18.55/5.64	YES
18.55/5.65	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
18.55/5.65	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
18.55/5.65	
18.55/5.65	
18.55/5.65	Left Termination of the query pattern
18.55/5.65	
18.55/5.65	ways(g,g,a)
18.55/5.65	
18.55/5.65	w.r.t. the given Prolog program could successfully be proven:
18.55/5.65	
18.55/5.65	(0) Prolog
18.55/5.65	(1) UnifyTransformerProof [EQUIVALENT, 0 ms]
18.55/5.65	(2) Prolog
18.55/5.65	(3) PrologToPiTRSProof [SOUND, 34 ms]
18.55/5.65	(4) PiTRS
18.55/5.65	(5) DependencyPairsProof [EQUIVALENT, 32 ms]
18.55/5.65	(6) PiDP
18.55/5.65	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
18.55/5.65	(8) AND
18.55/5.65	    (9) PiDP
18.55/5.65	        (10) UsableRulesProof [EQUIVALENT, 0 ms]
18.55/5.65	        (11) PiDP
18.55/5.65	        (12) PiDPToQDPProof [EQUIVALENT, 0 ms]
18.55/5.65	        (13) QDP
18.55/5.65	        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
18.55/5.65	        (15) YES
18.55/5.65	    (16) PiDP
18.55/5.65	        (17) UsableRulesProof [EQUIVALENT, 0 ms]
18.55/5.65	        (18) PiDP
18.55/5.65	        (19) PiDPToQDPProof [SOUND, 0 ms]
18.55/5.65	        (20) QDP
18.55/5.65	        (21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
18.55/5.65	        (22) YES
18.55/5.65	    (23) PiDP
18.55/5.65	        (24) UsableRulesProof [EQUIVALENT, 0 ms]
18.55/5.65	        (25) PiDP
18.55/5.65	        (26) PiDPToQDPProof [SOUND, 0 ms]
18.55/5.65	        (27) QDP
18.55/5.65	        (28) QDPSizeChangeProof [EQUIVALENT, 0 ms]
18.55/5.65	        (29) YES
18.55/5.65	    (30) PiDP
18.55/5.65	        (31) PiDPToQDPProof [SOUND, 0 ms]
18.55/5.65	        (32) QDP
18.55/5.65	        (33) TransformationProof [EQUIVALENT, 0 ms]
18.55/5.65	        (34) QDP
18.55/5.65	        (35) TransformationProof [EQUIVALENT, 0 ms]
18.55/5.65	        (36) QDP
18.55/5.65	        (37) TransformationProof [EQUIVALENT, 0 ms]
18.55/5.65	        (38) QDP
18.55/5.65	        (39) TransformationProof [EQUIVALENT, 0 ms]
18.55/5.65	        (40) QDP
18.55/5.65	        (41) TransformationProof [EQUIVALENT, 0 ms]
18.55/5.65	        (42) QDP
18.55/5.65	        (43) TransformationProof [EQUIVALENT, 0 ms]
18.55/5.65	        (44) QDP
18.55/5.65	        (45) QDPOrderProof [EQUIVALENT, 169 ms]
18.55/5.65	        (46) QDP
18.55/5.65	        (47) DependencyGraphProof [EQUIVALENT, 0 ms]
18.55/5.65	        (48) QDP
18.55/5.65	        (49) QDPOrderProof [EQUIVALENT, 49 ms]
18.55/5.65	        (50) QDP
18.55/5.65	        (51) DependencyGraphProof [EQUIVALENT, 0 ms]
18.55/5.65	        (52) TRUE
18.55/5.65	
18.55/5.65	
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(0)
18.55/5.65	Obligation:
18.55/5.65	Clauses:
18.55/5.65	
18.55/5.65	nat(0).
18.55/5.65	nat(s(X)) :- nat(X).
18.55/5.65	plus(0, X, X) :- nat(X).
18.55/5.65	plus(s(X), Y, s(Z)) :- plus(X, Y, Z).
18.55/5.65	ways(0, X1, s(0)).
18.55/5.65	ways(X2, [], 0).
18.55/5.65	ways(Amount, .(s(C), Coins), N) :- ','(=(Amount, s(X3)), ','(plus(s(C), NewAmount, Amount), ','(ways(Amount, Coins, N1), ','(ways(NewAmount, .(s(C), Coins), N2), plus(N1, N2, N))))).
18.55/5.65	ways(Amount, .(s(C), Coins), N) :- ','(=(Amount, s(X4)), ','(plus(Amount, s(X5), s(C)), ways(Amount, Coins, N))).
18.55/5.65	
18.55/5.65	
18.55/5.65	Query: ways(g,g,a)
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(1) UnifyTransformerProof (EQUIVALENT)
18.55/5.65	Added a fact for the built-in = predicate [PROLOG].
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(2)
18.55/5.65	Obligation:
18.55/5.65	Clauses:
18.55/5.65	
18.55/5.65	nat(0).
18.55/5.65	nat(s(X)) :- nat(X).
18.55/5.65	plus(0, X, X) :- nat(X).
18.55/5.65	plus(s(X), Y, s(Z)) :- plus(X, Y, Z).
18.55/5.65	ways(0, X1, s(0)).
18.55/5.65	ways(X2, [], 0).
18.55/5.65	ways(Amount, .(s(C), Coins), N) :- ','(=(Amount, s(X3)), ','(plus(s(C), NewAmount, Amount), ','(ways(Amount, Coins, N1), ','(ways(NewAmount, .(s(C), Coins), N2), plus(N1, N2, N))))).
18.55/5.65	ways(Amount, .(s(C), Coins), N) :- ','(=(Amount, s(X4)), ','(plus(Amount, s(X5), s(C)), ways(Amount, Coins, N))).
18.55/5.65	=(X, X).
18.55/5.65	
18.55/5.65	
18.55/5.65	Query: ways(g,g,a)
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(3) PrologToPiTRSProof (SOUND)
18.55/5.65	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
18.55/5.65	
18.55/5.65	ways_in_3: (b,b,f)
18.55/5.65	
18.55/5.65	plus_in_3: (b,f,b) (b,b,f)
18.55/5.65	
18.55/5.65	nat_in_1: (b)
18.55/5.65	
18.55/5.65	Transforming Prolog into the following Term Rewriting System:
18.55/5.65	
18.55/5.65	Pi-finite rewrite system:
18.55/5.65	The TRS R consists of the following rules:
18.55/5.65	
18.55/5.65	   ways_in_gga(0, X1, s(0)) -> ways_out_gga(0, X1, s(0))
18.55/5.65	   ways_in_gga(X2, [], 0) -> ways_out_gga(X2, [], 0)
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins), N) -> U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
18.55/5.65	   =_in_ga(X, X) -> =_out_ga(X, X)
18.55/5.65	   U4_gga(Amount, C, Coins, N, =_out_ga(Amount, s(X3))) -> U5_gga(Amount, C, Coins, N, X3, plus_in_gag(s(C), NewAmount, Amount))
18.55/5.65	   plus_in_gag(0, X, X) -> U2_gag(X, nat_in_g(X))
18.55/5.65	   nat_in_g(0) -> nat_out_g(0)
18.55/5.65	   nat_in_g(s(X)) -> U1_g(X, nat_in_g(X))
18.55/5.65	   U1_g(X, nat_out_g(X)) -> nat_out_g(s(X))
18.55/5.65	   U2_gag(X, nat_out_g(X)) -> plus_out_gag(0, X, X)
18.55/5.65	   plus_in_gag(s(X), Y, s(Z)) -> U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
18.55/5.65	   U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) -> plus_out_gag(s(X), Y, s(Z))
18.55/5.65	   U5_gga(Amount, C, Coins, N, X3, plus_out_gag(s(C), NewAmount, Amount)) -> U6_gga(Amount, C, Coins, N, X3, NewAmount, ways_in_gga(Amount, Coins, N1))
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins), N) -> U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
18.55/5.65	   U9_gga(Amount, C, Coins, N, =_out_ga(Amount, s(X4))) -> U10_gga(Amount, C, Coins, N, X4, plus_in_gag(Amount, s(X5), s(C)))
18.55/5.65	   U10_gga(Amount, C, Coins, N, X4, plus_out_gag(Amount, s(X5), s(C))) -> U11_gga(Amount, C, Coins, N, ways_in_gga(Amount, Coins, N))
18.55/5.65	   U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) -> ways_out_gga(Amount, .(s(C), Coins), N)
18.55/5.65	   U6_gga(Amount, C, Coins, N, X3, NewAmount, ways_out_gga(Amount, Coins, N1)) -> U7_gga(Amount, C, Coins, N, X3, NewAmount, N1, ways_in_gga(NewAmount, .(s(C), Coins), N2))
18.55/5.65	   U7_gga(Amount, C, Coins, N, X3, NewAmount, N1, ways_out_gga(NewAmount, .(s(C), Coins), N2)) -> U8_gga(Amount, C, Coins, N, plus_in_gga(N1, N2, N))
18.55/5.65	   plus_in_gga(0, X, X) -> U2_gga(X, nat_in_g(X))
18.55/5.65	   U2_gga(X, nat_out_g(X)) -> plus_out_gga(0, X, X)
18.55/5.65	   plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
18.55/5.65	   U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
18.55/5.65	   U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) -> ways_out_gga(Amount, .(s(C), Coins), N)
18.55/5.65	
18.55/5.65	The argument filtering Pi contains the following mapping:
18.55/5.65	ways_in_gga(x1, x2, x3)  =  ways_in_gga(x1, x2)
18.55/5.65	
18.55/5.65	0  =  0
18.55/5.65	
18.55/5.65	ways_out_gga(x1, x2, x3)  =  ways_out_gga(x3)
18.55/5.65	
18.55/5.65	[]  =  []
18.55/5.65	
18.55/5.65	.(x1, x2)  =  .(x1, x2)
18.55/5.65	
18.55/5.65	s(x1)  =  s(x1)
18.55/5.65	
18.55/5.65	U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	=_in_ga(x1, x2)  =  =_in_ga(x1)
18.55/5.65	
18.55/5.65	=_out_ga(x1, x2)  =  =_out_ga(x2)
18.55/5.65	
18.55/5.65	U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x6)
18.55/5.65	
18.55/5.65	plus_in_gag(x1, x2, x3)  =  plus_in_gag(x1, x3)
18.55/5.65	
18.55/5.65	U2_gag(x1, x2)  =  U2_gag(x1, x2)
18.55/5.65	
18.55/5.65	nat_in_g(x1)  =  nat_in_g(x1)
18.55/5.65	
18.55/5.65	nat_out_g(x1)  =  nat_out_g
18.55/5.65	
18.55/5.65	U1_g(x1, x2)  =  U1_g(x2)
18.55/5.65	
18.55/5.65	plus_out_gag(x1, x2, x3)  =  plus_out_gag(x2)
18.55/5.65	
18.55/5.65	U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
18.55/5.65	
18.55/5.65	U6_gga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gga(x2, x3, x6, x7)
18.55/5.65	
18.55/5.65	U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	U10_gga(x1, x2, x3, x4, x5, x6)  =  U10_gga(x1, x3, x6)
18.55/5.65	
18.55/5.65	U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x5)
18.55/5.65	
18.55/5.65	U7_gga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U7_gga(x7, x8)
18.55/5.65	
18.55/5.65	U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x5)
18.55/5.65	
18.55/5.65	plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
18.55/5.65	
18.55/5.65	U2_gga(x1, x2)  =  U2_gga(x1, x2)
18.55/5.65	
18.55/5.65	plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
18.55/5.65	
18.55/5.65	U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
18.55/5.65	
18.55/5.65	
18.55/5.65	
18.55/5.65	
18.55/5.65	
18.55/5.65	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
18.55/5.65	
18.55/5.65	
18.55/5.65	
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(4)
18.55/5.65	Obligation:
18.55/5.65	Pi-finite rewrite system:
18.55/5.65	The TRS R consists of the following rules:
18.55/5.65	
18.55/5.65	   ways_in_gga(0, X1, s(0)) -> ways_out_gga(0, X1, s(0))
18.55/5.65	   ways_in_gga(X2, [], 0) -> ways_out_gga(X2, [], 0)
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins), N) -> U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
18.55/5.65	   =_in_ga(X, X) -> =_out_ga(X, X)
18.55/5.65	   U4_gga(Amount, C, Coins, N, =_out_ga(Amount, s(X3))) -> U5_gga(Amount, C, Coins, N, X3, plus_in_gag(s(C), NewAmount, Amount))
18.55/5.65	   plus_in_gag(0, X, X) -> U2_gag(X, nat_in_g(X))
18.55/5.65	   nat_in_g(0) -> nat_out_g(0)
18.55/5.65	   nat_in_g(s(X)) -> U1_g(X, nat_in_g(X))
18.55/5.65	   U1_g(X, nat_out_g(X)) -> nat_out_g(s(X))
18.55/5.65	   U2_gag(X, nat_out_g(X)) -> plus_out_gag(0, X, X)
18.55/5.65	   plus_in_gag(s(X), Y, s(Z)) -> U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
18.55/5.65	   U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) -> plus_out_gag(s(X), Y, s(Z))
18.55/5.65	   U5_gga(Amount, C, Coins, N, X3, plus_out_gag(s(C), NewAmount, Amount)) -> U6_gga(Amount, C, Coins, N, X3, NewAmount, ways_in_gga(Amount, Coins, N1))
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins), N) -> U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
18.55/5.65	   U9_gga(Amount, C, Coins, N, =_out_ga(Amount, s(X4))) -> U10_gga(Amount, C, Coins, N, X4, plus_in_gag(Amount, s(X5), s(C)))
18.55/5.65	   U10_gga(Amount, C, Coins, N, X4, plus_out_gag(Amount, s(X5), s(C))) -> U11_gga(Amount, C, Coins, N, ways_in_gga(Amount, Coins, N))
18.55/5.65	   U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) -> ways_out_gga(Amount, .(s(C), Coins), N)
18.55/5.65	   U6_gga(Amount, C, Coins, N, X3, NewAmount, ways_out_gga(Amount, Coins, N1)) -> U7_gga(Amount, C, Coins, N, X3, NewAmount, N1, ways_in_gga(NewAmount, .(s(C), Coins), N2))
18.55/5.65	   U7_gga(Amount, C, Coins, N, X3, NewAmount, N1, ways_out_gga(NewAmount, .(s(C), Coins), N2)) -> U8_gga(Amount, C, Coins, N, plus_in_gga(N1, N2, N))
18.55/5.65	   plus_in_gga(0, X, X) -> U2_gga(X, nat_in_g(X))
18.55/5.65	   U2_gga(X, nat_out_g(X)) -> plus_out_gga(0, X, X)
18.55/5.65	   plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
18.55/5.65	   U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
18.55/5.65	   U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) -> ways_out_gga(Amount, .(s(C), Coins), N)
18.55/5.65	
18.55/5.65	The argument filtering Pi contains the following mapping:
18.55/5.65	ways_in_gga(x1, x2, x3)  =  ways_in_gga(x1, x2)
18.55/5.65	
18.55/5.65	0  =  0
18.55/5.65	
18.55/5.65	ways_out_gga(x1, x2, x3)  =  ways_out_gga(x3)
18.55/5.65	
18.55/5.65	[]  =  []
18.55/5.65	
18.55/5.65	.(x1, x2)  =  .(x1, x2)
18.55/5.65	
18.55/5.65	s(x1)  =  s(x1)
18.55/5.65	
18.55/5.65	U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	=_in_ga(x1, x2)  =  =_in_ga(x1)
18.55/5.65	
18.55/5.65	=_out_ga(x1, x2)  =  =_out_ga(x2)
18.55/5.65	
18.55/5.65	U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x6)
18.55/5.65	
18.55/5.65	plus_in_gag(x1, x2, x3)  =  plus_in_gag(x1, x3)
18.55/5.65	
18.55/5.65	U2_gag(x1, x2)  =  U2_gag(x1, x2)
18.55/5.65	
18.55/5.65	nat_in_g(x1)  =  nat_in_g(x1)
18.55/5.65	
18.55/5.65	nat_out_g(x1)  =  nat_out_g
18.55/5.65	
18.55/5.65	U1_g(x1, x2)  =  U1_g(x2)
18.55/5.65	
18.55/5.65	plus_out_gag(x1, x2, x3)  =  plus_out_gag(x2)
18.55/5.65	
18.55/5.65	U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
18.55/5.65	
18.55/5.65	U6_gga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gga(x2, x3, x6, x7)
18.55/5.65	
18.55/5.65	U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	U10_gga(x1, x2, x3, x4, x5, x6)  =  U10_gga(x1, x3, x6)
18.55/5.65	
18.55/5.65	U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x5)
18.55/5.65	
18.55/5.65	U7_gga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U7_gga(x7, x8)
18.55/5.65	
18.55/5.65	U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x5)
18.55/5.65	
18.55/5.65	plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
18.55/5.65	
18.55/5.65	U2_gga(x1, x2)  =  U2_gga(x1, x2)
18.55/5.65	
18.55/5.65	plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
18.55/5.65	
18.55/5.65	U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
18.55/5.65	
18.55/5.65	
18.55/5.65	
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(5) DependencyPairsProof (EQUIVALENT)
18.55/5.65	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
18.55/5.65	Pi DP problem:
18.55/5.65	The TRS P consists of the following rules:
18.55/5.65	
18.55/5.65	   WAYS_IN_GGA(Amount, .(s(C), Coins), N) -> U4_GGA(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
18.55/5.65	   WAYS_IN_GGA(Amount, .(s(C), Coins), N) -> =_IN_GA(Amount, s(X3))
18.55/5.65	   U4_GGA(Amount, C, Coins, N, =_out_ga(Amount, s(X3))) -> U5_GGA(Amount, C, Coins, N, X3, plus_in_gag(s(C), NewAmount, Amount))
18.55/5.65	   U4_GGA(Amount, C, Coins, N, =_out_ga(Amount, s(X3))) -> PLUS_IN_GAG(s(C), NewAmount, Amount)
18.55/5.65	   PLUS_IN_GAG(0, X, X) -> U2_GAG(X, nat_in_g(X))
18.55/5.65	   PLUS_IN_GAG(0, X, X) -> NAT_IN_G(X)
18.55/5.65	   NAT_IN_G(s(X)) -> U1_G(X, nat_in_g(X))
18.55/5.65	   NAT_IN_G(s(X)) -> NAT_IN_G(X)
18.55/5.65	   PLUS_IN_GAG(s(X), Y, s(Z)) -> U3_GAG(X, Y, Z, plus_in_gag(X, Y, Z))
18.55/5.65	   PLUS_IN_GAG(s(X), Y, s(Z)) -> PLUS_IN_GAG(X, Y, Z)
18.55/5.65	   U5_GGA(Amount, C, Coins, N, X3, plus_out_gag(s(C), NewAmount, Amount)) -> U6_GGA(Amount, C, Coins, N, X3, NewAmount, ways_in_gga(Amount, Coins, N1))
18.55/5.65	   U5_GGA(Amount, C, Coins, N, X3, plus_out_gag(s(C), NewAmount, Amount)) -> WAYS_IN_GGA(Amount, Coins, N1)
18.55/5.65	   WAYS_IN_GGA(Amount, .(s(C), Coins), N) -> U9_GGA(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
18.55/5.65	   U9_GGA(Amount, C, Coins, N, =_out_ga(Amount, s(X4))) -> U10_GGA(Amount, C, Coins, N, X4, plus_in_gag(Amount, s(X5), s(C)))
18.55/5.65	   U9_GGA(Amount, C, Coins, N, =_out_ga(Amount, s(X4))) -> PLUS_IN_GAG(Amount, s(X5), s(C))
18.55/5.65	   U10_GGA(Amount, C, Coins, N, X4, plus_out_gag(Amount, s(X5), s(C))) -> U11_GGA(Amount, C, Coins, N, ways_in_gga(Amount, Coins, N))
18.55/5.65	   U10_GGA(Amount, C, Coins, N, X4, plus_out_gag(Amount, s(X5), s(C))) -> WAYS_IN_GGA(Amount, Coins, N)
18.55/5.65	   U6_GGA(Amount, C, Coins, N, X3, NewAmount, ways_out_gga(Amount, Coins, N1)) -> U7_GGA(Amount, C, Coins, N, X3, NewAmount, N1, ways_in_gga(NewAmount, .(s(C), Coins), N2))
18.55/5.65	   U6_GGA(Amount, C, Coins, N, X3, NewAmount, ways_out_gga(Amount, Coins, N1)) -> WAYS_IN_GGA(NewAmount, .(s(C), Coins), N2)
18.55/5.65	   U7_GGA(Amount, C, Coins, N, X3, NewAmount, N1, ways_out_gga(NewAmount, .(s(C), Coins), N2)) -> U8_GGA(Amount, C, Coins, N, plus_in_gga(N1, N2, N))
18.55/5.65	   U7_GGA(Amount, C, Coins, N, X3, NewAmount, N1, ways_out_gga(NewAmount, .(s(C), Coins), N2)) -> PLUS_IN_GGA(N1, N2, N)
18.55/5.65	   PLUS_IN_GGA(0, X, X) -> U2_GGA(X, nat_in_g(X))
18.55/5.65	   PLUS_IN_GGA(0, X, X) -> NAT_IN_G(X)
18.55/5.65	   PLUS_IN_GGA(s(X), Y, s(Z)) -> U3_GGA(X, Y, Z, plus_in_gga(X, Y, Z))
18.55/5.65	   PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z)
18.55/5.65	
18.55/5.65	The TRS R consists of the following rules:
18.55/5.65	
18.55/5.65	   ways_in_gga(0, X1, s(0)) -> ways_out_gga(0, X1, s(0))
18.55/5.65	   ways_in_gga(X2, [], 0) -> ways_out_gga(X2, [], 0)
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins), N) -> U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
18.55/5.65	   =_in_ga(X, X) -> =_out_ga(X, X)
18.55/5.65	   U4_gga(Amount, C, Coins, N, =_out_ga(Amount, s(X3))) -> U5_gga(Amount, C, Coins, N, X3, plus_in_gag(s(C), NewAmount, Amount))
18.55/5.65	   plus_in_gag(0, X, X) -> U2_gag(X, nat_in_g(X))
18.55/5.65	   nat_in_g(0) -> nat_out_g(0)
18.55/5.65	   nat_in_g(s(X)) -> U1_g(X, nat_in_g(X))
18.55/5.65	   U1_g(X, nat_out_g(X)) -> nat_out_g(s(X))
18.55/5.65	   U2_gag(X, nat_out_g(X)) -> plus_out_gag(0, X, X)
18.55/5.65	   plus_in_gag(s(X), Y, s(Z)) -> U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
18.55/5.65	   U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) -> plus_out_gag(s(X), Y, s(Z))
18.55/5.65	   U5_gga(Amount, C, Coins, N, X3, plus_out_gag(s(C), NewAmount, Amount)) -> U6_gga(Amount, C, Coins, N, X3, NewAmount, ways_in_gga(Amount, Coins, N1))
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins), N) -> U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
18.55/5.65	   U9_gga(Amount, C, Coins, N, =_out_ga(Amount, s(X4))) -> U10_gga(Amount, C, Coins, N, X4, plus_in_gag(Amount, s(X5), s(C)))
18.55/5.65	   U10_gga(Amount, C, Coins, N, X4, plus_out_gag(Amount, s(X5), s(C))) -> U11_gga(Amount, C, Coins, N, ways_in_gga(Amount, Coins, N))
18.55/5.65	   U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) -> ways_out_gga(Amount, .(s(C), Coins), N)
18.55/5.65	   U6_gga(Amount, C, Coins, N, X3, NewAmount, ways_out_gga(Amount, Coins, N1)) -> U7_gga(Amount, C, Coins, N, X3, NewAmount, N1, ways_in_gga(NewAmount, .(s(C), Coins), N2))
18.55/5.65	   U7_gga(Amount, C, Coins, N, X3, NewAmount, N1, ways_out_gga(NewAmount, .(s(C), Coins), N2)) -> U8_gga(Amount, C, Coins, N, plus_in_gga(N1, N2, N))
18.55/5.65	   plus_in_gga(0, X, X) -> U2_gga(X, nat_in_g(X))
18.55/5.65	   U2_gga(X, nat_out_g(X)) -> plus_out_gga(0, X, X)
18.55/5.65	   plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
18.55/5.65	   U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
18.55/5.65	   U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) -> ways_out_gga(Amount, .(s(C), Coins), N)
18.55/5.65	
18.55/5.65	The argument filtering Pi contains the following mapping:
18.55/5.65	ways_in_gga(x1, x2, x3)  =  ways_in_gga(x1, x2)
18.55/5.65	
18.55/5.65	0  =  0
18.55/5.65	
18.55/5.65	ways_out_gga(x1, x2, x3)  =  ways_out_gga(x3)
18.55/5.65	
18.55/5.65	[]  =  []
18.55/5.65	
18.55/5.65	.(x1, x2)  =  .(x1, x2)
18.55/5.65	
18.55/5.65	s(x1)  =  s(x1)
18.55/5.65	
18.55/5.65	U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	=_in_ga(x1, x2)  =  =_in_ga(x1)
18.55/5.65	
18.55/5.65	=_out_ga(x1, x2)  =  =_out_ga(x2)
18.55/5.65	
18.55/5.65	U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x6)
18.55/5.65	
18.55/5.65	plus_in_gag(x1, x2, x3)  =  plus_in_gag(x1, x3)
18.55/5.65	
18.55/5.65	U2_gag(x1, x2)  =  U2_gag(x1, x2)
18.55/5.65	
18.55/5.65	nat_in_g(x1)  =  nat_in_g(x1)
18.55/5.65	
18.55/5.65	nat_out_g(x1)  =  nat_out_g
18.55/5.65	
18.55/5.65	U1_g(x1, x2)  =  U1_g(x2)
18.55/5.65	
18.55/5.65	plus_out_gag(x1, x2, x3)  =  plus_out_gag(x2)
18.55/5.65	
18.55/5.65	U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
18.55/5.65	
18.55/5.65	U6_gga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gga(x2, x3, x6, x7)
18.55/5.65	
18.55/5.65	U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	U10_gga(x1, x2, x3, x4, x5, x6)  =  U10_gga(x1, x3, x6)
18.55/5.65	
18.55/5.65	U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x5)
18.55/5.65	
18.55/5.65	U7_gga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U7_gga(x7, x8)
18.55/5.65	
18.55/5.65	U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x5)
18.55/5.65	
18.55/5.65	plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
18.55/5.65	
18.55/5.65	U2_gga(x1, x2)  =  U2_gga(x1, x2)
18.55/5.65	
18.55/5.65	plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
18.55/5.65	
18.55/5.65	U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
18.55/5.65	
18.55/5.65	WAYS_IN_GGA(x1, x2, x3)  =  WAYS_IN_GGA(x1, x2)
18.55/5.65	
18.55/5.65	U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	=_IN_GA(x1, x2)  =  =_IN_GA(x1)
18.55/5.65	
18.55/5.65	U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x1, x2, x3, x6)
18.55/5.65	
18.55/5.65	PLUS_IN_GAG(x1, x2, x3)  =  PLUS_IN_GAG(x1, x3)
18.55/5.65	
18.55/5.65	U2_GAG(x1, x2)  =  U2_GAG(x1, x2)
18.55/5.65	
18.55/5.65	NAT_IN_G(x1)  =  NAT_IN_G(x1)
18.55/5.65	
18.55/5.65	U1_G(x1, x2)  =  U1_G(x2)
18.55/5.65	
18.55/5.65	U3_GAG(x1, x2, x3, x4)  =  U3_GAG(x4)
18.55/5.65	
18.55/5.65	U6_GGA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GGA(x2, x3, x6, x7)
18.55/5.65	
18.55/5.65	U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	U10_GGA(x1, x2, x3, x4, x5, x6)  =  U10_GGA(x1, x3, x6)
18.55/5.65	
18.55/5.65	U11_GGA(x1, x2, x3, x4, x5)  =  U11_GGA(x5)
18.55/5.65	
18.55/5.65	U7_GGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U7_GGA(x7, x8)
18.55/5.65	
18.55/5.65	U8_GGA(x1, x2, x3, x4, x5)  =  U8_GGA(x5)
18.55/5.65	
18.55/5.65	PLUS_IN_GGA(x1, x2, x3)  =  PLUS_IN_GGA(x1, x2)
18.55/5.65	
18.55/5.65	U2_GGA(x1, x2)  =  U2_GGA(x1, x2)
18.55/5.65	
18.55/5.65	U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
18.55/5.65	
18.55/5.65	
18.55/5.65	We have to consider all (P,R,Pi)-chains
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(6)
18.55/5.65	Obligation:
18.55/5.65	Pi DP problem:
18.55/5.65	The TRS P consists of the following rules:
18.55/5.65	
18.55/5.65	   WAYS_IN_GGA(Amount, .(s(C), Coins), N) -> U4_GGA(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
18.55/5.65	   WAYS_IN_GGA(Amount, .(s(C), Coins), N) -> =_IN_GA(Amount, s(X3))
18.55/5.65	   U4_GGA(Amount, C, Coins, N, =_out_ga(Amount, s(X3))) -> U5_GGA(Amount, C, Coins, N, X3, plus_in_gag(s(C), NewAmount, Amount))
18.55/5.65	   U4_GGA(Amount, C, Coins, N, =_out_ga(Amount, s(X3))) -> PLUS_IN_GAG(s(C), NewAmount, Amount)
18.55/5.65	   PLUS_IN_GAG(0, X, X) -> U2_GAG(X, nat_in_g(X))
18.55/5.65	   PLUS_IN_GAG(0, X, X) -> NAT_IN_G(X)
18.55/5.65	   NAT_IN_G(s(X)) -> U1_G(X, nat_in_g(X))
18.55/5.65	   NAT_IN_G(s(X)) -> NAT_IN_G(X)
18.55/5.65	   PLUS_IN_GAG(s(X), Y, s(Z)) -> U3_GAG(X, Y, Z, plus_in_gag(X, Y, Z))
18.55/5.65	   PLUS_IN_GAG(s(X), Y, s(Z)) -> PLUS_IN_GAG(X, Y, Z)
18.55/5.65	   U5_GGA(Amount, C, Coins, N, X3, plus_out_gag(s(C), NewAmount, Amount)) -> U6_GGA(Amount, C, Coins, N, X3, NewAmount, ways_in_gga(Amount, Coins, N1))
18.55/5.65	   U5_GGA(Amount, C, Coins, N, X3, plus_out_gag(s(C), NewAmount, Amount)) -> WAYS_IN_GGA(Amount, Coins, N1)
18.55/5.65	   WAYS_IN_GGA(Amount, .(s(C), Coins), N) -> U9_GGA(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
18.55/5.65	   U9_GGA(Amount, C, Coins, N, =_out_ga(Amount, s(X4))) -> U10_GGA(Amount, C, Coins, N, X4, plus_in_gag(Amount, s(X5), s(C)))
18.55/5.65	   U9_GGA(Amount, C, Coins, N, =_out_ga(Amount, s(X4))) -> PLUS_IN_GAG(Amount, s(X5), s(C))
18.55/5.65	   U10_GGA(Amount, C, Coins, N, X4, plus_out_gag(Amount, s(X5), s(C))) -> U11_GGA(Amount, C, Coins, N, ways_in_gga(Amount, Coins, N))
18.55/5.65	   U10_GGA(Amount, C, Coins, N, X4, plus_out_gag(Amount, s(X5), s(C))) -> WAYS_IN_GGA(Amount, Coins, N)
18.55/5.65	   U6_GGA(Amount, C, Coins, N, X3, NewAmount, ways_out_gga(Amount, Coins, N1)) -> U7_GGA(Amount, C, Coins, N, X3, NewAmount, N1, ways_in_gga(NewAmount, .(s(C), Coins), N2))
18.55/5.65	   U6_GGA(Amount, C, Coins, N, X3, NewAmount, ways_out_gga(Amount, Coins, N1)) -> WAYS_IN_GGA(NewAmount, .(s(C), Coins), N2)
18.55/5.65	   U7_GGA(Amount, C, Coins, N, X3, NewAmount, N1, ways_out_gga(NewAmount, .(s(C), Coins), N2)) -> U8_GGA(Amount, C, Coins, N, plus_in_gga(N1, N2, N))
18.55/5.65	   U7_GGA(Amount, C, Coins, N, X3, NewAmount, N1, ways_out_gga(NewAmount, .(s(C), Coins), N2)) -> PLUS_IN_GGA(N1, N2, N)
18.55/5.65	   PLUS_IN_GGA(0, X, X) -> U2_GGA(X, nat_in_g(X))
18.55/5.65	   PLUS_IN_GGA(0, X, X) -> NAT_IN_G(X)
18.55/5.65	   PLUS_IN_GGA(s(X), Y, s(Z)) -> U3_GGA(X, Y, Z, plus_in_gga(X, Y, Z))
18.55/5.65	   PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z)
18.55/5.65	
18.55/5.65	The TRS R consists of the following rules:
18.55/5.65	
18.55/5.65	   ways_in_gga(0, X1, s(0)) -> ways_out_gga(0, X1, s(0))
18.55/5.65	   ways_in_gga(X2, [], 0) -> ways_out_gga(X2, [], 0)
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins), N) -> U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
18.55/5.65	   =_in_ga(X, X) -> =_out_ga(X, X)
18.55/5.65	   U4_gga(Amount, C, Coins, N, =_out_ga(Amount, s(X3))) -> U5_gga(Amount, C, Coins, N, X3, plus_in_gag(s(C), NewAmount, Amount))
18.55/5.65	   plus_in_gag(0, X, X) -> U2_gag(X, nat_in_g(X))
18.55/5.65	   nat_in_g(0) -> nat_out_g(0)
18.55/5.65	   nat_in_g(s(X)) -> U1_g(X, nat_in_g(X))
18.55/5.65	   U1_g(X, nat_out_g(X)) -> nat_out_g(s(X))
18.55/5.65	   U2_gag(X, nat_out_g(X)) -> plus_out_gag(0, X, X)
18.55/5.65	   plus_in_gag(s(X), Y, s(Z)) -> U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
18.55/5.65	   U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) -> plus_out_gag(s(X), Y, s(Z))
18.55/5.65	   U5_gga(Amount, C, Coins, N, X3, plus_out_gag(s(C), NewAmount, Amount)) -> U6_gga(Amount, C, Coins, N, X3, NewAmount, ways_in_gga(Amount, Coins, N1))
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins), N) -> U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
18.55/5.65	   U9_gga(Amount, C, Coins, N, =_out_ga(Amount, s(X4))) -> U10_gga(Amount, C, Coins, N, X4, plus_in_gag(Amount, s(X5), s(C)))
18.55/5.65	   U10_gga(Amount, C, Coins, N, X4, plus_out_gag(Amount, s(X5), s(C))) -> U11_gga(Amount, C, Coins, N, ways_in_gga(Amount, Coins, N))
18.55/5.65	   U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) -> ways_out_gga(Amount, .(s(C), Coins), N)
18.55/5.65	   U6_gga(Amount, C, Coins, N, X3, NewAmount, ways_out_gga(Amount, Coins, N1)) -> U7_gga(Amount, C, Coins, N, X3, NewAmount, N1, ways_in_gga(NewAmount, .(s(C), Coins), N2))
18.55/5.65	   U7_gga(Amount, C, Coins, N, X3, NewAmount, N1, ways_out_gga(NewAmount, .(s(C), Coins), N2)) -> U8_gga(Amount, C, Coins, N, plus_in_gga(N1, N2, N))
18.55/5.65	   plus_in_gga(0, X, X) -> U2_gga(X, nat_in_g(X))
18.55/5.65	   U2_gga(X, nat_out_g(X)) -> plus_out_gga(0, X, X)
18.55/5.65	   plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
18.55/5.65	   U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
18.55/5.65	   U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) -> ways_out_gga(Amount, .(s(C), Coins), N)
18.55/5.65	
18.55/5.65	The argument filtering Pi contains the following mapping:
18.55/5.65	ways_in_gga(x1, x2, x3)  =  ways_in_gga(x1, x2)
18.55/5.65	
18.55/5.65	0  =  0
18.55/5.65	
18.55/5.65	ways_out_gga(x1, x2, x3)  =  ways_out_gga(x3)
18.55/5.65	
18.55/5.65	[]  =  []
18.55/5.65	
18.55/5.65	.(x1, x2)  =  .(x1, x2)
18.55/5.65	
18.55/5.65	s(x1)  =  s(x1)
18.55/5.65	
18.55/5.65	U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	=_in_ga(x1, x2)  =  =_in_ga(x1)
18.55/5.65	
18.55/5.65	=_out_ga(x1, x2)  =  =_out_ga(x2)
18.55/5.65	
18.55/5.65	U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x6)
18.55/5.65	
18.55/5.65	plus_in_gag(x1, x2, x3)  =  plus_in_gag(x1, x3)
18.55/5.65	
18.55/5.65	U2_gag(x1, x2)  =  U2_gag(x1, x2)
18.55/5.65	
18.55/5.65	nat_in_g(x1)  =  nat_in_g(x1)
18.55/5.65	
18.55/5.65	nat_out_g(x1)  =  nat_out_g
18.55/5.65	
18.55/5.65	U1_g(x1, x2)  =  U1_g(x2)
18.55/5.65	
18.55/5.65	plus_out_gag(x1, x2, x3)  =  plus_out_gag(x2)
18.55/5.65	
18.55/5.65	U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
18.55/5.65	
18.55/5.65	U6_gga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gga(x2, x3, x6, x7)
18.55/5.65	
18.55/5.65	U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	U10_gga(x1, x2, x3, x4, x5, x6)  =  U10_gga(x1, x3, x6)
18.55/5.65	
18.55/5.65	U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x5)
18.55/5.65	
18.55/5.65	U7_gga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U7_gga(x7, x8)
18.55/5.65	
18.55/5.65	U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x5)
18.55/5.65	
18.55/5.65	plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
18.55/5.65	
18.55/5.65	U2_gga(x1, x2)  =  U2_gga(x1, x2)
18.55/5.65	
18.55/5.65	plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
18.55/5.65	
18.55/5.65	U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
18.55/5.65	
18.55/5.65	WAYS_IN_GGA(x1, x2, x3)  =  WAYS_IN_GGA(x1, x2)
18.55/5.65	
18.55/5.65	U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	=_IN_GA(x1, x2)  =  =_IN_GA(x1)
18.55/5.65	
18.55/5.65	U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x1, x2, x3, x6)
18.55/5.65	
18.55/5.65	PLUS_IN_GAG(x1, x2, x3)  =  PLUS_IN_GAG(x1, x3)
18.55/5.65	
18.55/5.65	U2_GAG(x1, x2)  =  U2_GAG(x1, x2)
18.55/5.65	
18.55/5.65	NAT_IN_G(x1)  =  NAT_IN_G(x1)
18.55/5.65	
18.55/5.65	U1_G(x1, x2)  =  U1_G(x2)
18.55/5.65	
18.55/5.65	U3_GAG(x1, x2, x3, x4)  =  U3_GAG(x4)
18.55/5.65	
18.55/5.65	U6_GGA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GGA(x2, x3, x6, x7)
18.55/5.65	
18.55/5.65	U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	U10_GGA(x1, x2, x3, x4, x5, x6)  =  U10_GGA(x1, x3, x6)
18.55/5.65	
18.55/5.65	U11_GGA(x1, x2, x3, x4, x5)  =  U11_GGA(x5)
18.55/5.65	
18.55/5.65	U7_GGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U7_GGA(x7, x8)
18.55/5.65	
18.55/5.65	U8_GGA(x1, x2, x3, x4, x5)  =  U8_GGA(x5)
18.55/5.65	
18.55/5.65	PLUS_IN_GGA(x1, x2, x3)  =  PLUS_IN_GGA(x1, x2)
18.55/5.65	
18.55/5.65	U2_GGA(x1, x2)  =  U2_GGA(x1, x2)
18.55/5.65	
18.55/5.65	U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
18.55/5.65	
18.55/5.65	
18.55/5.65	We have to consider all (P,R,Pi)-chains
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(7) DependencyGraphProof (EQUIVALENT)
18.55/5.65	The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 14 less nodes.
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(8)
18.55/5.65	Complex Obligation (AND)
18.55/5.65	
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(9)
18.55/5.65	Obligation:
18.55/5.65	Pi DP problem:
18.55/5.65	The TRS P consists of the following rules:
18.55/5.65	
18.55/5.65	   NAT_IN_G(s(X)) -> NAT_IN_G(X)
18.55/5.65	
18.55/5.65	The TRS R consists of the following rules:
18.55/5.65	
18.55/5.65	   ways_in_gga(0, X1, s(0)) -> ways_out_gga(0, X1, s(0))
18.55/5.65	   ways_in_gga(X2, [], 0) -> ways_out_gga(X2, [], 0)
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins), N) -> U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
18.55/5.65	   =_in_ga(X, X) -> =_out_ga(X, X)
18.55/5.65	   U4_gga(Amount, C, Coins, N, =_out_ga(Amount, s(X3))) -> U5_gga(Amount, C, Coins, N, X3, plus_in_gag(s(C), NewAmount, Amount))
18.55/5.65	   plus_in_gag(0, X, X) -> U2_gag(X, nat_in_g(X))
18.55/5.65	   nat_in_g(0) -> nat_out_g(0)
18.55/5.65	   nat_in_g(s(X)) -> U1_g(X, nat_in_g(X))
18.55/5.65	   U1_g(X, nat_out_g(X)) -> nat_out_g(s(X))
18.55/5.65	   U2_gag(X, nat_out_g(X)) -> plus_out_gag(0, X, X)
18.55/5.65	   plus_in_gag(s(X), Y, s(Z)) -> U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
18.55/5.65	   U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) -> plus_out_gag(s(X), Y, s(Z))
18.55/5.65	   U5_gga(Amount, C, Coins, N, X3, plus_out_gag(s(C), NewAmount, Amount)) -> U6_gga(Amount, C, Coins, N, X3, NewAmount, ways_in_gga(Amount, Coins, N1))
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins), N) -> U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
18.55/5.65	   U9_gga(Amount, C, Coins, N, =_out_ga(Amount, s(X4))) -> U10_gga(Amount, C, Coins, N, X4, plus_in_gag(Amount, s(X5), s(C)))
18.55/5.65	   U10_gga(Amount, C, Coins, N, X4, plus_out_gag(Amount, s(X5), s(C))) -> U11_gga(Amount, C, Coins, N, ways_in_gga(Amount, Coins, N))
18.55/5.65	   U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) -> ways_out_gga(Amount, .(s(C), Coins), N)
18.55/5.65	   U6_gga(Amount, C, Coins, N, X3, NewAmount, ways_out_gga(Amount, Coins, N1)) -> U7_gga(Amount, C, Coins, N, X3, NewAmount, N1, ways_in_gga(NewAmount, .(s(C), Coins), N2))
18.55/5.65	   U7_gga(Amount, C, Coins, N, X3, NewAmount, N1, ways_out_gga(NewAmount, .(s(C), Coins), N2)) -> U8_gga(Amount, C, Coins, N, plus_in_gga(N1, N2, N))
18.55/5.65	   plus_in_gga(0, X, X) -> U2_gga(X, nat_in_g(X))
18.55/5.65	   U2_gga(X, nat_out_g(X)) -> plus_out_gga(0, X, X)
18.55/5.65	   plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
18.55/5.65	   U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
18.55/5.65	   U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) -> ways_out_gga(Amount, .(s(C), Coins), N)
18.55/5.65	
18.55/5.65	The argument filtering Pi contains the following mapping:
18.55/5.65	ways_in_gga(x1, x2, x3)  =  ways_in_gga(x1, x2)
18.55/5.65	
18.55/5.65	0  =  0
18.55/5.65	
18.55/5.65	ways_out_gga(x1, x2, x3)  =  ways_out_gga(x3)
18.55/5.65	
18.55/5.65	[]  =  []
18.55/5.65	
18.55/5.65	.(x1, x2)  =  .(x1, x2)
18.55/5.65	
18.55/5.65	s(x1)  =  s(x1)
18.55/5.65	
18.55/5.65	U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	=_in_ga(x1, x2)  =  =_in_ga(x1)
18.55/5.65	
18.55/5.65	=_out_ga(x1, x2)  =  =_out_ga(x2)
18.55/5.65	
18.55/5.65	U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x6)
18.55/5.65	
18.55/5.65	plus_in_gag(x1, x2, x3)  =  plus_in_gag(x1, x3)
18.55/5.65	
18.55/5.65	U2_gag(x1, x2)  =  U2_gag(x1, x2)
18.55/5.65	
18.55/5.65	nat_in_g(x1)  =  nat_in_g(x1)
18.55/5.65	
18.55/5.65	nat_out_g(x1)  =  nat_out_g
18.55/5.65	
18.55/5.65	U1_g(x1, x2)  =  U1_g(x2)
18.55/5.65	
18.55/5.65	plus_out_gag(x1, x2, x3)  =  plus_out_gag(x2)
18.55/5.65	
18.55/5.65	U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
18.55/5.65	
18.55/5.65	U6_gga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gga(x2, x3, x6, x7)
18.55/5.65	
18.55/5.65	U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	U10_gga(x1, x2, x3, x4, x5, x6)  =  U10_gga(x1, x3, x6)
18.55/5.65	
18.55/5.65	U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x5)
18.55/5.65	
18.55/5.65	U7_gga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U7_gga(x7, x8)
18.55/5.65	
18.55/5.65	U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x5)
18.55/5.65	
18.55/5.65	plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
18.55/5.65	
18.55/5.65	U2_gga(x1, x2)  =  U2_gga(x1, x2)
18.55/5.65	
18.55/5.65	plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
18.55/5.65	
18.55/5.65	U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
18.55/5.65	
18.55/5.65	NAT_IN_G(x1)  =  NAT_IN_G(x1)
18.55/5.65	
18.55/5.65	
18.55/5.65	We have to consider all (P,R,Pi)-chains
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(10) UsableRulesProof (EQUIVALENT)
18.55/5.65	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(11)
18.55/5.65	Obligation:
18.55/5.65	Pi DP problem:
18.55/5.65	The TRS P consists of the following rules:
18.55/5.65	
18.55/5.65	   NAT_IN_G(s(X)) -> NAT_IN_G(X)
18.55/5.65	
18.55/5.65	R is empty.
18.55/5.65	Pi is empty.
18.55/5.65	We have to consider all (P,R,Pi)-chains
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(12) PiDPToQDPProof (EQUIVALENT)
18.55/5.65	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(13)
18.55/5.65	Obligation:
18.55/5.65	Q DP problem:
18.55/5.65	The TRS P consists of the following rules:
18.55/5.65	
18.55/5.65	   NAT_IN_G(s(X)) -> NAT_IN_G(X)
18.55/5.65	
18.55/5.65	R is empty.
18.55/5.65	Q is empty.
18.55/5.65	We have to consider all (P,Q,R)-chains.
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(14) QDPSizeChangeProof (EQUIVALENT)
18.55/5.65	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. 
18.55/5.65	
18.55/5.65	From the DPs we obtained the following set of size-change graphs:
18.55/5.65	*NAT_IN_G(s(X)) -> NAT_IN_G(X)
18.55/5.65	The graph contains the following edges 1 > 1
18.55/5.65	
18.55/5.65	
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(15)
18.55/5.65	YES
18.55/5.65	
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(16)
18.55/5.65	Obligation:
18.55/5.65	Pi DP problem:
18.55/5.65	The TRS P consists of the following rules:
18.55/5.65	
18.55/5.65	   PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z)
18.55/5.65	
18.55/5.65	The TRS R consists of the following rules:
18.55/5.65	
18.55/5.65	   ways_in_gga(0, X1, s(0)) -> ways_out_gga(0, X1, s(0))
18.55/5.65	   ways_in_gga(X2, [], 0) -> ways_out_gga(X2, [], 0)
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins), N) -> U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
18.55/5.65	   =_in_ga(X, X) -> =_out_ga(X, X)
18.55/5.65	   U4_gga(Amount, C, Coins, N, =_out_ga(Amount, s(X3))) -> U5_gga(Amount, C, Coins, N, X3, plus_in_gag(s(C), NewAmount, Amount))
18.55/5.65	   plus_in_gag(0, X, X) -> U2_gag(X, nat_in_g(X))
18.55/5.65	   nat_in_g(0) -> nat_out_g(0)
18.55/5.65	   nat_in_g(s(X)) -> U1_g(X, nat_in_g(X))
18.55/5.65	   U1_g(X, nat_out_g(X)) -> nat_out_g(s(X))
18.55/5.65	   U2_gag(X, nat_out_g(X)) -> plus_out_gag(0, X, X)
18.55/5.65	   plus_in_gag(s(X), Y, s(Z)) -> U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
18.55/5.65	   U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) -> plus_out_gag(s(X), Y, s(Z))
18.55/5.65	   U5_gga(Amount, C, Coins, N, X3, plus_out_gag(s(C), NewAmount, Amount)) -> U6_gga(Amount, C, Coins, N, X3, NewAmount, ways_in_gga(Amount, Coins, N1))
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins), N) -> U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
18.55/5.65	   U9_gga(Amount, C, Coins, N, =_out_ga(Amount, s(X4))) -> U10_gga(Amount, C, Coins, N, X4, plus_in_gag(Amount, s(X5), s(C)))
18.55/5.65	   U10_gga(Amount, C, Coins, N, X4, plus_out_gag(Amount, s(X5), s(C))) -> U11_gga(Amount, C, Coins, N, ways_in_gga(Amount, Coins, N))
18.55/5.65	   U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) -> ways_out_gga(Amount, .(s(C), Coins), N)
18.55/5.65	   U6_gga(Amount, C, Coins, N, X3, NewAmount, ways_out_gga(Amount, Coins, N1)) -> U7_gga(Amount, C, Coins, N, X3, NewAmount, N1, ways_in_gga(NewAmount, .(s(C), Coins), N2))
18.55/5.65	   U7_gga(Amount, C, Coins, N, X3, NewAmount, N1, ways_out_gga(NewAmount, .(s(C), Coins), N2)) -> U8_gga(Amount, C, Coins, N, plus_in_gga(N1, N2, N))
18.55/5.65	   plus_in_gga(0, X, X) -> U2_gga(X, nat_in_g(X))
18.55/5.65	   U2_gga(X, nat_out_g(X)) -> plus_out_gga(0, X, X)
18.55/5.65	   plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
18.55/5.65	   U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
18.55/5.65	   U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) -> ways_out_gga(Amount, .(s(C), Coins), N)
18.55/5.65	
18.55/5.65	The argument filtering Pi contains the following mapping:
18.55/5.65	ways_in_gga(x1, x2, x3)  =  ways_in_gga(x1, x2)
18.55/5.65	
18.55/5.65	0  =  0
18.55/5.65	
18.55/5.65	ways_out_gga(x1, x2, x3)  =  ways_out_gga(x3)
18.55/5.65	
18.55/5.65	[]  =  []
18.55/5.65	
18.55/5.65	.(x1, x2)  =  .(x1, x2)
18.55/5.65	
18.55/5.65	s(x1)  =  s(x1)
18.55/5.65	
18.55/5.65	U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	=_in_ga(x1, x2)  =  =_in_ga(x1)
18.55/5.65	
18.55/5.65	=_out_ga(x1, x2)  =  =_out_ga(x2)
18.55/5.65	
18.55/5.65	U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x6)
18.55/5.65	
18.55/5.65	plus_in_gag(x1, x2, x3)  =  plus_in_gag(x1, x3)
18.55/5.65	
18.55/5.65	U2_gag(x1, x2)  =  U2_gag(x1, x2)
18.55/5.65	
18.55/5.65	nat_in_g(x1)  =  nat_in_g(x1)
18.55/5.65	
18.55/5.65	nat_out_g(x1)  =  nat_out_g
18.55/5.65	
18.55/5.65	U1_g(x1, x2)  =  U1_g(x2)
18.55/5.65	
18.55/5.65	plus_out_gag(x1, x2, x3)  =  plus_out_gag(x2)
18.55/5.65	
18.55/5.65	U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
18.55/5.65	
18.55/5.65	U6_gga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gga(x2, x3, x6, x7)
18.55/5.65	
18.55/5.65	U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	U10_gga(x1, x2, x3, x4, x5, x6)  =  U10_gga(x1, x3, x6)
18.55/5.65	
18.55/5.65	U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x5)
18.55/5.65	
18.55/5.65	U7_gga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U7_gga(x7, x8)
18.55/5.65	
18.55/5.65	U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x5)
18.55/5.65	
18.55/5.65	plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
18.55/5.65	
18.55/5.65	U2_gga(x1, x2)  =  U2_gga(x1, x2)
18.55/5.65	
18.55/5.65	plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
18.55/5.65	
18.55/5.65	U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
18.55/5.65	
18.55/5.65	PLUS_IN_GGA(x1, x2, x3)  =  PLUS_IN_GGA(x1, x2)
18.55/5.65	
18.55/5.65	
18.55/5.65	We have to consider all (P,R,Pi)-chains
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(17) UsableRulesProof (EQUIVALENT)
18.55/5.65	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(18)
18.55/5.65	Obligation:
18.55/5.65	Pi DP problem:
18.55/5.65	The TRS P consists of the following rules:
18.55/5.65	
18.55/5.65	   PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z)
18.55/5.65	
18.55/5.65	R is empty.
18.55/5.65	The argument filtering Pi contains the following mapping:
18.55/5.65	s(x1)  =  s(x1)
18.55/5.65	
18.55/5.65	PLUS_IN_GGA(x1, x2, x3)  =  PLUS_IN_GGA(x1, x2)
18.55/5.65	
18.55/5.65	
18.55/5.65	We have to consider all (P,R,Pi)-chains
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(19) PiDPToQDPProof (SOUND)
18.55/5.65	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(20)
18.55/5.65	Obligation:
18.55/5.65	Q DP problem:
18.55/5.65	The TRS P consists of the following rules:
18.55/5.65	
18.55/5.65	   PLUS_IN_GGA(s(X), Y) -> PLUS_IN_GGA(X, Y)
18.55/5.65	
18.55/5.65	R is empty.
18.55/5.65	Q is empty.
18.55/5.65	We have to consider all (P,Q,R)-chains.
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(21) QDPSizeChangeProof (EQUIVALENT)
18.55/5.65	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. 
18.55/5.65	
18.55/5.65	From the DPs we obtained the following set of size-change graphs:
18.55/5.65	*PLUS_IN_GGA(s(X), Y) -> PLUS_IN_GGA(X, Y)
18.55/5.65	The graph contains the following edges 1 > 1, 2 >= 2
18.55/5.65	
18.55/5.65	
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(22)
18.55/5.65	YES
18.55/5.65	
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(23)
18.55/5.65	Obligation:
18.55/5.65	Pi DP problem:
18.55/5.65	The TRS P consists of the following rules:
18.55/5.65	
18.55/5.65	   PLUS_IN_GAG(s(X), Y, s(Z)) -> PLUS_IN_GAG(X, Y, Z)
18.55/5.65	
18.55/5.65	The TRS R consists of the following rules:
18.55/5.65	
18.55/5.65	   ways_in_gga(0, X1, s(0)) -> ways_out_gga(0, X1, s(0))
18.55/5.65	   ways_in_gga(X2, [], 0) -> ways_out_gga(X2, [], 0)
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins), N) -> U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
18.55/5.65	   =_in_ga(X, X) -> =_out_ga(X, X)
18.55/5.65	   U4_gga(Amount, C, Coins, N, =_out_ga(Amount, s(X3))) -> U5_gga(Amount, C, Coins, N, X3, plus_in_gag(s(C), NewAmount, Amount))
18.55/5.65	   plus_in_gag(0, X, X) -> U2_gag(X, nat_in_g(X))
18.55/5.65	   nat_in_g(0) -> nat_out_g(0)
18.55/5.65	   nat_in_g(s(X)) -> U1_g(X, nat_in_g(X))
18.55/5.65	   U1_g(X, nat_out_g(X)) -> nat_out_g(s(X))
18.55/5.65	   U2_gag(X, nat_out_g(X)) -> plus_out_gag(0, X, X)
18.55/5.65	   plus_in_gag(s(X), Y, s(Z)) -> U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
18.55/5.65	   U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) -> plus_out_gag(s(X), Y, s(Z))
18.55/5.65	   U5_gga(Amount, C, Coins, N, X3, plus_out_gag(s(C), NewAmount, Amount)) -> U6_gga(Amount, C, Coins, N, X3, NewAmount, ways_in_gga(Amount, Coins, N1))
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins), N) -> U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
18.55/5.65	   U9_gga(Amount, C, Coins, N, =_out_ga(Amount, s(X4))) -> U10_gga(Amount, C, Coins, N, X4, plus_in_gag(Amount, s(X5), s(C)))
18.55/5.65	   U10_gga(Amount, C, Coins, N, X4, plus_out_gag(Amount, s(X5), s(C))) -> U11_gga(Amount, C, Coins, N, ways_in_gga(Amount, Coins, N))
18.55/5.65	   U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) -> ways_out_gga(Amount, .(s(C), Coins), N)
18.55/5.65	   U6_gga(Amount, C, Coins, N, X3, NewAmount, ways_out_gga(Amount, Coins, N1)) -> U7_gga(Amount, C, Coins, N, X3, NewAmount, N1, ways_in_gga(NewAmount, .(s(C), Coins), N2))
18.55/5.65	   U7_gga(Amount, C, Coins, N, X3, NewAmount, N1, ways_out_gga(NewAmount, .(s(C), Coins), N2)) -> U8_gga(Amount, C, Coins, N, plus_in_gga(N1, N2, N))
18.55/5.65	   plus_in_gga(0, X, X) -> U2_gga(X, nat_in_g(X))
18.55/5.65	   U2_gga(X, nat_out_g(X)) -> plus_out_gga(0, X, X)
18.55/5.65	   plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
18.55/5.65	   U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
18.55/5.65	   U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) -> ways_out_gga(Amount, .(s(C), Coins), N)
18.55/5.65	
18.55/5.65	The argument filtering Pi contains the following mapping:
18.55/5.65	ways_in_gga(x1, x2, x3)  =  ways_in_gga(x1, x2)
18.55/5.65	
18.55/5.65	0  =  0
18.55/5.65	
18.55/5.65	ways_out_gga(x1, x2, x3)  =  ways_out_gga(x3)
18.55/5.65	
18.55/5.65	[]  =  []
18.55/5.65	
18.55/5.65	.(x1, x2)  =  .(x1, x2)
18.55/5.65	
18.55/5.65	s(x1)  =  s(x1)
18.55/5.65	
18.55/5.65	U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	=_in_ga(x1, x2)  =  =_in_ga(x1)
18.55/5.65	
18.55/5.65	=_out_ga(x1, x2)  =  =_out_ga(x2)
18.55/5.65	
18.55/5.65	U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x6)
18.55/5.65	
18.55/5.65	plus_in_gag(x1, x2, x3)  =  plus_in_gag(x1, x3)
18.55/5.65	
18.55/5.65	U2_gag(x1, x2)  =  U2_gag(x1, x2)
18.55/5.65	
18.55/5.65	nat_in_g(x1)  =  nat_in_g(x1)
18.55/5.65	
18.55/5.65	nat_out_g(x1)  =  nat_out_g
18.55/5.65	
18.55/5.65	U1_g(x1, x2)  =  U1_g(x2)
18.55/5.65	
18.55/5.65	plus_out_gag(x1, x2, x3)  =  plus_out_gag(x2)
18.55/5.65	
18.55/5.65	U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
18.55/5.65	
18.55/5.65	U6_gga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gga(x2, x3, x6, x7)
18.55/5.65	
18.55/5.65	U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	U10_gga(x1, x2, x3, x4, x5, x6)  =  U10_gga(x1, x3, x6)
18.55/5.65	
18.55/5.65	U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x5)
18.55/5.65	
18.55/5.65	U7_gga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U7_gga(x7, x8)
18.55/5.65	
18.55/5.65	U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x5)
18.55/5.65	
18.55/5.65	plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
18.55/5.65	
18.55/5.65	U2_gga(x1, x2)  =  U2_gga(x1, x2)
18.55/5.65	
18.55/5.65	plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
18.55/5.65	
18.55/5.65	U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
18.55/5.65	
18.55/5.65	PLUS_IN_GAG(x1, x2, x3)  =  PLUS_IN_GAG(x1, x3)
18.55/5.65	
18.55/5.65	
18.55/5.65	We have to consider all (P,R,Pi)-chains
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(24) UsableRulesProof (EQUIVALENT)
18.55/5.65	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(25)
18.55/5.65	Obligation:
18.55/5.65	Pi DP problem:
18.55/5.65	The TRS P consists of the following rules:
18.55/5.65	
18.55/5.65	   PLUS_IN_GAG(s(X), Y, s(Z)) -> PLUS_IN_GAG(X, Y, Z)
18.55/5.65	
18.55/5.65	R is empty.
18.55/5.65	The argument filtering Pi contains the following mapping:
18.55/5.65	s(x1)  =  s(x1)
18.55/5.65	
18.55/5.65	PLUS_IN_GAG(x1, x2, x3)  =  PLUS_IN_GAG(x1, x3)
18.55/5.65	
18.55/5.65	
18.55/5.65	We have to consider all (P,R,Pi)-chains
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(26) PiDPToQDPProof (SOUND)
18.55/5.65	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(27)
18.55/5.65	Obligation:
18.55/5.65	Q DP problem:
18.55/5.65	The TRS P consists of the following rules:
18.55/5.65	
18.55/5.65	   PLUS_IN_GAG(s(X), s(Z)) -> PLUS_IN_GAG(X, Z)
18.55/5.65	
18.55/5.65	R is empty.
18.55/5.65	Q is empty.
18.55/5.65	We have to consider all (P,Q,R)-chains.
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(28) QDPSizeChangeProof (EQUIVALENT)
18.55/5.65	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. 
18.55/5.65	
18.55/5.65	From the DPs we obtained the following set of size-change graphs:
18.55/5.65	*PLUS_IN_GAG(s(X), s(Z)) -> PLUS_IN_GAG(X, Z)
18.55/5.65	The graph contains the following edges 1 > 1, 2 > 2
18.55/5.65	
18.55/5.65	
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(29)
18.55/5.65	YES
18.55/5.65	
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(30)
18.55/5.65	Obligation:
18.55/5.65	Pi DP problem:
18.55/5.65	The TRS P consists of the following rules:
18.55/5.65	
18.55/5.65	   U4_GGA(Amount, C, Coins, N, =_out_ga(Amount, s(X3))) -> U5_GGA(Amount, C, Coins, N, X3, plus_in_gag(s(C), NewAmount, Amount))
18.55/5.65	   U5_GGA(Amount, C, Coins, N, X3, plus_out_gag(s(C), NewAmount, Amount)) -> U6_GGA(Amount, C, Coins, N, X3, NewAmount, ways_in_gga(Amount, Coins, N1))
18.55/5.65	   U6_GGA(Amount, C, Coins, N, X3, NewAmount, ways_out_gga(Amount, Coins, N1)) -> WAYS_IN_GGA(NewAmount, .(s(C), Coins), N2)
18.55/5.65	   WAYS_IN_GGA(Amount, .(s(C), Coins), N) -> U4_GGA(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
18.55/5.65	   WAYS_IN_GGA(Amount, .(s(C), Coins), N) -> U9_GGA(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
18.55/5.65	   U9_GGA(Amount, C, Coins, N, =_out_ga(Amount, s(X4))) -> U10_GGA(Amount, C, Coins, N, X4, plus_in_gag(Amount, s(X5), s(C)))
18.55/5.65	   U10_GGA(Amount, C, Coins, N, X4, plus_out_gag(Amount, s(X5), s(C))) -> WAYS_IN_GGA(Amount, Coins, N)
18.55/5.65	   U5_GGA(Amount, C, Coins, N, X3, plus_out_gag(s(C), NewAmount, Amount)) -> WAYS_IN_GGA(Amount, Coins, N1)
18.55/5.65	
18.55/5.65	The TRS R consists of the following rules:
18.55/5.65	
18.55/5.65	   ways_in_gga(0, X1, s(0)) -> ways_out_gga(0, X1, s(0))
18.55/5.65	   ways_in_gga(X2, [], 0) -> ways_out_gga(X2, [], 0)
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins), N) -> U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
18.55/5.65	   =_in_ga(X, X) -> =_out_ga(X, X)
18.55/5.65	   U4_gga(Amount, C, Coins, N, =_out_ga(Amount, s(X3))) -> U5_gga(Amount, C, Coins, N, X3, plus_in_gag(s(C), NewAmount, Amount))
18.55/5.65	   plus_in_gag(0, X, X) -> U2_gag(X, nat_in_g(X))
18.55/5.65	   nat_in_g(0) -> nat_out_g(0)
18.55/5.65	   nat_in_g(s(X)) -> U1_g(X, nat_in_g(X))
18.55/5.65	   U1_g(X, nat_out_g(X)) -> nat_out_g(s(X))
18.55/5.65	   U2_gag(X, nat_out_g(X)) -> plus_out_gag(0, X, X)
18.55/5.65	   plus_in_gag(s(X), Y, s(Z)) -> U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
18.55/5.65	   U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) -> plus_out_gag(s(X), Y, s(Z))
18.55/5.65	   U5_gga(Amount, C, Coins, N, X3, plus_out_gag(s(C), NewAmount, Amount)) -> U6_gga(Amount, C, Coins, N, X3, NewAmount, ways_in_gga(Amount, Coins, N1))
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins), N) -> U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
18.55/5.65	   U9_gga(Amount, C, Coins, N, =_out_ga(Amount, s(X4))) -> U10_gga(Amount, C, Coins, N, X4, plus_in_gag(Amount, s(X5), s(C)))
18.55/5.65	   U10_gga(Amount, C, Coins, N, X4, plus_out_gag(Amount, s(X5), s(C))) -> U11_gga(Amount, C, Coins, N, ways_in_gga(Amount, Coins, N))
18.55/5.65	   U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) -> ways_out_gga(Amount, .(s(C), Coins), N)
18.55/5.65	   U6_gga(Amount, C, Coins, N, X3, NewAmount, ways_out_gga(Amount, Coins, N1)) -> U7_gga(Amount, C, Coins, N, X3, NewAmount, N1, ways_in_gga(NewAmount, .(s(C), Coins), N2))
18.55/5.65	   U7_gga(Amount, C, Coins, N, X3, NewAmount, N1, ways_out_gga(NewAmount, .(s(C), Coins), N2)) -> U8_gga(Amount, C, Coins, N, plus_in_gga(N1, N2, N))
18.55/5.65	   plus_in_gga(0, X, X) -> U2_gga(X, nat_in_g(X))
18.55/5.65	   U2_gga(X, nat_out_g(X)) -> plus_out_gga(0, X, X)
18.55/5.65	   plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
18.55/5.65	   U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
18.55/5.65	   U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) -> ways_out_gga(Amount, .(s(C), Coins), N)
18.55/5.65	
18.55/5.65	The argument filtering Pi contains the following mapping:
18.55/5.65	ways_in_gga(x1, x2, x3)  =  ways_in_gga(x1, x2)
18.55/5.65	
18.55/5.65	0  =  0
18.55/5.65	
18.55/5.65	ways_out_gga(x1, x2, x3)  =  ways_out_gga(x3)
18.55/5.65	
18.55/5.65	[]  =  []
18.55/5.65	
18.55/5.65	.(x1, x2)  =  .(x1, x2)
18.55/5.65	
18.55/5.65	s(x1)  =  s(x1)
18.55/5.65	
18.55/5.65	U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	=_in_ga(x1, x2)  =  =_in_ga(x1)
18.55/5.65	
18.55/5.65	=_out_ga(x1, x2)  =  =_out_ga(x2)
18.55/5.65	
18.55/5.65	U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x6)
18.55/5.65	
18.55/5.65	plus_in_gag(x1, x2, x3)  =  plus_in_gag(x1, x3)
18.55/5.65	
18.55/5.65	U2_gag(x1, x2)  =  U2_gag(x1, x2)
18.55/5.65	
18.55/5.65	nat_in_g(x1)  =  nat_in_g(x1)
18.55/5.65	
18.55/5.65	nat_out_g(x1)  =  nat_out_g
18.55/5.65	
18.55/5.65	U1_g(x1, x2)  =  U1_g(x2)
18.55/5.65	
18.55/5.65	plus_out_gag(x1, x2, x3)  =  plus_out_gag(x2)
18.55/5.65	
18.55/5.65	U3_gag(x1, x2, x3, x4)  =  U3_gag(x4)
18.55/5.65	
18.55/5.65	U6_gga(x1, x2, x3, x4, x5, x6, x7)  =  U6_gga(x2, x3, x6, x7)
18.55/5.65	
18.55/5.65	U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	U10_gga(x1, x2, x3, x4, x5, x6)  =  U10_gga(x1, x3, x6)
18.55/5.65	
18.55/5.65	U11_gga(x1, x2, x3, x4, x5)  =  U11_gga(x5)
18.55/5.65	
18.55/5.65	U7_gga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U7_gga(x7, x8)
18.55/5.65	
18.55/5.65	U8_gga(x1, x2, x3, x4, x5)  =  U8_gga(x5)
18.55/5.65	
18.55/5.65	plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)
18.55/5.65	
18.55/5.65	U2_gga(x1, x2)  =  U2_gga(x1, x2)
18.55/5.65	
18.55/5.65	plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)
18.55/5.65	
18.55/5.65	U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
18.55/5.65	
18.55/5.65	WAYS_IN_GGA(x1, x2, x3)  =  WAYS_IN_GGA(x1, x2)
18.55/5.65	
18.55/5.65	U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x1, x2, x3, x6)
18.55/5.65	
18.55/5.65	U6_GGA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GGA(x2, x3, x6, x7)
18.55/5.65	
18.55/5.65	U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x2, x3, x5)
18.55/5.65	
18.55/5.65	U10_GGA(x1, x2, x3, x4, x5, x6)  =  U10_GGA(x1, x3, x6)
18.55/5.65	
18.55/5.65	
18.55/5.65	We have to consider all (P,R,Pi)-chains
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(31) PiDPToQDPProof (SOUND)
18.55/5.65	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(32)
18.55/5.65	Obligation:
18.55/5.65	Q DP problem:
18.55/5.65	The TRS P consists of the following rules:
18.55/5.65	
18.55/5.65	   U4_GGA(Amount, C, Coins, =_out_ga(s(X3))) -> U5_GGA(Amount, C, Coins, plus_in_gag(s(C), Amount))
18.55/5.65	   U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) -> U6_GGA(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
18.55/5.65	   U6_GGA(C, Coins, NewAmount, ways_out_gga(N1)) -> WAYS_IN_GGA(NewAmount, .(s(C), Coins))
18.55/5.65	   WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U4_GGA(Amount, C, Coins, =_in_ga(Amount))
18.55/5.65	   WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U9_GGA(Amount, C, Coins, =_in_ga(Amount))
18.55/5.65	   U9_GGA(Amount, C, Coins, =_out_ga(s(X4))) -> U10_GGA(Amount, Coins, plus_in_gag(Amount, s(C)))
18.55/5.65	   U10_GGA(Amount, Coins, plus_out_gag(s(X5))) -> WAYS_IN_GGA(Amount, Coins)
18.55/5.65	   U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) -> WAYS_IN_GGA(Amount, Coins)
18.55/5.65	
18.55/5.65	The TRS R consists of the following rules:
18.55/5.65	
18.55/5.65	   ways_in_gga(0, X1) -> ways_out_gga(s(0))
18.55/5.65	   ways_in_gga(X2, []) -> ways_out_gga(0)
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins)) -> U4_gga(Amount, C, Coins, =_in_ga(Amount))
18.55/5.65	   =_in_ga(X) -> =_out_ga(X)
18.55/5.65	   U4_gga(Amount, C, Coins, =_out_ga(s(X3))) -> U5_gga(Amount, C, Coins, plus_in_gag(s(C), Amount))
18.55/5.65	   plus_in_gag(0, X) -> U2_gag(X, nat_in_g(X))
18.55/5.65	   nat_in_g(0) -> nat_out_g
18.55/5.65	   nat_in_g(s(X)) -> U1_g(nat_in_g(X))
18.55/5.65	   U1_g(nat_out_g) -> nat_out_g
18.55/5.65	   U2_gag(X, nat_out_g) -> plus_out_gag(X)
18.55/5.65	   plus_in_gag(s(X), s(Z)) -> U3_gag(plus_in_gag(X, Z))
18.55/5.65	   U3_gag(plus_out_gag(Y)) -> plus_out_gag(Y)
18.55/5.65	   U5_gga(Amount, C, Coins, plus_out_gag(NewAmount)) -> U6_gga(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins)) -> U9_gga(Amount, C, Coins, =_in_ga(Amount))
18.55/5.65	   U9_gga(Amount, C, Coins, =_out_ga(s(X4))) -> U10_gga(Amount, Coins, plus_in_gag(Amount, s(C)))
18.55/5.65	   U10_gga(Amount, Coins, plus_out_gag(s(X5))) -> U11_gga(ways_in_gga(Amount, Coins))
18.55/5.65	   U11_gga(ways_out_gga(N)) -> ways_out_gga(N)
18.55/5.65	   U6_gga(C, Coins, NewAmount, ways_out_gga(N1)) -> U7_gga(N1, ways_in_gga(NewAmount, .(s(C), Coins)))
18.55/5.65	   U7_gga(N1, ways_out_gga(N2)) -> U8_gga(plus_in_gga(N1, N2))
18.55/5.65	   plus_in_gga(0, X) -> U2_gga(X, nat_in_g(X))
18.55/5.65	   U2_gga(X, nat_out_g) -> plus_out_gga(X)
18.55/5.65	   plus_in_gga(s(X), Y) -> U3_gga(plus_in_gga(X, Y))
18.55/5.65	   U3_gga(plus_out_gga(Z)) -> plus_out_gga(s(Z))
18.55/5.65	   U8_gga(plus_out_gga(N)) -> ways_out_gga(N)
18.55/5.65	
18.55/5.65	The set Q consists of the following terms:
18.55/5.65	
18.55/5.65	   ways_in_gga(x0, x1)
18.55/5.65	   =_in_ga(x0)
18.55/5.65	   U4_gga(x0, x1, x2, x3)
18.55/5.65	   plus_in_gag(x0, x1)
18.55/5.65	   nat_in_g(x0)
18.55/5.65	   U1_g(x0)
18.55/5.65	   U2_gag(x0, x1)
18.55/5.65	   U3_gag(x0)
18.55/5.65	   U5_gga(x0, x1, x2, x3)
18.55/5.65	   U9_gga(x0, x1, x2, x3)
18.55/5.65	   U10_gga(x0, x1, x2)
18.55/5.65	   U11_gga(x0)
18.55/5.65	   U6_gga(x0, x1, x2, x3)
18.55/5.65	   U7_gga(x0, x1)
18.55/5.65	   plus_in_gga(x0, x1)
18.55/5.65	   U2_gga(x0, x1)
18.55/5.65	   U3_gga(x0)
18.55/5.65	   U8_gga(x0)
18.55/5.65	
18.55/5.65	We have to consider all (P,Q,R)-chains.
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(33) TransformationProof (EQUIVALENT)
18.55/5.65	By rewriting [LPAR04] the rule WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U4_GGA(Amount, C, Coins, =_in_ga(Amount)) at position [3] we obtained the following new rules [LPAR04]:
18.55/5.65	
18.55/5.65	   (WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U4_GGA(Amount, C, Coins, =_out_ga(Amount)),WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U4_GGA(Amount, C, Coins, =_out_ga(Amount)))
18.55/5.65	
18.55/5.65	
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(34)
18.55/5.65	Obligation:
18.55/5.65	Q DP problem:
18.55/5.65	The TRS P consists of the following rules:
18.55/5.65	
18.55/5.65	   U4_GGA(Amount, C, Coins, =_out_ga(s(X3))) -> U5_GGA(Amount, C, Coins, plus_in_gag(s(C), Amount))
18.55/5.65	   U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) -> U6_GGA(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
18.55/5.65	   U6_GGA(C, Coins, NewAmount, ways_out_gga(N1)) -> WAYS_IN_GGA(NewAmount, .(s(C), Coins))
18.55/5.65	   WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U9_GGA(Amount, C, Coins, =_in_ga(Amount))
18.55/5.65	   U9_GGA(Amount, C, Coins, =_out_ga(s(X4))) -> U10_GGA(Amount, Coins, plus_in_gag(Amount, s(C)))
18.55/5.65	   U10_GGA(Amount, Coins, plus_out_gag(s(X5))) -> WAYS_IN_GGA(Amount, Coins)
18.55/5.65	   U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) -> WAYS_IN_GGA(Amount, Coins)
18.55/5.65	   WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U4_GGA(Amount, C, Coins, =_out_ga(Amount))
18.55/5.65	
18.55/5.65	The TRS R consists of the following rules:
18.55/5.65	
18.55/5.65	   ways_in_gga(0, X1) -> ways_out_gga(s(0))
18.55/5.65	   ways_in_gga(X2, []) -> ways_out_gga(0)
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins)) -> U4_gga(Amount, C, Coins, =_in_ga(Amount))
18.55/5.65	   =_in_ga(X) -> =_out_ga(X)
18.55/5.65	   U4_gga(Amount, C, Coins, =_out_ga(s(X3))) -> U5_gga(Amount, C, Coins, plus_in_gag(s(C), Amount))
18.55/5.65	   plus_in_gag(0, X) -> U2_gag(X, nat_in_g(X))
18.55/5.65	   nat_in_g(0) -> nat_out_g
18.55/5.65	   nat_in_g(s(X)) -> U1_g(nat_in_g(X))
18.55/5.65	   U1_g(nat_out_g) -> nat_out_g
18.55/5.65	   U2_gag(X, nat_out_g) -> plus_out_gag(X)
18.55/5.65	   plus_in_gag(s(X), s(Z)) -> U3_gag(plus_in_gag(X, Z))
18.55/5.65	   U3_gag(plus_out_gag(Y)) -> plus_out_gag(Y)
18.55/5.65	   U5_gga(Amount, C, Coins, plus_out_gag(NewAmount)) -> U6_gga(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins)) -> U9_gga(Amount, C, Coins, =_in_ga(Amount))
18.55/5.65	   U9_gga(Amount, C, Coins, =_out_ga(s(X4))) -> U10_gga(Amount, Coins, plus_in_gag(Amount, s(C)))
18.55/5.65	   U10_gga(Amount, Coins, plus_out_gag(s(X5))) -> U11_gga(ways_in_gga(Amount, Coins))
18.55/5.65	   U11_gga(ways_out_gga(N)) -> ways_out_gga(N)
18.55/5.65	   U6_gga(C, Coins, NewAmount, ways_out_gga(N1)) -> U7_gga(N1, ways_in_gga(NewAmount, .(s(C), Coins)))
18.55/5.65	   U7_gga(N1, ways_out_gga(N2)) -> U8_gga(plus_in_gga(N1, N2))
18.55/5.65	   plus_in_gga(0, X) -> U2_gga(X, nat_in_g(X))
18.55/5.65	   U2_gga(X, nat_out_g) -> plus_out_gga(X)
18.55/5.65	   plus_in_gga(s(X), Y) -> U3_gga(plus_in_gga(X, Y))
18.55/5.65	   U3_gga(plus_out_gga(Z)) -> plus_out_gga(s(Z))
18.55/5.65	   U8_gga(plus_out_gga(N)) -> ways_out_gga(N)
18.55/5.65	
18.55/5.65	The set Q consists of the following terms:
18.55/5.65	
18.55/5.65	   ways_in_gga(x0, x1)
18.55/5.65	   =_in_ga(x0)
18.55/5.65	   U4_gga(x0, x1, x2, x3)
18.55/5.65	   plus_in_gag(x0, x1)
18.55/5.65	   nat_in_g(x0)
18.55/5.65	   U1_g(x0)
18.55/5.65	   U2_gag(x0, x1)
18.55/5.65	   U3_gag(x0)
18.55/5.65	   U5_gga(x0, x1, x2, x3)
18.55/5.65	   U9_gga(x0, x1, x2, x3)
18.55/5.65	   U10_gga(x0, x1, x2)
18.55/5.65	   U11_gga(x0)
18.55/5.65	   U6_gga(x0, x1, x2, x3)
18.55/5.65	   U7_gga(x0, x1)
18.55/5.65	   plus_in_gga(x0, x1)
18.55/5.65	   U2_gga(x0, x1)
18.55/5.65	   U3_gga(x0)
18.55/5.65	   U8_gga(x0)
18.55/5.65	
18.55/5.65	We have to consider all (P,Q,R)-chains.
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(35) TransformationProof (EQUIVALENT)
18.55/5.65	By rewriting [LPAR04] the rule WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U9_GGA(Amount, C, Coins, =_in_ga(Amount)) at position [3] we obtained the following new rules [LPAR04]:
18.55/5.65	
18.55/5.65	   (WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U9_GGA(Amount, C, Coins, =_out_ga(Amount)),WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U9_GGA(Amount, C, Coins, =_out_ga(Amount)))
18.55/5.65	
18.55/5.65	
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(36)
18.55/5.65	Obligation:
18.55/5.65	Q DP problem:
18.55/5.65	The TRS P consists of the following rules:
18.55/5.65	
18.55/5.65	   U4_GGA(Amount, C, Coins, =_out_ga(s(X3))) -> U5_GGA(Amount, C, Coins, plus_in_gag(s(C), Amount))
18.55/5.65	   U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) -> U6_GGA(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
18.55/5.65	   U6_GGA(C, Coins, NewAmount, ways_out_gga(N1)) -> WAYS_IN_GGA(NewAmount, .(s(C), Coins))
18.55/5.65	   U9_GGA(Amount, C, Coins, =_out_ga(s(X4))) -> U10_GGA(Amount, Coins, plus_in_gag(Amount, s(C)))
18.55/5.65	   U10_GGA(Amount, Coins, plus_out_gag(s(X5))) -> WAYS_IN_GGA(Amount, Coins)
18.55/5.65	   U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) -> WAYS_IN_GGA(Amount, Coins)
18.55/5.65	   WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U4_GGA(Amount, C, Coins, =_out_ga(Amount))
18.55/5.65	   WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U9_GGA(Amount, C, Coins, =_out_ga(Amount))
18.55/5.65	
18.55/5.65	The TRS R consists of the following rules:
18.55/5.65	
18.55/5.65	   ways_in_gga(0, X1) -> ways_out_gga(s(0))
18.55/5.65	   ways_in_gga(X2, []) -> ways_out_gga(0)
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins)) -> U4_gga(Amount, C, Coins, =_in_ga(Amount))
18.55/5.65	   =_in_ga(X) -> =_out_ga(X)
18.55/5.65	   U4_gga(Amount, C, Coins, =_out_ga(s(X3))) -> U5_gga(Amount, C, Coins, plus_in_gag(s(C), Amount))
18.55/5.65	   plus_in_gag(0, X) -> U2_gag(X, nat_in_g(X))
18.55/5.65	   nat_in_g(0) -> nat_out_g
18.55/5.65	   nat_in_g(s(X)) -> U1_g(nat_in_g(X))
18.55/5.65	   U1_g(nat_out_g) -> nat_out_g
18.55/5.65	   U2_gag(X, nat_out_g) -> plus_out_gag(X)
18.55/5.65	   plus_in_gag(s(X), s(Z)) -> U3_gag(plus_in_gag(X, Z))
18.55/5.65	   U3_gag(plus_out_gag(Y)) -> plus_out_gag(Y)
18.55/5.65	   U5_gga(Amount, C, Coins, plus_out_gag(NewAmount)) -> U6_gga(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins)) -> U9_gga(Amount, C, Coins, =_in_ga(Amount))
18.55/5.65	   U9_gga(Amount, C, Coins, =_out_ga(s(X4))) -> U10_gga(Amount, Coins, plus_in_gag(Amount, s(C)))
18.55/5.65	   U10_gga(Amount, Coins, plus_out_gag(s(X5))) -> U11_gga(ways_in_gga(Amount, Coins))
18.55/5.65	   U11_gga(ways_out_gga(N)) -> ways_out_gga(N)
18.55/5.65	   U6_gga(C, Coins, NewAmount, ways_out_gga(N1)) -> U7_gga(N1, ways_in_gga(NewAmount, .(s(C), Coins)))
18.55/5.65	   U7_gga(N1, ways_out_gga(N2)) -> U8_gga(plus_in_gga(N1, N2))
18.55/5.65	   plus_in_gga(0, X) -> U2_gga(X, nat_in_g(X))
18.55/5.65	   U2_gga(X, nat_out_g) -> plus_out_gga(X)
18.55/5.65	   plus_in_gga(s(X), Y) -> U3_gga(plus_in_gga(X, Y))
18.55/5.65	   U3_gga(plus_out_gga(Z)) -> plus_out_gga(s(Z))
18.55/5.65	   U8_gga(plus_out_gga(N)) -> ways_out_gga(N)
18.55/5.65	
18.55/5.65	The set Q consists of the following terms:
18.55/5.65	
18.55/5.65	   ways_in_gga(x0, x1)
18.55/5.65	   =_in_ga(x0)
18.55/5.65	   U4_gga(x0, x1, x2, x3)
18.55/5.65	   plus_in_gag(x0, x1)
18.55/5.65	   nat_in_g(x0)
18.55/5.65	   U1_g(x0)
18.55/5.65	   U2_gag(x0, x1)
18.55/5.65	   U3_gag(x0)
18.55/5.65	   U5_gga(x0, x1, x2, x3)
18.55/5.65	   U9_gga(x0, x1, x2, x3)
18.55/5.65	   U10_gga(x0, x1, x2)
18.55/5.65	   U11_gga(x0)
18.55/5.65	   U6_gga(x0, x1, x2, x3)
18.55/5.65	   U7_gga(x0, x1)
18.55/5.65	   plus_in_gga(x0, x1)
18.55/5.65	   U2_gga(x0, x1)
18.55/5.65	   U3_gga(x0)
18.55/5.65	   U8_gga(x0)
18.55/5.65	
18.55/5.65	We have to consider all (P,Q,R)-chains.
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(37) TransformationProof (EQUIVALENT)
18.55/5.65	By instantiating [LPAR04] the rule U4_GGA(Amount, C, Coins, =_out_ga(s(X3))) -> U5_GGA(Amount, C, Coins, plus_in_gag(s(C), Amount)) we obtained the following new rules [LPAR04]:
18.55/5.65	
18.55/5.65	   (U4_GGA(s(x3), z1, z2, =_out_ga(s(x3))) -> U5_GGA(s(x3), z1, z2, plus_in_gag(s(z1), s(x3))),U4_GGA(s(x3), z1, z2, =_out_ga(s(x3))) -> U5_GGA(s(x3), z1, z2, plus_in_gag(s(z1), s(x3))))
18.55/5.65	
18.55/5.65	
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(38)
18.55/5.65	Obligation:
18.55/5.65	Q DP problem:
18.55/5.65	The TRS P consists of the following rules:
18.55/5.65	
18.55/5.65	   U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) -> U6_GGA(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
18.55/5.65	   U6_GGA(C, Coins, NewAmount, ways_out_gga(N1)) -> WAYS_IN_GGA(NewAmount, .(s(C), Coins))
18.55/5.65	   U9_GGA(Amount, C, Coins, =_out_ga(s(X4))) -> U10_GGA(Amount, Coins, plus_in_gag(Amount, s(C)))
18.55/5.65	   U10_GGA(Amount, Coins, plus_out_gag(s(X5))) -> WAYS_IN_GGA(Amount, Coins)
18.55/5.65	   U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) -> WAYS_IN_GGA(Amount, Coins)
18.55/5.65	   WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U4_GGA(Amount, C, Coins, =_out_ga(Amount))
18.55/5.65	   WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U9_GGA(Amount, C, Coins, =_out_ga(Amount))
18.55/5.65	   U4_GGA(s(x3), z1, z2, =_out_ga(s(x3))) -> U5_GGA(s(x3), z1, z2, plus_in_gag(s(z1), s(x3)))
18.55/5.65	
18.55/5.65	The TRS R consists of the following rules:
18.55/5.65	
18.55/5.65	   ways_in_gga(0, X1) -> ways_out_gga(s(0))
18.55/5.65	   ways_in_gga(X2, []) -> ways_out_gga(0)
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins)) -> U4_gga(Amount, C, Coins, =_in_ga(Amount))
18.55/5.65	   =_in_ga(X) -> =_out_ga(X)
18.55/5.65	   U4_gga(Amount, C, Coins, =_out_ga(s(X3))) -> U5_gga(Amount, C, Coins, plus_in_gag(s(C), Amount))
18.55/5.65	   plus_in_gag(0, X) -> U2_gag(X, nat_in_g(X))
18.55/5.65	   nat_in_g(0) -> nat_out_g
18.55/5.65	   nat_in_g(s(X)) -> U1_g(nat_in_g(X))
18.55/5.65	   U1_g(nat_out_g) -> nat_out_g
18.55/5.65	   U2_gag(X, nat_out_g) -> plus_out_gag(X)
18.55/5.65	   plus_in_gag(s(X), s(Z)) -> U3_gag(plus_in_gag(X, Z))
18.55/5.65	   U3_gag(plus_out_gag(Y)) -> plus_out_gag(Y)
18.55/5.65	   U5_gga(Amount, C, Coins, plus_out_gag(NewAmount)) -> U6_gga(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins)) -> U9_gga(Amount, C, Coins, =_in_ga(Amount))
18.55/5.65	   U9_gga(Amount, C, Coins, =_out_ga(s(X4))) -> U10_gga(Amount, Coins, plus_in_gag(Amount, s(C)))
18.55/5.65	   U10_gga(Amount, Coins, plus_out_gag(s(X5))) -> U11_gga(ways_in_gga(Amount, Coins))
18.55/5.65	   U11_gga(ways_out_gga(N)) -> ways_out_gga(N)
18.55/5.65	   U6_gga(C, Coins, NewAmount, ways_out_gga(N1)) -> U7_gga(N1, ways_in_gga(NewAmount, .(s(C), Coins)))
18.55/5.65	   U7_gga(N1, ways_out_gga(N2)) -> U8_gga(plus_in_gga(N1, N2))
18.55/5.65	   plus_in_gga(0, X) -> U2_gga(X, nat_in_g(X))
18.55/5.65	   U2_gga(X, nat_out_g) -> plus_out_gga(X)
18.55/5.65	   plus_in_gga(s(X), Y) -> U3_gga(plus_in_gga(X, Y))
18.55/5.65	   U3_gga(plus_out_gga(Z)) -> plus_out_gga(s(Z))
18.55/5.65	   U8_gga(plus_out_gga(N)) -> ways_out_gga(N)
18.55/5.65	
18.55/5.65	The set Q consists of the following terms:
18.55/5.65	
18.55/5.65	   ways_in_gga(x0, x1)
18.55/5.65	   =_in_ga(x0)
18.55/5.65	   U4_gga(x0, x1, x2, x3)
18.55/5.65	   plus_in_gag(x0, x1)
18.55/5.65	   nat_in_g(x0)
18.55/5.65	   U1_g(x0)
18.55/5.65	   U2_gag(x0, x1)
18.55/5.65	   U3_gag(x0)
18.55/5.65	   U5_gga(x0, x1, x2, x3)
18.55/5.65	   U9_gga(x0, x1, x2, x3)
18.55/5.65	   U10_gga(x0, x1, x2)
18.55/5.65	   U11_gga(x0)
18.55/5.65	   U6_gga(x0, x1, x2, x3)
18.55/5.65	   U7_gga(x0, x1)
18.55/5.65	   plus_in_gga(x0, x1)
18.55/5.65	   U2_gga(x0, x1)
18.55/5.65	   U3_gga(x0)
18.55/5.65	   U8_gga(x0)
18.55/5.65	
18.55/5.65	We have to consider all (P,Q,R)-chains.
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(39) TransformationProof (EQUIVALENT)
18.55/5.65	By rewriting [LPAR04] the rule U4_GGA(s(x3), z1, z2, =_out_ga(s(x3))) -> U5_GGA(s(x3), z1, z2, plus_in_gag(s(z1), s(x3))) at position [3] we obtained the following new rules [LPAR04]:
18.55/5.65	
18.55/5.65	   (U4_GGA(s(x3), z1, z2, =_out_ga(s(x3))) -> U5_GGA(s(x3), z1, z2, U3_gag(plus_in_gag(z1, x3))),U4_GGA(s(x3), z1, z2, =_out_ga(s(x3))) -> U5_GGA(s(x3), z1, z2, U3_gag(plus_in_gag(z1, x3))))
18.55/5.65	
18.55/5.65	
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(40)
18.55/5.65	Obligation:
18.55/5.65	Q DP problem:
18.55/5.65	The TRS P consists of the following rules:
18.55/5.65	
18.55/5.65	   U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) -> U6_GGA(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
18.55/5.65	   U6_GGA(C, Coins, NewAmount, ways_out_gga(N1)) -> WAYS_IN_GGA(NewAmount, .(s(C), Coins))
18.55/5.65	   U9_GGA(Amount, C, Coins, =_out_ga(s(X4))) -> U10_GGA(Amount, Coins, plus_in_gag(Amount, s(C)))
18.55/5.65	   U10_GGA(Amount, Coins, plus_out_gag(s(X5))) -> WAYS_IN_GGA(Amount, Coins)
18.55/5.65	   U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) -> WAYS_IN_GGA(Amount, Coins)
18.55/5.65	   WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U4_GGA(Amount, C, Coins, =_out_ga(Amount))
18.55/5.65	   WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U9_GGA(Amount, C, Coins, =_out_ga(Amount))
18.55/5.65	   U4_GGA(s(x3), z1, z2, =_out_ga(s(x3))) -> U5_GGA(s(x3), z1, z2, U3_gag(plus_in_gag(z1, x3)))
18.55/5.65	
18.55/5.65	The TRS R consists of the following rules:
18.55/5.65	
18.55/5.65	   ways_in_gga(0, X1) -> ways_out_gga(s(0))
18.55/5.65	   ways_in_gga(X2, []) -> ways_out_gga(0)
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins)) -> U4_gga(Amount, C, Coins, =_in_ga(Amount))
18.55/5.65	   =_in_ga(X) -> =_out_ga(X)
18.55/5.65	   U4_gga(Amount, C, Coins, =_out_ga(s(X3))) -> U5_gga(Amount, C, Coins, plus_in_gag(s(C), Amount))
18.55/5.65	   plus_in_gag(0, X) -> U2_gag(X, nat_in_g(X))
18.55/5.65	   nat_in_g(0) -> nat_out_g
18.55/5.65	   nat_in_g(s(X)) -> U1_g(nat_in_g(X))
18.55/5.65	   U1_g(nat_out_g) -> nat_out_g
18.55/5.65	   U2_gag(X, nat_out_g) -> plus_out_gag(X)
18.55/5.65	   plus_in_gag(s(X), s(Z)) -> U3_gag(plus_in_gag(X, Z))
18.55/5.65	   U3_gag(plus_out_gag(Y)) -> plus_out_gag(Y)
18.55/5.65	   U5_gga(Amount, C, Coins, plus_out_gag(NewAmount)) -> U6_gga(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
18.55/5.65	   ways_in_gga(Amount, .(s(C), Coins)) -> U9_gga(Amount, C, Coins, =_in_ga(Amount))
18.55/5.65	   U9_gga(Amount, C, Coins, =_out_ga(s(X4))) -> U10_gga(Amount, Coins, plus_in_gag(Amount, s(C)))
18.55/5.65	   U10_gga(Amount, Coins, plus_out_gag(s(X5))) -> U11_gga(ways_in_gga(Amount, Coins))
18.55/5.65	   U11_gga(ways_out_gga(N)) -> ways_out_gga(N)
18.55/5.65	   U6_gga(C, Coins, NewAmount, ways_out_gga(N1)) -> U7_gga(N1, ways_in_gga(NewAmount, .(s(C), Coins)))
18.55/5.65	   U7_gga(N1, ways_out_gga(N2)) -> U8_gga(plus_in_gga(N1, N2))
18.55/5.65	   plus_in_gga(0, X) -> U2_gga(X, nat_in_g(X))
18.55/5.65	   U2_gga(X, nat_out_g) -> plus_out_gga(X)
18.55/5.65	   plus_in_gga(s(X), Y) -> U3_gga(plus_in_gga(X, Y))
18.55/5.65	   U3_gga(plus_out_gga(Z)) -> plus_out_gga(s(Z))
18.55/5.65	   U8_gga(plus_out_gga(N)) -> ways_out_gga(N)
18.55/5.65	
18.55/5.65	The set Q consists of the following terms:
18.55/5.65	
18.55/5.65	   ways_in_gga(x0, x1)
18.55/5.65	   =_in_ga(x0)
18.55/5.65	   U4_gga(x0, x1, x2, x3)
18.55/5.65	   plus_in_gag(x0, x1)
18.55/5.65	   nat_in_g(x0)
18.55/5.65	   U1_g(x0)
18.55/5.65	   U2_gag(x0, x1)
18.55/5.65	   U3_gag(x0)
18.55/5.65	   U5_gga(x0, x1, x2, x3)
18.55/5.65	   U9_gga(x0, x1, x2, x3)
18.55/5.65	   U10_gga(x0, x1, x2)
18.55/5.65	   U11_gga(x0)
18.55/5.65	   U6_gga(x0, x1, x2, x3)
18.55/5.65	   U7_gga(x0, x1)
18.55/5.65	   plus_in_gga(x0, x1)
18.55/5.65	   U2_gga(x0, x1)
18.55/5.65	   U3_gga(x0)
18.55/5.65	   U8_gga(x0)
18.55/5.65	
18.55/5.65	We have to consider all (P,Q,R)-chains.
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(41) TransformationProof (EQUIVALENT)
18.55/5.65	By instantiating [LPAR04] the rule U9_GGA(Amount, C, Coins, =_out_ga(s(X4))) -> U10_GGA(Amount, Coins, plus_in_gag(Amount, s(C))) we obtained the following new rules [LPAR04]:
18.55/5.65	
18.55/5.65	   (U9_GGA(s(x3), z1, z2, =_out_ga(s(x3))) -> U10_GGA(s(x3), z2, plus_in_gag(s(x3), s(z1))),U9_GGA(s(x3), z1, z2, =_out_ga(s(x3))) -> U10_GGA(s(x3), z2, plus_in_gag(s(x3), s(z1))))
18.55/5.65	
18.55/5.65	
18.55/5.65	----------------------------------------
18.55/5.65	
18.55/5.65	(42)
18.55/5.65	Obligation:
18.55/5.65	Q DP problem:
18.55/5.65	The TRS P consists of the following rules:
18.55/5.65	
18.55/5.65	   U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) -> U6_GGA(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
18.55/5.65	   U6_GGA(C, Coins, NewAmount, ways_out_gga(N1)) -> WAYS_IN_GGA(NewAmount, .(s(C), Coins))
18.55/5.65	   U10_GGA(Amount, Coins, plus_out_gag(s(X5))) -> WAYS_IN_GGA(Amount, Coins)
18.55/5.65	   U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) -> WAYS_IN_GGA(Amount, Coins)
18.55/5.65	   WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U4_GGA(Amount, C, Coins, =_out_ga(Amount))
18.55/5.65	   WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U9_GGA(Amount, C, Coins, =_out_ga(Amount))
18.55/5.65	   U4_GGA(s(x3), z1, z2, =_out_ga(s(x3))) -> U5_GGA(s(x3), z1, z2, U3_gag(plus_in_gag(z1, x3)))
18.55/5.65	   U9_GGA(s(x3), z1, z2, =_out_ga(s(x3))) -> U10_GGA(s(x3), z2, plus_in_gag(s(x3), s(z1)))
18.55/5.65	
18.55/5.65	The TRS R consists of the following rules:
18.55/5.66	
18.55/5.66	   ways_in_gga(0, X1) -> ways_out_gga(s(0))
18.55/5.66	   ways_in_gga(X2, []) -> ways_out_gga(0)
18.55/5.66	   ways_in_gga(Amount, .(s(C), Coins)) -> U4_gga(Amount, C, Coins, =_in_ga(Amount))
18.55/5.66	   =_in_ga(X) -> =_out_ga(X)
18.55/5.66	   U4_gga(Amount, C, Coins, =_out_ga(s(X3))) -> U5_gga(Amount, C, Coins, plus_in_gag(s(C), Amount))
18.55/5.66	   plus_in_gag(0, X) -> U2_gag(X, nat_in_g(X))
18.55/5.66	   nat_in_g(0) -> nat_out_g
18.55/5.66	   nat_in_g(s(X)) -> U1_g(nat_in_g(X))
18.55/5.66	   U1_g(nat_out_g) -> nat_out_g
18.55/5.66	   U2_gag(X, nat_out_g) -> plus_out_gag(X)
18.55/5.66	   plus_in_gag(s(X), s(Z)) -> U3_gag(plus_in_gag(X, Z))
18.55/5.66	   U3_gag(plus_out_gag(Y)) -> plus_out_gag(Y)
18.55/5.66	   U5_gga(Amount, C, Coins, plus_out_gag(NewAmount)) -> U6_gga(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
18.55/5.66	   ways_in_gga(Amount, .(s(C), Coins)) -> U9_gga(Amount, C, Coins, =_in_ga(Amount))
18.55/5.66	   U9_gga(Amount, C, Coins, =_out_ga(s(X4))) -> U10_gga(Amount, Coins, plus_in_gag(Amount, s(C)))
18.55/5.66	   U10_gga(Amount, Coins, plus_out_gag(s(X5))) -> U11_gga(ways_in_gga(Amount, Coins))
18.55/5.66	   U11_gga(ways_out_gga(N)) -> ways_out_gga(N)
18.55/5.66	   U6_gga(C, Coins, NewAmount, ways_out_gga(N1)) -> U7_gga(N1, ways_in_gga(NewAmount, .(s(C), Coins)))
18.55/5.66	   U7_gga(N1, ways_out_gga(N2)) -> U8_gga(plus_in_gga(N1, N2))
18.55/5.66	   plus_in_gga(0, X) -> U2_gga(X, nat_in_g(X))
18.55/5.66	   U2_gga(X, nat_out_g) -> plus_out_gga(X)
18.55/5.66	   plus_in_gga(s(X), Y) -> U3_gga(plus_in_gga(X, Y))
18.55/5.66	   U3_gga(plus_out_gga(Z)) -> plus_out_gga(s(Z))
18.55/5.66	   U8_gga(plus_out_gga(N)) -> ways_out_gga(N)
18.55/5.66	
18.55/5.66	The set Q consists of the following terms:
18.55/5.66	
18.55/5.66	   ways_in_gga(x0, x1)
18.55/5.66	   =_in_ga(x0)
18.55/5.66	   U4_gga(x0, x1, x2, x3)
18.55/5.66	   plus_in_gag(x0, x1)
18.55/5.66	   nat_in_g(x0)
18.55/5.66	   U1_g(x0)
18.55/5.66	   U2_gag(x0, x1)
18.55/5.66	   U3_gag(x0)
18.55/5.66	   U5_gga(x0, x1, x2, x3)
18.55/5.66	   U9_gga(x0, x1, x2, x3)
18.55/5.66	   U10_gga(x0, x1, x2)
18.55/5.66	   U11_gga(x0)
18.55/5.66	   U6_gga(x0, x1, x2, x3)
18.55/5.66	   U7_gga(x0, x1)
18.55/5.66	   plus_in_gga(x0, x1)
18.55/5.66	   U2_gga(x0, x1)
18.55/5.66	   U3_gga(x0)
18.55/5.66	   U8_gga(x0)
18.55/5.66	
18.55/5.66	We have to consider all (P,Q,R)-chains.
18.55/5.66	----------------------------------------
18.55/5.66	
18.55/5.66	(43) TransformationProof (EQUIVALENT)
18.55/5.66	By rewriting [LPAR04] the rule U9_GGA(s(x3), z1, z2, =_out_ga(s(x3))) -> U10_GGA(s(x3), z2, plus_in_gag(s(x3), s(z1))) at position [2] we obtained the following new rules [LPAR04]:
18.55/5.66	
18.55/5.66	   (U9_GGA(s(x3), z1, z2, =_out_ga(s(x3))) -> U10_GGA(s(x3), z2, U3_gag(plus_in_gag(x3, z1))),U9_GGA(s(x3), z1, z2, =_out_ga(s(x3))) -> U10_GGA(s(x3), z2, U3_gag(plus_in_gag(x3, z1))))
18.55/5.66	
18.55/5.66	
18.55/5.66	----------------------------------------
18.55/5.66	
18.55/5.66	(44)
18.55/5.66	Obligation:
18.55/5.66	Q DP problem:
18.55/5.66	The TRS P consists of the following rules:
18.55/5.66	
18.55/5.66	   U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) -> U6_GGA(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
18.55/5.66	   U6_GGA(C, Coins, NewAmount, ways_out_gga(N1)) -> WAYS_IN_GGA(NewAmount, .(s(C), Coins))
18.55/5.66	   U10_GGA(Amount, Coins, plus_out_gag(s(X5))) -> WAYS_IN_GGA(Amount, Coins)
18.55/5.66	   U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) -> WAYS_IN_GGA(Amount, Coins)
18.55/5.66	   WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U4_GGA(Amount, C, Coins, =_out_ga(Amount))
18.55/5.66	   WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U9_GGA(Amount, C, Coins, =_out_ga(Amount))
18.55/5.66	   U4_GGA(s(x3), z1, z2, =_out_ga(s(x3))) -> U5_GGA(s(x3), z1, z2, U3_gag(plus_in_gag(z1, x3)))
18.55/5.66	   U9_GGA(s(x3), z1, z2, =_out_ga(s(x3))) -> U10_GGA(s(x3), z2, U3_gag(plus_in_gag(x3, z1)))
18.55/5.66	
18.55/5.66	The TRS R consists of the following rules:
18.55/5.66	
18.55/5.66	   ways_in_gga(0, X1) -> ways_out_gga(s(0))
18.55/5.66	   ways_in_gga(X2, []) -> ways_out_gga(0)
18.55/5.66	   ways_in_gga(Amount, .(s(C), Coins)) -> U4_gga(Amount, C, Coins, =_in_ga(Amount))
18.55/5.66	   =_in_ga(X) -> =_out_ga(X)
18.55/5.66	   U4_gga(Amount, C, Coins, =_out_ga(s(X3))) -> U5_gga(Amount, C, Coins, plus_in_gag(s(C), Amount))
18.55/5.66	   plus_in_gag(0, X) -> U2_gag(X, nat_in_g(X))
18.55/5.66	   nat_in_g(0) -> nat_out_g
18.55/5.66	   nat_in_g(s(X)) -> U1_g(nat_in_g(X))
18.55/5.66	   U1_g(nat_out_g) -> nat_out_g
18.55/5.66	   U2_gag(X, nat_out_g) -> plus_out_gag(X)
18.55/5.66	   plus_in_gag(s(X), s(Z)) -> U3_gag(plus_in_gag(X, Z))
18.55/5.66	   U3_gag(plus_out_gag(Y)) -> plus_out_gag(Y)
18.55/5.66	   U5_gga(Amount, C, Coins, plus_out_gag(NewAmount)) -> U6_gga(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
18.55/5.66	   ways_in_gga(Amount, .(s(C), Coins)) -> U9_gga(Amount, C, Coins, =_in_ga(Amount))
18.55/5.66	   U9_gga(Amount, C, Coins, =_out_ga(s(X4))) -> U10_gga(Amount, Coins, plus_in_gag(Amount, s(C)))
18.55/5.66	   U10_gga(Amount, Coins, plus_out_gag(s(X5))) -> U11_gga(ways_in_gga(Amount, Coins))
18.55/5.66	   U11_gga(ways_out_gga(N)) -> ways_out_gga(N)
18.55/5.66	   U6_gga(C, Coins, NewAmount, ways_out_gga(N1)) -> U7_gga(N1, ways_in_gga(NewAmount, .(s(C), Coins)))
18.55/5.66	   U7_gga(N1, ways_out_gga(N2)) -> U8_gga(plus_in_gga(N1, N2))
18.55/5.66	   plus_in_gga(0, X) -> U2_gga(X, nat_in_g(X))
18.55/5.66	   U2_gga(X, nat_out_g) -> plus_out_gga(X)
18.55/5.66	   plus_in_gga(s(X), Y) -> U3_gga(plus_in_gga(X, Y))
18.55/5.66	   U3_gga(plus_out_gga(Z)) -> plus_out_gga(s(Z))
18.55/5.66	   U8_gga(plus_out_gga(N)) -> ways_out_gga(N)
18.55/5.66	
18.55/5.66	The set Q consists of the following terms:
18.55/5.66	
18.55/5.66	   ways_in_gga(x0, x1)
18.55/5.66	   =_in_ga(x0)
18.55/5.66	   U4_gga(x0, x1, x2, x3)
18.55/5.66	   plus_in_gag(x0, x1)
18.55/5.66	   nat_in_g(x0)
18.55/5.66	   U1_g(x0)
18.55/5.66	   U2_gag(x0, x1)
18.55/5.66	   U3_gag(x0)
18.55/5.66	   U5_gga(x0, x1, x2, x3)
18.55/5.66	   U9_gga(x0, x1, x2, x3)
18.55/5.66	   U10_gga(x0, x1, x2)
18.55/5.66	   U11_gga(x0)
18.55/5.66	   U6_gga(x0, x1, x2, x3)
18.55/5.66	   U7_gga(x0, x1)
18.55/5.66	   plus_in_gga(x0, x1)
18.55/5.66	   U2_gga(x0, x1)
18.55/5.66	   U3_gga(x0)
18.55/5.66	   U8_gga(x0)
18.55/5.66	
18.55/5.66	We have to consider all (P,Q,R)-chains.
18.55/5.66	----------------------------------------
18.55/5.66	
18.55/5.66	(45) QDPOrderProof (EQUIVALENT)
18.55/5.66	We use the reduction pair processor [LPAR04,JAR06].
18.55/5.66	
18.55/5.66	
18.55/5.66	The following pairs can be oriented strictly and are deleted.
18.55/5.66	
18.55/5.66	   U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) -> WAYS_IN_GGA(Amount, Coins)
18.55/5.66	   U9_GGA(s(x3), z1, z2, =_out_ga(s(x3))) -> U10_GGA(s(x3), z2, U3_gag(plus_in_gag(x3, z1)))
18.55/5.66	The remaining pairs can at least be oriented weakly.
18.55/5.66	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
18.55/5.66	
18.55/5.66	POL( U6_GGA_4(x_1, ..., x_4) ) = 2x_2 + 1
18.55/5.66	POL( ways_in_gga_2(x_1, x_2) ) = max{0, 2x_1 - 2}
18.55/5.66	POL( 0 ) = 0
18.55/5.66	POL( ways_out_gga_1(x_1) ) = max{0, x_1 - 2}
18.55/5.66	POL( s_1(x_1) ) = 0
18.55/5.66	POL( [] ) = 2
18.55/5.66	POL( ._2(x_1, x_2) ) = x_2 + 1
18.55/5.66	POL( U4_gga_4(x_1, ..., x_4) ) = max{0, 2x_1 + x_2 + x_3 + x_4 - 2}
18.55/5.66	POL( =_in_ga_1(x_1) ) = 2x_1
18.55/5.66	POL( U9_gga_4(x_1, ..., x_4) ) = max{0, x_3 - 2}
18.55/5.66	POL( U10_GGA_3(x_1, ..., x_3) ) = 2x_2
18.55/5.66	POL( U3_gag_1(x_1) ) = max{0, -2}
18.55/5.66	POL( U10_gga_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
18.55/5.66	POL( U5_GGA_4(x_1, ..., x_4) ) = 2x_3 + 1
18.55/5.66	POL( U5_gga_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_3 - 2}
18.55/5.66	POL( plus_in_gag_2(x_1, x_2) ) = max{0, -2}
18.55/5.66	POL( U2_gag_2(x_1, x_2) ) = max{0, 2x_2 - 2}
18.55/5.66	POL( nat_in_g_1(x_1) ) = 2x_1
18.55/5.66	POL( plus_out_gag_1(x_1) ) = 0
18.55/5.66	POL( =_out_ga_1(x_1) ) = 0
18.55/5.66	POL( U6_gga_4(x_1, ..., x_4) ) = x_3 + 2
18.55/5.66	POL( U11_gga_1(x_1) ) = max{0, 2x_1 - 2}
18.55/5.66	POL( U7_gga_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2}
18.55/5.66	POL( U8_gga_1(x_1) ) = 2
18.55/5.66	POL( plus_in_gga_2(x_1, x_2) ) = 1
18.55/5.66	POL( U2_gga_2(x_1, x_2) ) = max{0, 2x_1 - 2}
18.55/5.66	POL( nat_out_g ) = 0
18.55/5.66	POL( U1_g_1(x_1) ) = x_1 + 1
18.55/5.66	POL( U3_gga_1(x_1) ) = max{0, -2}
18.55/5.66	POL( plus_out_gga_1(x_1) ) = max{0, -2}
18.55/5.66	POL( WAYS_IN_GGA_2(x_1, x_2) ) = max{0, 2x_2 - 1}
18.55/5.66	POL( U4_GGA_4(x_1, ..., x_4) ) = 2x_3 + 1
18.55/5.66	POL( U9_GGA_4(x_1, ..., x_4) ) = 2x_3 + 1
18.55/5.66	
18.55/5.66	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
18.55/5.66	none
18.55/5.66	
18.55/5.66	
18.55/5.66	----------------------------------------
18.55/5.66	
18.55/5.66	(46)
18.55/5.66	Obligation:
18.55/5.66	Q DP problem:
18.55/5.66	The TRS P consists of the following rules:
18.55/5.66	
18.55/5.66	   U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) -> U6_GGA(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
18.55/5.66	   U6_GGA(C, Coins, NewAmount, ways_out_gga(N1)) -> WAYS_IN_GGA(NewAmount, .(s(C), Coins))
18.55/5.66	   U10_GGA(Amount, Coins, plus_out_gag(s(X5))) -> WAYS_IN_GGA(Amount, Coins)
18.55/5.66	   WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U4_GGA(Amount, C, Coins, =_out_ga(Amount))
18.55/5.66	   WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U9_GGA(Amount, C, Coins, =_out_ga(Amount))
18.55/5.66	   U4_GGA(s(x3), z1, z2, =_out_ga(s(x3))) -> U5_GGA(s(x3), z1, z2, U3_gag(plus_in_gag(z1, x3)))
18.55/5.66	
18.55/5.66	The TRS R consists of the following rules:
18.55/5.66	
18.55/5.66	   ways_in_gga(0, X1) -> ways_out_gga(s(0))
18.55/5.66	   ways_in_gga(X2, []) -> ways_out_gga(0)
18.55/5.66	   ways_in_gga(Amount, .(s(C), Coins)) -> U4_gga(Amount, C, Coins, =_in_ga(Amount))
18.55/5.66	   =_in_ga(X) -> =_out_ga(X)
18.55/5.66	   U4_gga(Amount, C, Coins, =_out_ga(s(X3))) -> U5_gga(Amount, C, Coins, plus_in_gag(s(C), Amount))
18.55/5.66	   plus_in_gag(0, X) -> U2_gag(X, nat_in_g(X))
18.55/5.66	   nat_in_g(0) -> nat_out_g
18.55/5.66	   nat_in_g(s(X)) -> U1_g(nat_in_g(X))
18.55/5.66	   U1_g(nat_out_g) -> nat_out_g
18.55/5.66	   U2_gag(X, nat_out_g) -> plus_out_gag(X)
18.55/5.66	   plus_in_gag(s(X), s(Z)) -> U3_gag(plus_in_gag(X, Z))
18.55/5.66	   U3_gag(plus_out_gag(Y)) -> plus_out_gag(Y)
18.55/5.66	   U5_gga(Amount, C, Coins, plus_out_gag(NewAmount)) -> U6_gga(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
18.55/5.66	   ways_in_gga(Amount, .(s(C), Coins)) -> U9_gga(Amount, C, Coins, =_in_ga(Amount))
18.55/5.66	   U9_gga(Amount, C, Coins, =_out_ga(s(X4))) -> U10_gga(Amount, Coins, plus_in_gag(Amount, s(C)))
18.55/5.66	   U10_gga(Amount, Coins, plus_out_gag(s(X5))) -> U11_gga(ways_in_gga(Amount, Coins))
18.55/5.66	   U11_gga(ways_out_gga(N)) -> ways_out_gga(N)
18.55/5.66	   U6_gga(C, Coins, NewAmount, ways_out_gga(N1)) -> U7_gga(N1, ways_in_gga(NewAmount, .(s(C), Coins)))
18.55/5.66	   U7_gga(N1, ways_out_gga(N2)) -> U8_gga(plus_in_gga(N1, N2))
18.55/5.66	   plus_in_gga(0, X) -> U2_gga(X, nat_in_g(X))
18.55/5.66	   U2_gga(X, nat_out_g) -> plus_out_gga(X)
18.55/5.66	   plus_in_gga(s(X), Y) -> U3_gga(plus_in_gga(X, Y))
18.55/5.66	   U3_gga(plus_out_gga(Z)) -> plus_out_gga(s(Z))
18.55/5.66	   U8_gga(plus_out_gga(N)) -> ways_out_gga(N)
18.55/5.66	
18.55/5.66	The set Q consists of the following terms:
18.55/5.66	
18.55/5.66	   ways_in_gga(x0, x1)
18.55/5.66	   =_in_ga(x0)
18.55/5.66	   U4_gga(x0, x1, x2, x3)
18.55/5.66	   plus_in_gag(x0, x1)
18.55/5.66	   nat_in_g(x0)
18.55/5.66	   U1_g(x0)
18.55/5.66	   U2_gag(x0, x1)
18.55/5.66	   U3_gag(x0)
18.55/5.66	   U5_gga(x0, x1, x2, x3)
18.55/5.66	   U9_gga(x0, x1, x2, x3)
18.55/5.66	   U10_gga(x0, x1, x2)
18.55/5.66	   U11_gga(x0)
18.55/5.66	   U6_gga(x0, x1, x2, x3)
18.55/5.66	   U7_gga(x0, x1)
18.55/5.66	   plus_in_gga(x0, x1)
18.55/5.66	   U2_gga(x0, x1)
18.55/5.66	   U3_gga(x0)
18.55/5.66	   U8_gga(x0)
18.55/5.66	
18.55/5.66	We have to consider all (P,Q,R)-chains.
18.55/5.66	----------------------------------------
18.55/5.66	
18.55/5.66	(47) DependencyGraphProof (EQUIVALENT)
18.55/5.66	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
18.55/5.66	----------------------------------------
18.55/5.66	
18.55/5.66	(48)
18.55/5.66	Obligation:
18.55/5.66	Q DP problem:
18.55/5.66	The TRS P consists of the following rules:
18.55/5.66	
18.55/5.66	   U6_GGA(C, Coins, NewAmount, ways_out_gga(N1)) -> WAYS_IN_GGA(NewAmount, .(s(C), Coins))
18.55/5.66	   WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U4_GGA(Amount, C, Coins, =_out_ga(Amount))
18.55/5.66	   U4_GGA(s(x3), z1, z2, =_out_ga(s(x3))) -> U5_GGA(s(x3), z1, z2, U3_gag(plus_in_gag(z1, x3)))
18.55/5.66	   U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) -> U6_GGA(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
18.55/5.66	
18.55/5.66	The TRS R consists of the following rules:
18.55/5.66	
18.55/5.66	   ways_in_gga(0, X1) -> ways_out_gga(s(0))
18.55/5.66	   ways_in_gga(X2, []) -> ways_out_gga(0)
18.55/5.66	   ways_in_gga(Amount, .(s(C), Coins)) -> U4_gga(Amount, C, Coins, =_in_ga(Amount))
18.55/5.66	   =_in_ga(X) -> =_out_ga(X)
18.55/5.66	   U4_gga(Amount, C, Coins, =_out_ga(s(X3))) -> U5_gga(Amount, C, Coins, plus_in_gag(s(C), Amount))
18.55/5.66	   plus_in_gag(0, X) -> U2_gag(X, nat_in_g(X))
18.55/5.66	   nat_in_g(0) -> nat_out_g
18.55/5.66	   nat_in_g(s(X)) -> U1_g(nat_in_g(X))
18.55/5.66	   U1_g(nat_out_g) -> nat_out_g
18.55/5.66	   U2_gag(X, nat_out_g) -> plus_out_gag(X)
18.55/5.66	   plus_in_gag(s(X), s(Z)) -> U3_gag(plus_in_gag(X, Z))
18.55/5.66	   U3_gag(plus_out_gag(Y)) -> plus_out_gag(Y)
18.55/5.66	   U5_gga(Amount, C, Coins, plus_out_gag(NewAmount)) -> U6_gga(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
18.55/5.66	   ways_in_gga(Amount, .(s(C), Coins)) -> U9_gga(Amount, C, Coins, =_in_ga(Amount))
18.55/5.66	   U9_gga(Amount, C, Coins, =_out_ga(s(X4))) -> U10_gga(Amount, Coins, plus_in_gag(Amount, s(C)))
18.55/5.66	   U10_gga(Amount, Coins, plus_out_gag(s(X5))) -> U11_gga(ways_in_gga(Amount, Coins))
18.55/5.66	   U11_gga(ways_out_gga(N)) -> ways_out_gga(N)
18.55/5.66	   U6_gga(C, Coins, NewAmount, ways_out_gga(N1)) -> U7_gga(N1, ways_in_gga(NewAmount, .(s(C), Coins)))
18.55/5.66	   U7_gga(N1, ways_out_gga(N2)) -> U8_gga(plus_in_gga(N1, N2))
18.55/5.66	   plus_in_gga(0, X) -> U2_gga(X, nat_in_g(X))
18.55/5.66	   U2_gga(X, nat_out_g) -> plus_out_gga(X)
18.55/5.66	   plus_in_gga(s(X), Y) -> U3_gga(plus_in_gga(X, Y))
18.55/5.66	   U3_gga(plus_out_gga(Z)) -> plus_out_gga(s(Z))
18.55/5.66	   U8_gga(plus_out_gga(N)) -> ways_out_gga(N)
18.55/5.66	
18.55/5.66	The set Q consists of the following terms:
18.55/5.66	
18.55/5.66	   ways_in_gga(x0, x1)
18.55/5.66	   =_in_ga(x0)
18.55/5.66	   U4_gga(x0, x1, x2, x3)
18.55/5.66	   plus_in_gag(x0, x1)
18.55/5.66	   nat_in_g(x0)
18.55/5.66	   U1_g(x0)
18.55/5.66	   U2_gag(x0, x1)
18.55/5.66	   U3_gag(x0)
18.55/5.66	   U5_gga(x0, x1, x2, x3)
18.55/5.66	   U9_gga(x0, x1, x2, x3)
18.55/5.66	   U10_gga(x0, x1, x2)
18.55/5.66	   U11_gga(x0)
18.55/5.66	   U6_gga(x0, x1, x2, x3)
18.55/5.66	   U7_gga(x0, x1)
18.55/5.66	   plus_in_gga(x0, x1)
18.55/5.66	   U2_gga(x0, x1)
18.55/5.66	   U3_gga(x0)
18.55/5.66	   U8_gga(x0)
18.55/5.66	
18.55/5.66	We have to consider all (P,Q,R)-chains.
18.55/5.66	----------------------------------------
18.55/5.66	
18.55/5.66	(49) QDPOrderProof (EQUIVALENT)
18.55/5.66	We use the reduction pair processor [LPAR04,JAR06].
18.55/5.66	
18.55/5.66	
18.55/5.66	The following pairs can be oriented strictly and are deleted.
18.55/5.66	
18.55/5.66	   U4_GGA(s(x3), z1, z2, =_out_ga(s(x3))) -> U5_GGA(s(x3), z1, z2, U3_gag(plus_in_gag(z1, x3)))
18.55/5.66	The remaining pairs can at least be oriented weakly.
18.55/5.66	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
18.55/5.66	
18.55/5.66	POL( U10_gga_3(x_1, ..., x_3) ) = max{0, x_1 - 2}
18.55/5.66	POL( U6_GGA_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2
18.55/5.66	POL( U3_gag_1(x_1) ) = x_1
18.55/5.66	POL( U5_GGA_4(x_1, ..., x_4) ) = max{0, 2x_3 + 2x_4 - 2}
18.55/5.66	POL( U5_gga_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_3 - 2}
18.55/5.66	POL( plus_in_gag_2(x_1, x_2) ) = 2x_2
18.55/5.66	POL( 0 ) = 2
18.55/5.66	POL( U2_gag_2(x_1, x_2) ) = x_1 + x_2
18.55/5.66	POL( nat_in_g_1(x_1) ) = x_1
18.55/5.66	POL( s_1(x_1) ) = 2x_1 + 2
18.55/5.66	POL( plus_out_gag_1(x_1) ) = x_1 + 2
18.55/5.66	POL( ways_in_gga_2(x_1, x_2) ) = max{0, x_1 - 2}
18.55/5.66	POL( ways_out_gga_1(x_1) ) = max{0, x_1 - 2}
18.55/5.66	POL( [] ) = 2
18.55/5.66	POL( ._2(x_1, x_2) ) = max{0, x_2 - 1}
18.55/5.66	POL( U4_gga_4(x_1, ..., x_4) ) = x_2 + 2x_3 + 1
18.55/5.66	POL( =_in_ga_1(x_1) ) = 0
18.55/5.66	POL( U9_gga_4(x_1, ..., x_4) ) = max{0, 2x_2 + x_3 - 2}
18.55/5.66	POL( =_out_ga_1(x_1) ) = 2x_1
18.55/5.66	POL( U6_gga_4(x_1, ..., x_4) ) = max{0, 2x_3 - 2}
18.55/5.66	POL( U11_gga_1(x_1) ) = 2
18.55/5.66	POL( U7_gga_2(x_1, x_2) ) = x_1 + 2
18.55/5.66	POL( U8_gga_1(x_1) ) = max{0, 2x_1 - 2}
18.55/5.66	POL( plus_in_gga_2(x_1, x_2) ) = 0
18.55/5.66	POL( U2_gga_2(x_1, x_2) ) = 2
18.55/5.66	POL( nat_out_g ) = 2
18.55/5.66	POL( U1_g_1(x_1) ) = 2x_1 + 2
18.55/5.66	POL( U3_gga_1(x_1) ) = max{0, x_1 - 2}
18.55/5.66	POL( plus_out_gga_1(x_1) ) = max{0, x_1 - 2}
18.55/5.66	POL( WAYS_IN_GGA_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2
18.55/5.66	POL( U4_GGA_4(x_1, ..., x_4) ) = max{0, 2x_3 + x_4 - 1}
18.55/5.66	
18.55/5.66	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
18.55/5.66	
18.55/5.66	   plus_in_gag(0, X) -> U2_gag(X, nat_in_g(X))
18.55/5.66	   plus_in_gag(s(X), s(Z)) -> U3_gag(plus_in_gag(X, Z))
18.55/5.66	   U3_gag(plus_out_gag(Y)) -> plus_out_gag(Y)
18.55/5.66	   nat_in_g(0) -> nat_out_g
18.55/5.66	   nat_in_g(s(X)) -> U1_g(nat_in_g(X))
18.55/5.66	   U2_gag(X, nat_out_g) -> plus_out_gag(X)
18.55/5.66	   U1_g(nat_out_g) -> nat_out_g
18.55/5.66	
18.55/5.66	
18.55/5.66	----------------------------------------
18.55/5.66	
18.55/5.66	(50)
18.55/5.66	Obligation:
18.55/5.66	Q DP problem:
18.55/5.66	The TRS P consists of the following rules:
18.55/5.66	
18.55/5.66	   U6_GGA(C, Coins, NewAmount, ways_out_gga(N1)) -> WAYS_IN_GGA(NewAmount, .(s(C), Coins))
18.55/5.66	   WAYS_IN_GGA(Amount, .(s(C), Coins)) -> U4_GGA(Amount, C, Coins, =_out_ga(Amount))
18.55/5.66	   U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) -> U6_GGA(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
18.55/5.66	
18.55/5.66	The TRS R consists of the following rules:
18.55/5.66	
18.55/5.66	   ways_in_gga(0, X1) -> ways_out_gga(s(0))
18.55/5.66	   ways_in_gga(X2, []) -> ways_out_gga(0)
18.55/5.66	   ways_in_gga(Amount, .(s(C), Coins)) -> U4_gga(Amount, C, Coins, =_in_ga(Amount))
18.55/5.66	   =_in_ga(X) -> =_out_ga(X)
18.55/5.66	   U4_gga(Amount, C, Coins, =_out_ga(s(X3))) -> U5_gga(Amount, C, Coins, plus_in_gag(s(C), Amount))
18.55/5.66	   plus_in_gag(0, X) -> U2_gag(X, nat_in_g(X))
18.55/5.66	   nat_in_g(0) -> nat_out_g
18.55/5.66	   nat_in_g(s(X)) -> U1_g(nat_in_g(X))
18.55/5.66	   U1_g(nat_out_g) -> nat_out_g
18.55/5.66	   U2_gag(X, nat_out_g) -> plus_out_gag(X)
18.55/5.66	   plus_in_gag(s(X), s(Z)) -> U3_gag(plus_in_gag(X, Z))
18.55/5.66	   U3_gag(plus_out_gag(Y)) -> plus_out_gag(Y)
18.55/5.66	   U5_gga(Amount, C, Coins, plus_out_gag(NewAmount)) -> U6_gga(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
18.55/5.66	   ways_in_gga(Amount, .(s(C), Coins)) -> U9_gga(Amount, C, Coins, =_in_ga(Amount))
18.55/5.66	   U9_gga(Amount, C, Coins, =_out_ga(s(X4))) -> U10_gga(Amount, Coins, plus_in_gag(Amount, s(C)))
18.55/5.66	   U10_gga(Amount, Coins, plus_out_gag(s(X5))) -> U11_gga(ways_in_gga(Amount, Coins))
18.55/5.66	   U11_gga(ways_out_gga(N)) -> ways_out_gga(N)
18.55/5.66	   U6_gga(C, Coins, NewAmount, ways_out_gga(N1)) -> U7_gga(N1, ways_in_gga(NewAmount, .(s(C), Coins)))
18.55/5.66	   U7_gga(N1, ways_out_gga(N2)) -> U8_gga(plus_in_gga(N1, N2))
18.55/5.66	   plus_in_gga(0, X) -> U2_gga(X, nat_in_g(X))
18.55/5.66	   U2_gga(X, nat_out_g) -> plus_out_gga(X)
18.55/5.66	   plus_in_gga(s(X), Y) -> U3_gga(plus_in_gga(X, Y))
18.55/5.66	   U3_gga(plus_out_gga(Z)) -> plus_out_gga(s(Z))
18.55/5.66	   U8_gga(plus_out_gga(N)) -> ways_out_gga(N)
18.55/5.66	
18.55/5.66	The set Q consists of the following terms:
18.55/5.66	
18.55/5.66	   ways_in_gga(x0, x1)
18.55/5.66	   =_in_ga(x0)
18.55/5.66	   U4_gga(x0, x1, x2, x3)
18.55/5.66	   plus_in_gag(x0, x1)
18.55/5.66	   nat_in_g(x0)
18.55/5.66	   U1_g(x0)
18.55/5.66	   U2_gag(x0, x1)
18.55/5.66	   U3_gag(x0)
18.55/5.66	   U5_gga(x0, x1, x2, x3)
18.55/5.66	   U9_gga(x0, x1, x2, x3)
18.55/5.66	   U10_gga(x0, x1, x2)
18.55/5.66	   U11_gga(x0)
18.55/5.66	   U6_gga(x0, x1, x2, x3)
18.55/5.66	   U7_gga(x0, x1)
18.55/5.66	   plus_in_gga(x0, x1)
18.55/5.66	   U2_gga(x0, x1)
18.55/5.66	   U3_gga(x0)
18.55/5.66	   U8_gga(x0)
18.55/5.66	
18.55/5.66	We have to consider all (P,Q,R)-chains.
18.55/5.66	----------------------------------------
18.55/5.66	
18.55/5.66	(51) DependencyGraphProof (EQUIVALENT)
18.55/5.66	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
18.55/5.66	----------------------------------------
18.55/5.66	
18.55/5.66	(52)
18.55/5.66	TRUE
18.75/5.71	EOF