BL0009

binary_power_two_successor_double

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

A successor power of two is exactly the sum of two predecessor powers.

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 e p q. (exists pa_b_bl_power pa_c_bl_power. ((forall pa_i_bl_power_repeat. (exists pa_lt_bl_power_repeat_bound. pa_lt_bl_power_repeat_bound + S pa_i_bl_power_repeat = e) -> (((exists pa_h_bl_power_repeat_decoded. pa_h_bl_power_repeat_decoded + S (2) = S ((S (pa_i_bl_power_repeat)) * pa_c_bl_power)) /\ exists pa_q_bl_power_repeat_decoded. pa_b_bl_power = pa_q_bl_power_repeat_decoded * S ((S (pa_i_bl_power_repeat)) * pa_c_bl_power) + (2)))) /\ (exists pa_u_bl_power_product pa_v_bl_power_product. ((((exists pa_h_bl_power_product_start. pa_h_bl_power_product_start + S (1) = S ((S (0)) * pa_v_bl_power_product)) /\ exists pa_q_bl_power_product_start. pa_u_bl_power_product = pa_q_bl_power_product_start * S ((S (0)) * pa_v_bl_power_product) + (1))) /\ ((((exists pa_h_bl_power_product_terminal. pa_h_bl_power_product_terminal + S (p) = S ((S (e)) * pa_v_bl_power_product)) /\ exists pa_q_bl_power_product_terminal. pa_u_bl_power_product = pa_q_bl_power_product_terminal * S ((S (e)) * pa_v_bl_power_product) + (p))) /\ forall pa_i_bl_power_product. (exists pa_lt_bl_power_product_bound. pa_lt_bl_power_product_bound + S pa_i_bl_power_product = e) -> exists pa_p_bl_power_product pa_r_bl_power_product pa_s_bl_power_product. ((((exists pa_h_bl_power_product_factor. pa_h_bl_power_product_factor + S (pa_p_bl_power_product) = S ((S (pa_i_bl_power_product)) * pa_c_bl_power)) /\ exists pa_q_bl_power_product_factor. pa_b_bl_power = pa_q_bl_power_product_factor * S ((S (pa_i_bl_power_product)) * pa_c_bl_power) + (pa_p_bl_power_product))) /\ ((((exists pa_h_bl_power_product_partial. pa_h_bl_power_product_partial + S (pa_r_bl_power_product) = S ((S (pa_i_bl_power_product)) * pa_v_bl_power_product)) /\ exists pa_q_bl_power_product_partial. pa_u_bl_power_product = pa_q_bl_power_product_partial * S ((S (pa_i_bl_power_product)) * pa_v_bl_power_product) + (pa_r_bl_power_product))) /\ ((((exists pa_h_bl_power_product_successor. pa_h_bl_power_product_successor + S (pa_s_bl_power_product) = S ((S (S pa_i_bl_power_product)) * pa_v_bl_power_product)) /\ exists pa_q_bl_power_product_successor. pa_u_bl_power_product = pa_q_bl_power_product_successor * S ((S (S pa_i_bl_power_product)) * pa_v_bl_power_product) + (pa_s_bl_power_product))) /\ pa_s_bl_power_product = pa_r_bl_power_product * pa_p_bl_power_product)))))))) -> (exists pa_b_bl_next_power pa_c_bl_next_power. ((forall pa_i_bl_next_power_repeat. (exists pa_lt_bl_next_power_repeat_bound. pa_lt_bl_next_power_repeat_bound + S pa_i_bl_next_power_repeat = S e) -> (((exists pa_h_bl_next_power_repeat_decoded. pa_h_bl_next_power_repeat_decoded + S (2) = S ((S (pa_i_bl_next_power_repeat)) * pa_c_bl_next_power)) /\ exists pa_q_bl_next_power_repeat_decoded. pa_b_bl_next_power = pa_q_bl_next_power_repeat_decoded * S ((S (pa_i_bl_next_power_repeat)) * pa_c_bl_next_power) + (2)))) /\ (exists pa_u_bl_next_power_product pa_v_bl_next_power_product. ((((exists pa_h_bl_next_power_product_start. pa_h_bl_next_power_product_start + S (1) = S ((S (0)) * pa_v_bl_next_power_product)) /\ exists pa_q_bl_next_power_product_start. pa_u_bl_next_power_product = pa_q_bl_next_power_product_start * S ((S (0)) * pa_v_bl_next_power_product) + (1))) /\ ((((exists pa_h_bl_next_power_product_terminal. pa_h_bl_next_power_product_terminal + S (q) = S ((S (S e)) * pa_v_bl_next_power_product)) /\ exists pa_q_bl_next_power_product_terminal. pa_u_bl_next_power_product = pa_q_bl_next_power_product_terminal * S ((S (S e)) * pa_v_bl_next_power_product) + (q))) /\ forall pa_i_bl_next_power_product. (exists pa_lt_bl_next_power_product_bound. pa_lt_bl_next_power_product_bound + S pa_i_bl_next_power_product = S e) -> exists pa_p_bl_next_power_product pa_r_bl_next_power_product pa_s_bl_next_power_product. ((((exists pa_h_bl_next_power_product_factor. pa_h_bl_next_power_product_factor + S (pa_p_bl_next_power_product) = S ((S (pa_i_bl_next_power_product)) * pa_c_bl_next_power)) /\ exists pa_q_bl_next_power_product_factor. pa_b_bl_next_power = pa_q_bl_next_power_product_factor * S ((S (pa_i_bl_next_power_product)) * pa_c_bl_next_power) + (pa_p_bl_next_power_product))) /\ ((((exists pa_h_bl_next_power_product_partial. pa_h_bl_next_power_product_partial + S (pa_r_bl_next_power_product) = S ((S (pa_i_bl_next_power_product)) * pa_v_bl_next_power_product)) /\ exists pa_q_bl_next_power_product_partial. pa_u_bl_next_power_product = pa_q_bl_next_power_product_partial * S ((S (pa_i_bl_next_power_product)) * pa_v_bl_next_power_product) + (pa_r_bl_next_power_product))) /\ ((((exists pa_h_bl_next_power_product_successor. pa_h_bl_next_power_product_successor + S (pa_s_bl_next_power_product) = S ((S (S pa_i_bl_next_power_product)) * pa_v_bl_next_power_product)) /\ exists pa_q_bl_next_power_product_successor. pa_u_bl_next_power_product = pa_q_bl_next_power_product_successor * S ((S (S pa_i_bl_next_power_product)) * pa_v_bl_next_power_product) + (pa_s_bl_next_power_product))) /\ pa_s_bl_next_power_product = pa_r_bl_next_power_product * pa_p_bl_next_power_product)))))))) -> q = p + p

Constructive proof overview

Generated structural guide

A successor power of two is exactly the sum of two predecessor powers.

The unchanged tactic script uses 3 declared prerequisites and contains 20 exact native proof lines.

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

Proof neighborhood

Direct dependencies

pow_successor_pair_mul Stable theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized two_mul_eq_add_self Alpha theorem; checked-use authorized

Direct dependents

BL000A binary_power_two_strict_growth BL000E binary_length_one

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

20 script commands · 6 reading checkpoints · 1 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 assumptionsL1–5

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

  1. L1
    intro e
  2. L2
    intro p
  3. L3
    intro q
  4. L4
    intro hp
  5. L5
    intro hq
02Establish hproductL6–15

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pow successor pair mul.

  1. L6
    have hproduct : q = p * 2
  2. L7
    specialize pow_successor_pair_mul 2
  3. L8
    specialize pow_successor_pair_mul e
  4. L9
    specialize pow_successor_pair_mul (S e)
  5. L10
    specialize pow_successor_pair_mul p
  6. L11
    specialize pow_successor_pair_mul q
  7. L12
    apply pow_successor_pair_mul
  8. L13
    refl
  9. L14
    exact hp
  10. L15
    exact hq
03Calculate and transport equalitiesL16–16

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

  1. L16
    trans p * 2
04Use earlier factsL17–17

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

  1. L17
    exact hproduct
05Calculate and transport equalitiesL18–18

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

  1. L18
    trans 2 * p
06Use earlier factsL19–20

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

  1. L19
    apply mul_comm
  2. L20
    apply two_mul_eq_add_self

Library-wide reading audit

Original exact command ledger · 20 lines
  1. 0001intro e
  2. 0002intro p
  3. 0003intro q
  4. 0004intro hp
  5. 0005intro hq
  6. 0006have hproduct : q = p * 2
  7. 0007specialize pow_successor_pair_mul 2
  8. 0008specialize pow_successor_pair_mul e
  9. 0009specialize pow_successor_pair_mul (S e)
  10. 0010specialize pow_successor_pair_mul p
  11. 0011specialize pow_successor_pair_mul q
  12. 0012apply pow_successor_pair_mul
  13. 0013refl
  14. 0014exact hp
  15. 0015exact hq
  16. 0016trans p * 2
  17. 0017exact hproduct
  18. 0018trans 2 * p
  19. 0019apply mul_comm
  20. 0020apply two_mul_eq_add_self