BL0009

binary_power_two_successor_double

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

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

∀ e. ∀ p. ∀ q. PowTwo(e,p) → PowTwo(S e,q) → q = p + p

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

Definition DAG

PowTwo(e,p) · 2

Actual proof prerequisites

pow_successor_pair_mul · checked external prerequisitemul_comm · checked external prerequisitetwo_mul_eq_add_self · checked external prerequisite
Original expanded first-order 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

Complete unchanged native tactic proof

All 20 lines are the exact independently kernel-checked original script.

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.

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 assumptionsL1–5

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 defined 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