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



↳ PROLOG
  ↳ UnrequestedClauseRemoverProof

concatenate3({}0, L, L).
concatenate3(.2(X, L1), L2, .2(X, L3)) :- concatenate3(L1, L2, L3).
member2(X, .2(X, L)).
member2(X, .2(Y, L)) :- member2(X, L).
reverse2(L, L1) :- reverseconcatenate3(L, {}0, L1).
reverseconcatenate3({}0, L, L).
reverseconcatenate3(.2(X, L1), L2, L3) :- reverseconcatenate3(L1, .2(X, L2), L3).


The clauses

concatenate3({}0, L, L).
concatenate3(.2(X, L1), L2, .2(X, L3)) :- concatenate3(L1, L2, L3).
member2(X, .2(X, L)).
member2(X, .2(Y, L)) :- member2(X, L).
reverse2(L, L1) :- reverseconcatenate3(L, {}0, L1).

can be ignored, as they are not needed by any of the given querys.

Deleting these clauses results in the following prolog program:

reverseconcatenate3({}0, L, L).
reverseconcatenate3(.2(X, L1), L2, L3) :- reverseconcatenate3(L1, .2(X, L2), L3).



↳ PROLOG
  ↳ UnrequestedClauseRemoverProof
    ↳ PROLOG
      ↳ PrologToPiTRSProof

reverseconcatenate3({}0, L, L).
reverseconcatenate3(.2(X, L1), L2, L3) :- reverseconcatenate3(L1, .2(X, L2), L3).


With regard to the inferred argument filtering the predicates were used in the following modes:
reverse_concatenate3: (b,b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:


reverse_concatenate_3_in_gga3([]_0, L, L) -> reverse_concatenate_3_out_gga3([]_0, L, L)
reverse_concatenate_3_in_gga3(._22(X, L1), L2, L3) -> if_reverse_concatenate_3_in_1_gga5(X, L1, L2, L3, reverse_concatenate_3_in_gga3(L1, ._22(X, L2), L3))
if_reverse_concatenate_3_in_1_gga5(X, L1, L2, L3, reverse_concatenate_3_out_gga3(L1, ._22(X, L2), L3)) -> reverse_concatenate_3_out_gga3(._22(X, L1), L2, L3)

The argument filtering Pi contains the following mapping:
reverse_concatenate_3_in_gga3(x1, x2, x3)  =  reverse_concatenate_3_in_gga2(x1, x2)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
reverse_concatenate_3_out_gga3(x1, x2, x3)  =  reverse_concatenate_3_out_gga1(x3)
if_reverse_concatenate_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_reverse_concatenate_3_in_1_gga1(x5)

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:

reverse_concatenate_3_in_gga3([]_0, L, L) -> reverse_concatenate_3_out_gga3([]_0, L, L)
reverse_concatenate_3_in_gga3(._22(X, L1), L2, L3) -> if_reverse_concatenate_3_in_1_gga5(X, L1, L2, L3, reverse_concatenate_3_in_gga3(L1, ._22(X, L2), L3))
if_reverse_concatenate_3_in_1_gga5(X, L1, L2, L3, reverse_concatenate_3_out_gga3(L1, ._22(X, L2), L3)) -> reverse_concatenate_3_out_gga3(._22(X, L1), L2, L3)

The argument filtering Pi contains the following mapping:
reverse_concatenate_3_in_gga3(x1, x2, x3)  =  reverse_concatenate_3_in_gga2(x1, x2)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
reverse_concatenate_3_out_gga3(x1, x2, x3)  =  reverse_concatenate_3_out_gga1(x3)
if_reverse_concatenate_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_reverse_concatenate_3_in_1_gga1(x5)


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

REVERSE_CONCATENATE_3_IN_GGA3(._22(X, L1), L2, L3) -> IF_REVERSE_CONCATENATE_3_IN_1_GGA5(X, L1, L2, L3, reverse_concatenate_3_in_gga3(L1, ._22(X, L2), L3))
REVERSE_CONCATENATE_3_IN_GGA3(._22(X, L1), L2, L3) -> REVERSE_CONCATENATE_3_IN_GGA3(L1, ._22(X, L2), L3)

The TRS R consists of the following rules:

reverse_concatenate_3_in_gga3([]_0, L, L) -> reverse_concatenate_3_out_gga3([]_0, L, L)
reverse_concatenate_3_in_gga3(._22(X, L1), L2, L3) -> if_reverse_concatenate_3_in_1_gga5(X, L1, L2, L3, reverse_concatenate_3_in_gga3(L1, ._22(X, L2), L3))
if_reverse_concatenate_3_in_1_gga5(X, L1, L2, L3, reverse_concatenate_3_out_gga3(L1, ._22(X, L2), L3)) -> reverse_concatenate_3_out_gga3(._22(X, L1), L2, L3)

The argument filtering Pi contains the following mapping:
reverse_concatenate_3_in_gga3(x1, x2, x3)  =  reverse_concatenate_3_in_gga2(x1, x2)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
reverse_concatenate_3_out_gga3(x1, x2, x3)  =  reverse_concatenate_3_out_gga1(x3)
if_reverse_concatenate_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_reverse_concatenate_3_in_1_gga1(x5)
IF_REVERSE_CONCATENATE_3_IN_1_GGA5(x1, x2, x3, x4, x5)  =  IF_REVERSE_CONCATENATE_3_IN_1_GGA1(x5)
REVERSE_CONCATENATE_3_IN_GGA3(x1, x2, x3)  =  REVERSE_CONCATENATE_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:

REVERSE_CONCATENATE_3_IN_GGA3(._22(X, L1), L2, L3) -> IF_REVERSE_CONCATENATE_3_IN_1_GGA5(X, L1, L2, L3, reverse_concatenate_3_in_gga3(L1, ._22(X, L2), L3))
REVERSE_CONCATENATE_3_IN_GGA3(._22(X, L1), L2, L3) -> REVERSE_CONCATENATE_3_IN_GGA3(L1, ._22(X, L2), L3)

The TRS R consists of the following rules:

reverse_concatenate_3_in_gga3([]_0, L, L) -> reverse_concatenate_3_out_gga3([]_0, L, L)
reverse_concatenate_3_in_gga3(._22(X, L1), L2, L3) -> if_reverse_concatenate_3_in_1_gga5(X, L1, L2, L3, reverse_concatenate_3_in_gga3(L1, ._22(X, L2), L3))
if_reverse_concatenate_3_in_1_gga5(X, L1, L2, L3, reverse_concatenate_3_out_gga3(L1, ._22(X, L2), L3)) -> reverse_concatenate_3_out_gga3(._22(X, L1), L2, L3)

The argument filtering Pi contains the following mapping:
reverse_concatenate_3_in_gga3(x1, x2, x3)  =  reverse_concatenate_3_in_gga2(x1, x2)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
reverse_concatenate_3_out_gga3(x1, x2, x3)  =  reverse_concatenate_3_out_gga1(x3)
if_reverse_concatenate_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_reverse_concatenate_3_in_1_gga1(x5)
IF_REVERSE_CONCATENATE_3_IN_1_GGA5(x1, x2, x3, x4, x5)  =  IF_REVERSE_CONCATENATE_3_IN_1_GGA1(x5)
REVERSE_CONCATENATE_3_IN_GGA3(x1, x2, x3)  =  REVERSE_CONCATENATE_3_IN_GGA2(x1, x2)

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

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

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

REVERSE_CONCATENATE_3_IN_GGA3(._22(X, L1), L2, L3) -> REVERSE_CONCATENATE_3_IN_GGA3(L1, ._22(X, L2), L3)

The TRS R consists of the following rules:

reverse_concatenate_3_in_gga3([]_0, L, L) -> reverse_concatenate_3_out_gga3([]_0, L, L)
reverse_concatenate_3_in_gga3(._22(X, L1), L2, L3) -> if_reverse_concatenate_3_in_1_gga5(X, L1, L2, L3, reverse_concatenate_3_in_gga3(L1, ._22(X, L2), L3))
if_reverse_concatenate_3_in_1_gga5(X, L1, L2, L3, reverse_concatenate_3_out_gga3(L1, ._22(X, L2), L3)) -> reverse_concatenate_3_out_gga3(._22(X, L1), L2, L3)

The argument filtering Pi contains the following mapping:
reverse_concatenate_3_in_gga3(x1, x2, x3)  =  reverse_concatenate_3_in_gga2(x1, x2)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
reverse_concatenate_3_out_gga3(x1, x2, x3)  =  reverse_concatenate_3_out_gga1(x3)
if_reverse_concatenate_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_reverse_concatenate_3_in_1_gga1(x5)
REVERSE_CONCATENATE_3_IN_GGA3(x1, x2, x3)  =  REVERSE_CONCATENATE_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:

REVERSE_CONCATENATE_3_IN_GGA3(._22(X, L1), L2, L3) -> REVERSE_CONCATENATE_3_IN_GGA3(L1, ._22(X, L2), L3)

R is empty.
The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._22(x1, x2)
REVERSE_CONCATENATE_3_IN_GGA3(x1, x2, x3)  =  REVERSE_CONCATENATE_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:

REVERSE_CONCATENATE_3_IN_GGA2(._22(X, L1), L2) -> REVERSE_CONCATENATE_3_IN_GGA2(L1, ._22(X, L2))

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