BL0008

binary_power_two_nonzero

Every beta-coded power of two is constructively nonzero.

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) → ¬p = 0

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

Definition DAG

PowTwo(e,p) · 1

Actual proof prerequisites

pow_nonzero_of_one_le · checked external prerequisite
Original expanded first-order statement 
forall e 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)))))))) -> ~(p = 0)

Complete unchanged native tactic proof

All 12 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

12 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.

01Fix variables and assumptionsL1–4

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

  1. L1
    intro e
  2. L2
    intro p
  3. L3
    intro hp
  4. L4
    intro hzero
02Use earlier factsL5–8

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

  1. L5
    specialize pow_nonzero_of_one_le 2
  2. L6
    specialize pow_nonzero_of_one_le e
  3. L7
    specialize pow_nonzero_of_one_le p
  4. L8
    apply pow_nonzero_of_one_le
03Construct an explicit witnessL9–9

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

  1. L9
    exists 1
04Calculate and transport equalitiesL10–10

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

  1. L10
    simp
05Use earlier factsL11–12

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

  1. L11
    exact hp
  2. L12
    exact hzero

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro e
  2. 0002intro p
  3. 0003intro hp
  4. 0004intro hzero
  5. 0005specialize pow_nonzero_of_one_le 2
  6. 0006specialize pow_nonzero_of_one_le e
  7. 0007specialize pow_nonzero_of_one_le p
  8. 0008apply pow_nonzero_of_one_le
  9. 0009exists 1
  10. 0010simp
  11. 0011exact hp
  12. 0012exact hzero