BD000B

binary_exponent_digit_prefix_value_functional

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

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

∀ n. ∀ N. ∀ l. ∀ b. ∀ c. BinaryExponentDigitCode(n,l,b,c) → BinaryExponentDigitCode(N,l,b,c) → n = N

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

Definition DAG

BinaryExponentDigitCode(e,ell,b,c) · 2

Actual proof prerequisites

beta_horner_eval_functional · checked external prerequisite
Original expanded first-order 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

Complete unchanged native tactic proof

All 18 lines are the exact independently kernel-checked original script.

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.

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 assumptionsL1–7

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 defined 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