Left Termination proof of /tmp/tmpFzJfas/pl7.6.2b.pl

Left Termination of the query pattern reach_in_4(g, g, g, g) w.r.t. the given Prolog program could not be shown:

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

Clauses:

reach(X, Y, Edges, Visited) :- member(.(X, .(Y, [])), Edges).
reach(X, Z, Edges, Visited) :- ','(member1(.(X, .(Y, [])), Edges), ','(member(Y, Visited), reach(Y, Z, Edges, .(Y, Visited)))).
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,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,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_gggg(X, Y, Edges, Visited) → U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_gggg(X, Y, Edges, Visited)
reach_in_gggg(X, Z, Edges, Visited) → U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U6_ag(X, H, L, member1_in_ag(X, L))
U6_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited)))
U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) → reach_out_gggg(X, Z, Edges, Visited)

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_gggg(X, Y, Edges, Visited) → U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_gggg(X, Y, Edges, Visited)
reach_in_gggg(X, Z, Edges, Visited) → U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U6_ag(X, H, L, member1_in_ag(X, L))
U6_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited)))
U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) → reach_out_gggg(X, Z, Edges, Visited)

REACH_IN_GGGG(X, Y, Edges, Visited) → U1_GGGG(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_GGGG(X, Y, Edges, Visited) → MEMBER_IN_GG(.(X, .(Y, [])), Edges)
MEMBER_IN_GG(X, .(H, L)) → U5_GG(X, H, L, member_in_gg(X, L))
MEMBER_IN_GG(X, .(H, L)) → MEMBER_IN_GG(X, L)
REACH_IN_GGGG(X, Z, Edges, Visited) → U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
REACH_IN_GGGG(X, Z, Edges, Visited) → MEMBER1_IN_AG(.(X, .(Y, [])), Edges)
MEMBER1_IN_AG(X, .(H, L)) → U6_AG(X, H, L, member1_in_ag(X, L))
MEMBER1_IN_AG(X, .(H, L)) → MEMBER1_IN_AG(X, L)
U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → MEMBER_IN_GG(Y, Visited)
U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → U4_GGGG(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited)))
U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited))

The TRS R consists of the following rules:

reach_in_gggg(X, Y, Edges, Visited) → U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_gggg(X, Y, Edges, Visited)
reach_in_gggg(X, Z, Edges, Visited) → U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U6_ag(X, H, L, member1_in_ag(X, L))
U6_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited)))
U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) → reach_out_gggg(X, Z, Edges, Visited)

The argument filtering Pi contains the following mapping:
reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4)
U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x5)
member_in_gg(x1, x2) = member_in_gg(x1, x2)
.(x1, x2) = .(x1, x2)
member_out_gg(x1, x2) = member_out_gg
U5_gg(x1, x2, x3, x4) = U5_gg(x4)
[] = []
reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg
U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x2, x3, x4, x5)
member1_in_ag(x1, x2) = member1_in_ag(x2)
member1_out_ag(x1, x2) = member1_out_ag(x1)
U6_ag(x1, x2, x3, x4) = U6_ag(x4)
U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x2, x3, x4, x5, x6)
U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x5)
U5_GG(x1, x2, x3, x4) = U5_GG(x4)
U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x2, x3, x4, x5)
U6_AG(x1, x2, x3, x4) = U6_AG(x4)
MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2)
REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4)
U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x2, x3, x4, x5, x6)
U1_GGGG(x1, x2, x3, x4, x5) = U1_GGGG(x5)
MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2)
U4_GGGG(x1, x2, x3, x4, x5) = U4_GGGG(x5)

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_GGGG(X, Y, Edges, Visited) → U1_GGGG(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_GGGG(X, Y, Edges, Visited) → MEMBER_IN_GG(.(X, .(Y, [])), Edges)
MEMBER_IN_GG(X, .(H, L)) → U5_GG(X, H, L, member_in_gg(X, L))
MEMBER_IN_GG(X, .(H, L)) → MEMBER_IN_GG(X, L)
REACH_IN_GGGG(X, Z, Edges, Visited) → U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
REACH_IN_GGGG(X, Z, Edges, Visited) → MEMBER1_IN_AG(.(X, .(Y, [])), Edges)
MEMBER1_IN_AG(X, .(H, L)) → U6_AG(X, H, L, member1_in_ag(X, L))
MEMBER1_IN_AG(X, .(H, L)) → MEMBER1_IN_AG(X, L)
U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → MEMBER_IN_GG(Y, Visited)
U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → U4_GGGG(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited)))
U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited))

The TRS R consists of the following rules:

reach_in_gggg(X, Y, Edges, Visited) → U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_gggg(X, Y, Edges, Visited)
reach_in_gggg(X, Z, Edges, Visited) → U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U6_ag(X, H, L, member1_in_ag(X, L))
U6_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited)))
U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) → reach_out_gggg(X, Z, Edges, Visited)

↳ 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_gggg(X, Y, Edges, Visited) → U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_gggg(X, Y, Edges, Visited)
reach_in_gggg(X, Z, Edges, Visited) → U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U6_ag(X, H, L, member1_in_ag(X, L))
U6_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited)))
U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) → reach_out_gggg(X, Z, Edges, Visited)

↳ 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_gggg(X, Y, Edges, Visited) → U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_gggg(X, Y, Edges, Visited)
reach_in_gggg(X, Z, Edges, Visited) → U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U6_ag(X, H, L, member1_in_ag(X, L))
U6_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited)))
U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) → reach_out_gggg(X, Z, Edges, Visited)

↳ 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_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited))
REACH_IN_GGGG(X, Z, Edges, Visited) → U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))

The TRS R consists of the following rules:

reach_in_gggg(X, Y, Edges, Visited) → U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_gggg(X, Y, Edges, Visited)
reach_in_gggg(X, Z, Edges, Visited) → U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U6_ag(X, H, L, member1_in_ag(X, L))
U6_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited)))
U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) → reach_out_gggg(X, Z, Edges, Visited)

The argument filtering Pi contains the following mapping:
reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4)
U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x5)
member_in_gg(x1, x2) = member_in_gg(x1, x2)
.(x1, x2) = .(x1, x2)
member_out_gg(x1, x2) = member_out_gg
U5_gg(x1, x2, x3, x4) = U5_gg(x4)
[] = []
reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg
U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x2, x3, x4, x5)
member1_in_ag(x1, x2) = member1_in_ag(x2)
member1_out_ag(x1, x2) = member1_out_ag(x1)
U6_ag(x1, x2, x3, x4) = U6_ag(x4)
U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x2, x3, x4, x5, x6)
U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x5)
U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x2, x3, x4, x5)
REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4)
U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x2, x3, x4, x5, x6)

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_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited))
REACH_IN_GGGG(X, Z, Edges, Visited) → U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U6_ag(X, H, L, member1_in_ag(X, L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U6_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The argument filtering Pi contains the following mapping:
member_in_gg(x1, x2) = member_in_gg(x1, x2)
.(x1, x2) = .(x1, x2)
member_out_gg(x1, x2) = member_out_gg
U5_gg(x1, x2, x3, x4) = U5_gg(x4)
[] = []
member1_in_ag(x1, x2) = member1_in_ag(x2)
member1_out_ag(x1, x2) = member1_out_ag(x1)
U6_ag(x1, x2, x3, x4) = U6_ag(x4)
U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x2, x3, x4, x5)
REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4)
U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x2, x3, x4, x5, x6)

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:

U3_GGGG(Z, Edges, Visited, Y, member_out_gg) → REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited))
U2_GGGG(Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])))) → U3_GGGG(Z, Edges, Visited, Y, member_in_gg(Y, Visited))
REACH_IN_GGGG(X, Z, Edges, Visited) → U2_GGGG(Z, Edges, Visited, member1_in_ag(Edges))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg
member_in_gg(X, .(H, L)) → U5_gg(member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U6_ag(member1_in_ag(L))
U5_gg(member_out_gg) → member_out_gg
U6_ag(member1_out_ag(X)) → member1_out_ag(X)

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0)
U6_ag(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U2_GGGG(Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])))) → U3_GGGG(Z, Edges, Visited, Y, member_in_gg(Y, Visited)) at position [4] we obtained the following new rules:

U2_GGGG(y0, y1, .(x1, x2), member1_out_ag(.(y3, .(x0, [])))) → U3_GGGG(y0, y1, .(x1, x2), x0, U5_gg(member_in_gg(x0, x2)))
U2_GGGG(y0, y1, .(x0, x1), member1_out_ag(.(y3, .(x0, [])))) → U3_GGGG(y0, y1, .(x0, x1), x0, member_out_gg)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

  ↳ PrologToPiTRSProof

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

U3_GGGG(Z, Edges, Visited, Y, member_out_gg) → REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited))
U2_GGGG(y0, y1, .(x1, x2), member1_out_ag(.(y3, .(x0, [])))) → U3_GGGG(y0, y1, .(x1, x2), x0, U5_gg(member_in_gg(x0, x2)))
U2_GGGG(y0, y1, .(x0, x1), member1_out_ag(.(y3, .(x0, [])))) → U3_GGGG(y0, y1, .(x0, x1), x0, member_out_gg)
REACH_IN_GGGG(X, Z, Edges, Visited) → U2_GGGG(Z, Edges, Visited, member1_in_ag(Edges))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg
member_in_gg(X, .(H, L)) → U5_gg(member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U6_ag(member1_in_ag(L))
U5_gg(member_out_gg) → member_out_gg
U6_ag(member1_out_ag(X)) → member1_out_ag(X)

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0)
U6_ag(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule REACH_IN_GGGG(X, Z, Edges, Visited) → U2_GGGG(Z, Edges, Visited, member1_in_ag(Edges)) at position [3] we obtained the following new rules:

REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y1, .(x0, x1), y3, member1_out_ag(x0))
REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y1, .(x0, x1), y3, U6_ag(member1_in_ag(x1)))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

  ↳ PrologToPiTRSProof

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

U3_GGGG(Z, Edges, Visited, Y, member_out_gg) → REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited))
REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y1, .(x0, x1), y3, member1_out_ag(x0))
U2_GGGG(y0, y1, .(x1, x2), member1_out_ag(.(y3, .(x0, [])))) → U3_GGGG(y0, y1, .(x1, x2), x0, U5_gg(member_in_gg(x0, x2)))
U2_GGGG(y0, y1, .(x0, x1), member1_out_ag(.(y3, .(x0, [])))) → U3_GGGG(y0, y1, .(x0, x1), x0, member_out_gg)
REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y1, .(x0, x1), y3, U6_ag(member1_in_ag(x1)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg
member_in_gg(X, .(H, L)) → U5_gg(member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U6_ag(member1_in_ag(L))
U5_gg(member_out_gg) → member_out_gg
U6_ag(member1_out_ag(X)) → member1_out_ag(X)

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0)
U6_ag(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U3_GGGG(Z, Edges, Visited, Y, member_out_gg) → REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) we obtained the following new rules:

U3_GGGG(z0, z1, .(z2, z3), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3)))
U3_GGGG(z0, z1, .(z2, z3), z5, member_out_gg) → REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3)))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

  ↳ PrologToPiTRSProof

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

REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y1, .(x0, x1), y3, member1_out_ag(x0))
U2_GGGG(y0, y1, .(x1, x2), member1_out_ag(.(y3, .(x0, [])))) → U3_GGGG(y0, y1, .(x1, x2), x0, U5_gg(member_in_gg(x0, x2)))
U2_GGGG(y0, y1, .(x0, x1), member1_out_ag(.(y3, .(x0, [])))) → U3_GGGG(y0, y1, .(x0, x1), x0, member_out_gg)
U3_GGGG(z0, z1, .(z2, z3), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3)))
U3_GGGG(z0, z1, .(z2, z3), z5, member_out_gg) → REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3)))
REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y1, .(x0, x1), y3, U6_ag(member1_in_ag(x1)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg
member_in_gg(X, .(H, L)) → U5_gg(member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U6_ag(member1_in_ag(L))
U5_gg(member_out_gg) → member_out_gg
U6_ag(member1_out_ag(X)) → member1_out_ag(X)

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0)
U6_ag(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_GGGG(y0, y1, .(x1, x2), member1_out_ag(.(y3, .(x0, [])))) → U3_GGGG(y0, y1, .(x1, x2), x0, U5_gg(member_in_gg(x0, x2))) we obtained the following new rules:

U2_GGGG(z1, .(z2, z3), .(x2, x3), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(x2, x3), x5, U5_gg(member_in_gg(x5, x3)))
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(x2, x3), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(x2, x3), x5, U5_gg(member_in_gg(x5, x3)))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

  ↳ PrologToPiTRSProof

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

REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y1, .(x0, x1), y3, member1_out_ag(x0))
U2_GGGG(y0, y1, .(x0, x1), member1_out_ag(.(y3, .(x0, [])))) → U3_GGGG(y0, y1, .(x0, x1), x0, member_out_gg)
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(x2, x3), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(x2, x3), x5, U5_gg(member_in_gg(x5, x3)))
U2_GGGG(z1, .(z2, z3), .(x2, x3), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(x2, x3), x5, U5_gg(member_in_gg(x5, x3)))
U3_GGGG(z0, z1, .(z2, z3), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3)))
REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y1, .(x0, x1), y3, U6_ag(member1_in_ag(x1)))
U3_GGGG(z0, z1, .(z2, z3), z5, member_out_gg) → REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg
member_in_gg(X, .(H, L)) → U5_gg(member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U6_ag(member1_in_ag(L))
U5_gg(member_out_gg) → member_out_gg
U6_ag(member1_out_ag(X)) → member1_out_ag(X)

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0)
U6_ag(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_GGGG(y0, y1, .(x0, x1), member1_out_ag(.(y3, .(x0, [])))) → U3_GGGG(y0, y1, .(x0, x1), x0, member_out_gg) we obtained the following new rules:

U2_GGGG(z1, .(z2, z3), .(x2, x3), member1_out_ag(.(x4, .(x2, [])))) → U3_GGGG(z1, .(z2, z3), .(x2, x3), x2, member_out_gg)
U2_GGGG(z1, .(.(x4, .(x2, [])), z3), .(x2, x3), member1_out_ag(.(x4, .(x2, [])))) → U3_GGGG(z1, .(.(x4, .(x2, [])), z3), .(x2, x3), x2, member_out_gg)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

  ↳ PrologToPiTRSProof

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

U2_GGGG(z1, .(z2, z3), .(x2, x3), member1_out_ag(.(x4, .(x2, [])))) → U3_GGGG(z1, .(z2, z3), .(x2, x3), x2, member_out_gg)
REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y1, .(x0, x1), y3, member1_out_ag(x0))
U2_GGGG(z1, .(z2, z3), .(x2, x3), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(x2, x3), x5, U5_gg(member_in_gg(x5, x3)))
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(x2, x3), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(x2, x3), x5, U5_gg(member_in_gg(x5, x3)))
U3_GGGG(z0, z1, .(z2, z3), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3)))
U3_GGGG(z0, z1, .(z2, z3), z5, member_out_gg) → REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3)))
REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y1, .(x0, x1), y3, U6_ag(member1_in_ag(x1)))
U2_GGGG(z1, .(.(x4, .(x2, [])), z3), .(x2, x3), member1_out_ag(.(x4, .(x2, [])))) → U3_GGGG(z1, .(.(x4, .(x2, [])), z3), .(x2, x3), x2, member_out_gg)

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg
member_in_gg(X, .(H, L)) → U5_gg(member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U6_ag(member1_in_ag(L))
U5_gg(member_out_gg) → member_out_gg
U6_ag(member1_out_ag(X)) → member1_out_ag(X)

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0)
U6_ag(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y1, .(x0, x1), y3, member1_out_ag(x0)) we obtained the following new rules:

REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2))
REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

  ↳ PrologToPiTRSProof

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

REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2))
U2_GGGG(z1, .(z2, z3), .(x2, x3), member1_out_ag(.(x4, .(x2, [])))) → U3_GGGG(z1, .(z2, z3), .(x2, x3), x2, member_out_gg)
REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2))
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(x2, x3), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(x2, x3), x5, U5_gg(member_in_gg(x5, x3)))
U2_GGGG(z1, .(z2, z3), .(x2, x3), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(x2, x3), x5, U5_gg(member_in_gg(x5, x3)))
U3_GGGG(z0, z1, .(z2, z3), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3)))
REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y1, .(x0, x1), y3, U6_ag(member1_in_ag(x1)))
U3_GGGG(z0, z1, .(z2, z3), z5, member_out_gg) → REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3)))
U2_GGGG(z1, .(.(x4, .(x2, [])), z3), .(x2, x3), member1_out_ag(.(x4, .(x2, [])))) → U3_GGGG(z1, .(.(x4, .(x2, [])), z3), .(x2, x3), x2, member_out_gg)

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg
member_in_gg(X, .(H, L)) → U5_gg(member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U6_ag(member1_in_ag(L))
U5_gg(member_out_gg) → member_out_gg
U6_ag(member1_out_ag(X)) → member1_out_ag(X)

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0)
U6_ag(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y1, .(x0, x1), y3, U6_ag(member1_in_ag(x1))) we obtained the following new rules:

REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3)))
REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3)))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

  ↳ PrologToPiTRSProof

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

REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2))
REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3)))
U2_GGGG(z1, .(z2, z3), .(x2, x3), member1_out_ag(.(x4, .(x2, [])))) → U3_GGGG(z1, .(z2, z3), .(x2, x3), x2, member_out_gg)
REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3)))
U2_GGGG(z1, .(z2, z3), .(x2, x3), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(x2, x3), x5, U5_gg(member_in_gg(x5, x3)))
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(x2, x3), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(x2, x3), x5, U5_gg(member_in_gg(x5, x3)))
REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2))
U3_GGGG(z0, z1, .(z2, z3), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3)))
U3_GGGG(z0, z1, .(z2, z3), z5, member_out_gg) → REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3)))
U2_GGGG(z1, .(.(x4, .(x2, [])), z3), .(x2, x3), member1_out_ag(.(x4, .(x2, [])))) → U3_GGGG(z1, .(.(x4, .(x2, [])), z3), .(x2, x3), x2, member_out_gg)

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg
member_in_gg(X, .(H, L)) → U5_gg(member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U6_ag(member1_in_ag(L))
U5_gg(member_out_gg) → member_out_gg
U6_ag(member1_out_ag(X)) → member1_out_ag(X)

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0)
U6_ag(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U3_GGGG(z0, z1, .(z2, z3), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, z1, .(z2, .(z2, z3))) we obtained the following new rules:

U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, z5), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z5)))
U3_GGGG(z0, .(z1, z2), .(z3, z4), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, z4)))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

  ↳ PrologToPiTRSProof

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

REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2))
REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3)))
U2_GGGG(z1, .(z2, z3), .(x2, x3), member1_out_ag(.(x4, .(x2, [])))) → U3_GGGG(z1, .(z2, z3), .(x2, x3), x2, member_out_gg)
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, z5), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z5)))
U3_GGGG(z0, .(z1, z2), .(z3, z4), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, z4)))
REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3)))
REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2))
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(x2, x3), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(x2, x3), x5, U5_gg(member_in_gg(x5, x3)))
U2_GGGG(z1, .(z2, z3), .(x2, x3), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(x2, x3), x5, U5_gg(member_in_gg(x5, x3)))
U3_GGGG(z0, z1, .(z2, z3), z5, member_out_gg) → REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3)))
U2_GGGG(z1, .(.(x4, .(x2, [])), z3), .(x2, x3), member1_out_ag(.(x4, .(x2, [])))) → U3_GGGG(z1, .(.(x4, .(x2, [])), z3), .(x2, x3), x2, member_out_gg)

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg
member_in_gg(X, .(H, L)) → U5_gg(member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U6_ag(member1_in_ag(L))
U5_gg(member_out_gg) → member_out_gg
U6_ag(member1_out_ag(X)) → member1_out_ag(X)

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0)
U6_ag(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U3_GGGG(z0, z1, .(z2, z3), z5, member_out_gg) → REACH_IN_GGGG(z5, z0, z1, .(z5, .(z2, z3))) we obtained the following new rules:

U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, z4), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)))
U3_GGGG(z0, .(z1, z2), .(z3, z4), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, z4)))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, z5), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)))
U3_GGGG(z0, .(z1, z2), .(z3, z4), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, z4)))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

  ↳ PrologToPiTRSProof

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

REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2))
REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3)))
U2_GGGG(z1, .(z2, z3), .(x2, x3), member1_out_ag(.(x4, .(x2, [])))) → U3_GGGG(z1, .(z2, z3), .(x2, x3), x2, member_out_gg)
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, z5), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z5)))
U3_GGGG(z0, .(z1, z2), .(z3, z4), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, z4)))
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(x2, x3), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(x2, x3), x5, U5_gg(member_in_gg(x5, x3)))
U3_GGGG(z0, .(z1, z2), .(z3, z4), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, z4)))
REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3)))
REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2))
U2_GGGG(z1, .(z2, z3), .(x2, x3), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(x2, x3), x5, U5_gg(member_in_gg(x5, x3)))
U2_GGGG(z1, .(.(x4, .(x2, [])), z3), .(x2, x3), member1_out_ag(.(x4, .(x2, [])))) → U3_GGGG(z1, .(.(x4, .(x2, [])), z3), .(x2, x3), x2, member_out_gg)
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, z5), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg
member_in_gg(X, .(H, L)) → U5_gg(member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U6_ag(member1_in_ag(L))
U5_gg(member_out_gg) → member_out_gg
U6_ag(member1_out_ag(X)) → member1_out_ag(X)

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0)
U6_ag(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_GGGG(z1, .(z2, z3), .(x2, x3), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(x2, x3), x5, U5_gg(member_in_gg(x5, x3))) we obtained the following new rules:

U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x6, U5_gg(member_in_gg(x6, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x6, U5_gg(member_in_gg(x6, .(z4, z5))))
U2_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z4, z5)), x6, U5_gg(member_in_gg(x6, .(z4, z5))))
U2_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z0, z4)), x6, U5_gg(member_in_gg(x6, .(z0, z4))))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Instantiation

  ↳ PrologToPiTRSProof

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

REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x6, U5_gg(member_in_gg(x6, .(z0, z4))))
REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3)))
U2_GGGG(z1, .(z2, z3), .(x2, x3), member1_out_ag(.(x4, .(x2, [])))) → U3_GGGG(z1, .(z2, z3), .(x2, x3), x2, member_out_gg)
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, z5), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z5)))
U3_GGGG(z0, .(z1, z2), .(z3, z4), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, z4)))
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(x2, x3), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(x2, x3), x5, U5_gg(member_in_gg(x5, x3)))
U2_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z4, z5)), x6, U5_gg(member_in_gg(x6, .(z4, z5))))
U2_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z0, z4)), x6, U5_gg(member_in_gg(x6, .(z0, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, z4), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, z4)))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x6, U5_gg(member_in_gg(x6, .(z4, z5))))
REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3)))
REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2))
U2_GGGG(z1, .(.(x4, .(x2, [])), z3), .(x2, x3), member1_out_ag(.(x4, .(x2, [])))) → U3_GGGG(z1, .(.(x4, .(x2, [])), z3), .(x2, x3), x2, member_out_gg)
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, z5), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg
member_in_gg(X, .(H, L)) → U5_gg(member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U6_ag(member1_in_ag(L))
U5_gg(member_out_gg) → member_out_gg
U6_ag(member1_out_ag(X)) → member1_out_ag(X)

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0)
U6_ag(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(x2, x3), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(x2, x3), x5, U5_gg(member_in_gg(x5, x3))) we obtained the following new rules:

U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, z5)), x2, U5_gg(member_in_gg(x2, .(z4, z5))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, z4)), x2, U5_gg(member_in_gg(x2, .(z0, z4))))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Instantiation

                                                                  ↳ QDP

                                                                    ↳ Instantiation

  ↳ PrologToPiTRSProof

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

REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2))
REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3)))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x6, U5_gg(member_in_gg(x6, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(x2, x3), member1_out_ag(.(x4, .(x2, [])))) → U3_GGGG(z1, .(z2, z3), .(x2, x3), x2, member_out_gg)
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, z5), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z5)))
U3_GGGG(z0, .(z1, z2), .(z3, z4), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, z4)))
U2_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z4, z5)), x6, U5_gg(member_in_gg(x6, .(z4, z5))))
U3_GGGG(z0, .(z1, z2), .(z3, z4), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, z4)))
U2_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z0, z4)), x6, U5_gg(member_in_gg(x6, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x6, U5_gg(member_in_gg(x6, .(z4, z5))))
REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3)))
REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2))
U2_GGGG(z1, .(.(x4, .(x2, [])), z3), .(x2, x3), member1_out_ag(.(x4, .(x2, [])))) → U3_GGGG(z1, .(.(x4, .(x2, [])), z3), .(x2, x3), x2, member_out_gg)
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, z5), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg
member_in_gg(X, .(H, L)) → U5_gg(member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U6_ag(member1_in_ag(L))
U5_gg(member_out_gg) → member_out_gg
U6_ag(member1_out_ag(X)) → member1_out_ag(X)

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0)
U6_ag(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_GGGG(z1, .(z2, z3), .(x2, x3), member1_out_ag(.(x4, .(x2, [])))) → U3_GGGG(z1, .(z2, z3), .(x2, x3), x2, member_out_gg) we obtained the following new rules:

U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Instantiation

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

  ↳ PrologToPiTRSProof

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

REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x6, U5_gg(member_in_gg(x6, .(z0, z4))))
REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3)))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg)
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, z5), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z5)))
U2_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg)
U3_GGGG(z0, .(z1, z2), .(z3, z4), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, z4)))
U2_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z4, z5)), x6, U5_gg(member_in_gg(x6, .(z4, z5))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z0, z4)), x6, U5_gg(member_in_gg(x6, .(z0, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, z4), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, z4)))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x6, U5_gg(member_in_gg(x6, .(z4, z5))))
REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3)))
REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2))
U2_GGGG(z1, .(.(x4, .(x2, [])), z3), .(x2, x3), member1_out_ag(.(x4, .(x2, [])))) → U3_GGGG(z1, .(.(x4, .(x2, [])), z3), .(x2, x3), x2, member_out_gg)
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, z5), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg
member_in_gg(X, .(H, L)) → U5_gg(member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U6_ag(member1_in_ag(L))
U5_gg(member_out_gg) → member_out_gg
U6_ag(member1_out_ag(X)) → member1_out_ag(X)

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0)
U6_ag(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_GGGG(z1, .(.(x4, .(x2, [])), z3), .(x2, x3), member1_out_ag(.(x4, .(x2, [])))) → U3_GGGG(z1, .(.(x4, .(x2, [])), z3), .(x2, x3), x2, member_out_gg) we obtained the following new rules:

U2_GGGG(z1, .(.(x1, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x1, .(z0, [])))) → U3_GGGG(z1, .(.(x1, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(.(x1, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x1, .(z0, [])))) → U3_GGGG(z1, .(.(x1, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Instantiation

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ ForwardInstantiation

  ↳ PrologToPiTRSProof

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

REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2))
REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3)))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x6, U5_gg(member_in_gg(x6, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg)
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, z5), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z5)))
U3_GGGG(z0, .(z1, z2), .(z3, z4), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, z4)))
U2_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z4, z5)), x6, U5_gg(member_in_gg(x6, .(z4, z5))))
U3_GGGG(z0, .(z1, z2), .(z3, z4), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, z4)))
U2_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z0, z4)), x6, U5_gg(member_in_gg(x6, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x6, U5_gg(member_in_gg(x6, .(z4, z5))))
REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3)))
REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, z5), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg
member_in_gg(X, .(H, L)) → U5_gg(member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U6_ag(member1_in_ag(L))
U5_gg(member_out_gg) → member_out_gg
U6_ag(member1_out_ag(X)) → member1_out_ag(X)

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0)
U6_ag(x0)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), member1_out_ag(x2)) we obtained the following new rules:

REACH_IN_GGGG(x0, x1, .(.(y_6, .(y_7, [])), x3), .(x0, .(x4, x5))) → U2_GGGG(x1, .(.(y_6, .(y_7, [])), x3), .(x0, .(x4, x5)), member1_out_ag(.(y_6, .(y_7, []))))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Instantiation

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ ForwardInstantiation

                                                                              ↳ QDP

                                                                                ↳ ForwardInstantiation

  ↳ PrologToPiTRSProof

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

U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x6, U5_gg(member_in_gg(x6, .(z0, z4))))
REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3)))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg)
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, z5), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z5)))
U2_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg)
U3_GGGG(z0, .(z1, z2), .(z3, z4), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, z4)))
U2_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z4, z5)), x6, U5_gg(member_in_gg(x6, .(z4, z5))))
REACH_IN_GGGG(x0, x1, .(.(y_6, .(y_7, [])), x3), .(x0, .(x4, x5))) → U2_GGGG(x1, .(.(y_6, .(y_7, [])), x3), .(x0, .(x4, x5)), member1_out_ag(.(y_6, .(y_7, []))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z0, z4)), x6, U5_gg(member_in_gg(x6, .(z0, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, z4), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, z4)))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x6, U5_gg(member_in_gg(x6, .(z4, z5))))
REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3)))
REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, z5), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg
member_in_gg(X, .(H, L)) → U5_gg(member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U6_ag(member1_in_ag(L))
U5_gg(member_out_gg) → member_out_gg
U6_ag(member1_out_ag(X)) → member1_out_ag(X)

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0)
U6_ag(x0)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2)) we obtained the following new rules:

REACH_IN_GGGG(x0, x1, .(.(y_6, .(y_7, [])), x3), .(x0, .(x0, x4))) → U2_GGGG(x1, .(.(y_6, .(y_7, [])), x3), .(x0, .(x0, x4)), member1_out_ag(.(y_6, .(y_7, []))))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Instantiation

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ ForwardInstantiation

                                                                              ↳ QDP

                                                                                ↳ ForwardInstantiation

                                                                                  ↳ QDP

                                                                                    ↳ NonTerminationProof

  ↳ PrologToPiTRSProof

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

REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3)))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x6, U5_gg(member_in_gg(x6, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg)
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, z5), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z5)))
U3_GGGG(z0, .(z1, z2), .(z3, z4), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, z4)))
U2_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z4, z5)), x6, U5_gg(member_in_gg(x6, .(z4, z5))))
REACH_IN_GGGG(x0, x1, .(.(y_6, .(y_7, [])), x3), .(x0, .(x4, x5))) → U2_GGGG(x1, .(.(y_6, .(y_7, [])), x3), .(x0, .(x4, x5)), member1_out_ag(.(y_6, .(y_7, []))))
U3_GGGG(z0, .(z1, z2), .(z3, z4), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, z4)))
U2_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z0, z4)), x6, U5_gg(member_in_gg(x6, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x6, U5_gg(member_in_gg(x6, .(z4, z5))))
REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3)))
REACH_IN_GGGG(x0, x1, .(.(y_6, .(y_7, [])), x3), .(x0, .(x0, x4))) → U2_GGGG(x1, .(.(y_6, .(y_7, [])), x3), .(x0, .(x0, x4)), member1_out_ag(.(y_6, .(y_7, []))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, z5), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg
member_in_gg(X, .(H, L)) → U5_gg(member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U6_ag(member1_in_ag(L))
U5_gg(member_out_gg) → member_out_gg
U6_ag(member1_out_ag(X)) → member1_out_ag(X)

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0)
U6_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_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3)))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x6, U5_gg(member_in_gg(x6, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg)
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, z5), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z5)))
U3_GGGG(z0, .(z1, z2), .(z3, z4), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, z4)))
U2_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(x5, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z4, z5)), x6, U5_gg(member_in_gg(x6, .(z4, z5))))
REACH_IN_GGGG(x0, x1, .(.(y_6, .(y_7, [])), x3), .(x0, .(x4, x5))) → U2_GGGG(x1, .(.(y_6, .(y_7, [])), x3), .(x0, .(x4, x5)), member1_out_ag(.(y_6, .(y_7, []))))
U3_GGGG(z0, .(z1, z2), .(z3, z4), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, z4)))
U2_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(x5, .(x6, [])), z3), .(z0, .(z0, z4)), x6, U5_gg(member_in_gg(x6, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x6, U5_gg(member_in_gg(x6, .(z4, z5))))
REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3)))
REACH_IN_GGGG(x0, x1, .(.(y_6, .(y_7, [])), x3), .(x0, .(x0, x4))) → U2_GGGG(x1, .(.(y_6, .(y_7, [])), x3), .(x0, .(x0, x4)), member1_out_ag(.(y_6, .(y_7, []))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, z5), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg
member_in_gg(X, .(H, L)) → U5_gg(member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H)
member1_in_ag(.(H, L)) → U6_ag(member1_in_ag(L))
U5_gg(member_out_gg) → member_out_gg
U6_ag(member1_out_ag(X)) → member1_out_ag(X)

s = U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, z5), z2, member_out_gg) evaluates to t =U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z5)), z2, member_out_gg)

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

Matcher: [z5 / .(z2, z5)]
Semiunifier: [ ]

Rewriting sequence

U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, z5), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z5)))
with rule U3_GGGG(z0', .(.(z1', .(z2', [])), z3'), .(z2', z5'), z2', member_out_gg) → REACH_IN_GGGG(z2', z0', .(.(z1', .(z2', [])), z3'), .(z2', .(z2', z5'))) at position [] and matcher [z3' / z3, z5' / z5, z1' / z1, z2' / z2, z0' / z0]

REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z5))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z5)), member1_out_ag(.(z1, .(z2, []))))
with rule REACH_IN_GGGG(x0, x1, .(.(y_6, .(y_7, [])), x3), .(x0, .(x4, x5'))) → U2_GGGG(x1, .(.(y_6, .(y_7, [])), x3), .(x0, .(x4, x5')), member1_out_ag(.(y_6, .(y_7, [])))) at position [] and matcher [x1 / z0, x4 / z2, y_6 / z1, x3 / z3, x0 / z2, x5' / z5, y_7 / z2]

U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z5)), member1_out_ag(.(z1, .(z2, [])))) → U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z5)), z2, member_out_gg)
with rule U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg)

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,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_gggg(X, Y, Edges, Visited) → U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_gggg(X, Y, Edges, Visited)
reach_in_gggg(X, Z, Edges, Visited) → U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U6_ag(X, H, L, member1_in_ag(X, L))
U6_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited)))
U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) → reach_out_gggg(X, Z, Edges, Visited)

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_gggg(X, Y, Edges, Visited) → U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_gggg(X, Y, Edges, Visited)
reach_in_gggg(X, Z, Edges, Visited) → U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U6_ag(X, H, L, member1_in_ag(X, L))
U6_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited)))
U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) → reach_out_gggg(X, Z, Edges, Visited)

REACH_IN_GGGG(X, Y, Edges, Visited) → U1_GGGG(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_GGGG(X, Y, Edges, Visited) → MEMBER_IN_GG(.(X, .(Y, [])), Edges)
MEMBER_IN_GG(X, .(H, L)) → U5_GG(X, H, L, member_in_gg(X, L))
MEMBER_IN_GG(X, .(H, L)) → MEMBER_IN_GG(X, L)
REACH_IN_GGGG(X, Z, Edges, Visited) → U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
REACH_IN_GGGG(X, Z, Edges, Visited) → MEMBER1_IN_AG(.(X, .(Y, [])), Edges)
MEMBER1_IN_AG(X, .(H, L)) → U6_AG(X, H, L, member1_in_ag(X, L))
MEMBER1_IN_AG(X, .(H, L)) → MEMBER1_IN_AG(X, L)
U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → MEMBER_IN_GG(Y, Visited)
U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → U4_GGGG(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited)))
U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited))

The TRS R consists of the following rules:

reach_in_gggg(X, Y, Edges, Visited) → U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_gggg(X, Y, Edges, Visited)
reach_in_gggg(X, Z, Edges, Visited) → U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U6_ag(X, H, L, member1_in_ag(X, L))
U6_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited)))
U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) → reach_out_gggg(X, Z, Edges, Visited)

The argument filtering Pi contains the following mapping:
reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4)
U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x1, x2, x3, x4, x5)
member_in_gg(x1, x2) = member_in_gg(x1, x2)
.(x1, x2) = .(x1, x2)
member_out_gg(x1, x2) = member_out_gg(x1, x2)
U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4)
[] = []
reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg(x1, x2, x3, x4)
U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x1, x2, x3, x4, x5)
member1_in_ag(x1, x2) = member1_in_ag(x2)
member1_out_ag(x1, x2) = member1_out_ag(x1, x2)
U6_ag(x1, x2, x3, x4) = U6_ag(x2, x3, x4)
U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x1, x2, x3, x4, x5, x6)
U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x1, x2, x3, x4, x5)
U5_GG(x1, x2, x3, x4) = U5_GG(x1, x2, x3, x4)
U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x1, x2, x3, x4, x5)
U6_AG(x1, x2, x3, x4) = U6_AG(x2, x3, x4)
MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2)
REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4)
U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x1, x2, x3, x4, x5, x6)
U1_GGGG(x1, x2, x3, x4, x5) = U1_GGGG(x1, x2, x3, x4, x5)
MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2)
U4_GGGG(x1, x2, x3, x4, x5) = U4_GGGG(x1, x2, x3, x4, x5)

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_GGGG(X, Y, Edges, Visited) → U1_GGGG(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_GGGG(X, Y, Edges, Visited) → MEMBER_IN_GG(.(X, .(Y, [])), Edges)
MEMBER_IN_GG(X, .(H, L)) → U5_GG(X, H, L, member_in_gg(X, L))
MEMBER_IN_GG(X, .(H, L)) → MEMBER_IN_GG(X, L)
REACH_IN_GGGG(X, Z, Edges, Visited) → U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
REACH_IN_GGGG(X, Z, Edges, Visited) → MEMBER1_IN_AG(.(X, .(Y, [])), Edges)
MEMBER1_IN_AG(X, .(H, L)) → U6_AG(X, H, L, member1_in_ag(X, L))
MEMBER1_IN_AG(X, .(H, L)) → MEMBER1_IN_AG(X, L)
U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → MEMBER_IN_GG(Y, Visited)
U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → U4_GGGG(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited)))
U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited))

The TRS R consists of the following rules:

reach_in_gggg(X, Y, Edges, Visited) → U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_gggg(X, Y, Edges, Visited)
reach_in_gggg(X, Z, Edges, Visited) → U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U6_ag(X, H, L, member1_in_ag(X, L))
U6_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited)))
U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) → reach_out_gggg(X, Z, Edges, Visited)

↳ 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_gggg(X, Y, Edges, Visited) → U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_gggg(X, Y, Edges, Visited)
reach_in_gggg(X, Z, Edges, Visited) → U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U6_ag(X, H, L, member1_in_ag(X, L))
U6_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited)))
U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) → reach_out_gggg(X, Z, Edges, Visited)

↳ 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_gggg(X, Y, Edges, Visited) → U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_gggg(X, Y, Edges, Visited)
reach_in_gggg(X, Z, Edges, Visited) → U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U6_ag(X, H, L, member1_in_ag(X, L))
U6_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited)))
U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) → reach_out_gggg(X, Z, Edges, Visited)

↳ 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_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited))
REACH_IN_GGGG(X, Z, Edges, Visited) → U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))

The TRS R consists of the following rules:

reach_in_gggg(X, Y, Edges, Visited) → U1_gggg(X, Y, Edges, Visited, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_gggg(X, Y, Edges, Visited, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_gggg(X, Y, Edges, Visited)
reach_in_gggg(X, Z, Edges, Visited) → U2_gggg(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U6_ag(X, H, L, member1_in_ag(X, L))
U6_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U2_gggg(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_gggg(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U3_gggg(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → U4_gggg(X, Z, Edges, Visited, reach_in_gggg(Y, Z, Edges, .(Y, Visited)))
U4_gggg(X, Z, Edges, Visited, reach_out_gggg(Y, Z, Edges, .(Y, Visited))) → reach_out_gggg(X, Z, Edges, Visited)

The argument filtering Pi contains the following mapping:
reach_in_gggg(x1, x2, x3, x4) = reach_in_gggg(x1, x2, x3, x4)
U1_gggg(x1, x2, x3, x4, x5) = U1_gggg(x1, x2, x3, x4, x5)
member_in_gg(x1, x2) = member_in_gg(x1, x2)
.(x1, x2) = .(x1, x2)
member_out_gg(x1, x2) = member_out_gg(x1, x2)
U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4)
[] = []
reach_out_gggg(x1, x2, x3, x4) = reach_out_gggg(x1, x2, x3, x4)
U2_gggg(x1, x2, x3, x4, x5) = U2_gggg(x1, x2, x3, x4, x5)
member1_in_ag(x1, x2) = member1_in_ag(x2)
member1_out_ag(x1, x2) = member1_out_ag(x1, x2)
U6_ag(x1, x2, x3, x4) = U6_ag(x2, x3, x4)
U3_gggg(x1, x2, x3, x4, x5, x6) = U3_gggg(x1, x2, x3, x4, x5, x6)
U4_gggg(x1, x2, x3, x4, x5) = U4_gggg(x1, x2, x3, x4, x5)
U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x1, x2, x3, x4, x5)
REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4)
U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x1, x2, x3, x4, x5, x6)

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_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited))
REACH_IN_GGGG(X, Z, Edges, Visited) → U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U6_ag(X, H, L, member1_in_ag(X, L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U6_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The argument filtering Pi contains the following mapping:
member_in_gg(x1, x2) = member_in_gg(x1, x2)
.(x1, x2) = .(x1, x2)
member_out_gg(x1, x2) = member_out_gg(x1, x2)
U5_gg(x1, x2, x3, x4) = U5_gg(x1, x2, x3, x4)
[] = []
member1_in_ag(x1, x2) = member1_in_ag(x2)
member1_out_ag(x1, x2) = member1_out_ag(x1, x2)
U6_ag(x1, x2, x3, x4) = U6_ag(x2, x3, x4)
U2_GGGG(x1, x2, x3, x4, x5) = U2_GGGG(x1, x2, x3, x4, x5)
REACH_IN_GGGG(x1, x2, x3, x4) = REACH_IN_GGGG(x1, x2, x3, x4)
U3_GGGG(x1, x2, x3, x4, x5, x6) = U3_GGGG(x1, x2, x3, x4, x5, x6)

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_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited))
REACH_IN_GGGG(X, Z, Edges, Visited) → U2_GGGG(X, Z, Edges, Visited, member1_in_ag(Edges))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(.(H, L)) → U6_ag(H, L, member1_in_ag(L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U6_ag(H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0, x1, x2, x3)
U6_ag(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U2_GGGG(X, Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])), Edges)) → U3_GGGG(X, Z, Edges, Visited, Y, member_in_gg(Y, Visited)) at position [5] we obtained the following new rules:

U2_GGGG(y0, y1, y2, .(x1, x2), member1_out_ag(.(y0, .(x0, [])), y2)) → U3_GGGG(y0, y1, y2, .(x1, x2), x0, U5_gg(x0, x1, x2, member_in_gg(x0, x2)))
U2_GGGG(y0, y1, y2, .(x0, x1), member1_out_ag(.(y0, .(x0, [])), y2)) → U3_GGGG(y0, y1, y2, .(x0, x1), x0, member_out_gg(x0, .(x0, x1)))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

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

U2_GGGG(y0, y1, y2, .(x1, x2), member1_out_ag(.(y0, .(x0, [])), y2)) → U3_GGGG(y0, y1, y2, .(x1, x2), x0, U5_gg(x0, x1, x2, member_in_gg(x0, x2)))
U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited))
REACH_IN_GGGG(X, Z, Edges, Visited) → U2_GGGG(X, Z, Edges, Visited, member1_in_ag(Edges))
U2_GGGG(y0, y1, y2, .(x0, x1), member1_out_ag(.(y0, .(x0, [])), y2)) → U3_GGGG(y0, y1, y2, .(x0, x1), x0, member_out_gg(x0, .(x0, x1)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(.(H, L)) → U6_ag(H, L, member1_in_ag(L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U6_ag(H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0, x1, x2, x3)
U6_ag(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule REACH_IN_GGGG(X, Z, Edges, Visited) → U2_GGGG(X, Z, Edges, Visited, member1_in_ag(Edges)) at position [4] we obtained the following new rules:

REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y0, y1, .(x0, x1), y3, U6_ag(x0, x1, member1_in_ag(x1)))
REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y0, y1, .(x0, x1), y3, member1_out_ag(x0, .(x0, x1)))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

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

REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y0, y1, .(x0, x1), y3, U6_ag(x0, x1, member1_in_ag(x1)))
U2_GGGG(y0, y1, y2, .(x1, x2), member1_out_ag(.(y0, .(x0, [])), y2)) → U3_GGGG(y0, y1, y2, .(x1, x2), x0, U5_gg(x0, x1, x2, member_in_gg(x0, x2)))
U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited))
REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y0, y1, .(x0, x1), y3, member1_out_ag(x0, .(x0, x1)))
U2_GGGG(y0, y1, y2, .(x0, x1), member1_out_ag(.(y0, .(x0, [])), y2)) → U3_GGGG(y0, y1, y2, .(x0, x1), x0, member_out_gg(x0, .(x0, x1)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(.(H, L)) → U6_ag(H, L, member1_in_ag(L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U6_ag(H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0, x1, x2, x3)
U6_ag(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U3_GGGG(X, Z, Edges, Visited, Y, member_out_gg(Y, Visited)) → REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited)) we obtained the following new rules:

U3_GGGG(z0, z1, z2, .(z3, z4), z3, member_out_gg(z3, .(z3, z4))) → REACH_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4)))
U3_GGGG(z0, z1, z2, .(z3, z4), z5, member_out_gg(z5, .(z3, z4))) → REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4)))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

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

REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y0, y1, .(x0, x1), y3, U6_ag(x0, x1, member1_in_ag(x1)))
U2_GGGG(y0, y1, y2, .(x1, x2), member1_out_ag(.(y0, .(x0, [])), y2)) → U3_GGGG(y0, y1, y2, .(x1, x2), x0, U5_gg(x0, x1, x2, member_in_gg(x0, x2)))
REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y0, y1, .(x0, x1), y3, member1_out_ag(x0, .(x0, x1)))
U3_GGGG(z0, z1, z2, .(z3, z4), z3, member_out_gg(z3, .(z3, z4))) → REACH_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4)))
U2_GGGG(y0, y1, y2, .(x0, x1), member1_out_ag(.(y0, .(x0, [])), y2)) → U3_GGGG(y0, y1, y2, .(x0, x1), x0, member_out_gg(x0, .(x0, x1)))
U3_GGGG(z0, z1, z2, .(z3, z4), z5, member_out_gg(z5, .(z3, z4))) → REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(.(H, L)) → U6_ag(H, L, member1_in_ag(L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U6_ag(H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0, x1, x2, x3)
U6_ag(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_GGGG(y0, y1, y2, .(x1, x2), member1_out_ag(.(y0, .(x0, [])), y2)) → U3_GGGG(y0, y1, y2, .(x1, x2), x0, U5_gg(x0, x1, x2, member_in_gg(x0, x2))) we obtained the following new rules:

U2_GGGG(z0, z1, .(z2, z3), .(x3, x4), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(x3, x4), x5, U5_gg(x5, x3, x4, member_in_gg(x5, x4)))
U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(x3, x4), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(x3, x4), x5, U5_gg(x5, x3, x4, member_in_gg(x5, x4)))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

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

REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y0, y1, .(x0, x1), y3, U6_ag(x0, x1, member1_in_ag(x1)))
U2_GGGG(z0, z1, .(z2, z3), .(x3, x4), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(x3, x4), x5, U5_gg(x5, x3, x4, member_in_gg(x5, x4)))
REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y0, y1, .(x0, x1), y3, member1_out_ag(x0, .(x0, x1)))
U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(x3, x4), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(x3, x4), x5, U5_gg(x5, x3, x4, member_in_gg(x5, x4)))
U3_GGGG(z0, z1, z2, .(z3, z4), z3, member_out_gg(z3, .(z3, z4))) → REACH_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4)))
U2_GGGG(y0, y1, y2, .(x0, x1), member1_out_ag(.(y0, .(x0, [])), y2)) → U3_GGGG(y0, y1, y2, .(x0, x1), x0, member_out_gg(x0, .(x0, x1)))
U3_GGGG(z0, z1, z2, .(z3, z4), z5, member_out_gg(z5, .(z3, z4))) → REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(.(H, L)) → U6_ag(H, L, member1_in_ag(L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U6_ag(H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0, x1, x2, x3)
U6_ag(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_GGGG(y0, y1, y2, .(x0, x1), member1_out_ag(.(y0, .(x0, [])), y2)) → U3_GGGG(y0, y1, y2, .(x0, x1), x0, member_out_gg(x0, .(x0, x1))) we obtained the following new rules:

U2_GGGG(z0, z1, .(z2, z3), .(x3, x4), member1_out_ag(.(z0, .(x3, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(x3, x4), x3, member_out_gg(x3, .(x3, x4)))
U2_GGGG(z0, z1, .(.(z0, .(x3, [])), z3), .(x3, x4), member1_out_ag(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x3, [])), z3), .(x3, x4), x3, member_out_gg(x3, .(x3, x4)))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

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

REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y0, y1, .(x0, x1), y3, U6_ag(x0, x1, member1_in_ag(x1)))
U2_GGGG(z0, z1, .(z2, z3), .(x3, x4), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(x3, x4), x5, U5_gg(x5, x3, x4, member_in_gg(x5, x4)))
REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y0, y1, .(x0, x1), y3, member1_out_ag(x0, .(x0, x1)))
U3_GGGG(z0, z1, z2, .(z3, z4), z3, member_out_gg(z3, .(z3, z4))) → REACH_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4)))
U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(x3, x4), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(x3, x4), x5, U5_gg(x5, x3, x4, member_in_gg(x5, x4)))
U2_GGGG(z0, z1, .(z2, z3), .(x3, x4), member1_out_ag(.(z0, .(x3, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(x3, x4), x3, member_out_gg(x3, .(x3, x4)))
U2_GGGG(z0, z1, .(.(z0, .(x3, [])), z3), .(x3, x4), member1_out_ag(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x3, [])), z3), .(x3, x4), x3, member_out_gg(x3, .(x3, x4)))
U3_GGGG(z0, z1, z2, .(z3, z4), z5, member_out_gg(z5, .(z3, z4))) → REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(.(H, L)) → U6_ag(H, L, member1_in_ag(L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U6_ag(H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0, x1, x2, x3)
U6_ag(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y0, y1, .(x0, x1), y3, U6_ag(x0, x1, member1_in_ag(x1))) we obtained the following new rules:

REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))
REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

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

REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))
U2_GGGG(z0, z1, .(z2, z3), .(x3, x4), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(x3, x4), x5, U5_gg(x5, x3, x4, member_in_gg(x5, x4)))
REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y0, y1, .(x0, x1), y3, member1_out_ag(x0, .(x0, x1)))
REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))
U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(x3, x4), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(x3, x4), x5, U5_gg(x5, x3, x4, member_in_gg(x5, x4)))
U3_GGGG(z0, z1, z2, .(z3, z4), z3, member_out_gg(z3, .(z3, z4))) → REACH_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4)))
U2_GGGG(z0, z1, .(z2, z3), .(x3, x4), member1_out_ag(.(z0, .(x3, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(x3, x4), x3, member_out_gg(x3, .(x3, x4)))
U3_GGGG(z0, z1, z2, .(z3, z4), z5, member_out_gg(z5, .(z3, z4))) → REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4)))
U2_GGGG(z0, z1, .(.(z0, .(x3, [])), z3), .(x3, x4), member1_out_ag(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x3, [])), z3), .(x3, x4), x3, member_out_gg(x3, .(x3, x4)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(.(H, L)) → U6_ag(H, L, member1_in_ag(L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U6_ag(H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0, x1, x2, x3)
U6_ag(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y0, y1, .(x0, x1), y3, member1_out_ag(x0, .(x0, x1))) we obtained the following new rules:

REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), member1_out_ag(x2, .(x2, x3)))
REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), member1_out_ag(x2, .(x2, x3)))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

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

REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))
U2_GGGG(z0, z1, .(z2, z3), .(x3, x4), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(x3, x4), x5, U5_gg(x5, x3, x4, member_in_gg(x5, x4)))
REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))
U3_GGGG(z0, z1, z2, .(z3, z4), z3, member_out_gg(z3, .(z3, z4))) → REACH_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4)))
U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(x3, x4), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(x3, x4), x5, U5_gg(x5, x3, x4, member_in_gg(x5, x4)))
U2_GGGG(z0, z1, .(z2, z3), .(x3, x4), member1_out_ag(.(z0, .(x3, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(x3, x4), x3, member_out_gg(x3, .(x3, x4)))
REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), member1_out_ag(x2, .(x2, x3)))
U2_GGGG(z0, z1, .(.(z0, .(x3, [])), z3), .(x3, x4), member1_out_ag(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x3, [])), z3), .(x3, x4), x3, member_out_gg(x3, .(x3, x4)))
U3_GGGG(z0, z1, z2, .(z3, z4), z5, member_out_gg(z5, .(z3, z4))) → REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4)))
REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), member1_out_ag(x2, .(x2, x3)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(.(H, L)) → U6_ag(H, L, member1_in_ag(L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U6_ag(H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0, x1, x2, x3)
U6_ag(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U3_GGGG(z0, z1, z2, .(z3, z4), z3, member_out_gg(z3, .(z3, z4))) → REACH_IN_GGGG(z3, z1, z2, .(z3, .(z3, z4))) we obtained the following new rules:

U3_GGGG(z0, z1, .(z2, z3), .(z4, z5), z4, member_out_gg(z4, .(z4, z5))) → REACH_IN_GGGG(z4, z1, .(z2, z3), .(z4, .(z4, z5)))
U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z2, z5), z2, member_out_gg(z2, .(z2, z5))) → REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z2, z5)))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

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

REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))
U2_GGGG(z0, z1, .(z2, z3), .(x3, x4), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(x3, x4), x5, U5_gg(x5, x3, x4, member_in_gg(x5, x4)))
REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))
U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(x3, x4), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(x3, x4), x5, U5_gg(x5, x3, x4, member_in_gg(x5, x4)))
U3_GGGG(z0, z1, .(z2, z3), .(z4, z5), z4, member_out_gg(z4, .(z4, z5))) → REACH_IN_GGGG(z4, z1, .(z2, z3), .(z4, .(z4, z5)))
U2_GGGG(z0, z1, .(z2, z3), .(x3, x4), member1_out_ag(.(z0, .(x3, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(x3, x4), x3, member_out_gg(x3, .(x3, x4)))
U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z2, z5), z2, member_out_gg(z2, .(z2, z5))) → REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z2, z5)))
U3_GGGG(z0, z1, z2, .(z3, z4), z5, member_out_gg(z5, .(z3, z4))) → REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4)))
U2_GGGG(z0, z1, .(.(z0, .(x3, [])), z3), .(x3, x4), member1_out_ag(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x3, [])), z3), .(x3, x4), x3, member_out_gg(x3, .(x3, x4)))
REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), member1_out_ag(x2, .(x2, x3)))
REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), member1_out_ag(x2, .(x2, x3)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(.(H, L)) → U6_ag(H, L, member1_in_ag(L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U6_ag(H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0, x1, x2, x3)
U6_ag(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U3_GGGG(z0, z1, z2, .(z3, z4), z5, member_out_gg(z5, .(z3, z4))) → REACH_IN_GGGG(z5, z1, z2, .(z5, .(z3, z4))) we obtained the following new rules:

U3_GGGG(z0, z1, .(z2, z3), .(z4, z5), z6, member_out_gg(z6, .(z4, z5))) → REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z4, z5)))
U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z4, z5), z2, member_out_gg(z2, .(z4, z5))) → REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z4, z5)))
U3_GGGG(z0, z1, .(z2, z3), .(z4, z5), z4, member_out_gg(z4, .(z4, z5))) → REACH_IN_GGGG(z4, z1, .(z2, z3), .(z4, .(z4, z5)))
U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z2, z4), z2, member_out_gg(z2, .(z2, z4))) → REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z2, z4)))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

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

U2_GGGG(z0, z1, .(z2, z3), .(x3, x4), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(x3, x4), x5, U5_gg(x5, x3, x4, member_in_gg(x5, x4)))
U3_GGGG(z0, z1, .(z2, z3), .(z4, z5), z6, member_out_gg(z6, .(z4, z5))) → REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z4, z5)))
U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z4, z5), z2, member_out_gg(z2, .(z4, z5))) → REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z4, z5)))
U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(x3, x4), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(x3, x4), x5, U5_gg(x5, x3, x4, member_in_gg(x5, x4)))
U2_GGGG(z0, z1, .(.(z0, .(x3, [])), z3), .(x3, x4), member1_out_ag(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x3, [])), z3), .(x3, x4), x3, member_out_gg(x3, .(x3, x4)))
REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), member1_out_ag(x2, .(x2, x3)))
REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), member1_out_ag(x2, .(x2, x3)))
REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))
REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))
U3_GGGG(z0, z1, .(z2, z3), .(z4, z5), z4, member_out_gg(z4, .(z4, z5))) → REACH_IN_GGGG(z4, z1, .(z2, z3), .(z4, .(z4, z5)))
U2_GGGG(z0, z1, .(z2, z3), .(x3, x4), member1_out_ag(.(z0, .(x3, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(x3, x4), x3, member_out_gg(x3, .(x3, x4)))
U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z2, z5), z2, member_out_gg(z2, .(z2, z5))) → REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z2, z5)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(.(H, L)) → U6_ag(H, L, member1_in_ag(L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U6_ag(H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0, x1, x2, x3)
U6_ag(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_GGGG(z0, z1, .(z2, z3), .(x3, x4), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(x3, x4), x5, U5_gg(x5, x3, x4, member_in_gg(x5, x4))) we obtained the following new rules:

U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, z4)), x6, U5_gg(x6, z0, .(z0, z4), member_in_gg(x6, .(z0, z4))))
U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, z5)), x6, U5_gg(x6, z0, .(z4, z5), member_in_gg(x6, .(z4, z5))))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x6, U5_gg(x6, z0, .(z4, z5), member_in_gg(x6, .(z4, z5))))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x6, U5_gg(x6, z0, .(z0, z4), member_in_gg(x6, .(z0, z4))))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Instantiation

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

U3_GGGG(z0, z1, .(z2, z3), .(z4, z5), z6, member_out_gg(z6, .(z4, z5))) → REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z4, z5)))
U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z4, z5), z2, member_out_gg(z2, .(z4, z5))) → REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z4, z5)))
U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(x3, x4), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(x3, x4), x5, U5_gg(x5, x3, x4, member_in_gg(x5, x4)))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x6, U5_gg(x6, z0, .(z4, z5), member_in_gg(x6, .(z4, z5))))
U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, z5)), x6, U5_gg(x6, z0, .(z4, z5), member_in_gg(x6, .(z4, z5))))
REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), member1_out_ag(x2, .(x2, x3)))
U2_GGGG(z0, z1, .(.(z0, .(x3, [])), z3), .(x3, x4), member1_out_ag(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x3, [])), z3), .(x3, x4), x3, member_out_gg(x3, .(x3, x4)))
REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), member1_out_ag(x2, .(x2, x3)))
REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))
U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, z4)), x6, U5_gg(x6, z0, .(z0, z4), member_in_gg(x6, .(z0, z4))))
REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x6, U5_gg(x6, z0, .(z0, z4), member_in_gg(x6, .(z0, z4))))
U2_GGGG(z0, z1, .(z2, z3), .(x3, x4), member1_out_ag(.(z0, .(x3, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(x3, x4), x3, member_out_gg(x3, .(x3, x4)))
U3_GGGG(z0, z1, .(z2, z3), .(z4, z5), z4, member_out_gg(z4, .(z4, z5))) → REACH_IN_GGGG(z4, z1, .(z2, z3), .(z4, .(z4, z5)))
U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z2, z5), z2, member_out_gg(z2, .(z2, z5))) → REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z2, z5)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(.(H, L)) → U6_ag(H, L, member1_in_ag(L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U6_ag(H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0, x1, x2, x3)
U6_ag(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(x3, x4), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(x3, x4), x5, U5_gg(x5, x3, x4, member_in_gg(x5, x4))) we obtained the following new rules:

U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, z4)), x2, U5_gg(x2, z0, .(z0, z4), member_in_gg(x2, .(z0, z4))))
U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, z5)), x2, U5_gg(x2, z0, .(z4, z5), member_in_gg(x2, .(z4, z5))))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Instantiation

                                                                  ↳ QDP

                                                                    ↳ Instantiation

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

U3_GGGG(z0, z1, .(z2, z3), .(z4, z5), z6, member_out_gg(z6, .(z4, z5))) → REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z4, z5)))
U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z4, z5), z2, member_out_gg(z2, .(z4, z5))) → REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z4, z5)))
U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, z5)), x6, U5_gg(x6, z0, .(z4, z5), member_in_gg(x6, .(z4, z5))))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x6, U5_gg(x6, z0, .(z4, z5), member_in_gg(x6, .(z4, z5))))
U2_GGGG(z0, z1, .(.(z0, .(x3, [])), z3), .(x3, x4), member1_out_ag(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x3, [])), z3), .(x3, x4), x3, member_out_gg(x3, .(x3, x4)))
REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), member1_out_ag(x2, .(x2, x3)))
REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), member1_out_ag(x2, .(x2, x3)))
REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))
U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, z4)), x6, U5_gg(x6, z0, .(z0, z4), member_in_gg(x6, .(z0, z4))))
REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))
U3_GGGG(z0, z1, .(z2, z3), .(z4, z5), z4, member_out_gg(z4, .(z4, z5))) → REACH_IN_GGGG(z4, z1, .(z2, z3), .(z4, .(z4, z5)))
U2_GGGG(z0, z1, .(z2, z3), .(x3, x4), member1_out_ag(.(z0, .(x3, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(x3, x4), x3, member_out_gg(x3, .(x3, x4)))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x6, U5_gg(x6, z0, .(z0, z4), member_in_gg(x6, .(z0, z4))))
U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z2, z5), z2, member_out_gg(z2, .(z2, z5))) → REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z2, z5)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(.(H, L)) → U6_ag(H, L, member1_in_ag(L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U6_ag(H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0, x1, x2, x3)
U6_ag(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_GGGG(z0, z1, .(z2, z3), .(x3, x4), member1_out_ag(.(z0, .(x3, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(x3, x4), x3, member_out_gg(x3, .(x3, x4))) we obtained the following new rules:

U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5))))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4))))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5))))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4))))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Instantiation

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

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

U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5))))
U3_GGGG(z0, z1, .(z2, z3), .(z4, z5), z6, member_out_gg(z6, .(z4, z5))) → REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z4, z5)))
U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z4, z5), z2, member_out_gg(z2, .(z4, z5))) → REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z4, z5)))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x6, U5_gg(x6, z0, .(z4, z5), member_in_gg(x6, .(z4, z5))))
U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, z5)), x6, U5_gg(x6, z0, .(z4, z5), member_in_gg(x6, .(z4, z5))))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4))))
REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), member1_out_ag(x2, .(x2, x3)))
U2_GGGG(z0, z1, .(.(z0, .(x3, [])), z3), .(x3, x4), member1_out_ag(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x3, [])), z3), .(x3, x4), x3, member_out_gg(x3, .(x3, x4)))
REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), member1_out_ag(x2, .(x2, x3)))
REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))
U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, z4)), x6, U5_gg(x6, z0, .(z0, z4), member_in_gg(x6, .(z0, z4))))
REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4))))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5))))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x6, U5_gg(x6, z0, .(z0, z4), member_in_gg(x6, .(z0, z4))))
U3_GGGG(z0, z1, .(z2, z3), .(z4, z5), z4, member_out_gg(z4, .(z4, z5))) → REACH_IN_GGGG(z4, z1, .(z2, z3), .(z4, .(z4, z5)))
U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z2, z5), z2, member_out_gg(z2, .(z2, z5))) → REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z2, z5)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(.(H, L)) → U6_ag(H, L, member1_in_ag(L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U6_ag(H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0, x1, x2, x3)
U6_ag(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_GGGG(z0, z1, .(.(z0, .(x3, [])), z3), .(x3, x4), member1_out_ag(.(z0, .(x3, [])), .(.(z0, .(x3, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x3, [])), z3), .(x3, x4), x3, member_out_gg(x3, .(x3, x4))) we obtained the following new rules:

U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5))))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4))))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Instantiation

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ ForwardInstantiation

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

U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5))))
U3_GGGG(z0, z1, .(z2, z3), .(z4, z5), z6, member_out_gg(z6, .(z4, z5))) → REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z4, z5)))
U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z4, z5), z2, member_out_gg(z2, .(z4, z5))) → REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z4, z5)))
U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, z5)), x6, U5_gg(x6, z0, .(z4, z5), member_in_gg(x6, .(z4, z5))))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x6, U5_gg(x6, z0, .(z4, z5), member_in_gg(x6, .(z4, z5))))
REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), member1_out_ag(x2, .(x2, x3)))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4))))
REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), member1_out_ag(x2, .(x2, x3)))
REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))
U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, z4)), x6, U5_gg(x6, z0, .(z0, z4), member_in_gg(x6, .(z0, z4))))
REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4))))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5))))
U3_GGGG(z0, z1, .(z2, z3), .(z4, z5), z4, member_out_gg(z4, .(z4, z5))) → REACH_IN_GGGG(z4, z1, .(z2, z3), .(z4, .(z4, z5)))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x6, U5_gg(x6, z0, .(z0, z4), member_in_gg(x6, .(z0, z4))))
U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z2, z5), z2, member_out_gg(z2, .(z2, z5))) → REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z2, z5)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(.(H, L)) → U6_ag(H, L, member1_in_ag(L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U6_ag(H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0, x1, x2, x3)
U6_ag(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), member1_out_ag(x2, .(x2, x3))) we obtained the following new rules:

REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, x5))) → U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, x5)), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3)))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Instantiation

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ ForwardInstantiation

                                                                              ↳ QDP

                                                                                ↳ ForwardInstantiation

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

U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5))))
U3_GGGG(z0, z1, .(z2, z3), .(z4, z5), z6, member_out_gg(z6, .(z4, z5))) → REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z4, z5)))
U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z4, z5), z2, member_out_gg(z2, .(z4, z5))) → REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z4, z5)))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x6, U5_gg(x6, z0, .(z4, z5), member_in_gg(x6, .(z4, z5))))
U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, z5)), x6, U5_gg(x6, z0, .(z4, z5), member_in_gg(x6, .(z4, z5))))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4))))
REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), member1_out_ag(x2, .(x2, x3)))
REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))
U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, z4)), x6, U5_gg(x6, z0, .(z0, z4), member_in_gg(x6, .(z0, z4))))
REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4))))
REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, x5))) → U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, x5)), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3)))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5))))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x6, U5_gg(x6, z0, .(z0, z4), member_in_gg(x6, .(z0, z4))))
U3_GGGG(z0, z1, .(z2, z3), .(z4, z5), z4, member_out_gg(z4, .(z4, z5))) → REACH_IN_GGGG(z4, z1, .(z2, z3), .(z4, .(z4, z5)))
U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z2, z5), z2, member_out_gg(z2, .(z2, z5))) → REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z2, z5)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(.(H, L)) → U6_ag(H, L, member1_in_ag(L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U6_ag(H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0, x1, x2, x3)
U6_ag(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), member1_out_ag(x2, .(x2, x3))) we obtained the following new rules:

REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, x4))) → U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, x4)), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3)))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ Instantiation

                                                      ↳ QDP

                                                        ↳ Instantiation

                                                          ↳ QDP

                                                            ↳ Instantiation

                                                              ↳ QDP

                                                                ↳ Instantiation

                                                                  ↳ QDP

                                                                    ↳ Instantiation

                                                                      ↳ QDP

                                                                        ↳ Instantiation

                                                                          ↳ QDP

                                                                            ↳ ForwardInstantiation

                                                                              ↳ QDP

                                                                                ↳ ForwardInstantiation

                                                                                  ↳ QDP

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

U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5))))
U3_GGGG(z0, z1, .(z2, z3), .(z4, z5), z6, member_out_gg(z6, .(z4, z5))) → REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z4, z5)))
U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z4, z5), z2, member_out_gg(z2, .(z4, z5))) → REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z4, z5)))
U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, z5)), x6, U5_gg(x6, z0, .(z4, z5), member_in_gg(x6, .(z4, z5))))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x6, U5_gg(x6, z0, .(z4, z5), member_in_gg(x6, .(z4, z5))))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4))))
REACH_IN_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4))) → U2_GGGG(z5, z1, .(x2, x3), .(z5, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))
U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, z4)), x6, U5_gg(x6, z0, .(z0, z4), member_in_gg(x6, .(z0, z4))))
REACH_IN_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4))) → U2_GGGG(z3, z1, .(x2, x3), .(z3, .(z3, z4)), U6_ag(x2, x3, member1_in_ag(x3)))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4))))
REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, x4))) → U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x0, x4)), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3)))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5))))
REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, x5))) → U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, x5)), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3)))
U3_GGGG(z0, z1, .(z2, z3), .(z4, z5), z4, member_out_gg(z4, .(z4, z5))) → REACH_IN_GGGG(z4, z1, .(z2, z3), .(z4, .(z4, z5)))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x6, U5_gg(x6, z0, .(z0, z4), member_in_gg(x6, .(z0, z4))))
U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z2, z5), z2, member_out_gg(z2, .(z2, z5))) → REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z2, z5)))

The TRS R consists of the following rules:

member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U5_gg(X, H, L, member_in_gg(X, L))
member1_in_ag(.(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(.(H, L)) → U6_ag(H, L, member1_in_ag(L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U6_ag(H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))

The set Q consists of the following terms:

member_in_gg(x0, x1)
member1_in_ag(x0)
U5_gg(x0, x1, x2, x3)
U6_ag(x0, x1, x2)

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