DV0016

divisor_mask_entry_omitted_value

Every omitted index is exactly canonical zero, including the explicit d=0 branch for arbitrary F(0).

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

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

A genuine divisor mask has S n entries, indexed zero through n, and forces its zeroth entry to zero regardless of F(0). Möbius values remain positive-domain only. This family constructs divisor sums and Möbius tables; cancellation and the full G007 endpoint are separately proved later in the same release.

Exact theorem in conservative defined notation

∀ F. ∀ n. ∀ d. ∀ z. d = 0 ∨ ¬Dvd(d,n) → DivisorMaskEntry(F,n,d,z) → z = 0

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

DivisorMaskEntry(F,n,d,z) · 1Dvd(d,n) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall F n d z. (d=0 \/ ~(exists pvs_factor_zero_guard. (n) = (d) * pvs_factor_zero_guard)) -> ((((~((d)=0)) /\ (exists dm_quotient_zero_mask. (((n)=(d)*dm_quotient_zero_mask) /\ (exists dst_positive_code_zero_maskinput dst_positive_scale_zero_maskinput dst_negative_code_zero_maskinput dst_negative_scale_zero_maskinput dst_positive_zero_maskinput dst_negative_zero_maskinput. (((F) = (((((dst_positive_code_zero_maskinput) + (dst_positive_scale_zero_maskinput)) * S ((dst_positive_code_zero_maskinput) + (dst_positive_scale_zero_maskinput)) + ((dst_positive_scale_zero_maskinput) + (dst_positive_scale_zero_maskinput))) + (((dst_negative_code_zero_maskinput) + (dst_negative_scale_zero_maskinput)) * S ((dst_negative_code_zero_maskinput) + (dst_negative_scale_zero_maskinput)) + ((dst_negative_scale_zero_maskinput) + (dst_negative_scale_zero_maskinput)))) * S ((((dst_positive_code_zero_maskinput) + (dst_positive_scale_zero_maskinput)) * S ((dst_positive_code_zero_maskinput) + (dst_positive_scale_zero_maskinput)) + ((dst_positive_scale_zero_maskinput) + (dst_positive_scale_zero_maskinput))) + (((dst_negative_code_zero_maskinput) + (dst_negative_scale_zero_maskinput)) * S ((dst_negative_code_zero_maskinput) + (dst_negative_scale_zero_maskinput)) + ((dst_negative_scale_zero_maskinput) + (dst_negative_scale_zero_maskinput)))) + ((((dst_negative_code_zero_maskinput) + (dst_negative_scale_zero_maskinput)) * S ((dst_negative_code_zero_maskinput) + (dst_negative_scale_zero_maskinput)) + ((dst_negative_scale_zero_maskinput) + (dst_negative_scale_zero_maskinput))) + (((dst_negative_code_zero_maskinput) + (dst_negative_scale_zero_maskinput)) * S ((dst_negative_code_zero_maskinput) + (dst_negative_scale_zero_maskinput)) + ((dst_negative_scale_zero_maskinput) + (dst_negative_scale_zero_maskinput)))))) /\ (((((exists ff_h_pvs_zero_maskinputpositive. ff_h_pvs_zero_maskinputpositive + S (dst_positive_zero_maskinput) = S ((S (d)) * dst_positive_scale_zero_maskinput)) /\ exists ff_q_pvs_zero_maskinputpositive. dst_positive_code_zero_maskinput = ff_q_pvs_zero_maskinputpositive * S ((S (d)) * dst_positive_scale_zero_maskinput) + (dst_positive_zero_maskinput))) /\ (((((exists ff_h_pvs_zero_maskinputnegative. ff_h_pvs_zero_maskinputnegative + S (dst_negative_zero_maskinput) = S ((S (d)) * dst_negative_scale_zero_maskinput)) /\ exists ff_q_pvs_zero_maskinputnegative. dst_negative_code_zero_maskinput = ff_q_pvs_zero_maskinputnegative * S ((S (d)) * dst_negative_scale_zero_maskinput) + (dst_negative_zero_maskinput))) /\ (exists ge_balance_positive_zero_maskinputvalue ge_balance_negative_zero_maskinputvalue. (((((z) = 2 * (ge_balance_positive_zero_maskinputvalue) /\ (ge_balance_negative_zero_maskinputvalue) = 0) \/ exists ge_signed_half_zero_maskinputvaluedecode. (((z) = 2 * ge_signed_half_zero_maskinputvaluedecode + 1 /\ (ge_balance_positive_zero_maskinputvalue) = 0) /\ (ge_balance_negative_zero_maskinputvalue) = S ge_signed_half_zero_maskinputvaluedecode))) /\ ((dst_positive_zero_maskinput) + ge_balance_negative_zero_maskinputvalue = (dst_negative_zero_maskinput) + ge_balance_positive_zero_maskinputvalue))))))))))))) \/ ((((d)=0 \/ ~(exists pvs_factor_zero_masknondivisor. (n) = (d) * pvs_factor_zero_masknondivisor)) /\ ((z)=0)))) -> z=0

Complete tactic proof in conservative notation

All 19 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

19 script commands · 7 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–6

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro F
  2. L2
    intro n
  3. L3
    intro d
  4. L4
    intro z
  5. L5
    intro hc
  6. L6
    intro he
02Separate the logical casesL7–12

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L7
    cases he
  2. L8
    cases he_left
  3. L9
    cases he_left_right
  4. L10
    cases he_left_right_witness
  5. L11
    exfalso
  6. L12
    cases hc
03Use earlier factsL13–15

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L13
    apply he_left_left
  2. L14
    exact hc_left
  3. L15
    apply hc_right
04Construct an explicit witnessL16–16

Supply the displayed value, then prove that it has the required property.

  1. L16
    exists x
05Use earlier factsL17–17

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L17
    exact he_left_right_witness_left
06Separate the logical casesL18–18

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L18
    cases he_right
07Use earlier factsL19–19

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L19
    exact he_right_right

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro F
  2. 0002intro n
  3. 0003intro d
  4. 0004intro z
  5. 0005intro hc
  6. 0006intro he
  7. 0007cases he
  8. 0008cases he_left
  9. 0009cases he_left_right
  10. 0010cases he_left_right_witness
  11. 0011exfalso
  12. 0012cases hc
  13. 0013apply he_left_left
  14. 0014exact hc_left
  15. 0015apply hc_right
  16. 0016exists x
  17. 0017exact he_left_right_witness_left
  18. 0018cases he_right
  19. 0019exact he_right_right