DV0012

divisor_mask_entry_from_nondivisor

A proved nondivisor contributes zero independently of every signed input value.

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. ¬Dvd(d,n) → DivisorMaskEntry(F,n,d,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. ~(exists pvs_factor_omit_guard. (n) = (d) * pvs_factor_omit_guard) -> ((((~((d)=0)) /\ (exists dm_quotient_omit_result. (((n)=(d)*dm_quotient_omit_result) /\ (exists dst_positive_code_omit_resultinput dst_positive_scale_omit_resultinput dst_negative_code_omit_resultinput dst_negative_scale_omit_resultinput dst_positive_omit_resultinput dst_negative_omit_resultinput. (((F) = (((((dst_positive_code_omit_resultinput) + (dst_positive_scale_omit_resultinput)) * S ((dst_positive_code_omit_resultinput) + (dst_positive_scale_omit_resultinput)) + ((dst_positive_scale_omit_resultinput) + (dst_positive_scale_omit_resultinput))) + (((dst_negative_code_omit_resultinput) + (dst_negative_scale_omit_resultinput)) * S ((dst_negative_code_omit_resultinput) + (dst_negative_scale_omit_resultinput)) + ((dst_negative_scale_omit_resultinput) + (dst_negative_scale_omit_resultinput)))) * S ((((dst_positive_code_omit_resultinput) + (dst_positive_scale_omit_resultinput)) * S ((dst_positive_code_omit_resultinput) + (dst_positive_scale_omit_resultinput)) + ((dst_positive_scale_omit_resultinput) + (dst_positive_scale_omit_resultinput))) + (((dst_negative_code_omit_resultinput) + (dst_negative_scale_omit_resultinput)) * S ((dst_negative_code_omit_resultinput) + (dst_negative_scale_omit_resultinput)) + ((dst_negative_scale_omit_resultinput) + (dst_negative_scale_omit_resultinput)))) + ((((dst_negative_code_omit_resultinput) + (dst_negative_scale_omit_resultinput)) * S ((dst_negative_code_omit_resultinput) + (dst_negative_scale_omit_resultinput)) + ((dst_negative_scale_omit_resultinput) + (dst_negative_scale_omit_resultinput))) + (((dst_negative_code_omit_resultinput) + (dst_negative_scale_omit_resultinput)) * S ((dst_negative_code_omit_resultinput) + (dst_negative_scale_omit_resultinput)) + ((dst_negative_scale_omit_resultinput) + (dst_negative_scale_omit_resultinput)))))) /\ (((((exists ff_h_pvs_omit_resultinputpositive. ff_h_pvs_omit_resultinputpositive + S (dst_positive_omit_resultinput) = S ((S (d)) * dst_positive_scale_omit_resultinput)) /\ exists ff_q_pvs_omit_resultinputpositive. dst_positive_code_omit_resultinput = ff_q_pvs_omit_resultinputpositive * S ((S (d)) * dst_positive_scale_omit_resultinput) + (dst_positive_omit_resultinput))) /\ (((((exists ff_h_pvs_omit_resultinputnegative. ff_h_pvs_omit_resultinputnegative + S (dst_negative_omit_resultinput) = S ((S (d)) * dst_negative_scale_omit_resultinput)) /\ exists ff_q_pvs_omit_resultinputnegative. dst_negative_code_omit_resultinput = ff_q_pvs_omit_resultinputnegative * S ((S (d)) * dst_negative_scale_omit_resultinput) + (dst_negative_omit_resultinput))) /\ (exists ge_balance_positive_omit_resultinputvalue ge_balance_negative_omit_resultinputvalue. (((((0) = 2 * (ge_balance_positive_omit_resultinputvalue) /\ (ge_balance_negative_omit_resultinputvalue) = 0) \/ exists ge_signed_half_omit_resultinputvaluedecode. (((0) = 2 * ge_signed_half_omit_resultinputvaluedecode + 1 /\ (ge_balance_positive_omit_resultinputvalue) = 0) /\ (ge_balance_negative_omit_resultinputvalue) = S ge_signed_half_omit_resultinputvaluedecode))) /\ ((dst_positive_omit_resultinput) + ge_balance_negative_omit_resultinputvalue = (dst_negative_omit_resultinput) + ge_balance_positive_omit_resultinputvalue))))))))))))) \/ ((((d)=0 \/ ~(exists pvs_factor_omit_resultnondivisor. (n) = (d) * pvs_factor_omit_resultnondivisor)) /\ ((0)=0))))

Complete tactic proof in conservative notation

All 9 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

9 script commands · 4 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 assumptionsL1–4

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 hd
02Separate the logical casesL5–7

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

  1. L5
    right
  2. L6
    split
  3. L7
    right
03Use earlier factsL8–8

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

  1. L8
    exact hd
04Calculate and transport equalitiesL9–9

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L9
    refl

Library-wide reading audit

Original defined command ledger · 9 lines
  1. 0001intro F
  2. 0002intro n
  3. 0003intro d
  4. 0004intro hd
  5. 0005right
  6. 0006split
  7. 0007right
  8. 0008exact hd
  9. 0009refl