Left Termination proof of /tmp/tmpIMMox8/naive

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].Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

rev_in(.(X, Xs), Ys) → U1(X, Xs, Ys, rev_in(Xs, Zs))
rev_in([], []) → rev_out([], [])
U1(X, Xs, Ys, rev_out(Xs, Zs)) → U2(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U3(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Xs, Ys, app_out(Zs, .(X, []), Ys)) → rev_out(.(X, Xs), Ys)

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(.(X, Xs), Ys) → U1(X, Xs, Ys, rev_in(Xs, Zs))
rev_in([], []) → rev_out([], [])
U1(X, Xs, Ys, rev_out(Xs, Zs)) → U2(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U3(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Xs, Ys, app_out(Zs, .(X, []), Ys)) → rev_out(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in(x1, x2) = rev_in(x1)
.(x1, x2) = .(x1, x2)
U1(x1, x2, x3, x4) = U1(x1, x4)
[] = []
rev_out(x1, x2) = rev_out(x2)
U2(x1, x2, x3, x4) = U2(x4)
app_in(x1, x2, x3) = app_in(x1, x2)
U3(x1, x2, x3, x4, x5) = U3(x1, x5)
app_out(x1, x2, x3) = app_out(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:

REV_IN(.(X, Xs), Ys) → U1¹(X, Xs, Ys, rev_in(Xs, Zs))
REV_IN(.(X, Xs), Ys) → REV_IN(Xs, Zs)
U1¹(X, Xs, Ys, rev_out(Xs, Zs)) → U2¹(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
U1¹(X, Xs, Ys, rev_out(Xs, Zs)) → APP_IN(Zs, .(X, []), Ys)
APP_IN(.(X, Xs), Ys, .(X, Zs)) → U3¹(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)

The TRS R consists of the following rules:

rev_in(.(X, Xs), Ys) → U1(X, Xs, Ys, rev_in(Xs, Zs))
rev_in([], []) → rev_out([], [])
U1(X, Xs, Ys, rev_out(Xs, Zs)) → U2(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U3(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Xs, Ys, app_out(Zs, .(X, []), Ys)) → rev_out(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in(x1, x2) = rev_in(x1)
.(x1, x2) = .(x1, x2)
U1(x1, x2, x3, x4) = U1(x1, x4)
[] = []
rev_out(x1, x2) = rev_out(x2)
U2(x1, x2, x3, x4) = U2(x4)
app_in(x1, x2, x3) = app_in(x1, x2)
U3(x1, x2, x3, x4, x5) = U3(x1, x5)
app_out(x1, x2, x3) = app_out(x3)
REV_IN(x1, x2) = REV_IN(x1)
U3¹(x1, x2, x3, x4, x5) = U3¹(x1, x5)
U2¹(x1, x2, x3, x4) = U2¹(x4)
APP_IN(x1, x2, x3) = APP_IN(x1, x2)
U1¹(x1, x2, x3, x4) = U1¹(x1, x4)

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(.(X, Xs), Ys) → U1¹(X, Xs, Ys, rev_in(Xs, Zs))
REV_IN(.(X, Xs), Ys) → REV_IN(Xs, Zs)
U1¹(X, Xs, Ys, rev_out(Xs, Zs)) → U2¹(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
U1¹(X, Xs, Ys, rev_out(Xs, Zs)) → APP_IN(Zs, .(X, []), Ys)
APP_IN(.(X, Xs), Ys, .(X, Zs)) → U3¹(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)

The TRS R consists of the following rules:

rev_in(.(X, Xs), Ys) → U1(X, Xs, Ys, rev_in(Xs, Zs))
rev_in([], []) → rev_out([], [])
U1(X, Xs, Ys, rev_out(Xs, Zs)) → U2(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U3(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Xs, Ys, app_out(Zs, .(X, []), Ys)) → rev_out(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in(x1, x2) = rev_in(x1)
.(x1, x2) = .(x1, x2)
U1(x1, x2, x3, x4) = U1(x1, x4)
[] = []
rev_out(x1, x2) = rev_out(x2)
U2(x1, x2, x3, x4) = U2(x4)
app_in(x1, x2, x3) = app_in(x1, x2)
U3(x1, x2, x3, x4, x5) = U3(x1, x5)
app_out(x1, x2, x3) = app_out(x3)
REV_IN(x1, x2) = REV_IN(x1)
U3¹(x1, x2, x3, x4, x5) = U3¹(x1, x5)
U2¹(x1, x2, x3, x4) = U2¹(x4)
APP_IN(x1, x2, x3) = APP_IN(x1, x2)
U1¹(x1, x2, x3, x4) = U1¹(x1, x4)

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

The TRS R consists of the following rules:

rev_in(.(X, Xs), Ys) → U1(X, Xs, Ys, rev_in(Xs, Zs))
rev_in([], []) → rev_out([], [])
U1(X, Xs, Ys, rev_out(Xs, Zs)) → U2(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U3(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Xs, Ys, app_out(Zs, .(X, []), Ys)) → rev_out(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in(x1, x2) = rev_in(x1)
.(x1, x2) = .(x1, x2)
U1(x1, x2, x3, x4) = U1(x1, x4)
[] = []
rev_out(x1, x2) = rev_out(x2)
U2(x1, x2, x3, x4) = U2(x4)
app_in(x1, x2, x3) = app_in(x1, x2)
U3(x1, x2, x3, x4, x5) = U3(x1, x5)
app_out(x1, x2, x3) = app_out(x3)
APP_IN(x1, x2, x3) = APP_IN(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(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP_IN(x1, x2, x3) = APP_IN(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(.(X, Xs), Ys) → APP_IN(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:

APP_IN(.(X, Xs), Ys) → APP_IN(Xs, Ys)
The graph contains the following edges 1 > 1, 2 >= 2

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

REV_IN(.(X, Xs), Ys) → REV_IN(Xs, Zs)

The TRS R consists of the following rules:

rev_in(.(X, Xs), Ys) → U1(X, Xs, Ys, rev_in(Xs, Zs))
rev_in([], []) → rev_out([], [])
U1(X, Xs, Ys, rev_out(Xs, Zs)) → U2(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U3(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Xs, Ys, app_out(Zs, .(X, []), Ys)) → rev_out(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in(x1, x2) = rev_in(x1)
.(x1, x2) = .(x1, x2)
U1(x1, x2, x3, x4) = U1(x1, x4)
[] = []
rev_out(x1, x2) = rev_out(x2)
U2(x1, x2, x3, x4) = U2(x4)
app_in(x1, x2, x3) = app_in(x1, x2)
U3(x1, x2, x3, x4, x5) = U3(x1, x5)
app_out(x1, x2, x3) = app_out(x3)
REV_IN(x1, x2) = REV_IN(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(.(X, Xs), Ys) → REV_IN(Xs, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
REV_IN(x1, x2) = REV_IN(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(.(X, Xs)) → REV_IN(Xs)

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

REV_IN(.(X, Xs)) → REV_IN(Xs)
The graph contains the following edges 1 > 1