0 Prolog
↳1 PrologToPiTRSProof (⇒, 0 ms)
↳2 PiTRS
↳3 DependencyPairsProof (⇔, 99 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 AND
↳7 PiDP
↳8 UsableRulesProof (⇔, 0 ms)
↳9 PiDP
↳10 PiDPToQDPProof (⇒, 0 ms)
↳11 QDP
↳12 QDPSizeChangeProof (⇔, 0 ms)
↳13 YES
↳14 PiDP
↳15 UsableRulesProof (⇔, 0 ms)
↳16 PiDP
↳17 PiDPToQDPProof (⇒, 0 ms)
↳18 QDP
↳19 QDPOrderProof (⇔, 612 ms)
↳20 QDP
↳21 DependencyGraphProof (⇔, 0 ms)
↳22 TRUE
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
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)
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)
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)
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)
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)
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)
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)
From the DPs we obtained the following set of size-change graphs:
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))
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)
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))
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))
U1_GA(transform_out_ga(Z)) → CNFEQUIV_IN_GA(Z)
CNFEQUIV_IN_GA(X) → U1_GA(transform_in_ga(X))
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))
transform_in_ga(x0)
U3_ga(x0, x1)
U4_ga(x0, x1)
U5_ga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
U1_GA(transform_out_ga(Z)) → CNFEQUIV_IN_GA(Z)
[U1GA1, CNFEQUIVINGA1]
[n1, U7ga1] > [o2, U3ga2, U4ga2] > [a2, U5ga2, U6ga2] > transformoutga1
U1GA1: [1]
transformoutga1: multiset
CNFEQUIVINGA1: [1]
n1: multiset
a2: multiset
o2: multiset
U3ga2: multiset
U4ga2: multiset
U5ga2: multiset
U6ga2: multiset
U7ga1: multiset
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))
CNFEQUIV_IN_GA(X) → U1_GA(transform_in_ga(X))
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))
transform_in_ga(x0)
U3_ga(x0, x1)
U4_ga(x0, x1)
U5_ga(x0, x1)
U6_ga(x0, x1)
U7_ga(x0)