Left Termination of the query pattern rev_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:



↳ Prolog
  ↳ PrologToPiTRSProof

Clauses:

rev([], []).
rev(.(X, Xs), Ys) :- ','(rev(Xs, Zs), app(Zs, .(X, []), Ys)).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Queries:

rev(g,a).

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

rev_in_ga([], []) → rev_out_ga([], [])
rev_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, rev_in_ga(Xs, Zs))
U1_ga(X, Xs, Ys, rev_out_ga(Xs, Zs)) → U2_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) → rev_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2)  =  rev_in_ga(x1)
[]  =  []
rev_out_ga(x1, x2)  =  rev_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)

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:

rev_in_ga([], []) → rev_out_ga([], [])
rev_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, rev_in_ga(Xs, Zs))
U1_ga(X, Xs, Ys, rev_out_ga(Xs, Zs)) → U2_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) → rev_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2)  =  rev_in_ga(x1)
[]  =  []
rev_out_ga(x1, x2)  =  rev_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)


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

REV_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, rev_in_ga(Xs, Zs))
REV_IN_GA(.(X, Xs), Ys) → REV_IN_GA(Xs, Zs)
U1_GA(X, Xs, Ys, rev_out_ga(Xs, Zs)) → U2_GA(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
U1_GA(X, Xs, Ys, rev_out_ga(Xs, Zs)) → APP_IN_GGA(Zs, .(X, []), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U3_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

rev_in_ga([], []) → rev_out_ga([], [])
rev_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, rev_in_ga(Xs, Zs))
U1_ga(X, Xs, Ys, rev_out_ga(Xs, Zs)) → U2_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) → rev_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2)  =  rev_in_ga(x1)
[]  =  []
rev_out_ga(x1, x2)  =  rev_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
REV_IN_GA(x1, x2)  =  REV_IN_GA(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:

REV_IN_GA(.(X, Xs), Ys) → U1_GA(X, Xs, Ys, rev_in_ga(Xs, Zs))
REV_IN_GA(.(X, Xs), Ys) → REV_IN_GA(Xs, Zs)
U1_GA(X, Xs, Ys, rev_out_ga(Xs, Zs)) → U2_GA(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
U1_GA(X, Xs, Ys, rev_out_ga(Xs, Zs)) → APP_IN_GGA(Zs, .(X, []), Ys)
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U3_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

rev_in_ga([], []) → rev_out_ga([], [])
rev_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, rev_in_ga(Xs, Zs))
U1_ga(X, Xs, Ys, rev_out_ga(Xs, Zs)) → U2_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) → rev_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2)  =  rev_in_ga(x1)
[]  =  []
rev_out_ga(x1, x2)  =  rev_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
REV_IN_GA(x1, x2)  =  REV_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 4 less nodes.

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

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

APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

rev_in_ga([], []) → rev_out_ga([], [])
rev_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, rev_in_ga(Xs, Zs))
U1_ga(X, Xs, Ys, rev_out_ga(Xs, Zs)) → U2_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) → rev_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2)  =  rev_in_ga(x1)
[]  =  []
rev_out_ga(x1, x2)  =  rev_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

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
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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

APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_GGA(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

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
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP

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

APP_IN_GGA(.(X, Xs), Ys) → APP_IN_GGA(Xs, Ys)

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: 



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

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

REV_IN_GA(.(X, Xs), Ys) → REV_IN_GA(Xs, Zs)

The TRS R consists of the following rules:

rev_in_ga([], []) → rev_out_ga([], [])
rev_in_ga(.(X, Xs), Ys) → U1_ga(X, Xs, Ys, rev_in_ga(Xs, Zs))
U1_ga(X, Xs, Ys, rev_out_ga(Xs, Zs)) → U2_ga(X, Xs, Ys, app_in_gga(Zs, .(X, []), Ys))
app_in_gga([], X, X) → app_out_gga([], X, X)
app_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
U3_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) → app_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Xs, Ys, app_out_gga(Zs, .(X, []), Ys)) → rev_out_ga(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in_ga(x1, x2)  =  rev_in_ga(x1)
[]  =  []
rev_out_ga(x1, x2)  =  rev_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)
app_out_gga(x1, x2, x3)  =  app_out_gga(x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)
REV_IN_GA(x1, x2)  =  REV_IN_GA(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
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof

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

REV_IN_GA(.(X, Xs), Ys) → REV_IN_GA(Xs, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
REV_IN_GA(x1, x2)  =  REV_IN_GA(x1)

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
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof

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

REV_IN_GA(.(X, Xs)) → REV_IN_GA(Xs)

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: