LU000R

lucas_divisible_implies_zero_mod

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

An explicit divisibility witness gives a balanced natural congruence to zero.

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

Exact expanded first-order arithmetic statement

forall p C. (exists lcv_quotient_zero_source. (C) = (p) * lcv_quotient_zero_source) -> (exists lcv_left_zero_result lcv_right_zero_result. (C) + (p) * lcv_left_zero_result = (0) + (p) * lcv_right_zero_result)

Constructive proof overview

Generated structural guide

An explicit divisibility witness gives a balanced natural congruence to zero.

The unchanged tactic script uses 1 declared prerequisite and contains 11 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Historical empty-context replay experiment only; that experiment persisted no certificate and granted no release authority. Current checked use follows separately sealed, independently verified proof bundles; there is no Stable promotion.

Proof neighborhood

Direct dependencies

zero_add Stable theorem; checked-use authorized

Direct dependents

LU000T lucas_prime_row_interior_zero_mod

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

11 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.

01Fix variables and assumptionsL1–3

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

  1. L1
    intro p
  2. L2
    intro C
  3. L3
    intro hdivides
02Separate the logical casesL4–4

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

  1. L4
    cases hdivides
03Construct an explicit witnessL5–6

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

  1. L5
    exists 0
  2. L6
    exists x
04Calculate and transport equalitiesL7–8

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

  1. L7
    simp
  2. L8
    trans p * x
05Use earlier factsL9–9

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

  1. L9
    exact hdivides_witness
06Calculate and transport equalitiesL10–10

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

  1. L10
    symm
07Use earlier factsL11–11

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

  1. L11
    apply zero_add

Library-wide reading audit

Original exact command ledger · 11 lines
  1. 0001intro p
  2. 0002intro C
  3. 0003intro hdivides
  4. 0004cases hdivides
  5. 0005exists 0
  6. 0006exists x
  7. 0007simp
  8. 0008trans p * x
  9. 0009exact hdivides_witness
  10. 0010symm
  11. 0011apply zero_add