Left Termination proof of /tmp/tmpvaQZoY/pl7.6.2a.pl

Left Termination of the query pattern reach_in_3(g, g, g) w.r.t. the given Prolog program could not be shown:

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

Clauses:

reach(X, Y, Edges) :- member(.(X, .(Y, [])), Edges).
reach(X, Z, Edges) :- ','(member1(.(X, .(Y, [])), Edges), reach(Y, Z, Edges)).
member(H, .(H, L)).
member(X, .(H, L)) :- member(X, L).
member1(H, .(H, L)).
member1(X, .(H, L)) :- member1(X, L).

Queries:

reach(g,g,g).

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:

reach_in(X, Z, Edges) → U2(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
member1_in(X, .(H, L)) → U5(X, H, L, member1_in(X, L))
member1_in(H, .(H, L)) → member1_out(H, .(H, L))
U5(X, H, L, member1_out(X, L)) → member1_out(X, .(H, L))
U2(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → U3(X, Z, Edges, reach_in(Y, Z, Edges))
reach_in(X, Y, Edges) → U1(X, Y, Edges, member_in(.(X, .(Y, [])), Edges))
member_in(X, .(H, L)) → U4(X, H, L, member_in(X, L))
member_in(H, .(H, L)) → member_out(H, .(H, L))
U4(X, H, L, member_out(X, L)) → member_out(X, .(H, L))
U1(X, Y, Edges, member_out(.(X, .(Y, [])), Edges)) → reach_out(X, Y, Edges)
U3(X, Z, Edges, reach_out(Y, Z, Edges)) → reach_out(X, Z, Edges)

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:

reach_in(X, Z, Edges) → U2(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
member1_in(X, .(H, L)) → U5(X, H, L, member1_in(X, L))
member1_in(H, .(H, L)) → member1_out(H, .(H, L))
U5(X, H, L, member1_out(X, L)) → member1_out(X, .(H, L))
U2(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → U3(X, Z, Edges, reach_in(Y, Z, Edges))
reach_in(X, Y, Edges) → U1(X, Y, Edges, member_in(.(X, .(Y, [])), Edges))
member_in(X, .(H, L)) → U4(X, H, L, member_in(X, L))
member_in(H, .(H, L)) → member_out(H, .(H, L))
U4(X, H, L, member_out(X, L)) → member_out(X, .(H, L))
U1(X, Y, Edges, member_out(.(X, .(Y, [])), Edges)) → reach_out(X, Y, Edges)
U3(X, Z, Edges, reach_out(Y, Z, Edges)) → reach_out(X, Z, Edges)

REACH_IN(X, Z, Edges) → U2¹(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
REACH_IN(X, Z, Edges) → MEMBER1_IN(.(X, .(Y, [])), Edges)
MEMBER1_IN(X, .(H, L)) → U5¹(X, H, L, member1_in(X, L))
MEMBER1_IN(X, .(H, L)) → MEMBER1_IN(X, L)
U2¹(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → U3¹(X, Z, Edges, reach_in(Y, Z, Edges))
U2¹(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → REACH_IN(Y, Z, Edges)
REACH_IN(X, Y, Edges) → U1¹(X, Y, Edges, member_in(.(X, .(Y, [])), Edges))
REACH_IN(X, Y, Edges) → MEMBER_IN(.(X, .(Y, [])), Edges)
MEMBER_IN(X, .(H, L)) → U4¹(X, H, L, member_in(X, L))
MEMBER_IN(X, .(H, L)) → MEMBER_IN(X, L)

The TRS R consists of the following rules:

reach_in(X, Z, Edges) → U2(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
member1_in(X, .(H, L)) → U5(X, H, L, member1_in(X, L))
member1_in(H, .(H, L)) → member1_out(H, .(H, L))
U5(X, H, L, member1_out(X, L)) → member1_out(X, .(H, L))
U2(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → U3(X, Z, Edges, reach_in(Y, Z, Edges))
reach_in(X, Y, Edges) → U1(X, Y, Edges, member_in(.(X, .(Y, [])), Edges))
member_in(X, .(H, L)) → U4(X, H, L, member_in(X, L))
member_in(H, .(H, L)) → member_out(H, .(H, L))
U4(X, H, L, member_out(X, L)) → member_out(X, .(H, L))
U1(X, Y, Edges, member_out(.(X, .(Y, [])), Edges)) → reach_out(X, Y, Edges)
U3(X, Z, Edges, reach_out(Y, Z, Edges)) → reach_out(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_in(x1, x2, x3) = reach_in(x1, x2, x3)
U2(x1, x2, x3, x4) = U2(x2, x3, x4)
member1_in(x1, x2) = member1_in(x2)
.(x1, x2) = .(x1, x2)
[] = []
U5(x1, x2, x3, x4) = U5(x4)
member1_out(x1, x2) = member1_out(x1)
U3(x1, x2, x3, x4) = U3(x4)
U1(x1, x2, x3, x4) = U1(x4)
member_in(x1, x2) = member_in(x1, x2)
U4(x1, x2, x3, x4) = U4(x4)
member_out(x1, x2) = member_out
reach_out(x1, x2, x3) = reach_out
U5¹(x1, x2, x3, x4) = U5¹(x4)
U3¹(x1, x2, x3, x4) = U3¹(x4)
REACH_IN(x1, x2, x3) = REACH_IN(x1, x2, x3)
U4¹(x1, x2, x3, x4) = U4¹(x4)
MEMBER1_IN(x1, x2) = MEMBER1_IN(x2)
MEMBER_IN(x1, x2) = MEMBER_IN(x1, x2)
U2¹(x1, x2, x3, x4) = U2¹(x2, x3, x4)
U1¹(x1, x2, x3, x4) = U1¹(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:

REACH_IN(X, Z, Edges) → U2¹(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
REACH_IN(X, Z, Edges) → MEMBER1_IN(.(X, .(Y, [])), Edges)
MEMBER1_IN(X, .(H, L)) → U5¹(X, H, L, member1_in(X, L))
MEMBER1_IN(X, .(H, L)) → MEMBER1_IN(X, L)
U2¹(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → U3¹(X, Z, Edges, reach_in(Y, Z, Edges))
U2¹(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → REACH_IN(Y, Z, Edges)
REACH_IN(X, Y, Edges) → U1¹(X, Y, Edges, member_in(.(X, .(Y, [])), Edges))
REACH_IN(X, Y, Edges) → MEMBER_IN(.(X, .(Y, [])), Edges)
MEMBER_IN(X, .(H, L)) → U4¹(X, H, L, member_in(X, L))
MEMBER_IN(X, .(H, L)) → MEMBER_IN(X, L)

The TRS R consists of the following rules:

reach_in(X, Z, Edges) → U2(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
member1_in(X, .(H, L)) → U5(X, H, L, member1_in(X, L))
member1_in(H, .(H, L)) → member1_out(H, .(H, L))
U5(X, H, L, member1_out(X, L)) → member1_out(X, .(H, L))
U2(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → U3(X, Z, Edges, reach_in(Y, Z, Edges))
reach_in(X, Y, Edges) → U1(X, Y, Edges, member_in(.(X, .(Y, [])), Edges))
member_in(X, .(H, L)) → U4(X, H, L, member_in(X, L))
member_in(H, .(H, L)) → member_out(H, .(H, L))
U4(X, H, L, member_out(X, L)) → member_out(X, .(H, L))
U1(X, Y, Edges, member_out(.(X, .(Y, [])), Edges)) → reach_out(X, Y, Edges)
U3(X, Z, Edges, reach_out(Y, Z, Edges)) → reach_out(X, Z, Edges)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

MEMBER_IN(X, .(H, L)) → MEMBER_IN(X, L)

The TRS R consists of the following rules:

reach_in(X, Z, Edges) → U2(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
member1_in(X, .(H, L)) → U5(X, H, L, member1_in(X, L))
member1_in(H, .(H, L)) → member1_out(H, .(H, L))
U5(X, H, L, member1_out(X, L)) → member1_out(X, .(H, L))
U2(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → U3(X, Z, Edges, reach_in(Y, Z, Edges))
reach_in(X, Y, Edges) → U1(X, Y, Edges, member_in(.(X, .(Y, [])), Edges))
member_in(X, .(H, L)) → U4(X, H, L, member_in(X, L))
member_in(H, .(H, L)) → member_out(H, .(H, L))
U4(X, H, L, member_out(X, L)) → member_out(X, .(H, L))
U1(X, Y, Edges, member_out(.(X, .(Y, [])), Edges)) → reach_out(X, Y, Edges)
U3(X, Z, Edges, reach_out(Y, Z, Edges)) → reach_out(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_in(x1, x2, x3) = reach_in(x1, x2, x3)
U2(x1, x2, x3, x4) = U2(x2, x3, x4)
member1_in(x1, x2) = member1_in(x2)
.(x1, x2) = .(x1, x2)
[] = []
U5(x1, x2, x3, x4) = U5(x4)
member1_out(x1, x2) = member1_out(x1)
U3(x1, x2, x3, x4) = U3(x4)
U1(x1, x2, x3, x4) = U1(x4)
member_in(x1, x2) = member_in(x1, x2)
U4(x1, x2, x3, x4) = U4(x4)
member_out(x1, x2) = member_out
reach_out(x1, x2, x3) = reach_out
MEMBER_IN(x1, x2) = MEMBER_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

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

MEMBER_IN(X, .(H, L)) → MEMBER_IN(X, L)

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

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

MEMBER_IN(X, .(H, L)) → MEMBER_IN(X, L)

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:

MEMBER_IN(X, .(H, L)) → MEMBER_IN(X, L)
The graph contains the following edges 1 >= 1, 2 > 2

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

MEMBER1_IN(X, .(H, L)) → MEMBER1_IN(X, L)

The TRS R consists of the following rules:

reach_in(X, Z, Edges) → U2(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
member1_in(X, .(H, L)) → U5(X, H, L, member1_in(X, L))
member1_in(H, .(H, L)) → member1_out(H, .(H, L))
U5(X, H, L, member1_out(X, L)) → member1_out(X, .(H, L))
U2(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → U3(X, Z, Edges, reach_in(Y, Z, Edges))
reach_in(X, Y, Edges) → U1(X, Y, Edges, member_in(.(X, .(Y, [])), Edges))
member_in(X, .(H, L)) → U4(X, H, L, member_in(X, L))
member_in(H, .(H, L)) → member_out(H, .(H, L))
U4(X, H, L, member_out(X, L)) → member_out(X, .(H, L))
U1(X, Y, Edges, member_out(.(X, .(Y, [])), Edges)) → reach_out(X, Y, Edges)
U3(X, Z, Edges, reach_out(Y, Z, Edges)) → reach_out(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_in(x1, x2, x3) = reach_in(x1, x2, x3)
U2(x1, x2, x3, x4) = U2(x2, x3, x4)
member1_in(x1, x2) = member1_in(x2)
.(x1, x2) = .(x1, x2)
[] = []
U5(x1, x2, x3, x4) = U5(x4)
member1_out(x1, x2) = member1_out(x1)
U3(x1, x2, x3, x4) = U3(x4)
U1(x1, x2, x3, x4) = U1(x4)
member_in(x1, x2) = member_in(x1, x2)
U4(x1, x2, x3, x4) = U4(x4)
member_out(x1, x2) = member_out
reach_out(x1, x2, x3) = reach_out
MEMBER1_IN(x1, x2) = MEMBER1_IN(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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

MEMBER1_IN(X, .(H, L)) → MEMBER1_IN(X, L)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
MEMBER1_IN(x1, x2) = MEMBER1_IN(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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

MEMBER1_IN(.(H, L)) → MEMBER1_IN(L)

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

MEMBER1_IN(.(H, L)) → MEMBER1_IN(L)
The graph contains the following edges 1 > 1

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

REACH_IN(X, Z, Edges) → U2¹(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
U2¹(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → REACH_IN(Y, Z, Edges)

The TRS R consists of the following rules:

reach_in(X, Z, Edges) → U2(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
member1_in(X, .(H, L)) → U5(X, H, L, member1_in(X, L))
member1_in(H, .(H, L)) → member1_out(H, .(H, L))
U5(X, H, L, member1_out(X, L)) → member1_out(X, .(H, L))
U2(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → U3(X, Z, Edges, reach_in(Y, Z, Edges))
reach_in(X, Y, Edges) → U1(X, Y, Edges, member_in(.(X, .(Y, [])), Edges))
member_in(X, .(H, L)) → U4(X, H, L, member_in(X, L))
member_in(H, .(H, L)) → member_out(H, .(H, L))
U4(X, H, L, member_out(X, L)) → member_out(X, .(H, L))
U1(X, Y, Edges, member_out(.(X, .(Y, [])), Edges)) → reach_out(X, Y, Edges)
U3(X, Z, Edges, reach_out(Y, Z, Edges)) → reach_out(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_in(x1, x2, x3) = reach_in(x1, x2, x3)
U2(x1, x2, x3, x4) = U2(x2, x3, x4)
member1_in(x1, x2) = member1_in(x2)
.(x1, x2) = .(x1, x2)
[] = []
U5(x1, x2, x3, x4) = U5(x4)
member1_out(x1, x2) = member1_out(x1)
U3(x1, x2, x3, x4) = U3(x4)
U1(x1, x2, x3, x4) = U1(x4)
member_in(x1, x2) = member_in(x1, x2)
U4(x1, x2, x3, x4) = U4(x4)
member_out(x1, x2) = member_out
reach_out(x1, x2, x3) = reach_out
REACH_IN(x1, x2, x3) = REACH_IN(x1, x2, x3)
U2¹(x1, x2, x3, x4) = U2¹(x2, x3, x4)

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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

  ↳ PrologToPiTRSProof

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

REACH_IN(X, Z, Edges) → U2¹(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
U2¹(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → REACH_IN(Y, Z, Edges)

The TRS R consists of the following rules:

member1_in(X, .(H, L)) → U5(X, H, L, member1_in(X, L))
member1_in(H, .(H, L)) → member1_out(H, .(H, L))
U5(X, H, L, member1_out(X, L)) → member1_out(X, .(H, L))

The argument filtering Pi contains the following mapping:
member1_in(x1, x2) = member1_in(x2)
.(x1, x2) = .(x1, x2)
[] = []
U5(x1, x2, x3, x4) = U5(x4)
member1_out(x1, x2) = member1_out(x1)
REACH_IN(x1, x2, x3) = REACH_IN(x1, x2, x3)
U2¹(x1, x2, x3, x4) = U2¹(x2, x3, x4)

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

                        ↳ Narrowing

  ↳ PrologToPiTRSProof

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

REACH_IN(X, Z, Edges) → U2¹(Z, Edges, member1_in(Edges))
U2¹(Z, Edges, member1_out(.(X, .(Y, [])))) → REACH_IN(Y, Z, Edges)

The TRS R consists of the following rules:

member1_in(.(H, L)) → U5(member1_in(L))
member1_in(.(H, L)) → member1_out(H)
U5(member1_out(X)) → member1_out(X)

The set Q consists of the following terms:

member1_in(x0)
U5(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule REACH_IN(X, Z, Edges) → U2¹(Z, Edges, member1_in(Edges)) at position [2] we obtained the following new rules:

REACH_IN(y0, y1, .(x0, x1)) → U2¹(y1, .(x0, x1), U5(member1_in(x1)))
REACH_IN(y0, y1, .(x0, x1)) → U2¹(y1, .(x0, x1), member1_out(x0))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Instantiation

  ↳ PrologToPiTRSProof

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

REACH_IN(y0, y1, .(x0, x1)) → U2¹(y1, .(x0, x1), U5(member1_in(x1)))
U2¹(Z, Edges, member1_out(.(X, .(Y, [])))) → REACH_IN(Y, Z, Edges)
REACH_IN(y0, y1, .(x0, x1)) → U2¹(y1, .(x0, x1), member1_out(x0))

The TRS R consists of the following rules:

member1_in(.(H, L)) → U5(member1_in(L))
member1_in(.(H, L)) → member1_out(H)
U5(member1_out(X)) → member1_out(X)

The set Q consists of the following terms:

member1_in(x0)
U5(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2¹(Z, Edges, member1_out(.(X, .(Y, [])))) → REACH_IN(Y, Z, Edges) we obtained the following new rules:

U2¹(z1, .(.(x2, .(x3, [])), z3), member1_out(.(x2, .(x3, [])))) → REACH_IN(x3, z1, .(.(x2, .(x3, [])), z3))
U2¹(z1, .(z2, z3), member1_out(.(x2, .(x3, [])))) → REACH_IN(x3, z1, .(z2, z3))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Instantiation

                              ↳ QDP

                                ↳ ForwardInstantiation

  ↳ PrologToPiTRSProof

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

U2¹(z1, .(.(x2, .(x3, [])), z3), member1_out(.(x2, .(x3, [])))) → REACH_IN(x3, z1, .(.(x2, .(x3, [])), z3))
U2¹(z1, .(z2, z3), member1_out(.(x2, .(x3, [])))) → REACH_IN(x3, z1, .(z2, z3))
REACH_IN(y0, y1, .(x0, x1)) → U2¹(y1, .(x0, x1), U5(member1_in(x1)))
REACH_IN(y0, y1, .(x0, x1)) → U2¹(y1, .(x0, x1), member1_out(x0))

The TRS R consists of the following rules:

member1_in(.(H, L)) → U5(member1_in(L))
member1_in(.(H, L)) → member1_out(H)
U5(member1_out(X)) → member1_out(X)

The set Q consists of the following terms:

member1_in(x0)
U5(x0)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule REACH_IN(y0, y1, .(x0, x1)) → U2¹(y1, .(x0, x1), member1_out(x0)) we obtained the following new rules:

REACH_IN(x0, x1, .(.(y_1, .(y_2, [])), x3)) → U2¹(x1, .(.(y_1, .(y_2, [])), x3), member1_out(.(y_1, .(y_2, []))))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Instantiation

                              ↳ QDP

                                ↳ ForwardInstantiation

                                  ↳ QDP

                                    ↳ NonTerminationProof

  ↳ PrologToPiTRSProof

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

U2¹(z1, .(.(x2, .(x3, [])), z3), member1_out(.(x2, .(x3, [])))) → REACH_IN(x3, z1, .(.(x2, .(x3, [])), z3))
REACH_IN(x0, x1, .(.(y_1, .(y_2, [])), x3)) → U2¹(x1, .(.(y_1, .(y_2, [])), x3), member1_out(.(y_1, .(y_2, []))))
U2¹(z1, .(z2, z3), member1_out(.(x2, .(x3, [])))) → REACH_IN(x3, z1, .(z2, z3))
REACH_IN(y0, y1, .(x0, x1)) → U2¹(y1, .(x0, x1), U5(member1_in(x1)))

The TRS R consists of the following rules:

member1_in(.(H, L)) → U5(member1_in(L))
member1_in(.(H, L)) → member1_out(H)
U5(member1_out(X)) → member1_out(X)

The set Q consists of the following terms:

member1_in(x0)
U5(x0)

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 narrowing to the left:

The TRS P consists of the following rules:

U2¹(z1, .(.(x2, .(x3, [])), z3), member1_out(.(x2, .(x3, [])))) → REACH_IN(x3, z1, .(.(x2, .(x3, [])), z3))
REACH_IN(x0, x1, .(.(y_1, .(y_2, [])), x3)) → U2¹(x1, .(.(y_1, .(y_2, [])), x3), member1_out(.(y_1, .(y_2, []))))
U2¹(z1, .(z2, z3), member1_out(.(x2, .(x3, [])))) → REACH_IN(x3, z1, .(z2, z3))
REACH_IN(y0, y1, .(x0, x1)) → U2¹(y1, .(x0, x1), U5(member1_in(x1)))

The TRS R consists of the following rules:

member1_in(.(H, L)) → U5(member1_in(L))
member1_in(.(H, L)) → member1_out(H)
U5(member1_out(X)) → member1_out(X)

s = REACH_IN(x0, x1, .(.(y_1, .(y_2, [])), x3')) evaluates to t =REACH_IN(y_2, x1, .(.(y_1, .(y_2, [])), x3'))

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

Matcher: [x0 / y_2]
Semiunifier: [ ]

Rewriting sequence

REACH_IN(x0, x1, .(.(y_1, .(y_2, [])), x3')) → U2¹(x1, .(.(y_1, .(y_2, [])), x3'), member1_out(.(y_1, .(y_2, []))))
with rule REACH_IN(x0', x1', .(.(y_1', .(y_2', [])), x3'')) → U2¹(x1', .(.(y_1', .(y_2', [])), x3''), member1_out(.(y_1', .(y_2', [])))) at position [] and matcher [y_2' / y_2, x1' / x1, x3'' / x3', y_1' / y_1, x0' / x0]

U2¹(x1, .(.(y_1, .(y_2, [])), x3'), member1_out(.(y_1, .(y_2, [])))) → REACH_IN(y_2, x1, .(.(y_1, .(y_2, [])), x3'))
with rule U2¹(z1, .(.(x2, .(x3, [])), z3), member1_out(.(x2, .(x3, [])))) → REACH_IN(x3, z1, .(.(x2, .(x3, [])), z3))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

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:

reach_in(X, Z, Edges) → U2(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
member1_in(X, .(H, L)) → U5(X, H, L, member1_in(X, L))
member1_in(H, .(H, L)) → member1_out(H, .(H, L))
U5(X, H, L, member1_out(X, L)) → member1_out(X, .(H, L))
U2(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → U3(X, Z, Edges, reach_in(Y, Z, Edges))
reach_in(X, Y, Edges) → U1(X, Y, Edges, member_in(.(X, .(Y, [])), Edges))
member_in(X, .(H, L)) → U4(X, H, L, member_in(X, L))
member_in(H, .(H, L)) → member_out(H, .(H, L))
U4(X, H, L, member_out(X, L)) → member_out(X, .(H, L))
U1(X, Y, Edges, member_out(.(X, .(Y, [])), Edges)) → reach_out(X, Y, Edges)
U3(X, Z, Edges, reach_out(Y, Z, Edges)) → reach_out(X, Z, Edges)

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:

reach_in(X, Z, Edges) → U2(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
member1_in(X, .(H, L)) → U5(X, H, L, member1_in(X, L))
member1_in(H, .(H, L)) → member1_out(H, .(H, L))
U5(X, H, L, member1_out(X, L)) → member1_out(X, .(H, L))
U2(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → U3(X, Z, Edges, reach_in(Y, Z, Edges))
reach_in(X, Y, Edges) → U1(X, Y, Edges, member_in(.(X, .(Y, [])), Edges))
member_in(X, .(H, L)) → U4(X, H, L, member_in(X, L))
member_in(H, .(H, L)) → member_out(H, .(H, L))
U4(X, H, L, member_out(X, L)) → member_out(X, .(H, L))
U1(X, Y, Edges, member_out(.(X, .(Y, [])), Edges)) → reach_out(X, Y, Edges)
U3(X, Z, Edges, reach_out(Y, Z, Edges)) → reach_out(X, Z, Edges)

REACH_IN(X, Z, Edges) → U2¹(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
REACH_IN(X, Z, Edges) → MEMBER1_IN(.(X, .(Y, [])), Edges)
MEMBER1_IN(X, .(H, L)) → U5¹(X, H, L, member1_in(X, L))
MEMBER1_IN(X, .(H, L)) → MEMBER1_IN(X, L)
U2¹(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → U3¹(X, Z, Edges, reach_in(Y, Z, Edges))
U2¹(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → REACH_IN(Y, Z, Edges)
REACH_IN(X, Y, Edges) → U1¹(X, Y, Edges, member_in(.(X, .(Y, [])), Edges))
REACH_IN(X, Y, Edges) → MEMBER_IN(.(X, .(Y, [])), Edges)
MEMBER_IN(X, .(H, L)) → U4¹(X, H, L, member_in(X, L))
MEMBER_IN(X, .(H, L)) → MEMBER_IN(X, L)

The TRS R consists of the following rules:

reach_in(X, Z, Edges) → U2(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
member1_in(X, .(H, L)) → U5(X, H, L, member1_in(X, L))
member1_in(H, .(H, L)) → member1_out(H, .(H, L))
U5(X, H, L, member1_out(X, L)) → member1_out(X, .(H, L))
U2(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → U3(X, Z, Edges, reach_in(Y, Z, Edges))
reach_in(X, Y, Edges) → U1(X, Y, Edges, member_in(.(X, .(Y, [])), Edges))
member_in(X, .(H, L)) → U4(X, H, L, member_in(X, L))
member_in(H, .(H, L)) → member_out(H, .(H, L))
U4(X, H, L, member_out(X, L)) → member_out(X, .(H, L))
U1(X, Y, Edges, member_out(.(X, .(Y, [])), Edges)) → reach_out(X, Y, Edges)
U3(X, Z, Edges, reach_out(Y, Z, Edges)) → reach_out(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_in(x1, x2, x3) = reach_in(x1, x2, x3)
U2(x1, x2, x3, x4) = U2(x1, x2, x3, x4)
member1_in(x1, x2) = member1_in(x2)
.(x1, x2) = .(x1, x2)
[] = []
U5(x1, x2, x3, x4) = U5(x2, x3, x4)
member1_out(x1, x2) = member1_out(x1, x2)
U3(x1, x2, x3, x4) = U3(x1, x2, x3, x4)
U1(x1, x2, x3, x4) = U1(x1, x2, x3, x4)
member_in(x1, x2) = member_in(x1, x2)
U4(x1, x2, x3, x4) = U4(x1, x2, x3, x4)
member_out(x1, x2) = member_out(x1, x2)
reach_out(x1, x2, x3) = reach_out(x1, x2, x3)
U5¹(x1, x2, x3, x4) = U5¹(x2, x3, x4)
U3¹(x1, x2, x3, x4) = U3¹(x1, x2, x3, x4)
REACH_IN(x1, x2, x3) = REACH_IN(x1, x2, x3)
U4¹(x1, x2, x3, x4) = U4¹(x1, x2, x3, x4)
MEMBER1_IN(x1, x2) = MEMBER1_IN(x2)
MEMBER_IN(x1, x2) = MEMBER_IN(x1, x2)
U2¹(x1, x2, x3, x4) = U2¹(x1, x2, x3, x4)
U1¹(x1, x2, x3, x4) = U1¹(x1, x2, x3, 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:

REACH_IN(X, Z, Edges) → U2¹(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
REACH_IN(X, Z, Edges) → MEMBER1_IN(.(X, .(Y, [])), Edges)
MEMBER1_IN(X, .(H, L)) → U5¹(X, H, L, member1_in(X, L))
MEMBER1_IN(X, .(H, L)) → MEMBER1_IN(X, L)
U2¹(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → U3¹(X, Z, Edges, reach_in(Y, Z, Edges))
U2¹(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → REACH_IN(Y, Z, Edges)
REACH_IN(X, Y, Edges) → U1¹(X, Y, Edges, member_in(.(X, .(Y, [])), Edges))
REACH_IN(X, Y, Edges) → MEMBER_IN(.(X, .(Y, [])), Edges)
MEMBER_IN(X, .(H, L)) → U4¹(X, H, L, member_in(X, L))
MEMBER_IN(X, .(H, L)) → MEMBER_IN(X, L)

The TRS R consists of the following rules:

reach_in(X, Z, Edges) → U2(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
member1_in(X, .(H, L)) → U5(X, H, L, member1_in(X, L))
member1_in(H, .(H, L)) → member1_out(H, .(H, L))
U5(X, H, L, member1_out(X, L)) → member1_out(X, .(H, L))
U2(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → U3(X, Z, Edges, reach_in(Y, Z, Edges))
reach_in(X, Y, Edges) → U1(X, Y, Edges, member_in(.(X, .(Y, [])), Edges))
member_in(X, .(H, L)) → U4(X, H, L, member_in(X, L))
member_in(H, .(H, L)) → member_out(H, .(H, L))
U4(X, H, L, member_out(X, L)) → member_out(X, .(H, L))
U1(X, Y, Edges, member_out(.(X, .(Y, [])), Edges)) → reach_out(X, Y, Edges)
U3(X, Z, Edges, reach_out(Y, Z, Edges)) → reach_out(X, Z, Edges)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

MEMBER_IN(X, .(H, L)) → MEMBER_IN(X, L)

The TRS R consists of the following rules:

reach_in(X, Z, Edges) → U2(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
member1_in(X, .(H, L)) → U5(X, H, L, member1_in(X, L))
member1_in(H, .(H, L)) → member1_out(H, .(H, L))
U5(X, H, L, member1_out(X, L)) → member1_out(X, .(H, L))
U2(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → U3(X, Z, Edges, reach_in(Y, Z, Edges))
reach_in(X, Y, Edges) → U1(X, Y, Edges, member_in(.(X, .(Y, [])), Edges))
member_in(X, .(H, L)) → U4(X, H, L, member_in(X, L))
member_in(H, .(H, L)) → member_out(H, .(H, L))
U4(X, H, L, member_out(X, L)) → member_out(X, .(H, L))
U1(X, Y, Edges, member_out(.(X, .(Y, [])), Edges)) → reach_out(X, Y, Edges)
U3(X, Z, Edges, reach_out(Y, Z, Edges)) → reach_out(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_in(x1, x2, x3) = reach_in(x1, x2, x3)
U2(x1, x2, x3, x4) = U2(x1, x2, x3, x4)
member1_in(x1, x2) = member1_in(x2)
.(x1, x2) = .(x1, x2)
[] = []
U5(x1, x2, x3, x4) = U5(x2, x3, x4)
member1_out(x1, x2) = member1_out(x1, x2)
U3(x1, x2, x3, x4) = U3(x1, x2, x3, x4)
U1(x1, x2, x3, x4) = U1(x1, x2, x3, x4)
member_in(x1, x2) = member_in(x1, x2)
U4(x1, x2, x3, x4) = U4(x1, x2, x3, x4)
member_out(x1, x2) = member_out(x1, x2)
reach_out(x1, x2, x3) = reach_out(x1, x2, x3)
MEMBER_IN(x1, x2) = MEMBER_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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

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

MEMBER_IN(X, .(H, L)) → MEMBER_IN(X, L)

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

  ↳ 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:

MEMBER_IN(X, .(H, L)) → MEMBER_IN(X, L)

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

MEMBER_IN(X, .(H, L)) → MEMBER_IN(X, L)
The graph contains the following edges 1 >= 1, 2 > 2

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

MEMBER1_IN(X, .(H, L)) → MEMBER1_IN(X, L)

The TRS R consists of the following rules:

reach_in(X, Z, Edges) → U2(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
member1_in(X, .(H, L)) → U5(X, H, L, member1_in(X, L))
member1_in(H, .(H, L)) → member1_out(H, .(H, L))
U5(X, H, L, member1_out(X, L)) → member1_out(X, .(H, L))
U2(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → U3(X, Z, Edges, reach_in(Y, Z, Edges))
reach_in(X, Y, Edges) → U1(X, Y, Edges, member_in(.(X, .(Y, [])), Edges))
member_in(X, .(H, L)) → U4(X, H, L, member_in(X, L))
member_in(H, .(H, L)) → member_out(H, .(H, L))
U4(X, H, L, member_out(X, L)) → member_out(X, .(H, L))
U1(X, Y, Edges, member_out(.(X, .(Y, [])), Edges)) → reach_out(X, Y, Edges)
U3(X, Z, Edges, reach_out(Y, Z, Edges)) → reach_out(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_in(x1, x2, x3) = reach_in(x1, x2, x3)
U2(x1, x2, x3, x4) = U2(x1, x2, x3, x4)
member1_in(x1, x2) = member1_in(x2)
.(x1, x2) = .(x1, x2)
[] = []
U5(x1, x2, x3, x4) = U5(x2, x3, x4)
member1_out(x1, x2) = member1_out(x1, x2)
U3(x1, x2, x3, x4) = U3(x1, x2, x3, x4)
U1(x1, x2, x3, x4) = U1(x1, x2, x3, x4)
member_in(x1, x2) = member_in(x1, x2)
U4(x1, x2, x3, x4) = U4(x1, x2, x3, x4)
member_out(x1, x2) = member_out(x1, x2)
reach_out(x1, x2, x3) = reach_out(x1, x2, x3)
MEMBER1_IN(x1, x2) = MEMBER1_IN(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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

MEMBER1_IN(X, .(H, L)) → MEMBER1_IN(X, L)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

MEMBER1_IN(.(H, L)) → MEMBER1_IN(L)

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

MEMBER1_IN(.(H, L)) → MEMBER1_IN(L)
The graph contains the following edges 1 > 1

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

REACH_IN(X, Z, Edges) → U2¹(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
U2¹(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → REACH_IN(Y, Z, Edges)

The TRS R consists of the following rules:

reach_in(X, Z, Edges) → U2(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
member1_in(X, .(H, L)) → U5(X, H, L, member1_in(X, L))
member1_in(H, .(H, L)) → member1_out(H, .(H, L))
U5(X, H, L, member1_out(X, L)) → member1_out(X, .(H, L))
U2(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → U3(X, Z, Edges, reach_in(Y, Z, Edges))
reach_in(X, Y, Edges) → U1(X, Y, Edges, member_in(.(X, .(Y, [])), Edges))
member_in(X, .(H, L)) → U4(X, H, L, member_in(X, L))
member_in(H, .(H, L)) → member_out(H, .(H, L))
U4(X, H, L, member_out(X, L)) → member_out(X, .(H, L))
U1(X, Y, Edges, member_out(.(X, .(Y, [])), Edges)) → reach_out(X, Y, Edges)
U3(X, Z, Edges, reach_out(Y, Z, Edges)) → reach_out(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_in(x1, x2, x3) = reach_in(x1, x2, x3)
U2(x1, x2, x3, x4) = U2(x1, x2, x3, x4)
member1_in(x1, x2) = member1_in(x2)
.(x1, x2) = .(x1, x2)
[] = []
U5(x1, x2, x3, x4) = U5(x2, x3, x4)
member1_out(x1, x2) = member1_out(x1, x2)
U3(x1, x2, x3, x4) = U3(x1, x2, x3, x4)
U1(x1, x2, x3, x4) = U1(x1, x2, x3, x4)
member_in(x1, x2) = member_in(x1, x2)
U4(x1, x2, x3, x4) = U4(x1, x2, x3, x4)
member_out(x1, x2) = member_out(x1, x2)
reach_out(x1, x2, x3) = reach_out(x1, x2, x3)
REACH_IN(x1, x2, x3) = REACH_IN(x1, x2, x3)
U2¹(x1, x2, x3, x4) = U2¹(x1, x2, x3, x4)

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:

REACH_IN(X, Z, Edges) → U2¹(X, Z, Edges, member1_in(.(X, .(Y, [])), Edges))
U2¹(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → REACH_IN(Y, Z, Edges)

The TRS R consists of the following rules:

member1_in(X, .(H, L)) → U5(X, H, L, member1_in(X, L))
member1_in(H, .(H, L)) → member1_out(H, .(H, L))
U5(X, H, L, member1_out(X, L)) → member1_out(X, .(H, L))

The argument filtering Pi contains the following mapping:
member1_in(x1, x2) = member1_in(x2)
.(x1, x2) = .(x1, x2)
[] = []
U5(x1, x2, x3, x4) = U5(x2, x3, x4)
member1_out(x1, x2) = member1_out(x1, x2)
REACH_IN(x1, x2, x3) = REACH_IN(x1, x2, x3)
U2¹(x1, x2, x3, x4) = U2¹(x1, x2, x3, x4)

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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

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

REACH_IN(X, Z, Edges) → U2¹(X, Z, Edges, member1_in(Edges))
U2¹(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → REACH_IN(Y, Z, Edges)

The TRS R consists of the following rules:

member1_in(.(H, L)) → U5(H, L, member1_in(L))
member1_in(.(H, L)) → member1_out(H, .(H, L))
U5(H, L, member1_out(X, L)) → member1_out(X, .(H, L))

The set Q consists of the following terms:

member1_in(x0)
U5(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule REACH_IN(X, Z, Edges) → U2¹(X, Z, Edges, member1_in(Edges)) at position [3] we obtained the following new rules:

REACH_IN(y0, y1, .(x0, x1)) → U2¹(y0, y1, .(x0, x1), member1_out(x0, .(x0, x1)))
REACH_IN(y0, y1, .(x0, x1)) → U2¹(y0, y1, .(x0, x1), U5(x0, x1, member1_in(x1)))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Instantiation

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

U2¹(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → REACH_IN(Y, Z, Edges)
REACH_IN(y0, y1, .(x0, x1)) → U2¹(y0, y1, .(x0, x1), member1_out(x0, .(x0, x1)))
REACH_IN(y0, y1, .(x0, x1)) → U2¹(y0, y1, .(x0, x1), U5(x0, x1, member1_in(x1)))

The TRS R consists of the following rules:

member1_in(.(H, L)) → U5(H, L, member1_in(L))
member1_in(.(H, L)) → member1_out(H, .(H, L))
U5(H, L, member1_out(X, L)) → member1_out(X, .(H, L))

The set Q consists of the following terms:

member1_in(x0)
U5(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2¹(X, Z, Edges, member1_out(.(X, .(Y, [])), Edges)) → REACH_IN(Y, Z, Edges) we obtained the following new rules:

U2¹(z0, z1, .(z2, z3), member1_out(.(z0, .(x3, [])), .(z2, z3))) → REACH_IN(x3, z1, .(z2, z3))
U2¹(z0, z1, .(.(z0, .(x3, [])), z3), member1_out(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) → REACH_IN(x3, z1, .(.(z0, .(x3, [])), z3))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Instantiation

                              ↳ QDP

                                ↳ ForwardInstantiation

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

U2¹(z0, z1, .(z2, z3), member1_out(.(z0, .(x3, [])), .(z2, z3))) → REACH_IN(x3, z1, .(z2, z3))
REACH_IN(y0, y1, .(x0, x1)) → U2¹(y0, y1, .(x0, x1), member1_out(x0, .(x0, x1)))
U2¹(z0, z1, .(.(z0, .(x3, [])), z3), member1_out(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) → REACH_IN(x3, z1, .(.(z0, .(x3, [])), z3))
REACH_IN(y0, y1, .(x0, x1)) → U2¹(y0, y1, .(x0, x1), U5(x0, x1, member1_in(x1)))

The TRS R consists of the following rules:

member1_in(.(H, L)) → U5(H, L, member1_in(L))
member1_in(.(H, L)) → member1_out(H, .(H, L))
U5(H, L, member1_out(X, L)) → member1_out(X, .(H, L))

The set Q consists of the following terms:

member1_in(x0)
U5(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule REACH_IN(y0, y1, .(x0, x1)) → U2¹(y0, y1, .(x0, x1), member1_out(x0, .(x0, x1))) we obtained the following new rules:

REACH_IN(x0, x1, .(.(y_4, .(y_5, [])), x3)) → U2¹(x0, x1, .(.(y_4, .(y_5, [])), x3), member1_out(.(y_4, .(y_5, [])), .(.(y_4, .(y_5, [])), x3)))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Instantiation

                              ↳ QDP

                                ↳ ForwardInstantiation

                                  ↳ QDP

                                    ↳ NonTerminationProof

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

U2¹(z0, z1, .(z2, z3), member1_out(.(z0, .(x3, [])), .(z2, z3))) → REACH_IN(x3, z1, .(z2, z3))
REACH_IN(x0, x1, .(.(y_4, .(y_5, [])), x3)) → U2¹(x0, x1, .(.(y_4, .(y_5, [])), x3), member1_out(.(y_4, .(y_5, [])), .(.(y_4, .(y_5, [])), x3)))
U2¹(z0, z1, .(.(z0, .(x3, [])), z3), member1_out(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) → REACH_IN(x3, z1, .(.(z0, .(x3, [])), z3))
REACH_IN(y0, y1, .(x0, x1)) → U2¹(y0, y1, .(x0, x1), U5(x0, x1, member1_in(x1)))

The TRS R consists of the following rules:

member1_in(.(H, L)) → U5(H, L, member1_in(L))
member1_in(.(H, L)) → member1_out(H, .(H, L))
U5(H, L, member1_out(X, L)) → member1_out(X, .(H, L))

The set Q consists of the following terms:

member1_in(x0)
U5(x0, x1, x2)

U2¹(z0, z1, .(z2, z3), member1_out(.(z0, .(x3, [])), .(z2, z3))) → REACH_IN(x3, z1, .(z2, z3))
REACH_IN(x0, x1, .(.(y_4, .(y_5, [])), x3)) → U2¹(x0, x1, .(.(y_4, .(y_5, [])), x3), member1_out(.(y_4, .(y_5, [])), .(.(y_4, .(y_5, [])), x3)))
U2¹(z0, z1, .(.(z0, .(x3, [])), z3), member1_out(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) → REACH_IN(x3, z1, .(.(z0, .(x3, [])), z3))
REACH_IN(y0, y1, .(x0, x1)) → U2¹(y0, y1, .(x0, x1), U5(x0, x1, member1_in(x1)))

The TRS R consists of the following rules:

member1_in(.(H, L)) → U5(H, L, member1_in(L))
member1_in(.(H, L)) → member1_out(H, .(H, L))
U5(H, L, member1_out(X, L)) → member1_out(X, .(H, L))

s = REACH_IN(x0, x1, .(.(x0, .(y_5, [])), x3')) evaluates to t =REACH_IN(y_5, x1, .(.(x0, .(y_5, [])), x3'))

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

Semiunifier: [y_5 / x0]
Matcher: [ ]

Rewriting sequence

REACH_IN(x0, x1, .(.(x0, .(x0, [])), x3')) → U2¹(x0, x1, .(.(x0, .(x0, [])), x3'), member1_out(.(x0, .(x0, [])), .(.(x0, .(x0, [])), x3')))
with rule REACH_IN(x0', x1', .(.(y_4, .(y_5, [])), x3'')) → U2¹(x0', x1', .(.(y_4, .(y_5, [])), x3''), member1_out(.(y_4, .(y_5, [])), .(.(y_4, .(y_5, [])), x3''))) at position [] and matcher [y_4 / x0, x1' / x1, x3'' / x3', x0' / x0, y_5 / x0]

U2¹(x0, x1, .(.(x0, .(x0, [])), x3'), member1_out(.(x0, .(x0, [])), .(.(x0, .(x0, [])), x3'))) → REACH_IN(x0, x1, .(.(x0, .(x0, [])), x3'))
with rule U2¹(z0, z1, .(z2, z3), member1_out(.(z0, .(x3, [])), .(z2, z3))) → REACH_IN(x3, z1, .(z2, z3))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.