BD000B

binary_exponent_digit_prefix_value_functional

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

A fixed actual beta-coded binary digit sequence represents exactly one natural Horner exponent.

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 n N l b c. (((forall ff_index_be_bd_value_left_digits ff_digit_be_bd_value_left_digits. (exists ff_lt_be_bd_value_left_digits_bound. ff_lt_be_bd_value_left_digits_bound + S ff_index_be_bd_value_left_digits = l) -> (((exists ff_h_be_bd_value_left_digits_digit. ff_h_be_bd_value_left_digits_digit + S (ff_digit_be_bd_value_left_digits) = S ((S (ff_index_be_bd_value_left_digits)) * c)) /\ exists ff_q_be_bd_value_left_digits_digit. b = ff_q_be_bd_value_left_digits_digit * S ((S (ff_index_be_bd_value_left_digits)) * c) + (ff_digit_be_bd_value_left_digits))) -> (ff_digit_be_bd_value_left_digits = 0 \/ ff_digit_be_bd_value_left_digits = 1)) /\ (exists ff_u_ph_bd_value_left_horner ff_v_ph_bd_value_left_horner. ((((exists fs_h_ph_bd_value_left_horner_body_start. fs_h_ph_bd_value_left_horner_body_start + S (0) = S ((S (0)) * ff_v_ph_bd_value_left_horner)) /\ exists fs_q_ph_bd_value_left_horner_body_start. ff_u_ph_bd_value_left_horner = fs_q_ph_bd_value_left_horner_body_start * S ((S (0)) * ff_v_ph_bd_value_left_horner) + (0))) /\ ((((exists fs_h_ph_bd_value_left_horner_body_terminal. fs_h_ph_bd_value_left_horner_body_terminal + S (n) = S ((S (l)) * ff_v_ph_bd_value_left_horner)) /\ exists fs_q_ph_bd_value_left_horner_body_terminal. ff_u_ph_bd_value_left_horner = fs_q_ph_bd_value_left_horner_body_terminal * S ((S (l)) * ff_v_ph_bd_value_left_horner) + (n))) /\ forall ff_i_ph_bd_value_left_horner_body_steps. (exists ph_bound_bd_value_left_horner_body_steps. ph_bound_bd_value_left_horner_body_steps + S ff_i_ph_bd_value_left_horner_body_steps = l) -> exists ff_coefficient_ph_bd_value_left_horner_body_steps ff_previous_ph_bd_value_left_horner_body_steps ff_current_ph_bd_value_left_horner_body_steps. ((((exists fs_h_ph_bd_value_left_horner_body_steps_coefficient. fs_h_ph_bd_value_left_horner_body_steps_coefficient + S (ff_coefficient_ph_bd_value_left_horner_body_steps) = S ((S (ff_i_ph_bd_value_left_horner_body_steps)) * c)) /\ exists fs_q_ph_bd_value_left_horner_body_steps_coefficient. b = fs_q_ph_bd_value_left_horner_body_steps_coefficient * S ((S (ff_i_ph_bd_value_left_horner_body_steps)) * c) + (ff_coefficient_ph_bd_value_left_horner_body_steps))) /\ ((((exists fs_h_ph_bd_value_left_horner_body_steps_before. fs_h_ph_bd_value_left_horner_body_steps_before + S (ff_previous_ph_bd_value_left_horner_body_steps) = S ((S (ff_i_ph_bd_value_left_horner_body_steps)) * ff_v_ph_bd_value_left_horner)) /\ exists fs_q_ph_bd_value_left_horner_body_steps_before. ff_u_ph_bd_value_left_horner = fs_q_ph_bd_value_left_horner_body_steps_before * S ((S (ff_i_ph_bd_value_left_horner_body_steps)) * ff_v_ph_bd_value_left_horner) + (ff_previous_ph_bd_value_left_horner_body_steps))) /\ ((((exists fs_h_ph_bd_value_left_horner_body_steps_after. fs_h_ph_bd_value_left_horner_body_steps_after + S (ff_current_ph_bd_value_left_horner_body_steps) = S ((S (S ff_i_ph_bd_value_left_horner_body_steps)) * ff_v_ph_bd_value_left_horner)) /\ exists fs_q_ph_bd_value_left_horner_body_steps_after. ff_u_ph_bd_value_left_horner = fs_q_ph_bd_value_left_horner_body_steps_after * S ((S (S ff_i_ph_bd_value_left_horner_body_steps)) * ff_v_ph_bd_value_left_horner) + (ff_current_ph_bd_value_left_horner_body_steps))) /\ ff_current_ph_bd_value_left_horner_body_steps = ff_previous_ph_bd_value_left_horner_body_steps * 2 + ff_coefficient_ph_bd_value_left_horner_body_steps)))))))) -> (((forall ff_index_be_bd_value_right_digits ff_digit_be_bd_value_right_digits. (exists ff_lt_be_bd_value_right_digits_bound. ff_lt_be_bd_value_right_digits_bound + S ff_index_be_bd_value_right_digits = l) -> (((exists ff_h_be_bd_value_right_digits_digit. ff_h_be_bd_value_right_digits_digit + S (ff_digit_be_bd_value_right_digits) = S ((S (ff_index_be_bd_value_right_digits)) * c)) /\ exists ff_q_be_bd_value_right_digits_digit. b = ff_q_be_bd_value_right_digits_digit * S ((S (ff_index_be_bd_value_right_digits)) * c) + (ff_digit_be_bd_value_right_digits))) -> (ff_digit_be_bd_value_right_digits = 0 \/ ff_digit_be_bd_value_right_digits = 1)) /\ (exists ff_u_ph_bd_value_right_horner ff_v_ph_bd_value_right_horner. ((((exists fs_h_ph_bd_value_right_horner_body_start. fs_h_ph_bd_value_right_horner_body_start + S (0) = S ((S (0)) * ff_v_ph_bd_value_right_horner)) /\ exists fs_q_ph_bd_value_right_horner_body_start. ff_u_ph_bd_value_right_horner = fs_q_ph_bd_value_right_horner_body_start * S ((S (0)) * ff_v_ph_bd_value_right_horner) + (0))) /\ ((((exists fs_h_ph_bd_value_right_horner_body_terminal. fs_h_ph_bd_value_right_horner_body_terminal + S (N) = S ((S (l)) * ff_v_ph_bd_value_right_horner)) /\ exists fs_q_ph_bd_value_right_horner_body_terminal. ff_u_ph_bd_value_right_horner = fs_q_ph_bd_value_right_horner_body_terminal * S ((S (l)) * ff_v_ph_bd_value_right_horner) + (N))) /\ forall ff_i_ph_bd_value_right_horner_body_steps. (exists ph_bound_bd_value_right_horner_body_steps. ph_bound_bd_value_right_horner_body_steps + S ff_i_ph_bd_value_right_horner_body_steps = l) -> exists ff_coefficient_ph_bd_value_right_horner_body_steps ff_previous_ph_bd_value_right_horner_body_steps ff_current_ph_bd_value_right_horner_body_steps. ((((exists fs_h_ph_bd_value_right_horner_body_steps_coefficient. fs_h_ph_bd_value_right_horner_body_steps_coefficient + S (ff_coefficient_ph_bd_value_right_horner_body_steps) = S ((S (ff_i_ph_bd_value_right_horner_body_steps)) * c)) /\ exists fs_q_ph_bd_value_right_horner_body_steps_coefficient. b = fs_q_ph_bd_value_right_horner_body_steps_coefficient * S ((S (ff_i_ph_bd_value_right_horner_body_steps)) * c) + (ff_coefficient_ph_bd_value_right_horner_body_steps))) /\ ((((exists fs_h_ph_bd_value_right_horner_body_steps_before. fs_h_ph_bd_value_right_horner_body_steps_before + S (ff_previous_ph_bd_value_right_horner_body_steps) = S ((S (ff_i_ph_bd_value_right_horner_body_steps)) * ff_v_ph_bd_value_right_horner)) /\ exists fs_q_ph_bd_value_right_horner_body_steps_before. ff_u_ph_bd_value_right_horner = fs_q_ph_bd_value_right_horner_body_steps_before * S ((S (ff_i_ph_bd_value_right_horner_body_steps)) * ff_v_ph_bd_value_right_horner) + (ff_previous_ph_bd_value_right_horner_body_steps))) /\ ((((exists fs_h_ph_bd_value_right_horner_body_steps_after. fs_h_ph_bd_value_right_horner_body_steps_after + S (ff_current_ph_bd_value_right_horner_body_steps) = S ((S (S ff_i_ph_bd_value_right_horner_body_steps)) * ff_v_ph_bd_value_right_horner)) /\ exists fs_q_ph_bd_value_right_horner_body_steps_after. ff_u_ph_bd_value_right_horner = fs_q_ph_bd_value_right_horner_body_steps_after * S ((S (S ff_i_ph_bd_value_right_horner_body_steps)) * ff_v_ph_bd_value_right_horner) + (ff_current_ph_bd_value_right_horner_body_steps))) /\ ff_current_ph_bd_value_right_horner_body_steps = ff_previous_ph_bd_value_right_horner_body_steps * 2 + ff_coefficient_ph_bd_value_right_horner_body_steps)))))))) -> n = N

Constructive proof overview

Generated structural guide

A fixed actual beta-coded binary digit sequence represents exactly one natural Horner exponent.

The unchanged tactic script uses 1 declared prerequisite and contains 18 exact native proof lines.

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

Proof neighborhood

Direct dependencies

beta_horner_eval_functional Alpha theorem; checked-use authorized

Direct dependents

none

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

18 script commands · 3 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 assumptionsL1–7

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

  1. L1
    intro n
  2. L2
    intro N
  3. L3
    intro l
  4. L4
    intro b
  5. L5
    intro c
  6. L6
    intro hleft
  7. L7
    intro hright
02Separate the logical casesL8–9

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

  1. L8
    cases hleft
  2. L9
    cases hright
03Use earlier factsL10–18

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

  1. L10
    specialize beta_horner_eval_functional b
  2. L11
    specialize beta_horner_eval_functional c
  3. L12
    specialize beta_horner_eval_functional 2
  4. L13
    specialize beta_horner_eval_functional l
  5. L14
    specialize beta_horner_eval_functional n
  6. L15
    specialize beta_horner_eval_functional N
  7. L16
    apply beta_horner_eval_functional
  8. L17
    exact hleft_right
  9. L18
    exact hright_right

Library-wide reading audit

Original exact command ledger · 18 lines
  1. 0001intro n
  2. 0002intro N
  3. 0003intro l
  4. 0004intro b
  5. 0005intro c
  6. 0006intro hleft
  7. 0007intro hright
  8. 0008cases hleft
  9. 0009cases hright
  10. 0010specialize beta_horner_eval_functional b
  11. 0011specialize beta_horner_eval_functional c
  12. 0012specialize beta_horner_eval_functional 2
  13. 0013specialize beta_horner_eval_functional l
  14. 0014specialize beta_horner_eval_functional n
  15. 0015specialize beta_horner_eval_functional N
  16. 0016apply beta_horner_eval_functional
  17. 0017exact hleft_right
  18. 0018exact hright_right

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