Left Termination of the query pattern app3_b(b,b,b,f) w.r.t. the given Prolog program could successfully be proven:



PROLOG
  ↳ UnrequestedClauseRemoverProof

app3a4(Xs, Ys, Zs, Us) :- app3(Xs, Ys, Vs), app3(Vs, Zs, Us).
app3b4(Xs, Ys, Zs, Us) :- app3(Ys, Zs, Vs), app3(Xs, Vs, Us).
app3({}0, Ys, Ys).
app3(.2(X, Xs), Ys, .2(X, Zs)) :- app3(Xs, Ys, Zs).


The clause

app3a4(Xs, Ys, Zs, Us) :- app3(Xs, Ys, Vs), app3(Vs, Zs, Us).

can be ignored, as it is not needed by any of the given querys.

Deleting this clauses results in the following prolog program:

app3b4(Xs, Ys, Zs, Us) :- app3(Ys, Zs, Vs), app3(Xs, Vs, Us).
app3({}0, Ys, Ys).
app3(.2(X, Xs), Ys, .2(X, Zs)) :- app3(Xs, Ys, Zs).



↳ PROLOG
  ↳ UnrequestedClauseRemoverProof
PROLOG
      ↳ PrologToPiTRSProof

app3b4(Xs, Ys, Zs, Us) :- app3(Ys, Zs, Vs), app3(Xs, Vs, Us).
app3({}0, Ys, Ys).
app3(.2(X, Xs), Ys, .2(X, Zs)) :- app3(Xs, Ys, Zs).


With regard to the inferred argument filtering the predicates were used in the following modes:
app3_b4: (b,b,b,f)
app3: (b,b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:


app3_b_4_in_ggga4(Xs, Ys, Zs, Us) -> if_app3_b_4_in_1_ggga5(Xs, Ys, Zs, Us, app_3_in_gga3(Ys, Zs, Vs))
app_3_in_gga3([]_0, Ys, Ys) -> app_3_out_gga3([]_0, Ys, Ys)
app_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_app3_b_4_in_1_ggga5(Xs, Ys, Zs, Us, app_3_out_gga3(Ys, Zs, Vs)) -> if_app3_b_4_in_2_ggga6(Xs, Ys, Zs, Us, Vs, app_3_in_gga3(Xs, Vs, Us))
if_app3_b_4_in_2_ggga6(Xs, Ys, Zs, Us, Vs, app_3_out_gga3(Xs, Vs, Us)) -> app3_b_4_out_ggga4(Xs, Ys, Zs, Us)

The argument filtering Pi contains the following mapping:
app3_b_4_in_ggga4(x1, x2, x3, x4)  =  app3_b_4_in_ggga3(x1, x2, x3)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
if_app3_b_4_in_1_ggga5(x1, x2, x3, x4, x5)  =  if_app3_b_4_in_1_ggga2(x1, x5)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
if_app3_b_4_in_2_ggga6(x1, x2, x3, x4, x5, x6)  =  if_app3_b_4_in_2_ggga1(x6)
app3_b_4_out_ggga4(x1, x2, x3, x4)  =  app3_b_4_out_ggga1(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG



↳ PROLOG
  ↳ UnrequestedClauseRemoverProof
    ↳ PROLOG
      ↳ PrologToPiTRSProof
PiTRS
          ↳ DependencyPairsProof

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

app3_b_4_in_ggga4(Xs, Ys, Zs, Us) -> if_app3_b_4_in_1_ggga5(Xs, Ys, Zs, Us, app_3_in_gga3(Ys, Zs, Vs))
app_3_in_gga3([]_0, Ys, Ys) -> app_3_out_gga3([]_0, Ys, Ys)
app_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_app3_b_4_in_1_ggga5(Xs, Ys, Zs, Us, app_3_out_gga3(Ys, Zs, Vs)) -> if_app3_b_4_in_2_ggga6(Xs, Ys, Zs, Us, Vs, app_3_in_gga3(Xs, Vs, Us))
if_app3_b_4_in_2_ggga6(Xs, Ys, Zs, Us, Vs, app_3_out_gga3(Xs, Vs, Us)) -> app3_b_4_out_ggga4(Xs, Ys, Zs, Us)

The argument filtering Pi contains the following mapping:
app3_b_4_in_ggga4(x1, x2, x3, x4)  =  app3_b_4_in_ggga3(x1, x2, x3)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
if_app3_b_4_in_1_ggga5(x1, x2, x3, x4, x5)  =  if_app3_b_4_in_1_ggga2(x1, x5)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
if_app3_b_4_in_2_ggga6(x1, x2, x3, x4, x5, x6)  =  if_app3_b_4_in_2_ggga1(x6)
app3_b_4_out_ggga4(x1, x2, x3, x4)  =  app3_b_4_out_ggga1(x4)


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

APP3_B_4_IN_GGGA4(Xs, Ys, Zs, Us) -> IF_APP3_B_4_IN_1_GGGA5(Xs, Ys, Zs, Us, app_3_in_gga3(Ys, Zs, Vs))
APP3_B_4_IN_GGGA4(Xs, Ys, Zs, Us) -> APP_3_IN_GGA3(Ys, Zs, Vs)
APP_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APP_3_IN_1_GGA5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
APP_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GGA3(Xs, Ys, Zs)
IF_APP3_B_4_IN_1_GGGA5(Xs, Ys, Zs, Us, app_3_out_gga3(Ys, Zs, Vs)) -> IF_APP3_B_4_IN_2_GGGA6(Xs, Ys, Zs, Us, Vs, app_3_in_gga3(Xs, Vs, Us))
IF_APP3_B_4_IN_1_GGGA5(Xs, Ys, Zs, Us, app_3_out_gga3(Ys, Zs, Vs)) -> APP_3_IN_GGA3(Xs, Vs, Us)

The TRS R consists of the following rules:

app3_b_4_in_ggga4(Xs, Ys, Zs, Us) -> if_app3_b_4_in_1_ggga5(Xs, Ys, Zs, Us, app_3_in_gga3(Ys, Zs, Vs))
app_3_in_gga3([]_0, Ys, Ys) -> app_3_out_gga3([]_0, Ys, Ys)
app_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_app3_b_4_in_1_ggga5(Xs, Ys, Zs, Us, app_3_out_gga3(Ys, Zs, Vs)) -> if_app3_b_4_in_2_ggga6(Xs, Ys, Zs, Us, Vs, app_3_in_gga3(Xs, Vs, Us))
if_app3_b_4_in_2_ggga6(Xs, Ys, Zs, Us, Vs, app_3_out_gga3(Xs, Vs, Us)) -> app3_b_4_out_ggga4(Xs, Ys, Zs, Us)

The argument filtering Pi contains the following mapping:
app3_b_4_in_ggga4(x1, x2, x3, x4)  =  app3_b_4_in_ggga3(x1, x2, x3)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
if_app3_b_4_in_1_ggga5(x1, x2, x3, x4, x5)  =  if_app3_b_4_in_1_ggga2(x1, x5)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
if_app3_b_4_in_2_ggga6(x1, x2, x3, x4, x5, x6)  =  if_app3_b_4_in_2_ggga1(x6)
app3_b_4_out_ggga4(x1, x2, x3, x4)  =  app3_b_4_out_ggga1(x4)
IF_APP3_B_4_IN_1_GGGA5(x1, x2, x3, x4, x5)  =  IF_APP3_B_4_IN_1_GGGA2(x1, x5)
IF_APP3_B_4_IN_2_GGGA6(x1, x2, x3, x4, x5, x6)  =  IF_APP3_B_4_IN_2_GGGA1(x6)
APP3_B_4_IN_GGGA4(x1, x2, x3, x4)  =  APP3_B_4_IN_GGGA3(x1, x2, x3)
IF_APP_3_IN_1_GGA5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_GGA2(x1, x5)
APP_3_IN_GGA3(x1, x2, x3)  =  APP_3_IN_GGA2(x1, x2)

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

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

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

APP3_B_4_IN_GGGA4(Xs, Ys, Zs, Us) -> IF_APP3_B_4_IN_1_GGGA5(Xs, Ys, Zs, Us, app_3_in_gga3(Ys, Zs, Vs))
APP3_B_4_IN_GGGA4(Xs, Ys, Zs, Us) -> APP_3_IN_GGA3(Ys, Zs, Vs)
APP_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APP_3_IN_1_GGA5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
APP_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GGA3(Xs, Ys, Zs)
IF_APP3_B_4_IN_1_GGGA5(Xs, Ys, Zs, Us, app_3_out_gga3(Ys, Zs, Vs)) -> IF_APP3_B_4_IN_2_GGGA6(Xs, Ys, Zs, Us, Vs, app_3_in_gga3(Xs, Vs, Us))
IF_APP3_B_4_IN_1_GGGA5(Xs, Ys, Zs, Us, app_3_out_gga3(Ys, Zs, Vs)) -> APP_3_IN_GGA3(Xs, Vs, Us)

The TRS R consists of the following rules:

app3_b_4_in_ggga4(Xs, Ys, Zs, Us) -> if_app3_b_4_in_1_ggga5(Xs, Ys, Zs, Us, app_3_in_gga3(Ys, Zs, Vs))
app_3_in_gga3([]_0, Ys, Ys) -> app_3_out_gga3([]_0, Ys, Ys)
app_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_app3_b_4_in_1_ggga5(Xs, Ys, Zs, Us, app_3_out_gga3(Ys, Zs, Vs)) -> if_app3_b_4_in_2_ggga6(Xs, Ys, Zs, Us, Vs, app_3_in_gga3(Xs, Vs, Us))
if_app3_b_4_in_2_ggga6(Xs, Ys, Zs, Us, Vs, app_3_out_gga3(Xs, Vs, Us)) -> app3_b_4_out_ggga4(Xs, Ys, Zs, Us)

The argument filtering Pi contains the following mapping:
app3_b_4_in_ggga4(x1, x2, x3, x4)  =  app3_b_4_in_ggga3(x1, x2, x3)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
if_app3_b_4_in_1_ggga5(x1, x2, x3, x4, x5)  =  if_app3_b_4_in_1_ggga2(x1, x5)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
if_app3_b_4_in_2_ggga6(x1, x2, x3, x4, x5, x6)  =  if_app3_b_4_in_2_ggga1(x6)
app3_b_4_out_ggga4(x1, x2, x3, x4)  =  app3_b_4_out_ggga1(x4)
IF_APP3_B_4_IN_1_GGGA5(x1, x2, x3, x4, x5)  =  IF_APP3_B_4_IN_1_GGGA2(x1, x5)
IF_APP3_B_4_IN_2_GGGA6(x1, x2, x3, x4, x5, x6)  =  IF_APP3_B_4_IN_2_GGGA1(x6)
APP3_B_4_IN_GGGA4(x1, x2, x3, x4)  =  APP3_B_4_IN_GGGA3(x1, x2, x3)
IF_APP_3_IN_1_GGA5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_GGA2(x1, x5)
APP_3_IN_GGA3(x1, x2, x3)  =  APP_3_IN_GGA2(x1, x2)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 1 SCC with 5 less nodes.

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

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

APP_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GGA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

app3_b_4_in_ggga4(Xs, Ys, Zs, Us) -> if_app3_b_4_in_1_ggga5(Xs, Ys, Zs, Us, app_3_in_gga3(Ys, Zs, Vs))
app_3_in_gga3([]_0, Ys, Ys) -> app_3_out_gga3([]_0, Ys, Ys)
app_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_app3_b_4_in_1_ggga5(Xs, Ys, Zs, Us, app_3_out_gga3(Ys, Zs, Vs)) -> if_app3_b_4_in_2_ggga6(Xs, Ys, Zs, Us, Vs, app_3_in_gga3(Xs, Vs, Us))
if_app3_b_4_in_2_ggga6(Xs, Ys, Zs, Us, Vs, app_3_out_gga3(Xs, Vs, Us)) -> app3_b_4_out_ggga4(Xs, Ys, Zs, Us)

The argument filtering Pi contains the following mapping:
app3_b_4_in_ggga4(x1, x2, x3, x4)  =  app3_b_4_in_ggga3(x1, x2, x3)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
if_app3_b_4_in_1_ggga5(x1, x2, x3, x4, x5)  =  if_app3_b_4_in_1_ggga2(x1, x5)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
if_app3_b_4_in_2_ggga6(x1, x2, x3, x4, x5, x6)  =  if_app3_b_4_in_2_ggga1(x6)
app3_b_4_out_ggga4(x1, x2, x3, x4)  =  app3_b_4_out_ggga1(x4)
APP_3_IN_GGA3(x1, x2, x3)  =  APP_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
  ↳ UnrequestedClauseRemoverProof
    ↳ PROLOG
      ↳ PrologToPiTRSProof
        ↳ PiTRS
          ↳ DependencyPairsProof
            ↳ PiDP
              ↳ DependencyGraphProof
                ↳ PiDP
                  ↳ UsableRulesProof
PiDP
                      ↳ PiDPToQDPProof

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

APP_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GGA3(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._22(x1, x2)
APP_3_IN_GGA3(x1, x2, x3)  =  APP_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
  ↳ UnrequestedClauseRemoverProof
    ↳ PROLOG
      ↳ PrologToPiTRSProof
        ↳ PiTRS
          ↳ DependencyPairsProof
            ↳ PiDP
              ↳ DependencyGraphProof
                ↳ PiDP
                  ↳ UsableRulesProof
                    ↳ PiDP
                      ↳ PiDPToQDPProof
QDP
                          ↳ QDPSizeChangeProof

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

APP_3_IN_GGA2(._22(X, Xs), Ys) -> APP_3_IN_GGA2(Xs, Ys)

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