Left Termination proof of /tmp/tmpQ0Hbsg/naive

Left Termination of the query pattern reverse_in_2(a, g) w.r.t. the given Prolog program could not be shown:

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

Clauses:

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

Queries:

reverse(a,g).

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

reverse_in_ag(.(X, Xs), Ys) → U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa(.(X, Xs), Ys) → U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa([], []) → reverse_out_aa([], [])
U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_aa(X, Xs, Ys, app_in_aaa(Zs, .(X, []), Ys))
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U1_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
app_in_aaa([], Ys, Ys) → app_out_aaa([], Ys, Ys)
U1_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Xs, Ys, app_out_aaa(Zs, .(X, []), Ys)) → reverse_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_ag(X, Xs, Ys, app_in_aag(Zs, .(X, []), Ys))
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U1_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U1_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Xs, Ys, app_out_aag(Zs, .(X, []), Ys)) → reverse_out_ag(.(X, Xs), Ys)
reverse_in_ag([], []) → reverse_out_ag([], [])

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

  ↳ PrologToPiTRSProof

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

reverse_in_ag(.(X, Xs), Ys) → U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa(.(X, Xs), Ys) → U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa([], []) → reverse_out_aa([], [])
U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_aa(X, Xs, Ys, app_in_aaa(Zs, .(X, []), Ys))
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U1_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
app_in_aaa([], Ys, Ys) → app_out_aaa([], Ys, Ys)
U1_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Xs, Ys, app_out_aaa(Zs, .(X, []), Ys)) → reverse_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_ag(X, Xs, Ys, app_in_aag(Zs, .(X, []), Ys))
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U1_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U1_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Xs, Ys, app_out_aag(Zs, .(X, []), Ys)) → reverse_out_ag(.(X, Xs), Ys)
reverse_in_ag([], []) → reverse_out_ag([], [])

REVERSE_IN_AG(.(X, Xs), Ys) → U2_AG(X, Xs, Ys, reverse_in_aa(Xs, Zs))
REVERSE_IN_AG(.(X, Xs), Ys) → REVERSE_IN_AA(Xs, Zs)
REVERSE_IN_AA(.(X, Xs), Ys) → U2_AA(X, Xs, Ys, reverse_in_aa(Xs, Zs))
REVERSE_IN_AA(.(X, Xs), Ys) → REVERSE_IN_AA(Xs, Zs)
U2_AA(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_AA(X, Xs, Ys, app_in_aaa(Zs, .(X, []), Ys))
U2_AA(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → APP_IN_AAA(Zs, .(X, []), Ys)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U1_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U2_AG(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_AG(X, Xs, Ys, app_in_aag(Zs, .(X, []), Ys))
U2_AG(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → APP_IN_AAG(Zs, .(X, []), Ys)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U1_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

reverse_in_ag(.(X, Xs), Ys) → U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa(.(X, Xs), Ys) → U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa([], []) → reverse_out_aa([], [])
U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_aa(X, Xs, Ys, app_in_aaa(Zs, .(X, []), Ys))
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U1_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
app_in_aaa([], Ys, Ys) → app_out_aaa([], Ys, Ys)
U1_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Xs, Ys, app_out_aaa(Zs, .(X, []), Ys)) → reverse_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_ag(X, Xs, Ys, app_in_aag(Zs, .(X, []), Ys))
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U1_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U1_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Xs, Ys, app_out_aag(Zs, .(X, []), Ys)) → reverse_out_ag(.(X, Xs), Ys)
reverse_in_ag([], []) → reverse_out_ag([], [])

The argument filtering Pi contains the following mapping:
reverse_in_ag(x1, x2) = reverse_in_ag(x2)
U2_ag(x1, x2, x3, x4) = U2_ag(x3, x4)
reverse_in_aa(x1, x2) = reverse_in_aa
U2_aa(x1, x2, x3, x4) = U2_aa(x4)
reverse_out_aa(x1, x2) = reverse_out_aa
U3_aa(x1, x2, x3, x4) = U3_aa(x4)
.(x1, x2) = .(x1, x2)
[] = []
app_in_aaa(x1, x2, x3) = app_in_aaa
U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5)
app_out_aaa(x1, x2, x3) = app_out_aaa
U3_ag(x1, x2, x3, x4) = U3_ag(x4)
app_in_aag(x1, x2, x3) = app_in_aag(x3)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x5)
app_out_aag(x1, x2, x3) = app_out_aag(x1, x2)
reverse_out_ag(x1, x2) = reverse_out_ag
APP_IN_AAA(x1, x2, x3) = APP_IN_AAA
U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x1, x5)
REVERSE_IN_AG(x1, x2) = REVERSE_IN_AG(x2)
APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3)
U3_AG(x1, x2, x3, x4) = U3_AG(x4)
U2_AA(x1, x2, x3, x4) = U2_AA(x4)
REVERSE_IN_AA(x1, x2) = REVERSE_IN_AA
U2_AG(x1, x2, x3, x4) = U2_AG(x3, x4)
U1_AAA(x1, x2, x3, x4, x5) = U1_AAA(x5)
U3_AA(x1, x2, x3, x4) = U3_AA(x4)

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

REVERSE_IN_AG(.(X, Xs), Ys) → U2_AG(X, Xs, Ys, reverse_in_aa(Xs, Zs))
REVERSE_IN_AG(.(X, Xs), Ys) → REVERSE_IN_AA(Xs, Zs)
REVERSE_IN_AA(.(X, Xs), Ys) → U2_AA(X, Xs, Ys, reverse_in_aa(Xs, Zs))
REVERSE_IN_AA(.(X, Xs), Ys) → REVERSE_IN_AA(Xs, Zs)
U2_AA(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_AA(X, Xs, Ys, app_in_aaa(Zs, .(X, []), Ys))
U2_AA(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → APP_IN_AAA(Zs, .(X, []), Ys)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U1_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U2_AG(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_AG(X, Xs, Ys, app_in_aag(Zs, .(X, []), Ys))
U2_AG(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → APP_IN_AAG(Zs, .(X, []), Ys)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U1_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

reverse_in_ag(.(X, Xs), Ys) → U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa(.(X, Xs), Ys) → U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa([], []) → reverse_out_aa([], [])
U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_aa(X, Xs, Ys, app_in_aaa(Zs, .(X, []), Ys))
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U1_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
app_in_aaa([], Ys, Ys) → app_out_aaa([], Ys, Ys)
U1_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Xs, Ys, app_out_aaa(Zs, .(X, []), Ys)) → reverse_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_ag(X, Xs, Ys, app_in_aag(Zs, .(X, []), Ys))
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U1_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U1_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Xs, Ys, app_out_aag(Zs, .(X, []), Ys)) → reverse_out_ag(.(X, Xs), Ys)
reverse_in_ag([], []) → reverse_out_ag([], [])

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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

The TRS R consists of the following rules:

reverse_in_ag(.(X, Xs), Ys) → U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa(.(X, Xs), Ys) → U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa([], []) → reverse_out_aa([], [])
U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_aa(X, Xs, Ys, app_in_aaa(Zs, .(X, []), Ys))
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U1_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
app_in_aaa([], Ys, Ys) → app_out_aaa([], Ys, Ys)
U1_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Xs, Ys, app_out_aaa(Zs, .(X, []), Ys)) → reverse_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_ag(X, Xs, Ys, app_in_aag(Zs, .(X, []), Ys))
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U1_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U1_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Xs, Ys, app_out_aag(Zs, .(X, []), Ys)) → reverse_out_ag(.(X, Xs), Ys)
reverse_in_ag([], []) → reverse_out_ag([], [])

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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

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

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

APP_IN_AAG(.(X, Zs)) → APP_IN_AAG(Zs)

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_AAG(.(X, Zs)) → APP_IN_AAG(Zs)
The graph contains the following edges 1 > 1

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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

The TRS R consists of the following rules:

reverse_in_ag(.(X, Xs), Ys) → U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa(.(X, Xs), Ys) → U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa([], []) → reverse_out_aa([], [])
U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_aa(X, Xs, Ys, app_in_aaa(Zs, .(X, []), Ys))
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U1_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
app_in_aaa([], Ys, Ys) → app_out_aaa([], Ys, Ys)
U1_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Xs, Ys, app_out_aaa(Zs, .(X, []), Ys)) → reverse_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_ag(X, Xs, Ys, app_in_aag(Zs, .(X, []), Ys))
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U1_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U1_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Xs, Ys, app_out_aag(Zs, .(X, []), Ys)) → reverse_out_ag(.(X, Xs), Ys)
reverse_in_ag([], []) → reverse_out_ag([], [])

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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

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

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

                        ↳ NonTerminationProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

APP_IN_AAA → APP_IN_AAA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

APP_IN_AAA → APP_IN_AAA

The TRS R consists of the following rules:none

s = APP_IN_AAA evaluates to t =APP_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP_IN_AAA to APP_IN_AAA.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

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

The TRS R consists of the following rules:

reverse_in_ag(.(X, Xs), Ys) → U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa(.(X, Xs), Ys) → U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa([], []) → reverse_out_aa([], [])
U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_aa(X, Xs, Ys, app_in_aaa(Zs, .(X, []), Ys))
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U1_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
app_in_aaa([], Ys, Ys) → app_out_aaa([], Ys, Ys)
U1_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Xs, Ys, app_out_aaa(Zs, .(X, []), Ys)) → reverse_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_ag(X, Xs, Ys, app_in_aag(Zs, .(X, []), Ys))
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U1_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U1_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Xs, Ys, app_out_aag(Zs, .(X, []), Ys)) → reverse_out_ag(.(X, Xs), Ys)
reverse_in_ag([], []) → reverse_out_ag([], [])

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

  ↳ PrologToPiTRSProof

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

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

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

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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

  ↳ PrologToPiTRSProof

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

REVERSE_IN_AA → REVERSE_IN_AA

The TRS R consists of the following rules:none

s = REVERSE_IN_AA evaluates to t =REVERSE_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from REVERSE_IN_AA to REVERSE_IN_AA.

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

reverse_in_ag(.(X, Xs), Ys) → U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa(.(X, Xs), Ys) → U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa([], []) → reverse_out_aa([], [])
U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_aa(X, Xs, Ys, app_in_aaa(Zs, .(X, []), Ys))
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U1_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
app_in_aaa([], Ys, Ys) → app_out_aaa([], Ys, Ys)
U1_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Xs, Ys, app_out_aaa(Zs, .(X, []), Ys)) → reverse_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_ag(X, Xs, Ys, app_in_aag(Zs, .(X, []), Ys))
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U1_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U1_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Xs, Ys, app_out_aag(Zs, .(X, []), Ys)) → reverse_out_ag(.(X, Xs), Ys)
reverse_in_ag([], []) → reverse_out_ag([], [])

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

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

reverse_in_ag(.(X, Xs), Ys) → U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa(.(X, Xs), Ys) → U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa([], []) → reverse_out_aa([], [])
U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_aa(X, Xs, Ys, app_in_aaa(Zs, .(X, []), Ys))
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U1_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
app_in_aaa([], Ys, Ys) → app_out_aaa([], Ys, Ys)
U1_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Xs, Ys, app_out_aaa(Zs, .(X, []), Ys)) → reverse_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_ag(X, Xs, Ys, app_in_aag(Zs, .(X, []), Ys))
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U1_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U1_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Xs, Ys, app_out_aag(Zs, .(X, []), Ys)) → reverse_out_ag(.(X, Xs), Ys)
reverse_in_ag([], []) → reverse_out_ag([], [])

REVERSE_IN_AG(.(X, Xs), Ys) → U2_AG(X, Xs, Ys, reverse_in_aa(Xs, Zs))
REVERSE_IN_AG(.(X, Xs), Ys) → REVERSE_IN_AA(Xs, Zs)
REVERSE_IN_AA(.(X, Xs), Ys) → U2_AA(X, Xs, Ys, reverse_in_aa(Xs, Zs))
REVERSE_IN_AA(.(X, Xs), Ys) → REVERSE_IN_AA(Xs, Zs)
U2_AA(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_AA(X, Xs, Ys, app_in_aaa(Zs, .(X, []), Ys))
U2_AA(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → APP_IN_AAA(Zs, .(X, []), Ys)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U1_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U2_AG(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_AG(X, Xs, Ys, app_in_aag(Zs, .(X, []), Ys))
U2_AG(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → APP_IN_AAG(Zs, .(X, []), Ys)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U1_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

reverse_in_ag(.(X, Xs), Ys) → U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa(.(X, Xs), Ys) → U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa([], []) → reverse_out_aa([], [])
U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_aa(X, Xs, Ys, app_in_aaa(Zs, .(X, []), Ys))
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U1_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
app_in_aaa([], Ys, Ys) → app_out_aaa([], Ys, Ys)
U1_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Xs, Ys, app_out_aaa(Zs, .(X, []), Ys)) → reverse_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_ag(X, Xs, Ys, app_in_aag(Zs, .(X, []), Ys))
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U1_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U1_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Xs, Ys, app_out_aag(Zs, .(X, []), Ys)) → reverse_out_ag(.(X, Xs), Ys)
reverse_in_ag([], []) → reverse_out_ag([], [])

The argument filtering Pi contains the following mapping:
reverse_in_ag(x1, x2) = reverse_in_ag(x2)
U2_ag(x1, x2, x3, x4) = U2_ag(x3, x4)
reverse_in_aa(x1, x2) = reverse_in_aa
U2_aa(x1, x2, x3, x4) = U2_aa(x4)
reverse_out_aa(x1, x2) = reverse_out_aa
U3_aa(x1, x2, x3, x4) = U3_aa(x4)
.(x1, x2) = .(x1, x2)
[] = []
app_in_aaa(x1, x2, x3) = app_in_aaa
U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5)
app_out_aaa(x1, x2, x3) = app_out_aaa
U3_ag(x1, x2, x3, x4) = U3_ag(x3, x4)
app_in_aag(x1, x2, x3) = app_in_aag(x3)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x4, x5)
app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3)
reverse_out_ag(x1, x2) = reverse_out_ag(x2)
APP_IN_AAA(x1, x2, x3) = APP_IN_AAA
U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x1, x4, x5)
REVERSE_IN_AG(x1, x2) = REVERSE_IN_AG(x2)
APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3)
U3_AG(x1, x2, x3, x4) = U3_AG(x3, x4)
U2_AA(x1, x2, x3, x4) = U2_AA(x4)
REVERSE_IN_AA(x1, x2) = REVERSE_IN_AA
U2_AG(x1, x2, x3, x4) = U2_AG(x3, x4)
U1_AAA(x1, x2, x3, x4, x5) = U1_AAA(x5)
U3_AA(x1, x2, x3, x4) = U3_AA(x4)

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

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

REVERSE_IN_AG(.(X, Xs), Ys) → U2_AG(X, Xs, Ys, reverse_in_aa(Xs, Zs))
REVERSE_IN_AG(.(X, Xs), Ys) → REVERSE_IN_AA(Xs, Zs)
REVERSE_IN_AA(.(X, Xs), Ys) → U2_AA(X, Xs, Ys, reverse_in_aa(Xs, Zs))
REVERSE_IN_AA(.(X, Xs), Ys) → REVERSE_IN_AA(Xs, Zs)
U2_AA(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_AA(X, Xs, Ys, app_in_aaa(Zs, .(X, []), Ys))
U2_AA(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → APP_IN_AAA(Zs, .(X, []), Ys)
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → U1_AAA(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
APP_IN_AAA(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAA(Xs, Ys, Zs)
U2_AG(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_AG(X, Xs, Ys, app_in_aag(Zs, .(X, []), Ys))
U2_AG(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → APP_IN_AAG(Zs, .(X, []), Ys)
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U1_AAG(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
APP_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APP_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

reverse_in_ag(.(X, Xs), Ys) → U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa(.(X, Xs), Ys) → U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa([], []) → reverse_out_aa([], [])
U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_aa(X, Xs, Ys, app_in_aaa(Zs, .(X, []), Ys))
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U1_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
app_in_aaa([], Ys, Ys) → app_out_aaa([], Ys, Ys)
U1_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Xs, Ys, app_out_aaa(Zs, .(X, []), Ys)) → reverse_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_ag(X, Xs, Ys, app_in_aag(Zs, .(X, []), Ys))
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U1_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U1_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Xs, Ys, app_out_aag(Zs, .(X, []), Ys)) → reverse_out_ag(.(X, Xs), Ys)
reverse_in_ag([], []) → reverse_out_ag([], [])

The argument filtering Pi contains the following mapping:
reverse_in_ag(x1, x2) = reverse_in_ag(x2)
U2_ag(x1, x2, x3, x4) = U2_ag(x3, x4)
reverse_in_aa(x1, x2) = reverse_in_aa
U2_aa(x1, x2, x3, x4) = U2_aa(x4)
reverse_out_aa(x1, x2) = reverse_out_aa
U3_aa(x1, x2, x3, x4) = U3_aa(x4)
.(x1, x2) = .(x1, x2)
[] = []
app_in_aaa(x1, x2, x3) = app_in_aaa
U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5)
app_out_aaa(x1, x2, x3) = app_out_aaa
U3_ag(x1, x2, x3, x4) = U3_ag(x3, x4)
app_in_aag(x1, x2, x3) = app_in_aag(x3)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x4, x5)
app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3)
reverse_out_ag(x1, x2) = reverse_out_ag(x2)
APP_IN_AAA(x1, x2, x3) = APP_IN_AAA
U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x1, x4, x5)
REVERSE_IN_AG(x1, x2) = REVERSE_IN_AG(x2)
APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(x3)
U3_AG(x1, x2, x3, x4) = U3_AG(x3, x4)
U2_AA(x1, x2, x3, x4) = U2_AA(x4)
REVERSE_IN_AA(x1, x2) = REVERSE_IN_AA
U2_AG(x1, x2, x3, x4) = U2_AG(x3, x4)
U1_AAA(x1, x2, x3, x4, x5) = U1_AAA(x5)
U3_AA(x1, x2, x3, x4) = U3_AA(x4)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 3 SCCs with 9 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

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

The TRS R consists of the following rules:

reverse_in_ag(.(X, Xs), Ys) → U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa(.(X, Xs), Ys) → U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa([], []) → reverse_out_aa([], [])
U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_aa(X, Xs, Ys, app_in_aaa(Zs, .(X, []), Ys))
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U1_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
app_in_aaa([], Ys, Ys) → app_out_aaa([], Ys, Ys)
U1_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Xs, Ys, app_out_aaa(Zs, .(X, []), Ys)) → reverse_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_ag(X, Xs, Ys, app_in_aag(Zs, .(X, []), Ys))
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U1_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U1_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Xs, Ys, app_out_aag(Zs, .(X, []), Ys)) → reverse_out_ag(.(X, Xs), Ys)
reverse_in_ag([], []) → reverse_out_ag([], [])

The argument filtering Pi contains the following mapping:
reverse_in_ag(x1, x2) = reverse_in_ag(x2)
U2_ag(x1, x2, x3, x4) = U2_ag(x3, x4)
reverse_in_aa(x1, x2) = reverse_in_aa
U2_aa(x1, x2, x3, x4) = U2_aa(x4)
reverse_out_aa(x1, x2) = reverse_out_aa
U3_aa(x1, x2, x3, x4) = U3_aa(x4)
.(x1, x2) = .(x1, x2)
[] = []
app_in_aaa(x1, x2, x3) = app_in_aaa
U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5)
app_out_aaa(x1, x2, x3) = app_out_aaa
U3_ag(x1, x2, x3, x4) = U3_ag(x3, x4)
app_in_aag(x1, x2, x3) = app_in_aag(x3)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x4, x5)
app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3)
reverse_out_ag(x1, x2) = reverse_out_ag(x2)
APP_IN_AAG(x1, x2, x3) = APP_IN_AAG(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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

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

APP_IN_AAG(.(X, Zs)) → APP_IN_AAG(Zs)

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

APP_IN_AAG(.(X, Zs)) → APP_IN_AAG(Zs)
The graph contains the following edges 1 > 1

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

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

The TRS R consists of the following rules:

reverse_in_ag(.(X, Xs), Ys) → U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa(.(X, Xs), Ys) → U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa([], []) → reverse_out_aa([], [])
U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_aa(X, Xs, Ys, app_in_aaa(Zs, .(X, []), Ys))
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U1_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
app_in_aaa([], Ys, Ys) → app_out_aaa([], Ys, Ys)
U1_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Xs, Ys, app_out_aaa(Zs, .(X, []), Ys)) → reverse_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_ag(X, Xs, Ys, app_in_aag(Zs, .(X, []), Ys))
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U1_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U1_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Xs, Ys, app_out_aag(Zs, .(X, []), Ys)) → reverse_out_ag(.(X, Xs), Ys)
reverse_in_ag([], []) → reverse_out_ag([], [])

The argument filtering Pi contains the following mapping:
reverse_in_ag(x1, x2) = reverse_in_ag(x2)
U2_ag(x1, x2, x3, x4) = U2_ag(x3, x4)
reverse_in_aa(x1, x2) = reverse_in_aa
U2_aa(x1, x2, x3, x4) = U2_aa(x4)
reverse_out_aa(x1, x2) = reverse_out_aa
U3_aa(x1, x2, x3, x4) = U3_aa(x4)
.(x1, x2) = .(x1, x2)
[] = []
app_in_aaa(x1, x2, x3) = app_in_aaa
U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5)
app_out_aaa(x1, x2, x3) = app_out_aaa
U3_ag(x1, x2, x3, x4) = U3_ag(x3, x4)
app_in_aag(x1, x2, x3) = app_in_aag(x3)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x4, x5)
app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3)
reverse_out_ag(x1, x2) = reverse_out_ag(x2)
APP_IN_AAA(x1, x2, x3) = APP_IN_AAA

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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

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

APP_IN_AAA → APP_IN_AAA

The TRS R consists of the following rules:none

s = APP_IN_AAA evaluates to t =APP_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP_IN_AAA to APP_IN_AAA.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

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

The TRS R consists of the following rules:

reverse_in_ag(.(X, Xs), Ys) → U2_ag(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa(.(X, Xs), Ys) → U2_aa(X, Xs, Ys, reverse_in_aa(Xs, Zs))
reverse_in_aa([], []) → reverse_out_aa([], [])
U2_aa(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_aa(X, Xs, Ys, app_in_aaa(Zs, .(X, []), Ys))
app_in_aaa(.(X, Xs), Ys, .(X, Zs)) → U1_aaa(X, Xs, Ys, Zs, app_in_aaa(Xs, Ys, Zs))
app_in_aaa([], Ys, Ys) → app_out_aaa([], Ys, Ys)
U1_aaa(X, Xs, Ys, Zs, app_out_aaa(Xs, Ys, Zs)) → app_out_aaa(.(X, Xs), Ys, .(X, Zs))
U3_aa(X, Xs, Ys, app_out_aaa(Zs, .(X, []), Ys)) → reverse_out_aa(.(X, Xs), Ys)
U2_ag(X, Xs, Ys, reverse_out_aa(Xs, Zs)) → U3_ag(X, Xs, Ys, app_in_aag(Zs, .(X, []), Ys))
app_in_aag(.(X, Xs), Ys, .(X, Zs)) → U1_aag(X, Xs, Ys, Zs, app_in_aag(Xs, Ys, Zs))
app_in_aag([], Ys, Ys) → app_out_aag([], Ys, Ys)
U1_aag(X, Xs, Ys, Zs, app_out_aag(Xs, Ys, Zs)) → app_out_aag(.(X, Xs), Ys, .(X, Zs))
U3_ag(X, Xs, Ys, app_out_aag(Zs, .(X, []), Ys)) → reverse_out_ag(.(X, Xs), Ys)
reverse_in_ag([], []) → reverse_out_ag([], [])

The argument filtering Pi contains the following mapping:
reverse_in_ag(x1, x2) = reverse_in_ag(x2)
U2_ag(x1, x2, x3, x4) = U2_ag(x3, x4)
reverse_in_aa(x1, x2) = reverse_in_aa
U2_aa(x1, x2, x3, x4) = U2_aa(x4)
reverse_out_aa(x1, x2) = reverse_out_aa
U3_aa(x1, x2, x3, x4) = U3_aa(x4)
.(x1, x2) = .(x1, x2)
[] = []
app_in_aaa(x1, x2, x3) = app_in_aaa
U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5)
app_out_aaa(x1, x2, x3) = app_out_aaa
U3_ag(x1, x2, x3, x4) = U3_ag(x3, x4)
app_in_aag(x1, x2, x3) = app_in_aag(x3)
U1_aag(x1, x2, x3, x4, x5) = U1_aag(x1, x4, x5)
app_out_aag(x1, x2, x3) = app_out_aag(x1, x2, x3)
reverse_out_ag(x1, x2) = reverse_out_ag(x2)
REVERSE_IN_AA(x1, x2) = REVERSE_IN_AA

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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

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

REVERSE_IN_AA → REVERSE_IN_AA

The TRS R consists of the following rules:none

s = REVERSE_IN_AA evaluates to t =REVERSE_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from REVERSE_IN_AA to REVERSE_IN_AA.