BD0001

binary_digit_code_recode_exists

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

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

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.

The exact G102 milestone is fully proved for every natural exponent and every modulus greater than one, including actual canonical digits, a beta-coded accumulator execution, modular-power correctness, and the formal bound k≤3·BitLen(e)+2. The independent T13 determinant/rank/integer-span substrate is now closed in the separate Alpha-v27 integer-linear-algebra branch. Full T13 proof · Alpha v27

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ l. ∀ d. ∃ x. ∃ y. Beta(x,y,l,d) ∧ (∀ z. ∀ n. Lt(z,l) → Beta(b,c,z,n) → Beta(x,y,z,n))

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

Beta(b,c,i,x) · 3Lt(a,b) · 1

Actual proof prerequisites

beta_prefix_extend · checked external prerequisite
Original expanded first-order 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))))))

Complete unchanged native tactic proof

All 9 lines are the exact independently kernel-checked original script.

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.

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 assumptionsL1–4

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 defined 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