Left Termination of the query pattern fib_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 AND
      ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇐)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 TRUE
      ↳14 PiDP
  ↳15 PrologToPiTRSProof (⇐)
    ↳16 PiTRS
    ↳17 DependencyPairsProof (⇔)
    ↳18 PiDP
    ↳19 DependencyGraphProof (⇔)
    ↳20 AND
      ↳21 PiDP
        ↳22 UsableRulesProof (⇔)
        ↳23 PiDP
        ↳24 PiDPToQDPProof (⇐)
        ↳25 QDP
        ↳26 QDPSizeChangeProof (⇔)
        ↳27 TRUE
      ↳28 PiDP
        ↳29 PiDPToQDPProof (⇐)
        ↳30 QDP
        ↳31 QDPSizeChangeProof (⇔)
        ↳32 TRUE

(0) Obligation:

Clauses:

add(0, 0, 0).
add(s(X), Y, s(N)) :- add(X, Y, N).
add(X, s(Y), s(N)) :- add(X, Y, N).
fib(0, 0).
fib(s(0), s(0)).
fib(s(s(X)), N) :- ','(fib(s(X), N1), ','(fib(X, N2), add(N1, N2, N))).

Queries:

fib(g,a).

(1) PrologToPiTRSProof (SOUND transformation)

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

fib_in_ga(0, 0) → fib_out_ga(0, 0)
fib_in_ga(s(0), s(0)) → fib_out_ga(s(0), s(0))
fib_in_ga(s(s(X)), N) → U3_ga(X, N, fib_in_ga(s(X), N1))
U3_ga(X, N, fib_out_ga(s(X), N1)) → U4_ga(X, N, N1, fib_in_ga(X, N2))
U4_ga(X, N, N1, fib_out_ga(X, N2)) → U5_ga(X, N, add_in_gga(N1, N2, N))
add_in_gga(0, 0, 0) → add_out_gga(0, 0, 0)
add_in_gga(s(X), Y, s(N)) → U1_gga(X, Y, N, add_in_gga(X, Y, N))
add_in_gga(X, s(Y), s(N)) → U2_gga(X, Y, N, add_in_gga(X, Y, N))
U2_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(X, s(Y), s(N))
U1_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(s(X), Y, s(N))
U5_ga(X, N, add_out_gga(N1, N2, N)) → fib_out_ga(s(s(X)), N)

The argument filtering Pi contains the following mapping:
fib_in_ga(x1, x2)  =  fib_in_ga(x1)
0  =  0
fib_out_ga(x1, x2)  =  fib_out_ga(x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x3, x4)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

fib_in_ga(0, 0) → fib_out_ga(0, 0)
fib_in_ga(s(0), s(0)) → fib_out_ga(s(0), s(0))
fib_in_ga(s(s(X)), N) → U3_ga(X, N, fib_in_ga(s(X), N1))
U3_ga(X, N, fib_out_ga(s(X), N1)) → U4_ga(X, N, N1, fib_in_ga(X, N2))
U4_ga(X, N, N1, fib_out_ga(X, N2)) → U5_ga(X, N, add_in_gga(N1, N2, N))
add_in_gga(0, 0, 0) → add_out_gga(0, 0, 0)
add_in_gga(s(X), Y, s(N)) → U1_gga(X, Y, N, add_in_gga(X, Y, N))
add_in_gga(X, s(Y), s(N)) → U2_gga(X, Y, N, add_in_gga(X, Y, N))
U2_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(X, s(Y), s(N))
U1_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(s(X), Y, s(N))
U5_ga(X, N, add_out_gga(N1, N2, N)) → fib_out_ga(s(s(X)), N)

The argument filtering Pi contains the following mapping:
fib_in_ga(x1, x2)  =  fib_in_ga(x1)
0  =  0
fib_out_ga(x1, x2)  =  fib_out_ga(x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x3, x4)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)

(3) 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:

FIB_IN_GA(s(s(X)), N) → U3_GA(X, N, fib_in_ga(s(X), N1))
FIB_IN_GA(s(s(X)), N) → FIB_IN_GA(s(X), N1)
U3_GA(X, N, fib_out_ga(s(X), N1)) → U4_GA(X, N, N1, fib_in_ga(X, N2))
U3_GA(X, N, fib_out_ga(s(X), N1)) → FIB_IN_GA(X, N2)
U4_GA(X, N, N1, fib_out_ga(X, N2)) → U5_GA(X, N, add_in_gga(N1, N2, N))
U4_GA(X, N, N1, fib_out_ga(X, N2)) → ADD_IN_GGA(N1, N2, N)
ADD_IN_GGA(s(X), Y, s(N)) → U1_GGA(X, Y, N, add_in_gga(X, Y, N))
ADD_IN_GGA(s(X), Y, s(N)) → ADD_IN_GGA(X, Y, N)
ADD_IN_GGA(X, s(Y), s(N)) → U2_GGA(X, Y, N, add_in_gga(X, Y, N))
ADD_IN_GGA(X, s(Y), s(N)) → ADD_IN_GGA(X, Y, N)

The TRS R consists of the following rules:

fib_in_ga(0, 0) → fib_out_ga(0, 0)
fib_in_ga(s(0), s(0)) → fib_out_ga(s(0), s(0))
fib_in_ga(s(s(X)), N) → U3_ga(X, N, fib_in_ga(s(X), N1))
U3_ga(X, N, fib_out_ga(s(X), N1)) → U4_ga(X, N, N1, fib_in_ga(X, N2))
U4_ga(X, N, N1, fib_out_ga(X, N2)) → U5_ga(X, N, add_in_gga(N1, N2, N))
add_in_gga(0, 0, 0) → add_out_gga(0, 0, 0)
add_in_gga(s(X), Y, s(N)) → U1_gga(X, Y, N, add_in_gga(X, Y, N))
add_in_gga(X, s(Y), s(N)) → U2_gga(X, Y, N, add_in_gga(X, Y, N))
U2_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(X, s(Y), s(N))
U1_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(s(X), Y, s(N))
U5_ga(X, N, add_out_gga(N1, N2, N)) → fib_out_ga(s(s(X)), N)

The argument filtering Pi contains the following mapping:
fib_in_ga(x1, x2)  =  fib_in_ga(x1)
0  =  0
fib_out_ga(x1, x2)  =  fib_out_ga(x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x3, x4)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
FIB_IN_GA(x1, x2)  =  FIB_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x3, x4)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x4)

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

(4) Obligation:

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

FIB_IN_GA(s(s(X)), N) → U3_GA(X, N, fib_in_ga(s(X), N1))
FIB_IN_GA(s(s(X)), N) → FIB_IN_GA(s(X), N1)
U3_GA(X, N, fib_out_ga(s(X), N1)) → U4_GA(X, N, N1, fib_in_ga(X, N2))
U3_GA(X, N, fib_out_ga(s(X), N1)) → FIB_IN_GA(X, N2)
U4_GA(X, N, N1, fib_out_ga(X, N2)) → U5_GA(X, N, add_in_gga(N1, N2, N))
U4_GA(X, N, N1, fib_out_ga(X, N2)) → ADD_IN_GGA(N1, N2, N)
ADD_IN_GGA(s(X), Y, s(N)) → U1_GGA(X, Y, N, add_in_gga(X, Y, N))
ADD_IN_GGA(s(X), Y, s(N)) → ADD_IN_GGA(X, Y, N)
ADD_IN_GGA(X, s(Y), s(N)) → U2_GGA(X, Y, N, add_in_gga(X, Y, N))
ADD_IN_GGA(X, s(Y), s(N)) → ADD_IN_GGA(X, Y, N)

The TRS R consists of the following rules:

fib_in_ga(0, 0) → fib_out_ga(0, 0)
fib_in_ga(s(0), s(0)) → fib_out_ga(s(0), s(0))
fib_in_ga(s(s(X)), N) → U3_ga(X, N, fib_in_ga(s(X), N1))
U3_ga(X, N, fib_out_ga(s(X), N1)) → U4_ga(X, N, N1, fib_in_ga(X, N2))
U4_ga(X, N, N1, fib_out_ga(X, N2)) → U5_ga(X, N, add_in_gga(N1, N2, N))
add_in_gga(0, 0, 0) → add_out_gga(0, 0, 0)
add_in_gga(s(X), Y, s(N)) → U1_gga(X, Y, N, add_in_gga(X, Y, N))
add_in_gga(X, s(Y), s(N)) → U2_gga(X, Y, N, add_in_gga(X, Y, N))
U2_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(X, s(Y), s(N))
U1_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(s(X), Y, s(N))
U5_ga(X, N, add_out_gga(N1, N2, N)) → fib_out_ga(s(s(X)), N)

The argument filtering Pi contains the following mapping:
fib_in_ga(x1, x2)  =  fib_in_ga(x1)
0  =  0
fib_out_ga(x1, x2)  =  fib_out_ga(x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x3, x4)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
FIB_IN_GA(x1, x2)  =  FIB_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x3, x4)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

ADD_IN_GGA(X, s(Y), s(N)) → ADD_IN_GGA(X, Y, N)
ADD_IN_GGA(s(X), Y, s(N)) → ADD_IN_GGA(X, Y, N)

The TRS R consists of the following rules:

fib_in_ga(0, 0) → fib_out_ga(0, 0)
fib_in_ga(s(0), s(0)) → fib_out_ga(s(0), s(0))
fib_in_ga(s(s(X)), N) → U3_ga(X, N, fib_in_ga(s(X), N1))
U3_ga(X, N, fib_out_ga(s(X), N1)) → U4_ga(X, N, N1, fib_in_ga(X, N2))
U4_ga(X, N, N1, fib_out_ga(X, N2)) → U5_ga(X, N, add_in_gga(N1, N2, N))
add_in_gga(0, 0, 0) → add_out_gga(0, 0, 0)
add_in_gga(s(X), Y, s(N)) → U1_gga(X, Y, N, add_in_gga(X, Y, N))
add_in_gga(X, s(Y), s(N)) → U2_gga(X, Y, N, add_in_gga(X, Y, N))
U2_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(X, s(Y), s(N))
U1_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(s(X), Y, s(N))
U5_ga(X, N, add_out_gga(N1, N2, N)) → fib_out_ga(s(s(X)), N)

The argument filtering Pi contains the following mapping:
fib_in_ga(x1, x2)  =  fib_in_ga(x1)
0  =  0
fib_out_ga(x1, x2)  =  fib_out_ga(x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x3, x4)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

ADD_IN_GGA(X, s(Y), s(N)) → ADD_IN_GGA(X, Y, N)
ADD_IN_GGA(s(X), Y, s(N)) → ADD_IN_GGA(X, Y, N)

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

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

ADD_IN_GGA(X, s(Y)) → ADD_IN_GGA(X, Y)
ADD_IN_GGA(s(X), Y) → ADD_IN_GGA(X, Y)

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

(12) 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:

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

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

(13) TRUE

(14) Obligation:

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

U3_GA(X, N, fib_out_ga(s(X), N1)) → FIB_IN_GA(X, N2)
FIB_IN_GA(s(s(X)), N) → U3_GA(X, N, fib_in_ga(s(X), N1))
FIB_IN_GA(s(s(X)), N) → FIB_IN_GA(s(X), N1)

The TRS R consists of the following rules:

fib_in_ga(0, 0) → fib_out_ga(0, 0)
fib_in_ga(s(0), s(0)) → fib_out_ga(s(0), s(0))
fib_in_ga(s(s(X)), N) → U3_ga(X, N, fib_in_ga(s(X), N1))
U3_ga(X, N, fib_out_ga(s(X), N1)) → U4_ga(X, N, N1, fib_in_ga(X, N2))
U4_ga(X, N, N1, fib_out_ga(X, N2)) → U5_ga(X, N, add_in_gga(N1, N2, N))
add_in_gga(0, 0, 0) → add_out_gga(0, 0, 0)
add_in_gga(s(X), Y, s(N)) → U1_gga(X, Y, N, add_in_gga(X, Y, N))
add_in_gga(X, s(Y), s(N)) → U2_gga(X, Y, N, add_in_gga(X, Y, N))
U2_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(X, s(Y), s(N))
U1_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(s(X), Y, s(N))
U5_ga(X, N, add_out_gga(N1, N2, N)) → fib_out_ga(s(s(X)), N)

The argument filtering Pi contains the following mapping:
fib_in_ga(x1, x2)  =  fib_in_ga(x1)
0  =  0
fib_out_ga(x1, x2)  =  fib_out_ga(x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x3, x4)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
FIB_IN_GA(x1, x2)  =  FIB_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)

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

(15) PrologToPiTRSProof (SOUND transformation)

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

fib_in_ga(0, 0) → fib_out_ga(0, 0)
fib_in_ga(s(0), s(0)) → fib_out_ga(s(0), s(0))
fib_in_ga(s(s(X)), N) → U3_ga(X, N, fib_in_ga(s(X), N1))
U3_ga(X, N, fib_out_ga(s(X), N1)) → U4_ga(X, N, N1, fib_in_ga(X, N2))
U4_ga(X, N, N1, fib_out_ga(X, N2)) → U5_ga(X, N, add_in_gga(N1, N2, N))
add_in_gga(0, 0, 0) → add_out_gga(0, 0, 0)
add_in_gga(s(X), Y, s(N)) → U1_gga(X, Y, N, add_in_gga(X, Y, N))
add_in_gga(X, s(Y), s(N)) → U2_gga(X, Y, N, add_in_gga(X, Y, N))
U2_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(X, s(Y), s(N))
U1_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(s(X), Y, s(N))
U5_ga(X, N, add_out_gga(N1, N2, N)) → fib_out_ga(s(s(X)), N)

The argument filtering Pi contains the following mapping:
fib_in_ga(x1, x2)  =  fib_in_ga(x1)
0  =  0
fib_out_ga(x1, x2)  =  fib_out_ga(x1, x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x3, x4)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(16) Obligation:

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

fib_in_ga(0, 0) → fib_out_ga(0, 0)
fib_in_ga(s(0), s(0)) → fib_out_ga(s(0), s(0))
fib_in_ga(s(s(X)), N) → U3_ga(X, N, fib_in_ga(s(X), N1))
U3_ga(X, N, fib_out_ga(s(X), N1)) → U4_ga(X, N, N1, fib_in_ga(X, N2))
U4_ga(X, N, N1, fib_out_ga(X, N2)) → U5_ga(X, N, add_in_gga(N1, N2, N))
add_in_gga(0, 0, 0) → add_out_gga(0, 0, 0)
add_in_gga(s(X), Y, s(N)) → U1_gga(X, Y, N, add_in_gga(X, Y, N))
add_in_gga(X, s(Y), s(N)) → U2_gga(X, Y, N, add_in_gga(X, Y, N))
U2_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(X, s(Y), s(N))
U1_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(s(X), Y, s(N))
U5_ga(X, N, add_out_gga(N1, N2, N)) → fib_out_ga(s(s(X)), N)

The argument filtering Pi contains the following mapping:
fib_in_ga(x1, x2)  =  fib_in_ga(x1)
0  =  0
fib_out_ga(x1, x2)  =  fib_out_ga(x1, x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x3, x4)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)

(17) 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:

FIB_IN_GA(s(s(X)), N) → U3_GA(X, N, fib_in_ga(s(X), N1))
FIB_IN_GA(s(s(X)), N) → FIB_IN_GA(s(X), N1)
U3_GA(X, N, fib_out_ga(s(X), N1)) → U4_GA(X, N, N1, fib_in_ga(X, N2))
U3_GA(X, N, fib_out_ga(s(X), N1)) → FIB_IN_GA(X, N2)
U4_GA(X, N, N1, fib_out_ga(X, N2)) → U5_GA(X, N, add_in_gga(N1, N2, N))
U4_GA(X, N, N1, fib_out_ga(X, N2)) → ADD_IN_GGA(N1, N2, N)
ADD_IN_GGA(s(X), Y, s(N)) → U1_GGA(X, Y, N, add_in_gga(X, Y, N))
ADD_IN_GGA(s(X), Y, s(N)) → ADD_IN_GGA(X, Y, N)
ADD_IN_GGA(X, s(Y), s(N)) → U2_GGA(X, Y, N, add_in_gga(X, Y, N))
ADD_IN_GGA(X, s(Y), s(N)) → ADD_IN_GGA(X, Y, N)

The TRS R consists of the following rules:

fib_in_ga(0, 0) → fib_out_ga(0, 0)
fib_in_ga(s(0), s(0)) → fib_out_ga(s(0), s(0))
fib_in_ga(s(s(X)), N) → U3_ga(X, N, fib_in_ga(s(X), N1))
U3_ga(X, N, fib_out_ga(s(X), N1)) → U4_ga(X, N, N1, fib_in_ga(X, N2))
U4_ga(X, N, N1, fib_out_ga(X, N2)) → U5_ga(X, N, add_in_gga(N1, N2, N))
add_in_gga(0, 0, 0) → add_out_gga(0, 0, 0)
add_in_gga(s(X), Y, s(N)) → U1_gga(X, Y, N, add_in_gga(X, Y, N))
add_in_gga(X, s(Y), s(N)) → U2_gga(X, Y, N, add_in_gga(X, Y, N))
U2_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(X, s(Y), s(N))
U1_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(s(X), Y, s(N))
U5_ga(X, N, add_out_gga(N1, N2, N)) → fib_out_ga(s(s(X)), N)

The argument filtering Pi contains the following mapping:
fib_in_ga(x1, x2)  =  fib_in_ga(x1)
0  =  0
fib_out_ga(x1, x2)  =  fib_out_ga(x1, x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x3, x4)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
FIB_IN_GA(x1, x2)  =  FIB_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x3, x4)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)

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

(18) Obligation:

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

FIB_IN_GA(s(s(X)), N) → U3_GA(X, N, fib_in_ga(s(X), N1))
FIB_IN_GA(s(s(X)), N) → FIB_IN_GA(s(X), N1)
U3_GA(X, N, fib_out_ga(s(X), N1)) → U4_GA(X, N, N1, fib_in_ga(X, N2))
U3_GA(X, N, fib_out_ga(s(X), N1)) → FIB_IN_GA(X, N2)
U4_GA(X, N, N1, fib_out_ga(X, N2)) → U5_GA(X, N, add_in_gga(N1, N2, N))
U4_GA(X, N, N1, fib_out_ga(X, N2)) → ADD_IN_GGA(N1, N2, N)
ADD_IN_GGA(s(X), Y, s(N)) → U1_GGA(X, Y, N, add_in_gga(X, Y, N))
ADD_IN_GGA(s(X), Y, s(N)) → ADD_IN_GGA(X, Y, N)
ADD_IN_GGA(X, s(Y), s(N)) → U2_GGA(X, Y, N, add_in_gga(X, Y, N))
ADD_IN_GGA(X, s(Y), s(N)) → ADD_IN_GGA(X, Y, N)

The TRS R consists of the following rules:

fib_in_ga(0, 0) → fib_out_ga(0, 0)
fib_in_ga(s(0), s(0)) → fib_out_ga(s(0), s(0))
fib_in_ga(s(s(X)), N) → U3_ga(X, N, fib_in_ga(s(X), N1))
U3_ga(X, N, fib_out_ga(s(X), N1)) → U4_ga(X, N, N1, fib_in_ga(X, N2))
U4_ga(X, N, N1, fib_out_ga(X, N2)) → U5_ga(X, N, add_in_gga(N1, N2, N))
add_in_gga(0, 0, 0) → add_out_gga(0, 0, 0)
add_in_gga(s(X), Y, s(N)) → U1_gga(X, Y, N, add_in_gga(X, Y, N))
add_in_gga(X, s(Y), s(N)) → U2_gga(X, Y, N, add_in_gga(X, Y, N))
U2_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(X, s(Y), s(N))
U1_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(s(X), Y, s(N))
U5_ga(X, N, add_out_gga(N1, N2, N)) → fib_out_ga(s(s(X)), N)

The argument filtering Pi contains the following mapping:
fib_in_ga(x1, x2)  =  fib_in_ga(x1)
0  =  0
fib_out_ga(x1, x2)  =  fib_out_ga(x1, x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x3, x4)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
FIB_IN_GA(x1, x2)  =  FIB_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x3, x4)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)

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

(19) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.

(20) Complex Obligation (AND)

(21) Obligation:

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

ADD_IN_GGA(X, s(Y), s(N)) → ADD_IN_GGA(X, Y, N)
ADD_IN_GGA(s(X), Y, s(N)) → ADD_IN_GGA(X, Y, N)

The TRS R consists of the following rules:

fib_in_ga(0, 0) → fib_out_ga(0, 0)
fib_in_ga(s(0), s(0)) → fib_out_ga(s(0), s(0))
fib_in_ga(s(s(X)), N) → U3_ga(X, N, fib_in_ga(s(X), N1))
U3_ga(X, N, fib_out_ga(s(X), N1)) → U4_ga(X, N, N1, fib_in_ga(X, N2))
U4_ga(X, N, N1, fib_out_ga(X, N2)) → U5_ga(X, N, add_in_gga(N1, N2, N))
add_in_gga(0, 0, 0) → add_out_gga(0, 0, 0)
add_in_gga(s(X), Y, s(N)) → U1_gga(X, Y, N, add_in_gga(X, Y, N))
add_in_gga(X, s(Y), s(N)) → U2_gga(X, Y, N, add_in_gga(X, Y, N))
U2_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(X, s(Y), s(N))
U1_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(s(X), Y, s(N))
U5_ga(X, N, add_out_gga(N1, N2, N)) → fib_out_ga(s(s(X)), N)

The argument filtering Pi contains the following mapping:
fib_in_ga(x1, x2)  =  fib_in_ga(x1)
0  =  0
fib_out_ga(x1, x2)  =  fib_out_ga(x1, x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x3, x4)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

ADD_IN_GGA(X, s(Y), s(N)) → ADD_IN_GGA(X, Y, N)
ADD_IN_GGA(s(X), Y, s(N)) → ADD_IN_GGA(X, Y, N)

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

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

ADD_IN_GGA(X, s(Y)) → ADD_IN_GGA(X, Y)
ADD_IN_GGA(s(X), Y) → ADD_IN_GGA(X, Y)

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

(26) 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:

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

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

(27) TRUE

(28) Obligation:

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

U3_GA(X, N, fib_out_ga(s(X), N1)) → FIB_IN_GA(X, N2)
FIB_IN_GA(s(s(X)), N) → U3_GA(X, N, fib_in_ga(s(X), N1))
FIB_IN_GA(s(s(X)), N) → FIB_IN_GA(s(X), N1)

The TRS R consists of the following rules:

fib_in_ga(0, 0) → fib_out_ga(0, 0)
fib_in_ga(s(0), s(0)) → fib_out_ga(s(0), s(0))
fib_in_ga(s(s(X)), N) → U3_ga(X, N, fib_in_ga(s(X), N1))
U3_ga(X, N, fib_out_ga(s(X), N1)) → U4_ga(X, N, N1, fib_in_ga(X, N2))
U4_ga(X, N, N1, fib_out_ga(X, N2)) → U5_ga(X, N, add_in_gga(N1, N2, N))
add_in_gga(0, 0, 0) → add_out_gga(0, 0, 0)
add_in_gga(s(X), Y, s(N)) → U1_gga(X, Y, N, add_in_gga(X, Y, N))
add_in_gga(X, s(Y), s(N)) → U2_gga(X, Y, N, add_in_gga(X, Y, N))
U2_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(X, s(Y), s(N))
U1_gga(X, Y, N, add_out_gga(X, Y, N)) → add_out_gga(s(X), Y, s(N))
U5_ga(X, N, add_out_gga(N1, N2, N)) → fib_out_ga(s(s(X)), N)

The argument filtering Pi contains the following mapping:
fib_in_ga(x1, x2)  =  fib_in_ga(x1)
0  =  0
fib_out_ga(x1, x2)  =  fib_out_ga(x1, x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x3, x4)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)
add_out_gga(x1, x2, x3)  =  add_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
FIB_IN_GA(x1, x2)  =  FIB_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)

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

(29) PiDPToQDPProof (SOUND transformation)

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

(30) Obligation:

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

U3_GA(X, fib_out_ga(s(X), N1)) → FIB_IN_GA(X)
FIB_IN_GA(s(s(X))) → U3_GA(X, fib_in_ga(s(X)))
FIB_IN_GA(s(s(X))) → FIB_IN_GA(s(X))

The TRS R consists of the following rules:

fib_in_ga(0) → fib_out_ga(0, 0)
fib_in_ga(s(0)) → fib_out_ga(s(0), s(0))
fib_in_ga(s(s(X))) → U3_ga(X, fib_in_ga(s(X)))
U3_ga(X, fib_out_ga(s(X), N1)) → U4_ga(X, N1, fib_in_ga(X))
U4_ga(X, N1, fib_out_ga(X, N2)) → U5_ga(X, add_in_gga(N1, N2))
add_in_gga(0, 0) → add_out_gga(0, 0, 0)
add_in_gga(s(X), Y) → U1_gga(X, Y, add_in_gga(X, Y))
add_in_gga(X, s(Y)) → U2_gga(X, Y, add_in_gga(X, Y))
U2_gga(X, Y, add_out_gga(X, Y, N)) → add_out_gga(X, s(Y), s(N))
U1_gga(X, Y, add_out_gga(X, Y, N)) → add_out_gga(s(X), Y, s(N))
U5_ga(X, add_out_gga(N1, N2, N)) → fib_out_ga(s(s(X)), N)

The set Q consists of the following terms:

fib_in_ga(x0)
U3_ga(x0, x1)
U4_ga(x0, x1, x2)
add_in_gga(x0, x1)
U2_gga(x0, x1, x2)
U1_gga(x0, x1, x2)
U5_ga(x0, x1)

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

(31) 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:

  • FIB_IN_GA(s(s(X))) → U3_GA(X, fib_in_ga(s(X)))
    The graph contains the following edges 1 > 1

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

  • U3_GA(X, fib_out_ga(s(X), N1)) → FIB_IN_GA(X)
    The graph contains the following edges 1 >= 1, 2 > 1

(32) TRUE