LU0015

lucas_digit_chain_initial_code_exists

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

Every natural is the zeroth decoded entry of an actual beta-coded quotient stream.

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 n. exists qb qc. (((exists ff_h_lmd_initial_source. ff_h_lmd_initial_source + S (n) = S ((S (0)) * qc)) /\ exists ff_q_lmd_initial_source. qb = ff_q_lmd_initial_source * S ((S (0)) * qc) + (n)))

Constructive proof overview

Generated structural guide

Every natural is the zeroth decoded entry of an actual beta-coded quotient stream.

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

beta_prefix_extend Stable theorem; checked-use authorized

Direct dependents

LU0017 lucas_digit_chain_empty_exists

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 · 5 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 assumptionsL1–1

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

  1. L1
    intro n
02Use earlier factsL2–5

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

  1. L2
    specialize beta_prefix_extend 0
  2. L3
    specialize beta_prefix_extend 0
  3. L4
    specialize beta_prefix_extend 0
  4. L5
    specialize beta_prefix_extend n
03Separate the logical casesL6–8

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

  1. L6
    cases beta_prefix_extend
  2. L7
    cases beta_prefix_extend_witness
  3. L8
    cases beta_prefix_extend_witness_witness
04Construct an explicit witnessL9–10

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

  1. L9
    exists x
  2. L10
    exists x1
05Use earlier factsL11–11

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

  1. L11
    exact beta_prefix_extend_witness_witness_left

Library-wide reading audit

Original exact command ledger · 11 lines
  1. 0001intro n
  2. 0002specialize beta_prefix_extend 0
  3. 0003specialize beta_prefix_extend 0
  4. 0004specialize beta_prefix_extend 0
  5. 0005specialize beta_prefix_extend n
  6. 0006cases beta_prefix_extend
  7. 0007cases beta_prefix_extend_witness
  8. 0008cases beta_prefix_extend_witness_witness
  9. 0009exists x
  10. 0010exists x1
  11. 0011exact beta_prefix_extend_witness_witness_left