Left Termination of the query pattern tc_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 PrologToPiTRSProof (⇐)
    ↳12 PiTRS
    ↳13 DependencyPairsProof (⇔)
    ↳14 PiDP
    ↳15 DependencyGraphProof (⇔)
    ↳16 PiDP
    ↳17 UsableRulesProof (⇔)
    ↳18 PiDP
    ↳19 PiDPToQDPProof (⇐)
    ↳20 QDP
    ↳21 UsableRulesReductionPairsProof (⇔)
    ↳22 QDP
    ↳23 UsableRulesReductionPairsProof (⇔)
    ↳24 QDP
    ↳25 DependencyGraphProof (⇔)
    ↳26 TRUE

(0) Obligation:

Clauses:

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

Queries:

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

tc_in_ga(X, X) → tc_out_ga(X, X)
tc_in_ga(X, Y) → U1_ga(X, Y, p_in_ga(X, Z))
p_in_ga(a, b) → p_out_ga(a, b)
p_in_ga(b, c) → p_out_ga(b, c)
U1_ga(X, Y, p_out_ga(X, Z)) → U2_ga(X, Y, Z, tc_in_ga(Z, Y))
U2_ga(X, Y, Z, tc_out_ga(Z, Y)) → tc_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
tc_in_ga(x1, x2)  =  tc_in_ga(x1)
tc_out_ga(x1, x2)  =  tc_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
p_in_ga(x1, x2)  =  p_in_ga(x1)
a  =  a
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
b  =  b
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, 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:

tc_in_ga(X, X) → tc_out_ga(X, X)
tc_in_ga(X, Y) → U1_ga(X, Y, p_in_ga(X, Z))
p_in_ga(a, b) → p_out_ga(a, b)
p_in_ga(b, c) → p_out_ga(b, c)
U1_ga(X, Y, p_out_ga(X, Z)) → U2_ga(X, Y, Z, tc_in_ga(Z, Y))
U2_ga(X, Y, Z, tc_out_ga(Z, Y)) → tc_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
tc_in_ga(x1, x2)  =  tc_in_ga(x1)
tc_out_ga(x1, x2)  =  tc_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
p_in_ga(x1, x2)  =  p_in_ga(x1)
a  =  a
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
b  =  b
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, 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:

TC_IN_GA(X, Y) → U1_GA(X, Y, p_in_ga(X, Z))
TC_IN_GA(X, Y) → P_IN_GA(X, Z)
U1_GA(X, Y, p_out_ga(X, Z)) → U2_GA(X, Y, Z, tc_in_ga(Z, Y))
U1_GA(X, Y, p_out_ga(X, Z)) → TC_IN_GA(Z, Y)

The TRS R consists of the following rules:

tc_in_ga(X, X) → tc_out_ga(X, X)
tc_in_ga(X, Y) → U1_ga(X, Y, p_in_ga(X, Z))
p_in_ga(a, b) → p_out_ga(a, b)
p_in_ga(b, c) → p_out_ga(b, c)
U1_ga(X, Y, p_out_ga(X, Z)) → U2_ga(X, Y, Z, tc_in_ga(Z, Y))
U2_ga(X, Y, Z, tc_out_ga(Z, Y)) → tc_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
tc_in_ga(x1, x2)  =  tc_in_ga(x1)
tc_out_ga(x1, x2)  =  tc_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
p_in_ga(x1, x2)  =  p_in_ga(x1)
a  =  a
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
b  =  b
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
TC_IN_GA(x1, x2)  =  TC_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)

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

(4) Obligation:

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

TC_IN_GA(X, Y) → U1_GA(X, Y, p_in_ga(X, Z))
TC_IN_GA(X, Y) → P_IN_GA(X, Z)
U1_GA(X, Y, p_out_ga(X, Z)) → U2_GA(X, Y, Z, tc_in_ga(Z, Y))
U1_GA(X, Y, p_out_ga(X, Z)) → TC_IN_GA(Z, Y)

The TRS R consists of the following rules:

tc_in_ga(X, X) → tc_out_ga(X, X)
tc_in_ga(X, Y) → U1_ga(X, Y, p_in_ga(X, Z))
p_in_ga(a, b) → p_out_ga(a, b)
p_in_ga(b, c) → p_out_ga(b, c)
U1_ga(X, Y, p_out_ga(X, Z)) → U2_ga(X, Y, Z, tc_in_ga(Z, Y))
U2_ga(X, Y, Z, tc_out_ga(Z, Y)) → tc_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
tc_in_ga(x1, x2)  =  tc_in_ga(x1)
tc_out_ga(x1, x2)  =  tc_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
p_in_ga(x1, x2)  =  p_in_ga(x1)
a  =  a
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
b  =  b
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
TC_IN_GA(x1, x2)  =  TC_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)

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, Y, p_out_ga(X, Z)) → TC_IN_GA(Z, Y)
TC_IN_GA(X, Y) → U1_GA(X, Y, p_in_ga(X, Z))

The TRS R consists of the following rules:

tc_in_ga(X, X) → tc_out_ga(X, X)
tc_in_ga(X, Y) → U1_ga(X, Y, p_in_ga(X, Z))
p_in_ga(a, b) → p_out_ga(a, b)
p_in_ga(b, c) → p_out_ga(b, c)
U1_ga(X, Y, p_out_ga(X, Z)) → U2_ga(X, Y, Z, tc_in_ga(Z, Y))
U2_ga(X, Y, Z, tc_out_ga(Z, Y)) → tc_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
tc_in_ga(x1, x2)  =  tc_in_ga(x1)
tc_out_ga(x1, x2)  =  tc_out_ga(x1, x2)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
p_in_ga(x1, x2)  =  p_in_ga(x1)
a  =  a
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
b  =  b
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
TC_IN_GA(x1, x2)  =  TC_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, 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, Y, p_out_ga(X, Z)) → TC_IN_GA(Z, Y)
TC_IN_GA(X, Y) → U1_GA(X, Y, p_in_ga(X, Z))

The TRS R consists of the following rules:

p_in_ga(a, b) → p_out_ga(a, b)
p_in_ga(b, c) → p_out_ga(b, c)

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
a  =  a
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
b  =  b
TC_IN_GA(x1, x2)  =  TC_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, 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(X, p_out_ga(X, Z)) → TC_IN_GA(Z)
TC_IN_GA(X) → U1_GA(X, p_in_ga(X))

The TRS R consists of the following rules:

p_in_ga(a) → p_out_ga(a, b)
p_in_ga(b) → p_out_ga(b, c)

The set Q consists of the following terms:

p_in_ga(x0)

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

(11) PrologToPiTRSProof (SOUND transformation)

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

tc_in_ga(X, X) → tc_out_ga(X, X)
tc_in_ga(X, Y) → U1_ga(X, Y, p_in_ga(X, Z))
p_in_ga(a, b) → p_out_ga(a, b)
p_in_ga(b, c) → p_out_ga(b, c)
U1_ga(X, Y, p_out_ga(X, Z)) → U2_ga(X, Y, Z, tc_in_ga(Z, Y))
U2_ga(X, Y, Z, tc_out_ga(Z, Y)) → tc_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
tc_in_ga(x1, x2)  =  tc_in_ga(x1)
tc_out_ga(x1, x2)  =  tc_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
p_in_ga(x1, x2)  =  p_in_ga(x1)
a  =  a
p_out_ga(x1, x2)  =  p_out_ga(x2)
b  =  b
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(12) Obligation:

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

tc_in_ga(X, X) → tc_out_ga(X, X)
tc_in_ga(X, Y) → U1_ga(X, Y, p_in_ga(X, Z))
p_in_ga(a, b) → p_out_ga(a, b)
p_in_ga(b, c) → p_out_ga(b, c)
U1_ga(X, Y, p_out_ga(X, Z)) → U2_ga(X, Y, Z, tc_in_ga(Z, Y))
U2_ga(X, Y, Z, tc_out_ga(Z, Y)) → tc_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
tc_in_ga(x1, x2)  =  tc_in_ga(x1)
tc_out_ga(x1, x2)  =  tc_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
p_in_ga(x1, x2)  =  p_in_ga(x1)
a  =  a
p_out_ga(x1, x2)  =  p_out_ga(x2)
b  =  b
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)

(13) 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:

TC_IN_GA(X, Y) → U1_GA(X, Y, p_in_ga(X, Z))
TC_IN_GA(X, Y) → P_IN_GA(X, Z)
U1_GA(X, Y, p_out_ga(X, Z)) → U2_GA(X, Y, Z, tc_in_ga(Z, Y))
U1_GA(X, Y, p_out_ga(X, Z)) → TC_IN_GA(Z, Y)

The TRS R consists of the following rules:

tc_in_ga(X, X) → tc_out_ga(X, X)
tc_in_ga(X, Y) → U1_ga(X, Y, p_in_ga(X, Z))
p_in_ga(a, b) → p_out_ga(a, b)
p_in_ga(b, c) → p_out_ga(b, c)
U1_ga(X, Y, p_out_ga(X, Z)) → U2_ga(X, Y, Z, tc_in_ga(Z, Y))
U2_ga(X, Y, Z, tc_out_ga(Z, Y)) → tc_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
tc_in_ga(x1, x2)  =  tc_in_ga(x1)
tc_out_ga(x1, x2)  =  tc_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
p_in_ga(x1, x2)  =  p_in_ga(x1)
a  =  a
p_out_ga(x1, x2)  =  p_out_ga(x2)
b  =  b
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
TC_IN_GA(x1, x2)  =  TC_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)

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

(14) Obligation:

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

TC_IN_GA(X, Y) → U1_GA(X, Y, p_in_ga(X, Z))
TC_IN_GA(X, Y) → P_IN_GA(X, Z)
U1_GA(X, Y, p_out_ga(X, Z)) → U2_GA(X, Y, Z, tc_in_ga(Z, Y))
U1_GA(X, Y, p_out_ga(X, Z)) → TC_IN_GA(Z, Y)

The TRS R consists of the following rules:

tc_in_ga(X, X) → tc_out_ga(X, X)
tc_in_ga(X, Y) → U1_ga(X, Y, p_in_ga(X, Z))
p_in_ga(a, b) → p_out_ga(a, b)
p_in_ga(b, c) → p_out_ga(b, c)
U1_ga(X, Y, p_out_ga(X, Z)) → U2_ga(X, Y, Z, tc_in_ga(Z, Y))
U2_ga(X, Y, Z, tc_out_ga(Z, Y)) → tc_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
tc_in_ga(x1, x2)  =  tc_in_ga(x1)
tc_out_ga(x1, x2)  =  tc_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
p_in_ga(x1, x2)  =  p_in_ga(x1)
a  =  a
p_out_ga(x1, x2)  =  p_out_ga(x2)
b  =  b
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
TC_IN_GA(x1, x2)  =  TC_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)

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

(15) DependencyGraphProof (EQUIVALENT transformation)

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

(16) Obligation:

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

U1_GA(X, Y, p_out_ga(X, Z)) → TC_IN_GA(Z, Y)
TC_IN_GA(X, Y) → U1_GA(X, Y, p_in_ga(X, Z))

The TRS R consists of the following rules:

tc_in_ga(X, X) → tc_out_ga(X, X)
tc_in_ga(X, Y) → U1_ga(X, Y, p_in_ga(X, Z))
p_in_ga(a, b) → p_out_ga(a, b)
p_in_ga(b, c) → p_out_ga(b, c)
U1_ga(X, Y, p_out_ga(X, Z)) → U2_ga(X, Y, Z, tc_in_ga(Z, Y))
U2_ga(X, Y, Z, tc_out_ga(Z, Y)) → tc_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
tc_in_ga(x1, x2)  =  tc_in_ga(x1)
tc_out_ga(x1, x2)  =  tc_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
p_in_ga(x1, x2)  =  p_in_ga(x1)
a  =  a
p_out_ga(x1, x2)  =  p_out_ga(x2)
b  =  b
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
TC_IN_GA(x1, x2)  =  TC_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

U1_GA(X, Y, p_out_ga(X, Z)) → TC_IN_GA(Z, Y)
TC_IN_GA(X, Y) → U1_GA(X, Y, p_in_ga(X, Z))

The TRS R consists of the following rules:

p_in_ga(a, b) → p_out_ga(a, b)
p_in_ga(b, c) → p_out_ga(b, c)

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2)  =  p_in_ga(x1)
a  =  a
p_out_ga(x1, x2)  =  p_out_ga(x2)
b  =  b
TC_IN_GA(x1, x2)  =  TC_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

U1_GA(p_out_ga(Z)) → TC_IN_GA(Z)
TC_IN_GA(X) → U1_GA(p_in_ga(X))

The TRS R consists of the following rules:

p_in_ga(a) → p_out_ga(b)
p_in_ga(b) → p_out_ga(c)

The set Q consists of the following terms:

p_in_ga(x0)

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

(21) 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.

No dependency pairs are removed.

The following rules are removed from R:

p_in_ga(a) → p_out_ga(b)
Used ordering: POLO with Polynomial interpretation [POLO]:

POL(TC_IN_GA(x1)) = 2 + 2·x1   
POL(U1_GA(x1)) = 2 + x1   
POL(a) = 1   
POL(b) = 0   
POL(c) = 0   
POL(p_in_ga(x1)) = x1   
POL(p_out_ga(x1)) = 2·x1   

(22) Obligation:

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

U1_GA(p_out_ga(Z)) → TC_IN_GA(Z)
TC_IN_GA(X) → U1_GA(p_in_ga(X))

The TRS R consists of the following rules:

p_in_ga(b) → p_out_ga(c)

The set Q consists of the following terms:

p_in_ga(x0)

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

(23) 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(p_out_ga(Z)) → TC_IN_GA(Z)
The following rules are removed from R:

p_in_ga(b) → p_out_ga(c)
Used ordering: POLO with Polynomial interpretation [POLO]:

POL(TC_IN_GA(x1)) = 2 + 2·x1   
POL(U1_GA(x1)) = 2 + x1   
POL(b) = 2   
POL(c) = 0   
POL(p_in_ga(x1)) = x1   
POL(p_out_ga(x1)) = 1 + 2·x1   

(24) Obligation:

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

TC_IN_GA(X) → U1_GA(p_in_ga(X))

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

p_in_ga(x0)

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

(25) DependencyGraphProof (EQUIVALENT transformation)

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

(26) TRUE