Left Termination of the query pattern reach(b,b,b,b) w.r.t. the given Prolog program could successfully be proven:



PROLOG
  ↳ PrologToPiTRSProof

reach4(X, Y, Edges, NotVisited) :- member2(.2(X, .2(Y, {}0)), Edges).
reach4(X, Z, Edges, NotVisited) :- member2(.2(X, .2(Y, {}0)), Edges), member2(Y, NotVisited), delete3(Y, NotVisited, V1), reach4(Y, Z, Edges, V1).
member2(H, .2(H, L)).
member2(X, .2(H, L)) :- member2(X, L).
delete3(X, .2(X, Y), Y).
delete3(X, .2(H, T1), .2(H, T2)) :- delete3(X, T1, T2).


With regard to the inferred argument filtering the predicates were used in the following modes:
reach4: (b,b,b,b)
member2: (b,b) (f,b)
delete3: (b,b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:


reach_4_in_gggg4(X, Y, Edges, Not_Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Not_Visited, 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_4_in_1_gggg5(X, Y, Edges, Not_Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Not_Visited)
reach_4_in_gggg4(X, Z, Edges, Not_Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Not_Visited, member_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_ag2(H, ._22(H, L)) -> member_2_out_ag2(H, ._22(H, L))
member_2_in_ag2(X, ._22(H, L)) -> if_member_2_in_1_ag4(X, H, L, member_2_in_ag2(X, L))
if_member_2_in_1_ag4(X, H, L, member_2_out_ag2(X, L)) -> member_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Not_Visited, member_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Not_Visited, Y, member_2_in_gg2(Y, Not_Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Not_Visited, Y, member_2_out_gg2(Y, Not_Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Not_Visited, Y, delete_3_in_gga3(Y, Not_Visited, V1))
delete_3_in_gga3(X, ._22(X, Y), Y) -> delete_3_out_gga3(X, ._22(X, Y), Y)
delete_3_in_gga3(X, ._22(H, T1), ._22(H, T2)) -> if_delete_3_in_1_gga5(X, H, T1, T2, delete_3_in_gga3(X, T1, T2))
if_delete_3_in_1_gga5(X, H, T1, T2, delete_3_out_gga3(X, T1, T2)) -> delete_3_out_gga3(X, ._22(H, T1), ._22(H, T2))
if_reach_4_in_4_gggg6(X, Z, Edges, Not_Visited, Y, delete_3_out_gga3(Y, Not_Visited, V1)) -> if_reach_4_in_5_gggg7(X, Z, Edges, Not_Visited, Y, V1, reach_4_in_gggg4(Y, Z, Edges, V1))
if_reach_4_in_5_gggg7(X, Z, Edges, Not_Visited, Y, V1, reach_4_out_gggg4(Y, Z, Edges, V1)) -> reach_4_out_gggg4(X, Z, Edges, Not_Visited)

The argument filtering Pi contains the following mapping:
reach_4_in_gggg4(x1, x2, x3, x4)  =  reach_4_in_gggg4(x1, x2, x3, x4)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_4_in_1_gggg5(x1, x2, x3, x4, x5)  =  if_reach_4_in_1_gggg1(x5)
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_4_out_gggg4(x1, x2, x3, x4)  =  reach_4_out_gggg
if_reach_4_in_2_gggg5(x1, x2, x3, x4, x5)  =  if_reach_4_in_2_gggg4(x2, x3, x4, x5)
member_2_in_ag2(x1, x2)  =  member_2_in_ag1(x2)
member_2_out_ag2(x1, x2)  =  member_2_out_ag1(x1)
if_member_2_in_1_ag4(x1, x2, x3, x4)  =  if_member_2_in_1_ag1(x4)
if_reach_4_in_3_gggg6(x1, x2, x3, x4, x5, x6)  =  if_reach_4_in_3_gggg5(x2, x3, x4, x5, x6)
if_reach_4_in_4_gggg6(x1, x2, x3, x4, x5, x6)  =  if_reach_4_in_4_gggg4(x2, x3, x5, x6)
delete_3_in_gga3(x1, x2, x3)  =  delete_3_in_gga2(x1, x2)
delete_3_out_gga3(x1, x2, x3)  =  delete_3_out_gga1(x3)
if_delete_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_delete_3_in_1_gga2(x2, x5)
if_reach_4_in_5_gggg7(x1, x2, x3, x4, x5, x6, x7)  =  if_reach_4_in_5_gggg1(x7)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG



↳ PROLOG
  ↳ PrologToPiTRSProof
PiTRS
      ↳ DependencyPairsProof

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

reach_4_in_gggg4(X, Y, Edges, Not_Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Not_Visited, 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_4_in_1_gggg5(X, Y, Edges, Not_Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Not_Visited)
reach_4_in_gggg4(X, Z, Edges, Not_Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Not_Visited, member_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_ag2(H, ._22(H, L)) -> member_2_out_ag2(H, ._22(H, L))
member_2_in_ag2(X, ._22(H, L)) -> if_member_2_in_1_ag4(X, H, L, member_2_in_ag2(X, L))
if_member_2_in_1_ag4(X, H, L, member_2_out_ag2(X, L)) -> member_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Not_Visited, member_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Not_Visited, Y, member_2_in_gg2(Y, Not_Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Not_Visited, Y, member_2_out_gg2(Y, Not_Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Not_Visited, Y, delete_3_in_gga3(Y, Not_Visited, V1))
delete_3_in_gga3(X, ._22(X, Y), Y) -> delete_3_out_gga3(X, ._22(X, Y), Y)
delete_3_in_gga3(X, ._22(H, T1), ._22(H, T2)) -> if_delete_3_in_1_gga5(X, H, T1, T2, delete_3_in_gga3(X, T1, T2))
if_delete_3_in_1_gga5(X, H, T1, T2, delete_3_out_gga3(X, T1, T2)) -> delete_3_out_gga3(X, ._22(H, T1), ._22(H, T2))
if_reach_4_in_4_gggg6(X, Z, Edges, Not_Visited, Y, delete_3_out_gga3(Y, Not_Visited, V1)) -> if_reach_4_in_5_gggg7(X, Z, Edges, Not_Visited, Y, V1, reach_4_in_gggg4(Y, Z, Edges, V1))
if_reach_4_in_5_gggg7(X, Z, Edges, Not_Visited, Y, V1, reach_4_out_gggg4(Y, Z, Edges, V1)) -> reach_4_out_gggg4(X, Z, Edges, Not_Visited)

The argument filtering Pi contains the following mapping:
reach_4_in_gggg4(x1, x2, x3, x4)  =  reach_4_in_gggg4(x1, x2, x3, x4)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_4_in_1_gggg5(x1, x2, x3, x4, x5)  =  if_reach_4_in_1_gggg1(x5)
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_4_out_gggg4(x1, x2, x3, x4)  =  reach_4_out_gggg
if_reach_4_in_2_gggg5(x1, x2, x3, x4, x5)  =  if_reach_4_in_2_gggg4(x2, x3, x4, x5)
member_2_in_ag2(x1, x2)  =  member_2_in_ag1(x2)
member_2_out_ag2(x1, x2)  =  member_2_out_ag1(x1)
if_member_2_in_1_ag4(x1, x2, x3, x4)  =  if_member_2_in_1_ag1(x4)
if_reach_4_in_3_gggg6(x1, x2, x3, x4, x5, x6)  =  if_reach_4_in_3_gggg5(x2, x3, x4, x5, x6)
if_reach_4_in_4_gggg6(x1, x2, x3, x4, x5, x6)  =  if_reach_4_in_4_gggg4(x2, x3, x5, x6)
delete_3_in_gga3(x1, x2, x3)  =  delete_3_in_gga2(x1, x2)
delete_3_out_gga3(x1, x2, x3)  =  delete_3_out_gga1(x3)
if_delete_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_delete_3_in_1_gga2(x2, x5)
if_reach_4_in_5_gggg7(x1, x2, x3, x4, x5, x6, x7)  =  if_reach_4_in_5_gggg1(x7)


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

REACH_4_IN_GGGG4(X, Y, Edges, Not_Visited) -> IF_REACH_4_IN_1_GGGG5(X, Y, Edges, Not_Visited, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
REACH_4_IN_GGGG4(X, Y, Edges, Not_Visited) -> 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_4_IN_GGGG4(X, Z, Edges, Not_Visited) -> IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Not_Visited, member_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
REACH_4_IN_GGGG4(X, Z, Edges, Not_Visited) -> MEMBER_2_IN_AG2(._22(X, ._22(Y, []_0)), Edges)
MEMBER_2_IN_AG2(X, ._22(H, L)) -> IF_MEMBER_2_IN_1_AG4(X, H, L, member_2_in_ag2(X, L))
MEMBER_2_IN_AG2(X, ._22(H, L)) -> MEMBER_2_IN_AG2(X, L)
IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Not_Visited, member_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Not_Visited, Y, member_2_in_gg2(Y, Not_Visited))
IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Not_Visited, member_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> MEMBER_2_IN_GG2(Y, Not_Visited)
IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Not_Visited, Y, member_2_out_gg2(Y, Not_Visited)) -> IF_REACH_4_IN_4_GGGG6(X, Z, Edges, Not_Visited, Y, delete_3_in_gga3(Y, Not_Visited, V1))
IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Not_Visited, Y, member_2_out_gg2(Y, Not_Visited)) -> DELETE_3_IN_GGA3(Y, Not_Visited, V1)
DELETE_3_IN_GGA3(X, ._22(H, T1), ._22(H, T2)) -> IF_DELETE_3_IN_1_GGA5(X, H, T1, T2, delete_3_in_gga3(X, T1, T2))
DELETE_3_IN_GGA3(X, ._22(H, T1), ._22(H, T2)) -> DELETE_3_IN_GGA3(X, T1, T2)
IF_REACH_4_IN_4_GGGG6(X, Z, Edges, Not_Visited, Y, delete_3_out_gga3(Y, Not_Visited, V1)) -> IF_REACH_4_IN_5_GGGG7(X, Z, Edges, Not_Visited, Y, V1, reach_4_in_gggg4(Y, Z, Edges, V1))
IF_REACH_4_IN_4_GGGG6(X, Z, Edges, Not_Visited, Y, delete_3_out_gga3(Y, Not_Visited, V1)) -> REACH_4_IN_GGGG4(Y, Z, Edges, V1)

The TRS R consists of the following rules:

reach_4_in_gggg4(X, Y, Edges, Not_Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Not_Visited, 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_4_in_1_gggg5(X, Y, Edges, Not_Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Not_Visited)
reach_4_in_gggg4(X, Z, Edges, Not_Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Not_Visited, member_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_ag2(H, ._22(H, L)) -> member_2_out_ag2(H, ._22(H, L))
member_2_in_ag2(X, ._22(H, L)) -> if_member_2_in_1_ag4(X, H, L, member_2_in_ag2(X, L))
if_member_2_in_1_ag4(X, H, L, member_2_out_ag2(X, L)) -> member_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Not_Visited, member_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Not_Visited, Y, member_2_in_gg2(Y, Not_Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Not_Visited, Y, member_2_out_gg2(Y, Not_Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Not_Visited, Y, delete_3_in_gga3(Y, Not_Visited, V1))
delete_3_in_gga3(X, ._22(X, Y), Y) -> delete_3_out_gga3(X, ._22(X, Y), Y)
delete_3_in_gga3(X, ._22(H, T1), ._22(H, T2)) -> if_delete_3_in_1_gga5(X, H, T1, T2, delete_3_in_gga3(X, T1, T2))
if_delete_3_in_1_gga5(X, H, T1, T2, delete_3_out_gga3(X, T1, T2)) -> delete_3_out_gga3(X, ._22(H, T1), ._22(H, T2))
if_reach_4_in_4_gggg6(X, Z, Edges, Not_Visited, Y, delete_3_out_gga3(Y, Not_Visited, V1)) -> if_reach_4_in_5_gggg7(X, Z, Edges, Not_Visited, Y, V1, reach_4_in_gggg4(Y, Z, Edges, V1))
if_reach_4_in_5_gggg7(X, Z, Edges, Not_Visited, Y, V1, reach_4_out_gggg4(Y, Z, Edges, V1)) -> reach_4_out_gggg4(X, Z, Edges, Not_Visited)

The argument filtering Pi contains the following mapping:
reach_4_in_gggg4(x1, x2, x3, x4)  =  reach_4_in_gggg4(x1, x2, x3, x4)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_4_in_1_gggg5(x1, x2, x3, x4, x5)  =  if_reach_4_in_1_gggg1(x5)
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_4_out_gggg4(x1, x2, x3, x4)  =  reach_4_out_gggg
if_reach_4_in_2_gggg5(x1, x2, x3, x4, x5)  =  if_reach_4_in_2_gggg4(x2, x3, x4, x5)
member_2_in_ag2(x1, x2)  =  member_2_in_ag1(x2)
member_2_out_ag2(x1, x2)  =  member_2_out_ag1(x1)
if_member_2_in_1_ag4(x1, x2, x3, x4)  =  if_member_2_in_1_ag1(x4)
if_reach_4_in_3_gggg6(x1, x2, x3, x4, x5, x6)  =  if_reach_4_in_3_gggg5(x2, x3, x4, x5, x6)
if_reach_4_in_4_gggg6(x1, x2, x3, x4, x5, x6)  =  if_reach_4_in_4_gggg4(x2, x3, x5, x6)
delete_3_in_gga3(x1, x2, x3)  =  delete_3_in_gga2(x1, x2)
delete_3_out_gga3(x1, x2, x3)  =  delete_3_out_gga1(x3)
if_delete_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_delete_3_in_1_gga2(x2, x5)
if_reach_4_in_5_gggg7(x1, x2, x3, x4, x5, x6, x7)  =  if_reach_4_in_5_gggg1(x7)
DELETE_3_IN_GGA3(x1, x2, x3)  =  DELETE_3_IN_GGA2(x1, x2)
IF_MEMBER_2_IN_1_GG4(x1, x2, x3, x4)  =  IF_MEMBER_2_IN_1_GG1(x4)
IF_REACH_4_IN_4_GGGG6(x1, x2, x3, x4, x5, x6)  =  IF_REACH_4_IN_4_GGGG4(x2, x3, x5, x6)
IF_MEMBER_2_IN_1_AG4(x1, x2, x3, x4)  =  IF_MEMBER_2_IN_1_AG1(x4)
REACH_4_IN_GGGG4(x1, x2, x3, x4)  =  REACH_4_IN_GGGG4(x1, x2, x3, x4)
IF_DELETE_3_IN_1_GGA5(x1, x2, x3, x4, x5)  =  IF_DELETE_3_IN_1_GGA2(x2, x5)
IF_REACH_4_IN_1_GGGG5(x1, x2, x3, x4, x5)  =  IF_REACH_4_IN_1_GGGG1(x5)
MEMBER_2_IN_GG2(x1, x2)  =  MEMBER_2_IN_GG2(x1, x2)
IF_REACH_4_IN_5_GGGG7(x1, x2, x3, x4, x5, x6, x7)  =  IF_REACH_4_IN_5_GGGG1(x7)
MEMBER_2_IN_AG2(x1, x2)  =  MEMBER_2_IN_AG1(x2)
IF_REACH_4_IN_2_GGGG5(x1, x2, x3, x4, x5)  =  IF_REACH_4_IN_2_GGGG4(x2, x3, x4, x5)
IF_REACH_4_IN_3_GGGG6(x1, x2, x3, x4, x5, x6)  =  IF_REACH_4_IN_3_GGGG5(x2, x3, x4, x5, x6)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
PiDP
          ↳ DependencyGraphProof

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

REACH_4_IN_GGGG4(X, Y, Edges, Not_Visited) -> IF_REACH_4_IN_1_GGGG5(X, Y, Edges, Not_Visited, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
REACH_4_IN_GGGG4(X, Y, Edges, Not_Visited) -> 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_4_IN_GGGG4(X, Z, Edges, Not_Visited) -> IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Not_Visited, member_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
REACH_4_IN_GGGG4(X, Z, Edges, Not_Visited) -> MEMBER_2_IN_AG2(._22(X, ._22(Y, []_0)), Edges)
MEMBER_2_IN_AG2(X, ._22(H, L)) -> IF_MEMBER_2_IN_1_AG4(X, H, L, member_2_in_ag2(X, L))
MEMBER_2_IN_AG2(X, ._22(H, L)) -> MEMBER_2_IN_AG2(X, L)
IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Not_Visited, member_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Not_Visited, Y, member_2_in_gg2(Y, Not_Visited))
IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Not_Visited, member_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> MEMBER_2_IN_GG2(Y, Not_Visited)
IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Not_Visited, Y, member_2_out_gg2(Y, Not_Visited)) -> IF_REACH_4_IN_4_GGGG6(X, Z, Edges, Not_Visited, Y, delete_3_in_gga3(Y, Not_Visited, V1))
IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Not_Visited, Y, member_2_out_gg2(Y, Not_Visited)) -> DELETE_3_IN_GGA3(Y, Not_Visited, V1)
DELETE_3_IN_GGA3(X, ._22(H, T1), ._22(H, T2)) -> IF_DELETE_3_IN_1_GGA5(X, H, T1, T2, delete_3_in_gga3(X, T1, T2))
DELETE_3_IN_GGA3(X, ._22(H, T1), ._22(H, T2)) -> DELETE_3_IN_GGA3(X, T1, T2)
IF_REACH_4_IN_4_GGGG6(X, Z, Edges, Not_Visited, Y, delete_3_out_gga3(Y, Not_Visited, V1)) -> IF_REACH_4_IN_5_GGGG7(X, Z, Edges, Not_Visited, Y, V1, reach_4_in_gggg4(Y, Z, Edges, V1))
IF_REACH_4_IN_4_GGGG6(X, Z, Edges, Not_Visited, Y, delete_3_out_gga3(Y, Not_Visited, V1)) -> REACH_4_IN_GGGG4(Y, Z, Edges, V1)

The TRS R consists of the following rules:

reach_4_in_gggg4(X, Y, Edges, Not_Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Not_Visited, 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_4_in_1_gggg5(X, Y, Edges, Not_Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Not_Visited)
reach_4_in_gggg4(X, Z, Edges, Not_Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Not_Visited, member_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_ag2(H, ._22(H, L)) -> member_2_out_ag2(H, ._22(H, L))
member_2_in_ag2(X, ._22(H, L)) -> if_member_2_in_1_ag4(X, H, L, member_2_in_ag2(X, L))
if_member_2_in_1_ag4(X, H, L, member_2_out_ag2(X, L)) -> member_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Not_Visited, member_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Not_Visited, Y, member_2_in_gg2(Y, Not_Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Not_Visited, Y, member_2_out_gg2(Y, Not_Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Not_Visited, Y, delete_3_in_gga3(Y, Not_Visited, V1))
delete_3_in_gga3(X, ._22(X, Y), Y) -> delete_3_out_gga3(X, ._22(X, Y), Y)
delete_3_in_gga3(X, ._22(H, T1), ._22(H, T2)) -> if_delete_3_in_1_gga5(X, H, T1, T2, delete_3_in_gga3(X, T1, T2))
if_delete_3_in_1_gga5(X, H, T1, T2, delete_3_out_gga3(X, T1, T2)) -> delete_3_out_gga3(X, ._22(H, T1), ._22(H, T2))
if_reach_4_in_4_gggg6(X, Z, Edges, Not_Visited, Y, delete_3_out_gga3(Y, Not_Visited, V1)) -> if_reach_4_in_5_gggg7(X, Z, Edges, Not_Visited, Y, V1, reach_4_in_gggg4(Y, Z, Edges, V1))
if_reach_4_in_5_gggg7(X, Z, Edges, Not_Visited, Y, V1, reach_4_out_gggg4(Y, Z, Edges, V1)) -> reach_4_out_gggg4(X, Z, Edges, Not_Visited)

The argument filtering Pi contains the following mapping:
reach_4_in_gggg4(x1, x2, x3, x4)  =  reach_4_in_gggg4(x1, x2, x3, x4)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_4_in_1_gggg5(x1, x2, x3, x4, x5)  =  if_reach_4_in_1_gggg1(x5)
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_4_out_gggg4(x1, x2, x3, x4)  =  reach_4_out_gggg
if_reach_4_in_2_gggg5(x1, x2, x3, x4, x5)  =  if_reach_4_in_2_gggg4(x2, x3, x4, x5)
member_2_in_ag2(x1, x2)  =  member_2_in_ag1(x2)
member_2_out_ag2(x1, x2)  =  member_2_out_ag1(x1)
if_member_2_in_1_ag4(x1, x2, x3, x4)  =  if_member_2_in_1_ag1(x4)
if_reach_4_in_3_gggg6(x1, x2, x3, x4, x5, x6)  =  if_reach_4_in_3_gggg5(x2, x3, x4, x5, x6)
if_reach_4_in_4_gggg6(x1, x2, x3, x4, x5, x6)  =  if_reach_4_in_4_gggg4(x2, x3, x5, x6)
delete_3_in_gga3(x1, x2, x3)  =  delete_3_in_gga2(x1, x2)
delete_3_out_gga3(x1, x2, x3)  =  delete_3_out_gga1(x3)
if_delete_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_delete_3_in_1_gga2(x2, x5)
if_reach_4_in_5_gggg7(x1, x2, x3, x4, x5, x6, x7)  =  if_reach_4_in_5_gggg1(x7)
DELETE_3_IN_GGA3(x1, x2, x3)  =  DELETE_3_IN_GGA2(x1, x2)
IF_MEMBER_2_IN_1_GG4(x1, x2, x3, x4)  =  IF_MEMBER_2_IN_1_GG1(x4)
IF_REACH_4_IN_4_GGGG6(x1, x2, x3, x4, x5, x6)  =  IF_REACH_4_IN_4_GGGG4(x2, x3, x5, x6)
IF_MEMBER_2_IN_1_AG4(x1, x2, x3, x4)  =  IF_MEMBER_2_IN_1_AG1(x4)
REACH_4_IN_GGGG4(x1, x2, x3, x4)  =  REACH_4_IN_GGGG4(x1, x2, x3, x4)
IF_DELETE_3_IN_1_GGA5(x1, x2, x3, x4, x5)  =  IF_DELETE_3_IN_1_GGA2(x2, x5)
IF_REACH_4_IN_1_GGGG5(x1, x2, x3, x4, x5)  =  IF_REACH_4_IN_1_GGGG1(x5)
MEMBER_2_IN_GG2(x1, x2)  =  MEMBER_2_IN_GG2(x1, x2)
IF_REACH_4_IN_5_GGGG7(x1, x2, x3, x4, x5, x6, x7)  =  IF_REACH_4_IN_5_GGGG1(x7)
MEMBER_2_IN_AG2(x1, x2)  =  MEMBER_2_IN_AG1(x2)
IF_REACH_4_IN_2_GGGG5(x1, x2, x3, x4, x5)  =  IF_REACH_4_IN_2_GGGG4(x2, x3, x4, x5)
IF_REACH_4_IN_3_GGGG6(x1, x2, x3, x4, x5, x6)  =  IF_REACH_4_IN_3_GGGG5(x2, x3, x4, x5, x6)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 4 SCCs with 9 less nodes.

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

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

DELETE_3_IN_GGA3(X, ._22(H, T1), ._22(H, T2)) -> DELETE_3_IN_GGA3(X, T1, T2)

The TRS R consists of the following rules:

reach_4_in_gggg4(X, Y, Edges, Not_Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Not_Visited, 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_4_in_1_gggg5(X, Y, Edges, Not_Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Not_Visited)
reach_4_in_gggg4(X, Z, Edges, Not_Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Not_Visited, member_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_ag2(H, ._22(H, L)) -> member_2_out_ag2(H, ._22(H, L))
member_2_in_ag2(X, ._22(H, L)) -> if_member_2_in_1_ag4(X, H, L, member_2_in_ag2(X, L))
if_member_2_in_1_ag4(X, H, L, member_2_out_ag2(X, L)) -> member_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Not_Visited, member_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Not_Visited, Y, member_2_in_gg2(Y, Not_Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Not_Visited, Y, member_2_out_gg2(Y, Not_Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Not_Visited, Y, delete_3_in_gga3(Y, Not_Visited, V1))
delete_3_in_gga3(X, ._22(X, Y), Y) -> delete_3_out_gga3(X, ._22(X, Y), Y)
delete_3_in_gga3(X, ._22(H, T1), ._22(H, T2)) -> if_delete_3_in_1_gga5(X, H, T1, T2, delete_3_in_gga3(X, T1, T2))
if_delete_3_in_1_gga5(X, H, T1, T2, delete_3_out_gga3(X, T1, T2)) -> delete_3_out_gga3(X, ._22(H, T1), ._22(H, T2))
if_reach_4_in_4_gggg6(X, Z, Edges, Not_Visited, Y, delete_3_out_gga3(Y, Not_Visited, V1)) -> if_reach_4_in_5_gggg7(X, Z, Edges, Not_Visited, Y, V1, reach_4_in_gggg4(Y, Z, Edges, V1))
if_reach_4_in_5_gggg7(X, Z, Edges, Not_Visited, Y, V1, reach_4_out_gggg4(Y, Z, Edges, V1)) -> reach_4_out_gggg4(X, Z, Edges, Not_Visited)

The argument filtering Pi contains the following mapping:
reach_4_in_gggg4(x1, x2, x3, x4)  =  reach_4_in_gggg4(x1, x2, x3, x4)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_4_in_1_gggg5(x1, x2, x3, x4, x5)  =  if_reach_4_in_1_gggg1(x5)
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_4_out_gggg4(x1, x2, x3, x4)  =  reach_4_out_gggg
if_reach_4_in_2_gggg5(x1, x2, x3, x4, x5)  =  if_reach_4_in_2_gggg4(x2, x3, x4, x5)
member_2_in_ag2(x1, x2)  =  member_2_in_ag1(x2)
member_2_out_ag2(x1, x2)  =  member_2_out_ag1(x1)
if_member_2_in_1_ag4(x1, x2, x3, x4)  =  if_member_2_in_1_ag1(x4)
if_reach_4_in_3_gggg6(x1, x2, x3, x4, x5, x6)  =  if_reach_4_in_3_gggg5(x2, x3, x4, x5, x6)
if_reach_4_in_4_gggg6(x1, x2, x3, x4, x5, x6)  =  if_reach_4_in_4_gggg4(x2, x3, x5, x6)
delete_3_in_gga3(x1, x2, x3)  =  delete_3_in_gga2(x1, x2)
delete_3_out_gga3(x1, x2, x3)  =  delete_3_out_gga1(x3)
if_delete_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_delete_3_in_1_gga2(x2, x5)
if_reach_4_in_5_gggg7(x1, x2, x3, x4, x5, x6, x7)  =  if_reach_4_in_5_gggg1(x7)
DELETE_3_IN_GGA3(x1, x2, x3)  =  DELETE_3_IN_GGA2(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
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

DELETE_3_IN_GGA3(X, ._22(H, T1), ._22(H, T2)) -> DELETE_3_IN_GGA3(X, T1, T2)

R is empty.
The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._22(x1, x2)
DELETE_3_IN_GGA3(x1, x2, x3)  =  DELETE_3_IN_GGA2(x1, 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
              ↳ PiDP

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

DELETE_3_IN_GGA2(X, ._22(H, T1)) -> DELETE_3_IN_GGA2(X, T1)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {DELETE_3_IN_GGA2}.
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
              ↳ PiDP

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

MEMBER_2_IN_AG2(X, ._22(H, L)) -> MEMBER_2_IN_AG2(X, L)

The TRS R consists of the following rules:

reach_4_in_gggg4(X, Y, Edges, Not_Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Not_Visited, 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_4_in_1_gggg5(X, Y, Edges, Not_Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Not_Visited)
reach_4_in_gggg4(X, Z, Edges, Not_Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Not_Visited, member_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_ag2(H, ._22(H, L)) -> member_2_out_ag2(H, ._22(H, L))
member_2_in_ag2(X, ._22(H, L)) -> if_member_2_in_1_ag4(X, H, L, member_2_in_ag2(X, L))
if_member_2_in_1_ag4(X, H, L, member_2_out_ag2(X, L)) -> member_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Not_Visited, member_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Not_Visited, Y, member_2_in_gg2(Y, Not_Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Not_Visited, Y, member_2_out_gg2(Y, Not_Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Not_Visited, Y, delete_3_in_gga3(Y, Not_Visited, V1))
delete_3_in_gga3(X, ._22(X, Y), Y) -> delete_3_out_gga3(X, ._22(X, Y), Y)
delete_3_in_gga3(X, ._22(H, T1), ._22(H, T2)) -> if_delete_3_in_1_gga5(X, H, T1, T2, delete_3_in_gga3(X, T1, T2))
if_delete_3_in_1_gga5(X, H, T1, T2, delete_3_out_gga3(X, T1, T2)) -> delete_3_out_gga3(X, ._22(H, T1), ._22(H, T2))
if_reach_4_in_4_gggg6(X, Z, Edges, Not_Visited, Y, delete_3_out_gga3(Y, Not_Visited, V1)) -> if_reach_4_in_5_gggg7(X, Z, Edges, Not_Visited, Y, V1, reach_4_in_gggg4(Y, Z, Edges, V1))
if_reach_4_in_5_gggg7(X, Z, Edges, Not_Visited, Y, V1, reach_4_out_gggg4(Y, Z, Edges, V1)) -> reach_4_out_gggg4(X, Z, Edges, Not_Visited)

The argument filtering Pi contains the following mapping:
reach_4_in_gggg4(x1, x2, x3, x4)  =  reach_4_in_gggg4(x1, x2, x3, x4)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_4_in_1_gggg5(x1, x2, x3, x4, x5)  =  if_reach_4_in_1_gggg1(x5)
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_4_out_gggg4(x1, x2, x3, x4)  =  reach_4_out_gggg
if_reach_4_in_2_gggg5(x1, x2, x3, x4, x5)  =  if_reach_4_in_2_gggg4(x2, x3, x4, x5)
member_2_in_ag2(x1, x2)  =  member_2_in_ag1(x2)
member_2_out_ag2(x1, x2)  =  member_2_out_ag1(x1)
if_member_2_in_1_ag4(x1, x2, x3, x4)  =  if_member_2_in_1_ag1(x4)
if_reach_4_in_3_gggg6(x1, x2, x3, x4, x5, x6)  =  if_reach_4_in_3_gggg5(x2, x3, x4, x5, x6)
if_reach_4_in_4_gggg6(x1, x2, x3, x4, x5, x6)  =  if_reach_4_in_4_gggg4(x2, x3, x5, x6)
delete_3_in_gga3(x1, x2, x3)  =  delete_3_in_gga2(x1, x2)
delete_3_out_gga3(x1, x2, x3)  =  delete_3_out_gga1(x3)
if_delete_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_delete_3_in_1_gga2(x2, x5)
if_reach_4_in_5_gggg7(x1, x2, x3, x4, x5, x6, x7)  =  if_reach_4_in_5_gggg1(x7)
MEMBER_2_IN_AG2(x1, x2)  =  MEMBER_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
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP

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

MEMBER_2_IN_AG2(X, ._22(H, L)) -> MEMBER_2_IN_AG2(X, L)

R is empty.
The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._22(x1, x2)
MEMBER_2_IN_AG2(x1, x2)  =  MEMBER_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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP

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

MEMBER_2_IN_AG1(._22(H, L)) -> MEMBER_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 {MEMBER_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
PiDP
                ↳ UsableRulesProof
              ↳ 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_4_in_gggg4(X, Y, Edges, Not_Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Not_Visited, 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_4_in_1_gggg5(X, Y, Edges, Not_Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Not_Visited)
reach_4_in_gggg4(X, Z, Edges, Not_Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Not_Visited, member_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_ag2(H, ._22(H, L)) -> member_2_out_ag2(H, ._22(H, L))
member_2_in_ag2(X, ._22(H, L)) -> if_member_2_in_1_ag4(X, H, L, member_2_in_ag2(X, L))
if_member_2_in_1_ag4(X, H, L, member_2_out_ag2(X, L)) -> member_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Not_Visited, member_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Not_Visited, Y, member_2_in_gg2(Y, Not_Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Not_Visited, Y, member_2_out_gg2(Y, Not_Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Not_Visited, Y, delete_3_in_gga3(Y, Not_Visited, V1))
delete_3_in_gga3(X, ._22(X, Y), Y) -> delete_3_out_gga3(X, ._22(X, Y), Y)
delete_3_in_gga3(X, ._22(H, T1), ._22(H, T2)) -> if_delete_3_in_1_gga5(X, H, T1, T2, delete_3_in_gga3(X, T1, T2))
if_delete_3_in_1_gga5(X, H, T1, T2, delete_3_out_gga3(X, T1, T2)) -> delete_3_out_gga3(X, ._22(H, T1), ._22(H, T2))
if_reach_4_in_4_gggg6(X, Z, Edges, Not_Visited, Y, delete_3_out_gga3(Y, Not_Visited, V1)) -> if_reach_4_in_5_gggg7(X, Z, Edges, Not_Visited, Y, V1, reach_4_in_gggg4(Y, Z, Edges, V1))
if_reach_4_in_5_gggg7(X, Z, Edges, Not_Visited, Y, V1, reach_4_out_gggg4(Y, Z, Edges, V1)) -> reach_4_out_gggg4(X, Z, Edges, Not_Visited)

The argument filtering Pi contains the following mapping:
reach_4_in_gggg4(x1, x2, x3, x4)  =  reach_4_in_gggg4(x1, x2, x3, x4)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_4_in_1_gggg5(x1, x2, x3, x4, x5)  =  if_reach_4_in_1_gggg1(x5)
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_4_out_gggg4(x1, x2, x3, x4)  =  reach_4_out_gggg
if_reach_4_in_2_gggg5(x1, x2, x3, x4, x5)  =  if_reach_4_in_2_gggg4(x2, x3, x4, x5)
member_2_in_ag2(x1, x2)  =  member_2_in_ag1(x2)
member_2_out_ag2(x1, x2)  =  member_2_out_ag1(x1)
if_member_2_in_1_ag4(x1, x2, x3, x4)  =  if_member_2_in_1_ag1(x4)
if_reach_4_in_3_gggg6(x1, x2, x3, x4, x5, x6)  =  if_reach_4_in_3_gggg5(x2, x3, x4, x5, x6)
if_reach_4_in_4_gggg6(x1, x2, x3, x4, x5, x6)  =  if_reach_4_in_4_gggg4(x2, x3, x5, x6)
delete_3_in_gga3(x1, x2, x3)  =  delete_3_in_gga2(x1, x2)
delete_3_out_gga3(x1, x2, x3)  =  delete_3_out_gga1(x3)
if_delete_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_delete_3_in_1_gga2(x2, x5)
if_reach_4_in_5_gggg7(x1, x2, x3, x4, x5, x6, x7)  =  if_reach_4_in_5_gggg1(x7)
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
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
              ↳ 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)

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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP

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
PiDP
                ↳ UsableRulesProof

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

IF_REACH_4_IN_4_GGGG6(X, Z, Edges, Not_Visited, Y, delete_3_out_gga3(Y, Not_Visited, V1)) -> REACH_4_IN_GGGG4(Y, Z, Edges, V1)
IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Not_Visited, Y, member_2_out_gg2(Y, Not_Visited)) -> IF_REACH_4_IN_4_GGGG6(X, Z, Edges, Not_Visited, Y, delete_3_in_gga3(Y, Not_Visited, V1))
REACH_4_IN_GGGG4(X, Z, Edges, Not_Visited) -> IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Not_Visited, member_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Not_Visited, member_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Not_Visited, Y, member_2_in_gg2(Y, Not_Visited))

The TRS R consists of the following rules:

reach_4_in_gggg4(X, Y, Edges, Not_Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Not_Visited, 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_4_in_1_gggg5(X, Y, Edges, Not_Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Not_Visited)
reach_4_in_gggg4(X, Z, Edges, Not_Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Not_Visited, member_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_ag2(H, ._22(H, L)) -> member_2_out_ag2(H, ._22(H, L))
member_2_in_ag2(X, ._22(H, L)) -> if_member_2_in_1_ag4(X, H, L, member_2_in_ag2(X, L))
if_member_2_in_1_ag4(X, H, L, member_2_out_ag2(X, L)) -> member_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Not_Visited, member_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Not_Visited, Y, member_2_in_gg2(Y, Not_Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Not_Visited, Y, member_2_out_gg2(Y, Not_Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Not_Visited, Y, delete_3_in_gga3(Y, Not_Visited, V1))
delete_3_in_gga3(X, ._22(X, Y), Y) -> delete_3_out_gga3(X, ._22(X, Y), Y)
delete_3_in_gga3(X, ._22(H, T1), ._22(H, T2)) -> if_delete_3_in_1_gga5(X, H, T1, T2, delete_3_in_gga3(X, T1, T2))
if_delete_3_in_1_gga5(X, H, T1, T2, delete_3_out_gga3(X, T1, T2)) -> delete_3_out_gga3(X, ._22(H, T1), ._22(H, T2))
if_reach_4_in_4_gggg6(X, Z, Edges, Not_Visited, Y, delete_3_out_gga3(Y, Not_Visited, V1)) -> if_reach_4_in_5_gggg7(X, Z, Edges, Not_Visited, Y, V1, reach_4_in_gggg4(Y, Z, Edges, V1))
if_reach_4_in_5_gggg7(X, Z, Edges, Not_Visited, Y, V1, reach_4_out_gggg4(Y, Z, Edges, V1)) -> reach_4_out_gggg4(X, Z, Edges, Not_Visited)

The argument filtering Pi contains the following mapping:
reach_4_in_gggg4(x1, x2, x3, x4)  =  reach_4_in_gggg4(x1, x2, x3, x4)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_reach_4_in_1_gggg5(x1, x2, x3, x4, x5)  =  if_reach_4_in_1_gggg1(x5)
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_4_out_gggg4(x1, x2, x3, x4)  =  reach_4_out_gggg
if_reach_4_in_2_gggg5(x1, x2, x3, x4, x5)  =  if_reach_4_in_2_gggg4(x2, x3, x4, x5)
member_2_in_ag2(x1, x2)  =  member_2_in_ag1(x2)
member_2_out_ag2(x1, x2)  =  member_2_out_ag1(x1)
if_member_2_in_1_ag4(x1, x2, x3, x4)  =  if_member_2_in_1_ag1(x4)
if_reach_4_in_3_gggg6(x1, x2, x3, x4, x5, x6)  =  if_reach_4_in_3_gggg5(x2, x3, x4, x5, x6)
if_reach_4_in_4_gggg6(x1, x2, x3, x4, x5, x6)  =  if_reach_4_in_4_gggg4(x2, x3, x5, x6)
delete_3_in_gga3(x1, x2, x3)  =  delete_3_in_gga2(x1, x2)
delete_3_out_gga3(x1, x2, x3)  =  delete_3_out_gga1(x3)
if_delete_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_delete_3_in_1_gga2(x2, x5)
if_reach_4_in_5_gggg7(x1, x2, x3, x4, x5, x6, x7)  =  if_reach_4_in_5_gggg1(x7)
IF_REACH_4_IN_4_GGGG6(x1, x2, x3, x4, x5, x6)  =  IF_REACH_4_IN_4_GGGG4(x2, x3, x5, x6)
REACH_4_IN_GGGG4(x1, x2, x3, x4)  =  REACH_4_IN_GGGG4(x1, x2, x3, x4)
IF_REACH_4_IN_2_GGGG5(x1, x2, x3, x4, x5)  =  IF_REACH_4_IN_2_GGGG4(x2, x3, x4, x5)
IF_REACH_4_IN_3_GGGG6(x1, x2, x3, x4, x5, x6)  =  IF_REACH_4_IN_3_GGGG5(x2, x3, x4, x5, x6)

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
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof

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

IF_REACH_4_IN_4_GGGG6(X, Z, Edges, Not_Visited, Y, delete_3_out_gga3(Y, Not_Visited, V1)) -> REACH_4_IN_GGGG4(Y, Z, Edges, V1)
IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Not_Visited, Y, member_2_out_gg2(Y, Not_Visited)) -> IF_REACH_4_IN_4_GGGG6(X, Z, Edges, Not_Visited, Y, delete_3_in_gga3(Y, Not_Visited, V1))
REACH_4_IN_GGGG4(X, Z, Edges, Not_Visited) -> IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Not_Visited, member_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Not_Visited, member_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Not_Visited, Y, member_2_in_gg2(Y, Not_Visited))

The TRS R consists of the following rules:

delete_3_in_gga3(X, ._22(X, Y), Y) -> delete_3_out_gga3(X, ._22(X, Y), Y)
delete_3_in_gga3(X, ._22(H, T1), ._22(H, T2)) -> if_delete_3_in_1_gga5(X, H, T1, T2, delete_3_in_gga3(X, T1, T2))
member_2_in_ag2(H, ._22(H, L)) -> member_2_out_ag2(H, ._22(H, L))
member_2_in_ag2(X, ._22(H, L)) -> if_member_2_in_1_ag4(X, H, L, member_2_in_ag2(X, L))
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_delete_3_in_1_gga5(X, H, T1, T2, delete_3_out_gga3(X, T1, T2)) -> delete_3_out_gga3(X, ._22(H, T1), ._22(H, T2))
if_member_2_in_1_ag4(X, H, L, member_2_out_ag2(X, L)) -> member_2_out_ag2(X, ._22(H, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))

The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
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)
member_2_in_ag2(x1, x2)  =  member_2_in_ag1(x2)
member_2_out_ag2(x1, x2)  =  member_2_out_ag1(x1)
if_member_2_in_1_ag4(x1, x2, x3, x4)  =  if_member_2_in_1_ag1(x4)
delete_3_in_gga3(x1, x2, x3)  =  delete_3_in_gga2(x1, x2)
delete_3_out_gga3(x1, x2, x3)  =  delete_3_out_gga1(x3)
if_delete_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_delete_3_in_1_gga2(x2, x5)
IF_REACH_4_IN_4_GGGG6(x1, x2, x3, x4, x5, x6)  =  IF_REACH_4_IN_4_GGGG4(x2, x3, x5, x6)
REACH_4_IN_GGGG4(x1, x2, x3, x4)  =  REACH_4_IN_GGGG4(x1, x2, x3, x4)
IF_REACH_4_IN_2_GGGG5(x1, x2, x3, x4, x5)  =  IF_REACH_4_IN_2_GGGG4(x2, x3, x4, x5)
IF_REACH_4_IN_3_GGGG6(x1, x2, x3, x4, x5, x6)  =  IF_REACH_4_IN_3_GGGG5(x2, x3, x4, x5, x6)

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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ QDPSizeChangeProof

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

IF_REACH_4_IN_4_GGGG4(Z, Edges, Y, delete_3_out_gga1(V1)) -> REACH_4_IN_GGGG4(Y, Z, Edges, V1)
IF_REACH_4_IN_3_GGGG5(Z, Edges, Not_Visited, Y, member_2_out_gg) -> IF_REACH_4_IN_4_GGGG4(Z, Edges, Y, delete_3_in_gga2(Y, Not_Visited))
REACH_4_IN_GGGG4(X, Z, Edges, Not_Visited) -> IF_REACH_4_IN_2_GGGG4(Z, Edges, Not_Visited, member_2_in_ag1(Edges))
IF_REACH_4_IN_2_GGGG4(Z, Edges, Not_Visited, member_2_out_ag1(._22(X, ._22(Y, []_0)))) -> IF_REACH_4_IN_3_GGGG5(Z, Edges, Not_Visited, Y, member_2_in_gg2(Y, Not_Visited))

The TRS R consists of the following rules:

delete_3_in_gga2(X, ._22(X, Y)) -> delete_3_out_gga1(Y)
delete_3_in_gga2(X, ._22(H, T1)) -> if_delete_3_in_1_gga2(H, delete_3_in_gga2(X, T1))
member_2_in_ag1(._22(H, L)) -> member_2_out_ag1(H)
member_2_in_ag1(._22(H, L)) -> if_member_2_in_1_ag1(member_2_in_ag1(L))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg1(member_2_in_gg2(X, L))
if_delete_3_in_1_gga2(H, delete_3_out_gga1(T2)) -> delete_3_out_gga1(._22(H, T2))
if_member_2_in_1_ag1(member_2_out_ag1(X)) -> member_2_out_ag1(X)
if_member_2_in_1_gg1(member_2_out_gg) -> member_2_out_gg

The set Q consists of the following terms:

delete_3_in_gga2(x0, x1)
member_2_in_ag1(x0)
member_2_in_gg2(x0, x1)
if_delete_3_in_1_gga2(x0, x1)
if_member_2_in_1_ag1(x0)
if_member_2_in_1_gg1(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {REACH_4_IN_GGGG4, IF_REACH_4_IN_4_GGGG4, IF_REACH_4_IN_3_GGGG5, IF_REACH_4_IN_2_GGGG4}.
We used the following order together with the size-change analysis to show that there are no infinite chains for this DP problem.

Order:Polynomial interpretation:


POL(if_delete_3_in_1_gga2(x1, x2)) = 1 + x2   
POL(._22(x1, x2)) = 1 + x2   
POL([]_0) = 1   
POL(member_2_out_gg) = 1   
POL(member_2_out_ag1(x1)) = 1   
POL(delete_3_in_gga2(x1, x2)) = x2   
POL(delete_3_out_gga1(x1)) = 1 + x1   

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

We oriented the following set of usable rules.


if_delete_3_in_1_gga2(H, delete_3_out_gga1(T2)) -> delete_3_out_gga1(._22(H, T2))
delete_3_in_gga2(X, ._22(X, Y)) -> delete_3_out_gga1(Y)
delete_3_in_gga2(X, ._22(H, T1)) -> if_delete_3_in_1_gga2(H, delete_3_in_gga2(X, T1))