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

(0) Obligation:

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).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. 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)

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)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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)

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)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
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)

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)
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)
U5_GG(x1, x2, x3, x4)  =  U5_GG(x4)
U2_GGGG(x1, x2, x3, x4, x5)  =  U2_GGGG(x2, x3, x4, x5)
MEMBER1_IN_AG(x1, x2)  =  MEMBER1_IN_AG(x2)
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)

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

(4) Obligation:

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)

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)
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)
U5_GG(x1, x2, x3, x4)  =  U5_GG(x4)
U2_GGGG(x1, x2, x3, x4, x5)  =  U2_GGGG(x2, x3, x4, x5)
MEMBER1_IN_AG(x1, x2)  =  MEMBER1_IN_AG(x2)
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)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 7 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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)

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)
MEMBER1_IN_AG(x1, x2)  =  MEMBER1_IN_AG(x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(9) Obligation:

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

(10) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(11) Obligation:

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.

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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

(13) TRUE

(14) Obligation:

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)

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)
MEMBER_IN_GG(x1, x2)  =  MEMBER_IN_GG(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(16) Obligation:

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

(17) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(18) Obligation:

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

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

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

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

(20) TRUE

(21) Obligation:

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

REACH_IN_GGGG(X, Z, Edges, Visited) → U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
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))

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)
REACH_IN_GGGG(x1, x2, x3, x4)  =  REACH_IN_GGGG(x1, x2, x3, x4)
U2_GGGG(x1, x2, x3, x4, x5)  =  U2_GGGG(x2, x3, x4, x5)
U3_GGGG(x1, x2, x3, x4, x5, x6)  =  U3_GGGG(x2, x3, x4, x5, x6)

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

(22) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(23) Obligation:

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

REACH_IN_GGGG(X, Z, Edges, Visited) → U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
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))

The TRS R consists of the following rules:

member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U6_ag(X, H, L, member1_in_ag(X, L))
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))
U6_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(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)
REACH_IN_GGGG(x1, x2, x3, x4)  =  REACH_IN_GGGG(x1, x2, x3, x4)
U2_GGGG(x1, x2, x3, x4, x5)  =  U2_GGGG(x2, x3, x4, x5)
U3_GGGG(x1, x2, x3, x4, x5, x6)  =  U3_GGGG(x2, x3, x4, x5, x6)

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

(24) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(25) Obligation:

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(26) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule REACH_IN_GGGG(X, Z, Edges, Visited) → U2_GGGG(Z, Edges, Visited, member1_in_ag(Edges)) we obtained the following new rules [LPAR04]:

REACH_IN_GGGG(z3, z0, z1, .(z3, z2)) → U2_GGGG(z0, z1, .(z3, z2), member1_in_ag(z1))

(27) Obligation:

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

U2_GGGG(Z, Edges, Visited, member1_out_ag(.(X, .(Y, [])))) → U3_GGGG(Z, Edges, Visited, Y, member_in_gg(Y, Visited))
U3_GGGG(Z, Edges, Visited, Y, member_out_gg) → REACH_IN_GGGG(Y, Z, Edges, .(Y, Visited))
REACH_IN_GGGG(z3, z0, z1, .(z3, z2)) → U2_GGGG(z0, z1, .(z3, z2), member1_in_ag(z1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(28) Narrowing (SOUND transformation)

By narrowing [LPAR04] 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 [LPAR04]:

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)))

(29) Obligation:

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

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

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

The set Q consists of the following terms:

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

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

(30) Narrowing (SOUND transformation)

By narrowing [LPAR04] 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 [LPAR04]:

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

(31) Obligation:

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))
REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y1, .(x0, x1), y3, U6_ag(member1_in_ag(x1)))
U2_GGGG(y0, y1, .(x0, x1), member1_out_ag(.(y3, .(x0, [])))) → U3_GGGG(y0, y1, .(x0, x1), x0, member_out_gg)
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)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(32) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] 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 [LPAR04]:

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)))

(33) Obligation:

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))
REACH_IN_GGGG(y0, y1, .(x0, x1), y3) → U2_GGGG(y1, .(x0, x1), y3, U6_ag(member1_in_ag(x1)))
U2_GGGG(y0, y1, .(x0, x1), member1_out_ag(.(y3, .(x0, [])))) → U3_GGGG(y0, y1, .(x0, x1), x0, member_out_gg)
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)))
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)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(34) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] 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 [LPAR04]:

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

(35) Obligation:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(36) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] 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 [LPAR04]:

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)))

(37) Obligation:

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(38) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] 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 [LPAR04]:

U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg)

(39) Obligation:

Q DP problem:
The TRS P consists of the following 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)))
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(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2))
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)))
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, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(40) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] 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 [LPAR04]:

U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))

(41) Obligation:

Q DP problem:
The TRS P consists of the following 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)))
REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), member1_out_ag(x2))
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)))
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, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(42) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] 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 [LPAR04]:

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

(43) Obligation:

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(44) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] 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 [LPAR04]:

U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) → REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5))))

(45) Obligation:

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)), member1_out_ag(x2))
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)))
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, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) → REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(46) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] 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 [LPAR04]:

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

(47) Obligation:

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)))
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, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) → REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(48) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] 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 [LPAR04]:

REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1))

(49) Obligation:

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)))
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, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) → REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(50) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule REACH_IN_GGGG(z2, z0, .(x2, x3), .(z2, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z2, .(z2, z3)), U6_ag(member1_in_ag(x3))) we obtained the following new rules [LPAR04]:

REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))

(51) Obligation:

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)), U6_ag(member1_in_ag(x3)))
U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) → REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(52) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule REACH_IN_GGGG(z4, z0, .(x2, x3), .(z4, .(z2, z3))) → U2_GGGG(z0, .(x2, x3), .(z4, .(z2, z3)), U6_ag(member1_in_ag(x3))) we obtained the following new rules [LPAR04]:

REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))

(53) Obligation:

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

U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) → REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(54) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z0, z4)), z0, member_out_gg) we obtained the following new rules [LPAR04]:

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

(55) Obligation:

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

U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) → REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(56) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(.(x4, .(z0, [])), z3), .(z0, .(z4, z5)), z0, member_out_gg) we obtained the following new rules [LPAR04]:

U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)

(57) Obligation:

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

U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) → REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(58) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg) we obtained the following new rules [LPAR04]:

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

(59) Obligation:

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

U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg)
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) → REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(60) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg) we obtained the following new rules [LPAR04]:

U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)

(61) Obligation:

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

U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) → REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(62) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) we obtained the following new rules [LPAR04]:

U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5)))))

(63) Obligation:

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

U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) → REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5)))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(64) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(.(x4, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) we obtained the following new rules [LPAR04]:

U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, U5_gg(member_in_gg(z0, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(member_in_gg(x2, .(z4, .(z5, z6)))))

(65) Obligation:

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

U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) → REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, U5_gg(member_in_gg(z0, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(member_in_gg(x2, .(z4, .(z5, z6)))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(66) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(member_in_gg(x5, .(z0, z4)))) we obtained the following new rules [LPAR04]:

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

(67) Obligation:

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

U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) → REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, U5_gg(member_in_gg(z0, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(member_in_gg(x2, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5)))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(68) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(x4, .(x5, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(member_in_gg(x5, .(z4, z5)))) we obtained the following new rules [LPAR04]:

U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(x6, .(x7, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(.(x6, .(x7, [])), z3), .(z0, .(z0, .(z0, z4))), x7, U5_gg(member_in_gg(x7, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(x6, .(x7, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(.(x6, .(x7, [])), z3), .(z0, .(z0, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, U5_gg(member_in_gg(z0, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(.(x6, .(x7, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(.(x6, .(x7, [])), z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(x6, .(x7, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(.(x6, .(x7, [])), z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), x7, U5_gg(member_in_gg(x7, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x7, U5_gg(member_in_gg(x7, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6)))))

(69) Obligation:

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

U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) → REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, U5_gg(member_in_gg(z0, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(member_in_gg(x2, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6)))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(70) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), member1_out_ag(z1)) we obtained the following new rules [LPAR04]:

REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4)))) → U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4))), member1_out_ag(.(y_1, .(y_2, []))))

(71) Obligation:

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

U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) → REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, U5_gg(member_in_gg(z0, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(member_in_gg(x2, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6)))))
REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4)))) → U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4))), member1_out_ag(.(y_1, .(y_2, []))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(72) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), member1_out_ag(z1)) we obtained the following new rules [LPAR04]:

REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x4, x5)))) → U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x4, x5))), member1_out_ag(.(y_1, .(y_2, []))))

(73) Obligation:

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

U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) → REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, U5_gg(member_in_gg(z0, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(member_in_gg(x2, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6)))))
REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4)))) → U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4))), member1_out_ag(.(y_1, .(y_2, []))))
REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x4, x5)))) → U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x4, x5))), member1_out_ag(.(y_1, .(y_2, []))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(74) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), member1_out_ag(z1)) we obtained the following new rules [LPAR04]:

REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x4, x5)))) → U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x4, x5))), member1_out_ag(.(y_1, .(y_2, []))))

(75) Obligation:

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

U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) → REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, U5_gg(member_in_gg(z0, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(member_in_gg(x2, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6)))))
REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4)))) → U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4))), member1_out_ag(.(y_1, .(y_2, []))))
REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x4, x5)))) → U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x4, x5))), member1_out_ag(.(y_1, .(y_2, []))))
REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x4, x5)))) → U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x4, x5))), member1_out_ag(.(y_1, .(y_2, []))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(76) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), member1_out_ag(z1)) we obtained the following new rules [LPAR04]:

REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x5, x6)))) → U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x5, x6))), member1_out_ag(.(y_1, .(y_2, []))))

(77) Obligation:

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

U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, z4)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z3, member_out_gg) → REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z4, z5)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))))
U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z4, .(z5, z6)), z2, member_out_gg) → REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z3, z4)), z6, member_out_gg) → REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z3, z4))))
U3_GGGG(z0, .(z1, z2), .(z3, .(z4, z5)), z7, member_out_gg) → REACH_IN_GGGG(z7, z0, .(z1, z2), .(z7, .(z3, .(z4, z5))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), member1_out_ag(.(z1, .(z2, []))))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z3, z0, .(z1, z2), .(z3, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z3, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z4, z5))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z4, .(z5, z6))), U6_ag(member1_in_ag(z3)))
REACH_IN_GGGG(z5, z0, .(z1, z2), .(z5, .(z3, .(z3, z4)))) → U2_GGGG(z0, .(z1, z2), .(z5, .(z3, .(z3, z4))), U6_ag(member1_in_ag(z2)))
REACH_IN_GGGG(z6, z0, .(z1, z2), .(z6, .(z3, .(z4, z5)))) → U2_GGGG(z0, .(z1, z2), .(z6, .(z3, .(z4, z5))), U6_ag(member1_in_ag(z2)))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg)
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(z0, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg)
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), z0, U5_gg(member_in_gg(z0, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(member_in_gg(x2, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, U5_gg(member_in_gg(z0, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z2, .(z0, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, U5_gg(member_in_gg(z0, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(member_in_gg(x2, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x1, .(x2, [])))) → U3_GGGG(z1, .(.(x1, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(member_in_gg(x2, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(member_in_gg(x6, .(z0, .(z0, z4)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(x5, .(x6, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(member_in_gg(x6, .(z0, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(.(z2, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x7, U5_gg(member_in_gg(x7, .(z4, .(z4, z5)))))
U2_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(x6, .(x7, [])))) → U3_GGGG(z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x7, U5_gg(member_in_gg(x7, .(z4, .(z5, z6)))))
REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4)))) → U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x0, x4))), member1_out_ag(.(y_1, .(y_2, []))))
REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x4, x5)))) → U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x0, .(x4, x5))), member1_out_ag(.(y_1, .(y_2, []))))
REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x4, x5)))) → U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x4, x5))), member1_out_ag(.(y_1, .(y_2, []))))
REACH_IN_GGGG(x0, x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x5, x6)))) → U2_GGGG(x1, .(.(y_1, .(y_2, [])), x3), .(x0, .(x4, .(x5, x6))), member1_out_ag(.(y_1, .(y_2, []))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(78) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) evaluates to t =REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, .(z2, z4)))))

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

• Matcher: [z4 / .(z2, z4)]
• Semiunifier: [ ]

---

Rewriting sequence

REACH_IN_GGGG(z2, z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4)))) → U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, []))))
with rule REACH_IN_GGGG(z2', z0', .(.(z1', .(z2', [])), z3'), .(z2', .(z2', .(z2', z4')))) → U2_GGGG(z0', .(.(z1', .(z2', [])), z3'), .(z2', .(z2', .(z2', z4'))), member1_out_ag(.(z1', .(z2', [])))) at position [] and matcher [z2' / z2, z0' / z0, z1' / z1, z3' / z3, z4' / z4]

U2_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), member1_out_ag(.(z1, .(z2, [])))) → U3_GGGG(z0, .(.(z1, .(z2, [])), z3), .(z2, .(z2, .(z2, z4))), z2, member_out_gg)
with rule U2_GGGG(z1', .(.(z2', .(z0', [])), z3'), .(z0', .(z0', .(z0', z4'))), member1_out_ag(.(z2', .(z0', [])))) → U3_GGGG(z1', .(.(z2', .(z0', [])), z3'), .(z0', .(z0', .(z0', z4'))), z0', member_out_gg) at position [] and matcher [z1' / z0, z2' / z1, z0' / z2, z3' / z3, z4' / z4]

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

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.

(79) FALSE

(80) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. 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)

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)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(81) Obligation:

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)

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)

(82) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
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)

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)
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)
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)
MEMBER1_IN_AG(x1, x2)  =  MEMBER1_IN_AG(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)

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

(83) Obligation:

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)

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)
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)
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)
MEMBER1_IN_AG(x1, x2)  =  MEMBER1_IN_AG(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)

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

(84) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 7 less nodes.

(85) Complex Obligation (AND)

(86) Obligation:

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)

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)
MEMBER1_IN_AG(x1, x2)  =  MEMBER1_IN_AG(x2)

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

(87) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(88) Obligation:

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

(89) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(90) Obligation:

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.

(91) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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

(92) TRUE

(93) Obligation:

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)

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)
MEMBER_IN_GG(x1, x2)  =  MEMBER_IN_GG(x1, x2)

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

(94) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(95) Obligation:

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

(96) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(97) Obligation:

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

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

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

(98) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

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

(99) TRUE

(100) Obligation:

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

REACH_IN_GGGG(X, Z, Edges, Visited) → U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
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))

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)
REACH_IN_GGGG(x1, x2, x3, x4)  =  REACH_IN_GGGG(x1, x2, x3, x4)
U2_GGGG(x1, x2, x3, x4, x5)  =  U2_GGGG(x1, x2, x3, x4, x5)
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

(101) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(102) Obligation:

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

REACH_IN_GGGG(X, Z, Edges, Visited) → U2_GGGG(X, Z, Edges, Visited, member1_in_ag(.(X, .(Y, [])), Edges))
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))

The TRS R consists of the following rules:

member1_in_ag(H, .(H, L)) → member1_out_ag(H, .(H, L))
member1_in_ag(X, .(H, L)) → U6_ag(X, H, L, member1_in_ag(X, L))
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))
U6_ag(X, H, L, member1_out_ag(X, L)) → member1_out_ag(X, .(H, L))
U5_gg(X, H, L, member_out_gg(X, L)) → member_out_gg(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)
REACH_IN_GGGG(x1, x2, x3, x4)  =  REACH_IN_GGGG(x1, x2, x3, x4)
U2_GGGG(x1, x2, x3, x4, x5)  =  U2_GGGG(x1, x2, x3, x4, x5)
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

(103) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(104) Obligation:

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

REACH_IN_GGGG(X, Z, Edges, Visited) → U2_GGGG(X, Z, Edges, Visited, member1_in_ag(Edges))
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))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(105) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule REACH_IN_GGGG(X, Z, Edges, Visited) → U2_GGGG(X, Z, Edges, Visited, member1_in_ag(Edges)) we obtained the following new rules [LPAR04]:

REACH_IN_GGGG(z4, z1, z2, .(z4, z3)) → U2_GGGG(z4, z1, z2, .(z4, z3), member1_in_ag(z2))

(106) Obligation:

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(z4, z1, z2, .(z4, z3)) → U2_GGGG(z4, z1, z2, .(z4, z3), member1_in_ag(z2))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(107) Narrowing (SOUND transformation)

By narrowing [LPAR04] 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 [LPAR04]:

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

(108) Obligation:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(109) Narrowing (SOUND transformation)

By narrowing [LPAR04] 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 [LPAR04]:

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)))
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)))

(110) Obligation:

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

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)))
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, .(x0, x1), member1_out_ag(.(y0, .(x0, [])), y2)) → U3_GGGG(y0, y1, y2, .(x0, x1), x0, member_out_gg(x0, .(x0, 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)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(111) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] 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 [LPAR04]:

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)))

(112) Obligation:

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, member1_out_ag(x0, .(x0, x1)))
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, .(x0, x1), member1_out_ag(.(y0, .(x0, [])), y2)) → U3_GGGG(y0, y1, y2, .(x0, x1), x0, member_out_gg(x0, .(x0, 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(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)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(113) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] 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 [LPAR04]:

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)), member1_out_ag(x2, .(x2, x3)))

(114) Obligation:

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, .(x0, x1), member1_out_ag(.(y0, .(x0, [])), y2)) → U3_GGGG(y0, y1, y2, .(x0, x1), x0, member_out_gg(x0, .(x0, 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(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)))
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)), member1_out_ag(x2, .(x2, x3)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(115) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] 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 [LPAR04]:

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)))
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)))

(116) Obligation:

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

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)))
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(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)))
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)), 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)), U6_ag(x2, x3, member1_in_ag(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)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(117) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] 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 [LPAR04]:

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

(118) Obligation:

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(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)))
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)), 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)), U6_ag(x2, x3, member1_in_ag(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, .(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))))
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, .(z0, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4))))
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))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(119) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] 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 [LPAR04]:

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

(120) Obligation:

Q DP problem:
The TRS P consists of the following 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)))
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)), 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)), U6_ag(x2, x3, member1_in_ag(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, .(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))))
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, .(z0, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4))))
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, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4))))
U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5))))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4))))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(121) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] 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 [LPAR04]:

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

(122) Obligation:

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(123) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] 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 [LPAR04]:

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

(124) Obligation:

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(125) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] 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 [LPAR04]:

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

(126) Obligation:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(127) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] 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 [LPAR04]:

REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2)))
REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2)))
REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) → U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3)))
REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) → U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3)))
REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) → U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3)))
REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) → U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3)))
REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) → U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3)))
REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) → U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3)))

(128) Obligation:

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(129) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule 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))) we obtained the following new rules [LPAR04]:

REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2)))
REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2)))
REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) → U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3)))
REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) → U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3)))

(130) Obligation:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(131) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule 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))) we obtained the following new rules [LPAR04]:

REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2)))
REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2)))
REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) → U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3)))
REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) → U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3)))
REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) → U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3)))
REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) → U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3)))
REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) → U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3)))
REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) → U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3)))

(132) Obligation:

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(133) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule 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)))) we obtained the following new rules [LPAR04]:

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

(134) Obligation:

Q DP problem:
The TRS P consists of the following 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, .(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, .(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, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4))))
U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5))))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4))))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5))))
U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) → REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))
U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z3, z4)), z0, member_out_gg(z0, .(z0, .(z3, z4)))) → REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))))
U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z0, member_out_gg(z0, .(z0, .(z0, z4)))) → REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))))
U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z0, member_out_gg(z0, .(z0, .(z4, z5)))) → REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))))
U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z0, z4)), z2, member_out_gg(z2, .(z0, .(z0, z4)))) → REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))))
U3_GGGG(z0, z1, .(.(z0, .(z2, [])), z3), .(z0, .(z4, z5)), z2, member_out_gg(z2, .(z0, .(z4, z5)))) → REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))))
U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), z5, member_out_gg(z5, .(z0, .(z0, z4)))) → REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))))
U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), z6, member_out_gg(z6, .(z0, .(z4, z5)))) → REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))))
REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2)))
REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2)))
REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) → U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3)))
REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) → U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3)))
REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) → U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3)))
REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) → U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z2, [])), .(.(z0, .(z2, [])), z3)))
REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) → U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3)))
REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) → U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3)))
REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2)))
REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4)))) → U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), U6_ag(.(z0, .(z0, [])), z2, member1_in_ag(z2)))
REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) → U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3)))
REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) → U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3)))
REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4)))) → U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z0, z4))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3)))
REACH_IN_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5)))) → U2_GGGG(z2, z1, .(.(z0, .(z2, [])), z3), .(z2, .(z0, .(z4, z5))), U6_ag(.(z0, .(z2, [])), z3, member1_in_ag(z3)))
REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) → U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), U6_ag(z2, z3, member1_in_ag(z3)))
REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) → U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), U6_ag(z2, z3, member1_in_ag(z3)))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4)))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(135) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule 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)))) we obtained the following new rules [LPAR04]:

U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6)))))

(136) Obligation:

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(137) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule 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)))) we obtained the following new rules [LPAR04]:

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

(138) Obligation:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(139) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule 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)))) we obtained the following new rules [LPAR04]:

U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z3, z4)))))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6)))))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z4)))))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z0, .(z4, z5)))))
U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) → U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), z0, member_out_gg(z0, .(z0, .(z2, .(z2, z4)))))
U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(.(z2, .(z0, [])), z3))) → U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z2, .(z4, z5)))))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), z0, member_out_gg(z0, .(z0, .(z4, .(z4, z5)))))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(z0, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), z0, member_out_gg(z0, .(z0, .(z4, .(z5, z6)))))

(140) Obligation:

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(141) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) we obtained the following new rules [LPAR04]:

U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U5_gg(z0, z0, .(z0, .(z0, z3)), member_in_gg(z0, .(z0, .(z0, z3)))))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U5_gg(z0, z0, .(z0, .(z3, z4)), member_in_gg(z0, .(z0, .(z3, z4)))))
U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(x2, z0, .(z0, .(z0, z4)), member_in_gg(x2, .(z0, .(z0, z4)))))
U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(x2, z0, .(z0, .(z4, z5)), member_in_gg(x2, .(z0, .(z4, z5)))))

(142) Obligation:

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(143) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(.(z0, .(x5, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x5, [])), z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) we obtained the following new rules [LPAR04]:

U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U5_gg(z0, z0, .(z0, .(z0, z3)), member_in_gg(z0, .(z0, .(z0, z3)))))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U5_gg(z0, z0, .(z0, .(z3, z4)), member_in_gg(z0, .(z0, .(z3, z4)))))
U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z0, z4))), x2, U5_gg(x2, z0, .(z0, .(z0, z4)), member_in_gg(x2, .(z0, .(z0, z4)))))
U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z0, .(z4, z5))), x2, U5_gg(x2, z0, .(z0, .(z4, z5)), member_in_gg(x2, .(z0, .(z4, z5)))))
U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z4, z5))), x2, U5_gg(x2, z0, .(z4, .(z4, z5)), member_in_gg(x2, .(z4, .(z4, z5)))))
U2_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(x2, [])), .(.(z0, .(x2, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x2, [])), z3), .(z0, .(z4, .(z5, z6))), x2, U5_gg(x2, z0, .(z4, .(z5, z6)), member_in_gg(x2, .(z4, .(z5, z6)))))

(144) Obligation:

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(145) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, z4)), x5, U5_gg(x5, z0, .(z0, z4), member_in_gg(x5, .(z0, z4)))) we obtained the following new rules [LPAR04]:

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

(146) Obligation:

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(147) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), member1_out_ag(.(z0, .(x5, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, z5)), x5, U5_gg(x5, z0, .(z4, z5), member_in_gg(x5, .(z4, z5)))) we obtained the following new rules [LPAR04]:

U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, U5_gg(z0, z0, .(z0, .(z0, z3)), member_in_gg(z0, .(z0, .(z0, z3)))))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), z0, U5_gg(z0, z0, .(z0, .(z3, z4)), member_in_gg(z0, .(z0, .(z3, z4)))))
U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(x6, z0, .(z0, .(z0, z4)), member_in_gg(x6, .(z0, .(z0, z4)))))
U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(x6, z0, .(z0, .(z4, z5)), member_in_gg(x6, .(z0, .(z4, z5)))))
U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, .(z4, z5))), x6, U5_gg(x6, z0, .(z4, .(z4, z5)), member_in_gg(x6, .(z4, .(z4, z5)))))
U2_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(x6, [])), z3))) → U3_GGGG(z0, z1, .(.(z0, .(x6, [])), z3), .(z0, .(z4, .(z5, z6))), x6, U5_gg(x6, z0, .(z4, .(z5, z6)), member_in_gg(x6, .(z4, .(z5, z6)))))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(z0, [])), z2))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), x6, U5_gg(x6, z0, .(z0, .(z0, z3)), member_in_gg(x6, .(z0, .(z0, z3)))))
U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), member1_out_ag(.(z0, .(x6, [])), .(.(z0, .(z0, [])), z2))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z3, z4))), x6, U5_gg(x6, z0, .(z0, .(z3, z4)), member_in_gg(x6, .(z0, .(z3, z4)))))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), x6, U5_gg(x6, z0, .(z0, .(z0, z4)), member_in_gg(x6, .(z0, .(z0, z4)))))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), x6, U5_gg(x6, z0, .(z0, .(z4, z5)), member_in_gg(x6, .(z0, .(z4, z5)))))
U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), member1_out_ag(.(z0, .(x6, [])), .(.(z2, .(z0, [])), z3))) → U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z2, z4))), x6, U5_gg(x6, z0, .(z2, .(z2, z4)), member_in_gg(x6, .(z2, .(z2, z4)))))
U2_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(.(z2, .(z0, [])), z3))) → U3_GGGG(z0, z1, .(.(z2, .(z0, [])), z3), .(z0, .(z2, .(z4, z5))), x6, U5_gg(x6, z0, .(z2, .(z4, z5)), member_in_gg(x6, .(z2, .(z4, z5)))))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z4, z5))), x6, U5_gg(x6, z0, .(z4, .(z4, z5)), member_in_gg(x6, .(z4, .(z4, z5)))))
U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), member1_out_ag(.(z0, .(x6, [])), .(z2, z3))) → U3_GGGG(z0, z1, .(z2, z3), .(z0, .(z4, .(z5, z6))), x6, U5_gg(x6, z0, .(z4, .(z5, z6)), member_in_gg(x6, .(z4, .(z5, z6)))))

(148) Obligation:

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(149) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4)))) → U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) we obtained the following new rules [LPAR04]:

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

(150) Obligation:

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(151) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule REACH_IN_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5)))) → U2_GGGG(z0, z1, .(z2, z3), .(z0, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) we obtained the following new rules [LPAR04]:

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

(152) Obligation:

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(153) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule REACH_IN_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4)))) → U2_GGGG(z5, z1, .(z2, z3), .(z5, .(z0, .(z0, z4))), member1_out_ag(z2, .(z2, z3))) we obtained the following new rules [LPAR04]:

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

(154) Obligation:

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(155) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule REACH_IN_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5)))) → U2_GGGG(z6, z1, .(z2, z3), .(z6, .(z0, .(z4, z5))), member1_out_ag(z2, .(z2, z3))) we obtained the following new rules [LPAR04]:

REACH_IN_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, .(x5, x6)))) → U2_GGGG(x0, x1, .(.(y_2, .(y_3, [])), x3), .(x0, .(x4, .(x5, x6))), member1_out_ag(.(y_2, .(y_3, [])), .(.(y_2, .(y_3, [])), x3)))

(156) Obligation:

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(157) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) evaluates to t =REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, .(z0, z3)))))

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

• Semiunifier: [ ]
• Matcher: [z3 / .(z0, z3)]

---

Rewriting sequence

REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3)))) → U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2)))
with rule REACH_IN_GGGG(z0', z1', .(.(z0', .(z0', [])), z2'), .(z0', .(z0', .(z0', z3')))) → U2_GGGG(z0', z1', .(.(z0', .(z0', [])), z2'), .(z0', .(z0', .(z0', z3'))), member1_out_ag(.(z0', .(z0', [])), .(.(z0', .(z0', [])), z2'))) at position [] and matcher [z0' / z0, z1' / z1, z2' / z2, z3' / z3]

U2_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), member1_out_ag(.(z0, .(z0, [])), .(.(z0, .(z0, [])), z2))) → U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3)))))
with rule U2_GGGG(z0', z1', .(.(z0', .(z0', [])), z2'), .(z0', .(z0', .(z0', z3'))), member1_out_ag(.(z0', .(z0', [])), .(.(z0', .(z0', [])), z2'))) → U3_GGGG(z0', z1', .(.(z0', .(z0', [])), z2'), .(z0', .(z0', .(z0', z3'))), z0', member_out_gg(z0', .(z0', .(z0', .(z0', z3'))))) at position [] and matcher [z0' / z0, z1' / z1, z2' / z2, z3' / z3]

U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))), z0, member_out_gg(z0, .(z0, .(z0, .(z0, z3))))) → REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, .(z0, z3)))))
with rule U3_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, z3)), z0, member_out_gg(z0, .(z0, .(z0, z3)))) → REACH_IN_GGGG(z0, z1, .(.(z0, .(z0, [])), z2), .(z0, .(z0, .(z0, z3))))

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


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

(158) FALSE