BL0005

binary_power_two_exists

Every natural exponent has a witnessed beta-coded power of two.

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. PowTwo(e,p)

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

Definition DAG

PowTwo(e,p) · 1

Actual proof prerequisites

pow_exists · checked external prerequisite
Original expanded first-order statement 
forall e. exists p. (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))))))))

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.

01Fix variables and assumptionsL1–1

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

  1. L1
    intro e
02Use earlier factsL2–4

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

  1. L2
    specialize pow_exists 2
  2. L3
    specialize pow_exists e
  3. L4
    exact pow_exists

Library-wide reading audit

Original defined command ledger · 4 lines
  1. 0001intro e
  2. 0002specialize pow_exists 2
  3. 0003specialize pow_exists e
  4. 0004exact pow_exists