EL000A

euclidean_log_zero_below_power

The zero Euclidean divisor is strictly below every witnessed power of two.

Alpha v34 checked-use · first admitted v23 · 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.

The exact G101 milestone is fully proved, including the stronger checked bound k≤2·BitLen(b), a real beta-coded execution, and its actual terminal gcd. The independent T13 determinant/rank/integer-span substrate is now closed in the separate Alpha-v27 integer-linear-algebra branch. Full T13 proof · Alpha v27

Exact theorem in conservative defined notation

∀ n. ∀ p. ∀ b. PowTwo(n,p) → b = 0 → Lt(b,p)

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

Definition DAG

PowTwo(e,p) · 1Lt(a,b) · 1

Actual proof prerequisites

binary_power_two_nonzero · checked external prerequisiteone_le_of_ne_zero · checked external prerequisite
Original expanded first-order statement 
forall n p b. (exists pa_b_bl_elb_power pa_c_bl_elb_power. ((forall pa_i_bl_elb_power_repeat. (exists pa_lt_bl_elb_power_repeat_bound. pa_lt_bl_elb_power_repeat_bound + S pa_i_bl_elb_power_repeat = n) -> (((exists pa_h_bl_elb_power_repeat_decoded. pa_h_bl_elb_power_repeat_decoded + S (2) = S ((S (pa_i_bl_elb_power_repeat)) * pa_c_bl_elb_power)) /\ exists pa_q_bl_elb_power_repeat_decoded. pa_b_bl_elb_power = pa_q_bl_elb_power_repeat_decoded * S ((S (pa_i_bl_elb_power_repeat)) * pa_c_bl_elb_power) + (2)))) /\ (exists pa_u_bl_elb_power_product pa_v_bl_elb_power_product. ((((exists pa_h_bl_elb_power_product_start. pa_h_bl_elb_power_product_start + S (1) = S ((S (0)) * pa_v_bl_elb_power_product)) /\ exists pa_q_bl_elb_power_product_start. pa_u_bl_elb_power_product = pa_q_bl_elb_power_product_start * S ((S (0)) * pa_v_bl_elb_power_product) + (1))) /\ ((((exists pa_h_bl_elb_power_product_terminal. pa_h_bl_elb_power_product_terminal + S (p) = S ((S (n)) * pa_v_bl_elb_power_product)) /\ exists pa_q_bl_elb_power_product_terminal. pa_u_bl_elb_power_product = pa_q_bl_elb_power_product_terminal * S ((S (n)) * pa_v_bl_elb_power_product) + (p))) /\ forall pa_i_bl_elb_power_product. (exists pa_lt_bl_elb_power_product_bound. pa_lt_bl_elb_power_product_bound + S pa_i_bl_elb_power_product = n) -> exists pa_p_bl_elb_power_product pa_r_bl_elb_power_product pa_s_bl_elb_power_product. ((((exists pa_h_bl_elb_power_product_factor. pa_h_bl_elb_power_product_factor + S (pa_p_bl_elb_power_product) = S ((S (pa_i_bl_elb_power_product)) * pa_c_bl_elb_power)) /\ exists pa_q_bl_elb_power_product_factor. pa_b_bl_elb_power = pa_q_bl_elb_power_product_factor * S ((S (pa_i_bl_elb_power_product)) * pa_c_bl_elb_power) + (pa_p_bl_elb_power_product))) /\ ((((exists pa_h_bl_elb_power_product_partial. pa_h_bl_elb_power_product_partial + S (pa_r_bl_elb_power_product) = S ((S (pa_i_bl_elb_power_product)) * pa_v_bl_elb_power_product)) /\ exists pa_q_bl_elb_power_product_partial. pa_u_bl_elb_power_product = pa_q_bl_elb_power_product_partial * S ((S (pa_i_bl_elb_power_product)) * pa_v_bl_elb_power_product) + (pa_r_bl_elb_power_product))) /\ ((((exists pa_h_bl_elb_power_product_successor. pa_h_bl_elb_power_product_successor + S (pa_s_bl_elb_power_product) = S ((S (S pa_i_bl_elb_power_product)) * pa_v_bl_elb_power_product)) /\ exists pa_q_bl_elb_power_product_successor. pa_u_bl_elb_power_product = pa_q_bl_elb_power_product_successor * S ((S (S pa_i_bl_elb_power_product)) * pa_v_bl_elb_power_product) + (pa_s_bl_elb_power_product))) /\ pa_s_bl_elb_power_product = pa_r_bl_elb_power_product * pa_p_bl_elb_power_product)))))))) -> b = 0 -> (exists ff_lt_elb_below. ff_lt_elb_below + S b = 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 · 8 reading checkpoints · 2 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 n
  2. L2
    intro p
  3. L3
    intro b
  4. L4
    intro hpower
  5. L5
    intro hzero
02Use earlier factsL6–7

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

  1. L6
    specialize binary_power_two_nonzero n
  2. L7
    specialize binary_power_two_nonzero p
03Establish hnonzeroL8–13

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply binary power two nonzero.

  1. L8
    have hnonzero : ~(p = 0)
  2. L9
    intro hpzero
  3. L10
    apply binary_power_two_nonzero
  4. L11
    exact hpower
  5. L12
    exact hpzero
  6. L13
    specialize one_le_of_ne_zero p
04Establish hpositiveL14–16

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply one le of ne zero.

  1. L14
    have hpositive : exists gap. gap + 1 = p
  2. L15
    apply one_le_of_ne_zero
  3. L16
    exact hnonzero
05Separate the logical casesL17–17

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

  1. L17
    cases hpositive
06Construct an explicit witnessL18–18

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

  1. L18
    exists x
07Calculate and transport equalitiesL19–19

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

  1. L19
    rewrite hzero
08Use earlier factsL20–20

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

  1. L20
    exact hpositive_witness

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro n
  2. 0002intro p
  3. 0003intro b
  4. 0004intro hpower
  5. 0005intro hzero
  6. 0006specialize binary_power_two_nonzero n
  7. 0007specialize binary_power_two_nonzero p
  8. 0008have hnonzero : ~(p = 0)
  9. 0009intro hpzero
  10. 0010apply binary_power_two_nonzero
  11. 0011exact hpower
  12. 0012exact hpzero
  13. 0013specialize one_le_of_ne_zero p
  14. 0014have hpositive : exists gap. gap + 1 = p
  15. 0015apply one_le_of_ne_zero
  16. 0016exact hnonzero
  17. 0017cases hpositive
  18. 0018exists x
  19. 0019rewrite hzero
  20. 0020exact hpositive_witness