BD0001

binary_digit_code_recode_exists

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

Every genuine beta-coded prefix can be recoded while appending any exact last digit.

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 b c l d. (exists z e. ((((exists ff_h_bd_recode_terminal. ff_h_bd_recode_terminal + S (d) = S ((S (l)) * e)) /\ exists ff_q_bd_recode_terminal. z = ff_q_bd_recode_terminal * S ((S (l)) * e) + (d))) /\ (forall bd_index_recode_result bd_value_recode_result. (exists bd_gap_recode_result. bd_gap_recode_result + S bd_index_recode_result = l) -> (((exists ff_h_bd_recode_result_old. ff_h_bd_recode_result_old + S (bd_value_recode_result) = S ((S (bd_index_recode_result)) * c)) /\ exists ff_q_bd_recode_result_old. b = ff_q_bd_recode_result_old * S ((S (bd_index_recode_result)) * c) + (bd_value_recode_result))) -> (((exists ff_h_bd_recode_result_new. ff_h_bd_recode_result_new + S (bd_value_recode_result) = S ((S (bd_index_recode_result)) * e)) /\ exists ff_q_bd_recode_result_new. z = ff_q_bd_recode_result_new * S ((S (bd_index_recode_result)) * e) + (bd_value_recode_result))))))

Constructive proof overview

Generated structural guide

Every genuine beta-coded prefix can be recoded while appending any exact last digit.

The unchanged tactic script uses 1 declared prerequisite and contains 9 exact native proof lines.

Alpha v34 checked-use · first admitted v23 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

beta_prefix_extend Stable theorem; checked-use authorized

Direct dependents

BD0004 binary_digit_prefix_append

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

9 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 assumptionsL1–4

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro l
  4. L4
    intro d
02Use earlier factsL5–9

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

  1. L5
    specialize beta_prefix_extend l
  2. L6
    specialize beta_prefix_extend b
  3. L7
    specialize beta_prefix_extend c
  4. L8
    specialize beta_prefix_extend d
  5. L9
    exact beta_prefix_extend

Library-wide reading audit

Original exact command ledger · 9 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro d
  5. 0005specialize beta_prefix_extend l
  6. 0006specialize beta_prefix_extend b
  7. 0007specialize beta_prefix_extend c
  8. 0008specialize beta_prefix_extend d
  9. 0009exact beta_prefix_extend

Separate complete second-wave branches: Full T13 proof · Alpha v27.