LU0005

lucas_base_p_digit_total

Dependency-curried candidate body; not Alpha-enrolled; no checked-use authority

Every natural value has an actual quotient and strictly bounded base-p digit for every nonzero base.

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 n. ~(p = 0) -> exists q d. (((n) = (p) * (q) + (d)) /\ (exists ldc_lt_native_bound. ldc_lt_native_bound + S (d) = p))

Constructive proof overview

Generated structural guide

Every natural value has an actual quotient and strictly bounded base-p digit for every nonzero base.

The unchanged tactic script uses 1 declared prerequisite and contains 7 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

Proof neighborhood

Direct dependencies

division_remainder_exists Stable theorem; checked-use authorized

Direct dependents

LU0006 lucas_prime_base_digit_total LU000D lucas_base_p_two_digit_total

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This dependency-curried candidate body does not grant checked theorem use or Stable membership.

Read the argument

Proof checkpoints

7 script commands · 2 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 n
  3. L3
    intro hnonzero
02Use earlier factsL4–7

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

  1. L4
    specialize division_remainder_exists p
  2. L5
    specialize division_remainder_exists n
  3. L6
    apply division_remainder_exists
  4. L7
    exact hnonzero

Library-wide reading audit

Original exact command ledger · 7 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro hnonzero
  4. 0004specialize division_remainder_exists p
  5. 0005specialize division_remainder_exists n
  6. 0006apply division_remainder_exists
  7. 0007exact hnonzero