Left Termination of the query pattern tc_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:



↳ Prolog
  ↳ PrologToPiTRSProof

Clauses:

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

Queries:

tc(g,a).

We use the technique of [30]. 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



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

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)


Using Dependency Pairs [1,30] 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)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
TC_IN_GA(x1, x2)  =  TC_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
U1_GA(x1, x2, x3)  =  U1_GA(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:

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)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
TC_IN_GA(x1, x2)  =  TC_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
U1_GA(x1, x2, x3)  =  U1_GA(x3)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 2 less nodes.

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

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
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:

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
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
                      ↳ UsableRulesReductionPairsProof

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

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

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.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] 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 [25]:

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



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
                    ↳ QDP
                      ↳ UsableRulesReductionPairsProof
                        ↳ QDP
                          ↳ UsableRulesReductionPairsProof

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

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

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.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] 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(b) → p_out_ga(c)
Used ordering: POLO with Polynomial interpretation [25]:

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



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
                    ↳ QDP
                      ↳ UsableRulesReductionPairsProof
                        ↳ QDP
                          ↳ UsableRulesReductionPairsProof
                            ↳ QDP
                              ↳ DependencyGraphProof

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

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

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

p_in_ga(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.