6.69/2.52	YES
6.69/2.54	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
6.69/2.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.69/2.54	
6.69/2.54	
6.69/2.54	Left Termination of the query pattern
6.69/2.54	
6.69/2.54	factor(g,a)
6.69/2.54	
6.69/2.54	w.r.t. the given Prolog program could successfully be proven:
6.69/2.54	
6.69/2.54	(0) Prolog
6.69/2.54	(1) PrologToPiTRSProof [SOUND, 0 ms]
6.69/2.54	(2) PiTRS
6.69/2.54	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
6.69/2.54	(4) PiDP
6.69/2.54	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
6.69/2.54	(6) AND
6.69/2.54	    (7) PiDP
6.69/2.54	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
6.69/2.54	        (9) PiDP
6.69/2.54	        (10) PiDPToQDPProof [SOUND, 0 ms]
6.69/2.54	        (11) QDP
6.69/2.54	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.69/2.54	        (13) YES
6.69/2.54	    (14) PiDP
6.69/2.54	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
6.69/2.54	        (16) PiDP
6.69/2.54	        (17) PiDPToQDPProof [SOUND, 0 ms]
6.69/2.54	        (18) QDP
6.69/2.54	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.69/2.54	        (20) YES
6.69/2.54	    (21) PiDP
6.69/2.54	        (22) UsableRulesProof [EQUIVALENT, 0 ms]
6.69/2.54	        (23) PiDP
6.69/2.54	        (24) PiDPToQDPProof [SOUND, 0 ms]
6.69/2.54	        (25) QDP
6.69/2.54	        (26) QDPOrderProof [EQUIVALENT, 36 ms]
6.69/2.54	        (27) QDP
6.69/2.54	        (28) DependencyGraphProof [EQUIVALENT, 0 ms]
6.69/2.54	        (29) TRUE
6.69/2.54	
6.69/2.54	
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(0)
6.69/2.54	Obligation:
6.69/2.54	Clauses:
6.69/2.54	
6.69/2.54	factor(.(X, []), X).
6.69/2.54	factor(.(X, .(Y, Xs)), T) :- ','(times(X, Y, Z), factor(.(Z, Xs), T)).
6.69/2.54	times(0, X_, 0).
6.69/2.54	times(s(X), Y, Z) :- ','(times(X, Y, XY), plus(XY, Y, Z)).
6.69/2.54	plus(0, X, X).
6.69/2.54	plus(s(X), Y, s(Z)) :- plus(X, Y, Z).
6.69/2.54	
6.69/2.54	
6.69/2.54	Query: factor(g,a)
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(1) PrologToPiTRSProof (SOUND)
6.69/2.54	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.69/2.54	
6.69/2.54	factor_in_2: (b,f)
6.69/2.54	
6.69/2.54	times_in_3: (b,b,f)
6.69/2.54	
6.69/2.54	plus_in_3: (b,b,f)
6.69/2.54	
6.69/2.54	Transforming Prolog into the following Term Rewriting System:
6.69/2.54	
6.69/2.54	Pi-finite rewrite system:
6.69/2.54	The TRS R consists of the following rules:
6.69/2.54	
6.69/2.54	   factor_in_ga(.(X, []), X) -> factor_out_ga(.(X, []), X)
6.69/2.54	   factor_in_ga(.(X, .(Y, Xs)), T) -> U1_ga(X, Y, Xs, T, times_in_gga(X, Y, Z))
6.69/2.54	   times_in_gga(0, X_, 0) -> times_out_gga(0, X_, 0)
6.69/2.54	   times_in_gga(s(X), Y, Z) -> U3_gga(X, Y, Z, times_in_gga(X, Y, XY))
6.69/2.54	   U3_gga(X, Y, Z, times_out_gga(X, Y, XY)) -> U4_gga(X, Y, Z, plus_in_gga(XY, Y, Z))
6.69/2.54	   plus_in_gga(0, X, X) -> plus_out_gga(0, X, X)
6.69/2.54	   plus_in_gga(s(X), Y, s(Z)) -> U5_gga(X, Y, Z, plus_in_gga(X, Y, Z))
6.69/2.54	   U5_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
6.69/2.54	   U4_gga(X, Y, Z, plus_out_gga(XY, Y, Z)) -> times_out_gga(s(X), Y, Z)
6.69/2.54	   U1_ga(X, Y, Xs, T, times_out_gga(X, Y, Z)) -> U2_ga(X, Y, Xs, T, factor_in_ga(.(Z, Xs), T))
6.69/2.54	   U2_ga(X, Y, Xs, T, factor_out_ga(.(Z, Xs), T)) -> factor_out_ga(.(X, .(Y, Xs)), T)
6.69/2.54	
6.69/2.54	The argument filtering Pi contains the following mapping:
6.69/2.54	factor_in_ga(x1, x2)  =  factor_in_ga(x1)
6.69/2.54	
6.69/2.54	.(x1, x2)  =  .(x1, x2)
6.69/2.54	
6.69/2.54	[]  =  []
6.69/2.54	
6.69/2.54	factor_out_ga(x1, x2)  =  factor_out_ga(x2)
6.69/2.54	
6.69/2.54	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x3, x5)
6.69/2.54	
6.69/2.54	times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
6.69/2.54	
6.69/2.54	0  =  0
6.69/2.54	
6.69/2.54	times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
6.69/2.54	
6.69/2.54	s(x1)  =  s(x1)
6.69/2.54	
6.69/2.54	U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
6.69/2.54	
6.69/2.54	U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
6.69/2.54	
6.69/2.54	plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
6.69/2.54	
6.69/2.54	plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
6.69/2.54	
6.69/2.54	U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
6.69/2.54	
6.69/2.54	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
6.69/2.54	
6.69/2.54	
6.69/2.54	
6.69/2.54	
6.69/2.54	
6.69/2.54	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.69/2.54	
6.69/2.54	
6.69/2.54	
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(2)
6.69/2.54	Obligation:
6.69/2.54	Pi-finite rewrite system:
6.69/2.54	The TRS R consists of the following rules:
6.69/2.54	
6.69/2.54	   factor_in_ga(.(X, []), X) -> factor_out_ga(.(X, []), X)
6.69/2.54	   factor_in_ga(.(X, .(Y, Xs)), T) -> U1_ga(X, Y, Xs, T, times_in_gga(X, Y, Z))
6.69/2.54	   times_in_gga(0, X_, 0) -> times_out_gga(0, X_, 0)
6.69/2.54	   times_in_gga(s(X), Y, Z) -> U3_gga(X, Y, Z, times_in_gga(X, Y, XY))
6.69/2.54	   U3_gga(X, Y, Z, times_out_gga(X, Y, XY)) -> U4_gga(X, Y, Z, plus_in_gga(XY, Y, Z))
6.69/2.54	   plus_in_gga(0, X, X) -> plus_out_gga(0, X, X)
6.69/2.54	   plus_in_gga(s(X), Y, s(Z)) -> U5_gga(X, Y, Z, plus_in_gga(X, Y, Z))
6.69/2.54	   U5_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
6.69/2.54	   U4_gga(X, Y, Z, plus_out_gga(XY, Y, Z)) -> times_out_gga(s(X), Y, Z)
6.69/2.54	   U1_ga(X, Y, Xs, T, times_out_gga(X, Y, Z)) -> U2_ga(X, Y, Xs, T, factor_in_ga(.(Z, Xs), T))
6.69/2.54	   U2_ga(X, Y, Xs, T, factor_out_ga(.(Z, Xs), T)) -> factor_out_ga(.(X, .(Y, Xs)), T)
6.69/2.54	
6.69/2.54	The argument filtering Pi contains the following mapping:
6.69/2.54	factor_in_ga(x1, x2)  =  factor_in_ga(x1)
6.69/2.54	
6.69/2.54	.(x1, x2)  =  .(x1, x2)
6.69/2.54	
6.69/2.54	[]  =  []
6.69/2.54	
6.69/2.54	factor_out_ga(x1, x2)  =  factor_out_ga(x2)
6.69/2.54	
6.69/2.54	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x3, x5)
6.69/2.54	
6.69/2.54	times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
6.69/2.54	
6.69/2.54	0  =  0
6.69/2.54	
6.69/2.54	times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
6.69/2.54	
6.69/2.54	s(x1)  =  s(x1)
6.69/2.54	
6.69/2.54	U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
6.69/2.54	
6.69/2.54	U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
6.69/2.54	
6.69/2.54	plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
6.69/2.54	
6.69/2.54	plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
6.69/2.54	
6.69/2.54	U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
6.69/2.54	
6.69/2.54	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
6.69/2.54	
6.69/2.54	
6.69/2.54	
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(3) DependencyPairsProof (EQUIVALENT)
6.69/2.54	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.69/2.54	Pi DP problem:
6.69/2.54	The TRS P consists of the following rules:
6.69/2.54	
6.69/2.54	   FACTOR_IN_GA(.(X, .(Y, Xs)), T) -> U1_GA(X, Y, Xs, T, times_in_gga(X, Y, Z))
6.69/2.54	   FACTOR_IN_GA(.(X, .(Y, Xs)), T) -> TIMES_IN_GGA(X, Y, Z)
6.69/2.54	   TIMES_IN_GGA(s(X), Y, Z) -> U3_GGA(X, Y, Z, times_in_gga(X, Y, XY))
6.69/2.54	   TIMES_IN_GGA(s(X), Y, Z) -> TIMES_IN_GGA(X, Y, XY)
6.69/2.54	   U3_GGA(X, Y, Z, times_out_gga(X, Y, XY)) -> U4_GGA(X, Y, Z, plus_in_gga(XY, Y, Z))
6.69/2.54	   U3_GGA(X, Y, Z, times_out_gga(X, Y, XY)) -> PLUS_IN_GGA(XY, Y, Z)
6.69/2.54	   PLUS_IN_GGA(s(X), Y, s(Z)) -> U5_GGA(X, Y, Z, plus_in_gga(X, Y, Z))
6.69/2.54	   PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z)
6.69/2.54	   U1_GA(X, Y, Xs, T, times_out_gga(X, Y, Z)) -> U2_GA(X, Y, Xs, T, factor_in_ga(.(Z, Xs), T))
6.69/2.54	   U1_GA(X, Y, Xs, T, times_out_gga(X, Y, Z)) -> FACTOR_IN_GA(.(Z, Xs), T)
6.69/2.54	
6.69/2.54	The TRS R consists of the following rules:
6.69/2.54	
6.69/2.54	   factor_in_ga(.(X, []), X) -> factor_out_ga(.(X, []), X)
6.69/2.54	   factor_in_ga(.(X, .(Y, Xs)), T) -> U1_ga(X, Y, Xs, T, times_in_gga(X, Y, Z))
6.69/2.54	   times_in_gga(0, X_, 0) -> times_out_gga(0, X_, 0)
6.69/2.54	   times_in_gga(s(X), Y, Z) -> U3_gga(X, Y, Z, times_in_gga(X, Y, XY))
6.69/2.54	   U3_gga(X, Y, Z, times_out_gga(X, Y, XY)) -> U4_gga(X, Y, Z, plus_in_gga(XY, Y, Z))
6.69/2.54	   plus_in_gga(0, X, X) -> plus_out_gga(0, X, X)
6.69/2.54	   plus_in_gga(s(X), Y, s(Z)) -> U5_gga(X, Y, Z, plus_in_gga(X, Y, Z))
6.69/2.54	   U5_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
6.69/2.54	   U4_gga(X, Y, Z, plus_out_gga(XY, Y, Z)) -> times_out_gga(s(X), Y, Z)
6.69/2.54	   U1_ga(X, Y, Xs, T, times_out_gga(X, Y, Z)) -> U2_ga(X, Y, Xs, T, factor_in_ga(.(Z, Xs), T))
6.69/2.54	   U2_ga(X, Y, Xs, T, factor_out_ga(.(Z, Xs), T)) -> factor_out_ga(.(X, .(Y, Xs)), T)
6.69/2.54	
6.69/2.54	The argument filtering Pi contains the following mapping:
6.69/2.54	factor_in_ga(x1, x2)  =  factor_in_ga(x1)
6.69/2.54	
6.69/2.54	.(x1, x2)  =  .(x1, x2)
6.69/2.54	
6.69/2.54	[]  =  []
6.69/2.54	
6.69/2.54	factor_out_ga(x1, x2)  =  factor_out_ga(x2)
6.69/2.54	
6.69/2.54	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x3, x5)
6.69/2.54	
6.69/2.54	times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
6.69/2.54	
6.69/2.54	0  =  0
6.69/2.54	
6.69/2.54	times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
6.69/2.54	
6.69/2.54	s(x1)  =  s(x1)
6.69/2.54	
6.69/2.54	U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
6.69/2.54	
6.69/2.54	U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
6.69/2.54	
6.69/2.54	plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
6.69/2.54	
6.69/2.54	plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
6.69/2.54	
6.69/2.54	U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
6.69/2.54	
6.69/2.54	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
6.69/2.54	
6.69/2.54	FACTOR_IN_GA(x1, x2)  =  FACTOR_IN_GA(x1)
6.69/2.54	
6.69/2.54	U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x3, x5)
6.69/2.54	
6.69/2.54	TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)
6.69/2.54	
6.69/2.54	U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x2, x4)
6.69/2.54	
6.69/2.54	U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x4)
6.69/2.54	
6.69/2.54	PLUS_IN_GGA(x1, x2, x3)  =  PLUS_IN_GGA(x1, x2)
6.69/2.54	
6.69/2.54	U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)
6.69/2.54	
6.69/2.54	U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x5)
6.69/2.54	
6.69/2.54	
6.69/2.54	We have to consider all (P,R,Pi)-chains
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(4)
6.69/2.54	Obligation:
6.69/2.54	Pi DP problem:
6.69/2.54	The TRS P consists of the following rules:
6.69/2.54	
6.69/2.54	   FACTOR_IN_GA(.(X, .(Y, Xs)), T) -> U1_GA(X, Y, Xs, T, times_in_gga(X, Y, Z))
6.69/2.54	   FACTOR_IN_GA(.(X, .(Y, Xs)), T) -> TIMES_IN_GGA(X, Y, Z)
6.69/2.54	   TIMES_IN_GGA(s(X), Y, Z) -> U3_GGA(X, Y, Z, times_in_gga(X, Y, XY))
6.69/2.54	   TIMES_IN_GGA(s(X), Y, Z) -> TIMES_IN_GGA(X, Y, XY)
6.69/2.54	   U3_GGA(X, Y, Z, times_out_gga(X, Y, XY)) -> U4_GGA(X, Y, Z, plus_in_gga(XY, Y, Z))
6.69/2.54	   U3_GGA(X, Y, Z, times_out_gga(X, Y, XY)) -> PLUS_IN_GGA(XY, Y, Z)
6.69/2.54	   PLUS_IN_GGA(s(X), Y, s(Z)) -> U5_GGA(X, Y, Z, plus_in_gga(X, Y, Z))
6.69/2.54	   PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z)
6.69/2.54	   U1_GA(X, Y, Xs, T, times_out_gga(X, Y, Z)) -> U2_GA(X, Y, Xs, T, factor_in_ga(.(Z, Xs), T))
6.69/2.54	   U1_GA(X, Y, Xs, T, times_out_gga(X, Y, Z)) -> FACTOR_IN_GA(.(Z, Xs), T)
6.69/2.54	
6.69/2.54	The TRS R consists of the following rules:
6.69/2.54	
6.69/2.54	   factor_in_ga(.(X, []), X) -> factor_out_ga(.(X, []), X)
6.69/2.54	   factor_in_ga(.(X, .(Y, Xs)), T) -> U1_ga(X, Y, Xs, T, times_in_gga(X, Y, Z))
6.69/2.54	   times_in_gga(0, X_, 0) -> times_out_gga(0, X_, 0)
6.69/2.54	   times_in_gga(s(X), Y, Z) -> U3_gga(X, Y, Z, times_in_gga(X, Y, XY))
6.69/2.54	   U3_gga(X, Y, Z, times_out_gga(X, Y, XY)) -> U4_gga(X, Y, Z, plus_in_gga(XY, Y, Z))
6.69/2.54	   plus_in_gga(0, X, X) -> plus_out_gga(0, X, X)
6.69/2.54	   plus_in_gga(s(X), Y, s(Z)) -> U5_gga(X, Y, Z, plus_in_gga(X, Y, Z))
6.69/2.54	   U5_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
6.69/2.54	   U4_gga(X, Y, Z, plus_out_gga(XY, Y, Z)) -> times_out_gga(s(X), Y, Z)
6.69/2.54	   U1_ga(X, Y, Xs, T, times_out_gga(X, Y, Z)) -> U2_ga(X, Y, Xs, T, factor_in_ga(.(Z, Xs), T))
6.69/2.54	   U2_ga(X, Y, Xs, T, factor_out_ga(.(Z, Xs), T)) -> factor_out_ga(.(X, .(Y, Xs)), T)
6.69/2.54	
6.69/2.54	The argument filtering Pi contains the following mapping:
6.69/2.54	factor_in_ga(x1, x2)  =  factor_in_ga(x1)
6.69/2.54	
6.69/2.54	.(x1, x2)  =  .(x1, x2)
6.69/2.54	
6.69/2.54	[]  =  []
6.69/2.54	
6.69/2.54	factor_out_ga(x1, x2)  =  factor_out_ga(x2)
6.69/2.54	
6.69/2.54	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x3, x5)
6.69/2.54	
6.69/2.54	times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
6.69/2.54	
6.69/2.54	0  =  0
6.69/2.54	
6.69/2.54	times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
6.69/2.54	
6.69/2.54	s(x1)  =  s(x1)
6.69/2.54	
6.69/2.54	U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
6.69/2.54	
6.69/2.54	U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
6.69/2.54	
6.69/2.54	plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
6.69/2.54	
6.69/2.54	plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
6.69/2.54	
6.69/2.54	U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
6.69/2.54	
6.69/2.54	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
6.69/2.54	
6.69/2.54	FACTOR_IN_GA(x1, x2)  =  FACTOR_IN_GA(x1)
6.69/2.54	
6.69/2.54	U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x3, x5)
6.69/2.54	
6.69/2.54	TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)
6.69/2.54	
6.69/2.54	U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x2, x4)
6.69/2.54	
6.69/2.54	U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x4)
6.69/2.54	
6.69/2.54	PLUS_IN_GGA(x1, x2, x3)  =  PLUS_IN_GGA(x1, x2)
6.69/2.54	
6.69/2.54	U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)
6.69/2.54	
6.69/2.54	U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x5)
6.69/2.54	
6.69/2.54	
6.69/2.54	We have to consider all (P,R,Pi)-chains
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(5) DependencyGraphProof (EQUIVALENT)
6.69/2.54	The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 6 less nodes.
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(6)
6.69/2.54	Complex Obligation (AND)
6.69/2.54	
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(7)
6.69/2.54	Obligation:
6.69/2.54	Pi DP problem:
6.69/2.54	The TRS P consists of the following rules:
6.69/2.54	
6.69/2.54	   PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z)
6.69/2.54	
6.69/2.54	The TRS R consists of the following rules:
6.69/2.54	
6.69/2.54	   factor_in_ga(.(X, []), X) -> factor_out_ga(.(X, []), X)
6.69/2.54	   factor_in_ga(.(X, .(Y, Xs)), T) -> U1_ga(X, Y, Xs, T, times_in_gga(X, Y, Z))
6.69/2.54	   times_in_gga(0, X_, 0) -> times_out_gga(0, X_, 0)
6.69/2.54	   times_in_gga(s(X), Y, Z) -> U3_gga(X, Y, Z, times_in_gga(X, Y, XY))
6.69/2.54	   U3_gga(X, Y, Z, times_out_gga(X, Y, XY)) -> U4_gga(X, Y, Z, plus_in_gga(XY, Y, Z))
6.69/2.54	   plus_in_gga(0, X, X) -> plus_out_gga(0, X, X)
6.69/2.54	   plus_in_gga(s(X), Y, s(Z)) -> U5_gga(X, Y, Z, plus_in_gga(X, Y, Z))
6.69/2.54	   U5_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
6.69/2.54	   U4_gga(X, Y, Z, plus_out_gga(XY, Y, Z)) -> times_out_gga(s(X), Y, Z)
6.69/2.54	   U1_ga(X, Y, Xs, T, times_out_gga(X, Y, Z)) -> U2_ga(X, Y, Xs, T, factor_in_ga(.(Z, Xs), T))
6.69/2.54	   U2_ga(X, Y, Xs, T, factor_out_ga(.(Z, Xs), T)) -> factor_out_ga(.(X, .(Y, Xs)), T)
6.69/2.54	
6.69/2.54	The argument filtering Pi contains the following mapping:
6.69/2.54	factor_in_ga(x1, x2)  =  factor_in_ga(x1)
6.69/2.54	
6.69/2.54	.(x1, x2)  =  .(x1, x2)
6.69/2.54	
6.69/2.54	[]  =  []
6.69/2.54	
6.69/2.54	factor_out_ga(x1, x2)  =  factor_out_ga(x2)
6.69/2.54	
6.69/2.54	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x3, x5)
6.69/2.54	
6.69/2.54	times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
6.69/2.54	
6.69/2.54	0  =  0
6.69/2.54	
6.69/2.54	times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
6.69/2.54	
6.69/2.54	s(x1)  =  s(x1)
6.69/2.54	
6.69/2.54	U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
6.69/2.54	
6.69/2.54	U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
6.69/2.54	
6.69/2.54	plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
6.69/2.54	
6.69/2.54	plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
6.69/2.54	
6.69/2.54	U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
6.69/2.54	
6.69/2.54	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
6.69/2.54	
6.69/2.54	PLUS_IN_GGA(x1, x2, x3)  =  PLUS_IN_GGA(x1, x2)
6.69/2.54	
6.69/2.54	
6.69/2.54	We have to consider all (P,R,Pi)-chains
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(8) UsableRulesProof (EQUIVALENT)
6.69/2.54	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(9)
6.69/2.54	Obligation:
6.69/2.54	Pi DP problem:
6.69/2.54	The TRS P consists of the following rules:
6.69/2.54	
6.69/2.54	   PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z)
6.69/2.54	
6.69/2.54	R is empty.
6.69/2.54	The argument filtering Pi contains the following mapping:
6.69/2.54	s(x1)  =  s(x1)
6.69/2.54	
6.69/2.54	PLUS_IN_GGA(x1, x2, x3)  =  PLUS_IN_GGA(x1, x2)
6.69/2.54	
6.69/2.54	
6.69/2.54	We have to consider all (P,R,Pi)-chains
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(10) PiDPToQDPProof (SOUND)
6.69/2.54	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(11)
6.69/2.54	Obligation:
6.69/2.54	Q DP problem:
6.69/2.54	The TRS P consists of the following rules:
6.69/2.54	
6.69/2.54	   PLUS_IN_GGA(s(X), Y) -> PLUS_IN_GGA(X, Y)
6.69/2.54	
6.69/2.54	R is empty.
6.69/2.54	Q is empty.
6.69/2.54	We have to consider all (P,Q,R)-chains.
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(12) QDPSizeChangeProof (EQUIVALENT)
6.69/2.54	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. 
6.69/2.54	
6.69/2.54	From the DPs we obtained the following set of size-change graphs:
6.69/2.54	*PLUS_IN_GGA(s(X), Y) -> PLUS_IN_GGA(X, Y)
6.69/2.54	The graph contains the following edges 1 > 1, 2 >= 2
6.69/2.54	
6.69/2.54	
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(13)
6.69/2.54	YES
6.69/2.54	
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(14)
6.69/2.54	Obligation:
6.69/2.54	Pi DP problem:
6.69/2.54	The TRS P consists of the following rules:
6.69/2.54	
6.69/2.54	   TIMES_IN_GGA(s(X), Y, Z) -> TIMES_IN_GGA(X, Y, XY)
6.69/2.54	
6.69/2.54	The TRS R consists of the following rules:
6.69/2.54	
6.69/2.54	   factor_in_ga(.(X, []), X) -> factor_out_ga(.(X, []), X)
6.69/2.54	   factor_in_ga(.(X, .(Y, Xs)), T) -> U1_ga(X, Y, Xs, T, times_in_gga(X, Y, Z))
6.69/2.54	   times_in_gga(0, X_, 0) -> times_out_gga(0, X_, 0)
6.69/2.54	   times_in_gga(s(X), Y, Z) -> U3_gga(X, Y, Z, times_in_gga(X, Y, XY))
6.69/2.54	   U3_gga(X, Y, Z, times_out_gga(X, Y, XY)) -> U4_gga(X, Y, Z, plus_in_gga(XY, Y, Z))
6.69/2.54	   plus_in_gga(0, X, X) -> plus_out_gga(0, X, X)
6.69/2.54	   plus_in_gga(s(X), Y, s(Z)) -> U5_gga(X, Y, Z, plus_in_gga(X, Y, Z))
6.69/2.54	   U5_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
6.69/2.54	   U4_gga(X, Y, Z, plus_out_gga(XY, Y, Z)) -> times_out_gga(s(X), Y, Z)
6.69/2.54	   U1_ga(X, Y, Xs, T, times_out_gga(X, Y, Z)) -> U2_ga(X, Y, Xs, T, factor_in_ga(.(Z, Xs), T))
6.69/2.54	   U2_ga(X, Y, Xs, T, factor_out_ga(.(Z, Xs), T)) -> factor_out_ga(.(X, .(Y, Xs)), T)
6.69/2.54	
6.69/2.54	The argument filtering Pi contains the following mapping:
6.69/2.54	factor_in_ga(x1, x2)  =  factor_in_ga(x1)
6.69/2.54	
6.69/2.54	.(x1, x2)  =  .(x1, x2)
6.69/2.54	
6.69/2.54	[]  =  []
6.69/2.54	
6.69/2.54	factor_out_ga(x1, x2)  =  factor_out_ga(x2)
6.69/2.54	
6.69/2.54	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x3, x5)
6.69/2.54	
6.69/2.54	times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
6.69/2.54	
6.69/2.54	0  =  0
6.69/2.54	
6.69/2.54	times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
6.69/2.54	
6.69/2.54	s(x1)  =  s(x1)
6.69/2.54	
6.69/2.54	U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
6.69/2.54	
6.69/2.54	U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
6.69/2.54	
6.69/2.54	plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
6.69/2.54	
6.69/2.54	plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
6.69/2.54	
6.69/2.54	U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
6.69/2.54	
6.69/2.54	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
6.69/2.54	
6.69/2.54	TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)
6.69/2.54	
6.69/2.54	
6.69/2.54	We have to consider all (P,R,Pi)-chains
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(15) UsableRulesProof (EQUIVALENT)
6.69/2.54	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(16)
6.69/2.54	Obligation:
6.69/2.54	Pi DP problem:
6.69/2.54	The TRS P consists of the following rules:
6.69/2.54	
6.69/2.54	   TIMES_IN_GGA(s(X), Y, Z) -> TIMES_IN_GGA(X, Y, XY)
6.69/2.54	
6.69/2.54	R is empty.
6.69/2.54	The argument filtering Pi contains the following mapping:
6.69/2.54	s(x1)  =  s(x1)
6.69/2.54	
6.69/2.54	TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)
6.69/2.54	
6.69/2.54	
6.69/2.54	We have to consider all (P,R,Pi)-chains
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(17) PiDPToQDPProof (SOUND)
6.69/2.54	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(18)
6.69/2.54	Obligation:
6.69/2.54	Q DP problem:
6.69/2.54	The TRS P consists of the following rules:
6.69/2.54	
6.69/2.54	   TIMES_IN_GGA(s(X), Y) -> TIMES_IN_GGA(X, Y)
6.69/2.54	
6.69/2.54	R is empty.
6.69/2.54	Q is empty.
6.69/2.54	We have to consider all (P,Q,R)-chains.
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(19) QDPSizeChangeProof (EQUIVALENT)
6.69/2.54	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. 
6.69/2.54	
6.69/2.54	From the DPs we obtained the following set of size-change graphs:
6.69/2.54	*TIMES_IN_GGA(s(X), Y) -> TIMES_IN_GGA(X, Y)
6.69/2.54	The graph contains the following edges 1 > 1, 2 >= 2
6.69/2.54	
6.69/2.54	
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(20)
6.69/2.54	YES
6.69/2.54	
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(21)
6.69/2.54	Obligation:
6.69/2.54	Pi DP problem:
6.69/2.54	The TRS P consists of the following rules:
6.69/2.54	
6.69/2.54	   U1_GA(X, Y, Xs, T, times_out_gga(X, Y, Z)) -> FACTOR_IN_GA(.(Z, Xs), T)
6.69/2.54	   FACTOR_IN_GA(.(X, .(Y, Xs)), T) -> U1_GA(X, Y, Xs, T, times_in_gga(X, Y, Z))
6.69/2.54	
6.69/2.54	The TRS R consists of the following rules:
6.69/2.54	
6.69/2.54	   factor_in_ga(.(X, []), X) -> factor_out_ga(.(X, []), X)
6.69/2.54	   factor_in_ga(.(X, .(Y, Xs)), T) -> U1_ga(X, Y, Xs, T, times_in_gga(X, Y, Z))
6.69/2.54	   times_in_gga(0, X_, 0) -> times_out_gga(0, X_, 0)
6.69/2.54	   times_in_gga(s(X), Y, Z) -> U3_gga(X, Y, Z, times_in_gga(X, Y, XY))
6.69/2.54	   U3_gga(X, Y, Z, times_out_gga(X, Y, XY)) -> U4_gga(X, Y, Z, plus_in_gga(XY, Y, Z))
6.69/2.54	   plus_in_gga(0, X, X) -> plus_out_gga(0, X, X)
6.69/2.54	   plus_in_gga(s(X), Y, s(Z)) -> U5_gga(X, Y, Z, plus_in_gga(X, Y, Z))
6.69/2.54	   U5_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
6.69/2.54	   U4_gga(X, Y, Z, plus_out_gga(XY, Y, Z)) -> times_out_gga(s(X), Y, Z)
6.69/2.54	   U1_ga(X, Y, Xs, T, times_out_gga(X, Y, Z)) -> U2_ga(X, Y, Xs, T, factor_in_ga(.(Z, Xs), T))
6.69/2.54	   U2_ga(X, Y, Xs, T, factor_out_ga(.(Z, Xs), T)) -> factor_out_ga(.(X, .(Y, Xs)), T)
6.69/2.54	
6.69/2.54	The argument filtering Pi contains the following mapping:
6.69/2.54	factor_in_ga(x1, x2)  =  factor_in_ga(x1)
6.69/2.54	
6.69/2.54	.(x1, x2)  =  .(x1, x2)
6.69/2.54	
6.69/2.54	[]  =  []
6.69/2.54	
6.69/2.54	factor_out_ga(x1, x2)  =  factor_out_ga(x2)
6.69/2.54	
6.69/2.54	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x3, x5)
6.69/2.54	
6.69/2.54	times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
6.69/2.54	
6.69/2.54	0  =  0
6.69/2.54	
6.69/2.54	times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
6.69/2.54	
6.69/2.54	s(x1)  =  s(x1)
6.69/2.54	
6.69/2.54	U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
6.69/2.54	
6.69/2.54	U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
6.69/2.54	
6.69/2.54	plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
6.69/2.54	
6.69/2.54	plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
6.69/2.54	
6.69/2.54	U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
6.69/2.54	
6.69/2.54	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
6.69/2.54	
6.69/2.54	FACTOR_IN_GA(x1, x2)  =  FACTOR_IN_GA(x1)
6.69/2.54	
6.69/2.54	U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x3, x5)
6.69/2.54	
6.69/2.54	
6.69/2.54	We have to consider all (P,R,Pi)-chains
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(22) UsableRulesProof (EQUIVALENT)
6.69/2.54	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(23)
6.69/2.54	Obligation:
6.69/2.54	Pi DP problem:
6.69/2.54	The TRS P consists of the following rules:
6.69/2.54	
6.69/2.54	   U1_GA(X, Y, Xs, T, times_out_gga(X, Y, Z)) -> FACTOR_IN_GA(.(Z, Xs), T)
6.69/2.54	   FACTOR_IN_GA(.(X, .(Y, Xs)), T) -> U1_GA(X, Y, Xs, T, times_in_gga(X, Y, Z))
6.69/2.54	
6.69/2.54	The TRS R consists of the following rules:
6.69/2.54	
6.69/2.54	   times_in_gga(0, X_, 0) -> times_out_gga(0, X_, 0)
6.69/2.54	   times_in_gga(s(X), Y, Z) -> U3_gga(X, Y, Z, times_in_gga(X, Y, XY))
6.69/2.54	   U3_gga(X, Y, Z, times_out_gga(X, Y, XY)) -> U4_gga(X, Y, Z, plus_in_gga(XY, Y, Z))
6.69/2.54	   U4_gga(X, Y, Z, plus_out_gga(XY, Y, Z)) -> times_out_gga(s(X), Y, Z)
6.69/2.54	   plus_in_gga(0, X, X) -> plus_out_gga(0, X, X)
6.69/2.54	   plus_in_gga(s(X), Y, s(Z)) -> U5_gga(X, Y, Z, plus_in_gga(X, Y, Z))
6.69/2.54	   U5_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
6.69/2.54	
6.69/2.54	The argument filtering Pi contains the following mapping:
6.69/2.54	.(x1, x2)  =  .(x1, x2)
6.69/2.54	
6.69/2.54	times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
6.69/2.54	
6.69/2.54	0  =  0
6.69/2.54	
6.69/2.54	times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
6.69/2.54	
6.69/2.54	s(x1)  =  s(x1)
6.69/2.54	
6.69/2.54	U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
6.69/2.54	
6.69/2.54	U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
6.69/2.54	
6.69/2.54	plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
6.69/2.54	
6.69/2.54	plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
6.69/2.54	
6.69/2.54	U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
6.69/2.54	
6.69/2.54	FACTOR_IN_GA(x1, x2)  =  FACTOR_IN_GA(x1)
6.69/2.54	
6.69/2.54	U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x3, x5)
6.69/2.54	
6.69/2.54	
6.69/2.54	We have to consider all (P,R,Pi)-chains
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(24) PiDPToQDPProof (SOUND)
6.69/2.54	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(25)
6.69/2.54	Obligation:
6.69/2.54	Q DP problem:
6.69/2.54	The TRS P consists of the following rules:
6.69/2.54	
6.69/2.54	   U1_GA(Xs, times_out_gga(Z)) -> FACTOR_IN_GA(.(Z, Xs))
6.69/2.54	   FACTOR_IN_GA(.(X, .(Y, Xs))) -> U1_GA(Xs, times_in_gga(X, Y))
6.69/2.54	
6.69/2.54	The TRS R consists of the following rules:
6.69/2.54	
6.69/2.54	   times_in_gga(0, X_) -> times_out_gga(0)
6.69/2.54	   times_in_gga(s(X), Y) -> U3_gga(Y, times_in_gga(X, Y))
6.69/2.54	   U3_gga(Y, times_out_gga(XY)) -> U4_gga(plus_in_gga(XY, Y))
6.69/2.54	   U4_gga(plus_out_gga(Z)) -> times_out_gga(Z)
6.69/2.54	   plus_in_gga(0, X) -> plus_out_gga(X)
6.69/2.54	   plus_in_gga(s(X), Y) -> U5_gga(plus_in_gga(X, Y))
6.69/2.54	   U5_gga(plus_out_gga(Z)) -> plus_out_gga(s(Z))
6.69/2.54	
6.69/2.54	The set Q consists of the following terms:
6.69/2.54	
6.69/2.54	   times_in_gga(x0, x1)
6.69/2.54	   U3_gga(x0, x1)
6.69/2.54	   U4_gga(x0)
6.69/2.54	   plus_in_gga(x0, x1)
6.69/2.54	   U5_gga(x0)
6.69/2.54	
6.69/2.54	We have to consider all (P,Q,R)-chains.
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(26) QDPOrderProof (EQUIVALENT)
6.69/2.54	We use the reduction pair processor [LPAR04,JAR06].
6.69/2.54	
6.69/2.54	
6.69/2.54	The following pairs can be oriented strictly and are deleted.
6.69/2.54	
6.69/2.54	   FACTOR_IN_GA(.(X, .(Y, Xs))) -> U1_GA(Xs, times_in_gga(X, Y))
6.69/2.54	The remaining pairs can at least be oriented weakly.
6.69/2.54	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
6.69/2.54	
6.69/2.54	POL( U1_GA_2(x_1, x_2) ) = 2x_1
6.69/2.54	POL( times_in_gga_2(x_1, x_2) ) = 0
6.69/2.54	POL( 0 ) = 2
6.69/2.54	POL( times_out_gga_1(x_1) ) = max{0, 2x_1 - 2}
6.69/2.54	POL( s_1(x_1) ) = 2x_1
6.69/2.54	POL( U3_gga_2(x_1, x_2) ) = max{0, x_1 - 2}
6.69/2.54	POL( U4_gga_1(x_1) ) = 2
6.69/2.54	POL( plus_in_gga_2(x_1, x_2) ) = 0
6.69/2.54	POL( plus_out_gga_1(x_1) ) = max{0, x_1 - 2}
6.69/2.54	POL( U5_gga_1(x_1) ) = max{0, -2}
6.69/2.54	POL( FACTOR_IN_GA_1(x_1) ) = max{0, 2x_1 - 2}
6.69/2.54	POL( ._2(x_1, x_2) ) = x_2 + 1
6.69/2.54	
6.69/2.54	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
6.69/2.54	none
6.69/2.54	
6.69/2.54	
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(27)
6.69/2.54	Obligation:
6.69/2.54	Q DP problem:
6.69/2.54	The TRS P consists of the following rules:
6.69/2.54	
6.69/2.54	   U1_GA(Xs, times_out_gga(Z)) -> FACTOR_IN_GA(.(Z, Xs))
6.69/2.54	
6.69/2.54	The TRS R consists of the following rules:
6.69/2.54	
6.69/2.54	   times_in_gga(0, X_) -> times_out_gga(0)
6.69/2.54	   times_in_gga(s(X), Y) -> U3_gga(Y, times_in_gga(X, Y))
6.69/2.54	   U3_gga(Y, times_out_gga(XY)) -> U4_gga(plus_in_gga(XY, Y))
6.69/2.54	   U4_gga(plus_out_gga(Z)) -> times_out_gga(Z)
6.69/2.54	   plus_in_gga(0, X) -> plus_out_gga(X)
6.69/2.54	   plus_in_gga(s(X), Y) -> U5_gga(plus_in_gga(X, Y))
6.69/2.54	   U5_gga(plus_out_gga(Z)) -> plus_out_gga(s(Z))
6.69/2.54	
6.69/2.54	The set Q consists of the following terms:
6.69/2.54	
6.69/2.54	   times_in_gga(x0, x1)
6.69/2.54	   U3_gga(x0, x1)
6.69/2.54	   U4_gga(x0)
6.69/2.54	   plus_in_gga(x0, x1)
6.69/2.54	   U5_gga(x0)
6.69/2.54	
6.69/2.54	We have to consider all (P,Q,R)-chains.
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(28) DependencyGraphProof (EQUIVALENT)
6.69/2.54	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
6.69/2.54	----------------------------------------
6.69/2.54	
6.69/2.54	(29)
6.69/2.54	TRUE
6.94/2.61	EOF