BL000F

binary_length_zero_input_value

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

Any binary-length witness for zero is necessarily one.

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 l. ((((0) = 0 /\ (l) = 1) \/ exists ff_exponent_bl_zero_input ff_lower_bl_zero_input ff_upper_bl_zero_input. (((l) = S ff_exponent_bl_zero_input) /\ ((exists ff_positive_bl_zero_input. ff_positive_bl_zero_input + 1 = (0)) /\ ((exists pa_b_bl_zero_input_lower pa_c_bl_zero_input_lower. ((forall pa_i_bl_zero_input_lower_repeat. (exists pa_lt_bl_zero_input_lower_repeat_bound. pa_lt_bl_zero_input_lower_repeat_bound + S pa_i_bl_zero_input_lower_repeat = ff_exponent_bl_zero_input) -> (((exists pa_h_bl_zero_input_lower_repeat_decoded. pa_h_bl_zero_input_lower_repeat_decoded + S (2) = S ((S (pa_i_bl_zero_input_lower_repeat)) * pa_c_bl_zero_input_lower)) /\ exists pa_q_bl_zero_input_lower_repeat_decoded. pa_b_bl_zero_input_lower = pa_q_bl_zero_input_lower_repeat_decoded * S ((S (pa_i_bl_zero_input_lower_repeat)) * pa_c_bl_zero_input_lower) + (2)))) /\ (exists pa_u_bl_zero_input_lower_product pa_v_bl_zero_input_lower_product. ((((exists pa_h_bl_zero_input_lower_product_start. pa_h_bl_zero_input_lower_product_start + S (1) = S ((S (0)) * pa_v_bl_zero_input_lower_product)) /\ exists pa_q_bl_zero_input_lower_product_start. pa_u_bl_zero_input_lower_product = pa_q_bl_zero_input_lower_product_start * S ((S (0)) * pa_v_bl_zero_input_lower_product) + (1))) /\ ((((exists pa_h_bl_zero_input_lower_product_terminal. pa_h_bl_zero_input_lower_product_terminal + S (ff_lower_bl_zero_input) = S ((S (ff_exponent_bl_zero_input)) * pa_v_bl_zero_input_lower_product)) /\ exists pa_q_bl_zero_input_lower_product_terminal. pa_u_bl_zero_input_lower_product = pa_q_bl_zero_input_lower_product_terminal * S ((S (ff_exponent_bl_zero_input)) * pa_v_bl_zero_input_lower_product) + (ff_lower_bl_zero_input))) /\ forall pa_i_bl_zero_input_lower_product. (exists pa_lt_bl_zero_input_lower_product_bound. pa_lt_bl_zero_input_lower_product_bound + S pa_i_bl_zero_input_lower_product = ff_exponent_bl_zero_input) -> exists pa_p_bl_zero_input_lower_product pa_r_bl_zero_input_lower_product pa_s_bl_zero_input_lower_product. ((((exists pa_h_bl_zero_input_lower_product_factor. pa_h_bl_zero_input_lower_product_factor + S (pa_p_bl_zero_input_lower_product) = S ((S (pa_i_bl_zero_input_lower_product)) * pa_c_bl_zero_input_lower)) /\ exists pa_q_bl_zero_input_lower_product_factor. pa_b_bl_zero_input_lower = pa_q_bl_zero_input_lower_product_factor * S ((S (pa_i_bl_zero_input_lower_product)) * pa_c_bl_zero_input_lower) + (pa_p_bl_zero_input_lower_product))) /\ ((((exists pa_h_bl_zero_input_lower_product_partial. pa_h_bl_zero_input_lower_product_partial + S (pa_r_bl_zero_input_lower_product) = S ((S (pa_i_bl_zero_input_lower_product)) * pa_v_bl_zero_input_lower_product)) /\ exists pa_q_bl_zero_input_lower_product_partial. pa_u_bl_zero_input_lower_product = pa_q_bl_zero_input_lower_product_partial * S ((S (pa_i_bl_zero_input_lower_product)) * pa_v_bl_zero_input_lower_product) + (pa_r_bl_zero_input_lower_product))) /\ ((((exists pa_h_bl_zero_input_lower_product_successor. pa_h_bl_zero_input_lower_product_successor + S (pa_s_bl_zero_input_lower_product) = S ((S (S pa_i_bl_zero_input_lower_product)) * pa_v_bl_zero_input_lower_product)) /\ exists pa_q_bl_zero_input_lower_product_successor. pa_u_bl_zero_input_lower_product = pa_q_bl_zero_input_lower_product_successor * S ((S (S pa_i_bl_zero_input_lower_product)) * pa_v_bl_zero_input_lower_product) + (pa_s_bl_zero_input_lower_product))) /\ pa_s_bl_zero_input_lower_product = pa_r_bl_zero_input_lower_product * pa_p_bl_zero_input_lower_product)))))))) /\ ((exists pa_b_bl_zero_input_upper pa_c_bl_zero_input_upper. ((forall pa_i_bl_zero_input_upper_repeat. (exists pa_lt_bl_zero_input_upper_repeat_bound. pa_lt_bl_zero_input_upper_repeat_bound + S pa_i_bl_zero_input_upper_repeat = l) -> (((exists pa_h_bl_zero_input_upper_repeat_decoded. pa_h_bl_zero_input_upper_repeat_decoded + S (2) = S ((S (pa_i_bl_zero_input_upper_repeat)) * pa_c_bl_zero_input_upper)) /\ exists pa_q_bl_zero_input_upper_repeat_decoded. pa_b_bl_zero_input_upper = pa_q_bl_zero_input_upper_repeat_decoded * S ((S (pa_i_bl_zero_input_upper_repeat)) * pa_c_bl_zero_input_upper) + (2)))) /\ (exists pa_u_bl_zero_input_upper_product pa_v_bl_zero_input_upper_product. ((((exists pa_h_bl_zero_input_upper_product_start. pa_h_bl_zero_input_upper_product_start + S (1) = S ((S (0)) * pa_v_bl_zero_input_upper_product)) /\ exists pa_q_bl_zero_input_upper_product_start. pa_u_bl_zero_input_upper_product = pa_q_bl_zero_input_upper_product_start * S ((S (0)) * pa_v_bl_zero_input_upper_product) + (1))) /\ ((((exists pa_h_bl_zero_input_upper_product_terminal. pa_h_bl_zero_input_upper_product_terminal + S (ff_upper_bl_zero_input) = S ((S (l)) * pa_v_bl_zero_input_upper_product)) /\ exists pa_q_bl_zero_input_upper_product_terminal. pa_u_bl_zero_input_upper_product = pa_q_bl_zero_input_upper_product_terminal * S ((S (l)) * pa_v_bl_zero_input_upper_product) + (ff_upper_bl_zero_input))) /\ forall pa_i_bl_zero_input_upper_product. (exists pa_lt_bl_zero_input_upper_product_bound. pa_lt_bl_zero_input_upper_product_bound + S pa_i_bl_zero_input_upper_product = l) -> exists pa_p_bl_zero_input_upper_product pa_r_bl_zero_input_upper_product pa_s_bl_zero_input_upper_product. ((((exists pa_h_bl_zero_input_upper_product_factor. pa_h_bl_zero_input_upper_product_factor + S (pa_p_bl_zero_input_upper_product) = S ((S (pa_i_bl_zero_input_upper_product)) * pa_c_bl_zero_input_upper)) /\ exists pa_q_bl_zero_input_upper_product_factor. pa_b_bl_zero_input_upper = pa_q_bl_zero_input_upper_product_factor * S ((S (pa_i_bl_zero_input_upper_product)) * pa_c_bl_zero_input_upper) + (pa_p_bl_zero_input_upper_product))) /\ ((((exists pa_h_bl_zero_input_upper_product_partial. pa_h_bl_zero_input_upper_product_partial + S (pa_r_bl_zero_input_upper_product) = S ((S (pa_i_bl_zero_input_upper_product)) * pa_v_bl_zero_input_upper_product)) /\ exists pa_q_bl_zero_input_upper_product_partial. pa_u_bl_zero_input_upper_product = pa_q_bl_zero_input_upper_product_partial * S ((S (pa_i_bl_zero_input_upper_product)) * pa_v_bl_zero_input_upper_product) + (pa_r_bl_zero_input_upper_product))) /\ ((((exists pa_h_bl_zero_input_upper_product_successor. pa_h_bl_zero_input_upper_product_successor + S (pa_s_bl_zero_input_upper_product) = S ((S (S pa_i_bl_zero_input_upper_product)) * pa_v_bl_zero_input_upper_product)) /\ exists pa_q_bl_zero_input_upper_product_successor. pa_u_bl_zero_input_upper_product = pa_q_bl_zero_input_upper_product_successor * S ((S (S pa_i_bl_zero_input_upper_product)) * pa_v_bl_zero_input_upper_product) + (pa_s_bl_zero_input_upper_product))) /\ pa_s_bl_zero_input_upper_product = pa_r_bl_zero_input_upper_product * pa_p_bl_zero_input_upper_product)))))))) /\ ((exists ff_lower_gap_bl_zero_input. ff_lower_gap_bl_zero_input + (ff_lower_bl_zero_input) = (0)) /\ (exists ff_upper_gap_bl_zero_input. ff_upper_gap_bl_zero_input + S (0) = (ff_upper_bl_zero_input))))))))) -> l = 1

Constructive proof overview

Generated structural guide

Any binary-length witness for zero is necessarily one.

The unchanged tactic script uses 2 declared prerequisites and contains 18 exact native proof lines.

Alpha v34 checked-use · first admitted v22 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

add_eq_zero_right Stable theorem; checked-use authorized succ_ne_zero Stable theorem; checked-use authorized

Direct dependents

BL0012 binary_length_zero_input_general

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 · 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.

01Fix variables and assumptionsL1–2

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

  1. L1
    intro l
  2. L2
    intro hlength
02Separate the logical casesL3–4

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

  1. L3
    cases hlength
  2. L4
    cases hlength_left
03Use earlier factsL5–5

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

  1. L5
    exact hlength_left_right
04Separate the logical casesL6–12

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

  1. L6
    cases hlength_right
  2. L7
    cases hlength_right_witness
  3. L8
    cases hlength_right_witness_witness
  4. L9
    cases hlength_right_witness_witness_witness
  5. L10
    cases hlength_right_witness_witness_witness_right
  6. L11
    cases hlength_right_witness_witness_witness_right_left
  7. L12
    exfalso
05Use earlier factsL13–18

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

  1. L13
    specialize succ_ne_zero 0
  2. L14
    apply succ_ne_zero
  3. L15
    specialize add_eq_zero_right x3
  4. L16
    specialize add_eq_zero_right 1
  5. L17
    apply add_eq_zero_right
  6. L18
    exact hlength_right_witness_witness_witness_right_left_witness

Library-wide reading audit

Original exact command ledger · 18 lines
  1. 0001intro l
  2. 0002intro hlength
  3. 0003cases hlength
  4. 0004cases hlength_left
  5. 0005exact hlength_left_right
  6. 0006cases hlength_right
  7. 0007cases hlength_right_witness
  8. 0008cases hlength_right_witness_witness
  9. 0009cases hlength_right_witness_witness_witness
  10. 0010cases hlength_right_witness_witness_witness_right
  11. 0011cases hlength_right_witness_witness_witness_right_left
  12. 0012exfalso
  13. 0013specialize succ_ne_zero 0
  14. 0014apply succ_ne_zero
  15. 0015specialize add_eq_zero_right x3
  16. 0016specialize add_eq_zero_right 1
  17. 0017apply add_eq_zero_right
  18. 0018exact hlength_right_witness_witness_witness_right_left_witness