Left Termination proof of /tmp/tmpNCYBqh/ways.pl

Left Termination of the query pattern ways_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PredefinedPredicateTransformerProof (⇐)

↳2 Prolog

  ↳3 PrologToPiTRSProof (⇐)

    ↳4 PiTRS

    ↳5 DependencyPairsProof (⇔)

    ↳6 PiDP

    ↳7 DependencyGraphProof (⇔)

    ↳8 AND

      ↳9 PiDP

        ↳10 UsableRulesProof (⇔)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇔)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔)

        ↳15 TRUE

      ↳16 PiDP

        ↳17 UsableRulesProof (⇔)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇐)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔)

        ↳22 TRUE

      ↳23 PiDP

        ↳24 UsableRulesProof (⇔)

        ↳25 PiDP

        ↳26 PiDPToQDPProof (⇐)

        ↳27 QDP

        ↳28 QDPSizeChangeProof (⇔)

        ↳29 TRUE

      ↳30 PiDP

        ↳31 PiDPToQDPProof (⇐)

        ↳32 QDP

        ↳33 QDPOrderProof (⇔)

        ↳34 QDP

        ↳35 DependencyGraphProof (⇔)

        ↳36 QDP

        ↳37 QDPOrderProof (⇔)

        ↳38 QDP

  ↳39 PrologToPiTRSProof (⇐)

    ↳40 PiTRS

    ↳41 DependencyPairsProof (⇔)

    ↳42 PiDP

    ↳43 DependencyGraphProof (⇔)

    ↳44 AND

      ↳45 PiDP

        ↳46 UsableRulesProof (⇔)

        ↳47 PiDP

        ↳48 PiDPToQDPProof (⇔)

        ↳49 QDP

        ↳50 QDPSizeChangeProof (⇔)

        ↳51 TRUE

      ↳52 PiDP

        ↳53 UsableRulesProof (⇔)

        ↳54 PiDP

        ↳55 PiDPToQDPProof (⇐)

        ↳56 QDP

        ↳57 QDPSizeChangeProof (⇔)

        ↳58 TRUE

      ↳59 PiDP

        ↳60 UsableRulesProof (⇔)

        ↳61 PiDP

        ↳62 PiDPToQDPProof (⇐)

        ↳63 QDP

        ↳64 QDPSizeChangeProof (⇔)

        ↳65 TRUE

      ↳66 PiDP

        ↳67 PiDPToQDPProof (⇐)

        ↳68 QDP

        ↳69 QDPOrderProof (⇔)

        ↳70 QDP

        ↳71 DependencyGraphProof (⇔)

        ↳72 QDP

        ↳73 QDPOrderProof (⇔)

        ↳74 QDP

        ↳75 QDPOrderProof (⇔)

        ↳76 QDP

        ↳77 DependencyGraphProof (⇔)

        ↳78 TRUE

(0) Obligation:

Clauses:

nat(0).
nat(s(X)) :- nat(X).
plus(0, X, X) :- nat(X).
plus(s(X), Y, s(Z)) :- plus(X, Y, Z).
ways(0, X1, s(0)).
ways(X2, [], 0).
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))))).
ways(Amount, .(s(C), Coins), N) :- ','(=(Amount, s(X4)), ','(plus(Amount, s(X5), s(C)), ways(Amount, Coins, N))).

Queries:

ways(g,g,a).

(1) PredefinedPredicateTransformerProof (SOUND transformation)

Added definitions of predefined predicates [PROLOG].

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
ways_in: (b,b,f)
plus_in: (b,f,b) (b,b,f)
nat_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

ways_in_gga(0, X1, s(0)) → ways_out_gga(0, X1, s(0))
ways_in_gga(X2, [], 0) → ways_out_gga(X2, [], 0)
ways_in_gga(Amount, .(s(C), Coins), N) → U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
=_in_ga(X, X) → =_out_ga(X, X)
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))
plus_in_gag(0, X, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g(0)
nat_in_g(s(X)) → U1_g(X, nat_in_g(X))
U1_g(X, nat_out_g(X)) → nat_out_g(s(X))
U2_gag(X, nat_out_g(X)) → plus_out_gag(0, X, X)
plus_in_gag(s(X), Y, s(Z)) → U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) → plus_out_gag(s(X), Y, s(Z))
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))
ways_in_gga(Amount, .(s(C), Coins), N) → U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
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))
U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) → ways_out_gga(Amount, .(s(C), Coins), N)
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))
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))
plus_in_gga(0, X, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g(X)) → plus_out_gga(0, X, X)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) → ways_out_gga(Amount, .(s(C), Coins), N)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

ways_in_gga(0, X1, s(0)) → ways_out_gga(0, X1, s(0))
ways_in_gga(X2, [], 0) → ways_out_gga(X2, [], 0)
ways_in_gga(Amount, .(s(C), Coins), N) → U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
=_in_ga(X, X) → =_out_ga(X, X)
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))
plus_in_gag(0, X, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g(0)
nat_in_g(s(X)) → U1_g(X, nat_in_g(X))
U1_g(X, nat_out_g(X)) → nat_out_g(s(X))
U2_gag(X, nat_out_g(X)) → plus_out_gag(0, X, X)
plus_in_gag(s(X), Y, s(Z)) → U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) → plus_out_gag(s(X), Y, s(Z))
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))
ways_in_gga(Amount, .(s(C), Coins), N) → U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
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))
U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) → ways_out_gga(Amount, .(s(C), Coins), N)
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))
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))
plus_in_gga(0, X, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g(X)) → plus_out_gga(0, X, X)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) → ways_out_gga(Amount, .(s(C), Coins), N)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

WAYS_IN_GGA(Amount, .(s(C), Coins), N) → U4_GGA(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
WAYS_IN_GGA(Amount, .(s(C), Coins), N) → =_IN_GA(Amount, s(X3))
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))
U4_GGA(Amount, C, Coins, N, =_out_ga(Amount, s(X3))) → PLUS_IN_GAG(s(C), NewAmount, Amount)
PLUS_IN_GAG(0, X, X) → U2_GAG(X, nat_in_g(X))
PLUS_IN_GAG(0, X, X) → NAT_IN_G(X)
NAT_IN_G(s(X)) → U1_G(X, nat_in_g(X))
NAT_IN_G(s(X)) → NAT_IN_G(X)
PLUS_IN_GAG(s(X), Y, s(Z)) → U3_GAG(X, Y, Z, plus_in_gag(X, Y, Z))
PLUS_IN_GAG(s(X), Y, s(Z)) → PLUS_IN_GAG(X, Y, Z)
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))
U5_GGA(Amount, C, Coins, N, X3, plus_out_gag(s(C), NewAmount, Amount)) → WAYS_IN_GGA(Amount, Coins, N1)
WAYS_IN_GGA(Amount, .(s(C), Coins), N) → U9_GGA(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
U9_GGA(Amount, C, Coins, N, =_out_ga(Amount, s(X4))) → PLUS_IN_GAG(Amount, s(X5), s(C))
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))
U10_GGA(Amount, C, Coins, N, X4, plus_out_gag(Amount, s(X5), s(C))) → WAYS_IN_GGA(Amount, Coins, N)
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))
U6_GGA(Amount, C, Coins, N, X3, NewAmount, ways_out_gga(Amount, Coins, N1)) → WAYS_IN_GGA(NewAmount, .(s(C), Coins), N2)
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))
U7_GGA(Amount, C, Coins, N, X3, NewAmount, N1, ways_out_gga(NewAmount, .(s(C), Coins), N2)) → PLUS_IN_GGA(N1, N2, N)
PLUS_IN_GGA(0, X, X) → U2_GGA(X, nat_in_g(X))
PLUS_IN_GGA(0, X, X) → NAT_IN_G(X)
PLUS_IN_GGA(s(X), Y, s(Z)) → U3_GGA(X, Y, Z, plus_in_gga(X, Y, Z))
PLUS_IN_GGA(s(X), Y, s(Z)) → PLUS_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

ways_in_gga(0, X1, s(0)) → ways_out_gga(0, X1, s(0))
ways_in_gga(X2, [], 0) → ways_out_gga(X2, [], 0)
ways_in_gga(Amount, .(s(C), Coins), N) → U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
=_in_ga(X, X) → =_out_ga(X, X)
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))
plus_in_gag(0, X, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g(0)
nat_in_g(s(X)) → U1_g(X, nat_in_g(X))
U1_g(X, nat_out_g(X)) → nat_out_g(s(X))
U2_gag(X, nat_out_g(X)) → plus_out_gag(0, X, X)
plus_in_gag(s(X), Y, s(Z)) → U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) → plus_out_gag(s(X), Y, s(Z))
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))
ways_in_gga(Amount, .(s(C), Coins), N) → U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
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))
U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) → ways_out_gga(Amount, .(s(C), Coins), N)
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))
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))
plus_in_gga(0, X, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g(X)) → plus_out_gga(0, X, X)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) → ways_out_gga(Amount, .(s(C), Coins), N)

The argument filtering Pi contains the following mapping:
ways_in_gga(x1, x2, x3) = ways_in_gga(x1, x2)
0 = 0
ways_out_gga(x1, x2, x3) = ways_out_gga(x1, x2, x3)
[] = []
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x3, x5)
=_in_ga(x1, x2) = =_in_ga(x1)
=_out_ga(x1, x2) = =_out_ga(x1, x2)
U5_gga(x1, x2, x3, x4, x5, x6) = U5_gga(x1, x2, x3, x6)
plus_in_gag(x1, x2, x3) = plus_in_gag(x1, x3)
U2_gag(x1, x2) = U2_gag(x1, x2)
nat_in_g(x1) = nat_in_g(x1)
nat_out_g(x1) = nat_out_g(x1)
U1_g(x1, x2) = U1_g(x1, x2)
plus_out_gag(x1, x2, x3) = plus_out_gag(x1, x2, x3)
U3_gag(x1, x2, x3, x4) = U3_gag(x1, x3, x4)
U6_gga(x1, x2, x3, x4, x5, x6, x7) = U6_gga(x1, x2, x3, x6, x7)
U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5)
U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x6)
U11_gga(x1, x2, x3, x4, x5) = U11_gga(x1, x2, x3, x5)
U7_gga(x1, x2, x3, x4, x5, x6, x7, x8) = U7_gga(x1, x2, x3, x7, x8)
U8_gga(x1, x2, x3, x4, x5) = U8_gga(x1, x2, x3, x5)
plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2)
U2_gga(x1, x2) = U2_gga(x1, x2)
plus_out_gga(x1, x2, x3) = plus_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
WAYS_IN_GGA(x1, x2, x3) = WAYS_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x2, x3, x5)
=_IN_GA(x1, x2) = =_IN_GA(x1)
U5_GGA(x1, x2, x3, x4, x5, x6) = U5_GGA(x1, x2, x3, x6)
PLUS_IN_GAG(x1, x2, x3) = PLUS_IN_GAG(x1, x3)
U2_GAG(x1, x2) = U2_GAG(x1, x2)
NAT_IN_G(x1) = NAT_IN_G(x1)
U1_G(x1, x2) = U1_G(x1, x2)
U3_GAG(x1, x2, x3, x4) = U3_GAG(x1, x3, x4)
U6_GGA(x1, x2, x3, x4, x5, x6, x7) = U6_GGA(x1, x2, x3, x6, x7)
U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x2, x3, x5)
U10_GGA(x1, x2, x3, x4, x5, x6) = U10_GGA(x1, x2, x3, x6)
U11_GGA(x1, x2, x3, x4, x5) = U11_GGA(x1, x2, x3, x5)
U7_GGA(x1, x2, x3, x4, x5, x6, x7, x8) = U7_GGA(x1, x2, x3, x7, x8)
U8_GGA(x1, x2, x3, x4, x5) = U8_GGA(x1, x2, x3, x5)
PLUS_IN_GGA(x1, x2, x3) = PLUS_IN_GGA(x1, x2)
U2_GGA(x1, x2) = U2_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)

We have to consider all (P,R,Pi)-chains

(6) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

WAYS_IN_GGA(Amount, .(s(C), Coins), N) → U4_GGA(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
WAYS_IN_GGA(Amount, .(s(C), Coins), N) → =_IN_GA(Amount, s(X3))
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))
U4_GGA(Amount, C, Coins, N, =_out_ga(Amount, s(X3))) → PLUS_IN_GAG(s(C), NewAmount, Amount)
PLUS_IN_GAG(0, X, X) → U2_GAG(X, nat_in_g(X))
PLUS_IN_GAG(0, X, X) → NAT_IN_G(X)
NAT_IN_G(s(X)) → U1_G(X, nat_in_g(X))
NAT_IN_G(s(X)) → NAT_IN_G(X)
PLUS_IN_GAG(s(X), Y, s(Z)) → U3_GAG(X, Y, Z, plus_in_gag(X, Y, Z))
PLUS_IN_GAG(s(X), Y, s(Z)) → PLUS_IN_GAG(X, Y, Z)
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))
U5_GGA(Amount, C, Coins, N, X3, plus_out_gag(s(C), NewAmount, Amount)) → WAYS_IN_GGA(Amount, Coins, N1)
WAYS_IN_GGA(Amount, .(s(C), Coins), N) → U9_GGA(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
U9_GGA(Amount, C, Coins, N, =_out_ga(Amount, s(X4))) → PLUS_IN_GAG(Amount, s(X5), s(C))
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))
U10_GGA(Amount, C, Coins, N, X4, plus_out_gag(Amount, s(X5), s(C))) → WAYS_IN_GGA(Amount, Coins, N)
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))
U6_GGA(Amount, C, Coins, N, X3, NewAmount, ways_out_gga(Amount, Coins, N1)) → WAYS_IN_GGA(NewAmount, .(s(C), Coins), N2)
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))
U7_GGA(Amount, C, Coins, N, X3, NewAmount, N1, ways_out_gga(NewAmount, .(s(C), Coins), N2)) → PLUS_IN_GGA(N1, N2, N)
PLUS_IN_GGA(0, X, X) → U2_GGA(X, nat_in_g(X))
PLUS_IN_GGA(0, X, X) → NAT_IN_G(X)
PLUS_IN_GGA(s(X), Y, s(Z)) → U3_GGA(X, Y, Z, plus_in_gga(X, Y, Z))
PLUS_IN_GGA(s(X), Y, s(Z)) → PLUS_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

ways_in_gga(0, X1, s(0)) → ways_out_gga(0, X1, s(0))
ways_in_gga(X2, [], 0) → ways_out_gga(X2, [], 0)
ways_in_gga(Amount, .(s(C), Coins), N) → U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
=_in_ga(X, X) → =_out_ga(X, X)
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))
plus_in_gag(0, X, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g(0)
nat_in_g(s(X)) → U1_g(X, nat_in_g(X))
U1_g(X, nat_out_g(X)) → nat_out_g(s(X))
U2_gag(X, nat_out_g(X)) → plus_out_gag(0, X, X)
plus_in_gag(s(X), Y, s(Z)) → U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) → plus_out_gag(s(X), Y, s(Z))
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))
ways_in_gga(Amount, .(s(C), Coins), N) → U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
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))
U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) → ways_out_gga(Amount, .(s(C), Coins), N)
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))
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))
plus_in_gga(0, X, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g(X)) → plus_out_gga(0, X, X)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) → ways_out_gga(Amount, .(s(C), Coins), N)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 14 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

NAT_IN_G(s(X)) → NAT_IN_G(X)

The TRS R consists of the following rules:

ways_in_gga(0, X1, s(0)) → ways_out_gga(0, X1, s(0))
ways_in_gga(X2, [], 0) → ways_out_gga(X2, [], 0)
ways_in_gga(Amount, .(s(C), Coins), N) → U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
=_in_ga(X, X) → =_out_ga(X, X)
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))
plus_in_gag(0, X, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g(0)
nat_in_g(s(X)) → U1_g(X, nat_in_g(X))
U1_g(X, nat_out_g(X)) → nat_out_g(s(X))
U2_gag(X, nat_out_g(X)) → plus_out_gag(0, X, X)
plus_in_gag(s(X), Y, s(Z)) → U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) → plus_out_gag(s(X), Y, s(Z))
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))
ways_in_gga(Amount, .(s(C), Coins), N) → U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
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))
U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) → ways_out_gga(Amount, .(s(C), Coins), N)
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))
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))
plus_in_gga(0, X, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g(X)) → plus_out_gga(0, X, X)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) → ways_out_gga(Amount, .(s(C), Coins), N)

(10) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(11) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

NAT_IN_G(s(X)) → NAT_IN_G(X)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains

(12) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(13) Obligation:

Q DP problem:
The TRS P consists of the following rules:

NAT_IN_G(s(X)) → NAT_IN_G(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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.

From the DPs we obtained the following set of size-change graphs:

NAT_IN_G(s(X)) → NAT_IN_G(X)
The graph contains the following edges 1 > 1

(15) TRUE

(16) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PLUS_IN_GGA(s(X), Y, s(Z)) → PLUS_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

ways_in_gga(0, X1, s(0)) → ways_out_gga(0, X1, s(0))
ways_in_gga(X2, [], 0) → ways_out_gga(X2, [], 0)
ways_in_gga(Amount, .(s(C), Coins), N) → U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
=_in_ga(X, X) → =_out_ga(X, X)
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))
plus_in_gag(0, X, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g(0)
nat_in_g(s(X)) → U1_g(X, nat_in_g(X))
U1_g(X, nat_out_g(X)) → nat_out_g(s(X))
U2_gag(X, nat_out_g(X)) → plus_out_gag(0, X, X)
plus_in_gag(s(X), Y, s(Z)) → U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) → plus_out_gag(s(X), Y, s(Z))
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))
ways_in_gga(Amount, .(s(C), Coins), N) → U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
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))
U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) → ways_out_gga(Amount, .(s(C), Coins), N)
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))
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))
plus_in_gga(0, X, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g(X)) → plus_out_gga(0, X, X)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) → ways_out_gga(Amount, .(s(C), Coins), N)

(17) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(18) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PLUS_IN_GGA(s(X), Y, s(Z)) → PLUS_IN_GGA(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
PLUS_IN_GGA(x1, x2, x3) = PLUS_IN_GGA(x1, x2)

We have to consider all (P,R,Pi)-chains

(19) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PLUS_IN_GGA(s(X), Y) → PLUS_IN_GGA(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(21) QDPSizeChangeProof (EQUIVALENT transformation)

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.

From the DPs we obtained the following set of size-change graphs:

PLUS_IN_GGA(s(X), Y) → PLUS_IN_GGA(X, Y)
The graph contains the following edges 1 > 1, 2 >= 2

(22) TRUE

(23) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PLUS_IN_GAG(s(X), Y, s(Z)) → PLUS_IN_GAG(X, Y, Z)

The TRS R consists of the following rules:

ways_in_gga(0, X1, s(0)) → ways_out_gga(0, X1, s(0))
ways_in_gga(X2, [], 0) → ways_out_gga(X2, [], 0)
ways_in_gga(Amount, .(s(C), Coins), N) → U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
=_in_ga(X, X) → =_out_ga(X, X)
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))
plus_in_gag(0, X, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g(0)
nat_in_g(s(X)) → U1_g(X, nat_in_g(X))
U1_g(X, nat_out_g(X)) → nat_out_g(s(X))
U2_gag(X, nat_out_g(X)) → plus_out_gag(0, X, X)
plus_in_gag(s(X), Y, s(Z)) → U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) → plus_out_gag(s(X), Y, s(Z))
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))
ways_in_gga(Amount, .(s(C), Coins), N) → U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
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))
U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) → ways_out_gga(Amount, .(s(C), Coins), N)
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))
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))
plus_in_gga(0, X, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g(X)) → plus_out_gga(0, X, X)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) → ways_out_gga(Amount, .(s(C), Coins), N)

(24) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(25) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PLUS_IN_GAG(s(X), Y, s(Z)) → PLUS_IN_GAG(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
PLUS_IN_GAG(x1, x2, x3) = PLUS_IN_GAG(x1, x3)

We have to consider all (P,R,Pi)-chains

(26) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(27) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PLUS_IN_GAG(s(X), s(Z)) → PLUS_IN_GAG(X, Z)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(28) QDPSizeChangeProof (EQUIVALENT transformation)

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.

From the DPs we obtained the following set of size-change graphs:

PLUS_IN_GAG(s(X), s(Z)) → PLUS_IN_GAG(X, Z)
The graph contains the following edges 1 > 1, 2 > 2

(29) TRUE

(30) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

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))
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))
U6_GGA(Amount, C, Coins, N, X3, NewAmount, ways_out_gga(Amount, Coins, N1)) → WAYS_IN_GGA(NewAmount, .(s(C), Coins), N2)
WAYS_IN_GGA(Amount, .(s(C), Coins), N) → U4_GGA(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
WAYS_IN_GGA(Amount, .(s(C), Coins), N) → U9_GGA(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
U10_GGA(Amount, C, Coins, N, X4, plus_out_gag(Amount, s(X5), s(C))) → WAYS_IN_GGA(Amount, Coins, N)
U5_GGA(Amount, C, Coins, N, X3, plus_out_gag(s(C), NewAmount, Amount)) → WAYS_IN_GGA(Amount, Coins, N1)

The TRS R consists of the following rules:

ways_in_gga(0, X1, s(0)) → ways_out_gga(0, X1, s(0))
ways_in_gga(X2, [], 0) → ways_out_gga(X2, [], 0)
ways_in_gga(Amount, .(s(C), Coins), N) → U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
=_in_ga(X, X) → =_out_ga(X, X)
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))
plus_in_gag(0, X, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g(0)
nat_in_g(s(X)) → U1_g(X, nat_in_g(X))
U1_g(X, nat_out_g(X)) → nat_out_g(s(X))
U2_gag(X, nat_out_g(X)) → plus_out_gag(0, X, X)
plus_in_gag(s(X), Y, s(Z)) → U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) → plus_out_gag(s(X), Y, s(Z))
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))
ways_in_gga(Amount, .(s(C), Coins), N) → U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
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))
U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) → ways_out_gga(Amount, .(s(C), Coins), N)
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))
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))
plus_in_gga(0, X, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g(X)) → plus_out_gga(0, X, X)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) → ways_out_gga(Amount, .(s(C), Coins), N)

The argument filtering Pi contains the following mapping:
ways_in_gga(x1, x2, x3) = ways_in_gga(x1, x2)
0 = 0
ways_out_gga(x1, x2, x3) = ways_out_gga(x1, x2, x3)
[] = []
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x3, x5)
=_in_ga(x1, x2) = =_in_ga(x1)
=_out_ga(x1, x2) = =_out_ga(x1, x2)
U5_gga(x1, x2, x3, x4, x5, x6) = U5_gga(x1, x2, x3, x6)
plus_in_gag(x1, x2, x3) = plus_in_gag(x1, x3)
U2_gag(x1, x2) = U2_gag(x1, x2)
nat_in_g(x1) = nat_in_g(x1)
nat_out_g(x1) = nat_out_g(x1)
U1_g(x1, x2) = U1_g(x1, x2)
plus_out_gag(x1, x2, x3) = plus_out_gag(x1, x2, x3)
U3_gag(x1, x2, x3, x4) = U3_gag(x1, x3, x4)
U6_gga(x1, x2, x3, x4, x5, x6, x7) = U6_gga(x1, x2, x3, x6, x7)
U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5)
U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x6)
U11_gga(x1, x2, x3, x4, x5) = U11_gga(x1, x2, x3, x5)
U7_gga(x1, x2, x3, x4, x5, x6, x7, x8) = U7_gga(x1, x2, x3, x7, x8)
U8_gga(x1, x2, x3, x4, x5) = U8_gga(x1, x2, x3, x5)
plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2)
U2_gga(x1, x2) = U2_gga(x1, x2)
plus_out_gga(x1, x2, x3) = plus_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
WAYS_IN_GGA(x1, x2, x3) = WAYS_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x2, x3, x5)
U5_GGA(x1, x2, x3, x4, x5, x6) = U5_GGA(x1, x2, x3, x6)
U6_GGA(x1, x2, x3, x4, x5, x6, x7) = U6_GGA(x1, x2, x3, x6, x7)
U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x2, x3, x5)
U10_GGA(x1, x2, x3, x4, x5, x6) = U10_GGA(x1, x2, x3, x6)

We have to consider all (P,R,Pi)-chains

(31) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(32) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U4_GGA(Amount, C, Coins, =_out_ga(Amount, s(X3))) → U5_GGA(Amount, C, Coins, plus_in_gag(s(C), Amount))
U5_GGA(Amount, C, Coins, plus_out_gag(s(C), NewAmount, Amount)) → U6_GGA(Amount, C, Coins, NewAmount, ways_in_gga(Amount, Coins))
U6_GGA(Amount, C, Coins, NewAmount, ways_out_gga(Amount, Coins, N1)) → WAYS_IN_GGA(NewAmount, .(s(C), Coins))
WAYS_IN_GGA(Amount, .(s(C), Coins)) → U4_GGA(Amount, C, Coins, =_in_ga(Amount))
WAYS_IN_GGA(Amount, .(s(C), Coins)) → U9_GGA(Amount, C, Coins, =_in_ga(Amount))
U9_GGA(Amount, C, Coins, =_out_ga(Amount, s(X4))) → U10_GGA(Amount, C, Coins, plus_in_gag(Amount, s(C)))
U10_GGA(Amount, C, Coins, plus_out_gag(Amount, s(X5), s(C))) → WAYS_IN_GGA(Amount, Coins)
U5_GGA(Amount, C, Coins, plus_out_gag(s(C), NewAmount, Amount)) → WAYS_IN_GGA(Amount, Coins)

The TRS R consists of the following rules:

ways_in_gga(0, X1) → ways_out_gga(0, X1, s(0))
ways_in_gga(X2, []) → ways_out_gga(X2, [], 0)
ways_in_gga(Amount, .(s(C), Coins)) → U4_gga(Amount, C, Coins, =_in_ga(Amount))
=_in_ga(X) → =_out_ga(X, X)
U4_gga(Amount, C, Coins, =_out_ga(Amount, s(X3))) → U5_gga(Amount, C, Coins, plus_in_gag(s(C), Amount))
plus_in_gag(0, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g(0)
nat_in_g(s(X)) → U1_g(X, nat_in_g(X))
U1_g(X, nat_out_g(X)) → nat_out_g(s(X))
U2_gag(X, nat_out_g(X)) → plus_out_gag(0, X, X)
plus_in_gag(s(X), s(Z)) → U3_gag(X, Z, plus_in_gag(X, Z))
U3_gag(X, Z, plus_out_gag(X, Y, Z)) → plus_out_gag(s(X), Y, s(Z))
U5_gga(Amount, C, Coins, plus_out_gag(s(C), NewAmount, Amount)) → U6_gga(Amount, C, Coins, NewAmount, ways_in_gga(Amount, Coins))
ways_in_gga(Amount, .(s(C), Coins)) → U9_gga(Amount, C, Coins, =_in_ga(Amount))
U9_gga(Amount, C, Coins, =_out_ga(Amount, s(X4))) → U10_gga(Amount, C, Coins, plus_in_gag(Amount, s(C)))
U10_gga(Amount, C, Coins, plus_out_gag(Amount, s(X5), s(C))) → U11_gga(Amount, C, Coins, ways_in_gga(Amount, Coins))
U11_gga(Amount, C, Coins, ways_out_gga(Amount, Coins, N)) → ways_out_gga(Amount, .(s(C), Coins), N)
U6_gga(Amount, C, Coins, NewAmount, ways_out_gga(Amount, Coins, N1)) → U7_gga(Amount, C, Coins, N1, ways_in_gga(NewAmount, .(s(C), Coins)))
U7_gga(Amount, C, Coins, N1, ways_out_gga(NewAmount, .(s(C), Coins), N2)) → U8_gga(Amount, C, Coins, plus_in_gga(N1, N2))
plus_in_gga(0, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g(X)) → plus_out_gga(0, X, X)
plus_in_gga(s(X), Y) → U3_gga(X, Y, plus_in_gga(X, Y))
U3_gga(X, Y, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_gga(Amount, C, Coins, plus_out_gga(N1, N2, N)) → ways_out_gga(Amount, .(s(C), Coins), N)

The set Q consists of the following terms:

ways_in_gga(x0, x1)
=_in_ga(x0)
U4_gga(x0, x1, x2, x3)
plus_in_gag(x0, x1)
nat_in_g(x0)
U1_g(x0, x1)
U2_gag(x0, x1)
U3_gag(x0, x1, x2)
U5_gga(x0, x1, x2, x3)
U9_gga(x0, x1, x2, x3)
U10_gga(x0, x1, x2, x3)
U11_gga(x0, x1, x2, x3)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4)
plus_in_gga(x0, x1)
U2_gga(x0, x1)
U3_gga(x0, x1, x2)
U8_gga(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U10_GGA(Amount, C, Coins, plus_out_gag(Amount, s(X5), s(C))) → WAYS_IN_GGA(Amount, Coins)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = x₁ + x₂
POL(0) = 1
POL(=_in_ga(x₁)) = 0
POL(=_out_ga(x₁, x₂)) = 0
POL(U10_GGA(x₁, x₂, x₃, x₄)) = x₃ + x₄
POL(U10_gga(x₁, x₂, x₃, x₄)) = 0
POL(U11_gga(x₁, x₂, x₃, x₄)) = 0
POL(U1_g(x₁, x₂)) = x₂
POL(U2_gag(x₁, x₂)) = x₂
POL(U2_gga(x₁, x₂)) = 0
POL(U3_gag(x₁, x₂, x₃)) = x₃
POL(U3_gga(x₁, x₂, x₃)) = 0
POL(U4_GGA(x₁, x₂, x₃, x₄)) = x₂ + x₃
POL(U4_gga(x₁, x₂, x₃, x₄)) = 0
POL(U5_GGA(x₁, x₂, x₃, x₄)) = x₂ + x₃
POL(U5_gga(x₁, x₂, x₃, x₄)) = 0
POL(U6_GGA(x₁, x₂, x₃, x₄, x₅)) = x₂ + x₃
POL(U6_gga(x₁, x₂, x₃, x₄, x₅)) = 0
POL(U7_gga(x₁, x₂, x₃, x₄, x₅)) = 0
POL(U8_gga(x₁, x₂, x₃, x₄)) = 0
POL(U9_GGA(x₁, x₂, x₃, x₄)) = x₂ + x₃
POL(U9_gga(x₁, x₂, x₃, x₄)) = 0
POL(WAYS_IN_GGA(x₁, x₂)) = x₂
POL([]) = 0
POL(nat_in_g(x₁)) = x₁
POL(nat_out_g(x₁)) = 1
POL(plus_in_gag(x₁, x₂)) = x₂
POL(plus_in_gga(x₁, x₂)) = 0
POL(plus_out_gag(x₁, x₂, x₃)) = 1
POL(plus_out_gga(x₁, x₂, x₃)) = 0
POL(s(x₁)) = x₁
POL(ways_in_gga(x₁, x₂)) = 0
POL(ways_out_gga(x₁, x₂, x₃)) = 0

The following usable rules [FROCOS05] were oriented:

plus_in_gag(s(X), s(Z)) → U3_gag(X, Z, plus_in_gag(X, Z))
plus_in_gag(0, X) → U2_gag(X, nat_in_g(X))
U3_gag(X, Z, plus_out_gag(X, Y, Z)) → plus_out_gag(s(X), Y, s(Z))
nat_in_g(0) → nat_out_g(0)
nat_in_g(s(X)) → U1_g(X, nat_in_g(X))
U2_gag(X, nat_out_g(X)) → plus_out_gag(0, X, X)
U1_g(X, nat_out_g(X)) → nat_out_g(s(X))

(34) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U4_GGA(Amount, C, Coins, =_out_ga(Amount, s(X3))) → U5_GGA(Amount, C, Coins, plus_in_gag(s(C), Amount))
U5_GGA(Amount, C, Coins, plus_out_gag(s(C), NewAmount, Amount)) → U6_GGA(Amount, C, Coins, NewAmount, ways_in_gga(Amount, Coins))
U6_GGA(Amount, C, Coins, NewAmount, ways_out_gga(Amount, Coins, N1)) → WAYS_IN_GGA(NewAmount, .(s(C), Coins))
WAYS_IN_GGA(Amount, .(s(C), Coins)) → U4_GGA(Amount, C, Coins, =_in_ga(Amount))
WAYS_IN_GGA(Amount, .(s(C), Coins)) → U9_GGA(Amount, C, Coins, =_in_ga(Amount))
U9_GGA(Amount, C, Coins, =_out_ga(Amount, s(X4))) → U10_GGA(Amount, C, Coins, plus_in_gag(Amount, s(C)))
U5_GGA(Amount, C, Coins, plus_out_gag(s(C), NewAmount, Amount)) → WAYS_IN_GGA(Amount, Coins)

The TRS R consists of the following rules:

ways_in_gga(0, X1) → ways_out_gga(0, X1, s(0))
ways_in_gga(X2, []) → ways_out_gga(X2, [], 0)
ways_in_gga(Amount, .(s(C), Coins)) → U4_gga(Amount, C, Coins, =_in_ga(Amount))
=_in_ga(X) → =_out_ga(X, X)
U4_gga(Amount, C, Coins, =_out_ga(Amount, s(X3))) → U5_gga(Amount, C, Coins, plus_in_gag(s(C), Amount))
plus_in_gag(0, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g(0)
nat_in_g(s(X)) → U1_g(X, nat_in_g(X))
U1_g(X, nat_out_g(X)) → nat_out_g(s(X))
U2_gag(X, nat_out_g(X)) → plus_out_gag(0, X, X)
plus_in_gag(s(X), s(Z)) → U3_gag(X, Z, plus_in_gag(X, Z))
U3_gag(X, Z, plus_out_gag(X, Y, Z)) → plus_out_gag(s(X), Y, s(Z))
U5_gga(Amount, C, Coins, plus_out_gag(s(C), NewAmount, Amount)) → U6_gga(Amount, C, Coins, NewAmount, ways_in_gga(Amount, Coins))
ways_in_gga(Amount, .(s(C), Coins)) → U9_gga(Amount, C, Coins, =_in_ga(Amount))
U9_gga(Amount, C, Coins, =_out_ga(Amount, s(X4))) → U10_gga(Amount, C, Coins, plus_in_gag(Amount, s(C)))
U10_gga(Amount, C, Coins, plus_out_gag(Amount, s(X5), s(C))) → U11_gga(Amount, C, Coins, ways_in_gga(Amount, Coins))
U11_gga(Amount, C, Coins, ways_out_gga(Amount, Coins, N)) → ways_out_gga(Amount, .(s(C), Coins), N)
U6_gga(Amount, C, Coins, NewAmount, ways_out_gga(Amount, Coins, N1)) → U7_gga(Amount, C, Coins, N1, ways_in_gga(NewAmount, .(s(C), Coins)))
U7_gga(Amount, C, Coins, N1, ways_out_gga(NewAmount, .(s(C), Coins), N2)) → U8_gga(Amount, C, Coins, plus_in_gga(N1, N2))
plus_in_gga(0, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g(X)) → plus_out_gga(0, X, X)
plus_in_gga(s(X), Y) → U3_gga(X, Y, plus_in_gga(X, Y))
U3_gga(X, Y, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_gga(Amount, C, Coins, plus_out_gga(N1, N2, N)) → ways_out_gga(Amount, .(s(C), Coins), N)

The set Q consists of the following terms:

ways_in_gga(x0, x1)
=_in_ga(x0)
U4_gga(x0, x1, x2, x3)
plus_in_gag(x0, x1)
nat_in_g(x0)
U1_g(x0, x1)
U2_gag(x0, x1)
U3_gag(x0, x1, x2)
U5_gga(x0, x1, x2, x3)
U9_gga(x0, x1, x2, x3)
U10_gga(x0, x1, x2, x3)
U11_gga(x0, x1, x2, x3)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4)
plus_in_gga(x0, x1)
U2_gga(x0, x1)
U3_gga(x0, x1, x2)
U8_gga(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.

(35) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(36) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U5_GGA(Amount, C, Coins, plus_out_gag(s(C), NewAmount, Amount)) → U6_GGA(Amount, C, Coins, NewAmount, ways_in_gga(Amount, Coins))
U6_GGA(Amount, C, Coins, NewAmount, ways_out_gga(Amount, Coins, N1)) → WAYS_IN_GGA(NewAmount, .(s(C), Coins))
WAYS_IN_GGA(Amount, .(s(C), Coins)) → U4_GGA(Amount, C, Coins, =_in_ga(Amount))
U4_GGA(Amount, C, Coins, =_out_ga(Amount, s(X3))) → U5_GGA(Amount, C, Coins, plus_in_gag(s(C), Amount))
U5_GGA(Amount, C, Coins, plus_out_gag(s(C), NewAmount, Amount)) → WAYS_IN_GGA(Amount, Coins)

The TRS R consists of the following rules:

ways_in_gga(0, X1) → ways_out_gga(0, X1, s(0))
ways_in_gga(X2, []) → ways_out_gga(X2, [], 0)
ways_in_gga(Amount, .(s(C), Coins)) → U4_gga(Amount, C, Coins, =_in_ga(Amount))
=_in_ga(X) → =_out_ga(X, X)
U4_gga(Amount, C, Coins, =_out_ga(Amount, s(X3))) → U5_gga(Amount, C, Coins, plus_in_gag(s(C), Amount))
plus_in_gag(0, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g(0)
nat_in_g(s(X)) → U1_g(X, nat_in_g(X))
U1_g(X, nat_out_g(X)) → nat_out_g(s(X))
U2_gag(X, nat_out_g(X)) → plus_out_gag(0, X, X)
plus_in_gag(s(X), s(Z)) → U3_gag(X, Z, plus_in_gag(X, Z))
U3_gag(X, Z, plus_out_gag(X, Y, Z)) → plus_out_gag(s(X), Y, s(Z))
U5_gga(Amount, C, Coins, plus_out_gag(s(C), NewAmount, Amount)) → U6_gga(Amount, C, Coins, NewAmount, ways_in_gga(Amount, Coins))
ways_in_gga(Amount, .(s(C), Coins)) → U9_gga(Amount, C, Coins, =_in_ga(Amount))
U9_gga(Amount, C, Coins, =_out_ga(Amount, s(X4))) → U10_gga(Amount, C, Coins, plus_in_gag(Amount, s(C)))
U10_gga(Amount, C, Coins, plus_out_gag(Amount, s(X5), s(C))) → U11_gga(Amount, C, Coins, ways_in_gga(Amount, Coins))
U11_gga(Amount, C, Coins, ways_out_gga(Amount, Coins, N)) → ways_out_gga(Amount, .(s(C), Coins), N)
U6_gga(Amount, C, Coins, NewAmount, ways_out_gga(Amount, Coins, N1)) → U7_gga(Amount, C, Coins, N1, ways_in_gga(NewAmount, .(s(C), Coins)))
U7_gga(Amount, C, Coins, N1, ways_out_gga(NewAmount, .(s(C), Coins), N2)) → U8_gga(Amount, C, Coins, plus_in_gga(N1, N2))
plus_in_gga(0, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g(X)) → plus_out_gga(0, X, X)
plus_in_gga(s(X), Y) → U3_gga(X, Y, plus_in_gga(X, Y))
U3_gga(X, Y, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_gga(Amount, C, Coins, plus_out_gga(N1, N2, N)) → ways_out_gga(Amount, .(s(C), Coins), N)

The set Q consists of the following terms:

ways_in_gga(x0, x1)
=_in_ga(x0)
U4_gga(x0, x1, x2, x3)
plus_in_gag(x0, x1)
nat_in_g(x0)
U1_g(x0, x1)
U2_gag(x0, x1)
U3_gag(x0, x1, x2)
U5_gga(x0, x1, x2, x3)
U9_gga(x0, x1, x2, x3)
U10_gga(x0, x1, x2, x3)
U11_gga(x0, x1, x2, x3)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4)
plus_in_gga(x0, x1)
U2_gga(x0, x1)
U3_gga(x0, x1, x2)
U8_gga(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U5_GGA(Amount, C, Coins, plus_out_gag(s(C), NewAmount, Amount)) → WAYS_IN_GGA(Amount, Coins)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = x₁ + x₂
POL(0) = 0
POL(=_in_ga(x₁)) = 0
POL(=_out_ga(x₁, x₂)) = 0
POL(U10_gga(x₁, x₂, x₃, x₄)) = 0
POL(U11_gga(x₁, x₂, x₃, x₄)) = 0
POL(U1_g(x₁, x₂)) = 0
POL(U2_gag(x₁, x₂)) = 0
POL(U2_gga(x₁, x₂)) = 0
POL(U3_gag(x₁, x₂, x₃)) = 0
POL(U3_gga(x₁, x₂, x₃)) = 0
POL(U4_GGA(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U4_gga(x₁, x₂, x₃, x₄)) = 0
POL(U5_GGA(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U5_gga(x₁, x₂, x₃, x₄)) = 0
POL(U6_GGA(x₁, x₂, x₃, x₄, x₅)) = 1 + x₃
POL(U6_gga(x₁, x₂, x₃, x₄, x₅)) = 0
POL(U7_gga(x₁, x₂, x₃, x₄, x₅)) = 0
POL(U8_gga(x₁, x₂, x₃, x₄)) = 0
POL(U9_gga(x₁, x₂, x₃, x₄)) = 0
POL(WAYS_IN_GGA(x₁, x₂)) = x₂
POL([]) = 0
POL(nat_in_g(x₁)) = 0
POL(nat_out_g(x₁)) = 0
POL(plus_in_gag(x₁, x₂)) = 0
POL(plus_in_gga(x₁, x₂)) = 0
POL(plus_out_gag(x₁, x₂, x₃)) = 0
POL(plus_out_gga(x₁, x₂, x₃)) = 0
POL(s(x₁)) = 1
POL(ways_in_gga(x₁, x₂)) = 0
POL(ways_out_gga(x₁, x₂, x₃)) = 0

The following usable rules [FROCOS05] were oriented: none

(38) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U5_GGA(Amount, C, Coins, plus_out_gag(s(C), NewAmount, Amount)) → U6_GGA(Amount, C, Coins, NewAmount, ways_in_gga(Amount, Coins))
U6_GGA(Amount, C, Coins, NewAmount, ways_out_gga(Amount, Coins, N1)) → WAYS_IN_GGA(NewAmount, .(s(C), Coins))
WAYS_IN_GGA(Amount, .(s(C), Coins)) → U4_GGA(Amount, C, Coins, =_in_ga(Amount))
U4_GGA(Amount, C, Coins, =_out_ga(Amount, s(X3))) → U5_GGA(Amount, C, Coins, plus_in_gag(s(C), Amount))

The TRS R consists of the following rules:

ways_in_gga(0, X1) → ways_out_gga(0, X1, s(0))
ways_in_gga(X2, []) → ways_out_gga(X2, [], 0)
ways_in_gga(Amount, .(s(C), Coins)) → U4_gga(Amount, C, Coins, =_in_ga(Amount))
=_in_ga(X) → =_out_ga(X, X)
U4_gga(Amount, C, Coins, =_out_ga(Amount, s(X3))) → U5_gga(Amount, C, Coins, plus_in_gag(s(C), Amount))
plus_in_gag(0, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g(0)
nat_in_g(s(X)) → U1_g(X, nat_in_g(X))
U1_g(X, nat_out_g(X)) → nat_out_g(s(X))
U2_gag(X, nat_out_g(X)) → plus_out_gag(0, X, X)
plus_in_gag(s(X), s(Z)) → U3_gag(X, Z, plus_in_gag(X, Z))
U3_gag(X, Z, plus_out_gag(X, Y, Z)) → plus_out_gag(s(X), Y, s(Z))
U5_gga(Amount, C, Coins, plus_out_gag(s(C), NewAmount, Amount)) → U6_gga(Amount, C, Coins, NewAmount, ways_in_gga(Amount, Coins))
ways_in_gga(Amount, .(s(C), Coins)) → U9_gga(Amount, C, Coins, =_in_ga(Amount))
U9_gga(Amount, C, Coins, =_out_ga(Amount, s(X4))) → U10_gga(Amount, C, Coins, plus_in_gag(Amount, s(C)))
U10_gga(Amount, C, Coins, plus_out_gag(Amount, s(X5), s(C))) → U11_gga(Amount, C, Coins, ways_in_gga(Amount, Coins))
U11_gga(Amount, C, Coins, ways_out_gga(Amount, Coins, N)) → ways_out_gga(Amount, .(s(C), Coins), N)
U6_gga(Amount, C, Coins, NewAmount, ways_out_gga(Amount, Coins, N1)) → U7_gga(Amount, C, Coins, N1, ways_in_gga(NewAmount, .(s(C), Coins)))
U7_gga(Amount, C, Coins, N1, ways_out_gga(NewAmount, .(s(C), Coins), N2)) → U8_gga(Amount, C, Coins, plus_in_gga(N1, N2))
plus_in_gga(0, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g(X)) → plus_out_gga(0, X, X)
plus_in_gga(s(X), Y) → U3_gga(X, Y, plus_in_gga(X, Y))
U3_gga(X, Y, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_gga(Amount, C, Coins, plus_out_gga(N1, N2, N)) → ways_out_gga(Amount, .(s(C), Coins), N)

The set Q consists of the following terms:

ways_in_gga(x0, x1)
=_in_ga(x0)
U4_gga(x0, x1, x2, x3)
plus_in_gag(x0, x1)
nat_in_g(x0)
U1_g(x0, x1)
U2_gag(x0, x1)
U3_gag(x0, x1, x2)
U5_gga(x0, x1, x2, x3)
U9_gga(x0, x1, x2, x3)
U10_gga(x0, x1, x2, x3)
U11_gga(x0, x1, x2, x3)
U6_gga(x0, x1, x2, x3, x4)
U7_gga(x0, x1, x2, x3, x4)
plus_in_gga(x0, x1)
U2_gga(x0, x1)
U3_gga(x0, x1, x2)
U8_gga(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.

(39) PrologToPiTRSProof (SOUND transformation)

ways_in_gga(0, X1, s(0)) → ways_out_gga(0, X1, s(0))
ways_in_gga(X2, [], 0) → ways_out_gga(X2, [], 0)
ways_in_gga(Amount, .(s(C), Coins), N) → U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
=_in_ga(X, X) → =_out_ga(X, X)
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))
plus_in_gag(0, X, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g(0)
nat_in_g(s(X)) → U1_g(X, nat_in_g(X))
U1_g(X, nat_out_g(X)) → nat_out_g(s(X))
U2_gag(X, nat_out_g(X)) → plus_out_gag(0, X, X)
plus_in_gag(s(X), Y, s(Z)) → U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) → plus_out_gag(s(X), Y, s(Z))
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))
ways_in_gga(Amount, .(s(C), Coins), N) → U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
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))
U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) → ways_out_gga(Amount, .(s(C), Coins), N)
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))
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))
plus_in_gga(0, X, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g(X)) → plus_out_gga(0, X, X)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) → ways_out_gga(Amount, .(s(C), Coins), N)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(40) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

ways_in_gga(0, X1, s(0)) → ways_out_gga(0, X1, s(0))
ways_in_gga(X2, [], 0) → ways_out_gga(X2, [], 0)
ways_in_gga(Amount, .(s(C), Coins), N) → U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
=_in_ga(X, X) → =_out_ga(X, X)
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))
plus_in_gag(0, X, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g(0)
nat_in_g(s(X)) → U1_g(X, nat_in_g(X))
U1_g(X, nat_out_g(X)) → nat_out_g(s(X))
U2_gag(X, nat_out_g(X)) → plus_out_gag(0, X, X)
plus_in_gag(s(X), Y, s(Z)) → U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) → plus_out_gag(s(X), Y, s(Z))
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))
ways_in_gga(Amount, .(s(C), Coins), N) → U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
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))
U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) → ways_out_gga(Amount, .(s(C), Coins), N)
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))
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))
plus_in_gga(0, X, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g(X)) → plus_out_gga(0, X, X)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) → ways_out_gga(Amount, .(s(C), Coins), N)

(41) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

WAYS_IN_GGA(Amount, .(s(C), Coins), N) → U4_GGA(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
WAYS_IN_GGA(Amount, .(s(C), Coins), N) → =_IN_GA(Amount, s(X3))
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))
U4_GGA(Amount, C, Coins, N, =_out_ga(Amount, s(X3))) → PLUS_IN_GAG(s(C), NewAmount, Amount)
PLUS_IN_GAG(0, X, X) → U2_GAG(X, nat_in_g(X))
PLUS_IN_GAG(0, X, X) → NAT_IN_G(X)
NAT_IN_G(s(X)) → U1_G(X, nat_in_g(X))
NAT_IN_G(s(X)) → NAT_IN_G(X)
PLUS_IN_GAG(s(X), Y, s(Z)) → U3_GAG(X, Y, Z, plus_in_gag(X, Y, Z))
PLUS_IN_GAG(s(X), Y, s(Z)) → PLUS_IN_GAG(X, Y, Z)
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))
U5_GGA(Amount, C, Coins, N, X3, plus_out_gag(s(C), NewAmount, Amount)) → WAYS_IN_GGA(Amount, Coins, N1)
WAYS_IN_GGA(Amount, .(s(C), Coins), N) → U9_GGA(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
U9_GGA(Amount, C, Coins, N, =_out_ga(Amount, s(X4))) → PLUS_IN_GAG(Amount, s(X5), s(C))
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))
U10_GGA(Amount, C, Coins, N, X4, plus_out_gag(Amount, s(X5), s(C))) → WAYS_IN_GGA(Amount, Coins, N)
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))
U6_GGA(Amount, C, Coins, N, X3, NewAmount, ways_out_gga(Amount, Coins, N1)) → WAYS_IN_GGA(NewAmount, .(s(C), Coins), N2)
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))
U7_GGA(Amount, C, Coins, N, X3, NewAmount, N1, ways_out_gga(NewAmount, .(s(C), Coins), N2)) → PLUS_IN_GGA(N1, N2, N)
PLUS_IN_GGA(0, X, X) → U2_GGA(X, nat_in_g(X))
PLUS_IN_GGA(0, X, X) → NAT_IN_G(X)
PLUS_IN_GGA(s(X), Y, s(Z)) → U3_GGA(X, Y, Z, plus_in_gga(X, Y, Z))
PLUS_IN_GGA(s(X), Y, s(Z)) → PLUS_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

ways_in_gga(0, X1, s(0)) → ways_out_gga(0, X1, s(0))
ways_in_gga(X2, [], 0) → ways_out_gga(X2, [], 0)
ways_in_gga(Amount, .(s(C), Coins), N) → U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
=_in_ga(X, X) → =_out_ga(X, X)
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))
plus_in_gag(0, X, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g(0)
nat_in_g(s(X)) → U1_g(X, nat_in_g(X))
U1_g(X, nat_out_g(X)) → nat_out_g(s(X))
U2_gag(X, nat_out_g(X)) → plus_out_gag(0, X, X)
plus_in_gag(s(X), Y, s(Z)) → U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) → plus_out_gag(s(X), Y, s(Z))
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))
ways_in_gga(Amount, .(s(C), Coins), N) → U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
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))
U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) → ways_out_gga(Amount, .(s(C), Coins), N)
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))
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))
plus_in_gga(0, X, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g(X)) → plus_out_gga(0, X, X)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) → ways_out_gga(Amount, .(s(C), Coins), N)

The argument filtering Pi contains the following mapping:
ways_in_gga(x1, x2, x3) = ways_in_gga(x1, x2)
0 = 0
ways_out_gga(x1, x2, x3) = ways_out_gga(x3)
[] = []
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x3, x5)
=_in_ga(x1, x2) = =_in_ga(x1)
=_out_ga(x1, x2) = =_out_ga(x2)
U5_gga(x1, x2, x3, x4, x5, x6) = U5_gga(x1, x2, x3, x6)
plus_in_gag(x1, x2, x3) = plus_in_gag(x1, x3)
U2_gag(x1, x2) = U2_gag(x1, x2)
nat_in_g(x1) = nat_in_g(x1)
nat_out_g(x1) = nat_out_g
U1_g(x1, x2) = U1_g(x2)
plus_out_gag(x1, x2, x3) = plus_out_gag(x2)
U3_gag(x1, x2, x3, x4) = U3_gag(x4)
U6_gga(x1, x2, x3, x4, x5, x6, x7) = U6_gga(x2, x3, x6, x7)
U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5)
U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x3, x6)
U11_gga(x1, x2, x3, x4, x5) = U11_gga(x5)
U7_gga(x1, x2, x3, x4, x5, x6, x7, x8) = U7_gga(x7, x8)
U8_gga(x1, x2, x3, x4, x5) = U8_gga(x5)
plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2)
U2_gga(x1, x2) = U2_gga(x1, x2)
plus_out_gga(x1, x2, x3) = plus_out_gga(x3)
U3_gga(x1, x2, x3, x4) = U3_gga(x4)
WAYS_IN_GGA(x1, x2, x3) = WAYS_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x2, x3, x5)
=_IN_GA(x1, x2) = =_IN_GA(x1)
U5_GGA(x1, x2, x3, x4, x5, x6) = U5_GGA(x1, x2, x3, x6)
PLUS_IN_GAG(x1, x2, x3) = PLUS_IN_GAG(x1, x3)
U2_GAG(x1, x2) = U2_GAG(x1, x2)
NAT_IN_G(x1) = NAT_IN_G(x1)
U1_G(x1, x2) = U1_G(x2)
U3_GAG(x1, x2, x3, x4) = U3_GAG(x4)
U6_GGA(x1, x2, x3, x4, x5, x6, x7) = U6_GGA(x2, x3, x6, x7)
U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x2, x3, x5)
U10_GGA(x1, x2, x3, x4, x5, x6) = U10_GGA(x1, x3, x6)
U11_GGA(x1, x2, x3, x4, x5) = U11_GGA(x5)
U7_GGA(x1, x2, x3, x4, x5, x6, x7, x8) = U7_GGA(x7, x8)
U8_GGA(x1, x2, x3, x4, x5) = U8_GGA(x5)
PLUS_IN_GGA(x1, x2, x3) = PLUS_IN_GGA(x1, x2)
U2_GGA(x1, x2) = U2_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x4)

We have to consider all (P,R,Pi)-chains

(42) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

WAYS_IN_GGA(Amount, .(s(C), Coins), N) → U4_GGA(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
WAYS_IN_GGA(Amount, .(s(C), Coins), N) → =_IN_GA(Amount, s(X3))
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))
U4_GGA(Amount, C, Coins, N, =_out_ga(Amount, s(X3))) → PLUS_IN_GAG(s(C), NewAmount, Amount)
PLUS_IN_GAG(0, X, X) → U2_GAG(X, nat_in_g(X))
PLUS_IN_GAG(0, X, X) → NAT_IN_G(X)
NAT_IN_G(s(X)) → U1_G(X, nat_in_g(X))
NAT_IN_G(s(X)) → NAT_IN_G(X)
PLUS_IN_GAG(s(X), Y, s(Z)) → U3_GAG(X, Y, Z, plus_in_gag(X, Y, Z))
PLUS_IN_GAG(s(X), Y, s(Z)) → PLUS_IN_GAG(X, Y, Z)
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))
U5_GGA(Amount, C, Coins, N, X3, plus_out_gag(s(C), NewAmount, Amount)) → WAYS_IN_GGA(Amount, Coins, N1)
WAYS_IN_GGA(Amount, .(s(C), Coins), N) → U9_GGA(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
U9_GGA(Amount, C, Coins, N, =_out_ga(Amount, s(X4))) → PLUS_IN_GAG(Amount, s(X5), s(C))
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))
U10_GGA(Amount, C, Coins, N, X4, plus_out_gag(Amount, s(X5), s(C))) → WAYS_IN_GGA(Amount, Coins, N)
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))
U6_GGA(Amount, C, Coins, N, X3, NewAmount, ways_out_gga(Amount, Coins, N1)) → WAYS_IN_GGA(NewAmount, .(s(C), Coins), N2)
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))
U7_GGA(Amount, C, Coins, N, X3, NewAmount, N1, ways_out_gga(NewAmount, .(s(C), Coins), N2)) → PLUS_IN_GGA(N1, N2, N)
PLUS_IN_GGA(0, X, X) → U2_GGA(X, nat_in_g(X))
PLUS_IN_GGA(0, X, X) → NAT_IN_G(X)
PLUS_IN_GGA(s(X), Y, s(Z)) → U3_GGA(X, Y, Z, plus_in_gga(X, Y, Z))
PLUS_IN_GGA(s(X), Y, s(Z)) → PLUS_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

ways_in_gga(0, X1, s(0)) → ways_out_gga(0, X1, s(0))
ways_in_gga(X2, [], 0) → ways_out_gga(X2, [], 0)
ways_in_gga(Amount, .(s(C), Coins), N) → U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
=_in_ga(X, X) → =_out_ga(X, X)
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))
plus_in_gag(0, X, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g(0)
nat_in_g(s(X)) → U1_g(X, nat_in_g(X))
U1_g(X, nat_out_g(X)) → nat_out_g(s(X))
U2_gag(X, nat_out_g(X)) → plus_out_gag(0, X, X)
plus_in_gag(s(X), Y, s(Z)) → U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) → plus_out_gag(s(X), Y, s(Z))
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))
ways_in_gga(Amount, .(s(C), Coins), N) → U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
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))
U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) → ways_out_gga(Amount, .(s(C), Coins), N)
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))
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))
plus_in_gga(0, X, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g(X)) → plus_out_gga(0, X, X)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) → ways_out_gga(Amount, .(s(C), Coins), N)

(43) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 14 less nodes.

(44) Complex Obligation (AND)

(45) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

NAT_IN_G(s(X)) → NAT_IN_G(X)

The TRS R consists of the following rules:

ways_in_gga(0, X1, s(0)) → ways_out_gga(0, X1, s(0))
ways_in_gga(X2, [], 0) → ways_out_gga(X2, [], 0)
ways_in_gga(Amount, .(s(C), Coins), N) → U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
=_in_ga(X, X) → =_out_ga(X, X)
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))
plus_in_gag(0, X, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g(0)
nat_in_g(s(X)) → U1_g(X, nat_in_g(X))
U1_g(X, nat_out_g(X)) → nat_out_g(s(X))
U2_gag(X, nat_out_g(X)) → plus_out_gag(0, X, X)
plus_in_gag(s(X), Y, s(Z)) → U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) → plus_out_gag(s(X), Y, s(Z))
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))
ways_in_gga(Amount, .(s(C), Coins), N) → U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
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))
U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) → ways_out_gga(Amount, .(s(C), Coins), N)
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))
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))
plus_in_gga(0, X, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g(X)) → plus_out_gga(0, X, X)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) → ways_out_gga(Amount, .(s(C), Coins), N)

(46) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(47) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

NAT_IN_G(s(X)) → NAT_IN_G(X)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains

(48) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(49) Obligation:

Q DP problem:
The TRS P consists of the following rules:

NAT_IN_G(s(X)) → NAT_IN_G(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(50) QDPSizeChangeProof (EQUIVALENT transformation)

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.

From the DPs we obtained the following set of size-change graphs:

NAT_IN_G(s(X)) → NAT_IN_G(X)
The graph contains the following edges 1 > 1

(51) TRUE

(52) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PLUS_IN_GGA(s(X), Y, s(Z)) → PLUS_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

ways_in_gga(0, X1, s(0)) → ways_out_gga(0, X1, s(0))
ways_in_gga(X2, [], 0) → ways_out_gga(X2, [], 0)
ways_in_gga(Amount, .(s(C), Coins), N) → U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
=_in_ga(X, X) → =_out_ga(X, X)
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))
plus_in_gag(0, X, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g(0)
nat_in_g(s(X)) → U1_g(X, nat_in_g(X))
U1_g(X, nat_out_g(X)) → nat_out_g(s(X))
U2_gag(X, nat_out_g(X)) → plus_out_gag(0, X, X)
plus_in_gag(s(X), Y, s(Z)) → U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) → plus_out_gag(s(X), Y, s(Z))
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))
ways_in_gga(Amount, .(s(C), Coins), N) → U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
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))
U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) → ways_out_gga(Amount, .(s(C), Coins), N)
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))
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))
plus_in_gga(0, X, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g(X)) → plus_out_gga(0, X, X)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) → ways_out_gga(Amount, .(s(C), Coins), N)

(53) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(54) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PLUS_IN_GGA(s(X), Y, s(Z)) → PLUS_IN_GGA(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
PLUS_IN_GGA(x1, x2, x3) = PLUS_IN_GGA(x1, x2)

We have to consider all (P,R,Pi)-chains

(55) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(56) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PLUS_IN_GGA(s(X), Y) → PLUS_IN_GGA(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(57) QDPSizeChangeProof (EQUIVALENT transformation)

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.

From the DPs we obtained the following set of size-change graphs:

PLUS_IN_GGA(s(X), Y) → PLUS_IN_GGA(X, Y)
The graph contains the following edges 1 > 1, 2 >= 2

(58) TRUE

(59) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PLUS_IN_GAG(s(X), Y, s(Z)) → PLUS_IN_GAG(X, Y, Z)

The TRS R consists of the following rules:

ways_in_gga(0, X1, s(0)) → ways_out_gga(0, X1, s(0))
ways_in_gga(X2, [], 0) → ways_out_gga(X2, [], 0)
ways_in_gga(Amount, .(s(C), Coins), N) → U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
=_in_ga(X, X) → =_out_ga(X, X)
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))
plus_in_gag(0, X, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g(0)
nat_in_g(s(X)) → U1_g(X, nat_in_g(X))
U1_g(X, nat_out_g(X)) → nat_out_g(s(X))
U2_gag(X, nat_out_g(X)) → plus_out_gag(0, X, X)
plus_in_gag(s(X), Y, s(Z)) → U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) → plus_out_gag(s(X), Y, s(Z))
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))
ways_in_gga(Amount, .(s(C), Coins), N) → U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
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))
U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) → ways_out_gga(Amount, .(s(C), Coins), N)
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))
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))
plus_in_gga(0, X, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g(X)) → plus_out_gga(0, X, X)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) → ways_out_gga(Amount, .(s(C), Coins), N)

(60) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(61) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

PLUS_IN_GAG(s(X), Y, s(Z)) → PLUS_IN_GAG(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
PLUS_IN_GAG(x1, x2, x3) = PLUS_IN_GAG(x1, x3)

We have to consider all (P,R,Pi)-chains

(62) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(63) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PLUS_IN_GAG(s(X), s(Z)) → PLUS_IN_GAG(X, Z)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(64) QDPSizeChangeProof (EQUIVALENT transformation)

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.

From the DPs we obtained the following set of size-change graphs:

PLUS_IN_GAG(s(X), s(Z)) → PLUS_IN_GAG(X, Z)
The graph contains the following edges 1 > 1, 2 > 2

(65) TRUE

(66) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

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))
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))
U6_GGA(Amount, C, Coins, N, X3, NewAmount, ways_out_gga(Amount, Coins, N1)) → WAYS_IN_GGA(NewAmount, .(s(C), Coins), N2)
WAYS_IN_GGA(Amount, .(s(C), Coins), N) → U4_GGA(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
WAYS_IN_GGA(Amount, .(s(C), Coins), N) → U9_GGA(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
U10_GGA(Amount, C, Coins, N, X4, plus_out_gag(Amount, s(X5), s(C))) → WAYS_IN_GGA(Amount, Coins, N)
U5_GGA(Amount, C, Coins, N, X3, plus_out_gag(s(C), NewAmount, Amount)) → WAYS_IN_GGA(Amount, Coins, N1)

The TRS R consists of the following rules:

ways_in_gga(0, X1, s(0)) → ways_out_gga(0, X1, s(0))
ways_in_gga(X2, [], 0) → ways_out_gga(X2, [], 0)
ways_in_gga(Amount, .(s(C), Coins), N) → U4_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X3)))
=_in_ga(X, X) → =_out_ga(X, X)
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))
plus_in_gag(0, X, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g(0)
nat_in_g(s(X)) → U1_g(X, nat_in_g(X))
U1_g(X, nat_out_g(X)) → nat_out_g(s(X))
U2_gag(X, nat_out_g(X)) → plus_out_gag(0, X, X)
plus_in_gag(s(X), Y, s(Z)) → U3_gag(X, Y, Z, plus_in_gag(X, Y, Z))
U3_gag(X, Y, Z, plus_out_gag(X, Y, Z)) → plus_out_gag(s(X), Y, s(Z))
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))
ways_in_gga(Amount, .(s(C), Coins), N) → U9_gga(Amount, C, Coins, N, =_in_ga(Amount, s(X4)))
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)))
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))
U11_gga(Amount, C, Coins, N, ways_out_gga(Amount, Coins, N)) → ways_out_gga(Amount, .(s(C), Coins), N)
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))
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))
plus_in_gga(0, X, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g(X)) → plus_out_gga(0, X, X)
plus_in_gga(s(X), Y, s(Z)) → U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) → plus_out_gga(s(X), Y, s(Z))
U8_gga(Amount, C, Coins, N, plus_out_gga(N1, N2, N)) → ways_out_gga(Amount, .(s(C), Coins), N)

The argument filtering Pi contains the following mapping:
ways_in_gga(x1, x2, x3) = ways_in_gga(x1, x2)
0 = 0
ways_out_gga(x1, x2, x3) = ways_out_gga(x3)
[] = []
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x2, x3, x5)
=_in_ga(x1, x2) = =_in_ga(x1)
=_out_ga(x1, x2) = =_out_ga(x2)
U5_gga(x1, x2, x3, x4, x5, x6) = U5_gga(x1, x2, x3, x6)
plus_in_gag(x1, x2, x3) = plus_in_gag(x1, x3)
U2_gag(x1, x2) = U2_gag(x1, x2)
nat_in_g(x1) = nat_in_g(x1)
nat_out_g(x1) = nat_out_g
U1_g(x1, x2) = U1_g(x2)
plus_out_gag(x1, x2, x3) = plus_out_gag(x2)
U3_gag(x1, x2, x3, x4) = U3_gag(x4)
U6_gga(x1, x2, x3, x4, x5, x6, x7) = U6_gga(x2, x3, x6, x7)
U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5)
U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x3, x6)
U11_gga(x1, x2, x3, x4, x5) = U11_gga(x5)
U7_gga(x1, x2, x3, x4, x5, x6, x7, x8) = U7_gga(x7, x8)
U8_gga(x1, x2, x3, x4, x5) = U8_gga(x5)
plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2)
U2_gga(x1, x2) = U2_gga(x1, x2)
plus_out_gga(x1, x2, x3) = plus_out_gga(x3)
U3_gga(x1, x2, x3, x4) = U3_gga(x4)
WAYS_IN_GGA(x1, x2, x3) = WAYS_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x2, x3, x5)
U5_GGA(x1, x2, x3, x4, x5, x6) = U5_GGA(x1, x2, x3, x6)
U6_GGA(x1, x2, x3, x4, x5, x6, x7) = U6_GGA(x2, x3, x6, x7)
U9_GGA(x1, x2, x3, x4, x5) = U9_GGA(x1, x2, x3, x5)
U10_GGA(x1, x2, x3, x4, x5, x6) = U10_GGA(x1, x3, x6)

We have to consider all (P,R,Pi)-chains

(67) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(68) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U4_GGA(Amount, C, Coins, =_out_ga(s(X3))) → U5_GGA(Amount, C, Coins, plus_in_gag(s(C), Amount))
U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) → U6_GGA(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
U6_GGA(C, Coins, NewAmount, ways_out_gga(N1)) → WAYS_IN_GGA(NewAmount, .(s(C), Coins))
WAYS_IN_GGA(Amount, .(s(C), Coins)) → U4_GGA(Amount, C, Coins, =_in_ga(Amount))
WAYS_IN_GGA(Amount, .(s(C), Coins)) → U9_GGA(Amount, C, Coins, =_in_ga(Amount))
U9_GGA(Amount, C, Coins, =_out_ga(s(X4))) → U10_GGA(Amount, Coins, plus_in_gag(Amount, s(C)))
U10_GGA(Amount, Coins, plus_out_gag(s(X5))) → WAYS_IN_GGA(Amount, Coins)
U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) → WAYS_IN_GGA(Amount, Coins)

The TRS R consists of the following rules:

ways_in_gga(0, X1) → ways_out_gga(s(0))
ways_in_gga(X2, []) → ways_out_gga(0)
ways_in_gga(Amount, .(s(C), Coins)) → U4_gga(Amount, C, Coins, =_in_ga(Amount))
=_in_ga(X) → =_out_ga(X)
U4_gga(Amount, C, Coins, =_out_ga(s(X3))) → U5_gga(Amount, C, Coins, plus_in_gag(s(C), Amount))
plus_in_gag(0, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g
nat_in_g(s(X)) → U1_g(nat_in_g(X))
U1_g(nat_out_g) → nat_out_g
U2_gag(X, nat_out_g) → plus_out_gag(X)
plus_in_gag(s(X), s(Z)) → U3_gag(plus_in_gag(X, Z))
U3_gag(plus_out_gag(Y)) → plus_out_gag(Y)
U5_gga(Amount, C, Coins, plus_out_gag(NewAmount)) → U6_gga(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
ways_in_gga(Amount, .(s(C), Coins)) → U9_gga(Amount, C, Coins, =_in_ga(Amount))
U9_gga(Amount, C, Coins, =_out_ga(s(X4))) → U10_gga(Amount, Coins, plus_in_gag(Amount, s(C)))
U10_gga(Amount, Coins, plus_out_gag(s(X5))) → U11_gga(ways_in_gga(Amount, Coins))
U11_gga(ways_out_gga(N)) → ways_out_gga(N)
U6_gga(C, Coins, NewAmount, ways_out_gga(N1)) → U7_gga(N1, ways_in_gga(NewAmount, .(s(C), Coins)))
U7_gga(N1, ways_out_gga(N2)) → U8_gga(plus_in_gga(N1, N2))
plus_in_gga(0, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g) → plus_out_gga(X)
plus_in_gga(s(X), Y) → U3_gga(plus_in_gga(X, Y))
U3_gga(plus_out_gga(Z)) → plus_out_gga(s(Z))
U8_gga(plus_out_gga(N)) → ways_out_gga(N)

The set Q consists of the following terms:

ways_in_gga(x0, x1)
=_in_ga(x0)
U4_gga(x0, x1, x2, x3)
plus_in_gag(x0, x1)
nat_in_g(x0)
U1_g(x0)
U2_gag(x0, x1)
U3_gag(x0)
U5_gga(x0, x1, x2, x3)
U9_gga(x0, x1, x2, x3)
U10_gga(x0, x1, x2)
U11_gga(x0)
U6_gga(x0, x1, x2, x3)
U7_gga(x0, x1)
plus_in_gga(x0, x1)
U2_gga(x0, x1)
U3_gga(x0)
U8_gga(x0)

We have to consider all (P,Q,R)-chains.

(69) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U10_GGA(Amount, Coins, plus_out_gag(s(X5))) → WAYS_IN_GGA(Amount, Coins)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = x₁ + x₂
POL(0) = 1
POL(=_in_ga(x₁)) = 0
POL(=_out_ga(x₁)) = 0
POL(U10_GGA(x₁, x₂, x₃)) = x₂ + x₃
POL(U10_gga(x₁, x₂, x₃)) = 0
POL(U11_gga(x₁)) = 0
POL(U1_g(x₁)) = x₁
POL(U2_gag(x₁, x₂)) = x₂
POL(U2_gga(x₁, x₂)) = 1 + x₁
POL(U3_gag(x₁)) = x₁
POL(U3_gga(x₁)) = 0
POL(U4_GGA(x₁, x₂, x₃, x₄)) = x₂ + x₃
POL(U4_gga(x₁, x₂, x₃, x₄)) = 0
POL(U5_GGA(x₁, x₂, x₃, x₄)) = x₂ + x₃
POL(U5_gga(x₁, x₂, x₃, x₄)) = 0
POL(U6_GGA(x₁, x₂, x₃, x₄)) = x₁ + x₂
POL(U6_gga(x₁, x₂, x₃, x₄)) = 0
POL(U7_gga(x₁, x₂)) = 0
POL(U8_gga(x₁)) = 0
POL(U9_GGA(x₁, x₂, x₃, x₄)) = x₂ + x₃
POL(U9_gga(x₁, x₂, x₃, x₄)) = 0
POL(WAYS_IN_GGA(x₁, x₂)) = x₂
POL([]) = 0
POL(nat_in_g(x₁)) = x₁
POL(nat_out_g) = 1
POL(plus_in_gag(x₁, x₂)) = x₂
POL(plus_in_gga(x₁, x₂)) = 1 + x₁ + x₂
POL(plus_out_gag(x₁)) = 1
POL(plus_out_gga(x₁)) = 0
POL(s(x₁)) = x₁
POL(ways_in_gga(x₁, x₂)) = 0
POL(ways_out_gga(x₁)) = 0

The following usable rules [FROCOS05] were oriented:

plus_in_gag(s(X), s(Z)) → U3_gag(plus_in_gag(X, Z))
plus_in_gag(0, X) → U2_gag(X, nat_in_g(X))
U3_gag(plus_out_gag(Y)) → plus_out_gag(Y)
nat_in_g(0) → nat_out_g
nat_in_g(s(X)) → U1_g(nat_in_g(X))
U2_gag(X, nat_out_g) → plus_out_gag(X)
U1_g(nat_out_g) → nat_out_g

(70) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U4_GGA(Amount, C, Coins, =_out_ga(s(X3))) → U5_GGA(Amount, C, Coins, plus_in_gag(s(C), Amount))
U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) → U6_GGA(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
U6_GGA(C, Coins, NewAmount, ways_out_gga(N1)) → WAYS_IN_GGA(NewAmount, .(s(C), Coins))
WAYS_IN_GGA(Amount, .(s(C), Coins)) → U4_GGA(Amount, C, Coins, =_in_ga(Amount))
WAYS_IN_GGA(Amount, .(s(C), Coins)) → U9_GGA(Amount, C, Coins, =_in_ga(Amount))
U9_GGA(Amount, C, Coins, =_out_ga(s(X4))) → U10_GGA(Amount, Coins, plus_in_gag(Amount, s(C)))
U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) → WAYS_IN_GGA(Amount, Coins)

The TRS R consists of the following rules:

ways_in_gga(0, X1) → ways_out_gga(s(0))
ways_in_gga(X2, []) → ways_out_gga(0)
ways_in_gga(Amount, .(s(C), Coins)) → U4_gga(Amount, C, Coins, =_in_ga(Amount))
=_in_ga(X) → =_out_ga(X)
U4_gga(Amount, C, Coins, =_out_ga(s(X3))) → U5_gga(Amount, C, Coins, plus_in_gag(s(C), Amount))
plus_in_gag(0, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g
nat_in_g(s(X)) → U1_g(nat_in_g(X))
U1_g(nat_out_g) → nat_out_g
U2_gag(X, nat_out_g) → plus_out_gag(X)
plus_in_gag(s(X), s(Z)) → U3_gag(plus_in_gag(X, Z))
U3_gag(plus_out_gag(Y)) → plus_out_gag(Y)
U5_gga(Amount, C, Coins, plus_out_gag(NewAmount)) → U6_gga(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
ways_in_gga(Amount, .(s(C), Coins)) → U9_gga(Amount, C, Coins, =_in_ga(Amount))
U9_gga(Amount, C, Coins, =_out_ga(s(X4))) → U10_gga(Amount, Coins, plus_in_gag(Amount, s(C)))
U10_gga(Amount, Coins, plus_out_gag(s(X5))) → U11_gga(ways_in_gga(Amount, Coins))
U11_gga(ways_out_gga(N)) → ways_out_gga(N)
U6_gga(C, Coins, NewAmount, ways_out_gga(N1)) → U7_gga(N1, ways_in_gga(NewAmount, .(s(C), Coins)))
U7_gga(N1, ways_out_gga(N2)) → U8_gga(plus_in_gga(N1, N2))
plus_in_gga(0, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g) → plus_out_gga(X)
plus_in_gga(s(X), Y) → U3_gga(plus_in_gga(X, Y))
U3_gga(plus_out_gga(Z)) → plus_out_gga(s(Z))
U8_gga(plus_out_gga(N)) → ways_out_gga(N)

The set Q consists of the following terms:

ways_in_gga(x0, x1)
=_in_ga(x0)
U4_gga(x0, x1, x2, x3)
plus_in_gag(x0, x1)
nat_in_g(x0)
U1_g(x0)
U2_gag(x0, x1)
U3_gag(x0)
U5_gga(x0, x1, x2, x3)
U9_gga(x0, x1, x2, x3)
U10_gga(x0, x1, x2)
U11_gga(x0)
U6_gga(x0, x1, x2, x3)
U7_gga(x0, x1)
plus_in_gga(x0, x1)
U2_gga(x0, x1)
U3_gga(x0)
U8_gga(x0)

We have to consider all (P,Q,R)-chains.

(71) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(72) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) → U6_GGA(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
U6_GGA(C, Coins, NewAmount, ways_out_gga(N1)) → WAYS_IN_GGA(NewAmount, .(s(C), Coins))
WAYS_IN_GGA(Amount, .(s(C), Coins)) → U4_GGA(Amount, C, Coins, =_in_ga(Amount))
U4_GGA(Amount, C, Coins, =_out_ga(s(X3))) → U5_GGA(Amount, C, Coins, plus_in_gag(s(C), Amount))
U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) → WAYS_IN_GGA(Amount, Coins)

The TRS R consists of the following rules:

ways_in_gga(0, X1) → ways_out_gga(s(0))
ways_in_gga(X2, []) → ways_out_gga(0)
ways_in_gga(Amount, .(s(C), Coins)) → U4_gga(Amount, C, Coins, =_in_ga(Amount))
=_in_ga(X) → =_out_ga(X)
U4_gga(Amount, C, Coins, =_out_ga(s(X3))) → U5_gga(Amount, C, Coins, plus_in_gag(s(C), Amount))
plus_in_gag(0, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g
nat_in_g(s(X)) → U1_g(nat_in_g(X))
U1_g(nat_out_g) → nat_out_g
U2_gag(X, nat_out_g) → plus_out_gag(X)
plus_in_gag(s(X), s(Z)) → U3_gag(plus_in_gag(X, Z))
U3_gag(plus_out_gag(Y)) → plus_out_gag(Y)
U5_gga(Amount, C, Coins, plus_out_gag(NewAmount)) → U6_gga(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
ways_in_gga(Amount, .(s(C), Coins)) → U9_gga(Amount, C, Coins, =_in_ga(Amount))
U9_gga(Amount, C, Coins, =_out_ga(s(X4))) → U10_gga(Amount, Coins, plus_in_gag(Amount, s(C)))
U10_gga(Amount, Coins, plus_out_gag(s(X5))) → U11_gga(ways_in_gga(Amount, Coins))
U11_gga(ways_out_gga(N)) → ways_out_gga(N)
U6_gga(C, Coins, NewAmount, ways_out_gga(N1)) → U7_gga(N1, ways_in_gga(NewAmount, .(s(C), Coins)))
U7_gga(N1, ways_out_gga(N2)) → U8_gga(plus_in_gga(N1, N2))
plus_in_gga(0, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g) → plus_out_gga(X)
plus_in_gga(s(X), Y) → U3_gga(plus_in_gga(X, Y))
U3_gga(plus_out_gga(Z)) → plus_out_gga(s(Z))
U8_gga(plus_out_gga(N)) → ways_out_gga(N)

The set Q consists of the following terms:

ways_in_gga(x0, x1)
=_in_ga(x0)
U4_gga(x0, x1, x2, x3)
plus_in_gag(x0, x1)
nat_in_g(x0)
U1_g(x0)
U2_gag(x0, x1)
U3_gag(x0)
U5_gga(x0, x1, x2, x3)
U9_gga(x0, x1, x2, x3)
U10_gga(x0, x1, x2)
U11_gga(x0)
U6_gga(x0, x1, x2, x3)
U7_gga(x0, x1)
plus_in_gga(x0, x1)
U2_gga(x0, x1)
U3_gga(x0)
U8_gga(x0)

We have to consider all (P,Q,R)-chains.

(73) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) → WAYS_IN_GGA(Amount, Coins)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₂
POL(0) = 0
POL(=_in_ga(x₁)) = 1
POL(=_out_ga(x₁)) = 1
POL(U10_gga(x₁, x₂, x₃)) = 1
POL(U11_gga(x₁)) = 1
POL(U1_g(x₁)) = 0
POL(U2_gag(x₁, x₂)) = 0
POL(U2_gga(x₁, x₂)) = 0
POL(U3_gag(x₁)) = 1
POL(U3_gga(x₁)) = 0
POL(U4_GGA(x₁, x₂, x₃, x₄)) = 1 + x₃ + x₄
POL(U4_gga(x₁, x₂, x₃, x₄)) = 1
POL(U5_GGA(x₁, x₂, x₃, x₄)) = 1 + x₃ + x₄
POL(U5_gga(x₁, x₂, x₃, x₄)) = x₄
POL(U6_GGA(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄
POL(U6_gga(x₁, x₂, x₃, x₄)) = 1
POL(U7_gga(x₁, x₂)) = x₂
POL(U8_gga(x₁)) = 1
POL(U9_gga(x₁, x₂, x₃, x₄)) = 1
POL(WAYS_IN_GGA(x₁, x₂)) = 1 + x₂
POL([]) = 0
POL(nat_in_g(x₁)) = 0
POL(nat_out_g) = 0
POL(plus_in_gag(x₁, x₂)) = 1
POL(plus_in_gga(x₁, x₂)) = 0
POL(plus_out_gag(x₁)) = 1
POL(plus_out_gga(x₁)) = 0
POL(s(x₁)) = 0
POL(ways_in_gga(x₁, x₂)) = 1
POL(ways_out_gga(x₁)) = 1

The following usable rules [FROCOS05] were oriented:

ways_in_gga(0, X1) → ways_out_gga(s(0))
ways_in_gga(X2, []) → ways_out_gga(0)
ways_in_gga(Amount, .(s(C), Coins)) → U4_gga(Amount, C, Coins, =_in_ga(Amount))
ways_in_gga(Amount, .(s(C), Coins)) → U9_gga(Amount, C, Coins, =_in_ga(Amount))
=_in_ga(X) → =_out_ga(X)
plus_in_gag(s(X), s(Z)) → U3_gag(plus_in_gag(X, Z))
U4_gga(Amount, C, Coins, =_out_ga(s(X3))) → U5_gga(Amount, C, Coins, plus_in_gag(s(C), Amount))
U5_gga(Amount, C, Coins, plus_out_gag(NewAmount)) → U6_gga(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
U9_gga(Amount, C, Coins, =_out_ga(s(X4))) → U10_gga(Amount, Coins, plus_in_gag(Amount, s(C)))
U10_gga(Amount, Coins, plus_out_gag(s(X5))) → U11_gga(ways_in_gga(Amount, Coins))
U11_gga(ways_out_gga(N)) → ways_out_gga(N)
U6_gga(C, Coins, NewAmount, ways_out_gga(N1)) → U7_gga(N1, ways_in_gga(NewAmount, .(s(C), Coins)))
U7_gga(N1, ways_out_gga(N2)) → U8_gga(plus_in_gga(N1, N2))
U3_gag(plus_out_gag(Y)) → plus_out_gag(Y)
U8_gga(plus_out_gga(N)) → ways_out_gga(N)

(74) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) → U6_GGA(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
U6_GGA(C, Coins, NewAmount, ways_out_gga(N1)) → WAYS_IN_GGA(NewAmount, .(s(C), Coins))
WAYS_IN_GGA(Amount, .(s(C), Coins)) → U4_GGA(Amount, C, Coins, =_in_ga(Amount))
U4_GGA(Amount, C, Coins, =_out_ga(s(X3))) → U5_GGA(Amount, C, Coins, plus_in_gag(s(C), Amount))

The TRS R consists of the following rules:

ways_in_gga(0, X1) → ways_out_gga(s(0))
ways_in_gga(X2, []) → ways_out_gga(0)
ways_in_gga(Amount, .(s(C), Coins)) → U4_gga(Amount, C, Coins, =_in_ga(Amount))
=_in_ga(X) → =_out_ga(X)
U4_gga(Amount, C, Coins, =_out_ga(s(X3))) → U5_gga(Amount, C, Coins, plus_in_gag(s(C), Amount))
plus_in_gag(0, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g
nat_in_g(s(X)) → U1_g(nat_in_g(X))
U1_g(nat_out_g) → nat_out_g
U2_gag(X, nat_out_g) → plus_out_gag(X)
plus_in_gag(s(X), s(Z)) → U3_gag(plus_in_gag(X, Z))
U3_gag(plus_out_gag(Y)) → plus_out_gag(Y)
U5_gga(Amount, C, Coins, plus_out_gag(NewAmount)) → U6_gga(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
ways_in_gga(Amount, .(s(C), Coins)) → U9_gga(Amount, C, Coins, =_in_ga(Amount))
U9_gga(Amount, C, Coins, =_out_ga(s(X4))) → U10_gga(Amount, Coins, plus_in_gag(Amount, s(C)))
U10_gga(Amount, Coins, plus_out_gag(s(X5))) → U11_gga(ways_in_gga(Amount, Coins))
U11_gga(ways_out_gga(N)) → ways_out_gga(N)
U6_gga(C, Coins, NewAmount, ways_out_gga(N1)) → U7_gga(N1, ways_in_gga(NewAmount, .(s(C), Coins)))
U7_gga(N1, ways_out_gga(N2)) → U8_gga(plus_in_gga(N1, N2))
plus_in_gga(0, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g) → plus_out_gga(X)
plus_in_gga(s(X), Y) → U3_gga(plus_in_gga(X, Y))
U3_gga(plus_out_gga(Z)) → plus_out_gga(s(Z))
U8_gga(plus_out_gga(N)) → ways_out_gga(N)

The set Q consists of the following terms:

ways_in_gga(x0, x1)
=_in_ga(x0)
U4_gga(x0, x1, x2, x3)
plus_in_gag(x0, x1)
nat_in_g(x0)
U1_g(x0)
U2_gag(x0, x1)
U3_gag(x0)
U5_gga(x0, x1, x2, x3)
U9_gga(x0, x1, x2, x3)
U10_gga(x0, x1, x2)
U11_gga(x0)
U6_gga(x0, x1, x2, x3)
U7_gga(x0, x1)
plus_in_gga(x0, x1)
U2_gga(x0, x1)
U3_gga(x0)
U8_gga(x0)

We have to consider all (P,Q,R)-chains.

(75) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U6_GGA(C, Coins, NewAmount, ways_out_gga(N1)) → WAYS_IN_GGA(NewAmount, .(s(C), Coins))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(U5_GGA(x₁, x₂, x₃, x₄)) = 0 +
[ 0, 0 ]

· x₁ +
[ 0, 0 ]

· x₂ +
[ 0, 0 ]

· x₃ +
[ 1, 0 ]

· x₄

POL(plus_out_gag(x₁)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 1 0 /

· x₁

POL(U6_GGA(x₁, x₂, x₃, x₄)) = 1 +
[ 0, 0 ]

· x₁ +
[ 0, 0 ]

· x₂ +
[ 1, 0 ]

· x₃ +
[ 0, 0 ]

· x₄

POL(ways_in_gga(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 1 \

\ 1 0 /

· x₂

POL(ways_out_gga(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 1 0 /

· x₁

POL(WAYS_IN_GGA(x₁, x₂)) = 0 +
[ 1, 0 ]

· x₁ +
[ 0, 0 ]

· x₂

POL(.(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(s(x₁)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁

POL(U4_GGA(x₁, x₂, x₃, x₄)) = 0 +
[ 1, 0 ]

· x₁ +
[ 0, 0 ]

· x₂ +
[ 0, 0 ]

· x₃ +
[ 0, 0 ]

· x₄

POL(=_in_ga(x₁)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 1 1 /

· x₁

POL(=_out_ga(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁

POL(plus_in_gag(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 1 0 /

· x₂

POL(0) =
/ 0 \

\ 1 /

POL([]) =
/ 0 \

\ 0 /

POL(U4_gga(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 1 /

+
/ 1 0 \

\ 0 0 /

· x₁ +
/ 1 1 \

\ 1 0 /

· x₂ +
/ 0 1 \

\ 1 1 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U9_gga(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 1 1 /

· x₂ +
/ 1 1 \

\ 1 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U3_gag(x₁)) =
/ 1 \

\ 0 /

+
/ 0 1 \

\ 0 1 /

· x₁

POL(U5_gga(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 1 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 1 0 \

\ 0 1 /

· x₃ +
/ 0 1 \

\ 0 0 /

· x₄

POL(U6_gga(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 1 1 /

· x₁ +
/ 0 0 \

\ 1 1 /

· x₂ +
/ 0 1 \

\ 1 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U10_gga(x₁, x₂, x₃)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 1 0 /

· x₁ +
/ 1 0 \

\ 0 1 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃

POL(U2_gag(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 1 0 /

· x₁ +
/ 1 0 \

\ 0 0 /

· x₂

POL(nat_in_g(x₁)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(U11_gga(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 1 1 /

· x₁

POL(U7_gga(x₁, x₂)) =
/ 1 \

\ 1 /

+
/ 0 1 \

\ 1 0 /

· x₁ +
/ 0 0 \

\ 1 1 /

· x₂

POL(U8_gga(x₁)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 1 0 /

· x₁

POL(plus_in_gga(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 1 0 /

· x₁ +
/ 1 0 \

\ 1 0 /

· x₂

POL(nat_out_g) =
/ 1 \

\ 0 /

POL(U1_g(x₁)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(U2_gga(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 1 \

\ 0 1 /

· x₁ +
/ 0 0 \

\ 1 0 /

· x₂

POL(U3_gga(x₁)) =
/ 1 \

\ 1 /

+
/ 0 0 \

\ 1 1 /

· x₁

POL(plus_out_gga(x₁)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 1 0 /

· x₁

The following usable rules [FROCOS05] were oriented:

plus_in_gag(s(X), s(Z)) → U3_gag(plus_in_gag(X, Z))
plus_in_gag(0, X) → U2_gag(X, nat_in_g(X))
U3_gag(plus_out_gag(Y)) → plus_out_gag(Y)
nat_in_g(0) → nat_out_g
nat_in_g(s(X)) → U1_g(nat_in_g(X))
U2_gag(X, nat_out_g) → plus_out_gag(X)
U1_g(nat_out_g) → nat_out_g

(76) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U5_GGA(Amount, C, Coins, plus_out_gag(NewAmount)) → U6_GGA(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
WAYS_IN_GGA(Amount, .(s(C), Coins)) → U4_GGA(Amount, C, Coins, =_in_ga(Amount))
U4_GGA(Amount, C, Coins, =_out_ga(s(X3))) → U5_GGA(Amount, C, Coins, plus_in_gag(s(C), Amount))

The TRS R consists of the following rules:

ways_in_gga(0, X1) → ways_out_gga(s(0))
ways_in_gga(X2, []) → ways_out_gga(0)
ways_in_gga(Amount, .(s(C), Coins)) → U4_gga(Amount, C, Coins, =_in_ga(Amount))
=_in_ga(X) → =_out_ga(X)
U4_gga(Amount, C, Coins, =_out_ga(s(X3))) → U5_gga(Amount, C, Coins, plus_in_gag(s(C), Amount))
plus_in_gag(0, X) → U2_gag(X, nat_in_g(X))
nat_in_g(0) → nat_out_g
nat_in_g(s(X)) → U1_g(nat_in_g(X))
U1_g(nat_out_g) → nat_out_g
U2_gag(X, nat_out_g) → plus_out_gag(X)
plus_in_gag(s(X), s(Z)) → U3_gag(plus_in_gag(X, Z))
U3_gag(plus_out_gag(Y)) → plus_out_gag(Y)
U5_gga(Amount, C, Coins, plus_out_gag(NewAmount)) → U6_gga(C, Coins, NewAmount, ways_in_gga(Amount, Coins))
ways_in_gga(Amount, .(s(C), Coins)) → U9_gga(Amount, C, Coins, =_in_ga(Amount))
U9_gga(Amount, C, Coins, =_out_ga(s(X4))) → U10_gga(Amount, Coins, plus_in_gag(Amount, s(C)))
U10_gga(Amount, Coins, plus_out_gag(s(X5))) → U11_gga(ways_in_gga(Amount, Coins))
U11_gga(ways_out_gga(N)) → ways_out_gga(N)
U6_gga(C, Coins, NewAmount, ways_out_gga(N1)) → U7_gga(N1, ways_in_gga(NewAmount, .(s(C), Coins)))
U7_gga(N1, ways_out_gga(N2)) → U8_gga(plus_in_gga(N1, N2))
plus_in_gga(0, X) → U2_gga(X, nat_in_g(X))
U2_gga(X, nat_out_g) → plus_out_gga(X)
plus_in_gga(s(X), Y) → U3_gga(plus_in_gga(X, Y))
U3_gga(plus_out_gga(Z)) → plus_out_gga(s(Z))
U8_gga(plus_out_gga(N)) → ways_out_gga(N)

The set Q consists of the following terms:

ways_in_gga(x0, x1)
=_in_ga(x0)
U4_gga(x0, x1, x2, x3)
plus_in_gag(x0, x1)
nat_in_g(x0)
U1_g(x0)
U2_gag(x0, x1)
U3_gag(x0)
U5_gga(x0, x1, x2, x3)
U9_gga(x0, x1, x2, x3)
U10_gga(x0, x1, x2)
U11_gga(x0)
U6_gga(x0, x1, x2, x3)
U7_gga(x0, x1)
plus_in_gga(x0, x1)
U2_gga(x0, x1)
U3_gga(x0)
U8_gga(x0)

We have to consider all (P,Q,R)-chains.

(77) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(0) Obligation:

(1) PredefinedPredicateTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (EQUIVALENT transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) TRUE

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (SOUND transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) TRUE

(23) Obligation:

(24) UsableRulesProof (EQUIVALENT transformation)

(25) Obligation:

(26) PiDPToQDPProof (SOUND transformation)

(27) Obligation:

(28) QDPSizeChangeProof (EQUIVALENT transformation)

(29) TRUE

(30) Obligation:

(31) PiDPToQDPProof (SOUND transformation)

(32) Obligation:

(33) QDPOrderProof (EQUIVALENT transformation)

(34) Obligation:

(35) DependencyGraphProof (EQUIVALENT transformation)

(36) Obligation:

(37) QDPOrderProof (EQUIVALENT transformation)

(38) Obligation:

(39) PrologToPiTRSProof (SOUND transformation)

(40) Obligation:

(41) DependencyPairsProof (EQUIVALENT transformation)

(42) Obligation:

(43) DependencyGraphProof (EQUIVALENT transformation)

(44) Complex Obligation (AND)

(45) Obligation:

(46) UsableRulesProof (EQUIVALENT transformation)

(47) Obligation:

(48) PiDPToQDPProof (EQUIVALENT transformation)

(49) Obligation:

(50) QDPSizeChangeProof (EQUIVALENT transformation)

(51) TRUE

(52) Obligation:

(53) UsableRulesProof (EQUIVALENT transformation)

(54) Obligation:

(55) PiDPToQDPProof (SOUND transformation)

(56) Obligation:

(57) QDPSizeChangeProof (EQUIVALENT transformation)

(58) TRUE

(59) Obligation:

(60) UsableRulesProof (EQUIVALENT transformation)

(61) Obligation:

(62) PiDPToQDPProof (SOUND transformation)

(63) Obligation:

(64) QDPSizeChangeProof (EQUIVALENT transformation)

(65) TRUE

(66) Obligation:

(67) PiDPToQDPProof (SOUND transformation)

(68) Obligation:

(69) QDPOrderProof (EQUIVALENT transformation)

(70) Obligation:

(71) DependencyGraphProof (EQUIVALENT transformation)

(72) Obligation:

(73) QDPOrderProof (EQUIVALENT transformation)

(74) Obligation:

(75) QDPOrderProof (EQUIVALENT transformation)

(76) Obligation:

(77) DependencyGraphProof (EQUIVALENT transformation)

(78) TRUE