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



↳ PROLOG
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof

reach3(X, Y, Edges) :- member2(.2(X, .2(Y, {}0)), Edges).
reach3(X, Z, Edges) :- member12(.2(X, .2(Y, {}0)), Edges), reach3(Y, Z, Edges).
member2(H, .2(H, L)).
member2(X, .2(H, L)) :- member2(X, L).
member12(H, .2(H, L)).
member12(X, .2(H, L)) :- member12(X, L).


With regard to the inferred argument filtering the predicates were used in the following modes:
reach3: (b,b,b)
member2: (b,b)
member12: (f,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:


reach_3_in_ggg3(X, Y, Edges) -> if_reach_3_in_1_ggg4(X, Y, Edges, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_3_in_1_ggg4(X, Y, Edges, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_3_out_ggg3(X, Y, Edges)
reach_3_in_ggg3(X, Z, Edges) -> if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_in_ggg3(Y, Z, Edges))
if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_out_ggg3(Y, Z, Edges)) -> reach_3_out_ggg3(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_3_in_ggg3(x1, x2, x3)  =  reach_3_in_ggg3(x1, x2, x3)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_3_in_1_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_1_ggg1(x4)
member_2_in_gg2(x1, x2)  =  member_2_in_gg2(x1, x2)
member_2_out_gg2(x1, x2)  =  member_2_out_gg
if_member_2_in_1_gg4(x1, x2, x3, x4)  =  if_member_2_in_1_gg1(x4)
reach_3_out_ggg3(x1, x2, x3)  =  reach_3_out_ggg
if_reach_3_in_2_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_2_ggg3(x2, x3, x4)
member1_2_in_ag2(x1, x2)  =  member1_2_in_ag1(x2)
member1_2_out_ag2(x1, x2)  =  member1_2_out_ag1(x1)
if_member1_2_in_1_ag4(x1, x2, x3, x4)  =  if_member1_2_in_1_ag1(x4)
if_reach_3_in_3_ggg5(x1, x2, x3, x4, x5)  =  if_reach_3_in_3_ggg1(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG



↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
  ↳ PrologToPiTRSProof

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

reach_3_in_ggg3(X, Y, Edges) -> if_reach_3_in_1_ggg4(X, Y, Edges, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_3_in_1_ggg4(X, Y, Edges, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_3_out_ggg3(X, Y, Edges)
reach_3_in_ggg3(X, Z, Edges) -> if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_in_ggg3(Y, Z, Edges))
if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_out_ggg3(Y, Z, Edges)) -> reach_3_out_ggg3(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_3_in_ggg3(x1, x2, x3)  =  reach_3_in_ggg3(x1, x2, x3)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_3_in_1_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_1_ggg1(x4)
member_2_in_gg2(x1, x2)  =  member_2_in_gg2(x1, x2)
member_2_out_gg2(x1, x2)  =  member_2_out_gg
if_member_2_in_1_gg4(x1, x2, x3, x4)  =  if_member_2_in_1_gg1(x4)
reach_3_out_ggg3(x1, x2, x3)  =  reach_3_out_ggg
if_reach_3_in_2_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_2_ggg3(x2, x3, x4)
member1_2_in_ag2(x1, x2)  =  member1_2_in_ag1(x2)
member1_2_out_ag2(x1, x2)  =  member1_2_out_ag1(x1)
if_member1_2_in_1_ag4(x1, x2, x3, x4)  =  if_member1_2_in_1_ag1(x4)
if_reach_3_in_3_ggg5(x1, x2, x3, x4, x5)  =  if_reach_3_in_3_ggg1(x5)


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

REACH_3_IN_GGG3(X, Y, Edges) -> IF_REACH_3_IN_1_GGG4(X, Y, Edges, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
REACH_3_IN_GGG3(X, Y, Edges) -> MEMBER_2_IN_GG2(._22(X, ._22(Y, []_0)), Edges)
MEMBER_2_IN_GG2(X, ._22(H, L)) -> IF_MEMBER_2_IN_1_GG4(X, H, L, member_2_in_gg2(X, L))
MEMBER_2_IN_GG2(X, ._22(H, L)) -> MEMBER_2_IN_GG2(X, L)
REACH_3_IN_GGG3(X, Z, Edges) -> IF_REACH_3_IN_2_GGG4(X, Z, Edges, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
REACH_3_IN_GGG3(X, Z, Edges) -> MEMBER1_2_IN_AG2(._22(X, ._22(Y, []_0)), Edges)
MEMBER1_2_IN_AG2(X, ._22(H, L)) -> IF_MEMBER1_2_IN_1_AG4(X, H, L, member1_2_in_ag2(X, L))
MEMBER1_2_IN_AG2(X, ._22(H, L)) -> MEMBER1_2_IN_AG2(X, L)
IF_REACH_3_IN_2_GGG4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> IF_REACH_3_IN_3_GGG5(X, Z, Edges, Y, reach_3_in_ggg3(Y, Z, Edges))
IF_REACH_3_IN_2_GGG4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> REACH_3_IN_GGG3(Y, Z, Edges)

The TRS R consists of the following rules:

reach_3_in_ggg3(X, Y, Edges) -> if_reach_3_in_1_ggg4(X, Y, Edges, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_3_in_1_ggg4(X, Y, Edges, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_3_out_ggg3(X, Y, Edges)
reach_3_in_ggg3(X, Z, Edges) -> if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_in_ggg3(Y, Z, Edges))
if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_out_ggg3(Y, Z, Edges)) -> reach_3_out_ggg3(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_3_in_ggg3(x1, x2, x3)  =  reach_3_in_ggg3(x1, x2, x3)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_3_in_1_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_1_ggg1(x4)
member_2_in_gg2(x1, x2)  =  member_2_in_gg2(x1, x2)
member_2_out_gg2(x1, x2)  =  member_2_out_gg
if_member_2_in_1_gg4(x1, x2, x3, x4)  =  if_member_2_in_1_gg1(x4)
reach_3_out_ggg3(x1, x2, x3)  =  reach_3_out_ggg
if_reach_3_in_2_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_2_ggg3(x2, x3, x4)
member1_2_in_ag2(x1, x2)  =  member1_2_in_ag1(x2)
member1_2_out_ag2(x1, x2)  =  member1_2_out_ag1(x1)
if_member1_2_in_1_ag4(x1, x2, x3, x4)  =  if_member1_2_in_1_ag1(x4)
if_reach_3_in_3_ggg5(x1, x2, x3, x4, x5)  =  if_reach_3_in_3_ggg1(x5)
IF_REACH_3_IN_3_GGG5(x1, x2, x3, x4, x5)  =  IF_REACH_3_IN_3_GGG1(x5)
IF_MEMBER_2_IN_1_GG4(x1, x2, x3, x4)  =  IF_MEMBER_2_IN_1_GG1(x4)
IF_REACH_3_IN_2_GGG4(x1, x2, x3, x4)  =  IF_REACH_3_IN_2_GGG3(x2, x3, x4)
MEMBER_2_IN_GG2(x1, x2)  =  MEMBER_2_IN_GG2(x1, x2)
MEMBER1_2_IN_AG2(x1, x2)  =  MEMBER1_2_IN_AG1(x2)
IF_MEMBER1_2_IN_1_AG4(x1, x2, x3, x4)  =  IF_MEMBER1_2_IN_1_AG1(x4)
REACH_3_IN_GGG3(x1, x2, x3)  =  REACH_3_IN_GGG3(x1, x2, x3)
IF_REACH_3_IN_1_GGG4(x1, x2, x3, x4)  =  IF_REACH_3_IN_1_GGG1(x4)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
  ↳ PrologToPiTRSProof

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

REACH_3_IN_GGG3(X, Y, Edges) -> IF_REACH_3_IN_1_GGG4(X, Y, Edges, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
REACH_3_IN_GGG3(X, Y, Edges) -> MEMBER_2_IN_GG2(._22(X, ._22(Y, []_0)), Edges)
MEMBER_2_IN_GG2(X, ._22(H, L)) -> IF_MEMBER_2_IN_1_GG4(X, H, L, member_2_in_gg2(X, L))
MEMBER_2_IN_GG2(X, ._22(H, L)) -> MEMBER_2_IN_GG2(X, L)
REACH_3_IN_GGG3(X, Z, Edges) -> IF_REACH_3_IN_2_GGG4(X, Z, Edges, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
REACH_3_IN_GGG3(X, Z, Edges) -> MEMBER1_2_IN_AG2(._22(X, ._22(Y, []_0)), Edges)
MEMBER1_2_IN_AG2(X, ._22(H, L)) -> IF_MEMBER1_2_IN_1_AG4(X, H, L, member1_2_in_ag2(X, L))
MEMBER1_2_IN_AG2(X, ._22(H, L)) -> MEMBER1_2_IN_AG2(X, L)
IF_REACH_3_IN_2_GGG4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> IF_REACH_3_IN_3_GGG5(X, Z, Edges, Y, reach_3_in_ggg3(Y, Z, Edges))
IF_REACH_3_IN_2_GGG4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> REACH_3_IN_GGG3(Y, Z, Edges)

The TRS R consists of the following rules:

reach_3_in_ggg3(X, Y, Edges) -> if_reach_3_in_1_ggg4(X, Y, Edges, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_3_in_1_ggg4(X, Y, Edges, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_3_out_ggg3(X, Y, Edges)
reach_3_in_ggg3(X, Z, Edges) -> if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_in_ggg3(Y, Z, Edges))
if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_out_ggg3(Y, Z, Edges)) -> reach_3_out_ggg3(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_3_in_ggg3(x1, x2, x3)  =  reach_3_in_ggg3(x1, x2, x3)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_3_in_1_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_1_ggg1(x4)
member_2_in_gg2(x1, x2)  =  member_2_in_gg2(x1, x2)
member_2_out_gg2(x1, x2)  =  member_2_out_gg
if_member_2_in_1_gg4(x1, x2, x3, x4)  =  if_member_2_in_1_gg1(x4)
reach_3_out_ggg3(x1, x2, x3)  =  reach_3_out_ggg
if_reach_3_in_2_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_2_ggg3(x2, x3, x4)
member1_2_in_ag2(x1, x2)  =  member1_2_in_ag1(x2)
member1_2_out_ag2(x1, x2)  =  member1_2_out_ag1(x1)
if_member1_2_in_1_ag4(x1, x2, x3, x4)  =  if_member1_2_in_1_ag1(x4)
if_reach_3_in_3_ggg5(x1, x2, x3, x4, x5)  =  if_reach_3_in_3_ggg1(x5)
IF_REACH_3_IN_3_GGG5(x1, x2, x3, x4, x5)  =  IF_REACH_3_IN_3_GGG1(x5)
IF_MEMBER_2_IN_1_GG4(x1, x2, x3, x4)  =  IF_MEMBER_2_IN_1_GG1(x4)
IF_REACH_3_IN_2_GGG4(x1, x2, x3, x4)  =  IF_REACH_3_IN_2_GGG3(x2, x3, x4)
MEMBER_2_IN_GG2(x1, x2)  =  MEMBER_2_IN_GG2(x1, x2)
MEMBER1_2_IN_AG2(x1, x2)  =  MEMBER1_2_IN_AG1(x2)
IF_MEMBER1_2_IN_1_AG4(x1, x2, x3, x4)  =  IF_MEMBER1_2_IN_1_AG1(x4)
REACH_3_IN_GGG3(x1, x2, x3)  =  REACH_3_IN_GGG3(x1, x2, x3)
IF_REACH_3_IN_1_GGG4(x1, x2, x3, x4)  =  IF_REACH_3_IN_1_GGG1(x4)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 3 SCCs with 6 less nodes.

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

MEMBER1_2_IN_AG2(X, ._22(H, L)) -> MEMBER1_2_IN_AG2(X, L)

The TRS R consists of the following rules:

reach_3_in_ggg3(X, Y, Edges) -> if_reach_3_in_1_ggg4(X, Y, Edges, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_3_in_1_ggg4(X, Y, Edges, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_3_out_ggg3(X, Y, Edges)
reach_3_in_ggg3(X, Z, Edges) -> if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_in_ggg3(Y, Z, Edges))
if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_out_ggg3(Y, Z, Edges)) -> reach_3_out_ggg3(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_3_in_ggg3(x1, x2, x3)  =  reach_3_in_ggg3(x1, x2, x3)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_3_in_1_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_1_ggg1(x4)
member_2_in_gg2(x1, x2)  =  member_2_in_gg2(x1, x2)
member_2_out_gg2(x1, x2)  =  member_2_out_gg
if_member_2_in_1_gg4(x1, x2, x3, x4)  =  if_member_2_in_1_gg1(x4)
reach_3_out_ggg3(x1, x2, x3)  =  reach_3_out_ggg
if_reach_3_in_2_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_2_ggg3(x2, x3, x4)
member1_2_in_ag2(x1, x2)  =  member1_2_in_ag1(x2)
member1_2_out_ag2(x1, x2)  =  member1_2_out_ag1(x1)
if_member1_2_in_1_ag4(x1, x2, x3, x4)  =  if_member1_2_in_1_ag1(x4)
if_reach_3_in_3_ggg5(x1, x2, x3, x4, x5)  =  if_reach_3_in_3_ggg1(x5)
MEMBER1_2_IN_AG2(x1, x2)  =  MEMBER1_2_IN_AG1(x2)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

MEMBER1_2_IN_AG2(X, ._22(H, L)) -> MEMBER1_2_IN_AG2(X, L)

R is empty.
The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._22(x1, x2)
MEMBER1_2_IN_AG2(x1, x2)  =  MEMBER1_2_IN_AG1(x2)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

MEMBER1_2_IN_AG1(._22(H, L)) -> MEMBER1_2_IN_AG1(L)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {MEMBER1_2_IN_AG1}.
By using the subterm criterion together with the size-change analysis 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: 



↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

MEMBER_2_IN_GG2(X, ._22(H, L)) -> MEMBER_2_IN_GG2(X, L)

The TRS R consists of the following rules:

reach_3_in_ggg3(X, Y, Edges) -> if_reach_3_in_1_ggg4(X, Y, Edges, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_3_in_1_ggg4(X, Y, Edges, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_3_out_ggg3(X, Y, Edges)
reach_3_in_ggg3(X, Z, Edges) -> if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_in_ggg3(Y, Z, Edges))
if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_out_ggg3(Y, Z, Edges)) -> reach_3_out_ggg3(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_3_in_ggg3(x1, x2, x3)  =  reach_3_in_ggg3(x1, x2, x3)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_3_in_1_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_1_ggg1(x4)
member_2_in_gg2(x1, x2)  =  member_2_in_gg2(x1, x2)
member_2_out_gg2(x1, x2)  =  member_2_out_gg
if_member_2_in_1_gg4(x1, x2, x3, x4)  =  if_member_2_in_1_gg1(x4)
reach_3_out_ggg3(x1, x2, x3)  =  reach_3_out_ggg
if_reach_3_in_2_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_2_ggg3(x2, x3, x4)
member1_2_in_ag2(x1, x2)  =  member1_2_in_ag1(x2)
member1_2_out_ag2(x1, x2)  =  member1_2_out_ag1(x1)
if_member1_2_in_1_ag4(x1, x2, x3, x4)  =  if_member1_2_in_1_ag1(x4)
if_reach_3_in_3_ggg5(x1, x2, x3, x4, x5)  =  if_reach_3_in_3_ggg1(x5)
MEMBER_2_IN_GG2(x1, x2)  =  MEMBER_2_IN_GG2(x1, x2)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

MEMBER_2_IN_GG2(X, ._22(H, L)) -> MEMBER_2_IN_GG2(X, L)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

MEMBER_2_IN_GG2(X, ._22(H, L)) -> MEMBER_2_IN_GG2(X, L)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {MEMBER_2_IN_GG2}.
By using the subterm criterion together with the size-change analysis 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: 



↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
  ↳ PrologToPiTRSProof

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

REACH_3_IN_GGG3(X, Z, Edges) -> IF_REACH_3_IN_2_GGG4(X, Z, Edges, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
IF_REACH_3_IN_2_GGG4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> REACH_3_IN_GGG3(Y, Z, Edges)

The TRS R consists of the following rules:

reach_3_in_ggg3(X, Y, Edges) -> if_reach_3_in_1_ggg4(X, Y, Edges, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_3_in_1_ggg4(X, Y, Edges, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_3_out_ggg3(X, Y, Edges)
reach_3_in_ggg3(X, Z, Edges) -> if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_in_ggg3(Y, Z, Edges))
if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_out_ggg3(Y, Z, Edges)) -> reach_3_out_ggg3(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_3_in_ggg3(x1, x2, x3)  =  reach_3_in_ggg3(x1, x2, x3)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_3_in_1_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_1_ggg1(x4)
member_2_in_gg2(x1, x2)  =  member_2_in_gg2(x1, x2)
member_2_out_gg2(x1, x2)  =  member_2_out_gg
if_member_2_in_1_gg4(x1, x2, x3, x4)  =  if_member_2_in_1_gg1(x4)
reach_3_out_ggg3(x1, x2, x3)  =  reach_3_out_ggg
if_reach_3_in_2_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_2_ggg3(x2, x3, x4)
member1_2_in_ag2(x1, x2)  =  member1_2_in_ag1(x2)
member1_2_out_ag2(x1, x2)  =  member1_2_out_ag1(x1)
if_member1_2_in_1_ag4(x1, x2, x3, x4)  =  if_member1_2_in_1_ag1(x4)
if_reach_3_in_3_ggg5(x1, x2, x3, x4, x5)  =  if_reach_3_in_3_ggg1(x5)
IF_REACH_3_IN_2_GGG4(x1, x2, x3, x4)  =  IF_REACH_3_IN_2_GGG3(x2, x3, x4)
REACH_3_IN_GGG3(x1, x2, x3)  =  REACH_3_IN_GGG3(x1, x2, x3)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
  ↳ PrologToPiTRSProof

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

REACH_3_IN_GGG3(X, Z, Edges) -> IF_REACH_3_IN_2_GGG4(X, Z, Edges, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
IF_REACH_3_IN_2_GGG4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> REACH_3_IN_GGG3(Y, Z, Edges)

The TRS R consists of the following rules:

member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))

The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
member1_2_in_ag2(x1, x2)  =  member1_2_in_ag1(x2)
member1_2_out_ag2(x1, x2)  =  member1_2_out_ag1(x1)
if_member1_2_in_1_ag4(x1, x2, x3, x4)  =  if_member1_2_in_1_ag1(x4)
IF_REACH_3_IN_2_GGG4(x1, x2, x3, x4)  =  IF_REACH_3_IN_2_GGG3(x2, x3, x4)
REACH_3_IN_GGG3(x1, x2, x3)  =  REACH_3_IN_GGG3(x1, x2, x3)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
  ↳ PrologToPiTRSProof

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

REACH_3_IN_GGG3(X, Z, Edges) -> IF_REACH_3_IN_2_GGG3(Z, Edges, member1_2_in_ag1(Edges))
IF_REACH_3_IN_2_GGG3(Z, Edges, member1_2_out_ag1(._22(X, ._22(Y, []_0)))) -> REACH_3_IN_GGG3(Y, Z, Edges)

The TRS R consists of the following rules:

member1_2_in_ag1(._22(H, L)) -> member1_2_out_ag1(H)
member1_2_in_ag1(._22(H, L)) -> if_member1_2_in_1_ag1(member1_2_in_ag1(L))
if_member1_2_in_1_ag1(member1_2_out_ag1(X)) -> member1_2_out_ag1(X)

The set Q consists of the following terms:

member1_2_in_ag1(x0)
if_member1_2_in_1_ag1(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_REACH_3_IN_2_GGG3, REACH_3_IN_GGG3}.
With regard to the inferred argument filtering the predicates were used in the following modes:
reach3: (b,b,b)
member2: (b,b)
member12: (f,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

reach_3_in_ggg3(X, Y, Edges) -> if_reach_3_in_1_ggg4(X, Y, Edges, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_3_in_1_ggg4(X, Y, Edges, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_3_out_ggg3(X, Y, Edges)
reach_3_in_ggg3(X, Z, Edges) -> if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_in_ggg3(Y, Z, Edges))
if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_out_ggg3(Y, Z, Edges)) -> reach_3_out_ggg3(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_3_in_ggg3(x1, x2, x3)  =  reach_3_in_ggg3(x1, x2, x3)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_3_in_1_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_1_ggg4(x1, x2, x3, x4)
member_2_in_gg2(x1, x2)  =  member_2_in_gg2(x1, x2)
member_2_out_gg2(x1, x2)  =  member_2_out_gg2(x1, x2)
if_member_2_in_1_gg4(x1, x2, x3, x4)  =  if_member_2_in_1_gg4(x1, x2, x3, x4)
reach_3_out_ggg3(x1, x2, x3)  =  reach_3_out_ggg3(x1, x2, x3)
if_reach_3_in_2_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_2_ggg4(x1, x2, x3, x4)
member1_2_in_ag2(x1, x2)  =  member1_2_in_ag1(x2)
member1_2_out_ag2(x1, x2)  =  member1_2_out_ag2(x1, x2)
if_member1_2_in_1_ag4(x1, x2, x3, x4)  =  if_member1_2_in_1_ag3(x2, x3, x4)
if_reach_3_in_3_ggg5(x1, x2, x3, x4, x5)  =  if_reach_3_in_3_ggg4(x1, x2, x3, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG



↳ PROLOG
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

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

reach_3_in_ggg3(X, Y, Edges) -> if_reach_3_in_1_ggg4(X, Y, Edges, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_3_in_1_ggg4(X, Y, Edges, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_3_out_ggg3(X, Y, Edges)
reach_3_in_ggg3(X, Z, Edges) -> if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_in_ggg3(Y, Z, Edges))
if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_out_ggg3(Y, Z, Edges)) -> reach_3_out_ggg3(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_3_in_ggg3(x1, x2, x3)  =  reach_3_in_ggg3(x1, x2, x3)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_3_in_1_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_1_ggg4(x1, x2, x3, x4)
member_2_in_gg2(x1, x2)  =  member_2_in_gg2(x1, x2)
member_2_out_gg2(x1, x2)  =  member_2_out_gg2(x1, x2)
if_member_2_in_1_gg4(x1, x2, x3, x4)  =  if_member_2_in_1_gg4(x1, x2, x3, x4)
reach_3_out_ggg3(x1, x2, x3)  =  reach_3_out_ggg3(x1, x2, x3)
if_reach_3_in_2_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_2_ggg4(x1, x2, x3, x4)
member1_2_in_ag2(x1, x2)  =  member1_2_in_ag1(x2)
member1_2_out_ag2(x1, x2)  =  member1_2_out_ag2(x1, x2)
if_member1_2_in_1_ag4(x1, x2, x3, x4)  =  if_member1_2_in_1_ag3(x2, x3, x4)
if_reach_3_in_3_ggg5(x1, x2, x3, x4, x5)  =  if_reach_3_in_3_ggg4(x1, x2, x3, x5)


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

REACH_3_IN_GGG3(X, Y, Edges) -> IF_REACH_3_IN_1_GGG4(X, Y, Edges, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
REACH_3_IN_GGG3(X, Y, Edges) -> MEMBER_2_IN_GG2(._22(X, ._22(Y, []_0)), Edges)
MEMBER_2_IN_GG2(X, ._22(H, L)) -> IF_MEMBER_2_IN_1_GG4(X, H, L, member_2_in_gg2(X, L))
MEMBER_2_IN_GG2(X, ._22(H, L)) -> MEMBER_2_IN_GG2(X, L)
REACH_3_IN_GGG3(X, Z, Edges) -> IF_REACH_3_IN_2_GGG4(X, Z, Edges, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
REACH_3_IN_GGG3(X, Z, Edges) -> MEMBER1_2_IN_AG2(._22(X, ._22(Y, []_0)), Edges)
MEMBER1_2_IN_AG2(X, ._22(H, L)) -> IF_MEMBER1_2_IN_1_AG4(X, H, L, member1_2_in_ag2(X, L))
MEMBER1_2_IN_AG2(X, ._22(H, L)) -> MEMBER1_2_IN_AG2(X, L)
IF_REACH_3_IN_2_GGG4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> IF_REACH_3_IN_3_GGG5(X, Z, Edges, Y, reach_3_in_ggg3(Y, Z, Edges))
IF_REACH_3_IN_2_GGG4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> REACH_3_IN_GGG3(Y, Z, Edges)

The TRS R consists of the following rules:

reach_3_in_ggg3(X, Y, Edges) -> if_reach_3_in_1_ggg4(X, Y, Edges, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_3_in_1_ggg4(X, Y, Edges, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_3_out_ggg3(X, Y, Edges)
reach_3_in_ggg3(X, Z, Edges) -> if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_in_ggg3(Y, Z, Edges))
if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_out_ggg3(Y, Z, Edges)) -> reach_3_out_ggg3(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_3_in_ggg3(x1, x2, x3)  =  reach_3_in_ggg3(x1, x2, x3)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_3_in_1_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_1_ggg4(x1, x2, x3, x4)
member_2_in_gg2(x1, x2)  =  member_2_in_gg2(x1, x2)
member_2_out_gg2(x1, x2)  =  member_2_out_gg2(x1, x2)
if_member_2_in_1_gg4(x1, x2, x3, x4)  =  if_member_2_in_1_gg4(x1, x2, x3, x4)
reach_3_out_ggg3(x1, x2, x3)  =  reach_3_out_ggg3(x1, x2, x3)
if_reach_3_in_2_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_2_ggg4(x1, x2, x3, x4)
member1_2_in_ag2(x1, x2)  =  member1_2_in_ag1(x2)
member1_2_out_ag2(x1, x2)  =  member1_2_out_ag2(x1, x2)
if_member1_2_in_1_ag4(x1, x2, x3, x4)  =  if_member1_2_in_1_ag3(x2, x3, x4)
if_reach_3_in_3_ggg5(x1, x2, x3, x4, x5)  =  if_reach_3_in_3_ggg4(x1, x2, x3, x5)
IF_REACH_3_IN_3_GGG5(x1, x2, x3, x4, x5)  =  IF_REACH_3_IN_3_GGG4(x1, x2, x3, x5)
IF_MEMBER_2_IN_1_GG4(x1, x2, x3, x4)  =  IF_MEMBER_2_IN_1_GG4(x1, x2, x3, x4)
IF_REACH_3_IN_2_GGG4(x1, x2, x3, x4)  =  IF_REACH_3_IN_2_GGG4(x1, x2, x3, x4)
MEMBER_2_IN_GG2(x1, x2)  =  MEMBER_2_IN_GG2(x1, x2)
MEMBER1_2_IN_AG2(x1, x2)  =  MEMBER1_2_IN_AG1(x2)
IF_MEMBER1_2_IN_1_AG4(x1, x2, x3, x4)  =  IF_MEMBER1_2_IN_1_AG3(x2, x3, x4)
REACH_3_IN_GGG3(x1, x2, x3)  =  REACH_3_IN_GGG3(x1, x2, x3)
IF_REACH_3_IN_1_GGG4(x1, x2, x3, x4)  =  IF_REACH_3_IN_1_GGG4(x1, x2, x3, x4)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof

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

REACH_3_IN_GGG3(X, Y, Edges) -> IF_REACH_3_IN_1_GGG4(X, Y, Edges, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
REACH_3_IN_GGG3(X, Y, Edges) -> MEMBER_2_IN_GG2(._22(X, ._22(Y, []_0)), Edges)
MEMBER_2_IN_GG2(X, ._22(H, L)) -> IF_MEMBER_2_IN_1_GG4(X, H, L, member_2_in_gg2(X, L))
MEMBER_2_IN_GG2(X, ._22(H, L)) -> MEMBER_2_IN_GG2(X, L)
REACH_3_IN_GGG3(X, Z, Edges) -> IF_REACH_3_IN_2_GGG4(X, Z, Edges, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
REACH_3_IN_GGG3(X, Z, Edges) -> MEMBER1_2_IN_AG2(._22(X, ._22(Y, []_0)), Edges)
MEMBER1_2_IN_AG2(X, ._22(H, L)) -> IF_MEMBER1_2_IN_1_AG4(X, H, L, member1_2_in_ag2(X, L))
MEMBER1_2_IN_AG2(X, ._22(H, L)) -> MEMBER1_2_IN_AG2(X, L)
IF_REACH_3_IN_2_GGG4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> IF_REACH_3_IN_3_GGG5(X, Z, Edges, Y, reach_3_in_ggg3(Y, Z, Edges))
IF_REACH_3_IN_2_GGG4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> REACH_3_IN_GGG3(Y, Z, Edges)

The TRS R consists of the following rules:

reach_3_in_ggg3(X, Y, Edges) -> if_reach_3_in_1_ggg4(X, Y, Edges, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_3_in_1_ggg4(X, Y, Edges, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_3_out_ggg3(X, Y, Edges)
reach_3_in_ggg3(X, Z, Edges) -> if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_in_ggg3(Y, Z, Edges))
if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_out_ggg3(Y, Z, Edges)) -> reach_3_out_ggg3(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_3_in_ggg3(x1, x2, x3)  =  reach_3_in_ggg3(x1, x2, x3)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_3_in_1_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_1_ggg4(x1, x2, x3, x4)
member_2_in_gg2(x1, x2)  =  member_2_in_gg2(x1, x2)
member_2_out_gg2(x1, x2)  =  member_2_out_gg2(x1, x2)
if_member_2_in_1_gg4(x1, x2, x3, x4)  =  if_member_2_in_1_gg4(x1, x2, x3, x4)
reach_3_out_ggg3(x1, x2, x3)  =  reach_3_out_ggg3(x1, x2, x3)
if_reach_3_in_2_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_2_ggg4(x1, x2, x3, x4)
member1_2_in_ag2(x1, x2)  =  member1_2_in_ag1(x2)
member1_2_out_ag2(x1, x2)  =  member1_2_out_ag2(x1, x2)
if_member1_2_in_1_ag4(x1, x2, x3, x4)  =  if_member1_2_in_1_ag3(x2, x3, x4)
if_reach_3_in_3_ggg5(x1, x2, x3, x4, x5)  =  if_reach_3_in_3_ggg4(x1, x2, x3, x5)
IF_REACH_3_IN_3_GGG5(x1, x2, x3, x4, x5)  =  IF_REACH_3_IN_3_GGG4(x1, x2, x3, x5)
IF_MEMBER_2_IN_1_GG4(x1, x2, x3, x4)  =  IF_MEMBER_2_IN_1_GG4(x1, x2, x3, x4)
IF_REACH_3_IN_2_GGG4(x1, x2, x3, x4)  =  IF_REACH_3_IN_2_GGG4(x1, x2, x3, x4)
MEMBER_2_IN_GG2(x1, x2)  =  MEMBER_2_IN_GG2(x1, x2)
MEMBER1_2_IN_AG2(x1, x2)  =  MEMBER1_2_IN_AG1(x2)
IF_MEMBER1_2_IN_1_AG4(x1, x2, x3, x4)  =  IF_MEMBER1_2_IN_1_AG3(x2, x3, x4)
REACH_3_IN_GGG3(x1, x2, x3)  =  REACH_3_IN_GGG3(x1, x2, x3)
IF_REACH_3_IN_1_GGG4(x1, x2, x3, x4)  =  IF_REACH_3_IN_1_GGG4(x1, x2, x3, x4)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 3 SCCs with 6 less nodes.

↳ PROLOG
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP

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

MEMBER1_2_IN_AG2(X, ._22(H, L)) -> MEMBER1_2_IN_AG2(X, L)

The TRS R consists of the following rules:

reach_3_in_ggg3(X, Y, Edges) -> if_reach_3_in_1_ggg4(X, Y, Edges, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_3_in_1_ggg4(X, Y, Edges, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_3_out_ggg3(X, Y, Edges)
reach_3_in_ggg3(X, Z, Edges) -> if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_in_ggg3(Y, Z, Edges))
if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_out_ggg3(Y, Z, Edges)) -> reach_3_out_ggg3(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_3_in_ggg3(x1, x2, x3)  =  reach_3_in_ggg3(x1, x2, x3)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_3_in_1_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_1_ggg4(x1, x2, x3, x4)
member_2_in_gg2(x1, x2)  =  member_2_in_gg2(x1, x2)
member_2_out_gg2(x1, x2)  =  member_2_out_gg2(x1, x2)
if_member_2_in_1_gg4(x1, x2, x3, x4)  =  if_member_2_in_1_gg4(x1, x2, x3, x4)
reach_3_out_ggg3(x1, x2, x3)  =  reach_3_out_ggg3(x1, x2, x3)
if_reach_3_in_2_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_2_ggg4(x1, x2, x3, x4)
member1_2_in_ag2(x1, x2)  =  member1_2_in_ag1(x2)
member1_2_out_ag2(x1, x2)  =  member1_2_out_ag2(x1, x2)
if_member1_2_in_1_ag4(x1, x2, x3, x4)  =  if_member1_2_in_1_ag3(x2, x3, x4)
if_reach_3_in_3_ggg5(x1, x2, x3, x4, x5)  =  if_reach_3_in_3_ggg4(x1, x2, x3, x5)
MEMBER1_2_IN_AG2(x1, x2)  =  MEMBER1_2_IN_AG1(x2)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

MEMBER1_2_IN_AG2(X, ._22(H, L)) -> MEMBER1_2_IN_AG2(X, L)

R is empty.
The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._22(x1, x2)
MEMBER1_2_IN_AG2(x1, x2)  =  MEMBER1_2_IN_AG1(x2)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

MEMBER_2_IN_GG2(X, ._22(H, L)) -> MEMBER_2_IN_GG2(X, L)

The TRS R consists of the following rules:

reach_3_in_ggg3(X, Y, Edges) -> if_reach_3_in_1_ggg4(X, Y, Edges, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_3_in_1_ggg4(X, Y, Edges, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_3_out_ggg3(X, Y, Edges)
reach_3_in_ggg3(X, Z, Edges) -> if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_in_ggg3(Y, Z, Edges))
if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_out_ggg3(Y, Z, Edges)) -> reach_3_out_ggg3(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_3_in_ggg3(x1, x2, x3)  =  reach_3_in_ggg3(x1, x2, x3)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_3_in_1_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_1_ggg4(x1, x2, x3, x4)
member_2_in_gg2(x1, x2)  =  member_2_in_gg2(x1, x2)
member_2_out_gg2(x1, x2)  =  member_2_out_gg2(x1, x2)
if_member_2_in_1_gg4(x1, x2, x3, x4)  =  if_member_2_in_1_gg4(x1, x2, x3, x4)
reach_3_out_ggg3(x1, x2, x3)  =  reach_3_out_ggg3(x1, x2, x3)
if_reach_3_in_2_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_2_ggg4(x1, x2, x3, x4)
member1_2_in_ag2(x1, x2)  =  member1_2_in_ag1(x2)
member1_2_out_ag2(x1, x2)  =  member1_2_out_ag2(x1, x2)
if_member1_2_in_1_ag4(x1, x2, x3, x4)  =  if_member1_2_in_1_ag3(x2, x3, x4)
if_reach_3_in_3_ggg5(x1, x2, x3, x4, x5)  =  if_reach_3_in_3_ggg4(x1, x2, x3, x5)
MEMBER_2_IN_GG2(x1, x2)  =  MEMBER_2_IN_GG2(x1, x2)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

REACH_3_IN_GGG3(X, Z, Edges) -> IF_REACH_3_IN_2_GGG4(X, Z, Edges, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
IF_REACH_3_IN_2_GGG4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> REACH_3_IN_GGG3(Y, Z, Edges)

The TRS R consists of the following rules:

reach_3_in_ggg3(X, Y, Edges) -> if_reach_3_in_1_ggg4(X, Y, Edges, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_3_in_1_ggg4(X, Y, Edges, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_3_out_ggg3(X, Y, Edges)
reach_3_in_ggg3(X, Z, Edges) -> if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_3_in_2_ggg4(X, Z, Edges, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_in_ggg3(Y, Z, Edges))
if_reach_3_in_3_ggg5(X, Z, Edges, Y, reach_3_out_ggg3(Y, Z, Edges)) -> reach_3_out_ggg3(X, Z, Edges)

The argument filtering Pi contains the following mapping:
reach_3_in_ggg3(x1, x2, x3)  =  reach_3_in_ggg3(x1, x2, x3)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_3_in_1_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_1_ggg4(x1, x2, x3, x4)
member_2_in_gg2(x1, x2)  =  member_2_in_gg2(x1, x2)
member_2_out_gg2(x1, x2)  =  member_2_out_gg2(x1, x2)
if_member_2_in_1_gg4(x1, x2, x3, x4)  =  if_member_2_in_1_gg4(x1, x2, x3, x4)
reach_3_out_ggg3(x1, x2, x3)  =  reach_3_out_ggg3(x1, x2, x3)
if_reach_3_in_2_ggg4(x1, x2, x3, x4)  =  if_reach_3_in_2_ggg4(x1, x2, x3, x4)
member1_2_in_ag2(x1, x2)  =  member1_2_in_ag1(x2)
member1_2_out_ag2(x1, x2)  =  member1_2_out_ag2(x1, x2)
if_member1_2_in_1_ag4(x1, x2, x3, x4)  =  if_member1_2_in_1_ag3(x2, x3, x4)
if_reach_3_in_3_ggg5(x1, x2, x3, x4, x5)  =  if_reach_3_in_3_ggg4(x1, x2, x3, x5)
IF_REACH_3_IN_2_GGG4(x1, x2, x3, x4)  =  IF_REACH_3_IN_2_GGG4(x1, x2, x3, x4)
REACH_3_IN_GGG3(x1, x2, x3)  =  REACH_3_IN_GGG3(x1, x2, x3)

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