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].Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

cnfequiv_in(X, X) → cnfequiv_out(X, X)
cnfequiv_in(X, Y) → U1(X, Y, transform_in(X, Z))
transform_in(n(X1), n(X2)) → U7(X1, X2, transform_in(X1, X2))
transform_in(a(X, Y1), a(X, Y2)) → U6(X, Y1, Y2, transform_in(Y1, Y2))
transform_in(a(X1, Y), a(X2, Y)) → U5(X1, Y, X2, transform_in(X1, X2))
transform_in(o(X, Y1), o(X, Y2)) → U4(X, Y1, Y2, transform_in(Y1, Y2))
transform_in(o(X1, Y), o(X2, Y)) → U3(X1, Y, X2, transform_in(X1, X2))
transform_in(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) → transform_out(o(a(X, Y), Z), a(o(X, Z), o(Y, Z)))
transform_in(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) → transform_out(o(X, a(Y, Z)), a(o(X, Y), o(X, Z)))
transform_in(n(o(X, Y)), a(n(X), n(Y))) → transform_out(n(o(X, Y)), a(n(X), n(Y)))
transform_in(n(a(X, Y)), o(n(X), n(Y))) → transform_out(n(a(X, Y)), o(n(X), n(Y)))
transform_in(n(n(X)), X) → transform_out(n(n(X)), X)
U3(X1, Y, X2, transform_out(X1, X2)) → transform_out(o(X1, Y), o(X2, Y))
U4(X, Y1, Y2, transform_out(Y1, Y2)) → transform_out(o(X, Y1), o(X, Y2))
U5(X1, Y, X2, transform_out(X1, X2)) → transform_out(a(X1, Y), a(X2, Y))
U6(X, Y1, Y2, transform_out(Y1, Y2)) → transform_out(a(X, Y1), a(X, Y2))
U7(X1, X2, transform_out(X1, X2)) → transform_out(n(X1), n(X2))
U1(X, Y, transform_out(X, Z)) → U2(X, Y, cnfequiv_in(Z, Y))
U2(X, Y, cnfequiv_out(Z, Y)) → cnfequiv_out(X, Y)

The argument filtering Pi contains the following mapping:
cnfequiv_in(x1, x2)  =  cnfequiv_in(x1)
cnfequiv_out(x1, x2)  =  cnfequiv_out(x2)
U1(x1, x2, x3)  =  U1(x3)
transform_in(x1, x2)  =  transform_in(x1)
n(x1)  =  n(x1)
U7(x1, x2, x3)  =  U7(x3)
a(x1, x2)  =  a(x1, x2)
U6(x1, x2, x3, x4)  =  U6(x1, x4)
U5(x1, x2, x3, x4)  =  U5(x2, x4)
o(x1, x2)  =  o(x1, x2)
U4(x1, x2, x3, x4)  =  U4(x1, x4)
U3(x1, x2, x3, x4)  =  U3(x2, x4)
transform_out(x1, x2)  =  transform_out(x2)
U2(x1, x2, x3)  =  U2(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:

cnfequiv_in(X, X) → cnfequiv_out(X, X)
cnfequiv_in(X, Y) → U1(X, Y, transform_in(X, Z))
transform_in(n(X1), n(X2)) → U7(X1, X2, transform_in(X1, X2))
transform_in(a(X, Y1), a(X, Y2)) → U6(X, Y1, Y2, transform_in(Y1, Y2))
transform_in(a(X1, Y), a(X2, Y)) → U5(X1, Y, X2, transform_in(X1, X2))
transform_in(o(X, Y1), o(X, Y2)) → U4(X, Y1, Y2, transform_in(Y1, Y2))
transform_in(o(X1, Y), o(X2, Y)) → U3(X1, Y, X2, transform_in(X1, X2))
transform_in(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) → transform_out(o(a(X, Y), Z), a(o(X, Z), o(Y, Z)))
transform_in(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) → transform_out(o(X, a(Y, Z)), a(o(X, Y), o(X, Z)))
transform_in(n(o(X, Y)), a(n(X), n(Y))) → transform_out(n(o(X, Y)), a(n(X), n(Y)))
transform_in(n(a(X, Y)), o(n(X), n(Y))) → transform_out(n(a(X, Y)), o(n(X), n(Y)))
transform_in(n(n(X)), X) → transform_out(n(n(X)), X)
U3(X1, Y, X2, transform_out(X1, X2)) → transform_out(o(X1, Y), o(X2, Y))
U4(X, Y1, Y2, transform_out(Y1, Y2)) → transform_out(o(X, Y1), o(X, Y2))
U5(X1, Y, X2, transform_out(X1, X2)) → transform_out(a(X1, Y), a(X2, Y))
U6(X, Y1, Y2, transform_out(Y1, Y2)) → transform_out(a(X, Y1), a(X, Y2))
U7(X1, X2, transform_out(X1, X2)) → transform_out(n(X1), n(X2))
U1(X, Y, transform_out(X, Z)) → U2(X, Y, cnfequiv_in(Z, Y))
U2(X, Y, cnfequiv_out(Z, Y)) → cnfequiv_out(X, Y)

The argument filtering Pi contains the following mapping:
cnfequiv_in(x1, x2)  =  cnfequiv_in(x1)
cnfequiv_out(x1, x2)  =  cnfequiv_out(x2)
U1(x1, x2, x3)  =  U1(x3)
transform_in(x1, x2)  =  transform_in(x1)
n(x1)  =  n(x1)
U7(x1, x2, x3)  =  U7(x3)
a(x1, x2)  =  a(x1, x2)
U6(x1, x2, x3, x4)  =  U6(x1, x4)
U5(x1, x2, x3, x4)  =  U5(x2, x4)
o(x1, x2)  =  o(x1, x2)
U4(x1, x2, x3, x4)  =  U4(x1, x4)
U3(x1, x2, x3, x4)  =  U3(x2, x4)
transform_out(x1, x2)  =  transform_out(x2)
U2(x1, x2, x3)  =  U2(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:

CNFEQUIV_IN(X, Y) → U11(X, Y, transform_in(X, Z))
CNFEQUIV_IN(X, Y) → TRANSFORM_IN(X, Z)
TRANSFORM_IN(n(X1), n(X2)) → U71(X1, X2, transform_in(X1, X2))
TRANSFORM_IN(n(X1), n(X2)) → TRANSFORM_IN(X1, X2)
TRANSFORM_IN(a(X, Y1), a(X, Y2)) → U61(X, Y1, Y2, transform_in(Y1, Y2))
TRANSFORM_IN(a(X, Y1), a(X, Y2)) → TRANSFORM_IN(Y1, Y2)
TRANSFORM_IN(a(X1, Y), a(X2, Y)) → U51(X1, Y, X2, transform_in(X1, X2))
TRANSFORM_IN(a(X1, Y), a(X2, Y)) → TRANSFORM_IN(X1, X2)
TRANSFORM_IN(o(X, Y1), o(X, Y2)) → U41(X, Y1, Y2, transform_in(Y1, Y2))
TRANSFORM_IN(o(X, Y1), o(X, Y2)) → TRANSFORM_IN(Y1, Y2)
TRANSFORM_IN(o(X1, Y), o(X2, Y)) → U31(X1, Y, X2, transform_in(X1, X2))
TRANSFORM_IN(o(X1, Y), o(X2, Y)) → TRANSFORM_IN(X1, X2)
U11(X, Y, transform_out(X, Z)) → U21(X, Y, cnfequiv_in(Z, Y))
U11(X, Y, transform_out(X, Z)) → CNFEQUIV_IN(Z, Y)

The TRS R consists of the following rules:

cnfequiv_in(X, X) → cnfequiv_out(X, X)
cnfequiv_in(X, Y) → U1(X, Y, transform_in(X, Z))
transform_in(n(X1), n(X2)) → U7(X1, X2, transform_in(X1, X2))
transform_in(a(X, Y1), a(X, Y2)) → U6(X, Y1, Y2, transform_in(Y1, Y2))
transform_in(a(X1, Y), a(X2, Y)) → U5(X1, Y, X2, transform_in(X1, X2))
transform_in(o(X, Y1), o(X, Y2)) → U4(X, Y1, Y2, transform_in(Y1, Y2))
transform_in(o(X1, Y), o(X2, Y)) → U3(X1, Y, X2, transform_in(X1, X2))
transform_in(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) → transform_out(o(a(X, Y), Z), a(o(X, Z), o(Y, Z)))
transform_in(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) → transform_out(o(X, a(Y, Z)), a(o(X, Y), o(X, Z)))
transform_in(n(o(X, Y)), a(n(X), n(Y))) → transform_out(n(o(X, Y)), a(n(X), n(Y)))
transform_in(n(a(X, Y)), o(n(X), n(Y))) → transform_out(n(a(X, Y)), o(n(X), n(Y)))
transform_in(n(n(X)), X) → transform_out(n(n(X)), X)
U3(X1, Y, X2, transform_out(X1, X2)) → transform_out(o(X1, Y), o(X2, Y))
U4(X, Y1, Y2, transform_out(Y1, Y2)) → transform_out(o(X, Y1), o(X, Y2))
U5(X1, Y, X2, transform_out(X1, X2)) → transform_out(a(X1, Y), a(X2, Y))
U6(X, Y1, Y2, transform_out(Y1, Y2)) → transform_out(a(X, Y1), a(X, Y2))
U7(X1, X2, transform_out(X1, X2)) → transform_out(n(X1), n(X2))
U1(X, Y, transform_out(X, Z)) → U2(X, Y, cnfequiv_in(Z, Y))
U2(X, Y, cnfequiv_out(Z, Y)) → cnfequiv_out(X, Y)

The argument filtering Pi contains the following mapping:
cnfequiv_in(x1, x2)  =  cnfequiv_in(x1)
cnfequiv_out(x1, x2)  =  cnfequiv_out(x2)
U1(x1, x2, x3)  =  U1(x3)
transform_in(x1, x2)  =  transform_in(x1)
n(x1)  =  n(x1)
U7(x1, x2, x3)  =  U7(x3)
a(x1, x2)  =  a(x1, x2)
U6(x1, x2, x3, x4)  =  U6(x1, x4)
U5(x1, x2, x3, x4)  =  U5(x2, x4)
o(x1, x2)  =  o(x1, x2)
U4(x1, x2, x3, x4)  =  U4(x1, x4)
U3(x1, x2, x3, x4)  =  U3(x2, x4)
transform_out(x1, x2)  =  transform_out(x2)
U2(x1, x2, x3)  =  U2(x3)
TRANSFORM_IN(x1, x2)  =  TRANSFORM_IN(x1)
U71(x1, x2, x3)  =  U71(x3)
U51(x1, x2, x3, x4)  =  U51(x2, x4)
U31(x1, x2, x3, x4)  =  U31(x2, x4)
U41(x1, x2, x3, x4)  =  U41(x1, x4)
CNFEQUIV_IN(x1, x2)  =  CNFEQUIV_IN(x1)
U21(x1, x2, x3)  =  U21(x3)
U61(x1, x2, x3, x4)  =  U61(x1, x4)
U11(x1, x2, x3)  =  U11(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(X, Y) → U11(X, Y, transform_in(X, Z))
CNFEQUIV_IN(X, Y) → TRANSFORM_IN(X, Z)
TRANSFORM_IN(n(X1), n(X2)) → U71(X1, X2, transform_in(X1, X2))
TRANSFORM_IN(n(X1), n(X2)) → TRANSFORM_IN(X1, X2)
TRANSFORM_IN(a(X, Y1), a(X, Y2)) → U61(X, Y1, Y2, transform_in(Y1, Y2))
TRANSFORM_IN(a(X, Y1), a(X, Y2)) → TRANSFORM_IN(Y1, Y2)
TRANSFORM_IN(a(X1, Y), a(X2, Y)) → U51(X1, Y, X2, transform_in(X1, X2))
TRANSFORM_IN(a(X1, Y), a(X2, Y)) → TRANSFORM_IN(X1, X2)
TRANSFORM_IN(o(X, Y1), o(X, Y2)) → U41(X, Y1, Y2, transform_in(Y1, Y2))
TRANSFORM_IN(o(X, Y1), o(X, Y2)) → TRANSFORM_IN(Y1, Y2)
TRANSFORM_IN(o(X1, Y), o(X2, Y)) → U31(X1, Y, X2, transform_in(X1, X2))
TRANSFORM_IN(o(X1, Y), o(X2, Y)) → TRANSFORM_IN(X1, X2)
U11(X, Y, transform_out(X, Z)) → U21(X, Y, cnfequiv_in(Z, Y))
U11(X, Y, transform_out(X, Z)) → CNFEQUIV_IN(Z, Y)

The TRS R consists of the following rules:

cnfequiv_in(X, X) → cnfequiv_out(X, X)
cnfequiv_in(X, Y) → U1(X, Y, transform_in(X, Z))
transform_in(n(X1), n(X2)) → U7(X1, X2, transform_in(X1, X2))
transform_in(a(X, Y1), a(X, Y2)) → U6(X, Y1, Y2, transform_in(Y1, Y2))
transform_in(a(X1, Y), a(X2, Y)) → U5(X1, Y, X2, transform_in(X1, X2))
transform_in(o(X, Y1), o(X, Y2)) → U4(X, Y1, Y2, transform_in(Y1, Y2))
transform_in(o(X1, Y), o(X2, Y)) → U3(X1, Y, X2, transform_in(X1, X2))
transform_in(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) → transform_out(o(a(X, Y), Z), a(o(X, Z), o(Y, Z)))
transform_in(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) → transform_out(o(X, a(Y, Z)), a(o(X, Y), o(X, Z)))
transform_in(n(o(X, Y)), a(n(X), n(Y))) → transform_out(n(o(X, Y)), a(n(X), n(Y)))
transform_in(n(a(X, Y)), o(n(X), n(Y))) → transform_out(n(a(X, Y)), o(n(X), n(Y)))
transform_in(n(n(X)), X) → transform_out(n(n(X)), X)
U3(X1, Y, X2, transform_out(X1, X2)) → transform_out(o(X1, Y), o(X2, Y))
U4(X, Y1, Y2, transform_out(Y1, Y2)) → transform_out(o(X, Y1), o(X, Y2))
U5(X1, Y, X2, transform_out(X1, X2)) → transform_out(a(X1, Y), a(X2, Y))
U6(X, Y1, Y2, transform_out(Y1, Y2)) → transform_out(a(X, Y1), a(X, Y2))
U7(X1, X2, transform_out(X1, X2)) → transform_out(n(X1), n(X2))
U1(X, Y, transform_out(X, Z)) → U2(X, Y, cnfequiv_in(Z, Y))
U2(X, Y, cnfequiv_out(Z, Y)) → cnfequiv_out(X, Y)

The argument filtering Pi contains the following mapping:
cnfequiv_in(x1, x2)  =  cnfequiv_in(x1)
cnfequiv_out(x1, x2)  =  cnfequiv_out(x2)
U1(x1, x2, x3)  =  U1(x3)
transform_in(x1, x2)  =  transform_in(x1)
n(x1)  =  n(x1)
U7(x1, x2, x3)  =  U7(x3)
a(x1, x2)  =  a(x1, x2)
U6(x1, x2, x3, x4)  =  U6(x1, x4)
U5(x1, x2, x3, x4)  =  U5(x2, x4)
o(x1, x2)  =  o(x1, x2)
U4(x1, x2, x3, x4)  =  U4(x1, x4)
U3(x1, x2, x3, x4)  =  U3(x2, x4)
transform_out(x1, x2)  =  transform_out(x2)
U2(x1, x2, x3)  =  U2(x3)
TRANSFORM_IN(x1, x2)  =  TRANSFORM_IN(x1)
U71(x1, x2, x3)  =  U71(x3)
U51(x1, x2, x3, x4)  =  U51(x2, x4)
U31(x1, x2, x3, x4)  =  U31(x2, x4)
U41(x1, x2, x3, x4)  =  U41(x1, x4)
CNFEQUIV_IN(x1, x2)  =  CNFEQUIV_IN(x1)
U21(x1, x2, x3)  =  U21(x3)
U61(x1, x2, x3, x4)  =  U61(x1, x4)
U11(x1, x2, x3)  =  U11(x3)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 7 less nodes.

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

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

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

The TRS R consists of the following rules:

cnfequiv_in(X, X) → cnfequiv_out(X, X)
cnfequiv_in(X, Y) → U1(X, Y, transform_in(X, Z))
transform_in(n(X1), n(X2)) → U7(X1, X2, transform_in(X1, X2))
transform_in(a(X, Y1), a(X, Y2)) → U6(X, Y1, Y2, transform_in(Y1, Y2))
transform_in(a(X1, Y), a(X2, Y)) → U5(X1, Y, X2, transform_in(X1, X2))
transform_in(o(X, Y1), o(X, Y2)) → U4(X, Y1, Y2, transform_in(Y1, Y2))
transform_in(o(X1, Y), o(X2, Y)) → U3(X1, Y, X2, transform_in(X1, X2))
transform_in(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) → transform_out(o(a(X, Y), Z), a(o(X, Z), o(Y, Z)))
transform_in(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) → transform_out(o(X, a(Y, Z)), a(o(X, Y), o(X, Z)))
transform_in(n(o(X, Y)), a(n(X), n(Y))) → transform_out(n(o(X, Y)), a(n(X), n(Y)))
transform_in(n(a(X, Y)), o(n(X), n(Y))) → transform_out(n(a(X, Y)), o(n(X), n(Y)))
transform_in(n(n(X)), X) → transform_out(n(n(X)), X)
U3(X1, Y, X2, transform_out(X1, X2)) → transform_out(o(X1, Y), o(X2, Y))
U4(X, Y1, Y2, transform_out(Y1, Y2)) → transform_out(o(X, Y1), o(X, Y2))
U5(X1, Y, X2, transform_out(X1, X2)) → transform_out(a(X1, Y), a(X2, Y))
U6(X, Y1, Y2, transform_out(Y1, Y2)) → transform_out(a(X, Y1), a(X, Y2))
U7(X1, X2, transform_out(X1, X2)) → transform_out(n(X1), n(X2))
U1(X, Y, transform_out(X, Z)) → U2(X, Y, cnfequiv_in(Z, Y))
U2(X, Y, cnfequiv_out(Z, Y)) → cnfequiv_out(X, Y)

The argument filtering Pi contains the following mapping:
cnfequiv_in(x1, x2)  =  cnfequiv_in(x1)
cnfequiv_out(x1, x2)  =  cnfequiv_out(x2)
U1(x1, x2, x3)  =  U1(x3)
transform_in(x1, x2)  =  transform_in(x1)
n(x1)  =  n(x1)
U7(x1, x2, x3)  =  U7(x3)
a(x1, x2)  =  a(x1, x2)
U6(x1, x2, x3, x4)  =  U6(x1, x4)
U5(x1, x2, x3, x4)  =  U5(x2, x4)
o(x1, x2)  =  o(x1, x2)
U4(x1, x2, x3, x4)  =  U4(x1, x4)
U3(x1, x2, x3, x4)  =  U3(x2, x4)
transform_out(x1, x2)  =  transform_out(x2)
U2(x1, x2, x3)  =  U2(x3)
TRANSFORM_IN(x1, x2)  =  TRANSFORM_IN(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(n(X1), n(X2)) → TRANSFORM_IN(X1, X2)
TRANSFORM_IN(a(X, Y1), a(X, Y2)) → TRANSFORM_IN(Y1, Y2)
TRANSFORM_IN(o(X1, Y), o(X2, Y)) → TRANSFORM_IN(X1, X2)
TRANSFORM_IN(a(X1, Y), a(X2, Y)) → TRANSFORM_IN(X1, X2)
TRANSFORM_IN(o(X, Y1), o(X, Y2)) → TRANSFORM_IN(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(x1, x2)  =  TRANSFORM_IN(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(o(X1, Y)) → TRANSFORM_IN(X1)
TRANSFORM_IN(a(X, Y1)) → TRANSFORM_IN(Y1)
TRANSFORM_IN(a(X1, Y)) → TRANSFORM_IN(X1)
TRANSFORM_IN(o(X, Y1)) → TRANSFORM_IN(Y1)
TRANSFORM_IN(n(X1)) → TRANSFORM_IN(X1)

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: 



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

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

U11(X, Y, transform_out(X, Z)) → CNFEQUIV_IN(Z, Y)
CNFEQUIV_IN(X, Y) → U11(X, Y, transform_in(X, Z))

The TRS R consists of the following rules:

cnfequiv_in(X, X) → cnfequiv_out(X, X)
cnfequiv_in(X, Y) → U1(X, Y, transform_in(X, Z))
transform_in(n(X1), n(X2)) → U7(X1, X2, transform_in(X1, X2))
transform_in(a(X, Y1), a(X, Y2)) → U6(X, Y1, Y2, transform_in(Y1, Y2))
transform_in(a(X1, Y), a(X2, Y)) → U5(X1, Y, X2, transform_in(X1, X2))
transform_in(o(X, Y1), o(X, Y2)) → U4(X, Y1, Y2, transform_in(Y1, Y2))
transform_in(o(X1, Y), o(X2, Y)) → U3(X1, Y, X2, transform_in(X1, X2))
transform_in(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) → transform_out(o(a(X, Y), Z), a(o(X, Z), o(Y, Z)))
transform_in(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) → transform_out(o(X, a(Y, Z)), a(o(X, Y), o(X, Z)))
transform_in(n(o(X, Y)), a(n(X), n(Y))) → transform_out(n(o(X, Y)), a(n(X), n(Y)))
transform_in(n(a(X, Y)), o(n(X), n(Y))) → transform_out(n(a(X, Y)), o(n(X), n(Y)))
transform_in(n(n(X)), X) → transform_out(n(n(X)), X)
U3(X1, Y, X2, transform_out(X1, X2)) → transform_out(o(X1, Y), o(X2, Y))
U4(X, Y1, Y2, transform_out(Y1, Y2)) → transform_out(o(X, Y1), o(X, Y2))
U5(X1, Y, X2, transform_out(X1, X2)) → transform_out(a(X1, Y), a(X2, Y))
U6(X, Y1, Y2, transform_out(Y1, Y2)) → transform_out(a(X, Y1), a(X, Y2))
U7(X1, X2, transform_out(X1, X2)) → transform_out(n(X1), n(X2))
U1(X, Y, transform_out(X, Z)) → U2(X, Y, cnfequiv_in(Z, Y))
U2(X, Y, cnfequiv_out(Z, Y)) → cnfequiv_out(X, Y)

The argument filtering Pi contains the following mapping:
cnfequiv_in(x1, x2)  =  cnfequiv_in(x1)
cnfequiv_out(x1, x2)  =  cnfequiv_out(x2)
U1(x1, x2, x3)  =  U1(x3)
transform_in(x1, x2)  =  transform_in(x1)
n(x1)  =  n(x1)
U7(x1, x2, x3)  =  U7(x3)
a(x1, x2)  =  a(x1, x2)
U6(x1, x2, x3, x4)  =  U6(x1, x4)
U5(x1, x2, x3, x4)  =  U5(x2, x4)
o(x1, x2)  =  o(x1, x2)
U4(x1, x2, x3, x4)  =  U4(x1, x4)
U3(x1, x2, x3, x4)  =  U3(x2, x4)
transform_out(x1, x2)  =  transform_out(x2)
U2(x1, x2, x3)  =  U2(x3)
CNFEQUIV_IN(x1, x2)  =  CNFEQUIV_IN(x1)
U11(x1, x2, x3)  =  U11(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:

U11(X, Y, transform_out(X, Z)) → CNFEQUIV_IN(Z, Y)
CNFEQUIV_IN(X, Y) → U11(X, Y, transform_in(X, Z))

The TRS R consists of the following rules:

transform_in(n(X1), n(X2)) → U7(X1, X2, transform_in(X1, X2))
transform_in(a(X, Y1), a(X, Y2)) → U6(X, Y1, Y2, transform_in(Y1, Y2))
transform_in(a(X1, Y), a(X2, Y)) → U5(X1, Y, X2, transform_in(X1, X2))
transform_in(o(X, Y1), o(X, Y2)) → U4(X, Y1, Y2, transform_in(Y1, Y2))
transform_in(o(X1, Y), o(X2, Y)) → U3(X1, Y, X2, transform_in(X1, X2))
transform_in(o(a(X, Y), Z), a(o(X, Z), o(Y, Z))) → transform_out(o(a(X, Y), Z), a(o(X, Z), o(Y, Z)))
transform_in(o(X, a(Y, Z)), a(o(X, Y), o(X, Z))) → transform_out(o(X, a(Y, Z)), a(o(X, Y), o(X, Z)))
transform_in(n(o(X, Y)), a(n(X), n(Y))) → transform_out(n(o(X, Y)), a(n(X), n(Y)))
transform_in(n(a(X, Y)), o(n(X), n(Y))) → transform_out(n(a(X, Y)), o(n(X), n(Y)))
transform_in(n(n(X)), X) → transform_out(n(n(X)), X)
U7(X1, X2, transform_out(X1, X2)) → transform_out(n(X1), n(X2))
U6(X, Y1, Y2, transform_out(Y1, Y2)) → transform_out(a(X, Y1), a(X, Y2))
U5(X1, Y, X2, transform_out(X1, X2)) → transform_out(a(X1, Y), a(X2, Y))
U4(X, Y1, Y2, transform_out(Y1, Y2)) → transform_out(o(X, Y1), o(X, Y2))
U3(X1, Y, X2, transform_out(X1, X2)) → transform_out(o(X1, Y), o(X2, Y))

The argument filtering Pi contains the following mapping:
transform_in(x1, x2)  =  transform_in(x1)
n(x1)  =  n(x1)
U7(x1, x2, x3)  =  U7(x3)
a(x1, x2)  =  a(x1, x2)
U6(x1, x2, x3, x4)  =  U6(x1, x4)
U5(x1, x2, x3, x4)  =  U5(x2, x4)
o(x1, x2)  =  o(x1, x2)
U4(x1, x2, x3, x4)  =  U4(x1, x4)
U3(x1, x2, x3, x4)  =  U3(x2, x4)
transform_out(x1, x2)  =  transform_out(x2)
CNFEQUIV_IN(x1, x2)  =  CNFEQUIV_IN(x1)
U11(x1, x2, x3)  =  U11(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:

U11(transform_out(Z)) → CNFEQUIV_IN(Z)
CNFEQUIV_IN(X) → U11(transform_in(X))

The TRS R consists of the following rules:

transform_in(n(X1)) → U7(transform_in(X1))
transform_in(a(X, Y1)) → U6(X, transform_in(Y1))
transform_in(a(X1, Y)) → U5(Y, transform_in(X1))
transform_in(o(X, Y1)) → U4(X, transform_in(Y1))
transform_in(o(X1, Y)) → U3(Y, transform_in(X1))
transform_in(o(a(X, Y), Z)) → transform_out(a(o(X, Z), o(Y, Z)))
transform_in(o(X, a(Y, Z))) → transform_out(a(o(X, Y), o(X, Z)))
transform_in(n(o(X, Y))) → transform_out(a(n(X), n(Y)))
transform_in(n(a(X, Y))) → transform_out(o(n(X), n(Y)))
transform_in(n(n(X))) → transform_out(X)
U7(transform_out(X2)) → transform_out(n(X2))
U6(X, transform_out(Y2)) → transform_out(a(X, Y2))
U5(Y, transform_out(X2)) → transform_out(a(X2, Y))
U4(X, transform_out(Y2)) → transform_out(o(X, Y2))
U3(Y, transform_out(X2)) → transform_out(o(X2, Y))

The set Q consists of the following terms:

transform_in(x0)
U7(x0)
U6(x0, x1)
U5(x0, x1)
U4(x0, x1)
U3(x0, x1)

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.


U11(transform_out(Z)) → CNFEQUIV_IN(Z)
CNFEQUIV_IN(X) → U11(transform_in(X))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
U11(x1)  =  U11(x1)
transform_out(x1)  =  transform_out(x1)
CNFEQUIV_IN(x1)  =  CNFEQUIV_IN(x1)
transform_in(x1)  =  x1
o(x1, x2)  =  o(x1, x2)
U4(x1, x2)  =  U4(x1, x2)
n(x1)  =  n(x1)
a(x1, x2)  =  a(x1, x2)
U6(x1, x2)  =  U6(x1, x2)
U5(x1, x2)  =  U5(x1, x2)
U7(x1)  =  U7(x1)
U3(x1, x2)  =  U3(x1, x2)

Recursive path order with status [2].
Quasi-Precedence:
[n1, U71] > [o2, U42, U32] > [a2, U62, U52] > transformout1 > CNFEQUIVIN1 > U1^11

Status:
U71: [1]
U52: multiset
CNFEQUIVIN1: [1]
U62: multiset
transformout1: multiset
U42: [1,2]
a2: multiset
U1^11: multiset
n1: [1]
o2: [1,2]
U32: [2,1]


The following usable rules [17] were oriented:

transform_in(o(X, Y1)) → U4(X, transform_in(Y1))
transform_in(n(a(X, Y))) → transform_out(o(n(X), n(Y)))
U6(X, transform_out(Y2)) → transform_out(a(X, Y2))
transform_in(a(X1, Y)) → U5(Y, transform_in(X1))
transform_in(n(o(X, Y))) → transform_out(a(n(X), n(Y)))
U5(Y, transform_out(X2)) → transform_out(a(X2, Y))
transform_in(a(X, Y1)) → U6(X, transform_in(Y1))
transform_in(o(X, a(Y, Z))) → transform_out(a(o(X, Y), o(X, Z)))
U4(X, transform_out(Y2)) → transform_out(o(X, Y2))
transform_in(n(n(X))) → transform_out(X)
transform_in(n(X1)) → U7(transform_in(X1))
transform_in(o(a(X, Y), Z)) → transform_out(a(o(X, Z), o(Y, Z)))
transform_in(o(X1, Y)) → U3(Y, transform_in(X1))
U3(Y, transform_out(X2)) → transform_out(o(X2, Y))
U7(transform_out(X2)) → transform_out(n(X2))



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ PisEmptyProof

Q DP problem:
P is empty.
The TRS R consists of the following rules:

transform_in(n(X1)) → U7(transform_in(X1))
transform_in(a(X, Y1)) → U6(X, transform_in(Y1))
transform_in(a(X1, Y)) → U5(Y, transform_in(X1))
transform_in(o(X, Y1)) → U4(X, transform_in(Y1))
transform_in(o(X1, Y)) → U3(Y, transform_in(X1))
transform_in(o(a(X, Y), Z)) → transform_out(a(o(X, Z), o(Y, Z)))
transform_in(o(X, a(Y, Z))) → transform_out(a(o(X, Y), o(X, Z)))
transform_in(n(o(X, Y))) → transform_out(a(n(X), n(Y)))
transform_in(n(a(X, Y))) → transform_out(o(n(X), n(Y)))
transform_in(n(n(X))) → transform_out(X)
U7(transform_out(X2)) → transform_out(n(X2))
U6(X, transform_out(Y2)) → transform_out(a(X, Y2))
U5(Y, transform_out(X2)) → transform_out(a(X2, Y))
U4(X, transform_out(Y2)) → transform_out(o(X, Y2))
U3(Y, transform_out(X2)) → transform_out(o(X2, Y))

The set Q consists of the following terms:

transform_in(x0)
U7(x0)
U6(x0, x1)
U5(x0, x1)
U4(x0, x1)
U3(x0, x1)

We have to consider all (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.