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



↳ Prolog
  ↳ PrologToPiTRSProof

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).

We use the technique of [30]. 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



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

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)


Using Dependency Pairs [1,30] 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)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x3, x4)
FIB_IN_GA(x1, x2)  =  FIB_IN_GA(x1)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x4)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)

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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof

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)
ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x3, x4)
FIB_IN_GA(x1, x2)  =  FIB_IN_GA(x1)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x4)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 5 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP

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
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP

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.
By using the subterm criterion [20] together with the size-change analysis [32] 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: 



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ PiDPToQDPProof

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)) → FIB_IN_GA(X, N2)

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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ PiDPToQDPProof
                  ↳ QDP
                    ↳ QDPSizeChangeProof

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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: