EL000B

euclidean_log_power_zero_divisor

A natural strictly below the witnessed zeroth power of two must be zero.

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

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

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_zero_value · checked external prerequisitele_of_succ_le_succ · checked external prerequisitele_zero · checked external prerequisite
Original expanded first-order statement 
forall p b. (exists pa_b_bl_elb_zero_power pa_c_bl_elb_zero_power. ((forall pa_i_bl_elb_zero_power_repeat. (exists pa_lt_bl_elb_zero_power_repeat_bound. pa_lt_bl_elb_zero_power_repeat_bound + S pa_i_bl_elb_zero_power_repeat = 0) -> (((exists pa_h_bl_elb_zero_power_repeat_decoded. pa_h_bl_elb_zero_power_repeat_decoded + S (2) = S ((S (pa_i_bl_elb_zero_power_repeat)) * pa_c_bl_elb_zero_power)) /\ exists pa_q_bl_elb_zero_power_repeat_decoded. pa_b_bl_elb_zero_power = pa_q_bl_elb_zero_power_repeat_decoded * S ((S (pa_i_bl_elb_zero_power_repeat)) * pa_c_bl_elb_zero_power) + (2)))) /\ (exists pa_u_bl_elb_zero_power_product pa_v_bl_elb_zero_power_product. ((((exists pa_h_bl_elb_zero_power_product_start. pa_h_bl_elb_zero_power_product_start + S (1) = S ((S (0)) * pa_v_bl_elb_zero_power_product)) /\ exists pa_q_bl_elb_zero_power_product_start. pa_u_bl_elb_zero_power_product = pa_q_bl_elb_zero_power_product_start * S ((S (0)) * pa_v_bl_elb_zero_power_product) + (1))) /\ ((((exists pa_h_bl_elb_zero_power_product_terminal. pa_h_bl_elb_zero_power_product_terminal + S (p) = S ((S (0)) * pa_v_bl_elb_zero_power_product)) /\ exists pa_q_bl_elb_zero_power_product_terminal. pa_u_bl_elb_zero_power_product = pa_q_bl_elb_zero_power_product_terminal * S ((S (0)) * pa_v_bl_elb_zero_power_product) + (p))) /\ forall pa_i_bl_elb_zero_power_product. (exists pa_lt_bl_elb_zero_power_product_bound. pa_lt_bl_elb_zero_power_product_bound + S pa_i_bl_elb_zero_power_product = 0) -> exists pa_p_bl_elb_zero_power_product pa_r_bl_elb_zero_power_product pa_s_bl_elb_zero_power_product. ((((exists pa_h_bl_elb_zero_power_product_factor. pa_h_bl_elb_zero_power_product_factor + S (pa_p_bl_elb_zero_power_product) = S ((S (pa_i_bl_elb_zero_power_product)) * pa_c_bl_elb_zero_power)) /\ exists pa_q_bl_elb_zero_power_product_factor. pa_b_bl_elb_zero_power = pa_q_bl_elb_zero_power_product_factor * S ((S (pa_i_bl_elb_zero_power_product)) * pa_c_bl_elb_zero_power) + (pa_p_bl_elb_zero_power_product))) /\ ((((exists pa_h_bl_elb_zero_power_product_partial. pa_h_bl_elb_zero_power_product_partial + S (pa_r_bl_elb_zero_power_product) = S ((S (pa_i_bl_elb_zero_power_product)) * pa_v_bl_elb_zero_power_product)) /\ exists pa_q_bl_elb_zero_power_product_partial. pa_u_bl_elb_zero_power_product = pa_q_bl_elb_zero_power_product_partial * S ((S (pa_i_bl_elb_zero_power_product)) * pa_v_bl_elb_zero_power_product) + (pa_r_bl_elb_zero_power_product))) /\ ((((exists pa_h_bl_elb_zero_power_product_successor. pa_h_bl_elb_zero_power_product_successor + S (pa_s_bl_elb_zero_power_product) = S ((S (S pa_i_bl_elb_zero_power_product)) * pa_v_bl_elb_zero_power_product)) /\ exists pa_q_bl_elb_zero_power_product_successor. pa_u_bl_elb_zero_power_product = pa_q_bl_elb_zero_power_product_successor * S ((S (S pa_i_bl_elb_zero_power_product)) * pa_v_bl_elb_zero_power_product) + (pa_s_bl_elb_zero_power_product))) /\ pa_s_bl_elb_zero_power_product = pa_r_bl_elb_zero_power_product * pa_p_bl_elb_zero_power_product)))))))) -> (exists ff_lt_elb_below. ff_lt_elb_below + S b = p) -> b = 0

Complete unchanged native tactic proof

All 17 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

17 script commands · 4 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 assumptionsL1–4

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

  1. L1
    intro p
  2. L2
    intro b
  3. L3
    intro hpower
  4. L4
    intro hbelow
02Use earlier factsL5–5

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

  1. L5
    specialize binary_power_two_zero_value p
03Establish hvalueL6–11

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

  1. L6
    have hvalue : p = 1
  2. L7
    apply binary_power_two_zero_value
  3. L8
    exact hpower
  4. L9
    rewrite hvalue at hbelow
  5. L10
    specialize le_of_succ_le_succ b
  6. L11
    specialize le_of_succ_le_succ 0
04Establish hzeroL12–17

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

  1. L12
    have hzero : exists gap. gap + b = 0
  2. L13
    apply le_of_succ_le_succ
  3. L14
    exact hbelow
  4. L15
    specialize le_zero b
  5. L16
    apply le_zero
  6. L17
    exact hzero

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro hpower
  4. 0004intro hbelow
  5. 0005specialize binary_power_two_zero_value p
  6. 0006have hvalue : p = 1
  7. 0007apply binary_power_two_zero_value
  8. 0008exact hpower
  9. 0009rewrite hvalue at hbelow
  10. 0010specialize le_of_succ_le_succ b
  11. 0011specialize le_of_succ_le_succ 0
  12. 0012have hzero : exists gap. gap + b = 0
  13. 0013apply le_of_succ_le_succ
  14. 0014exact hbelow
  15. 0015specialize le_zero b
  16. 0016apply le_zero
  17. 0017exact hzero