Left Termination of the query pattern p_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 PiDP
    ↳7 UsableRulesProof (⇔)
    ↳8 PiDP
    ↳9 PiDPToQDPProof (⇐)
    ↳10 QDP
    ↳11 UsableRulesReductionPairsProof (⇔)
    ↳12 QDP
    ↳13 DependencyGraphProof (⇔)
    ↳14 TRUE
  ↳15 PrologToPiTRSProof (⇐)
    ↳16 PiTRS
    ↳17 DependencyPairsProof (⇔)
    ↳18 PiDP
    ↳19 DependencyGraphProof (⇔)
    ↳20 PiDP
    ↳21 UsableRulesProof (⇔)
    ↳22 PiDP
    ↳23 PiDPToQDPProof (⇐)
    ↳24 QDP

(0) Obligation:

Clauses:

p(X, Z) :- ','(q(X, Y), p(Y, Z)).
p(X, X).
q(a, b).

Queries:

p(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:
p_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p_in_ga(X, Z) → U1_ga(X, Z, q_in_ga(X, Y))
q_in_ga(a, b) → q_out_ga(a, b)
U1_ga(X, Z, q_out_ga(X, Y)) → U2_ga(X, Z, p_in_ga(Y, Z))
p_in_ga(X, X) → p_out_ga(X, X)
U2_ga(X, Z, p_out_ga(Y, Z)) → p_out_ga(X, Z)

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
q_in_ga(x1, x2)  =  q_in_ga(x1)
a  =  a
q_out_ga(x1, x2)  =  q_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
p_out_ga(x1, x2)  =  p_out_ga(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

p_in_ga(X, Z) → U1_ga(X, Z, q_in_ga(X, Y))
q_in_ga(a, b) → q_out_ga(a, b)
U1_ga(X, Z, q_out_ga(X, Y)) → U2_ga(X, Z, p_in_ga(Y, Z))
p_in_ga(X, X) → p_out_ga(X, X)
U2_ga(X, Z, p_out_ga(Y, Z)) → p_out_ga(X, Z)

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
q_in_ga(x1, x2)  =  q_in_ga(x1)
a  =  a
q_out_ga(x1, x2)  =  q_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
p_out_ga(x1, x2)  =  p_out_ga(x2)

(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:

P_IN_GA(X, Z) → U1_GA(X, Z, q_in_ga(X, Y))
P_IN_GA(X, Z) → Q_IN_GA(X, Y)
U1_GA(X, Z, q_out_ga(X, Y)) → U2_GA(X, Z, p_in_ga(Y, Z))
U1_GA(X, Z, q_out_ga(X, Y)) → P_IN_GA(Y, Z)

The TRS R consists of the following rules:

p_in_ga(X, Z) → U1_ga(X, Z, q_in_ga(X, Y))
q_in_ga(a, b) → q_out_ga(a, b)
U1_ga(X, Z, q_out_ga(X, Y)) → U2_ga(X, Z, p_in_ga(Y, Z))
p_in_ga(X, X) → p_out_ga(X, X)
U2_ga(X, Z, p_out_ga(Y, Z)) → p_out_ga(X, Z)

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
q_in_ga(x1, x2)  =  q_in_ga(x1)
a  =  a
q_out_ga(x1, x2)  =  q_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
p_out_ga(x1, x2)  =  p_out_ga(x2)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
Q_IN_GA(x1, x2)  =  Q_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x3)

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

(4) Obligation:

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

P_IN_GA(X, Z) → U1_GA(X, Z, q_in_ga(X, Y))
P_IN_GA(X, Z) → Q_IN_GA(X, Y)
U1_GA(X, Z, q_out_ga(X, Y)) → U2_GA(X, Z, p_in_ga(Y, Z))
U1_GA(X, Z, q_out_ga(X, Y)) → P_IN_GA(Y, Z)

The TRS R consists of the following rules:

p_in_ga(X, Z) → U1_ga(X, Z, q_in_ga(X, Y))
q_in_ga(a, b) → q_out_ga(a, b)
U1_ga(X, Z, q_out_ga(X, Y)) → U2_ga(X, Z, p_in_ga(Y, Z))
p_in_ga(X, X) → p_out_ga(X, X)
U2_ga(X, Z, p_out_ga(Y, Z)) → p_out_ga(X, Z)

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
q_in_ga(x1, x2)  =  q_in_ga(x1)
a  =  a
q_out_ga(x1, x2)  =  q_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
p_out_ga(x1, x2)  =  p_out_ga(x2)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
Q_IN_GA(x1, x2)  =  Q_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x3)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(6) Obligation:

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

U1_GA(X, Z, q_out_ga(X, Y)) → P_IN_GA(Y, Z)
P_IN_GA(X, Z) → U1_GA(X, Z, q_in_ga(X, Y))

The TRS R consists of the following rules:

p_in_ga(X, Z) → U1_ga(X, Z, q_in_ga(X, Y))
q_in_ga(a, b) → q_out_ga(a, b)
U1_ga(X, Z, q_out_ga(X, Y)) → U2_ga(X, Z, p_in_ga(Y, Z))
p_in_ga(X, X) → p_out_ga(X, X)
U2_ga(X, Z, p_out_ga(Y, Z)) → p_out_ga(X, Z)

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
q_in_ga(x1, x2)  =  q_in_ga(x1)
a  =  a
q_out_ga(x1, x2)  =  q_out_ga(x2)
U2_ga(x1, x2, x3)  =  U2_ga(x3)
p_out_ga(x1, x2)  =  p_out_ga(x2)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)

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

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

U1_GA(X, Z, q_out_ga(X, Y)) → P_IN_GA(Y, Z)
P_IN_GA(X, Z) → U1_GA(X, Z, q_in_ga(X, Y))

The TRS R consists of the following rules:

q_in_ga(a, b) → q_out_ga(a, b)

The argument filtering Pi contains the following mapping:
q_in_ga(x1, x2)  =  q_in_ga(x1)
a  =  a
q_out_ga(x1, x2)  =  q_out_ga(x2)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)

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

(9) PiDPToQDPProof (SOUND transformation)

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

(10) Obligation:

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

U1_GA(q_out_ga(Y)) → P_IN_GA(Y)
P_IN_GA(X) → U1_GA(q_in_ga(X))

The TRS R consists of the following rules:

q_in_ga(a) → q_out_ga(b)

The set Q consists of the following terms:

q_in_ga(x0)

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

(11) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

U1_GA(q_out_ga(Y)) → P_IN_GA(Y)
The following rules are removed from R:

q_in_ga(a) → q_out_ga(b)
Used ordering: POLO with Polynomial interpretation [POLO]:

POL(P_IN_GA(x1)) = 2 + 2·x1   
POL(U1_GA(x1)) = 2 + x1   
POL(a) = 2   
POL(b) = 0   
POL(q_in_ga(x1)) = x1   
POL(q_out_ga(x1)) = 1 + 2·x1   

(12) Obligation:

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

P_IN_GA(X) → U1_GA(q_in_ga(X))

R is empty.
The set Q consists of the following terms:

q_in_ga(x0)

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

(13) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(14) TRUE

(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:
p_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p_in_ga(X, Z) → U1_ga(X, Z, q_in_ga(X, Y))
q_in_ga(a, b) → q_out_ga(a, b)
U1_ga(X, Z, q_out_ga(X, Y)) → U2_ga(X, Z, p_in_ga(Y, Z))
p_in_ga(X, X) → p_out_ga(X, X)
U2_ga(X, Z, p_out_ga(Y, Z)) → p_out_ga(X, Z)

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
q_in_ga(x1, x2)  =  q_in_ga(x1)
a  =  a
q_out_ga(x1, x2)  =  q_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(16) Obligation:

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

p_in_ga(X, Z) → U1_ga(X, Z, q_in_ga(X, Y))
q_in_ga(a, b) → q_out_ga(a, b)
U1_ga(X, Z, q_out_ga(X, Y)) → U2_ga(X, Z, p_in_ga(Y, Z))
p_in_ga(X, X) → p_out_ga(X, X)
U2_ga(X, Z, p_out_ga(Y, Z)) → p_out_ga(X, Z)

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
q_in_ga(x1, x2)  =  q_in_ga(x1)
a  =  a
q_out_ga(x1, x2)  =  q_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)

(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:

P_IN_GA(X, Z) → U1_GA(X, Z, q_in_ga(X, Y))
P_IN_GA(X, Z) → Q_IN_GA(X, Y)
U1_GA(X, Z, q_out_ga(X, Y)) → U2_GA(X, Z, p_in_ga(Y, Z))
U1_GA(X, Z, q_out_ga(X, Y)) → P_IN_GA(Y, Z)

The TRS R consists of the following rules:

p_in_ga(X, Z) → U1_ga(X, Z, q_in_ga(X, Y))
q_in_ga(a, b) → q_out_ga(a, b)
U1_ga(X, Z, q_out_ga(X, Y)) → U2_ga(X, Z, p_in_ga(Y, Z))
p_in_ga(X, X) → p_out_ga(X, X)
U2_ga(X, Z, p_out_ga(Y, Z)) → p_out_ga(X, Z)

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
q_in_ga(x1, x2)  =  q_in_ga(x1)
a  =  a
q_out_ga(x1, x2)  =  q_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
Q_IN_GA(x1, x2)  =  Q_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)

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

(18) Obligation:

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

P_IN_GA(X, Z) → U1_GA(X, Z, q_in_ga(X, Y))
P_IN_GA(X, Z) → Q_IN_GA(X, Y)
U1_GA(X, Z, q_out_ga(X, Y)) → U2_GA(X, Z, p_in_ga(Y, Z))
U1_GA(X, Z, q_out_ga(X, Y)) → P_IN_GA(Y, Z)

The TRS R consists of the following rules:

p_in_ga(X, Z) → U1_ga(X, Z, q_in_ga(X, Y))
q_in_ga(a, b) → q_out_ga(a, b)
U1_ga(X, Z, q_out_ga(X, Y)) → U2_ga(X, Z, p_in_ga(Y, Z))
p_in_ga(X, X) → p_out_ga(X, X)
U2_ga(X, Z, p_out_ga(Y, Z)) → p_out_ga(X, Z)

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
q_in_ga(x1, x2)  =  q_in_ga(x1)
a  =  a
q_out_ga(x1, x2)  =  q_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
Q_IN_GA(x1, x2)  =  Q_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)

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

(19) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(20) Obligation:

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

U1_GA(X, Z, q_out_ga(X, Y)) → P_IN_GA(Y, Z)
P_IN_GA(X, Z) → U1_GA(X, Z, q_in_ga(X, Y))

The TRS R consists of the following rules:

p_in_ga(X, Z) → U1_ga(X, Z, q_in_ga(X, Y))
q_in_ga(a, b) → q_out_ga(a, b)
U1_ga(X, Z, q_out_ga(X, Y)) → U2_ga(X, Z, p_in_ga(Y, Z))
p_in_ga(X, X) → p_out_ga(X, X)
U2_ga(X, Z, p_out_ga(Y, Z)) → p_out_ga(X, Z)

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
q_in_ga(x1, x2)  =  q_in_ga(x1)
a  =  a
q_out_ga(x1, x2)  =  q_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)

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

(21) UsableRulesProof (EQUIVALENT transformation)

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

(22) Obligation:

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

U1_GA(X, Z, q_out_ga(X, Y)) → P_IN_GA(Y, Z)
P_IN_GA(X, Z) → U1_GA(X, Z, q_in_ga(X, Y))

The TRS R consists of the following rules:

q_in_ga(a, b) → q_out_ga(a, b)

The argument filtering Pi contains the following mapping:
q_in_ga(x1, x2)  =  q_in_ga(x1)
a  =  a
q_out_ga(x1, x2)  =  q_out_ga(x1, x2)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)

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

(23) PiDPToQDPProof (SOUND transformation)

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

(24) Obligation:

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

U1_GA(X, q_out_ga(X, Y)) → P_IN_GA(Y)
P_IN_GA(X) → U1_GA(X, q_in_ga(X))

The TRS R consists of the following rules:

q_in_ga(a) → q_out_ga(a, b)

The set Q consists of the following terms:

q_in_ga(x0)

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