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

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 AND
      ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇐)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 TRUE
      ↳14 PiDP
        ↳15 UsableRulesProof (⇔)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇐)
        ↳18 QDP
        ↳19 QDPOrderProof (⇔)
        ↳20 QDP
        ↳21 DependencyGraphProof (⇔)
        ↳22 TRUE
  ↳23 PrologToPiTRSProof (⇐)
    ↳24 PiTRS
    ↳25 DependencyPairsProof (⇔)
    ↳26 PiDP
    ↳27 DependencyGraphProof (⇔)
    ↳28 AND
      ↳29 PiDP
        ↳30 UsableRulesProof (⇔)
        ↳31 PiDP
        ↳32 PiDPToQDPProof (⇐)
        ↳33 QDP
        ↳34 QDPSizeChangeProof (⇔)
        ↳35 TRUE
      ↳36 PiDP
        ↳37 UsableRulesProof (⇔)
        ↳38 PiDP
        ↳39 PiDPToQDPProof (⇐)
        ↳40 QDP

(0) Obligation:

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).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. 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)

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)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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)

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)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
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)

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)
TRANSFORM_IN_GA(x1, x2)  =  TRANSFORM_IN_GA(x1)
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)

We have to consider all (P,R,Pi)-chains

(4) Obligation:

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)

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)
TRANSFORM_IN_GA(x1, x2)  =  TRANSFORM_IN_GA(x1)
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)

We have to consider all (P,R,Pi)-chains

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 7 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

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

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

(8) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(9) Obligation:

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

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

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

(10) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(11) Obligation:

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

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

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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(o(X, Y1)) → TRANSFORM_IN_GA(Y1)
    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(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

  • TRANSFORM_IN_GA(n(X1)) → TRANSFORM_IN_GA(X1)
    The graph contains the following edges 1 > 1

(13) TRUE

(14) Obligation:

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

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

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

(15) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(16) Obligation:

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

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

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

(17) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(18) Obligation:

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

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

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


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

Recursive path order with status [RPO].
Quasi-Precedence:
[U1GA1, CNFEQUIVINGA1]
[n1, U7ga1] > [o2, U3ga2, U4ga2] > [a2, U5ga2, U6ga2] > transformoutga1

Status:
U1GA1: [1]
transformoutga1: multiset
CNFEQUIVINGA1: [1]
n1: multiset
a2: multiset
o2: multiset
U3ga2: multiset
U4ga2: multiset
U5ga2: multiset
U6ga2: multiset
U7ga1: multiset


The following usable rules [FROCOS05] were oriented:

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))
U4_ga(X, transform_out_ga(Y2)) → transform_out_ga(o(X, Y2))
U3_ga(Y, transform_out_ga(X2)) → transform_out_ga(o(X2, Y))
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))

(20) Obligation:

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.

(21) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(22) TRUE

(23) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. 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)

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

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(24) Obligation:

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)

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

(25) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
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)

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

We have to consider all (P,R,Pi)-chains

(26) Obligation:

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)

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

We have to consider all (P,R,Pi)-chains

(27) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 7 less nodes.

(28) Complex Obligation (AND)

(29) Obligation:

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

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

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(x1, x3)
transform_in_ga(x1, x2)  =  transform_in_ga(x1)
n(x1)  =  n(x1)
transform_out_ga(x1, x2)  =  transform_out_ga(x1, x2)
a(x1, x2)  =  a(x1, x2)
o(x1, x2)  =  o(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x2, x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x1, x2, x4)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
cnfequiv_out_ga(x1, x2)  =  cnfequiv_out_ga(x1, x2)
TRANSFORM_IN_GA(x1, x2)  =  TRANSFORM_IN_GA(x1)

We have to consider all (P,R,Pi)-chains

(30) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(31) Obligation:

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

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

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

(32) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(33) Obligation:

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

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

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(34) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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(o(X, Y1)) → TRANSFORM_IN_GA(Y1)
    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(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

  • TRANSFORM_IN_GA(n(X1)) → TRANSFORM_IN_GA(X1)
    The graph contains the following edges 1 > 1

(35) TRUE

(36) Obligation:

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

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

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(x1, x3)
transform_in_ga(x1, x2)  =  transform_in_ga(x1)
n(x1)  =  n(x1)
transform_out_ga(x1, x2)  =  transform_out_ga(x1, x2)
a(x1, x2)  =  a(x1, x2)
o(x1, x2)  =  o(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x2, x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x1, x2, x4)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
cnfequiv_out_ga(x1, x2)  =  cnfequiv_out_ga(x1, x2)
CNFEQUIV_IN_GA(x1, x2)  =  CNFEQUIV_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)

We have to consider all (P,R,Pi)-chains

(37) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(38) Obligation:

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

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

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(x1, x2)
a(x1, x2)  =  a(x1, x2)
o(x1, x2)  =  o(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x2, x4)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x2, x4)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x1, x2, x4)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
CNFEQUIV_IN_GA(x1, x2)  =  CNFEQUIV_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)

We have to consider all (P,R,Pi)-chains

(39) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(40) Obligation:

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

transform_in_ga(x0)
U3_ga(x0, x1, x2)
U4_ga(x0, x1, x2)
U5_ga(x0, x1, x2)
U6_ga(x0, x1, x2)
U7_ga(x0, x1)

We have to consider all (P,Q,R)-chains.