ND0314

DirichletUnitAtOne(F)

An actual lookup F(1) has canonical signed code 2 or 1. This direct disjunction contains ArithAt, not a SignedUnit or inverse subformula. Table validity, the finite window bound, and the inverse criterion are separate hypotheses or theorems.

Conservative notation; not a theorem, primitive, or axiom.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

Definition in prerequisite notation

ArithAt(F,1,2)ArithAt(F,1,1)

Only definitions earlier in this acyclic notation graph are used here.

Hygienic expanded first-order definition
(exists dst_positive_code_dirichlet_inversepositive dst_positive_scale_dirichlet_inversepositive dst_negative_code_dirichlet_inversepositive dst_negative_scale_dirichlet_inversepositive dst_positive_dirichlet_inversepositive dst_negative_dirichlet_inversepositive. ((((F)) = (((((dst_positive_code_dirichlet_inversepositive) + (dst_positive_scale_dirichlet_inversepositive)) * S ((dst_positive_code_dirichlet_inversepositive) + (dst_positive_scale_dirichlet_inversepositive)) + ((dst_positive_scale_dirichlet_inversepositive) + (dst_positive_scale_dirichlet_inversepositive))) + (((dst_negative_code_dirichlet_inversepositive) + (dst_negative_scale_dirichlet_inversepositive)) * S ((dst_negative_code_dirichlet_inversepositive) + (dst_negative_scale_dirichlet_inversepositive)) + ((dst_negative_scale_dirichlet_inversepositive) + (dst_negative_scale_dirichlet_inversepositive)))) * S ((((dst_positive_code_dirichlet_inversepositive) + (dst_positive_scale_dirichlet_inversepositive)) * S ((dst_positive_code_dirichlet_inversepositive) + (dst_positive_scale_dirichlet_inversepositive)) + ((dst_positive_scale_dirichlet_inversepositive) + (dst_positive_scale_dirichlet_inversepositive))) + (((dst_negative_code_dirichlet_inversepositive) + (dst_negative_scale_dirichlet_inversepositive)) * S ((dst_negative_code_dirichlet_inversepositive) + (dst_negative_scale_dirichlet_inversepositive)) + ((dst_negative_scale_dirichlet_inversepositive) + (dst_negative_scale_dirichlet_inversepositive)))) + ((((dst_negative_code_dirichlet_inversepositive) + (dst_negative_scale_dirichlet_inversepositive)) * S ((dst_negative_code_dirichlet_inversepositive) + (dst_negative_scale_dirichlet_inversepositive)) + ((dst_negative_scale_dirichlet_inversepositive) + (dst_negative_scale_dirichlet_inversepositive))) + (((dst_negative_code_dirichlet_inversepositive) + (dst_negative_scale_dirichlet_inversepositive)) * S ((dst_negative_code_dirichlet_inversepositive) + (dst_negative_scale_dirichlet_inversepositive)) + ((dst_negative_scale_dirichlet_inversepositive) + (dst_negative_scale_dirichlet_inversepositive)))))) /\ (((((exists ff_h_pvs_dirichlet_inversepositivepositive. ff_h_pvs_dirichlet_inversepositivepositive + S (dst_positive_dirichlet_inversepositive) = S ((S (1)) * dst_positive_scale_dirichlet_inversepositive)) /\ exists ff_q_pvs_dirichlet_inversepositivepositive. dst_positive_code_dirichlet_inversepositive = ff_q_pvs_dirichlet_inversepositivepositive * S ((S (1)) * dst_positive_scale_dirichlet_inversepositive) + (dst_positive_dirichlet_inversepositive))) /\ (((((exists ff_h_pvs_dirichlet_inversepositivenegative. ff_h_pvs_dirichlet_inversepositivenegative + S (dst_negative_dirichlet_inversepositive) = S ((S (1)) * dst_negative_scale_dirichlet_inversepositive)) /\ exists ff_q_pvs_dirichlet_inversepositivenegative. dst_negative_code_dirichlet_inversepositive = ff_q_pvs_dirichlet_inversepositivenegative * S ((S (1)) * dst_negative_scale_dirichlet_inversepositive) + (dst_negative_dirichlet_inversepositive))) /\ (exists ge_balance_positive_dirichlet_inversepositivevalue ge_balance_negative_dirichlet_inversepositivevalue. (((((2) = 2 * (ge_balance_positive_dirichlet_inversepositivevalue) /\ (ge_balance_negative_dirichlet_inversepositivevalue) = 0) \/ exists ge_signed_half_dirichlet_inversepositivevaluedecode. (((2) = 2 * ge_signed_half_dirichlet_inversepositivevaluedecode + 1 /\ (ge_balance_positive_dirichlet_inversepositivevalue) = 0) /\ (ge_balance_negative_dirichlet_inversepositivevalue) = S ge_signed_half_dirichlet_inversepositivevaluedecode))) /\ ((dst_positive_dirichlet_inversepositive) + ge_balance_negative_dirichlet_inversepositivevalue = (dst_negative_dirichlet_inversepositive) + ge_balance_positive_dirichlet_inversepositivevalue))))))))) \/ (exists dst_positive_code_dirichlet_inversenegative dst_positive_scale_dirichlet_inversenegative dst_negative_code_dirichlet_inversenegative dst_negative_scale_dirichlet_inversenegative dst_positive_dirichlet_inversenegative dst_negative_dirichlet_inversenegative. ((((F)) = (((((dst_positive_code_dirichlet_inversenegative) + (dst_positive_scale_dirichlet_inversenegative)) * S ((dst_positive_code_dirichlet_inversenegative) + (dst_positive_scale_dirichlet_inversenegative)) + ((dst_positive_scale_dirichlet_inversenegative) + (dst_positive_scale_dirichlet_inversenegative))) + (((dst_negative_code_dirichlet_inversenegative) + (dst_negative_scale_dirichlet_inversenegative)) * S ((dst_negative_code_dirichlet_inversenegative) + (dst_negative_scale_dirichlet_inversenegative)) + ((dst_negative_scale_dirichlet_inversenegative) + (dst_negative_scale_dirichlet_inversenegative)))) * S ((((dst_positive_code_dirichlet_inversenegative) + (dst_positive_scale_dirichlet_inversenegative)) * S ((dst_positive_code_dirichlet_inversenegative) + (dst_positive_scale_dirichlet_inversenegative)) + ((dst_positive_scale_dirichlet_inversenegative) + (dst_positive_scale_dirichlet_inversenegative))) + (((dst_negative_code_dirichlet_inversenegative) + (dst_negative_scale_dirichlet_inversenegative)) * S ((dst_negative_code_dirichlet_inversenegative) + (dst_negative_scale_dirichlet_inversenegative)) + ((dst_negative_scale_dirichlet_inversenegative) + (dst_negative_scale_dirichlet_inversenegative)))) + ((((dst_negative_code_dirichlet_inversenegative) + (dst_negative_scale_dirichlet_inversenegative)) * S ((dst_negative_code_dirichlet_inversenegative) + (dst_negative_scale_dirichlet_inversenegative)) + ((dst_negative_scale_dirichlet_inversenegative) + (dst_negative_scale_dirichlet_inversenegative))) + (((dst_negative_code_dirichlet_inversenegative) + (dst_negative_scale_dirichlet_inversenegative)) * S ((dst_negative_code_dirichlet_inversenegative) + (dst_negative_scale_dirichlet_inversenegative)) + ((dst_negative_scale_dirichlet_inversenegative) + (dst_negative_scale_dirichlet_inversenegative)))))) /\ (((((exists ff_h_pvs_dirichlet_inversenegativepositive. ff_h_pvs_dirichlet_inversenegativepositive + S (dst_positive_dirichlet_inversenegative) = S ((S (1)) * dst_positive_scale_dirichlet_inversenegative)) /\ exists ff_q_pvs_dirichlet_inversenegativepositive. dst_positive_code_dirichlet_inversenegative = ff_q_pvs_dirichlet_inversenegativepositive * S ((S (1)) * dst_positive_scale_dirichlet_inversenegative) + (dst_positive_dirichlet_inversenegative))) /\ (((((exists ff_h_pvs_dirichlet_inversenegativenegative. ff_h_pvs_dirichlet_inversenegativenegative + S (dst_negative_dirichlet_inversenegative) = S ((S (1)) * dst_negative_scale_dirichlet_inversenegative)) /\ exists ff_q_pvs_dirichlet_inversenegativenegative. dst_negative_code_dirichlet_inversenegative = ff_q_pvs_dirichlet_inversenegativenegative * S ((S (1)) * dst_negative_scale_dirichlet_inversenegative) + (dst_negative_dirichlet_inversenegative))) /\ (exists ge_balance_positive_dirichlet_inversenegativevalue ge_balance_negative_dirichlet_inversenegativevalue. (((((1) = 2 * (ge_balance_positive_dirichlet_inversenegativevalue) /\ (ge_balance_negative_dirichlet_inversenegativevalue) = 0) \/ exists ge_signed_half_dirichlet_inversenegativevaluedecode. (((1) = 2 * ge_signed_half_dirichlet_inversenegativevaluedecode + 1 /\ (ge_balance_positive_dirichlet_inversenegativevalue) = 0) /\ (ge_balance_negative_dirichlet_inversenegativevalue) = S ge_signed_half_dirichlet_inversenegativevaluedecode))) /\ ((dst_positive_dirichlet_inversenegative) + ge_balance_negative_dirichlet_inversenegativevalue = (dst_negative_dirichlet_inversenegative) + ge_balance_positive_dirichlet_inversenegativevalue)))))))))

The unchanged native kernel never receives this surface symbol. Binder-safe expansion produces only its existing first-order syntax.

Direct definition dependencies

Definitions depending on this notation

none

Checked theorems using this definition

none directly; see definition consumers