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



PROLOG
  ↳ PrologToPiTRSProof

rev2({}0, {}0).
rev2(.2(X, Xs), Ys) :- rev2(Xs, Zs), app3(Zs, .2(X, {}0), Ys).
app3({}0, X, X).
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:
rev2: (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:


rev_2_in_ga2([]_0, []_0) -> rev_2_out_ga2([]_0, []_0)
rev_2_in_ga2(._22(X, Xs), Ys) -> if_rev_2_in_1_ga4(X, Xs, Ys, rev_2_in_ga2(Xs, Zs))
if_rev_2_in_1_ga4(X, Xs, Ys, rev_2_out_ga2(Xs, Zs)) -> if_rev_2_in_2_ga5(X, Xs, Ys, Zs, app_3_in_gga3(Zs, ._22(X, []_0), Ys))
app_3_in_gga3([]_0, X, X) -> app_3_out_gga3([]_0, X, X)
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_rev_2_in_2_ga5(X, Xs, Ys, Zs, app_3_out_gga3(Zs, ._22(X, []_0), Ys)) -> rev_2_out_ga2(._22(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_2_in_ga2(x1, x2)  =  rev_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
rev_2_out_ga2(x1, x2)  =  rev_2_out_ga1(x2)
if_rev_2_in_1_ga4(x1, x2, x3, x4)  =  if_rev_2_in_1_ga2(x1, x4)
if_rev_2_in_2_ga5(x1, x2, x3, x4, x5)  =  if_rev_2_in_2_ga1(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)

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:

rev_2_in_ga2([]_0, []_0) -> rev_2_out_ga2([]_0, []_0)
rev_2_in_ga2(._22(X, Xs), Ys) -> if_rev_2_in_1_ga4(X, Xs, Ys, rev_2_in_ga2(Xs, Zs))
if_rev_2_in_1_ga4(X, Xs, Ys, rev_2_out_ga2(Xs, Zs)) -> if_rev_2_in_2_ga5(X, Xs, Ys, Zs, app_3_in_gga3(Zs, ._22(X, []_0), Ys))
app_3_in_gga3([]_0, X, X) -> app_3_out_gga3([]_0, X, X)
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_rev_2_in_2_ga5(X, Xs, Ys, Zs, app_3_out_gga3(Zs, ._22(X, []_0), Ys)) -> rev_2_out_ga2(._22(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_2_in_ga2(x1, x2)  =  rev_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
rev_2_out_ga2(x1, x2)  =  rev_2_out_ga1(x2)
if_rev_2_in_1_ga4(x1, x2, x3, x4)  =  if_rev_2_in_1_ga2(x1, x4)
if_rev_2_in_2_ga5(x1, x2, x3, x4, x5)  =  if_rev_2_in_2_ga1(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)


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

REV_2_IN_GA2(._22(X, Xs), Ys) -> IF_REV_2_IN_1_GA4(X, Xs, Ys, rev_2_in_ga2(Xs, Zs))
REV_2_IN_GA2(._22(X, Xs), Ys) -> REV_2_IN_GA2(Xs, Zs)
IF_REV_2_IN_1_GA4(X, Xs, Ys, rev_2_out_ga2(Xs, Zs)) -> IF_REV_2_IN_2_GA5(X, Xs, Ys, Zs, app_3_in_gga3(Zs, ._22(X, []_0), Ys))
IF_REV_2_IN_1_GA4(X, Xs, Ys, rev_2_out_ga2(Xs, Zs)) -> APP_3_IN_GGA3(Zs, ._22(X, []_0), 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))
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:

rev_2_in_ga2([]_0, []_0) -> rev_2_out_ga2([]_0, []_0)
rev_2_in_ga2(._22(X, Xs), Ys) -> if_rev_2_in_1_ga4(X, Xs, Ys, rev_2_in_ga2(Xs, Zs))
if_rev_2_in_1_ga4(X, Xs, Ys, rev_2_out_ga2(Xs, Zs)) -> if_rev_2_in_2_ga5(X, Xs, Ys, Zs, app_3_in_gga3(Zs, ._22(X, []_0), Ys))
app_3_in_gga3([]_0, X, X) -> app_3_out_gga3([]_0, X, X)
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_rev_2_in_2_ga5(X, Xs, Ys, Zs, app_3_out_gga3(Zs, ._22(X, []_0), Ys)) -> rev_2_out_ga2(._22(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_2_in_ga2(x1, x2)  =  rev_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
rev_2_out_ga2(x1, x2)  =  rev_2_out_ga1(x2)
if_rev_2_in_1_ga4(x1, x2, x3, x4)  =  if_rev_2_in_1_ga2(x1, x4)
if_rev_2_in_2_ga5(x1, x2, x3, x4, x5)  =  if_rev_2_in_2_ga1(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_REV_2_IN_2_GA5(x1, x2, x3, x4, x5)  =  IF_REV_2_IN_2_GA1(x5)
IF_REV_2_IN_1_GA4(x1, x2, x3, x4)  =  IF_REV_2_IN_1_GA2(x1, x4)
REV_2_IN_GA2(x1, x2)  =  REV_2_IN_GA1(x1)
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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
PiDP
          ↳ DependencyGraphProof

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

REV_2_IN_GA2(._22(X, Xs), Ys) -> IF_REV_2_IN_1_GA4(X, Xs, Ys, rev_2_in_ga2(Xs, Zs))
REV_2_IN_GA2(._22(X, Xs), Ys) -> REV_2_IN_GA2(Xs, Zs)
IF_REV_2_IN_1_GA4(X, Xs, Ys, rev_2_out_ga2(Xs, Zs)) -> IF_REV_2_IN_2_GA5(X, Xs, Ys, Zs, app_3_in_gga3(Zs, ._22(X, []_0), Ys))
IF_REV_2_IN_1_GA4(X, Xs, Ys, rev_2_out_ga2(Xs, Zs)) -> APP_3_IN_GGA3(Zs, ._22(X, []_0), 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))
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:

rev_2_in_ga2([]_0, []_0) -> rev_2_out_ga2([]_0, []_0)
rev_2_in_ga2(._22(X, Xs), Ys) -> if_rev_2_in_1_ga4(X, Xs, Ys, rev_2_in_ga2(Xs, Zs))
if_rev_2_in_1_ga4(X, Xs, Ys, rev_2_out_ga2(Xs, Zs)) -> if_rev_2_in_2_ga5(X, Xs, Ys, Zs, app_3_in_gga3(Zs, ._22(X, []_0), Ys))
app_3_in_gga3([]_0, X, X) -> app_3_out_gga3([]_0, X, X)
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_rev_2_in_2_ga5(X, Xs, Ys, Zs, app_3_out_gga3(Zs, ._22(X, []_0), Ys)) -> rev_2_out_ga2(._22(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_2_in_ga2(x1, x2)  =  rev_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
rev_2_out_ga2(x1, x2)  =  rev_2_out_ga1(x2)
if_rev_2_in_1_ga4(x1, x2, x3, x4)  =  if_rev_2_in_1_ga2(x1, x4)
if_rev_2_in_2_ga5(x1, x2, x3, x4, x5)  =  if_rev_2_in_2_ga1(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_REV_2_IN_2_GA5(x1, x2, x3, x4, x5)  =  IF_REV_2_IN_2_GA1(x5)
IF_REV_2_IN_1_GA4(x1, x2, x3, x4)  =  IF_REV_2_IN_1_GA2(x1, x4)
REV_2_IN_GA2(x1, x2)  =  REV_2_IN_GA1(x1)
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 2 SCCs with 4 less nodes.

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

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:

rev_2_in_ga2([]_0, []_0) -> rev_2_out_ga2([]_0, []_0)
rev_2_in_ga2(._22(X, Xs), Ys) -> if_rev_2_in_1_ga4(X, Xs, Ys, rev_2_in_ga2(Xs, Zs))
if_rev_2_in_1_ga4(X, Xs, Ys, rev_2_out_ga2(Xs, Zs)) -> if_rev_2_in_2_ga5(X, Xs, Ys, Zs, app_3_in_gga3(Zs, ._22(X, []_0), Ys))
app_3_in_gga3([]_0, X, X) -> app_3_out_gga3([]_0, X, X)
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_rev_2_in_2_ga5(X, Xs, Ys, Zs, app_3_out_gga3(Zs, ._22(X, []_0), Ys)) -> rev_2_out_ga2(._22(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_2_in_ga2(x1, x2)  =  rev_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
rev_2_out_ga2(x1, x2)  =  rev_2_out_ga1(x2)
if_rev_2_in_1_ga4(x1, x2, x3, x4)  =  if_rev_2_in_1_ga2(x1, x4)
if_rev_2_in_2_ga5(x1, x2, x3, x4, x5)  =  if_rev_2_in_2_ga1(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)
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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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

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:



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

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

REV_2_IN_GA2(._22(X, Xs), Ys) -> REV_2_IN_GA2(Xs, Zs)

The TRS R consists of the following rules:

rev_2_in_ga2([]_0, []_0) -> rev_2_out_ga2([]_0, []_0)
rev_2_in_ga2(._22(X, Xs), Ys) -> if_rev_2_in_1_ga4(X, Xs, Ys, rev_2_in_ga2(Xs, Zs))
if_rev_2_in_1_ga4(X, Xs, Ys, rev_2_out_ga2(Xs, Zs)) -> if_rev_2_in_2_ga5(X, Xs, Ys, Zs, app_3_in_gga3(Zs, ._22(X, []_0), Ys))
app_3_in_gga3([]_0, X, X) -> app_3_out_gga3([]_0, X, X)
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_rev_2_in_2_ga5(X, Xs, Ys, Zs, app_3_out_gga3(Zs, ._22(X, []_0), Ys)) -> rev_2_out_ga2(._22(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_2_in_ga2(x1, x2)  =  rev_2_in_ga1(x1)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
rev_2_out_ga2(x1, x2)  =  rev_2_out_ga1(x2)
if_rev_2_in_1_ga4(x1, x2, x3, x4)  =  if_rev_2_in_1_ga2(x1, x4)
if_rev_2_in_2_ga5(x1, x2, x3, x4, x5)  =  if_rev_2_in_2_ga1(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)
REV_2_IN_GA2(x1, x2)  =  REV_2_IN_GA1(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
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof

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

REV_2_IN_GA2(._22(X, Xs), Ys) -> REV_2_IN_GA2(Xs, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._22(x1, x2)
REV_2_IN_GA2(x1, x2)  =  REV_2_IN_GA1(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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ QDPSizeChangeProof

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

REV_2_IN_GA1(._22(X, Xs)) -> REV_2_IN_GA1(Xs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {REV_2_IN_GA1}.
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: