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



↳ PROLOG
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof

sublist2(X, Y) :- append3(U, X, V), append3(V, W, Y).
append3({}0, Ys, Ys).
append3(.2(X, Xs), Ys, .2(X, Zs)) :- append3(Xs, Ys, Zs).


With regard to the inferred argument filtering the predicates were used in the following modes:
sublist2: (b,f)
append3: (f,b,f) (b,f,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:


sublist_2_in_ga2(X, Y) -> if_sublist_2_in_1_ga3(X, Y, append_3_in_aga3(U, X, V))
append_3_in_aga3([]_0, Ys, Ys) -> append_3_out_aga3([]_0, Ys, Ys)
append_3_in_aga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_in_aga3(Xs, Ys, Zs))
if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_out_aga3(Xs, Ys, Zs)) -> append_3_out_aga3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_1_ga3(X, Y, append_3_out_aga3(U, X, V)) -> if_sublist_2_in_2_ga4(X, Y, V, append_3_in_gaa3(V, W, Y))
append_3_in_gaa3([]_0, Ys, Ys) -> append_3_out_gaa3([]_0, Ys, Ys)
append_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_in_gaa3(Xs, Ys, Zs))
if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_out_gaa3(Xs, Ys, Zs)) -> append_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_2_ga4(X, Y, V, append_3_out_gaa3(V, W, Y)) -> sublist_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
sublist_2_in_ga2(x1, x2)  =  sublist_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
if_sublist_2_in_1_ga3(x1, x2, x3)  =  if_sublist_2_in_1_ga1(x3)
append_3_in_aga3(x1, x2, x3)  =  append_3_in_aga1(x2)
append_3_out_aga3(x1, x2, x3)  =  append_3_out_aga2(x1, x3)
if_append_3_in_1_aga5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_aga1(x5)
if_sublist_2_in_2_ga4(x1, x2, x3, x4)  =  if_sublist_2_in_2_ga1(x4)
append_3_in_gaa3(x1, x2, x3)  =  append_3_in_gaa1(x1)
append_3_out_gaa3(x1, x2, x3)  =  append_3_out_gaa
if_append_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_gaa1(x5)
sublist_2_out_ga2(x1, x2)  =  sublist_2_out_ga

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:

sublist_2_in_ga2(X, Y) -> if_sublist_2_in_1_ga3(X, Y, append_3_in_aga3(U, X, V))
append_3_in_aga3([]_0, Ys, Ys) -> append_3_out_aga3([]_0, Ys, Ys)
append_3_in_aga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_in_aga3(Xs, Ys, Zs))
if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_out_aga3(Xs, Ys, Zs)) -> append_3_out_aga3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_1_ga3(X, Y, append_3_out_aga3(U, X, V)) -> if_sublist_2_in_2_ga4(X, Y, V, append_3_in_gaa3(V, W, Y))
append_3_in_gaa3([]_0, Ys, Ys) -> append_3_out_gaa3([]_0, Ys, Ys)
append_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_in_gaa3(Xs, Ys, Zs))
if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_out_gaa3(Xs, Ys, Zs)) -> append_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_2_ga4(X, Y, V, append_3_out_gaa3(V, W, Y)) -> sublist_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
sublist_2_in_ga2(x1, x2)  =  sublist_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
if_sublist_2_in_1_ga3(x1, x2, x3)  =  if_sublist_2_in_1_ga1(x3)
append_3_in_aga3(x1, x2, x3)  =  append_3_in_aga1(x2)
append_3_out_aga3(x1, x2, x3)  =  append_3_out_aga2(x1, x3)
if_append_3_in_1_aga5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_aga1(x5)
if_sublist_2_in_2_ga4(x1, x2, x3, x4)  =  if_sublist_2_in_2_ga1(x4)
append_3_in_gaa3(x1, x2, x3)  =  append_3_in_gaa1(x1)
append_3_out_gaa3(x1, x2, x3)  =  append_3_out_gaa
if_append_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_gaa1(x5)
sublist_2_out_ga2(x1, x2)  =  sublist_2_out_ga


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

SUBLIST_2_IN_GA2(X, Y) -> IF_SUBLIST_2_IN_1_GA3(X, Y, append_3_in_aga3(U, X, V))
SUBLIST_2_IN_GA2(X, Y) -> APPEND_3_IN_AGA3(U, X, V)
APPEND_3_IN_AGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APPEND_3_IN_1_AGA5(X, Xs, Ys, Zs, append_3_in_aga3(Xs, Ys, Zs))
APPEND_3_IN_AGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND_3_IN_AGA3(Xs, Ys, Zs)
IF_SUBLIST_2_IN_1_GA3(X, Y, append_3_out_aga3(U, X, V)) -> IF_SUBLIST_2_IN_2_GA4(X, Y, V, append_3_in_gaa3(V, W, Y))
IF_SUBLIST_2_IN_1_GA3(X, Y, append_3_out_aga3(U, X, V)) -> APPEND_3_IN_GAA3(V, W, Y)
APPEND_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APPEND_3_IN_1_GAA5(X, Xs, Ys, Zs, append_3_in_gaa3(Xs, Ys, Zs))
APPEND_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND_3_IN_GAA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_2_in_ga2(X, Y) -> if_sublist_2_in_1_ga3(X, Y, append_3_in_aga3(U, X, V))
append_3_in_aga3([]_0, Ys, Ys) -> append_3_out_aga3([]_0, Ys, Ys)
append_3_in_aga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_in_aga3(Xs, Ys, Zs))
if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_out_aga3(Xs, Ys, Zs)) -> append_3_out_aga3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_1_ga3(X, Y, append_3_out_aga3(U, X, V)) -> if_sublist_2_in_2_ga4(X, Y, V, append_3_in_gaa3(V, W, Y))
append_3_in_gaa3([]_0, Ys, Ys) -> append_3_out_gaa3([]_0, Ys, Ys)
append_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_in_gaa3(Xs, Ys, Zs))
if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_out_gaa3(Xs, Ys, Zs)) -> append_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_2_ga4(X, Y, V, append_3_out_gaa3(V, W, Y)) -> sublist_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
sublist_2_in_ga2(x1, x2)  =  sublist_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
if_sublist_2_in_1_ga3(x1, x2, x3)  =  if_sublist_2_in_1_ga1(x3)
append_3_in_aga3(x1, x2, x3)  =  append_3_in_aga1(x2)
append_3_out_aga3(x1, x2, x3)  =  append_3_out_aga2(x1, x3)
if_append_3_in_1_aga5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_aga1(x5)
if_sublist_2_in_2_ga4(x1, x2, x3, x4)  =  if_sublist_2_in_2_ga1(x4)
append_3_in_gaa3(x1, x2, x3)  =  append_3_in_gaa1(x1)
append_3_out_gaa3(x1, x2, x3)  =  append_3_out_gaa
if_append_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_gaa1(x5)
sublist_2_out_ga2(x1, x2)  =  sublist_2_out_ga
IF_APPEND_3_IN_1_AGA5(x1, x2, x3, x4, x5)  =  IF_APPEND_3_IN_1_AGA1(x5)
SUBLIST_2_IN_GA2(x1, x2)  =  SUBLIST_2_IN_GA1(x1)
APPEND_3_IN_GAA3(x1, x2, x3)  =  APPEND_3_IN_GAA1(x1)
APPEND_3_IN_AGA3(x1, x2, x3)  =  APPEND_3_IN_AGA1(x2)
IF_SUBLIST_2_IN_2_GA4(x1, x2, x3, x4)  =  IF_SUBLIST_2_IN_2_GA1(x4)
IF_APPEND_3_IN_1_GAA5(x1, x2, x3, x4, x5)  =  IF_APPEND_3_IN_1_GAA1(x5)
IF_SUBLIST_2_IN_1_GA3(x1, x2, x3)  =  IF_SUBLIST_2_IN_1_GA1(x3)

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:

SUBLIST_2_IN_GA2(X, Y) -> IF_SUBLIST_2_IN_1_GA3(X, Y, append_3_in_aga3(U, X, V))
SUBLIST_2_IN_GA2(X, Y) -> APPEND_3_IN_AGA3(U, X, V)
APPEND_3_IN_AGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APPEND_3_IN_1_AGA5(X, Xs, Ys, Zs, append_3_in_aga3(Xs, Ys, Zs))
APPEND_3_IN_AGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND_3_IN_AGA3(Xs, Ys, Zs)
IF_SUBLIST_2_IN_1_GA3(X, Y, append_3_out_aga3(U, X, V)) -> IF_SUBLIST_2_IN_2_GA4(X, Y, V, append_3_in_gaa3(V, W, Y))
IF_SUBLIST_2_IN_1_GA3(X, Y, append_3_out_aga3(U, X, V)) -> APPEND_3_IN_GAA3(V, W, Y)
APPEND_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APPEND_3_IN_1_GAA5(X, Xs, Ys, Zs, append_3_in_gaa3(Xs, Ys, Zs))
APPEND_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND_3_IN_GAA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_2_in_ga2(X, Y) -> if_sublist_2_in_1_ga3(X, Y, append_3_in_aga3(U, X, V))
append_3_in_aga3([]_0, Ys, Ys) -> append_3_out_aga3([]_0, Ys, Ys)
append_3_in_aga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_in_aga3(Xs, Ys, Zs))
if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_out_aga3(Xs, Ys, Zs)) -> append_3_out_aga3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_1_ga3(X, Y, append_3_out_aga3(U, X, V)) -> if_sublist_2_in_2_ga4(X, Y, V, append_3_in_gaa3(V, W, Y))
append_3_in_gaa3([]_0, Ys, Ys) -> append_3_out_gaa3([]_0, Ys, Ys)
append_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_in_gaa3(Xs, Ys, Zs))
if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_out_gaa3(Xs, Ys, Zs)) -> append_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_2_ga4(X, Y, V, append_3_out_gaa3(V, W, Y)) -> sublist_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
sublist_2_in_ga2(x1, x2)  =  sublist_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
if_sublist_2_in_1_ga3(x1, x2, x3)  =  if_sublist_2_in_1_ga1(x3)
append_3_in_aga3(x1, x2, x3)  =  append_3_in_aga1(x2)
append_3_out_aga3(x1, x2, x3)  =  append_3_out_aga2(x1, x3)
if_append_3_in_1_aga5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_aga1(x5)
if_sublist_2_in_2_ga4(x1, x2, x3, x4)  =  if_sublist_2_in_2_ga1(x4)
append_3_in_gaa3(x1, x2, x3)  =  append_3_in_gaa1(x1)
append_3_out_gaa3(x1, x2, x3)  =  append_3_out_gaa
if_append_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_gaa1(x5)
sublist_2_out_ga2(x1, x2)  =  sublist_2_out_ga
IF_APPEND_3_IN_1_AGA5(x1, x2, x3, x4, x5)  =  IF_APPEND_3_IN_1_AGA1(x5)
SUBLIST_2_IN_GA2(x1, x2)  =  SUBLIST_2_IN_GA1(x1)
APPEND_3_IN_GAA3(x1, x2, x3)  =  APPEND_3_IN_GAA1(x1)
APPEND_3_IN_AGA3(x1, x2, x3)  =  APPEND_3_IN_AGA1(x2)
IF_SUBLIST_2_IN_2_GA4(x1, x2, x3, x4)  =  IF_SUBLIST_2_IN_2_GA1(x4)
IF_APPEND_3_IN_1_GAA5(x1, x2, x3, x4, x5)  =  IF_APPEND_3_IN_1_GAA1(x5)
IF_SUBLIST_2_IN_1_GA3(x1, x2, x3)  =  IF_SUBLIST_2_IN_1_GA1(x3)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 2 SCCs with 6 less nodes.

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

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

APPEND_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND_3_IN_GAA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_2_in_ga2(X, Y) -> if_sublist_2_in_1_ga3(X, Y, append_3_in_aga3(U, X, V))
append_3_in_aga3([]_0, Ys, Ys) -> append_3_out_aga3([]_0, Ys, Ys)
append_3_in_aga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_in_aga3(Xs, Ys, Zs))
if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_out_aga3(Xs, Ys, Zs)) -> append_3_out_aga3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_1_ga3(X, Y, append_3_out_aga3(U, X, V)) -> if_sublist_2_in_2_ga4(X, Y, V, append_3_in_gaa3(V, W, Y))
append_3_in_gaa3([]_0, Ys, Ys) -> append_3_out_gaa3([]_0, Ys, Ys)
append_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_in_gaa3(Xs, Ys, Zs))
if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_out_gaa3(Xs, Ys, Zs)) -> append_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_2_ga4(X, Y, V, append_3_out_gaa3(V, W, Y)) -> sublist_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
sublist_2_in_ga2(x1, x2)  =  sublist_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
if_sublist_2_in_1_ga3(x1, x2, x3)  =  if_sublist_2_in_1_ga1(x3)
append_3_in_aga3(x1, x2, x3)  =  append_3_in_aga1(x2)
append_3_out_aga3(x1, x2, x3)  =  append_3_out_aga2(x1, x3)
if_append_3_in_1_aga5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_aga1(x5)
if_sublist_2_in_2_ga4(x1, x2, x3, x4)  =  if_sublist_2_in_2_ga1(x4)
append_3_in_gaa3(x1, x2, x3)  =  append_3_in_gaa1(x1)
append_3_out_gaa3(x1, x2, x3)  =  append_3_out_gaa
if_append_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_gaa1(x5)
sublist_2_out_ga2(x1, x2)  =  sublist_2_out_ga
APPEND_3_IN_GAA3(x1, x2, x3)  =  APPEND_3_IN_GAA1(x1)

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
  ↳ PrologToPiTRSProof

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

APPEND_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND_3_IN_GAA3(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._21(x2)
APPEND_3_IN_GAA3(x1, x2, x3)  =  APPEND_3_IN_GAA1(x1)

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
  ↳ PrologToPiTRSProof

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

APPEND_3_IN_GAA1(._21(Xs)) -> APPEND_3_IN_GAA1(Xs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {APPEND_3_IN_GAA1}.
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
  ↳ PrologToPiTRSProof

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

APPEND_3_IN_AGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND_3_IN_AGA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_2_in_ga2(X, Y) -> if_sublist_2_in_1_ga3(X, Y, append_3_in_aga3(U, X, V))
append_3_in_aga3([]_0, Ys, Ys) -> append_3_out_aga3([]_0, Ys, Ys)
append_3_in_aga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_in_aga3(Xs, Ys, Zs))
if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_out_aga3(Xs, Ys, Zs)) -> append_3_out_aga3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_1_ga3(X, Y, append_3_out_aga3(U, X, V)) -> if_sublist_2_in_2_ga4(X, Y, V, append_3_in_gaa3(V, W, Y))
append_3_in_gaa3([]_0, Ys, Ys) -> append_3_out_gaa3([]_0, Ys, Ys)
append_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_in_gaa3(Xs, Ys, Zs))
if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_out_gaa3(Xs, Ys, Zs)) -> append_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_2_ga4(X, Y, V, append_3_out_gaa3(V, W, Y)) -> sublist_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
sublist_2_in_ga2(x1, x2)  =  sublist_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
if_sublist_2_in_1_ga3(x1, x2, x3)  =  if_sublist_2_in_1_ga1(x3)
append_3_in_aga3(x1, x2, x3)  =  append_3_in_aga1(x2)
append_3_out_aga3(x1, x2, x3)  =  append_3_out_aga2(x1, x3)
if_append_3_in_1_aga5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_aga1(x5)
if_sublist_2_in_2_ga4(x1, x2, x3, x4)  =  if_sublist_2_in_2_ga1(x4)
append_3_in_gaa3(x1, x2, x3)  =  append_3_in_gaa1(x1)
append_3_out_gaa3(x1, x2, x3)  =  append_3_out_gaa
if_append_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_gaa1(x5)
sublist_2_out_ga2(x1, x2)  =  sublist_2_out_ga
APPEND_3_IN_AGA3(x1, x2, x3)  =  APPEND_3_IN_AGA1(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
  ↳ PrologToPiTRSProof

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

APPEND_3_IN_AGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND_3_IN_AGA3(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._21(x2)
APPEND_3_IN_AGA3(x1, x2, x3)  =  APPEND_3_IN_AGA1(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
  ↳ PrologToPiTRSProof

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

APPEND_3_IN_AGA1(Ys) -> APPEND_3_IN_AGA1(Ys)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {APPEND_3_IN_AGA1}.
With regard to the inferred argument filtering the predicates were used in the following modes:
sublist2: (b,f)
append3: (f,b,f) (b,f,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

sublist_2_in_ga2(X, Y) -> if_sublist_2_in_1_ga3(X, Y, append_3_in_aga3(U, X, V))
append_3_in_aga3([]_0, Ys, Ys) -> append_3_out_aga3([]_0, Ys, Ys)
append_3_in_aga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_in_aga3(Xs, Ys, Zs))
if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_out_aga3(Xs, Ys, Zs)) -> append_3_out_aga3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_1_ga3(X, Y, append_3_out_aga3(U, X, V)) -> if_sublist_2_in_2_ga4(X, Y, V, append_3_in_gaa3(V, W, Y))
append_3_in_gaa3([]_0, Ys, Ys) -> append_3_out_gaa3([]_0, Ys, Ys)
append_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_in_gaa3(Xs, Ys, Zs))
if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_out_gaa3(Xs, Ys, Zs)) -> append_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_2_ga4(X, Y, V, append_3_out_gaa3(V, W, Y)) -> sublist_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
sublist_2_in_ga2(x1, x2)  =  sublist_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
if_sublist_2_in_1_ga3(x1, x2, x3)  =  if_sublist_2_in_1_ga2(x1, x3)
append_3_in_aga3(x1, x2, x3)  =  append_3_in_aga1(x2)
append_3_out_aga3(x1, x2, x3)  =  append_3_out_aga3(x1, x2, x3)
if_append_3_in_1_aga5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_aga2(x3, x5)
if_sublist_2_in_2_ga4(x1, x2, x3, x4)  =  if_sublist_2_in_2_ga2(x1, x4)
append_3_in_gaa3(x1, x2, x3)  =  append_3_in_gaa1(x1)
append_3_out_gaa3(x1, x2, x3)  =  append_3_out_gaa1(x1)
if_append_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_gaa2(x2, x5)
sublist_2_out_ga2(x1, x2)  =  sublist_2_out_ga1(x1)

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:

sublist_2_in_ga2(X, Y) -> if_sublist_2_in_1_ga3(X, Y, append_3_in_aga3(U, X, V))
append_3_in_aga3([]_0, Ys, Ys) -> append_3_out_aga3([]_0, Ys, Ys)
append_3_in_aga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_in_aga3(Xs, Ys, Zs))
if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_out_aga3(Xs, Ys, Zs)) -> append_3_out_aga3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_1_ga3(X, Y, append_3_out_aga3(U, X, V)) -> if_sublist_2_in_2_ga4(X, Y, V, append_3_in_gaa3(V, W, Y))
append_3_in_gaa3([]_0, Ys, Ys) -> append_3_out_gaa3([]_0, Ys, Ys)
append_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_in_gaa3(Xs, Ys, Zs))
if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_out_gaa3(Xs, Ys, Zs)) -> append_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_2_ga4(X, Y, V, append_3_out_gaa3(V, W, Y)) -> sublist_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
sublist_2_in_ga2(x1, x2)  =  sublist_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
if_sublist_2_in_1_ga3(x1, x2, x3)  =  if_sublist_2_in_1_ga2(x1, x3)
append_3_in_aga3(x1, x2, x3)  =  append_3_in_aga1(x2)
append_3_out_aga3(x1, x2, x3)  =  append_3_out_aga3(x1, x2, x3)
if_append_3_in_1_aga5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_aga2(x3, x5)
if_sublist_2_in_2_ga4(x1, x2, x3, x4)  =  if_sublist_2_in_2_ga2(x1, x4)
append_3_in_gaa3(x1, x2, x3)  =  append_3_in_gaa1(x1)
append_3_out_gaa3(x1, x2, x3)  =  append_3_out_gaa1(x1)
if_append_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_gaa2(x2, x5)
sublist_2_out_ga2(x1, x2)  =  sublist_2_out_ga1(x1)


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

SUBLIST_2_IN_GA2(X, Y) -> IF_SUBLIST_2_IN_1_GA3(X, Y, append_3_in_aga3(U, X, V))
SUBLIST_2_IN_GA2(X, Y) -> APPEND_3_IN_AGA3(U, X, V)
APPEND_3_IN_AGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APPEND_3_IN_1_AGA5(X, Xs, Ys, Zs, append_3_in_aga3(Xs, Ys, Zs))
APPEND_3_IN_AGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND_3_IN_AGA3(Xs, Ys, Zs)
IF_SUBLIST_2_IN_1_GA3(X, Y, append_3_out_aga3(U, X, V)) -> IF_SUBLIST_2_IN_2_GA4(X, Y, V, append_3_in_gaa3(V, W, Y))
IF_SUBLIST_2_IN_1_GA3(X, Y, append_3_out_aga3(U, X, V)) -> APPEND_3_IN_GAA3(V, W, Y)
APPEND_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APPEND_3_IN_1_GAA5(X, Xs, Ys, Zs, append_3_in_gaa3(Xs, Ys, Zs))
APPEND_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND_3_IN_GAA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_2_in_ga2(X, Y) -> if_sublist_2_in_1_ga3(X, Y, append_3_in_aga3(U, X, V))
append_3_in_aga3([]_0, Ys, Ys) -> append_3_out_aga3([]_0, Ys, Ys)
append_3_in_aga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_in_aga3(Xs, Ys, Zs))
if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_out_aga3(Xs, Ys, Zs)) -> append_3_out_aga3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_1_ga3(X, Y, append_3_out_aga3(U, X, V)) -> if_sublist_2_in_2_ga4(X, Y, V, append_3_in_gaa3(V, W, Y))
append_3_in_gaa3([]_0, Ys, Ys) -> append_3_out_gaa3([]_0, Ys, Ys)
append_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_in_gaa3(Xs, Ys, Zs))
if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_out_gaa3(Xs, Ys, Zs)) -> append_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_2_ga4(X, Y, V, append_3_out_gaa3(V, W, Y)) -> sublist_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
sublist_2_in_ga2(x1, x2)  =  sublist_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
if_sublist_2_in_1_ga3(x1, x2, x3)  =  if_sublist_2_in_1_ga2(x1, x3)
append_3_in_aga3(x1, x2, x3)  =  append_3_in_aga1(x2)
append_3_out_aga3(x1, x2, x3)  =  append_3_out_aga3(x1, x2, x3)
if_append_3_in_1_aga5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_aga2(x3, x5)
if_sublist_2_in_2_ga4(x1, x2, x3, x4)  =  if_sublist_2_in_2_ga2(x1, x4)
append_3_in_gaa3(x1, x2, x3)  =  append_3_in_gaa1(x1)
append_3_out_gaa3(x1, x2, x3)  =  append_3_out_gaa1(x1)
if_append_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_gaa2(x2, x5)
sublist_2_out_ga2(x1, x2)  =  sublist_2_out_ga1(x1)
IF_APPEND_3_IN_1_AGA5(x1, x2, x3, x4, x5)  =  IF_APPEND_3_IN_1_AGA2(x3, x5)
SUBLIST_2_IN_GA2(x1, x2)  =  SUBLIST_2_IN_GA1(x1)
APPEND_3_IN_GAA3(x1, x2, x3)  =  APPEND_3_IN_GAA1(x1)
APPEND_3_IN_AGA3(x1, x2, x3)  =  APPEND_3_IN_AGA1(x2)
IF_SUBLIST_2_IN_2_GA4(x1, x2, x3, x4)  =  IF_SUBLIST_2_IN_2_GA2(x1, x4)
IF_APPEND_3_IN_1_GAA5(x1, x2, x3, x4, x5)  =  IF_APPEND_3_IN_1_GAA2(x2, x5)
IF_SUBLIST_2_IN_1_GA3(x1, x2, x3)  =  IF_SUBLIST_2_IN_1_GA2(x1, x3)

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:

SUBLIST_2_IN_GA2(X, Y) -> IF_SUBLIST_2_IN_1_GA3(X, Y, append_3_in_aga3(U, X, V))
SUBLIST_2_IN_GA2(X, Y) -> APPEND_3_IN_AGA3(U, X, V)
APPEND_3_IN_AGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APPEND_3_IN_1_AGA5(X, Xs, Ys, Zs, append_3_in_aga3(Xs, Ys, Zs))
APPEND_3_IN_AGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND_3_IN_AGA3(Xs, Ys, Zs)
IF_SUBLIST_2_IN_1_GA3(X, Y, append_3_out_aga3(U, X, V)) -> IF_SUBLIST_2_IN_2_GA4(X, Y, V, append_3_in_gaa3(V, W, Y))
IF_SUBLIST_2_IN_1_GA3(X, Y, append_3_out_aga3(U, X, V)) -> APPEND_3_IN_GAA3(V, W, Y)
APPEND_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APPEND_3_IN_1_GAA5(X, Xs, Ys, Zs, append_3_in_gaa3(Xs, Ys, Zs))
APPEND_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND_3_IN_GAA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_2_in_ga2(X, Y) -> if_sublist_2_in_1_ga3(X, Y, append_3_in_aga3(U, X, V))
append_3_in_aga3([]_0, Ys, Ys) -> append_3_out_aga3([]_0, Ys, Ys)
append_3_in_aga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_in_aga3(Xs, Ys, Zs))
if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_out_aga3(Xs, Ys, Zs)) -> append_3_out_aga3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_1_ga3(X, Y, append_3_out_aga3(U, X, V)) -> if_sublist_2_in_2_ga4(X, Y, V, append_3_in_gaa3(V, W, Y))
append_3_in_gaa3([]_0, Ys, Ys) -> append_3_out_gaa3([]_0, Ys, Ys)
append_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_in_gaa3(Xs, Ys, Zs))
if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_out_gaa3(Xs, Ys, Zs)) -> append_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_2_ga4(X, Y, V, append_3_out_gaa3(V, W, Y)) -> sublist_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
sublist_2_in_ga2(x1, x2)  =  sublist_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
if_sublist_2_in_1_ga3(x1, x2, x3)  =  if_sublist_2_in_1_ga2(x1, x3)
append_3_in_aga3(x1, x2, x3)  =  append_3_in_aga1(x2)
append_3_out_aga3(x1, x2, x3)  =  append_3_out_aga3(x1, x2, x3)
if_append_3_in_1_aga5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_aga2(x3, x5)
if_sublist_2_in_2_ga4(x1, x2, x3, x4)  =  if_sublist_2_in_2_ga2(x1, x4)
append_3_in_gaa3(x1, x2, x3)  =  append_3_in_gaa1(x1)
append_3_out_gaa3(x1, x2, x3)  =  append_3_out_gaa1(x1)
if_append_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_gaa2(x2, x5)
sublist_2_out_ga2(x1, x2)  =  sublist_2_out_ga1(x1)
IF_APPEND_3_IN_1_AGA5(x1, x2, x3, x4, x5)  =  IF_APPEND_3_IN_1_AGA2(x3, x5)
SUBLIST_2_IN_GA2(x1, x2)  =  SUBLIST_2_IN_GA1(x1)
APPEND_3_IN_GAA3(x1, x2, x3)  =  APPEND_3_IN_GAA1(x1)
APPEND_3_IN_AGA3(x1, x2, x3)  =  APPEND_3_IN_AGA1(x2)
IF_SUBLIST_2_IN_2_GA4(x1, x2, x3, x4)  =  IF_SUBLIST_2_IN_2_GA2(x1, x4)
IF_APPEND_3_IN_1_GAA5(x1, x2, x3, x4, x5)  =  IF_APPEND_3_IN_1_GAA2(x2, x5)
IF_SUBLIST_2_IN_1_GA3(x1, x2, x3)  =  IF_SUBLIST_2_IN_1_GA2(x1, x3)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 2 SCCs with 6 less nodes.

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

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

APPEND_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND_3_IN_GAA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_2_in_ga2(X, Y) -> if_sublist_2_in_1_ga3(X, Y, append_3_in_aga3(U, X, V))
append_3_in_aga3([]_0, Ys, Ys) -> append_3_out_aga3([]_0, Ys, Ys)
append_3_in_aga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_in_aga3(Xs, Ys, Zs))
if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_out_aga3(Xs, Ys, Zs)) -> append_3_out_aga3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_1_ga3(X, Y, append_3_out_aga3(U, X, V)) -> if_sublist_2_in_2_ga4(X, Y, V, append_3_in_gaa3(V, W, Y))
append_3_in_gaa3([]_0, Ys, Ys) -> append_3_out_gaa3([]_0, Ys, Ys)
append_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_in_gaa3(Xs, Ys, Zs))
if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_out_gaa3(Xs, Ys, Zs)) -> append_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_2_ga4(X, Y, V, append_3_out_gaa3(V, W, Y)) -> sublist_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
sublist_2_in_ga2(x1, x2)  =  sublist_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
if_sublist_2_in_1_ga3(x1, x2, x3)  =  if_sublist_2_in_1_ga2(x1, x3)
append_3_in_aga3(x1, x2, x3)  =  append_3_in_aga1(x2)
append_3_out_aga3(x1, x2, x3)  =  append_3_out_aga3(x1, x2, x3)
if_append_3_in_1_aga5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_aga2(x3, x5)
if_sublist_2_in_2_ga4(x1, x2, x3, x4)  =  if_sublist_2_in_2_ga2(x1, x4)
append_3_in_gaa3(x1, x2, x3)  =  append_3_in_gaa1(x1)
append_3_out_gaa3(x1, x2, x3)  =  append_3_out_gaa1(x1)
if_append_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_gaa2(x2, x5)
sublist_2_out_ga2(x1, x2)  =  sublist_2_out_ga1(x1)
APPEND_3_IN_GAA3(x1, x2, x3)  =  APPEND_3_IN_GAA1(x1)

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

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

APPEND_3_IN_GAA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND_3_IN_GAA3(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._21(x2)
APPEND_3_IN_GAA3(x1, x2, x3)  =  APPEND_3_IN_GAA1(x1)

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

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

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

APPEND_3_IN_AGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND_3_IN_AGA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_2_in_ga2(X, Y) -> if_sublist_2_in_1_ga3(X, Y, append_3_in_aga3(U, X, V))
append_3_in_aga3([]_0, Ys, Ys) -> append_3_out_aga3([]_0, Ys, Ys)
append_3_in_aga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_in_aga3(Xs, Ys, Zs))
if_append_3_in_1_aga5(X, Xs, Ys, Zs, append_3_out_aga3(Xs, Ys, Zs)) -> append_3_out_aga3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_1_ga3(X, Y, append_3_out_aga3(U, X, V)) -> if_sublist_2_in_2_ga4(X, Y, V, append_3_in_gaa3(V, W, Y))
append_3_in_gaa3([]_0, Ys, Ys) -> append_3_out_gaa3([]_0, Ys, Ys)
append_3_in_gaa3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_in_gaa3(Xs, Ys, Zs))
if_append_3_in_1_gaa5(X, Xs, Ys, Zs, append_3_out_gaa3(Xs, Ys, Zs)) -> append_3_out_gaa3(._22(X, Xs), Ys, ._22(X, Zs))
if_sublist_2_in_2_ga4(X, Y, V, append_3_out_gaa3(V, W, Y)) -> sublist_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
sublist_2_in_ga2(x1, x2)  =  sublist_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._21(x2)
if_sublist_2_in_1_ga3(x1, x2, x3)  =  if_sublist_2_in_1_ga2(x1, x3)
append_3_in_aga3(x1, x2, x3)  =  append_3_in_aga1(x2)
append_3_out_aga3(x1, x2, x3)  =  append_3_out_aga3(x1, x2, x3)
if_append_3_in_1_aga5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_aga2(x3, x5)
if_sublist_2_in_2_ga4(x1, x2, x3, x4)  =  if_sublist_2_in_2_ga2(x1, x4)
append_3_in_gaa3(x1, x2, x3)  =  append_3_in_gaa1(x1)
append_3_out_gaa3(x1, x2, x3)  =  append_3_out_gaa1(x1)
if_append_3_in_1_gaa5(x1, x2, x3, x4, x5)  =  if_append_3_in_1_gaa2(x2, x5)
sublist_2_out_ga2(x1, x2)  =  sublist_2_out_ga1(x1)
APPEND_3_IN_AGA3(x1, x2, x3)  =  APPEND_3_IN_AGA1(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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP

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

APPEND_3_IN_AGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND_3_IN_AGA3(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._21(x2)
APPEND_3_IN_AGA3(x1, x2, x3)  =  APPEND_3_IN_AGA1(x2)

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