Left Termination of the query pattern p1(b) w.r.t. the given Prolog program could successfully be proven:



↳ PROLOG
  ↳ UnrequestedClauseRemoverProof

p11(f1(X)) :- p11(X).
p21(f1(X)) :- p21(X).


The clause

p21(f1(X)) :- p21(X).

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

Deleting this clauses results in the following prolog program:

p11(f1(X)) :- p11(X).



↳ PROLOG
  ↳ UnrequestedClauseRemoverProof
    ↳ PROLOG
      ↳ PrologToPiTRSProof

p11(f1(X)) :- p11(X).


With regard to the inferred argument filtering the predicates were used in the following modes:
p11: (b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:


p1_1_in_g1(f_11(X)) -> if_p1_1_in_1_g2(X, p1_1_in_g1(X))
if_p1_1_in_1_g2(X, p1_1_out_g1(X)) -> p1_1_out_g1(f_11(X))

The argument filtering Pi contains the following mapping:
p1_1_in_g1(x1)  =  p1_1_in_g1(x1)
f_11(x1)  =  f_11(x1)
if_p1_1_in_1_g2(x1, x2)  =  if_p1_1_in_1_g1(x2)
p1_1_out_g1(x1)  =  p1_1_out_g

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:

p1_1_in_g1(f_11(X)) -> if_p1_1_in_1_g2(X, p1_1_in_g1(X))
if_p1_1_in_1_g2(X, p1_1_out_g1(X)) -> p1_1_out_g1(f_11(X))

The argument filtering Pi contains the following mapping:
p1_1_in_g1(x1)  =  p1_1_in_g1(x1)
f_11(x1)  =  f_11(x1)
if_p1_1_in_1_g2(x1, x2)  =  if_p1_1_in_1_g1(x2)
p1_1_out_g1(x1)  =  p1_1_out_g


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

P1_1_IN_G1(f_11(X)) -> IF_P1_1_IN_1_G2(X, p1_1_in_g1(X))
P1_1_IN_G1(f_11(X)) -> P1_1_IN_G1(X)

The TRS R consists of the following rules:

p1_1_in_g1(f_11(X)) -> if_p1_1_in_1_g2(X, p1_1_in_g1(X))
if_p1_1_in_1_g2(X, p1_1_out_g1(X)) -> p1_1_out_g1(f_11(X))

The argument filtering Pi contains the following mapping:
p1_1_in_g1(x1)  =  p1_1_in_g1(x1)
f_11(x1)  =  f_11(x1)
if_p1_1_in_1_g2(x1, x2)  =  if_p1_1_in_1_g1(x2)
p1_1_out_g1(x1)  =  p1_1_out_g
IF_P1_1_IN_1_G2(x1, x2)  =  IF_P1_1_IN_1_G1(x2)
P1_1_IN_G1(x1)  =  P1_1_IN_G1(x1)

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:

P1_1_IN_G1(f_11(X)) -> IF_P1_1_IN_1_G2(X, p1_1_in_g1(X))
P1_1_IN_G1(f_11(X)) -> P1_1_IN_G1(X)

The TRS R consists of the following rules:

p1_1_in_g1(f_11(X)) -> if_p1_1_in_1_g2(X, p1_1_in_g1(X))
if_p1_1_in_1_g2(X, p1_1_out_g1(X)) -> p1_1_out_g1(f_11(X))

The argument filtering Pi contains the following mapping:
p1_1_in_g1(x1)  =  p1_1_in_g1(x1)
f_11(x1)  =  f_11(x1)
if_p1_1_in_1_g2(x1, x2)  =  if_p1_1_in_1_g1(x2)
p1_1_out_g1(x1)  =  p1_1_out_g
IF_P1_1_IN_1_G2(x1, x2)  =  IF_P1_1_IN_1_G1(x2)
P1_1_IN_G1(x1)  =  P1_1_IN_G1(x1)

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:

P1_1_IN_G1(f_11(X)) -> P1_1_IN_G1(X)

The TRS R consists of the following rules:

p1_1_in_g1(f_11(X)) -> if_p1_1_in_1_g2(X, p1_1_in_g1(X))
if_p1_1_in_1_g2(X, p1_1_out_g1(X)) -> p1_1_out_g1(f_11(X))

The argument filtering Pi contains the following mapping:
p1_1_in_g1(x1)  =  p1_1_in_g1(x1)
f_11(x1)  =  f_11(x1)
if_p1_1_in_1_g2(x1, x2)  =  if_p1_1_in_1_g1(x2)
p1_1_out_g1(x1)  =  p1_1_out_g
P1_1_IN_G1(x1)  =  P1_1_IN_G1(x1)

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:

P1_1_IN_G1(f_11(X)) -> P1_1_IN_G1(X)

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
  ↳ UnrequestedClauseRemoverProof
    ↳ PROLOG
      ↳ PrologToPiTRSProof
        ↳ PiTRS
          ↳ DependencyPairsProof
            ↳ PiDP
              ↳ DependencyGraphProof
                ↳ PiDP
                  ↳ UsableRulesProof
                    ↳ PiDP
                      ↳ PiDPToQDPProof
                        ↳ QDP
                          ↳ QDPSizeChangeProof

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

P1_1_IN_G1(f_11(X)) -> P1_1_IN_G1(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {P1_1_IN_G1}.
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: