, B)) -> d_out_gga(times(U, V), X, +(times(B, U), times(A, V)))
10.61/3.59	
10.61/3.59	The set Q consists of the following terms:
10.61/3.59	
10.61/3.59	   d_in_gga(x0, x1)
10.61/3.59	   isnumber_in_g(x0)
10.61/3.59	   U8_g(x0, x1)
10.61/3.59	   U7_g(x0, x1)
10.61/3.59	   U1_gga(x0, x1, x2)
10.61/3.59	   U5_gga(x0, x1, x2, x3)
10.61/3.59	   U6_gga(x0, x1, x2, x3)
10.61/3.59	   U4_gga(x0, x1, x2, x3)
10.61/3.59	   U2_gga(x0, x1, x2, x3)
10.61/3.59	   U3_gga(x0, x1, x2, x3, x4)
10.61/3.59	
10.61/3.59	We have to consider all (P,Q,R)-chains.
10.61/3.59	----------------------------------------
10.61/3.59	
10.61/3.59	(19) QDPQMonotonicMRRProof (EQUIVALENT)
10.61/3.59	By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.
10.61/3.59	
10.61/3.59	Strictly oriented dependency pairs:
10.61/3.59	
10.61/3.59	   D_IN_GGA(div(U, V), X) -> D_IN_GGA(times(U, power(V, p(0))), X)
10.61/3.59	
10.61/3.59	
10.61/3.59	Used ordering: Polynomial interpretation [POLO]:
10.61/3.59	
10.61/3.59	   POL(+(x_1, x_2)) = 2
10.61/3.59	   POL(0) = 1
10.61/3.59	   POL(1) = 2
10.61/3.59	   POL(D_IN_GGA(x_1, x_2)) = 2*x_1
10.61/3.59	   POL(U1_gga(x_1, x_2, x_3)) = 2
10.61/3.59	   POL(U2_GGA(x_1, x_2, x_3, x_4)) = 2*x_2
10.61/3.59	   POL(U2_gga(x_1, x_2, x_3, x_4)) = 1
10.61/3.59	   POL(U3_gga(x_1, x_2, x_3, x_4, x_5)) = 1
10.61/3.59	   POL(U4_gga(x_1, x_2, x_3, x_4)) = 2 + x_2
10.61/3.59	   POL(U5_gga(x_1, x_2, x_3, x_4)) = 2
10.61/3.59	   POL(U6_gga(x_1, x_2, x_3, x_4)) = 2
10.61/3.59	   POL(U7_g(x_1, x_2)) = 2
10.61/3.59	   POL(U8_g(x_1, x_2)) = 0
10.61/3.59	   POL(d_in_gga(x_1, x_2)) = 2 + x_1
10.61/3.59	   POL(d_out_gga(x_1, x_2, x_3)) = 0
10.61/3.59	   POL(div(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
10.61/3.59	   POL(isnumber_in_g(x_1)) = 2*x_1
10.61/3.59	   POL(isnumber_out_g(x_1)) = 0
10.61/3.59	   POL(p(x_1)) = 0
10.61/3.59	   POL(power(x_1, x_2)) = 2*x_2
10.61/3.59	   POL(s(x_1)) = 1 + 2*x_1
10.61/3.59	   POL(times(x_1, x_2)) = 2*x_1 + 2*x_2
10.61/3.59	
10.61/3.59	
10.61/3.59	----------------------------------------
10.61/3.59	
10.61/3.59	(20)
10.61/3.59	Obligation:
10.61/3.59	Q DP problem:
10.61/3.59	The TRS P consists of the following rules:
10.61/3.59	
10.61/3.59	   D_IN_GGA(times(U, V), X) -> U2_GGA(U, V, X, d_in_gga(U, X))
10.61/3.59	   U2_GGA(U, V, X, d_out_gga(U, X, A)) -> D_IN_GGA(V, X)
10.61/3.59	   D_IN_GGA(times(U, V), X) -> D_IN_GGA(U, X)
10.61/3.59	
10.61/3.59	The TRS R consists of the following rules:
10.61/3.59	
10.61/3.59	   d_in_gga(X, X) -> d_out_gga(X, X, 1)
10.61/3.59	   d_in_gga(T, X1) -> U1_gga(T, X1, isnumber_in_g(T))
10.61/3.59	   isnumber_in_g(0) -> isnumber_out_g(0)
10.61/3.59	   isnumber_in_g(s(X)) -> U7_g(X, isnumber_in_g(X))
10.61/3.59	   isnumber_in_g(p(X)) -> U8_g(X, isnumber_in_g(X))
10.61/3.59	   U8_g(X, isnumber_out_g(X)) -> isnumber_out_g(p(X))
10.61/3.59	   U7_g(X, isnumber_out_g(X)) -> isnumber_out_g(s(X))
10.61/3.59	   U1_gga(T, X1, isnumber_out_g(T)) -> d_out_gga(T, X1, 0)
10.61/3.59	   d_in_gga(times(U, V), X) -> U2_gga(U, V, X, d_in_gga(U, X))
10.61/3.59	   d_in_gga(div(U, V), X) -> U4_gga(U, V, X, d_in_gga(times(U, power(V, p(0))), X))
10.61/3.59	   d_in_gga(power(U, V), X) -> U5_gga(U, V, X, isnumber_in_g(V))
10.61/3.59	   U5_gga(U, V, X, isnumber_out_g(V)) -> U6_gga(U, V, X, d_in_gga(U, X))
10.61/3.59	   U6_gga(U, V, X, d_out_gga(U, X, W)) -> d_out_gga(power(U, V), X, times(V, times(W, power(U, p(V)))))
10.61/3.59	   U4_gga(U, V, X, d_out_gga(times(U, power(V, p(0))), X, W)) -> d_out_gga(div(U, V), X, W)
10.61/3.59	   U2_gga(U, V, X, d_out_gga(U, X, A)) -> U3_gga(U, V, X, A, d_in_gga(V, X))
10.61/3.59	   U3_gga(U, V, X, A, d_out_gga(V, X, B)) -> d_out_gga(times(U, V), X, +(times(B, U), times(A, V)))
10.61/3.59	
10.61/3.59	The set Q consists of the following terms:
10.61/3.59	
10.61/3.59	   d_in_gga(x0, x1)
10.61/3.59	   isnumber_in_g(x0)
10.61/3.59	   U8_g(x0, x1)
10.61/3.59	   U7_g(x0, x1)
10.61/3.59	   U1_gga(x0, x1, x2)
10.61/3.59	   U5_gga(x0, x1, x2, x3)
10.61/3.59	   U6_gga(x0, x1, x2, x3)
10.61/3.59	   U4_gga(x0, x1, x2, x3)
10.61/3.59	   U2_gga(x0, x1, x2, x3)
10.61/3.59	   U3_gga(x0, x1, x2, x3, x4)
10.61/3.59	
10.61/3.59	We have to consider all (P,Q,R)-chains.
10.61/3.59	----------------------------------------
10.61/3.59	
10.61/3.59	(21) QDPSizeChangeProof (EQUIVALENT)
10.61/3.59	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. 
10.77/3.59	
10.77/3.59	From the DPs we obtained the following set of size-change graphs:
10.77/3.59	*U2_GGA(U, V, X, d_out_gga(U, X, A)) -> D_IN_GGA(V, X)
10.77/3.59	The graph contains the following edges 2 >= 1, 3 >= 2, 4 > 2
10.77/3.59	
10.77/3.59	
10.77/3.59	*D_IN_GGA(times(U, V), X) -> D_IN_GGA(U, X)
10.77/3.59	The graph contains the following edges 1 > 1, 2 >= 2
10.77/3.59	
10.77/3.59	
10.77/3.59	*D_IN_GGA(times(U, V), X) -> U2_GGA(U, V, X, d_in_gga(U, X))
10.77/3.59	The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3
10.77/3.59	
10.77/3.59	
10.77/3.59	----------------------------------------
10.77/3.59	
10.77/3.59	(22)
10.77/3.59	YES
10.87/3.66	EOF