BL000D

binary_length_zero

The blueprint convention gives zero exactly one displayed binary digit.

Alpha v34 checked-use · first admitted v22 · 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.

G101 and G102 were OPEN when these BitLen foundations were first admitted in Alpha v22. Both are now CLOSED in Alpha v23: complete canonical exponent digits and both exact logarithmic execution bounds are proved.

Exact theorem in conservative defined notation

BitLen(0,1)

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

Definition DAG

BitLen(n,ell) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
(((0) = 0 /\ (1) = 1) \/ exists ff_exponent_bl_zero ff_lower_bl_zero ff_upper_bl_zero. (((1) = S ff_exponent_bl_zero) /\ ((exists ff_positive_bl_zero. ff_positive_bl_zero + 1 = (0)) /\ ((exists pa_b_bl_zero_lower pa_c_bl_zero_lower. ((forall pa_i_bl_zero_lower_repeat. (exists pa_lt_bl_zero_lower_repeat_bound. pa_lt_bl_zero_lower_repeat_bound + S pa_i_bl_zero_lower_repeat = ff_exponent_bl_zero) -> (((exists pa_h_bl_zero_lower_repeat_decoded. pa_h_bl_zero_lower_repeat_decoded + S (2) = S ((S (pa_i_bl_zero_lower_repeat)) * pa_c_bl_zero_lower)) /\ exists pa_q_bl_zero_lower_repeat_decoded. pa_b_bl_zero_lower = pa_q_bl_zero_lower_repeat_decoded * S ((S (pa_i_bl_zero_lower_repeat)) * pa_c_bl_zero_lower) + (2)))) /\ (exists pa_u_bl_zero_lower_product pa_v_bl_zero_lower_product. ((((exists pa_h_bl_zero_lower_product_start. pa_h_bl_zero_lower_product_start + S (1) = S ((S (0)) * pa_v_bl_zero_lower_product)) /\ exists pa_q_bl_zero_lower_product_start. pa_u_bl_zero_lower_product = pa_q_bl_zero_lower_product_start * S ((S (0)) * pa_v_bl_zero_lower_product) + (1))) /\ ((((exists pa_h_bl_zero_lower_product_terminal. pa_h_bl_zero_lower_product_terminal + S (ff_lower_bl_zero) = S ((S (ff_exponent_bl_zero)) * pa_v_bl_zero_lower_product)) /\ exists pa_q_bl_zero_lower_product_terminal. pa_u_bl_zero_lower_product = pa_q_bl_zero_lower_product_terminal * S ((S (ff_exponent_bl_zero)) * pa_v_bl_zero_lower_product) + (ff_lower_bl_zero))) /\ forall pa_i_bl_zero_lower_product. (exists pa_lt_bl_zero_lower_product_bound. pa_lt_bl_zero_lower_product_bound + S pa_i_bl_zero_lower_product = ff_exponent_bl_zero) -> exists pa_p_bl_zero_lower_product pa_r_bl_zero_lower_product pa_s_bl_zero_lower_product. ((((exists pa_h_bl_zero_lower_product_factor. pa_h_bl_zero_lower_product_factor + S (pa_p_bl_zero_lower_product) = S ((S (pa_i_bl_zero_lower_product)) * pa_c_bl_zero_lower)) /\ exists pa_q_bl_zero_lower_product_factor. pa_b_bl_zero_lower = pa_q_bl_zero_lower_product_factor * S ((S (pa_i_bl_zero_lower_product)) * pa_c_bl_zero_lower) + (pa_p_bl_zero_lower_product))) /\ ((((exists pa_h_bl_zero_lower_product_partial. pa_h_bl_zero_lower_product_partial + S (pa_r_bl_zero_lower_product) = S ((S (pa_i_bl_zero_lower_product)) * pa_v_bl_zero_lower_product)) /\ exists pa_q_bl_zero_lower_product_partial. pa_u_bl_zero_lower_product = pa_q_bl_zero_lower_product_partial * S ((S (pa_i_bl_zero_lower_product)) * pa_v_bl_zero_lower_product) + (pa_r_bl_zero_lower_product))) /\ ((((exists pa_h_bl_zero_lower_product_successor. pa_h_bl_zero_lower_product_successor + S (pa_s_bl_zero_lower_product) = S ((S (S pa_i_bl_zero_lower_product)) * pa_v_bl_zero_lower_product)) /\ exists pa_q_bl_zero_lower_product_successor. pa_u_bl_zero_lower_product = pa_q_bl_zero_lower_product_successor * S ((S (S pa_i_bl_zero_lower_product)) * pa_v_bl_zero_lower_product) + (pa_s_bl_zero_lower_product))) /\ pa_s_bl_zero_lower_product = pa_r_bl_zero_lower_product * pa_p_bl_zero_lower_product)))))))) /\ ((exists pa_b_bl_zero_upper pa_c_bl_zero_upper. ((forall pa_i_bl_zero_upper_repeat. (exists pa_lt_bl_zero_upper_repeat_bound. pa_lt_bl_zero_upper_repeat_bound + S pa_i_bl_zero_upper_repeat = 1) -> (((exists pa_h_bl_zero_upper_repeat_decoded. pa_h_bl_zero_upper_repeat_decoded + S (2) = S ((S (pa_i_bl_zero_upper_repeat)) * pa_c_bl_zero_upper)) /\ exists pa_q_bl_zero_upper_repeat_decoded. pa_b_bl_zero_upper = pa_q_bl_zero_upper_repeat_decoded * S ((S (pa_i_bl_zero_upper_repeat)) * pa_c_bl_zero_upper) + (2)))) /\ (exists pa_u_bl_zero_upper_product pa_v_bl_zero_upper_product. ((((exists pa_h_bl_zero_upper_product_start. pa_h_bl_zero_upper_product_start + S (1) = S ((S (0)) * pa_v_bl_zero_upper_product)) /\ exists pa_q_bl_zero_upper_product_start. pa_u_bl_zero_upper_product = pa_q_bl_zero_upper_product_start * S ((S (0)) * pa_v_bl_zero_upper_product) + (1))) /\ ((((exists pa_h_bl_zero_upper_product_terminal. pa_h_bl_zero_upper_product_terminal + S (ff_upper_bl_zero) = S ((S (1)) * pa_v_bl_zero_upper_product)) /\ exists pa_q_bl_zero_upper_product_terminal. pa_u_bl_zero_upper_product = pa_q_bl_zero_upper_product_terminal * S ((S (1)) * pa_v_bl_zero_upper_product) + (ff_upper_bl_zero))) /\ forall pa_i_bl_zero_upper_product. (exists pa_lt_bl_zero_upper_product_bound. pa_lt_bl_zero_upper_product_bound + S pa_i_bl_zero_upper_product = 1) -> exists pa_p_bl_zero_upper_product pa_r_bl_zero_upper_product pa_s_bl_zero_upper_product. ((((exists pa_h_bl_zero_upper_product_factor. pa_h_bl_zero_upper_product_factor + S (pa_p_bl_zero_upper_product) = S ((S (pa_i_bl_zero_upper_product)) * pa_c_bl_zero_upper)) /\ exists pa_q_bl_zero_upper_product_factor. pa_b_bl_zero_upper = pa_q_bl_zero_upper_product_factor * S ((S (pa_i_bl_zero_upper_product)) * pa_c_bl_zero_upper) + (pa_p_bl_zero_upper_product))) /\ ((((exists pa_h_bl_zero_upper_product_partial. pa_h_bl_zero_upper_product_partial + S (pa_r_bl_zero_upper_product) = S ((S (pa_i_bl_zero_upper_product)) * pa_v_bl_zero_upper_product)) /\ exists pa_q_bl_zero_upper_product_partial. pa_u_bl_zero_upper_product = pa_q_bl_zero_upper_product_partial * S ((S (pa_i_bl_zero_upper_product)) * pa_v_bl_zero_upper_product) + (pa_r_bl_zero_upper_product))) /\ ((((exists pa_h_bl_zero_upper_product_successor. pa_h_bl_zero_upper_product_successor + S (pa_s_bl_zero_upper_product) = S ((S (S pa_i_bl_zero_upper_product)) * pa_v_bl_zero_upper_product)) /\ exists pa_q_bl_zero_upper_product_successor. pa_u_bl_zero_upper_product = pa_q_bl_zero_upper_product_successor * S ((S (S pa_i_bl_zero_upper_product)) * pa_v_bl_zero_upper_product) + (pa_s_bl_zero_upper_product))) /\ pa_s_bl_zero_upper_product = pa_r_bl_zero_upper_product * pa_p_bl_zero_upper_product)))))))) /\ ((exists ff_lower_gap_bl_zero. ff_lower_gap_bl_zero + (ff_lower_bl_zero) = (0)) /\ (exists ff_upper_gap_bl_zero. ff_upper_gap_bl_zero + S (0) = (ff_upper_bl_zero))))))))

Complete unchanged native tactic proof

All 4 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

4 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.

01Separate the logical casesL1–2

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

  1. L1
    left
  2. L2
    split
02Calculate and transport equalitiesL3–4

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L3
    refl
  2. L4
    refl

Library-wide reading audit

Original defined command ledger · 4 lines
  1. 0001left
  2. 0002split
  3. 0003refl
  4. 0004refl