Left Termination proof of /tmp/tmpwO84RS/cnfequiv.pl

Left Termination of the query pattern cnfequiv_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

cnfequiv(X, Y) :- ','(transform(X, Z), cnfequiv(Z, Y)).
cnfequiv(X, X).
transform(n(n(X)), X).
transform(n(a(X, Y)), o(n(X), n(Y))).
transform(n(o(X, Y)), a(n(X), n(Y))).
transform(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))).
transform(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))).
transform(o(X1, Y), o(X2, Y)) :- transform(X1, X2).
transform(o(X, Y1), o(X, Y2)) :- transform(Y1, Y2).
transform(a(X1, Y), a(X2, Y)) :- transform(X1, X2).
transform(a(X, Y1), a(X, Y2)) :- transform(Y1, Y2).
transform(n(X1), n(X2)) :- transform(X1, X2).

Queries:

cnfequiv(g,a).

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

cnfequiv_in_ga(X, Y) → U1_ga(X, Y, transform_in_ga(X, Z))
transform_in_ga(n(n(X)), X) → transform_out_ga(n(n(X)), X)
transform_in_ga(n(a(X, Y)), o(n(X), n(Y))) → transform_out_ga(n(a(X, Y)), o(n(X), n(Y)))
transform_in_ga(n(o(X, Y)), a(n(X), n(Y))) → transform_out_ga(n(o(X, Y)), a(n(X), n(Y)))
transform_in_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) → transform_out_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z)))
transform_in_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) → transform_out_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z)))
transform_in_ga(o(X1, Y), o(X2, Y)) → U3_ga(X1, Y, X2, transform_in_ga(X1, X2))
transform_in_ga(o(X, Y1), o(X, Y2)) → U4_ga(X, Y1, Y2, transform_in_ga(Y1, Y2))
transform_in_ga(a(X1, Y), a(X2, Y)) → U5_ga(X1, Y, X2, transform_in_ga(X1, X2))
transform_in_ga(a(X, Y1), a(X, Y2)) → U6_ga(X, Y1, Y2, transform_in_ga(Y1, Y2))
transform_in_ga(n(X1), n(X2)) → U7_ga(X1, X2, transform_in_ga(X1, X2))
U7_ga(X1, X2, transform_out_ga(X1, X2)) → transform_out_ga(n(X1), n(X2))
U6_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) → transform_out_ga(a(X, Y1), a(X, Y2))
U5_ga(X1, Y, X2, transform_out_ga(X1, X2)) → transform_out_ga(a(X1, Y), a(X2, Y))
U4_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) → transform_out_ga(o(X, Y1), o(X, Y2))
U3_ga(X1, Y, X2, transform_out_ga(X1, X2)) → transform_out_ga(o(X1, Y), o(X2, Y))
U1_ga(X, Y, transform_out_ga(X, Z)) → U2_ga(X, Y, cnfequiv_in_ga(Z, Y))
cnfequiv_in_ga(X, X) → cnfequiv_out_ga(X, X)
U2_ga(X, Y, cnfequiv_out_ga(Z, Y)) → cnfequiv_out_ga(X, Y)

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:

cnfequiv_in_ga(X, Y) → U1_ga(X, Y, transform_in_ga(X, Z))
transform_in_ga(n(n(X)), X) → transform_out_ga(n(n(X)), X)
transform_in_ga(n(a(X, Y)), o(n(X), n(Y))) → transform_out_ga(n(a(X, Y)), o(n(X), n(Y)))
transform_in_ga(n(o(X, Y)), a(n(X), n(Y))) → transform_out_ga(n(o(X, Y)), a(n(X), n(Y)))
transform_in_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) → transform_out_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z)))
transform_in_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) → transform_out_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z)))
transform_in_ga(o(X1, Y), o(X2, Y)) → U3_ga(X1, Y, X2, transform_in_ga(X1, X2))
transform_in_ga(o(X, Y1), o(X, Y2)) → U4_ga(X, Y1, Y2, transform_in_ga(Y1, Y2))
transform_in_ga(a(X1, Y), a(X2, Y)) → U5_ga(X1, Y, X2, transform_in_ga(X1, X2))
transform_in_ga(a(X, Y1), a(X, Y2)) → U6_ga(X, Y1, Y2, transform_in_ga(Y1, Y2))
transform_in_ga(n(X1), n(X2)) → U7_ga(X1, X2, transform_in_ga(X1, X2))
U7_ga(X1, X2, transform_out_ga(X1, X2)) → transform_out_ga(n(X1), n(X2))
U6_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) → transform_out_ga(a(X, Y1), a(X, Y2))
U5_ga(X1, Y, X2, transform_out_ga(X1, X2)) → transform_out_ga(a(X1, Y), a(X2, Y))
U4_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) → transform_out_ga(o(X, Y1), o(X, Y2))
U3_ga(X1, Y, X2, transform_out_ga(X1, X2)) → transform_out_ga(o(X1, Y), o(X2, Y))
U1_ga(X, Y, transform_out_ga(X, Z)) → U2_ga(X, Y, cnfequiv_in_ga(Z, Y))
cnfequiv_in_ga(X, X) → cnfequiv_out_ga(X, X)
U2_ga(X, Y, cnfequiv_out_ga(Z, Y)) → cnfequiv_out_ga(X, Y)

CNFEQUIV_IN_GA(X, Y) → U1_GA(X, Y, transform_in_ga(X, Z))
CNFEQUIV_IN_GA(X, Y) → TRANSFORM_IN_GA(X, Z)
TRANSFORM_IN_GA(o(X1, Y), o(X2, Y)) → U3_GA(X1, Y, X2, transform_in_ga(X1, X2))
TRANSFORM_IN_GA(o(X1, Y), o(X2, Y)) → TRANSFORM_IN_GA(X1, X2)
TRANSFORM_IN_GA(o(X, Y1), o(X, Y2)) → U4_GA(X, Y1, Y2, transform_in_ga(Y1, Y2))
TRANSFORM_IN_GA(o(X, Y1), o(X, Y2)) → TRANSFORM_IN_GA(Y1, Y2)
TRANSFORM_IN_GA(a(X1, Y), a(X2, Y)) → U5_GA(X1, Y, X2, transform_in_ga(X1, X2))
TRANSFORM_IN_GA(a(X1, Y), a(X2, Y)) → TRANSFORM_IN_GA(X1, X2)
TRANSFORM_IN_GA(a(X, Y1), a(X, Y2)) → U6_GA(X, Y1, Y2, transform_in_ga(Y1, Y2))
TRANSFORM_IN_GA(a(X, Y1), a(X, Y2)) → TRANSFORM_IN_GA(Y1, Y2)
TRANSFORM_IN_GA(n(X1), n(X2)) → U7_GA(X1, X2, transform_in_ga(X1, X2))
TRANSFORM_IN_GA(n(X1), n(X2)) → TRANSFORM_IN_GA(X1, X2)
U1_GA(X, Y, transform_out_ga(X, Z)) → U2_GA(X, Y, cnfequiv_in_ga(Z, Y))
U1_GA(X, Y, transform_out_ga(X, Z)) → CNFEQUIV_IN_GA(Z, Y)

The TRS R consists of the following rules:

cnfequiv_in_ga(X, Y) → U1_ga(X, Y, transform_in_ga(X, Z))
transform_in_ga(n(n(X)), X) → transform_out_ga(n(n(X)), X)
transform_in_ga(n(a(X, Y)), o(n(X), n(Y))) → transform_out_ga(n(a(X, Y)), o(n(X), n(Y)))
transform_in_ga(n(o(X, Y)), a(n(X), n(Y))) → transform_out_ga(n(o(X, Y)), a(n(X), n(Y)))
transform_in_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) → transform_out_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z)))
transform_in_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) → transform_out_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z)))
transform_in_ga(o(X1, Y), o(X2, Y)) → U3_ga(X1, Y, X2, transform_in_ga(X1, X2))
transform_in_ga(o(X, Y1), o(X, Y2)) → U4_ga(X, Y1, Y2, transform_in_ga(Y1, Y2))
transform_in_ga(a(X1, Y), a(X2, Y)) → U5_ga(X1, Y, X2, transform_in_ga(X1, X2))
transform_in_ga(a(X, Y1), a(X, Y2)) → U6_ga(X, Y1, Y2, transform_in_ga(Y1, Y2))
transform_in_ga(n(X1), n(X2)) → U7_ga(X1, X2, transform_in_ga(X1, X2))
U7_ga(X1, X2, transform_out_ga(X1, X2)) → transform_out_ga(n(X1), n(X2))
U6_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) → transform_out_ga(a(X, Y1), a(X, Y2))
U5_ga(X1, Y, X2, transform_out_ga(X1, X2)) → transform_out_ga(a(X1, Y), a(X2, Y))
U4_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) → transform_out_ga(o(X, Y1), o(X, Y2))
U3_ga(X1, Y, X2, transform_out_ga(X1, X2)) → transform_out_ga(o(X1, Y), o(X2, Y))
U1_ga(X, Y, transform_out_ga(X, Z)) → U2_ga(X, Y, cnfequiv_in_ga(Z, Y))
cnfequiv_in_ga(X, X) → cnfequiv_out_ga(X, X)
U2_ga(X, Y, cnfequiv_out_ga(Z, Y)) → cnfequiv_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
cnfequiv_in_ga(x1, x2) = cnfequiv_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
transform_in_ga(x1, x2) = transform_in_ga(x1)
n(x1) = n(x1)
transform_out_ga(x1, x2) = transform_out_ga(x2)
a(x1, x2) = a(x1, x2)
o(x1, x2) = o(x1, x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4)
U4_ga(x1, x2, x3, x4) = U4_ga(x1, x4)
U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4)
U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4)
U7_ga(x1, x2, x3) = U7_ga(x3)
U2_ga(x1, x2, x3) = U2_ga(x3)
cnfequiv_out_ga(x1, x2) = cnfequiv_out_ga(x2)
U6_GA(x1, x2, x3, x4) = U6_GA(x1, x4)
U5_GA(x1, x2, x3, x4) = U5_GA(x2, x4)
TRANSFORM_IN_GA(x1, x2) = TRANSFORM_IN_GA(x1)
U4_GA(x1, x2, x3, x4) = U4_GA(x1, x4)
U7_GA(x1, x2, x3) = U7_GA(x3)
U3_GA(x1, x2, x3, x4) = U3_GA(x2, x4)
CNFEQUIV_IN_GA(x1, x2) = CNFEQUIV_IN_GA(x1)
U2_GA(x1, x2, x3) = U2_GA(x3)
U1_GA(x1, x2, x3) = U1_GA(x3)

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:

CNFEQUIV_IN_GA(X, Y) → U1_GA(X, Y, transform_in_ga(X, Z))
CNFEQUIV_IN_GA(X, Y) → TRANSFORM_IN_GA(X, Z)
TRANSFORM_IN_GA(o(X1, Y), o(X2, Y)) → U3_GA(X1, Y, X2, transform_in_ga(X1, X2))
TRANSFORM_IN_GA(o(X1, Y), o(X2, Y)) → TRANSFORM_IN_GA(X1, X2)
TRANSFORM_IN_GA(o(X, Y1), o(X, Y2)) → U4_GA(X, Y1, Y2, transform_in_ga(Y1, Y2))
TRANSFORM_IN_GA(o(X, Y1), o(X, Y2)) → TRANSFORM_IN_GA(Y1, Y2)
TRANSFORM_IN_GA(a(X1, Y), a(X2, Y)) → U5_GA(X1, Y, X2, transform_in_ga(X1, X2))
TRANSFORM_IN_GA(a(X1, Y), a(X2, Y)) → TRANSFORM_IN_GA(X1, X2)
TRANSFORM_IN_GA(a(X, Y1), a(X, Y2)) → U6_GA(X, Y1, Y2, transform_in_ga(Y1, Y2))
TRANSFORM_IN_GA(a(X, Y1), a(X, Y2)) → TRANSFORM_IN_GA(Y1, Y2)
TRANSFORM_IN_GA(n(X1), n(X2)) → U7_GA(X1, X2, transform_in_ga(X1, X2))
TRANSFORM_IN_GA(n(X1), n(X2)) → TRANSFORM_IN_GA(X1, X2)
U1_GA(X, Y, transform_out_ga(X, Z)) → U2_GA(X, Y, cnfequiv_in_ga(Z, Y))
U1_GA(X, Y, transform_out_ga(X, Z)) → CNFEQUIV_IN_GA(Z, Y)

The TRS R consists of the following rules:

cnfequiv_in_ga(X, Y) → U1_ga(X, Y, transform_in_ga(X, Z))
transform_in_ga(n(n(X)), X) → transform_out_ga(n(n(X)), X)
transform_in_ga(n(a(X, Y)), o(n(X), n(Y))) → transform_out_ga(n(a(X, Y)), o(n(X), n(Y)))
transform_in_ga(n(o(X, Y)), a(n(X), n(Y))) → transform_out_ga(n(o(X, Y)), a(n(X), n(Y)))
transform_in_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) → transform_out_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z)))
transform_in_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) → transform_out_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z)))
transform_in_ga(o(X1, Y), o(X2, Y)) → U3_ga(X1, Y, X2, transform_in_ga(X1, X2))
transform_in_ga(o(X, Y1), o(X, Y2)) → U4_ga(X, Y1, Y2, transform_in_ga(Y1, Y2))
transform_in_ga(a(X1, Y), a(X2, Y)) → U5_ga(X1, Y, X2, transform_in_ga(X1, X2))
transform_in_ga(a(X, Y1), a(X, Y2)) → U6_ga(X, Y1, Y2, transform_in_ga(Y1, Y2))
transform_in_ga(n(X1), n(X2)) → U7_ga(X1, X2, transform_in_ga(X1, X2))
U7_ga(X1, X2, transform_out_ga(X1, X2)) → transform_out_ga(n(X1), n(X2))
U6_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) → transform_out_ga(a(X, Y1), a(X, Y2))
U5_ga(X1, Y, X2, transform_out_ga(X1, X2)) → transform_out_ga(a(X1, Y), a(X2, Y))
U4_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) → transform_out_ga(o(X, Y1), o(X, Y2))
U3_ga(X1, Y, X2, transform_out_ga(X1, X2)) → transform_out_ga(o(X1, Y), o(X2, Y))
U1_ga(X, Y, transform_out_ga(X, Z)) → U2_ga(X, Y, cnfequiv_in_ga(Z, Y))
cnfequiv_in_ga(X, X) → cnfequiv_out_ga(X, X)
U2_ga(X, Y, cnfequiv_out_ga(Z, Y)) → cnfequiv_out_ga(X, Y)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

TRANSFORM_IN_GA(n(X1), n(X2)) → TRANSFORM_IN_GA(X1, X2)
TRANSFORM_IN_GA(o(X, Y1), o(X, Y2)) → TRANSFORM_IN_GA(Y1, Y2)
TRANSFORM_IN_GA(a(X1, Y), a(X2, Y)) → TRANSFORM_IN_GA(X1, X2)
TRANSFORM_IN_GA(o(X1, Y), o(X2, Y)) → TRANSFORM_IN_GA(X1, X2)
TRANSFORM_IN_GA(a(X, Y1), a(X, Y2)) → TRANSFORM_IN_GA(Y1, Y2)

The TRS R consists of the following rules:

cnfequiv_in_ga(X, Y) → U1_ga(X, Y, transform_in_ga(X, Z))
transform_in_ga(n(n(X)), X) → transform_out_ga(n(n(X)), X)
transform_in_ga(n(a(X, Y)), o(n(X), n(Y))) → transform_out_ga(n(a(X, Y)), o(n(X), n(Y)))
transform_in_ga(n(o(X, Y)), a(n(X), n(Y))) → transform_out_ga(n(o(X, Y)), a(n(X), n(Y)))
transform_in_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) → transform_out_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z)))
transform_in_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) → transform_out_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z)))
transform_in_ga(o(X1, Y), o(X2, Y)) → U3_ga(X1, Y, X2, transform_in_ga(X1, X2))
transform_in_ga(o(X, Y1), o(X, Y2)) → U4_ga(X, Y1, Y2, transform_in_ga(Y1, Y2))
transform_in_ga(a(X1, Y), a(X2, Y)) → U5_ga(X1, Y, X2, transform_in_ga(X1, X2))
transform_in_ga(a(X, Y1), a(X, Y2)) → U6_ga(X, Y1, Y2, transform_in_ga(Y1, Y2))
transform_in_ga(n(X1), n(X2)) → U7_ga(X1, X2, transform_in_ga(X1, X2))
U7_ga(X1, X2, transform_out_ga(X1, X2)) → transform_out_ga(n(X1), n(X2))
U6_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) → transform_out_ga(a(X, Y1), a(X, Y2))
U5_ga(X1, Y, X2, transform_out_ga(X1, X2)) → transform_out_ga(a(X1, Y), a(X2, Y))
U4_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) → transform_out_ga(o(X, Y1), o(X, Y2))
U3_ga(X1, Y, X2, transform_out_ga(X1, X2)) → transform_out_ga(o(X1, Y), o(X2, Y))
U1_ga(X, Y, transform_out_ga(X, Z)) → U2_ga(X, Y, cnfequiv_in_ga(Z, Y))
cnfequiv_in_ga(X, X) → cnfequiv_out_ga(X, X)
U2_ga(X, Y, cnfequiv_out_ga(Z, Y)) → cnfequiv_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
cnfequiv_in_ga(x1, x2) = cnfequiv_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
transform_in_ga(x1, x2) = transform_in_ga(x1)
n(x1) = n(x1)
transform_out_ga(x1, x2) = transform_out_ga(x2)
a(x1, x2) = a(x1, x2)
o(x1, x2) = o(x1, x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4)
U4_ga(x1, x2, x3, x4) = U4_ga(x1, x4)
U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4)
U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4)
U7_ga(x1, x2, x3) = U7_ga(x3)
U2_ga(x1, x2, x3) = U2_ga(x3)
cnfequiv_out_ga(x1, x2) = cnfequiv_out_ga(x2)
TRANSFORM_IN_GA(x1, x2) = TRANSFORM_IN_GA(x1)

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

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

TRANSFORM_IN_GA(n(X1), n(X2)) → TRANSFORM_IN_GA(X1, X2)
TRANSFORM_IN_GA(o(X, Y1), o(X, Y2)) → TRANSFORM_IN_GA(Y1, Y2)
TRANSFORM_IN_GA(a(X1, Y), a(X2, Y)) → TRANSFORM_IN_GA(X1, X2)
TRANSFORM_IN_GA(o(X1, Y), o(X2, Y)) → TRANSFORM_IN_GA(X1, X2)
TRANSFORM_IN_GA(a(X, Y1), a(X, Y2)) → TRANSFORM_IN_GA(Y1, Y2)

R is empty.
The argument filtering Pi contains the following mapping:
n(x1) = n(x1)
a(x1, x2) = a(x1, x2)
o(x1, x2) = o(x1, x2)
TRANSFORM_IN_GA(x1, x2) = TRANSFORM_IN_GA(x1)

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

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

TRANSFORM_IN_GA(a(X1, Y)) → TRANSFORM_IN_GA(X1)
TRANSFORM_IN_GA(n(X1)) → TRANSFORM_IN_GA(X1)
TRANSFORM_IN_GA(o(X, Y1)) → TRANSFORM_IN_GA(Y1)
TRANSFORM_IN_GA(o(X1, Y)) → TRANSFORM_IN_GA(X1)
TRANSFORM_IN_GA(a(X, Y1)) → TRANSFORM_IN_GA(Y1)

R is empty.
Q is empty.
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:

TRANSFORM_IN_GA(n(X1)) → TRANSFORM_IN_GA(X1)
The graph contains the following edges 1 > 1
TRANSFORM_IN_GA(o(X, Y1)) → TRANSFORM_IN_GA(Y1)
The graph contains the following edges 1 > 1
TRANSFORM_IN_GA(a(X1, Y)) → TRANSFORM_IN_GA(X1)
The graph contains the following edges 1 > 1
TRANSFORM_IN_GA(o(X1, Y)) → TRANSFORM_IN_GA(X1)
The graph contains the following edges 1 > 1
TRANSFORM_IN_GA(a(X, Y1)) → TRANSFORM_IN_GA(Y1)
The graph contains the following edges 1 > 1

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

CNFEQUIV_IN_GA(X, Y) → U1_GA(X, Y, transform_in_ga(X, Z))
U1_GA(X, Y, transform_out_ga(X, Z)) → CNFEQUIV_IN_GA(Z, Y)

The TRS R consists of the following rules:

cnfequiv_in_ga(X, Y) → U1_ga(X, Y, transform_in_ga(X, Z))
transform_in_ga(n(n(X)), X) → transform_out_ga(n(n(X)), X)
transform_in_ga(n(a(X, Y)), o(n(X), n(Y))) → transform_out_ga(n(a(X, Y)), o(n(X), n(Y)))
transform_in_ga(n(o(X, Y)), a(n(X), n(Y))) → transform_out_ga(n(o(X, Y)), a(n(X), n(Y)))
transform_in_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) → transform_out_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z)))
transform_in_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) → transform_out_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z)))
transform_in_ga(o(X1, Y), o(X2, Y)) → U3_ga(X1, Y, X2, transform_in_ga(X1, X2))
transform_in_ga(o(X, Y1), o(X, Y2)) → U4_ga(X, Y1, Y2, transform_in_ga(Y1, Y2))
transform_in_ga(a(X1, Y), a(X2, Y)) → U5_ga(X1, Y, X2, transform_in_ga(X1, X2))
transform_in_ga(a(X, Y1), a(X, Y2)) → U6_ga(X, Y1, Y2, transform_in_ga(Y1, Y2))
transform_in_ga(n(X1), n(X2)) → U7_ga(X1, X2, transform_in_ga(X1, X2))
U7_ga(X1, X2, transform_out_ga(X1, X2)) → transform_out_ga(n(X1), n(X2))
U6_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) → transform_out_ga(a(X, Y1), a(X, Y2))
U5_ga(X1, Y, X2, transform_out_ga(X1, X2)) → transform_out_ga(a(X1, Y), a(X2, Y))
U4_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) → transform_out_ga(o(X, Y1), o(X, Y2))
U3_ga(X1, Y, X2, transform_out_ga(X1, X2)) → transform_out_ga(o(X1, Y), o(X2, Y))
U1_ga(X, Y, transform_out_ga(X, Z)) → U2_ga(X, Y, cnfequiv_in_ga(Z, Y))
cnfequiv_in_ga(X, X) → cnfequiv_out_ga(X, X)
U2_ga(X, Y, cnfequiv_out_ga(Z, Y)) → cnfequiv_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
cnfequiv_in_ga(x1, x2) = cnfequiv_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
transform_in_ga(x1, x2) = transform_in_ga(x1)
n(x1) = n(x1)
transform_out_ga(x1, x2) = transform_out_ga(x2)
a(x1, x2) = a(x1, x2)
o(x1, x2) = o(x1, x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4)
U4_ga(x1, x2, x3, x4) = U4_ga(x1, x4)
U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4)
U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4)
U7_ga(x1, x2, x3) = U7_ga(x3)
U2_ga(x1, x2, x3) = U2_ga(x3)
cnfequiv_out_ga(x1, x2) = cnfequiv_out_ga(x2)
CNFEQUIV_IN_GA(x1, x2) = CNFEQUIV_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x3)

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

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

CNFEQUIV_IN_GA(X, Y) → U1_GA(X, Y, transform_in_ga(X, Z))
U1_GA(X, Y, transform_out_ga(X, Z)) → CNFEQUIV_IN_GA(Z, Y)

The TRS R consists of the following rules:

transform_in_ga(n(n(X)), X) → transform_out_ga(n(n(X)), X)
transform_in_ga(n(a(X, Y)), o(n(X), n(Y))) → transform_out_ga(n(a(X, Y)), o(n(X), n(Y)))
transform_in_ga(n(o(X, Y)), a(n(X), n(Y))) → transform_out_ga(n(o(X, Y)), a(n(X), n(Y)))
transform_in_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) → transform_out_ga(o(X, a(Y, Z)), a(o(X, Y), o(X, Z)))
transform_in_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) → transform_out_ga(o(a(X, Y), Z), a(o(X, Z), o(Y, Z)))
transform_in_ga(o(X1, Y), o(X2, Y)) → U3_ga(X1, Y, X2, transform_in_ga(X1, X2))
transform_in_ga(o(X, Y1), o(X, Y2)) → U4_ga(X, Y1, Y2, transform_in_ga(Y1, Y2))
transform_in_ga(a(X1, Y), a(X2, Y)) → U5_ga(X1, Y, X2, transform_in_ga(X1, X2))
transform_in_ga(a(X, Y1), a(X, Y2)) → U6_ga(X, Y1, Y2, transform_in_ga(Y1, Y2))
transform_in_ga(n(X1), n(X2)) → U7_ga(X1, X2, transform_in_ga(X1, X2))
U3_ga(X1, Y, X2, transform_out_ga(X1, X2)) → transform_out_ga(o(X1, Y), o(X2, Y))
U4_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) → transform_out_ga(o(X, Y1), o(X, Y2))
U5_ga(X1, Y, X2, transform_out_ga(X1, X2)) → transform_out_ga(a(X1, Y), a(X2, Y))
U6_ga(X, Y1, Y2, transform_out_ga(Y1, Y2)) → transform_out_ga(a(X, Y1), a(X, Y2))
U7_ga(X1, X2, transform_out_ga(X1, X2)) → transform_out_ga(n(X1), n(X2))

The argument filtering Pi contains the following mapping:
transform_in_ga(x1, x2) = transform_in_ga(x1)
n(x1) = n(x1)
transform_out_ga(x1, x2) = transform_out_ga(x2)
a(x1, x2) = a(x1, x2)
o(x1, x2) = o(x1, x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4)
U4_ga(x1, x2, x3, x4) = U4_ga(x1, x4)
U5_ga(x1, x2, x3, x4) = U5_ga(x2, x4)
U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4)
U7_ga(x1, x2, x3) = U7_ga(x3)
CNFEQUIV_IN_GA(x1, x2) = CNFEQUIV_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x3)

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

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

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

CNFEQUIV_IN_GA(X) → U1_GA(transform_in_ga(X))
U1_GA(transform_out_ga(Z)) → CNFEQUIV_IN_GA(Z)

The TRS R consists of the following rules:

transform_in_ga(n(n(X))) → transform_out_ga(X)
transform_in_ga(n(a(X, Y))) → transform_out_ga(o(n(X), n(Y)))
transform_in_ga(n(o(X, Y))) → transform_out_ga(a(n(X), n(Y)))
transform_in_ga(o(X, a(Y, Z))) → transform_out_ga(a(o(X, Y), o(X, Z)))
transform_in_ga(o(a(X, Y), Z)) → transform_out_ga(a(o(X, Z), o(Y, Z)))
transform_in_ga(o(X1, Y)) → U3_ga(Y, transform_in_ga(X1))
transform_in_ga(o(X, Y1)) → U4_ga(X, transform_in_ga(Y1))
transform_in_ga(a(X1, Y)) → U5_ga(Y, transform_in_ga(X1))
transform_in_ga(a(X, Y1)) → U6_ga(X, transform_in_ga(Y1))
transform_in_ga(n(X1)) → U7_ga(transform_in_ga(X1))
U3_ga(Y, transform_out_ga(X2)) → transform_out_ga(o(X2, Y))
U4_ga(X, transform_out_ga(Y2)) → transform_out_ga(o(X, Y2))
U5_ga(Y, transform_out_ga(X2)) → transform_out_ga(a(X2, Y))
U6_ga(X, transform_out_ga(Y2)) → transform_out_ga(a(X, Y2))
U7_ga(transform_out_ga(X2)) → transform_out_ga(n(X2))

The set Q consists of the following terms:

transform_in_ga(x0)
U3_ga(x0, x1)
U4_ga(x0, x1)
U5_ga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0)

We have to consider all (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

U1_GA(transform_out_ga(Z)) → CNFEQUIV_IN_GA(Z)

The remaining pairs can at least be oriented weakly.

CNFEQUIV_IN_GA(X) → U1_GA(transform_in_ga(X))

Used ordering: Combined order from the following AFS and order.
CNFEQUIV_IN_GA(x1) = CNFEQUIV_IN_GA(x1)
U1_GA(x1) = U1_GA(x1)
transform_in_ga(x1) = x1
transform_out_ga(x1) = transform_out_ga(x1)
o(x1, x2) = o(x1, x2)
U3_ga(x1, x2) = U3_ga(x1, x2)
n(x1) = n(x1)
U7_ga(x1) = U7_ga(x1)
a(x1, x2) = a(x1, x2)
U4_ga(x1, x2) = U4_ga(x1, x2)
U6_ga(x1, x2) = U6_ga(x1, x2)
U5_ga(x1, x2) = U5_ga(x1, x2)

Recursive path order with status [2].
Quasi-Precedence:

[n₁, U7_{ga_1]} > [o₂, U3_ga₂, U4_{ga_2]} > [a₂, U6_ga₂, U5_{ga_2]} > transform_{out_ga₁} > [CNFEQUIV_{IN_GA₁}, U1_{GA_1]}

Status:

U6_ga₂: multiset
U5_ga₂: multiset
a₂: multiset
transform_{out_ga₁}: multiset
U4_ga₂: [1,2]
U3_ga₂: [2,1]
n₁: [1]
U7_ga₁: [1]
o₂: [1,2]
CNFEQUIV_{IN_GA₁}: multiset
U1_GA₁: multiset

The following usable rules [17] were oriented:

transform_in_ga(o(X1, Y)) → U3_ga(Y, transform_in_ga(X1))
transform_in_ga(n(X1)) → U7_ga(transform_in_ga(X1))
U7_ga(transform_out_ga(X2)) → transform_out_ga(n(X2))
transform_in_ga(n(o(X, Y))) → transform_out_ga(a(n(X), n(Y)))
U3_ga(Y, transform_out_ga(X2)) → transform_out_ga(o(X2, Y))
transform_in_ga(o(X, Y1)) → U4_ga(X, transform_in_ga(Y1))
U6_ga(X, transform_out_ga(Y2)) → transform_out_ga(a(X, Y2))
transform_in_ga(n(a(X, Y))) → transform_out_ga(o(n(X), n(Y)))
U4_ga(X, transform_out_ga(Y2)) → transform_out_ga(o(X, Y2))
transform_in_ga(a(X1, Y)) → U5_ga(Y, transform_in_ga(X1))
U5_ga(Y, transform_out_ga(X2)) → transform_out_ga(a(X2, Y))
transform_in_ga(n(n(X))) → transform_out_ga(X)
transform_in_ga(o(a(X, Y), Z)) → transform_out_ga(a(o(X, Z), o(Y, Z)))
transform_in_ga(a(X, Y1)) → U6_ga(X, transform_in_ga(Y1))
transform_in_ga(o(X, a(Y, Z))) → transform_out_ga(a(o(X, Y), o(X, Z)))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

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

CNFEQUIV_IN_GA(X) → U1_GA(transform_in_ga(X))

The TRS R consists of the following rules:

transform_in_ga(n(n(X))) → transform_out_ga(X)
transform_in_ga(n(a(X, Y))) → transform_out_ga(o(n(X), n(Y)))
transform_in_ga(n(o(X, Y))) → transform_out_ga(a(n(X), n(Y)))
transform_in_ga(o(X, a(Y, Z))) → transform_out_ga(a(o(X, Y), o(X, Z)))
transform_in_ga(o(a(X, Y), Z)) → transform_out_ga(a(o(X, Z), o(Y, Z)))
transform_in_ga(o(X1, Y)) → U3_ga(Y, transform_in_ga(X1))
transform_in_ga(o(X, Y1)) → U4_ga(X, transform_in_ga(Y1))
transform_in_ga(a(X1, Y)) → U5_ga(Y, transform_in_ga(X1))
transform_in_ga(a(X, Y1)) → U6_ga(X, transform_in_ga(Y1))
transform_in_ga(n(X1)) → U7_ga(transform_in_ga(X1))
U3_ga(Y, transform_out_ga(X2)) → transform_out_ga(o(X2, Y))
U4_ga(X, transform_out_ga(Y2)) → transform_out_ga(o(X, Y2))
U5_ga(Y, transform_out_ga(X2)) → transform_out_ga(a(X2, Y))
U6_ga(X, transform_out_ga(Y2)) → transform_out_ga(a(X, Y2))
U7_ga(transform_out_ga(X2)) → transform_out_ga(n(X2))

The set Q consists of the following terms:

transform_in_ga(x0)
U3_ga(x0, x1)
U4_ga(x0, x1)
U5_ga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.