Left Termination of the query pattern f_in_1(g) w.r.t. the given Prolog program could successfully be proven:



Prolog
  ↳ PrologToPiTRSProof

Clauses:

f(X) :- g(s(s(s(X)))).
f(s(X)) :- f(X).
g(s(s(s(s(X))))) :- f(X).

Queries:

f(g).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
f_in: (b)
g_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

f_in_g(X) → U1_g(X, g_in_g(s(s(s(X)))))
g_in_g(s(s(s(s(X))))) → U3_g(X, f_in_g(X))
f_in_g(s(X)) → U2_g(X, f_in_g(X))
U2_g(X, f_out_g(X)) → f_out_g(s(X))
U3_g(X, f_out_g(X)) → g_out_g(s(s(s(s(X)))))
U1_g(X, g_out_g(s(s(s(X))))) → f_out_g(X)

The argument filtering Pi contains the following mapping:
f_in_g(x1)  =  f_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
g_in_g(x1)  =  g_in_g(x1)
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
U2_g(x1, x2)  =  U2_g(x2)
f_out_g(x1)  =  f_out_g
g_out_g(x1)  =  g_out_g

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:

f_in_g(X) → U1_g(X, g_in_g(s(s(s(X)))))
g_in_g(s(s(s(s(X))))) → U3_g(X, f_in_g(X))
f_in_g(s(X)) → U2_g(X, f_in_g(X))
U2_g(X, f_out_g(X)) → f_out_g(s(X))
U3_g(X, f_out_g(X)) → g_out_g(s(s(s(s(X)))))
U1_g(X, g_out_g(s(s(s(X))))) → f_out_g(X)

The argument filtering Pi contains the following mapping:
f_in_g(x1)  =  f_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
g_in_g(x1)  =  g_in_g(x1)
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
U2_g(x1, x2)  =  U2_g(x2)
f_out_g(x1)  =  f_out_g
g_out_g(x1)  =  g_out_g


Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

F_IN_G(X) → U1_G(X, g_in_g(s(s(s(X)))))
F_IN_G(X) → G_IN_G(s(s(s(X))))
G_IN_G(s(s(s(s(X))))) → U3_G(X, f_in_g(X))
G_IN_G(s(s(s(s(X))))) → F_IN_G(X)
F_IN_G(s(X)) → U2_G(X, f_in_g(X))
F_IN_G(s(X)) → F_IN_G(X)

The TRS R consists of the following rules:

f_in_g(X) → U1_g(X, g_in_g(s(s(s(X)))))
g_in_g(s(s(s(s(X))))) → U3_g(X, f_in_g(X))
f_in_g(s(X)) → U2_g(X, f_in_g(X))
U2_g(X, f_out_g(X)) → f_out_g(s(X))
U3_g(X, f_out_g(X)) → g_out_g(s(s(s(s(X)))))
U1_g(X, g_out_g(s(s(s(X))))) → f_out_g(X)

The argument filtering Pi contains the following mapping:
f_in_g(x1)  =  f_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
g_in_g(x1)  =  g_in_g(x1)
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
U2_g(x1, x2)  =  U2_g(x2)
f_out_g(x1)  =  f_out_g
g_out_g(x1)  =  g_out_g
U2_G(x1, x2)  =  U2_G(x2)
F_IN_G(x1)  =  F_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U3_G(x1, x2)  =  U3_G(x2)
G_IN_G(x1)  =  G_IN_G(x1)

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:

F_IN_G(X) → U1_G(X, g_in_g(s(s(s(X)))))
F_IN_G(X) → G_IN_G(s(s(s(X))))
G_IN_G(s(s(s(s(X))))) → U3_G(X, f_in_g(X))
G_IN_G(s(s(s(s(X))))) → F_IN_G(X)
F_IN_G(s(X)) → U2_G(X, f_in_g(X))
F_IN_G(s(X)) → F_IN_G(X)

The TRS R consists of the following rules:

f_in_g(X) → U1_g(X, g_in_g(s(s(s(X)))))
g_in_g(s(s(s(s(X))))) → U3_g(X, f_in_g(X))
f_in_g(s(X)) → U2_g(X, f_in_g(X))
U2_g(X, f_out_g(X)) → f_out_g(s(X))
U3_g(X, f_out_g(X)) → g_out_g(s(s(s(s(X)))))
U1_g(X, g_out_g(s(s(s(X))))) → f_out_g(X)

The argument filtering Pi contains the following mapping:
f_in_g(x1)  =  f_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
g_in_g(x1)  =  g_in_g(x1)
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
U2_g(x1, x2)  =  U2_g(x2)
f_out_g(x1)  =  f_out_g
g_out_g(x1)  =  g_out_g
U2_G(x1, x2)  =  U2_G(x2)
F_IN_G(x1)  =  F_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U3_G(x1, x2)  =  U3_G(x2)
G_IN_G(x1)  =  G_IN_G(x1)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 3 less nodes.

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

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

G_IN_G(s(s(s(s(X))))) → F_IN_G(X)
F_IN_G(s(X)) → F_IN_G(X)
F_IN_G(X) → G_IN_G(s(s(s(X))))

The TRS R consists of the following rules:

f_in_g(X) → U1_g(X, g_in_g(s(s(s(X)))))
g_in_g(s(s(s(s(X))))) → U3_g(X, f_in_g(X))
f_in_g(s(X)) → U2_g(X, f_in_g(X))
U2_g(X, f_out_g(X)) → f_out_g(s(X))
U3_g(X, f_out_g(X)) → g_out_g(s(s(s(s(X)))))
U1_g(X, g_out_g(s(s(s(X))))) → f_out_g(X)

The argument filtering Pi contains the following mapping:
f_in_g(x1)  =  f_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
g_in_g(x1)  =  g_in_g(x1)
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
U2_g(x1, x2)  =  U2_g(x2)
f_out_g(x1)  =  f_out_g
g_out_g(x1)  =  g_out_g
F_IN_G(x1)  =  F_IN_G(x1)
G_IN_G(x1)  =  G_IN_G(x1)

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
            ↳ PiDP
              ↳ UsableRulesProof
PiDP
                  ↳ PiDPToQDPProof

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

G_IN_G(s(s(s(s(X))))) → F_IN_G(X)
F_IN_G(s(X)) → F_IN_G(X)
F_IN_G(X) → G_IN_G(s(s(s(X))))

R is empty.
Pi is empty.
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
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
QDP
                      ↳ RFCMatchBoundsDPProof

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

G_IN_G(s(s(s(s(X))))) → F_IN_G(X)
F_IN_G(s(X)) → F_IN_G(X)
F_IN_G(X) → G_IN_G(s(s(s(X))))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
Termination of the TRS P cup R can be shown by a matchbound [6,7] of 2. This implies finiteness of the given DP problem.
The following rules (P cup R) were used to construct the certificate:

G_IN_G(s(s(s(s(X))))) → F_IN_G(X)
F_IN_G(s(X)) → F_IN_G(X)
F_IN_G(X) → G_IN_G(s(s(s(X))))

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

38, 39, 40, 42, 41, 43, 45, 44, 48, 47, 46

Node 38 is start node and node 39 is final node.

Those nodes are connect through the following edges: