Left Termination of the query pattern p_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:



↳ Prolog
  ↳ PrologToPiTRSProof

Clauses:

p(X, g(X)).
p(X, f(Y)) :- p(X, g(Y)).

Queries:

p(g,a).

We use the technique of [30].Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p_in(X, f(Y)) → U1(X, Y, p_in(X, g(Y)))
p_in(X, g(X)) → p_out(X, g(X))
U1(X, Y, p_out(X, g(Y))) → p_out(X, f(Y))

The argument filtering Pi contains the following mapping:
p_in(x1, x2)  =  p_in(x1)
f(x1)  =  f(x1)
U1(x1, x2, x3)  =  U1(x3)
g(x1)  =  g(x1)
p_out(x1, x2)  =  p_out(x2)

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:

p_in(X, f(Y)) → U1(X, Y, p_in(X, g(Y)))
p_in(X, g(X)) → p_out(X, g(X))
U1(X, Y, p_out(X, g(Y))) → p_out(X, f(Y))

The argument filtering Pi contains the following mapping:
p_in(x1, x2)  =  p_in(x1)
f(x1)  =  f(x1)
U1(x1, x2, x3)  =  U1(x3)
g(x1)  =  g(x1)
p_out(x1, x2)  =  p_out(x2)


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

P_IN(X, f(Y)) → U11(X, Y, p_in(X, g(Y)))
P_IN(X, f(Y)) → P_IN(X, g(Y))

The TRS R consists of the following rules:

p_in(X, f(Y)) → U1(X, Y, p_in(X, g(Y)))
p_in(X, g(X)) → p_out(X, g(X))
U1(X, Y, p_out(X, g(Y))) → p_out(X, f(Y))

The argument filtering Pi contains the following mapping:
p_in(x1, x2)  =  p_in(x1)
f(x1)  =  f(x1)
U1(x1, x2, x3)  =  U1(x3)
g(x1)  =  g(x1)
p_out(x1, x2)  =  p_out(x2)
P_IN(x1, x2)  =  P_IN(x1)
U11(x1, x2, x3)  =  U11(x3)

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:

P_IN(X, f(Y)) → U11(X, Y, p_in(X, g(Y)))
P_IN(X, f(Y)) → P_IN(X, g(Y))

The TRS R consists of the following rules:

p_in(X, f(Y)) → U1(X, Y, p_in(X, g(Y)))
p_in(X, g(X)) → p_out(X, g(X))
U1(X, Y, p_out(X, g(Y))) → p_out(X, f(Y))

The argument filtering Pi contains the following mapping:
p_in(x1, x2)  =  p_in(x1)
f(x1)  =  f(x1)
U1(x1, x2, x3)  =  U1(x3)
g(x1)  =  g(x1)
p_out(x1, x2)  =  p_out(x2)
P_IN(x1, x2)  =  P_IN(x1)
U11(x1, x2, x3)  =  U11(x3)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 0 SCCs with 2 less nodes.