BL0011

binary_length_exists

Every natural number has a constructively witnessed canonical bit length.

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

∀ n. ∃ l. BitLen(n,l)

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

Definition DAG

BitLen(n,ell) · 1

Actual proof prerequisites

binary_length_zerobinary_length_successor_step
Original expanded first-order statement 
forall n. exists l. ((((n) = 0 /\ (l) = 1) \/ exists ff_exponent_bl_length ff_lower_bl_length ff_upper_bl_length. (((l) = S ff_exponent_bl_length) /\ ((exists ff_positive_bl_length. ff_positive_bl_length + 1 = (n)) /\ ((exists pa_b_bl_length_lower pa_c_bl_length_lower. ((forall pa_i_bl_length_lower_repeat. (exists pa_lt_bl_length_lower_repeat_bound. pa_lt_bl_length_lower_repeat_bound + S pa_i_bl_length_lower_repeat = ff_exponent_bl_length) -> (((exists pa_h_bl_length_lower_repeat_decoded. pa_h_bl_length_lower_repeat_decoded + S (2) = S ((S (pa_i_bl_length_lower_repeat)) * pa_c_bl_length_lower)) /\ exists pa_q_bl_length_lower_repeat_decoded. pa_b_bl_length_lower = pa_q_bl_length_lower_repeat_decoded * S ((S (pa_i_bl_length_lower_repeat)) * pa_c_bl_length_lower) + (2)))) /\ (exists pa_u_bl_length_lower_product pa_v_bl_length_lower_product. ((((exists pa_h_bl_length_lower_product_start. pa_h_bl_length_lower_product_start + S (1) = S ((S (0)) * pa_v_bl_length_lower_product)) /\ exists pa_q_bl_length_lower_product_start. pa_u_bl_length_lower_product = pa_q_bl_length_lower_product_start * S ((S (0)) * pa_v_bl_length_lower_product) + (1))) /\ ((((exists pa_h_bl_length_lower_product_terminal. pa_h_bl_length_lower_product_terminal + S (ff_lower_bl_length) = S ((S (ff_exponent_bl_length)) * pa_v_bl_length_lower_product)) /\ exists pa_q_bl_length_lower_product_terminal. pa_u_bl_length_lower_product = pa_q_bl_length_lower_product_terminal * S ((S (ff_exponent_bl_length)) * pa_v_bl_length_lower_product) + (ff_lower_bl_length))) /\ forall pa_i_bl_length_lower_product. (exists pa_lt_bl_length_lower_product_bound. pa_lt_bl_length_lower_product_bound + S pa_i_bl_length_lower_product = ff_exponent_bl_length) -> exists pa_p_bl_length_lower_product pa_r_bl_length_lower_product pa_s_bl_length_lower_product. ((((exists pa_h_bl_length_lower_product_factor. pa_h_bl_length_lower_product_factor + S (pa_p_bl_length_lower_product) = S ((S (pa_i_bl_length_lower_product)) * pa_c_bl_length_lower)) /\ exists pa_q_bl_length_lower_product_factor. pa_b_bl_length_lower = pa_q_bl_length_lower_product_factor * S ((S (pa_i_bl_length_lower_product)) * pa_c_bl_length_lower) + (pa_p_bl_length_lower_product))) /\ ((((exists pa_h_bl_length_lower_product_partial. pa_h_bl_length_lower_product_partial + S (pa_r_bl_length_lower_product) = S ((S (pa_i_bl_length_lower_product)) * pa_v_bl_length_lower_product)) /\ exists pa_q_bl_length_lower_product_partial. pa_u_bl_length_lower_product = pa_q_bl_length_lower_product_partial * S ((S (pa_i_bl_length_lower_product)) * pa_v_bl_length_lower_product) + (pa_r_bl_length_lower_product))) /\ ((((exists pa_h_bl_length_lower_product_successor. pa_h_bl_length_lower_product_successor + S (pa_s_bl_length_lower_product) = S ((S (S pa_i_bl_length_lower_product)) * pa_v_bl_length_lower_product)) /\ exists pa_q_bl_length_lower_product_successor. pa_u_bl_length_lower_product = pa_q_bl_length_lower_product_successor * S ((S (S pa_i_bl_length_lower_product)) * pa_v_bl_length_lower_product) + (pa_s_bl_length_lower_product))) /\ pa_s_bl_length_lower_product = pa_r_bl_length_lower_product * pa_p_bl_length_lower_product)))))))) /\ ((exists pa_b_bl_length_upper pa_c_bl_length_upper. ((forall pa_i_bl_length_upper_repeat. (exists pa_lt_bl_length_upper_repeat_bound. pa_lt_bl_length_upper_repeat_bound + S pa_i_bl_length_upper_repeat = l) -> (((exists pa_h_bl_length_upper_repeat_decoded. pa_h_bl_length_upper_repeat_decoded + S (2) = S ((S (pa_i_bl_length_upper_repeat)) * pa_c_bl_length_upper)) /\ exists pa_q_bl_length_upper_repeat_decoded. pa_b_bl_length_upper = pa_q_bl_length_upper_repeat_decoded * S ((S (pa_i_bl_length_upper_repeat)) * pa_c_bl_length_upper) + (2)))) /\ (exists pa_u_bl_length_upper_product pa_v_bl_length_upper_product. ((((exists pa_h_bl_length_upper_product_start. pa_h_bl_length_upper_product_start + S (1) = S ((S (0)) * pa_v_bl_length_upper_product)) /\ exists pa_q_bl_length_upper_product_start. pa_u_bl_length_upper_product = pa_q_bl_length_upper_product_start * S ((S (0)) * pa_v_bl_length_upper_product) + (1))) /\ ((((exists pa_h_bl_length_upper_product_terminal. pa_h_bl_length_upper_product_terminal + S (ff_upper_bl_length) = S ((S (l)) * pa_v_bl_length_upper_product)) /\ exists pa_q_bl_length_upper_product_terminal. pa_u_bl_length_upper_product = pa_q_bl_length_upper_product_terminal * S ((S (l)) * pa_v_bl_length_upper_product) + (ff_upper_bl_length))) /\ forall pa_i_bl_length_upper_product. (exists pa_lt_bl_length_upper_product_bound. pa_lt_bl_length_upper_product_bound + S pa_i_bl_length_upper_product = l) -> exists pa_p_bl_length_upper_product pa_r_bl_length_upper_product pa_s_bl_length_upper_product. ((((exists pa_h_bl_length_upper_product_factor. pa_h_bl_length_upper_product_factor + S (pa_p_bl_length_upper_product) = S ((S (pa_i_bl_length_upper_product)) * pa_c_bl_length_upper)) /\ exists pa_q_bl_length_upper_product_factor. pa_b_bl_length_upper = pa_q_bl_length_upper_product_factor * S ((S (pa_i_bl_length_upper_product)) * pa_c_bl_length_upper) + (pa_p_bl_length_upper_product))) /\ ((((exists pa_h_bl_length_upper_product_partial. pa_h_bl_length_upper_product_partial + S (pa_r_bl_length_upper_product) = S ((S (pa_i_bl_length_upper_product)) * pa_v_bl_length_upper_product)) /\ exists pa_q_bl_length_upper_product_partial. pa_u_bl_length_upper_product = pa_q_bl_length_upper_product_partial * S ((S (pa_i_bl_length_upper_product)) * pa_v_bl_length_upper_product) + (pa_r_bl_length_upper_product))) /\ ((((exists pa_h_bl_length_upper_product_successor. pa_h_bl_length_upper_product_successor + S (pa_s_bl_length_upper_product) = S ((S (S pa_i_bl_length_upper_product)) * pa_v_bl_length_upper_product)) /\ exists pa_q_bl_length_upper_product_successor. pa_u_bl_length_upper_product = pa_q_bl_length_upper_product_successor * S ((S (S pa_i_bl_length_upper_product)) * pa_v_bl_length_upper_product) + (pa_s_bl_length_upper_product))) /\ pa_s_bl_length_upper_product = pa_r_bl_length_upper_product * pa_p_bl_length_upper_product)))))))) /\ ((exists ff_lower_gap_bl_length. ff_lower_gap_bl_length + (ff_lower_bl_length) = (n)) /\ (exists ff_upper_gap_bl_length. ff_upper_gap_bl_length + S (n) = (ff_upper_bl_length)))))))))

Complete unchanged native tactic proof

All 8 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

8 script commands · 5 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.

Named ingredients (2)
binary length zerobinary length successor step
01Induction on nL1–1

Split the argument into the base and successor obligations. The induction hypothesis is available only in the successor branch.

  1. L1
    induction n
02Construct an explicit witnessL2–2

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

  1. L2
    exists 1
03Use earlier factsL3–3

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

  1. L3
    exact binary_length_zero
04Separate the logical casesL4–4

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

  1. L4
    cases IH
05Use earlier factsL5–8

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

  1. L5
    specialize binary_length_successor_step n
  2. L6
    specialize binary_length_successor_step x
  3. L7
    apply binary_length_successor_step
  4. L8
    exact IH_witness

Library-wide reading audit

Original defined command ledger · 8 lines
  1. 0001induction n
  2. 0002exists 1
  3. 0003exact binary_length_zero
  4. 0004cases IH
  5. 0005specialize binary_length_successor_step n
  6. 0006specialize binary_length_successor_step x
  7. 0007apply binary_length_successor_step
  8. 0008exact IH_witness