Left Termination of the query pattern list_in_1(g) w.r.t. the given Prolog program could successfully be proven:



↳ Prolog
  ↳ PrologToPiTRSProof

Clauses:

list([]).
list(.(X, Ts)) :- list(Ts).

Queries:

list(g).

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

list_in_g([]) → list_out_g([])
list_in_g(.(X, Ts)) → U1_g(X, Ts, list_in_g(Ts))
U1_g(X, Ts, list_out_g(Ts)) → list_out_g(.(X, Ts))

The argument filtering Pi contains the following mapping:
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
list_out_g(x1)  =  list_out_g
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3)  =  U1_g(x3)

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:

list_in_g([]) → list_out_g([])
list_in_g(.(X, Ts)) → U1_g(X, Ts, list_in_g(Ts))
U1_g(X, Ts, list_out_g(Ts)) → list_out_g(.(X, Ts))

The argument filtering Pi contains the following mapping:
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
list_out_g(x1)  =  list_out_g
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3)  =  U1_g(x3)


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

LIST_IN_G(.(X, Ts)) → U1_G(X, Ts, list_in_g(Ts))
LIST_IN_G(.(X, Ts)) → LIST_IN_G(Ts)

The TRS R consists of the following rules:

list_in_g([]) → list_out_g([])
list_in_g(.(X, Ts)) → U1_g(X, Ts, list_in_g(Ts))
U1_g(X, Ts, list_out_g(Ts)) → list_out_g(.(X, Ts))

The argument filtering Pi contains the following mapping:
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
list_out_g(x1)  =  list_out_g
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3)  =  U1_g(x3)
U1_G(x1, x2, x3)  =  U1_G(x3)
LIST_IN_G(x1)  =  LIST_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:

LIST_IN_G(.(X, Ts)) → U1_G(X, Ts, list_in_g(Ts))
LIST_IN_G(.(X, Ts)) → LIST_IN_G(Ts)

The TRS R consists of the following rules:

list_in_g([]) → list_out_g([])
list_in_g(.(X, Ts)) → U1_g(X, Ts, list_in_g(Ts))
U1_g(X, Ts, list_out_g(Ts)) → list_out_g(.(X, Ts))

The argument filtering Pi contains the following mapping:
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
list_out_g(x1)  =  list_out_g
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3)  =  U1_g(x3)
U1_G(x1, x2, x3)  =  U1_G(x3)
LIST_IN_G(x1)  =  LIST_IN_G(x1)

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

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

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

LIST_IN_G(.(X, Ts)) → LIST_IN_G(Ts)

The TRS R consists of the following rules:

list_in_g([]) → list_out_g([])
list_in_g(.(X, Ts)) → U1_g(X, Ts, list_in_g(Ts))
U1_g(X, Ts, list_out_g(Ts)) → list_out_g(.(X, Ts))

The argument filtering Pi contains the following mapping:
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
list_out_g(x1)  =  list_out_g
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3)  =  U1_g(x3)
LIST_IN_G(x1)  =  LIST_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:

LIST_IN_G(.(X, Ts)) → LIST_IN_G(Ts)

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
                      ↳ QDPSizeChangeProof

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

LIST_IN_G(.(X, Ts)) → LIST_IN_G(Ts)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: