Left Termination proof of /tmp/tmp0dtk8-/pl7.6.2a.pl

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

Clauses:

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

Queries:

reach(g,g,g).

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

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
member1_in_gg(H, .(H, L)) → member1_out_gg(H, .(H, L))
member1_in_gg(X, .(H, L)) → U5_gg(X, H, L, member1_in_gg(X, L))
U5_gg(X, H, L, member1_out_gg(X, L)) → member1_out_gg(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
reach_in_agg(X, Y, Edges) → U1_agg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
U1_agg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_agg(X, Y, Edges)
reach_in_agg(X, Z, Edges) → U2_agg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
U2_agg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_agg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U3_agg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_agg(X, Z, Edges)
U3_ggg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

  ↳ PrologToPiTRSProof

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

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
member1_in_gg(H, .(H, L)) → member1_out_gg(H, .(H, L))
member1_in_gg(X, .(H, L)) → U5_gg(X, H, L, member1_in_gg(X, L))
U5_gg(X, H, L, member1_out_gg(X, L)) → member1_out_gg(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
reach_in_agg(X, Y, Edges) → U1_agg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
U1_agg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_agg(X, Y, Edges)
reach_in_agg(X, Z, Edges) → U2_agg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
U2_agg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_agg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U3_agg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_agg(X, Z, Edges)
U3_ggg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

REACH_IN_GGG(X, Y, Edges) → U1_GGG(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_GGG(X, Y, Edges) → MEMBER_IN_GG(.(X, .(Y, [])), Edges)
MEMBER_IN_GG(X, .(H, L)) → U4_GG(X, H, L, member_in_gg(X, L))
MEMBER_IN_GG(X, .(H, L)) → MEMBER_IN_GG(X, L)
REACH_IN_GGG(X, Z, Edges) → U2_GGG(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_GGG(X, Z, Edges) → MEMBER1_IN_GG(.(X, .(Y, [])), Edges)
MEMBER1_IN_GG(X, .(H, L)) → U5_GG(X, H, L, member1_in_gg(X, L))
MEMBER1_IN_GG(X, .(H, L)) → MEMBER1_IN_GG(X, L)
U2_GGG(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_GGG(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U2_GGG(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → REACH_IN_AGG(Y, Z, Edges)
REACH_IN_AGG(X, Y, Edges) → U1_AGG(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_AGG(X, Y, Edges) → MEMBER_IN_GG(.(X, .(Y, [])), Edges)
REACH_IN_AGG(X, Z, Edges) → U2_AGG(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_AGG(X, Z, Edges) → MEMBER1_IN_GG(.(X, .(Y, [])), Edges)
U2_AGG(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_AGG(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U2_AGG(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → REACH_IN_AGG(Y, Z, Edges)

The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
member1_in_gg(H, .(H, L)) → member1_out_gg(H, .(H, L))
member1_in_gg(X, .(H, L)) → U5_gg(X, H, L, member1_in_gg(X, L))
U5_gg(X, H, L, member1_out_gg(X, L)) → member1_out_gg(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
reach_in_agg(X, Y, Edges) → U1_agg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
U1_agg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_agg(X, Y, Edges)
reach_in_agg(X, Z, Edges) → U2_agg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
U2_agg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_agg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U3_agg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_agg(X, Z, Edges)
U3_ggg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_in_ggg(x1, x2, x3) = reach_in_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4) = U1_ggg(x4)
member_in_gg(x1, x2) = member_in_gg(x1, x2)
.(x1, x2) = .(x2)
member_out_gg(x1, x2) = member_out_gg
U4_gg(x1, x2, x3, x4) = U4_gg(x4)
[] = []
reach_out_ggg(x1, x2, x3) = reach_out_ggg
U2_ggg(x1, x2, x3, x4) = U2_ggg(x2, x3, x4)
member1_in_gg(x1, x2) = member1_in_gg(x1, x2)
member1_out_gg(x1, x2) = member1_out_gg
U5_gg(x1, x2, x3, x4) = U5_gg(x4)
U3_ggg(x1, x2, x3, x4) = U3_ggg(x4)
reach_in_agg(x1, x2, x3) = reach_in_agg(x2, x3)
U1_agg(x1, x2, x3, x4) = U1_agg(x4)
reach_out_agg(x1, x2, x3) = reach_out_agg
U2_agg(x1, x2, x3, x4) = U2_agg(x2, x3, x4)
U3_agg(x1, x2, x3, x4) = U3_agg(x4)
U3_AGG(x1, x2, x3, x4) = U3_AGG(x4)
U5_GG(x1, x2, x3, x4) = U5_GG(x4)
U1_GGG(x1, x2, x3, x4) = U1_GGG(x4)
MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2)
U3_GGG(x1, x2, x3, x4) = U3_GGG(x4)
MEMBER1_IN_GG(x1, x2) = MEMBER1_IN_GG(x1, x2)
REACH_IN_GGG(x1, x2, x3) = REACH_IN_GGG(x1, x2, x3)
U2_AGG(x1, x2, x3, x4) = U2_AGG(x2, x3, x4)
U2_GGG(x1, x2, x3, x4) = U2_GGG(x2, x3, x4)
REACH_IN_AGG(x1, x2, x3) = REACH_IN_AGG(x2, x3)
U4_GG(x1, x2, x3, x4) = U4_GG(x4)
U1_AGG(x1, x2, x3, x4) = U1_AGG(x4)

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

REACH_IN_GGG(X, Y, Edges) → U1_GGG(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_GGG(X, Y, Edges) → MEMBER_IN_GG(.(X, .(Y, [])), Edges)
MEMBER_IN_GG(X, .(H, L)) → U4_GG(X, H, L, member_in_gg(X, L))
MEMBER_IN_GG(X, .(H, L)) → MEMBER_IN_GG(X, L)
REACH_IN_GGG(X, Z, Edges) → U2_GGG(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_GGG(X, Z, Edges) → MEMBER1_IN_GG(.(X, .(Y, [])), Edges)
MEMBER1_IN_GG(X, .(H, L)) → U5_GG(X, H, L, member1_in_gg(X, L))
MEMBER1_IN_GG(X, .(H, L)) → MEMBER1_IN_GG(X, L)
U2_GGG(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_GGG(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U2_GGG(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → REACH_IN_AGG(Y, Z, Edges)
REACH_IN_AGG(X, Y, Edges) → U1_AGG(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_AGG(X, Y, Edges) → MEMBER_IN_GG(.(X, .(Y, [])), Edges)
REACH_IN_AGG(X, Z, Edges) → U2_AGG(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_AGG(X, Z, Edges) → MEMBER1_IN_GG(.(X, .(Y, [])), Edges)
U2_AGG(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_AGG(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U2_AGG(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → REACH_IN_AGG(Y, Z, Edges)

The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
member1_in_gg(H, .(H, L)) → member1_out_gg(H, .(H, L))
member1_in_gg(X, .(H, L)) → U5_gg(X, H, L, member1_in_gg(X, L))
U5_gg(X, H, L, member1_out_gg(X, L)) → member1_out_gg(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
reach_in_agg(X, Y, Edges) → U1_agg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
U1_agg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_agg(X, Y, Edges)
reach_in_agg(X, Z, Edges) → U2_agg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
U2_agg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_agg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U3_agg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_agg(X, Z, Edges)
U3_ggg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

MEMBER1_IN_GG(X, .(H, L)) → MEMBER1_IN_GG(X, L)

The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
member1_in_gg(H, .(H, L)) → member1_out_gg(H, .(H, L))
member1_in_gg(X, .(H, L)) → U5_gg(X, H, L, member1_in_gg(X, L))
U5_gg(X, H, L, member1_out_gg(X, L)) → member1_out_gg(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
reach_in_agg(X, Y, Edges) → U1_agg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
U1_agg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_agg(X, Y, Edges)
reach_in_agg(X, Z, Edges) → U2_agg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
U2_agg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_agg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U3_agg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_agg(X, Z, Edges)
U3_ggg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

↳ 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_GG(X, .(H, L)) → MEMBER1_IN_GG(X, L)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
MEMBER1_IN_GG(x1, x2) = MEMBER1_IN_GG(x1, x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

MEMBER1_IN_GG(X, .(L)) → MEMBER1_IN_GG(X, L)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

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

MEMBER1_IN_GG(X, .(L)) → MEMBER1_IN_GG(X, L)
The graph contains the following edges 1 >= 1, 2 > 2

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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

The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
member1_in_gg(H, .(H, L)) → member1_out_gg(H, .(H, L))
member1_in_gg(X, .(H, L)) → U5_gg(X, H, L, member1_in_gg(X, L))
U5_gg(X, H, L, member1_out_gg(X, L)) → member1_out_gg(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
reach_in_agg(X, Y, Edges) → U1_agg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
U1_agg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_agg(X, Y, Edges)
reach_in_agg(X, Z, Edges) → U2_agg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
U2_agg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_agg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U3_agg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_agg(X, Z, Edges)
U3_ggg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

↳ 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.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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

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

MEMBER_IN_GG(X, .(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_AGG(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → REACH_IN_AGG(Y, Z, Edges)
REACH_IN_AGG(X, Z, Edges) → U2_AGG(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))

The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
member1_in_gg(H, .(H, L)) → member1_out_gg(H, .(H, L))
member1_in_gg(X, .(H, L)) → U5_gg(X, H, L, member1_in_gg(X, L))
U5_gg(X, H, L, member1_out_gg(X, L)) → member1_out_gg(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
reach_in_agg(X, Y, Edges) → U1_agg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
U1_agg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_agg(X, Y, Edges)
reach_in_agg(X, Z, Edges) → U2_agg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
U2_agg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_agg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U3_agg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_agg(X, Z, Edges)
U3_ggg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_in_ggg(x1, x2, x3) = reach_in_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4) = U1_ggg(x4)
member_in_gg(x1, x2) = member_in_gg(x1, x2)
.(x1, x2) = .(x2)
member_out_gg(x1, x2) = member_out_gg
U4_gg(x1, x2, x3, x4) = U4_gg(x4)
[] = []
reach_out_ggg(x1, x2, x3) = reach_out_ggg
U2_ggg(x1, x2, x3, x4) = U2_ggg(x2, x3, x4)
member1_in_gg(x1, x2) = member1_in_gg(x1, x2)
member1_out_gg(x1, x2) = member1_out_gg
U5_gg(x1, x2, x3, x4) = U5_gg(x4)
U3_ggg(x1, x2, x3, x4) = U3_ggg(x4)
reach_in_agg(x1, x2, x3) = reach_in_agg(x2, x3)
U1_agg(x1, x2, x3, x4) = U1_agg(x4)
reach_out_agg(x1, x2, x3) = reach_out_agg
U2_agg(x1, x2, x3, x4) = U2_agg(x2, x3, x4)
U3_agg(x1, x2, x3, x4) = U3_agg(x4)
U2_AGG(x1, x2, x3, x4) = U2_AGG(x2, x3, x4)
REACH_IN_AGG(x1, x2, x3) = REACH_IN_AGG(x2, x3)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

  ↳ PrologToPiTRSProof

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

U2_AGG(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → REACH_IN_AGG(Y, Z, Edges)
REACH_IN_AGG(X, Z, Edges) → U2_AGG(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))

The TRS R consists of the following rules:

member1_in_gg(H, .(H, L)) → member1_out_gg(H, .(H, L))
member1_in_gg(X, .(H, L)) → U5_gg(X, H, L, member1_in_gg(X, L))
U5_gg(X, H, L, member1_out_gg(X, L)) → member1_out_gg(X, .(H, L))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
[] = []
member1_in_gg(x1, x2) = member1_in_gg(x1, x2)
member1_out_gg(x1, x2) = member1_out_gg
U5_gg(x1, x2, x3, x4) = U5_gg(x4)
U2_AGG(x1, x2, x3, x4) = U2_AGG(x2, x3, x4)
REACH_IN_AGG(x1, x2, x3) = REACH_IN_AGG(x2, x3)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

  ↳ PrologToPiTRSProof

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

U2_AGG(Z, Edges, member1_out_gg) → REACH_IN_AGG(Z, Edges)
REACH_IN_AGG(Z, Edges) → U2_AGG(Z, Edges, member1_in_gg(.(.([])), Edges))

The TRS R consists of the following rules:

member1_in_gg(H, .(L)) → member1_out_gg
member1_in_gg(X, .(L)) → U5_gg(member1_in_gg(X, L))
U5_gg(member1_out_gg) → member1_out_gg

The set Q consists of the following terms:

member1_in_gg(x0, x1)
U5_gg(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule REACH_IN_AGG(Z, Edges) → U2_AGG(Z, Edges, member1_in_gg(.(.([])), Edges)) at position [2] we obtained the following new rules:

REACH_IN_AGG(y0, .(x1)) → U2_AGG(y0, .(x1), member1_out_gg)
REACH_IN_AGG(y0, .(x1)) → U2_AGG(y0, .(x1), U5_gg(member1_in_gg(.(.([])), x1)))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Instantiation

  ↳ PrologToPiTRSProof

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

REACH_IN_AGG(y0, .(x1)) → U2_AGG(y0, .(x1), member1_out_gg)
U2_AGG(Z, Edges, member1_out_gg) → REACH_IN_AGG(Z, Edges)
REACH_IN_AGG(y0, .(x1)) → U2_AGG(y0, .(x1), U5_gg(member1_in_gg(.(.([])), x1)))

The TRS R consists of the following rules:

member1_in_gg(H, .(L)) → member1_out_gg
member1_in_gg(X, .(L)) → U5_gg(member1_in_gg(X, L))
U5_gg(member1_out_gg) → member1_out_gg

The set Q consists of the following terms:

member1_in_gg(x0, x1)
U5_gg(x0)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_AGG(Z, Edges, member1_out_gg) → REACH_IN_AGG(Z, Edges) we obtained the following new rules:

U2_AGG(z0, .(z1), member1_out_gg) → REACH_IN_AGG(z0, .(z1))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Instantiation

                              ↳ QDP

                                ↳ NonTerminationProof

  ↳ PrologToPiTRSProof

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

REACH_IN_AGG(y0, .(x1)) → U2_AGG(y0, .(x1), member1_out_gg)
U2_AGG(z0, .(z1), member1_out_gg) → REACH_IN_AGG(z0, .(z1))
REACH_IN_AGG(y0, .(x1)) → U2_AGG(y0, .(x1), U5_gg(member1_in_gg(.(.([])), x1)))

The TRS R consists of the following rules:

member1_in_gg(H, .(L)) → member1_out_gg
member1_in_gg(X, .(L)) → U5_gg(member1_in_gg(X, L))
U5_gg(member1_out_gg) → member1_out_gg

The set Q consists of the following terms:

member1_in_gg(x0, x1)
U5_gg(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_AGG(y0, .(x1)) → U2_AGG(y0, .(x1), member1_out_gg)
U2_AGG(z0, .(z1), member1_out_gg) → REACH_IN_AGG(z0, .(z1))
REACH_IN_AGG(y0, .(x1)) → U2_AGG(y0, .(x1), U5_gg(member1_in_gg(.(.([])), x1)))

The TRS R consists of the following rules:

member1_in_gg(H, .(L)) → member1_out_gg
member1_in_gg(X, .(L)) → U5_gg(member1_in_gg(X, L))
U5_gg(member1_out_gg) → member1_out_gg

s = U2_AGG(z0, .(z1), member1_out_gg) evaluates to t =U2_AGG(z0, .(z1), member1_out_gg)

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

U2_AGG(z0, .(z1), member1_out_gg) → REACH_IN_AGG(z0, .(z1))
with rule U2_AGG(z0', .(z1'), member1_out_gg) → REACH_IN_AGG(z0', .(z1')) at position [] and matcher [z1' / z1, z0' / z0]

REACH_IN_AGG(z0, .(z1)) → U2_AGG(z0, .(z1), member1_out_gg)
with rule REACH_IN_AGG(y0, .(x1)) → U2_AGG(y0, .(x1), member1_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) (f,b,b)
member_in: (b,b)
member1_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
member1_in_gg(H, .(H, L)) → member1_out_gg(H, .(H, L))
member1_in_gg(X, .(H, L)) → U5_gg(X, H, L, member1_in_gg(X, L))
U5_gg(X, H, L, member1_out_gg(X, L)) → member1_out_gg(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
reach_in_agg(X, Y, Edges) → U1_agg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
U1_agg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_agg(X, Y, Edges)
reach_in_agg(X, Z, Edges) → U2_agg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
U2_agg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_agg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U3_agg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_agg(X, Z, Edges)
U3_ggg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

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

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
member1_in_gg(H, .(H, L)) → member1_out_gg(H, .(H, L))
member1_in_gg(X, .(H, L)) → U5_gg(X, H, L, member1_in_gg(X, L))
U5_gg(X, H, L, member1_out_gg(X, L)) → member1_out_gg(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
reach_in_agg(X, Y, Edges) → U1_agg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
U1_agg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_agg(X, Y, Edges)
reach_in_agg(X, Z, Edges) → U2_agg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
U2_agg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_agg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U3_agg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_agg(X, Z, Edges)
U3_ggg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

REACH_IN_GGG(X, Y, Edges) → U1_GGG(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_GGG(X, Y, Edges) → MEMBER_IN_GG(.(X, .(Y, [])), Edges)
MEMBER_IN_GG(X, .(H, L)) → U4_GG(X, H, L, member_in_gg(X, L))
MEMBER_IN_GG(X, .(H, L)) → MEMBER_IN_GG(X, L)
REACH_IN_GGG(X, Z, Edges) → U2_GGG(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_GGG(X, Z, Edges) → MEMBER1_IN_GG(.(X, .(Y, [])), Edges)
MEMBER1_IN_GG(X, .(H, L)) → U5_GG(X, H, L, member1_in_gg(X, L))
MEMBER1_IN_GG(X, .(H, L)) → MEMBER1_IN_GG(X, L)
U2_GGG(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_GGG(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U2_GGG(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → REACH_IN_AGG(Y, Z, Edges)
REACH_IN_AGG(X, Y, Edges) → U1_AGG(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_AGG(X, Y, Edges) → MEMBER_IN_GG(.(X, .(Y, [])), Edges)
REACH_IN_AGG(X, Z, Edges) → U2_AGG(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_AGG(X, Z, Edges) → MEMBER1_IN_GG(.(X, .(Y, [])), Edges)
U2_AGG(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_AGG(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U2_AGG(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → REACH_IN_AGG(Y, Z, Edges)

The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
member1_in_gg(H, .(H, L)) → member1_out_gg(H, .(H, L))
member1_in_gg(X, .(H, L)) → U5_gg(X, H, L, member1_in_gg(X, L))
U5_gg(X, H, L, member1_out_gg(X, L)) → member1_out_gg(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
reach_in_agg(X, Y, Edges) → U1_agg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
U1_agg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_agg(X, Y, Edges)
reach_in_agg(X, Z, Edges) → U2_agg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
U2_agg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_agg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U3_agg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_agg(X, Z, Edges)
U3_ggg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_in_ggg(x1, x2, x3) = reach_in_ggg(x1, x2, x3)
U1_ggg(x1, x2, x3, x4) = U1_ggg(x1, x2, x3, x4)
member_in_gg(x1, x2) = member_in_gg(x1, x2)
.(x1, x2) = .(x2)
member_out_gg(x1, x2) = member_out_gg(x1, x2)
U4_gg(x1, x2, x3, x4) = U4_gg(x1, x3, x4)
[] = []
reach_out_ggg(x1, x2, x3) = reach_out_ggg(x1, x2, x3)
U2_ggg(x1, x2, x3, x4) = U2_ggg(x1, x2, x3, x4)
member1_in_gg(x1, x2) = member1_in_gg(x1, x2)
member1_out_gg(x1, x2) = member1_out_gg(x1, x2)
U5_gg(x1, x2, x3, x4) = U5_gg(x1, x3, x4)
U3_ggg(x1, x2, x3, x4) = U3_ggg(x1, x2, x3, x4)
reach_in_agg(x1, x2, x3) = reach_in_agg(x2, x3)
U1_agg(x1, x2, x3, x4) = U1_agg(x2, x3, x4)
reach_out_agg(x1, x2, x3) = reach_out_agg(x2, x3)
U2_agg(x1, x2, x3, x4) = U2_agg(x2, x3, x4)
U3_agg(x1, x2, x3, x4) = U3_agg(x2, x3, x4)
U3_AGG(x1, x2, x3, x4) = U3_AGG(x2, x3, x4)
U5_GG(x1, x2, x3, x4) = U5_GG(x1, x3, x4)
U1_GGG(x1, x2, x3, x4) = U1_GGG(x1, x2, x3, x4)
MEMBER_IN_GG(x1, x2) = MEMBER_IN_GG(x1, x2)
U3_GGG(x1, x2, x3, x4) = U3_GGG(x1, x2, x3, x4)
MEMBER1_IN_GG(x1, x2) = MEMBER1_IN_GG(x1, x2)
REACH_IN_GGG(x1, x2, x3) = REACH_IN_GGG(x1, x2, x3)
U2_AGG(x1, x2, x3, x4) = U2_AGG(x2, x3, x4)
U2_GGG(x1, x2, x3, x4) = U2_GGG(x1, x2, x3, x4)
REACH_IN_AGG(x1, x2, x3) = REACH_IN_AGG(x2, x3)
U4_GG(x1, x2, x3, x4) = U4_GG(x1, x3, x4)
U1_AGG(x1, x2, x3, x4) = U1_AGG(x2, x3, x4)

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

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

REACH_IN_GGG(X, Y, Edges) → U1_GGG(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_GGG(X, Y, Edges) → MEMBER_IN_GG(.(X, .(Y, [])), Edges)
MEMBER_IN_GG(X, .(H, L)) → U4_GG(X, H, L, member_in_gg(X, L))
MEMBER_IN_GG(X, .(H, L)) → MEMBER_IN_GG(X, L)
REACH_IN_GGG(X, Z, Edges) → U2_GGG(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_GGG(X, Z, Edges) → MEMBER1_IN_GG(.(X, .(Y, [])), Edges)
MEMBER1_IN_GG(X, .(H, L)) → U5_GG(X, H, L, member1_in_gg(X, L))
MEMBER1_IN_GG(X, .(H, L)) → MEMBER1_IN_GG(X, L)
U2_GGG(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_GGG(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U2_GGG(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → REACH_IN_AGG(Y, Z, Edges)
REACH_IN_AGG(X, Y, Edges) → U1_AGG(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_AGG(X, Y, Edges) → MEMBER_IN_GG(.(X, .(Y, [])), Edges)
REACH_IN_AGG(X, Z, Edges) → U2_AGG(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
REACH_IN_AGG(X, Z, Edges) → MEMBER1_IN_GG(.(X, .(Y, [])), Edges)
U2_AGG(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_AGG(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U2_AGG(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → REACH_IN_AGG(Y, Z, Edges)

The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
member1_in_gg(H, .(H, L)) → member1_out_gg(H, .(H, L))
member1_in_gg(X, .(H, L)) → U5_gg(X, H, L, member1_in_gg(X, L))
U5_gg(X, H, L, member1_out_gg(X, L)) → member1_out_gg(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
reach_in_agg(X, Y, Edges) → U1_agg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
U1_agg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_agg(X, Y, Edges)
reach_in_agg(X, Z, Edges) → U2_agg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
U2_agg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_agg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U3_agg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_agg(X, Z, Edges)
U3_ggg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

MEMBER1_IN_GG(X, .(H, L)) → MEMBER1_IN_GG(X, L)

The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
member1_in_gg(H, .(H, L)) → member1_out_gg(H, .(H, L))
member1_in_gg(X, .(H, L)) → U5_gg(X, H, L, member1_in_gg(X, L))
U5_gg(X, H, L, member1_out_gg(X, L)) → member1_out_gg(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
reach_in_agg(X, Y, Edges) → U1_agg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
U1_agg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_agg(X, Y, Edges)
reach_in_agg(X, Z, Edges) → U2_agg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
U2_agg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_agg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U3_agg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_agg(X, Z, Edges)
U3_ggg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

↳ 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_GG(X, .(H, L)) → MEMBER1_IN_GG(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_GG(X, .(L)) → MEMBER1_IN_GG(X, L)

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

MEMBER1_IN_GG(X, .(L)) → MEMBER1_IN_GG(X, L)
The graph contains the following edges 1 >= 1, 2 > 2

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

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

The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
member1_in_gg(H, .(H, L)) → member1_out_gg(H, .(H, L))
member1_in_gg(X, .(H, L)) → U5_gg(X, H, L, member1_in_gg(X, L))
U5_gg(X, H, L, member1_out_gg(X, L)) → member1_out_gg(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
reach_in_agg(X, Y, Edges) → U1_agg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
U1_agg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_agg(X, Y, Edges)
reach_in_agg(X, Z, Edges) → U2_agg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
U2_agg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_agg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U3_agg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_agg(X, Z, Edges)
U3_ggg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

↳ 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)

↳ 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, .(L)) → MEMBER_IN_GG(X, L)

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

MEMBER_IN_GG(X, .(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_AGG(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → REACH_IN_AGG(Y, Z, Edges)
REACH_IN_AGG(X, Z, Edges) → U2_AGG(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))

The TRS R consists of the following rules:

reach_in_ggg(X, Y, Edges) → U1_ggg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
member_in_gg(H, .(H, L)) → member_out_gg(H, .(H, L))
member_in_gg(X, .(H, L)) → U4_gg(X, H, L, member_in_gg(X, L))
U4_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(X, .(H, L))
U1_ggg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_ggg(X, Y, Edges)
reach_in_ggg(X, Z, Edges) → U2_ggg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
member1_in_gg(H, .(H, L)) → member1_out_gg(H, .(H, L))
member1_in_gg(X, .(H, L)) → U5_gg(X, H, L, member1_in_gg(X, L))
U5_gg(X, H, L, member1_out_gg(X, L)) → member1_out_gg(X, .(H, L))
U2_ggg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_ggg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
reach_in_agg(X, Y, Edges) → U1_agg(X, Y, Edges, member_in_gg(.(X, .(Y, [])), Edges))
U1_agg(X, Y, Edges, member_out_gg(.(X, .(Y, [])), Edges)) → reach_out_agg(X, Y, Edges)
reach_in_agg(X, Z, Edges) → U2_agg(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))
U2_agg(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → U3_agg(X, Z, Edges, reach_in_agg(Y, Z, Edges))
U3_agg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_agg(X, Z, Edges)
U3_ggg(X, Z, Edges, reach_out_agg(Y, Z, Edges)) → reach_out_ggg(X, Z, Edges)

↳ 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_AGG(X, Z, Edges, member1_out_gg(.(X, .(Y, [])), Edges)) → REACH_IN_AGG(Y, Z, Edges)
REACH_IN_AGG(X, Z, Edges) → U2_AGG(X, Z, Edges, member1_in_gg(.(X, .(Y, [])), Edges))

The TRS R consists of the following rules:

member1_in_gg(H, .(H, L)) → member1_out_gg(H, .(H, L))
member1_in_gg(X, .(H, L)) → U5_gg(X, H, L, member1_in_gg(X, L))
U5_gg(X, H, L, member1_out_gg(X, L)) → member1_out_gg(X, .(H, L))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
[] = []
member1_in_gg(x1, x2) = member1_in_gg(x1, x2)
member1_out_gg(x1, x2) = member1_out_gg(x1, x2)
U5_gg(x1, x2, x3, x4) = U5_gg(x1, x3, x4)
U2_AGG(x1, x2, x3, x4) = U2_AGG(x2, x3, x4)
REACH_IN_AGG(x1, x2, x3) = REACH_IN_AGG(x2, x3)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

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

REACH_IN_AGG(Z, Edges) → U2_AGG(Z, Edges, member1_in_gg(.(.([])), Edges))
U2_AGG(Z, Edges, member1_out_gg(.(.([])), Edges)) → REACH_IN_AGG(Z, Edges)

The TRS R consists of the following rules:

member1_in_gg(H, .(L)) → member1_out_gg(H, .(L))
member1_in_gg(X, .(L)) → U5_gg(X, L, member1_in_gg(X, L))
U5_gg(X, L, member1_out_gg(X, L)) → member1_out_gg(X, .(L))

The set Q consists of the following terms:

member1_in_gg(x0, x1)
U5_gg(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule REACH_IN_AGG(Z, Edges) → U2_AGG(Z, Edges, member1_in_gg(.(.([])), Edges)) at position [2] we obtained the following new rules:

REACH_IN_AGG(y0, .(x1)) → U2_AGG(y0, .(x1), member1_out_gg(.(.([])), .(x1)))
REACH_IN_AGG(y0, .(x1)) → U2_AGG(y0, .(x1), U5_gg(.(.([])), x1, member1_in_gg(.(.([])), x1)))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Instantiation

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

U2_AGG(Z, Edges, member1_out_gg(.(.([])), Edges)) → REACH_IN_AGG(Z, Edges)
REACH_IN_AGG(y0, .(x1)) → U2_AGG(y0, .(x1), member1_out_gg(.(.([])), .(x1)))
REACH_IN_AGG(y0, .(x1)) → U2_AGG(y0, .(x1), U5_gg(.(.([])), x1, member1_in_gg(.(.([])), x1)))

The TRS R consists of the following rules:

member1_in_gg(H, .(L)) → member1_out_gg(H, .(L))
member1_in_gg(X, .(L)) → U5_gg(X, L, member1_in_gg(X, L))
U5_gg(X, L, member1_out_gg(X, L)) → member1_out_gg(X, .(L))

The set Q consists of the following terms:

member1_in_gg(x0, x1)
U5_gg(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule U2_AGG(Z, Edges, member1_out_gg(.(.([])), Edges)) → REACH_IN_AGG(Z, Edges) we obtained the following new rules:

U2_AGG(z0, .(z1), member1_out_gg(.(.([])), .(z1))) → REACH_IN_AGG(z0, .(z1))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Instantiation

                              ↳ QDP

                                ↳ NonTerminationProof

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

U2_AGG(z0, .(z1), member1_out_gg(.(.([])), .(z1))) → REACH_IN_AGG(z0, .(z1))
REACH_IN_AGG(y0, .(x1)) → U2_AGG(y0, .(x1), member1_out_gg(.(.([])), .(x1)))
REACH_IN_AGG(y0, .(x1)) → U2_AGG(y0, .(x1), U5_gg(.(.([])), x1, member1_in_gg(.(.([])), x1)))

The TRS R consists of the following rules:

member1_in_gg(H, .(L)) → member1_out_gg(H, .(L))
member1_in_gg(X, .(L)) → U5_gg(X, L, member1_in_gg(X, L))
U5_gg(X, L, member1_out_gg(X, L)) → member1_out_gg(X, .(L))

The set Q consists of the following terms:

member1_in_gg(x0, x1)
U5_gg(x0, x1, x2)

U2_AGG(z0, .(z1), member1_out_gg(.(.([])), .(z1))) → REACH_IN_AGG(z0, .(z1))
REACH_IN_AGG(y0, .(x1)) → U2_AGG(y0, .(x1), member1_out_gg(.(.([])), .(x1)))
REACH_IN_AGG(y0, .(x1)) → U2_AGG(y0, .(x1), U5_gg(.(.([])), x1, member1_in_gg(.(.([])), x1)))

The TRS R consists of the following rules:

member1_in_gg(H, .(L)) → member1_out_gg(H, .(L))
member1_in_gg(X, .(L)) → U5_gg(X, L, member1_in_gg(X, L))
U5_gg(X, L, member1_out_gg(X, L)) → member1_out_gg(X, .(L))

s = REACH_IN_AGG(y0, .(x1)) evaluates to t =REACH_IN_AGG(y0, .(x1))

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

REACH_IN_AGG(y0, .(x1)) → U2_AGG(y0, .(x1), member1_out_gg(.(.([])), .(x1)))
with rule REACH_IN_AGG(y0', .(x1')) → U2_AGG(y0', .(x1'), member1_out_gg(.(.([])), .(x1'))) at position [] and matcher [x1' / x1, y0' / y0]

U2_AGG(y0, .(x1), member1_out_gg(.(.([])), .(x1))) → REACH_IN_AGG(y0, .(x1))
with rule U2_AGG(z0, .(z1), member1_out_gg(.(.([])), .(z1))) → REACH_IN_AGG(z0, .(z1))

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.