BD000D

binary_digit_prefix_all_bits

Every universal genuine digit-prefix invariant yields actual beta-decoded zero-or-one witnesses at every position.

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. BinaryDigitPrefix(b,c,l) → AllBits(b,c,l)

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

Definition DAG

BinaryDigitPrefix(d,s,n) · 1AllBits(b,c,l) · 1

Actual proof prerequisites

beta_at_exists · checked external prerequisite
Original expanded first-order statement 
forall b c l. (forall ff_index_be_bd_old ff_digit_be_bd_old. (exists ff_lt_be_bd_old_bound. ff_lt_be_bd_old_bound + S ff_index_be_bd_old = l) -> (((exists ff_h_be_bd_old_digit. ff_h_be_bd_old_digit + S (ff_digit_be_bd_old) = S ((S (ff_index_be_bd_old)) * c)) /\ exists ff_q_be_bd_old_digit. b = ff_q_be_bd_old_digit * S ((S (ff_index_be_bd_old)) * c) + (ff_digit_be_bd_old))) -> (ff_digit_be_bd_old = 0 \/ ff_digit_be_bd_old = 1)) -> (forall ff_i_bd_all_bits. (exists ff_lt_bd_all_bits_bound. ff_lt_bd_all_bits_bound + S ff_i_bd_all_bits = l) -> exists ff_bit_bd_all_bits. ((((exists ff_h_bd_all_bits_decoded. ff_h_bd_all_bits_decoded + S (ff_bit_bd_all_bits) = S ((S (ff_i_bd_all_bits)) * c)) /\ exists ff_q_bd_all_bits_decoded. b = ff_q_bd_all_bits_decoded * S ((S (ff_i_bd_all_bits)) * c) + (ff_bit_bd_all_bits))) /\ (ff_bit_bd_all_bits = 0 \/ ff_bit_bd_all_bits = 1)))

Complete unchanged native tactic proof

All 20 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

20 script commands · 6 reading checkpoints · 1 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 assumptionsL1–6

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 hdigits
  5. L5
    intro i
  6. L6
    intro hbound
02Establish hentryL7–11

Establish this local claim before using it. It is not an additional assumption.

  1. L7
    have hentry : exists digit. (((exists ff_h_bd_all_bits_entry. ff_h_bd_all_bits_entry + S (digit) = S ((S (i)) * c)) /\ exists ff_q_bd_all_bits_entry. b = ff_q_bd_all_bits_entry * S ((S (i)) * c) + (digit)))
  2. L8
    specialize beta_at_exists b
  3. L9
    specialize beta_at_exists c
  4. L10
    specialize beta_at_exists i
  5. L11
    exact beta_at_exists
03Separate the logical casesL12–12

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

  1. L12
    cases hentry
04Construct an explicit witnessL13–13

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

  1. L13
    exists x
05Separate the logical casesL14–14

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

  1. L14
    split
06Use earlier factsL15–20

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

  1. L15
    exact hentry_witness
  2. L16
    specialize hdigits i
  3. L17
    specialize hdigits x
  4. L18
    apply hdigits
  5. L19
    exact hbound
  6. L20
    exact hentry_witness

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro hdigits
  5. 0005intro i
  6. 0006intro hbound
  7. 0007have hentry : exists digit. (((exists ff_h_bd_all_bits_entry. ff_h_bd_all_bits_entry + S (digit) = S ((S (i)) * c)) /\ exists ff_q_bd_all_bits_entry. b = ff_q_bd_all_bits_entry * S ((S (i)) * c) + (digit)))
  8. 0008specialize beta_at_exists b
  9. 0009specialize beta_at_exists c
  10. 0010specialize beta_at_exists i
  11. 0011exact beta_at_exists
  12. 0012cases hentry
  13. 0013exists x
  14. 0014split
  15. 0015exact hentry_witness
  16. 0016specialize hdigits i
  17. 0017specialize hdigits x
  18. 0018apply hdigits
  19. 0019exact hbound
  20. 0020exact hentry_witness