LU0015 · theorem body

lucas_digit_chain_initial_code_exists

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

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

historical independently replay-verified empty-context experiment; the experiment itself persisted no certificate and granted no release authority; current checked use follows separately sealed proof bundles; no Stable promotion.

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.

Statement with defined notation

∀ n. ∃ qb. ∃ qc. BetaAt(qb,qc,0,n)

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

PD0013 BetaAt

In local proof propositions

none
Exact expanded first-order 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)))

Proof neighborhood

Direct theorem prerequisites

beta_prefix_extend · Stable closed

Direct theorem dependents

LU0017 lucas_digit_chain_empty_exists

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

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.

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 assumptionsL1–1

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 defined 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