BD000E

binary_digit_prefix_bit_count_exists

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

Every actual beta-coded binary prefix has an independently beta-witnessed exact population count.

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. (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)) -> exists ones. (((exists ff_u_bd_prefix_count_sum ff_v_bd_prefix_count_sum. ((((exists ff_h_bd_prefix_count_sum_start. ff_h_bd_prefix_count_sum_start + S (0) = S ((S (0)) * ff_v_bd_prefix_count_sum)) /\ exists ff_q_bd_prefix_count_sum_start. ff_u_bd_prefix_count_sum = ff_q_bd_prefix_count_sum_start * S ((S (0)) * ff_v_bd_prefix_count_sum) + (0))) /\ ((((exists ff_h_bd_prefix_count_sum_terminal. ff_h_bd_prefix_count_sum_terminal + S (ones) = S ((S (l)) * ff_v_bd_prefix_count_sum)) /\ exists ff_q_bd_prefix_count_sum_terminal. ff_u_bd_prefix_count_sum = ff_q_bd_prefix_count_sum_terminal * S ((S (l)) * ff_v_bd_prefix_count_sum) + (ones))) /\ forall ff_i_bd_prefix_count_sum. (exists ff_lt_bd_prefix_count_sum_bound. ff_lt_bd_prefix_count_sum_bound + S ff_i_bd_prefix_count_sum = l) -> exists ff_a_bd_prefix_count_sum ff_r_bd_prefix_count_sum ff_s_bd_prefix_count_sum. ((((exists ff_h_bd_prefix_count_sum_summand. ff_h_bd_prefix_count_sum_summand + S (ff_a_bd_prefix_count_sum) = S ((S (ff_i_bd_prefix_count_sum)) * c)) /\ exists ff_q_bd_prefix_count_sum_summand. b = ff_q_bd_prefix_count_sum_summand * S ((S (ff_i_bd_prefix_count_sum)) * c) + (ff_a_bd_prefix_count_sum))) /\ ((((exists ff_h_bd_prefix_count_sum_partial. ff_h_bd_prefix_count_sum_partial + S (ff_r_bd_prefix_count_sum) = S ((S (ff_i_bd_prefix_count_sum)) * ff_v_bd_prefix_count_sum)) /\ exists ff_q_bd_prefix_count_sum_partial. ff_u_bd_prefix_count_sum = ff_q_bd_prefix_count_sum_partial * S ((S (ff_i_bd_prefix_count_sum)) * ff_v_bd_prefix_count_sum) + (ff_r_bd_prefix_count_sum))) /\ ((((exists ff_h_bd_prefix_count_sum_successor. ff_h_bd_prefix_count_sum_successor + S (ff_s_bd_prefix_count_sum) = S ((S (S ff_i_bd_prefix_count_sum)) * ff_v_bd_prefix_count_sum)) /\ exists ff_q_bd_prefix_count_sum_successor. ff_u_bd_prefix_count_sum = ff_q_bd_prefix_count_sum_successor * S ((S (S ff_i_bd_prefix_count_sum)) * ff_v_bd_prefix_count_sum) + (ff_s_bd_prefix_count_sum))) /\ ff_s_bd_prefix_count_sum = ff_r_bd_prefix_count_sum + ff_a_bd_prefix_count_sum)))))) /\ (forall ff_i_bd_prefix_count_bits. (exists ff_lt_bd_prefix_count_bits_bound. ff_lt_bd_prefix_count_bits_bound + S ff_i_bd_prefix_count_bits = l) -> exists ff_bit_bd_prefix_count_bits. ((((exists ff_h_bd_prefix_count_bits_decoded. ff_h_bd_prefix_count_bits_decoded + S (ff_bit_bd_prefix_count_bits) = S ((S (ff_i_bd_prefix_count_bits)) * c)) /\ exists ff_q_bd_prefix_count_bits_decoded. b = ff_q_bd_prefix_count_bits_decoded * S ((S (ff_i_bd_prefix_count_bits)) * c) + (ff_bit_bd_prefix_count_bits))) /\ (ff_bit_bd_prefix_count_bits = 0 \/ ff_bit_bd_prefix_count_bits = 1)))))

Constructive proof overview

Generated structural guide

Every actual beta-coded binary prefix has an independently beta-witnessed exact population count.

The unchanged tactic script uses 2 declared prerequisites and contains 15 exact native proof lines.

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

Proof neighborhood

Direct dependencies

BD000D binary_digit_prefix_all_bits bit_count_exists Stable theorem; checked-use authorized

Direct dependents

BD0010 binary_digit_operation_count_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

15 script commands · 3 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.

Named ingredients (1)
binary digit prefix all bits
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 hdigits
02Establish hallL5–14

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply binary digit prefix all bits.

  1. L5
    have hall : (forall ff_i_bd_count_all. (exists ff_lt_bd_count_all_bound. ff_lt_bd_count_all_bound + S ff_i_bd_count_all = l) -> exists ff_bit_bd_count_all. ((((exists ff_h_bd_count_all_decoded. ff_h_bd_count_all_decoded + S (ff_bit_bd_count_all) = S ((S (ff_i_bd_count_all)) * c)) /\ exists ff_q_bd_count_all_decoded. b = ff_q_bd_count_all_decoded * S ((S (ff_i_bd_count_all)) * c) + (ff_bit_bd_count_all))) /\ (ff_bit_bd_count_all = 0 \/ ff_bit_bd_count_all = 1)))
  2. L6
    specialize binary_digit_prefix_all_bits b
  3. L7
    specialize binary_digit_prefix_all_bits c
  4. L8
    specialize binary_digit_prefix_all_bits l
  5. L9
    apply binary_digit_prefix_all_bits
  6. L10
    exact hdigits
  7. L11
    specialize bit_count_exists b
  8. L12
    specialize bit_count_exists c
  9. L13
    specialize bit_count_exists l
  10. L14
    apply bit_count_exists
03Use earlier factsL15–15

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

  1. L15
    exact hall

Library-wide reading audit

Original exact command ledger · 15 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro hdigits
  5. 0005have hall : (forall ff_i_bd_count_all. (exists ff_lt_bd_count_all_bound. ff_lt_bd_count_all_bound + S ff_i_bd_count_all = l) -> exists ff_bit_bd_count_all. ((((exists ff_h_bd_count_all_decoded. ff_h_bd_count_all_decoded + S (ff_bit_bd_count_all) = S ((S (ff_i_bd_count_all)) * c)) /\ exists ff_q_bd_count_all_decoded. b = ff_q_bd_count_all_decoded * S ((S (ff_i_bd_count_all)) * c) + (ff_bit_bd_count_all))) /\ (ff_bit_bd_count_all = 0 \/ ff_bit_bd_count_all = 1)))
  6. 0006specialize binary_digit_prefix_all_bits b
  7. 0007specialize binary_digit_prefix_all_bits c
  8. 0008specialize binary_digit_prefix_all_bits l
  9. 0009apply binary_digit_prefix_all_bits
  10. 0010exact hdigits
  11. 0011specialize bit_count_exists b
  12. 0012specialize bit_count_exists c
  13. 0013specialize bit_count_exists l
  14. 0014apply bit_count_exists
  15. 0015exact hall

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