Left Termination of the query pattern fold_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:



↳ Prolog
  ↳ PrologToPiTRSProof

Clauses:

fold(X, .(Y, Ys), Z) :- ','(myop(X, Y, V), fold(V, Ys, Z)).
fold(X, [], X).
myop(a, b, c).

Queries:

fold(g,g,a).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
fold_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

fold_in_gga(X, .(Y, Ys), Z) → U1_gga(X, Y, Ys, Z, myop_in_gga(X, Y, V))
myop_in_gga(a, b, c) → myop_out_gga(a, b, c)
U1_gga(X, Y, Ys, Z, myop_out_gga(X, Y, V)) → U2_gga(X, Y, Ys, Z, fold_in_gga(V, Ys, Z))
fold_in_gga(X, [], X) → fold_out_gga(X, [], X)
U2_gga(X, Y, Ys, Z, fold_out_gga(V, Ys, Z)) → fold_out_gga(X, .(Y, Ys), Z)

The argument filtering Pi contains the following mapping:
fold_in_gga(x1, x2, x3)  =  fold_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x3, x5)
myop_in_gga(x1, x2, x3)  =  myop_in_gga(x1, x2)
a  =  a
b  =  b
myop_out_gga(x1, x2, x3)  =  myop_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
[]  =  []
fold_out_gga(x1, x2, x3)  =  fold_out_gga(x3)

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:

fold_in_gga(X, .(Y, Ys), Z) → U1_gga(X, Y, Ys, Z, myop_in_gga(X, Y, V))
myop_in_gga(a, b, c) → myop_out_gga(a, b, c)
U1_gga(X, Y, Ys, Z, myop_out_gga(X, Y, V)) → U2_gga(X, Y, Ys, Z, fold_in_gga(V, Ys, Z))
fold_in_gga(X, [], X) → fold_out_gga(X, [], X)
U2_gga(X, Y, Ys, Z, fold_out_gga(V, Ys, Z)) → fold_out_gga(X, .(Y, Ys), Z)

The argument filtering Pi contains the following mapping:
fold_in_gga(x1, x2, x3)  =  fold_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x3, x5)
myop_in_gga(x1, x2, x3)  =  myop_in_gga(x1, x2)
a  =  a
b  =  b
myop_out_gga(x1, x2, x3)  =  myop_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
[]  =  []
fold_out_gga(x1, x2, x3)  =  fold_out_gga(x3)


Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

FOLD_IN_GGA(X, .(Y, Ys), Z) → U1_GGA(X, Y, Ys, Z, myop_in_gga(X, Y, V))
FOLD_IN_GGA(X, .(Y, Ys), Z) → MYOP_IN_GGA(X, Y, V)
U1_GGA(X, Y, Ys, Z, myop_out_gga(X, Y, V)) → U2_GGA(X, Y, Ys, Z, fold_in_gga(V, Ys, Z))
U1_GGA(X, Y, Ys, Z, myop_out_gga(X, Y, V)) → FOLD_IN_GGA(V, Ys, Z)

The TRS R consists of the following rules:

fold_in_gga(X, .(Y, Ys), Z) → U1_gga(X, Y, Ys, Z, myop_in_gga(X, Y, V))
myop_in_gga(a, b, c) → myop_out_gga(a, b, c)
U1_gga(X, Y, Ys, Z, myop_out_gga(X, Y, V)) → U2_gga(X, Y, Ys, Z, fold_in_gga(V, Ys, Z))
fold_in_gga(X, [], X) → fold_out_gga(X, [], X)
U2_gga(X, Y, Ys, Z, fold_out_gga(V, Ys, Z)) → fold_out_gga(X, .(Y, Ys), Z)

The argument filtering Pi contains the following mapping:
fold_in_gga(x1, x2, x3)  =  fold_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x3, x5)
myop_in_gga(x1, x2, x3)  =  myop_in_gga(x1, x2)
a  =  a
b  =  b
myop_out_gga(x1, x2, x3)  =  myop_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
[]  =  []
fold_out_gga(x1, x2, x3)  =  fold_out_gga(x3)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x3, x5)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x5)
FOLD_IN_GGA(x1, x2, x3)  =  FOLD_IN_GGA(x1, x2)
MYOP_IN_GGA(x1, x2, x3)  =  MYOP_IN_GGA(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:

FOLD_IN_GGA(X, .(Y, Ys), Z) → U1_GGA(X, Y, Ys, Z, myop_in_gga(X, Y, V))
FOLD_IN_GGA(X, .(Y, Ys), Z) → MYOP_IN_GGA(X, Y, V)
U1_GGA(X, Y, Ys, Z, myop_out_gga(X, Y, V)) → U2_GGA(X, Y, Ys, Z, fold_in_gga(V, Ys, Z))
U1_GGA(X, Y, Ys, Z, myop_out_gga(X, Y, V)) → FOLD_IN_GGA(V, Ys, Z)

The TRS R consists of the following rules:

fold_in_gga(X, .(Y, Ys), Z) → U1_gga(X, Y, Ys, Z, myop_in_gga(X, Y, V))
myop_in_gga(a, b, c) → myop_out_gga(a, b, c)
U1_gga(X, Y, Ys, Z, myop_out_gga(X, Y, V)) → U2_gga(X, Y, Ys, Z, fold_in_gga(V, Ys, Z))
fold_in_gga(X, [], X) → fold_out_gga(X, [], X)
U2_gga(X, Y, Ys, Z, fold_out_gga(V, Ys, Z)) → fold_out_gga(X, .(Y, Ys), Z)

The argument filtering Pi contains the following mapping:
fold_in_gga(x1, x2, x3)  =  fold_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x3, x5)
myop_in_gga(x1, x2, x3)  =  myop_in_gga(x1, x2)
a  =  a
b  =  b
myop_out_gga(x1, x2, x3)  =  myop_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
[]  =  []
fold_out_gga(x1, x2, x3)  =  fold_out_gga(x3)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x3, x5)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x5)
FOLD_IN_GGA(x1, x2, x3)  =  FOLD_IN_GGA(x1, x2)
MYOP_IN_GGA(x1, x2, x3)  =  MYOP_IN_GGA(x1, x2)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 2 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof

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

U1_GGA(X, Y, Ys, Z, myop_out_gga(X, Y, V)) → FOLD_IN_GGA(V, Ys, Z)
FOLD_IN_GGA(X, .(Y, Ys), Z) → U1_GGA(X, Y, Ys, Z, myop_in_gga(X, Y, V))

The TRS R consists of the following rules:

fold_in_gga(X, .(Y, Ys), Z) → U1_gga(X, Y, Ys, Z, myop_in_gga(X, Y, V))
myop_in_gga(a, b, c) → myop_out_gga(a, b, c)
U1_gga(X, Y, Ys, Z, myop_out_gga(X, Y, V)) → U2_gga(X, Y, Ys, Z, fold_in_gga(V, Ys, Z))
fold_in_gga(X, [], X) → fold_out_gga(X, [], X)
U2_gga(X, Y, Ys, Z, fold_out_gga(V, Ys, Z)) → fold_out_gga(X, .(Y, Ys), Z)

The argument filtering Pi contains the following mapping:
fold_in_gga(x1, x2, x3)  =  fold_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x3, x5)
myop_in_gga(x1, x2, x3)  =  myop_in_gga(x1, x2)
a  =  a
b  =  b
myop_out_gga(x1, x2, x3)  =  myop_out_gga(x3)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)
[]  =  []
fold_out_gga(x1, x2, x3)  =  fold_out_gga(x3)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x3, x5)
FOLD_IN_GGA(x1, x2, x3)  =  FOLD_IN_GGA(x1, x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof

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

U1_GGA(X, Y, Ys, Z, myop_out_gga(X, Y, V)) → FOLD_IN_GGA(V, Ys, Z)
FOLD_IN_GGA(X, .(Y, Ys), Z) → U1_GGA(X, Y, Ys, Z, myop_in_gga(X, Y, V))

The TRS R consists of the following rules:

myop_in_gga(a, b, c) → myop_out_gga(a, b, c)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
myop_in_gga(x1, x2, x3)  =  myop_in_gga(x1, x2)
a  =  a
b  =  b
myop_out_gga(x1, x2, x3)  =  myop_out_gga(x3)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x3, x5)
FOLD_IN_GGA(x1, x2, x3)  =  FOLD_IN_GGA(x1, x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

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

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

FOLD_IN_GGA(X, .(Y, Ys)) → U1_GGA(Ys, myop_in_gga(X, Y))
U1_GGA(Ys, myop_out_gga(V)) → FOLD_IN_GGA(V, Ys)

The TRS R consists of the following rules:

myop_in_gga(a, b) → myop_out_gga(c)

The set Q consists of the following terms:

myop_in_gga(x0, x1)

We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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: