Left Termination proof of /tmp/tmp5BJpaT/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]. With regard to the inferred argument filtering the predicates were used in the following modes:
reach_in: (b,b,b)
member_in: (b,b)
member1_in: (f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U5_ag(X, H, L, member1_in_ag(X, L))
U5_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) → reach_out_ggg(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_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U5_ag(X, H, L, member1_in_ag(X, L))
U5_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

REACH_IN_GGG(X, Y, Edges) → U1_GGG(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_GGG(X, Y, Edges) → MEMBER_IN_GG(.(X, .(Y, [])), Edges)
MEMBER_IN_GG(X, .(H, L)) → U4_GG(X, H, L, member_in_gg(X, L))
MEMBER_IN_GG(X, .(H, L)) → MEMBER_IN_GG(X, L)
REACH_IN_GGG(X, Z, Edges) → U2_GGG(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
REACH_IN_GGG(X, Z, Edges) → MEMBER1_IN_AG(.(X, .(Y, [])), Edges)
MEMBER1_IN_AG(X, .(H, L)) → U5_AG(X, H, L, member1_in_ag(X, L))
MEMBER1_IN_AG(X, .(H, L)) → MEMBER1_IN_AG(X, L)
U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_GGG(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → REACH_IN_GGG(Y, Z, Edges)

The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U5_ag(X, H, L, member1_in_ag(X, L))
U5_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_in_ggg(x1, x2, x3) = reach_in_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4) = U1_ggg(x4)
member_in_gg(x1, x2) = member_in_gg(x1, x2)
.(x1, x2) = .(x1, x2)
member_out_gg(x1, x2) = member_out_gg
U4_gg(x1, x2, x3, x4) = U4_gg(x4)
[] = []
reach_out_ggg(x1, x2, x3) = reach_out_ggg
U2_ggg(x1, x2, x3, x4) = U2_ggg(x2, x3, x4)
member1_in_ag(x1, x2) = member1_in_ag(x2)
member1_out_ag(x1, x2) = member1_out_ag(x1)
U5_ag(x1, x2, x3, x4) = U5_ag(x4)
U3_ggg(x1, x2, x3, x4) = U3_ggg(x4)
U1_GGG(x1, x2, x3, x4) = U1_GGG(x4)
MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2)
U3_GGG(x1, x2, x3, x4) = U3_GGG(x4)
REACH_IN_GGG(x1, x2, x3) = REACH_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4) = U2_GGG(x2, x3, x4)
MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2)
U4_GG(x1, x2, x3, x4) = U4_GG(x4)
U5_AG(x1, x2, x3, x4) = U5_AG(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_GGG(X, Y, Edges) → U1_GGG(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_GGG(X, Y, Edges) → MEMBER_IN_GG(.(X, .(Y, [])), Edges)
MEMBER_IN_GG(X, .(H, L)) → U4_GG(X, H, L, member_in_gg(X, L))
MEMBER_IN_GG(X, .(H, L)) → MEMBER_IN_GG(X, L)
REACH_IN_GGG(X, Z, Edges) → U2_GGG(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
REACH_IN_GGG(X, Z, Edges) → MEMBER1_IN_AG(.(X, .(Y, [])), Edges)
MEMBER1_IN_AG(X, .(H, L)) → U5_AG(X, H, L, member1_in_ag(X, L))
MEMBER1_IN_AG(X, .(H, L)) → MEMBER1_IN_AG(X, L)
U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_GGG(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → REACH_IN_GGG(Y, Z, Edges)

The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U5_ag(X, H, L, member1_in_ag(X, L))
U5_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) → reach_out_ggg(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:

MEMBER1_IN_AG(X, .(H, L)) → MEMBER1_IN_AG(X, L)

The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U5_ag(X, H, L, member1_in_ag(X, L))
U5_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_in_ggg(x1, x2, x3) = reach_in_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4) = U1_ggg(x4)
member_in_gg(x1, x2) = member_in_gg(x1, x2)
.(x1, x2) = .(x1, x2)
member_out_gg(x1, x2) = member_out_gg
U4_gg(x1, x2, x3, x4) = U4_gg(x4)
[] = []
reach_out_ggg(x1, x2, x3) = reach_out_ggg
U2_ggg(x1, x2, x3, x4) = U2_ggg(x2, x3, x4)
member1_in_ag(x1, x2) = member1_in_ag(x2)
member1_out_ag(x1, x2) = member1_out_ag(x1)
U5_ag(x1, x2, x3, x4) = U5_ag(x4)
U3_ggg(x1, x2, x3, x4) = U3_ggg(x4)
MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(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:

MEMBER1_IN_AG(X, .(H, L)) → MEMBER1_IN_AG(X, L)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(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

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

MEMBER1_IN_AG(.(H, L)) → MEMBER1_IN_AG(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:

MEMBER1_IN_AG(.(H, L)) → MEMBER1_IN_AG(L)
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:

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

The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U5_ag(X, H, L, member1_in_ag(X, L))
U5_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_in_ggg(x1, x2, x3) = reach_in_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4) = U1_ggg(x4)
member_in_gg(x1, x2) = member_in_gg(x1, x2)
.(x1, x2) = .(x1, x2)
member_out_gg(x1, x2) = member_out_gg
U4_gg(x1, x2, x3, x4) = U4_gg(x4)
[] = []
reach_out_ggg(x1, x2, x3) = reach_out_ggg
U2_ggg(x1, x2, x3, x4) = U2_ggg(x2, x3, x4)
member1_in_ag(x1, x2) = member1_in_ag(x2)
member1_out_ag(x1, x2) = member1_out_ag(x1)
U5_ag(x1, x2, x3, x4) = U5_ag(x4)
U3_ggg(x1, x2, x3, x4) = U3_ggg(x4)
MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → REACH_IN_GGG(Y, Z, Edges)
REACH_IN_GGG(X, Z, Edges) → U2_GGG(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))

The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U5_ag(X, H, L, member1_in_ag(X, L))
U5_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_in_ggg(x1, x2, x3) = reach_in_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4) = U1_ggg(x4)
member_in_gg(x1, x2) = member_in_gg(x1, x2)
.(x1, x2) = .(x1, x2)
member_out_gg(x1, x2) = member_out_gg
U4_gg(x1, x2, x3, x4) = U4_gg(x4)
[] = []
reach_out_ggg(x1, x2, x3) = reach_out_ggg
U2_ggg(x1, x2, x3, x4) = U2_ggg(x2, x3, x4)
member1_in_ag(x1, x2) = member1_in_ag(x2)
member1_out_ag(x1, x2) = member1_out_ag(x1)
U5_ag(x1, x2, x3, x4) = U5_ag(x4)
U3_ggg(x1, x2, x3, x4) = U3_ggg(x4)
REACH_IN_GGG(x1, x2, x3) = REACH_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4) = U2_GGG(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:

U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → REACH_IN_GGG(Y, Z, Edges)
REACH_IN_GGG(X, Z, Edges) → U2_GGG(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))

The TRS R consists of the following rules:

member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U5_ag(X, H, L, member1_in_ag(X, L))
U5_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
member1_in_ag(x1, x2) = member1_in_ag(x2)
member1_out_ag(x1, x2) = member1_out_ag(x1)
U5_ag(x1, x2, x3, x4) = U5_ag(x4)
REACH_IN_GGG(x1, x2, x3) = REACH_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4) = U2_GGG(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_GGG(X, Z, Edges) → U2_GGG(Z, Edges, member1_in_ag(Edges))
U2_GGG(Z, Edges, member1_out_ag(.(X, .(Y, [])))) → REACH_IN_GGG(Y, Z, Edges)

The TRS R consists of the following rules:

member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U5_ag(member1_in_ag(L))
U5_ag(member1_out_ag(X)) → member1_out_ag(X)

The set Q consists of the following terms:

member1_in_ag(x0)
U5_ag(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule REACH_IN_GGG(X, Z, Edges) → U2_GGG(Z, Edges, member1_in_ag(Edges)) at position [2] we obtained the following new rules:

REACH_IN_GGG(y0, y1, .(x0, x1)) → U2_GGG(y1, .(x0, x1), member1_out_ag(x0))
REACH_IN_GGG(y0, y1, .(x0, x1)) → U2_GGG(y1, .(x0, x1), U5_ag(member1_in_ag(x1)))

↳ 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_GGG(y0, y1, .(x0, x1)) → U2_GGG(y1, .(x0, x1), member1_out_ag(x0))
REACH_IN_GGG(y0, y1, .(x0, x1)) → U2_GGG(y1, .(x0, x1), U5_ag(member1_in_ag(x1)))
U2_GGG(Z, Edges, member1_out_ag(.(X, .(Y, [])))) → REACH_IN_GGG(Y, Z, Edges)

The TRS R consists of the following rules:

member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U5_ag(member1_in_ag(L))
U5_ag(member1_out_ag(X)) → member1_out_ag(X)

The set Q consists of the following terms:

member1_in_ag(x0)
U5_ag(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_GGG(Z, Edges, member1_out_ag(.(X, .(Y, [])))) → REACH_IN_GGG(Y, Z, Edges) we obtained the following new rules:

U2_GGG(z1, .(z2, z3), member1_out_ag(.(x2, .(x3, [])))) → REACH_IN_GGG(x3, z1, .(z2, z3))
U2_GGG(z1, .(.(x2, .(x3, [])), z3), member1_out_ag(.(x2, .(x3, [])))) → REACH_IN_GGG(x3, z1, .(.(x2, .(x3, [])), 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:

REACH_IN_GGG(y0, y1, .(x0, x1)) → U2_GGG(y1, .(x0, x1), member1_out_ag(x0))
U2_GGG(z1, .(z2, z3), member1_out_ag(.(x2, .(x3, [])))) → REACH_IN_GGG(x3, z1, .(z2, z3))
REACH_IN_GGG(y0, y1, .(x0, x1)) → U2_GGG(y1, .(x0, x1), U5_ag(member1_in_ag(x1)))
U2_GGG(z1, .(.(x2, .(x3, [])), z3), member1_out_ag(.(x2, .(x3, [])))) → REACH_IN_GGG(x3, z1, .(.(x2, .(x3, [])), z3))

The TRS R consists of the following rules:

member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U5_ag(member1_in_ag(L))
U5_ag(member1_out_ag(X)) → member1_out_ag(X)

The set Q consists of the following terms:

member1_in_ag(x0)
U5_ag(x0)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule REACH_IN_GGG(y0, y1, .(x0, x1)) → U2_GGG(y1, .(x0, x1), member1_out_ag(x0)) we obtained the following new rules:

REACH_IN_GGG(x0, x1, .(.(y_3, .(y_4, [])), x3)) → U2_GGG(x1, .(.(y_3, .(y_4, [])), x3), member1_out_ag(.(y_3, .(y_4, []))))

↳ 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:

REACH_IN_GGG(y0, y1, .(x0, x1)) → U2_GGG(y1, .(x0, x1), U5_ag(member1_in_ag(x1)))
U2_GGG(z1, .(z2, z3), member1_out_ag(.(x2, .(x3, [])))) → REACH_IN_GGG(x3, z1, .(z2, z3))
U2_GGG(z1, .(.(x2, .(x3, [])), z3), member1_out_ag(.(x2, .(x3, [])))) → REACH_IN_GGG(x3, z1, .(.(x2, .(x3, [])), z3))
REACH_IN_GGG(x0, x1, .(.(y_3, .(y_4, [])), x3)) → U2_GGG(x1, .(.(y_3, .(y_4, [])), x3), member1_out_ag(.(y_3, .(y_4, []))))

The TRS R consists of the following rules:

member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U5_ag(member1_in_ag(L))
U5_ag(member1_out_ag(X)) → member1_out_ag(X)

The set Q consists of the following terms:

member1_in_ag(x0)
U5_ag(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:

REACH_IN_GGG(y0, y1, .(x0, x1)) → U2_GGG(y1, .(x0, x1), U5_ag(member1_in_ag(x1)))
U2_GGG(z1, .(z2, z3), member1_out_ag(.(x2, .(x3, [])))) → REACH_IN_GGG(x3, z1, .(z2, z3))
U2_GGG(z1, .(.(x2, .(x3, [])), z3), member1_out_ag(.(x2, .(x3, [])))) → REACH_IN_GGG(x3, z1, .(.(x2, .(x3, [])), z3))
REACH_IN_GGG(x0, x1, .(.(y_3, .(y_4, [])), x3)) → U2_GGG(x1, .(.(y_3, .(y_4, [])), x3), member1_out_ag(.(y_3, .(y_4, []))))

The TRS R consists of the following rules:

member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U5_ag(member1_in_ag(L))
U5_ag(member1_out_ag(X)) → member1_out_ag(X)

s = REACH_IN_GGG(x0, x1, .(.(y_3, .(y_4, [])), x3')) evaluates to t =REACH_IN_GGG(y_4, x1, .(.(y_3, .(y_4, [])), x3'))

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

Matcher: [x0 / y_4]
Semiunifier: [ ]

Rewriting sequence

REACH_IN_GGG(x0, x1, .(.(y_3, .(y_4, [])), x3')) → U2_GGG(x1, .(.(y_3, .(y_4, [])), x3'), member1_out_ag(.(y_3, .(y_4, []))))
with rule REACH_IN_GGG(x0', x1', .(.(y_3', .(y_4', [])), x3'')) → U2_GGG(x1', .(.(y_3', .(y_4', [])), x3''), member1_out_ag(.(y_3', .(y_4', [])))) at position [] and matcher [x1' / x1, y_4' / y_4, x3'' / x3', y_3' / y_3, x0' / x0]

U2_GGG(x1, .(.(y_3, .(y_4, [])), x3'), member1_out_ag(.(y_3, .(y_4, [])))) → REACH_IN_GGG(y_4, x1, .(.(y_3, .(y_4, [])), x3'))
with rule U2_GGG(z1, .(z2, z3), member1_out_ag(.(x2, .(x3, [])))) → REACH_IN_GGG(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.

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
reach_in: (b,b,b)
member_in: (b,b)
member1_in: (f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U5_ag(X, H, L, member1_in_ag(X, L))
U5_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) → reach_out_ggg(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_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U5_ag(X, H, L, member1_in_ag(X, L))
U5_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

REACH_IN_GGG(X, Y, Edges) → U1_GGG(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_GGG(X, Y, Edges) → MEMBER_IN_GG(.(X, .(Y, [])), Edges)
MEMBER_IN_GG(X, .(H, L)) → U4_GG(X, H, L, member_in_gg(X, L))
MEMBER_IN_GG(X, .(H, L)) → MEMBER_IN_GG(X, L)
REACH_IN_GGG(X, Z, Edges) → U2_GGG(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
REACH_IN_GGG(X, Z, Edges) → MEMBER1_IN_AG(.(X, .(Y, [])), Edges)
MEMBER1_IN_AG(X, .(H, L)) → U5_AG(X, H, L, member1_in_ag(X, L))
MEMBER1_IN_AG(X, .(H, L)) → MEMBER1_IN_AG(X, L)
U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_GGG(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → REACH_IN_GGG(Y, Z, Edges)

The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U5_ag(X, H, L, member1_in_ag(X, L))
U5_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_in_ggg(x1, x2, x3) = reach_in_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4) = U1_ggg(x1, x2, x3, x4)
member_in_gg(x1, x2) = member_in_gg(x1, x2)
.(x1, x2) = .(x1, x2)
member_out_gg(x1, x2) = member_out_gg(x1, x2)
U4_gg(x1, x2, x3, x4) = U4_gg(x1, x2, x3, x4)
[] = []
reach_out_ggg(x1, x2, x3) = reach_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4) = U2_ggg(x1, x2, x3, x4)
member1_in_ag(x1, x2) = member1_in_ag(x2)
member1_out_ag(x1, x2) = member1_out_ag(x1, x2)
U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4)
U3_ggg(x1, x2, x3, x4) = U3_ggg(x1, x2, x3, x4)
U1_GGG(x1, x2, x3, x4) = U1_GGG(x1, x2, x3, x4)
MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2)
U3_GGG(x1, x2, x3, x4) = U3_GGG(x1, x2, x3, x4)
REACH_IN_GGG(x1, x2, x3) = REACH_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4) = U2_GGG(x1, x2, x3, x4)
MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2)
U4_GG(x1, x2, x3, x4) = U4_GG(x1, x2, x3, x4)
U5_AG(x1, x2, x3, x4) = U5_AG(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_GGG(X, Y, Edges) → U1_GGG(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_GGG(X, Y, Edges) → MEMBER_IN_GG(.(X, .(Y, [])), Edges)
MEMBER_IN_GG(X, .(H, L)) → U4_GG(X, H, L, member_in_gg(X, L))
MEMBER_IN_GG(X, .(H, L)) → MEMBER_IN_GG(X, L)
REACH_IN_GGG(X, Z, Edges) → U2_GGG(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
REACH_IN_GGG(X, Z, Edges) → MEMBER1_IN_AG(.(X, .(Y, [])), Edges)
MEMBER1_IN_AG(X, .(H, L)) → U5_AG(X, H, L, member1_in_ag(X, L))
MEMBER1_IN_AG(X, .(H, L)) → MEMBER1_IN_AG(X, L)
U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_GGG(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → REACH_IN_GGG(Y, Z, Edges)

The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U5_ag(X, H, L, member1_in_ag(X, L))
U5_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) → reach_out_ggg(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:

MEMBER1_IN_AG(X, .(H, L)) → MEMBER1_IN_AG(X, L)

The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U5_ag(X, H, L, member1_in_ag(X, L))
U5_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_in_ggg(x1, x2, x3) = reach_in_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4) = U1_ggg(x1, x2, x3, x4)
member_in_gg(x1, x2) = member_in_gg(x1, x2)
.(x1, x2) = .(x1, x2)
member_out_gg(x1, x2) = member_out_gg(x1, x2)
U4_gg(x1, x2, x3, x4) = U4_gg(x1, x2, x3, x4)
[] = []
reach_out_ggg(x1, x2, x3) = reach_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4) = U2_ggg(x1, x2, x3, x4)
member1_in_ag(x1, x2) = member1_in_ag(x2)
member1_out_ag(x1, x2) = member1_out_ag(x1, x2)
U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4)
U3_ggg(x1, x2, x3, x4) = U3_ggg(x1, x2, x3, x4)
MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(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:

MEMBER1_IN_AG(X, .(H, L)) → MEMBER1_IN_AG(X, L)

↳ 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:

MEMBER1_IN_AG(.(H, L)) → MEMBER1_IN_AG(L)

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

MEMBER1_IN_AG(.(H, L)) → MEMBER1_IN_AG(L)
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:

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

The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U5_ag(X, H, L, member1_in_ag(X, L))
U5_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_in_ggg(x1, x2, x3) = reach_in_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4) = U1_ggg(x1, x2, x3, x4)
member_in_gg(x1, x2) = member_in_gg(x1, x2)
.(x1, x2) = .(x1, x2)
member_out_gg(x1, x2) = member_out_gg(x1, x2)
U4_gg(x1, x2, x3, x4) = U4_gg(x1, x2, x3, x4)
[] = []
reach_out_ggg(x1, x2, x3) = reach_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4) = U2_ggg(x1, x2, x3, x4)
member1_in_ag(x1, x2) = member1_in_ag(x2)
member1_out_ag(x1, x2) = member1_out_ag(x1, x2)
U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4)
U3_ggg(x1, x2, x3, x4) = U3_ggg(x1, x2, x3, x4)
MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

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

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → REACH_IN_GGG(Y, Z, Edges)
REACH_IN_GGG(X, Z, Edges) → U2_GGG(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))

The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U5_ag(X, H, L, member1_in_ag(X, L))
U5_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_ggg(Y, Z, Edges))
U3_ggg(X, Z, Edges, reach_out_ggg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_in_ggg(x1, x2, x3) = reach_in_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4) = U1_ggg(x1, x2, x3, x4)
member_in_gg(x1, x2) = member_in_gg(x1, x2)
.(x1, x2) = .(x1, x2)
member_out_gg(x1, x2) = member_out_gg(x1, x2)
U4_gg(x1, x2, x3, x4) = U4_gg(x1, x2, x3, x4)
[] = []
reach_out_ggg(x1, x2, x3) = reach_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4) = U2_ggg(x1, x2, x3, x4)
member1_in_ag(x1, x2) = member1_in_ag(x2)
member1_out_ag(x1, x2) = member1_out_ag(x1, x2)
U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4)
U3_ggg(x1, x2, x3, x4) = U3_ggg(x1, x2, x3, x4)
REACH_IN_GGG(x1, x2, x3) = REACH_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4) = U2_GGG(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:

U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → REACH_IN_GGG(Y, Z, Edges)
REACH_IN_GGG(X, Z, Edges) → U2_GGG(X, Z, Edges, member1_in_ag(.(X, .(Y, [])), Edges))

The TRS R consists of the following rules:

member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U5_ag(X, H, L, member1_in_ag(X, L))
U5_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
member1_in_ag(x1, x2) = member1_in_ag(x2)
member1_out_ag(x1, x2) = member1_out_ag(x1, x2)
U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4)
REACH_IN_GGG(x1, x2, x3) = REACH_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4) = U2_GGG(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:

U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → REACH_IN_GGG(Y, Z, Edges)
REACH_IN_GGG(X, Z, Edges) → U2_GGG(X, Z, Edges, member1_in_ag(Edges))

The TRS R consists of the following rules:

member1_in_ag(.(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(.(H, L)) → U5_ag(H, L, member1_in_ag(L))
U5_ag(H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The set Q consists of the following terms:

member1_in_ag(x0)
U5_ag(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule REACH_IN_GGG(X, Z, Edges) → U2_GGG(X, Z, Edges, member1_in_ag(Edges)) at position [3] we obtained the following new rules:

REACH_IN_GGG(y0, y1, .(x0, x1)) → U2_GGG(y0, y1, .(x0, x1), U5_ag(x0, x1, member1_in_ag(x1)))
REACH_IN_GGG(y0, y1, .(x0, x1)) → U2_GGG(y0, y1, .(x0, x1), member1_out_ag(x0, .(x0, 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:

U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → REACH_IN_GGG(Y, Z, Edges)
REACH_IN_GGG(y0, y1, .(x0, x1)) → U2_GGG(y0, y1, .(x0, x1), U5_ag(x0, x1, member1_in_ag(x1)))
REACH_IN_GGG(y0, y1, .(x0, x1)) → U2_GGG(y0, y1, .(x0, x1), member1_out_ag(x0, .(x0, x1)))

The TRS R consists of the following rules:

member1_in_ag(.(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(.(H, L)) → U5_ag(H, L, member1_in_ag(L))
U5_ag(H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The set Q consists of the following terms:

member1_in_ag(x0)
U5_ag(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_GGG(X, Z, Edges, member1_out_ag(.(X, .(Y, [])), Edges)) → REACH_IN_GGG(Y, Z, Edges) we obtained the following new rules:

U2_GGG(z0, z1, .(z2, z3), member1_out_ag(.(z0, .(x3, [])), .(z2, z3))) → REACH_IN_GGG(x3, z1, .(z2, z3))
U2_GGG(z0, z1, .(.(z0, .(x3, [])), z3), member1_out_ag(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) → REACH_IN_GGG(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:

U2_GGG(z0, z1, .(z2, z3), member1_out_ag(.(z0, .(x3, [])), .(z2, z3))) → REACH_IN_GGG(x3, z1, .(z2, z3))
U2_GGG(z0, z1, .(.(z0, .(x3, [])), z3), member1_out_ag(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) → REACH_IN_GGG(x3, z1, .(.(z0, .(x3, [])), z3))
REACH_IN_GGG(y0, y1, .(x0, x1)) → U2_GGG(y0, y1, .(x0, x1), U5_ag(x0, x1, member1_in_ag(x1)))
REACH_IN_GGG(y0, y1, .(x0, x1)) → U2_GGG(y0, y1, .(x0, x1), member1_out_ag(x0, .(x0, x1)))

The TRS R consists of the following rules:

member1_in_ag(.(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(.(H, L)) → U5_ag(H, L, member1_in_ag(L))
U5_ag(H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The set Q consists of the following terms:

member1_in_ag(x0)
U5_ag(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule REACH_IN_GGG(y0, y1, .(x0, x1)) → U2_GGG(y0, y1, .(x0, x1), member1_out_ag(x0, .(x0, x1))) we obtained the following new rules:

REACH_IN_GGG(x0, x1, .(.(y_4, .(y_5, [])), x3)) → U2_GGG(x0, x1, .(.(y_4, .(y_5, [])), x3), member1_out_ag(.(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:

REACH_IN_GGG(x0, x1, .(.(y_4, .(y_5, [])), x3)) → U2_GGG(x0, x1, .(.(y_4, .(y_5, [])), x3), member1_out_ag(.(y_4, .(y_5, [])), .(.(y_4, .(y_5, [])), x3)))
U2_GGG(z0, z1, .(z2, z3), member1_out_ag(.(z0, .(x3, [])), .(z2, z3))) → REACH_IN_GGG(x3, z1, .(z2, z3))
U2_GGG(z0, z1, .(.(z0, .(x3, [])), z3), member1_out_ag(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) → REACH_IN_GGG(x3, z1, .(.(z0, .(x3, [])), z3))
REACH_IN_GGG(y0, y1, .(x0, x1)) → U2_GGG(y0, y1, .(x0, x1), U5_ag(x0, x1, member1_in_ag(x1)))

The TRS R consists of the following rules:

member1_in_ag(.(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(.(H, L)) → U5_ag(H, L, member1_in_ag(L))
U5_ag(H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The set Q consists of the following terms:

member1_in_ag(x0)
U5_ag(x0, x1, x2)

REACH_IN_GGG(x0, x1, .(.(y_4, .(y_5, [])), x3)) → U2_GGG(x0, x1, .(.(y_4, .(y_5, [])), x3), member1_out_ag(.(y_4, .(y_5, [])), .(.(y_4, .(y_5, [])), x3)))
U2_GGG(z0, z1, .(z2, z3), member1_out_ag(.(z0, .(x3, [])), .(z2, z3))) → REACH_IN_GGG(x3, z1, .(z2, z3))
U2_GGG(z0, z1, .(.(z0, .(x3, [])), z3), member1_out_ag(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) → REACH_IN_GGG(x3, z1, .(.(z0, .(x3, [])), z3))
REACH_IN_GGG(y0, y1, .(x0, x1)) → U2_GGG(y0, y1, .(x0, x1), U5_ag(x0, x1, member1_in_ag(x1)))

The TRS R consists of the following rules:

member1_in_ag(.(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(.(H, L)) → U5_ag(H, L, member1_in_ag(L))
U5_ag(H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

s = U2_GGG(z0, z1, .(.(y_4, .(y_5, [])), z3), member1_out_ag(.(z0, .(x3', [])), .(.(y_4, .(y_5, [])), z3))) evaluates to t =U2_GGG(x3', z1, .(.(y_4, .(y_5, [])), z3), member1_out_ag(.(y_4, .(y_5, [])), .(.(y_4, .(y_5, [])), z3)))

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

Semiunifier: [y_4 / z0, x3' / z0, y_5 / z0]
Matcher: [ ]

Rewriting sequence

U2_GGG(z0, z1, .(.(z0, .(z0, [])), z3), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) → REACH_IN_GGG(z0, z1, .(.(z0, .(z0, [])), z3))
with rule U2_GGG(z0', z1', .(z2, z3'), member1_out_ag(.(z0', .(x3', [])), .(z2, z3'))) → REACH_IN_GGG(x3', z1', .(z2, z3')) at position [] and matcher [z3' / z3, x3' / z0, z1' / z1, z2 / .(z0, .(z0, [])), z0' / z0]

REACH_IN_GGG(z0, z1, .(.(z0, .(z0, [])), z3)) → U2_GGG(z0, z1, .(.(z0, .(z0, [])), z3), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3)))
with rule REACH_IN_GGG(x0, x1, .(.(y_4, .(y_5, [])), x3)) → U2_GGG(x0, x1, .(.(y_4, .(y_5, [])), x3), member1_out_ag(.(y_4, .(y_5, [])), .(.(y_4, .(y_5, [])), x3)))

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.