EL0007

lte_power_zero_exact

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Every natural base has an actual beta-coded zeroth power equal to one.

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.

Exact expanded first-order arithmetic statement

forall a. (exists pa_b_olte_zero pa_c_olte_zero. ((forall pa_i_olte_zero_repeat. (exists pa_lt_olte_zero_repeat_bound. pa_lt_olte_zero_repeat_bound + S pa_i_olte_zero_repeat = 0) -> (((exists pa_h_olte_zero_repeat_decoded. pa_h_olte_zero_repeat_decoded + S (a) = S ((S (pa_i_olte_zero_repeat)) * pa_c_olte_zero)) /\ exists pa_q_olte_zero_repeat_decoded. pa_b_olte_zero = pa_q_olte_zero_repeat_decoded * S ((S (pa_i_olte_zero_repeat)) * pa_c_olte_zero) + (a)))) /\ (exists pa_u_olte_zero_product pa_v_olte_zero_product. ((((exists pa_h_olte_zero_product_start. pa_h_olte_zero_product_start + S (1) = S ((S (0)) * pa_v_olte_zero_product)) /\ exists pa_q_olte_zero_product_start. pa_u_olte_zero_product = pa_q_olte_zero_product_start * S ((S (0)) * pa_v_olte_zero_product) + (1))) /\ ((((exists pa_h_olte_zero_product_terminal. pa_h_olte_zero_product_terminal + S (1) = S ((S (0)) * pa_v_olte_zero_product)) /\ exists pa_q_olte_zero_product_terminal. pa_u_olte_zero_product = pa_q_olte_zero_product_terminal * S ((S (0)) * pa_v_olte_zero_product) + (1))) /\ forall pa_i_olte_zero_product. (exists pa_lt_olte_zero_product_bound. pa_lt_olte_zero_product_bound + S pa_i_olte_zero_product = 0) -> exists pa_p_olte_zero_product pa_r_olte_zero_product pa_s_olte_zero_product. ((((exists pa_h_olte_zero_product_factor. pa_h_olte_zero_product_factor + S (pa_p_olte_zero_product) = S ((S (pa_i_olte_zero_product)) * pa_c_olte_zero)) /\ exists pa_q_olte_zero_product_factor. pa_b_olte_zero = pa_q_olte_zero_product_factor * S ((S (pa_i_olte_zero_product)) * pa_c_olte_zero) + (pa_p_olte_zero_product))) /\ ((((exists pa_h_olte_zero_product_partial. pa_h_olte_zero_product_partial + S (pa_r_olte_zero_product) = S ((S (pa_i_olte_zero_product)) * pa_v_olte_zero_product)) /\ exists pa_q_olte_zero_product_partial. pa_u_olte_zero_product = pa_q_olte_zero_product_partial * S ((S (pa_i_olte_zero_product)) * pa_v_olte_zero_product) + (pa_r_olte_zero_product))) /\ ((((exists pa_h_olte_zero_product_successor. pa_h_olte_zero_product_successor + S (pa_s_olte_zero_product) = S ((S (S pa_i_olte_zero_product)) * pa_v_olte_zero_product)) /\ exists pa_q_olte_zero_product_successor. pa_u_olte_zero_product = pa_q_olte_zero_product_successor * S ((S (S pa_i_olte_zero_product)) * pa_v_olte_zero_product) + (pa_s_olte_zero_product))) /\ pa_s_olte_zero_product = pa_r_olte_zero_product * pa_p_olte_zero_product))))))))

Constructive proof overview

Generated structural guide

Every natural base has an actual beta-coded zeroth power equal to one.

The unchanged tactic script uses 2 declared prerequisites and contains 16 exact native proof lines.

Alpha v34 checked-use · first admitted v29 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

pow_exists Stable theorem; checked-use authorized pow_zero Stable theorem; checked-use authorized

Direct dependents

EL0008 lte_power_one_exact EL000A lte_power_difference_second_order_exists EL0020 lte_prime_power_iteration

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

16 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.

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–1

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

  1. L1
    intro a
02Establish hexistsL2–5

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pow exists.

  1. L2
    have hexists : ∃ x. Pow(a,0,x)Definitions: Pow
  2. L3
    specialize pow_exists (a)
  3. L4
    specialize pow_exists (0)
  4. L5
    apply pow_exists
03Separate the logical casesL6–6

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

  1. L6
    cases hexists
04Establish hxL7–16

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

  1. L7
    have hx : x = 1
  2. L8
    specialize pow_zero (a)
  3. L9
    specialize pow_zero (0)
  4. L10
    specialize pow_zero (x)
  5. L11
    apply pow_zero
  6. L12
    refl
  7. L13
    exact hexists_witness
  8. L14
    rewrite hx at hexists_witness
  9. L15
    rewrite hx at hexists_witness
  10. L16
    exact hexists_witness

Library-wide reading audit

Original exact command ledger · 16 lines
  1. 0001intro a
  2. 0002have hexists : exists x. (exists pa_b_olte_zero_witness pa_c_olte_zero_witness. ((forall pa_i_olte_zero_witness_repeat. (exists pa_lt_olte_zero_witness_repeat_bound. pa_lt_olte_zero_witness_repeat_bound + S pa_i_olte_zero_witness_repeat = 0) -> (((exists pa_h_olte_zero_witness_repeat_decoded. pa_h_olte_zero_witness_repeat_decoded + S (a) = S ((S (pa_i_olte_zero_witness_repeat)) * pa_c_olte_zero_witness)) /\ exists pa_q_olte_zero_witness_repeat_decoded. pa_b_olte_zero_witness = pa_q_olte_zero_witness_repeat_decoded * S ((S (pa_i_olte_zero_witness_repeat)) * pa_c_olte_zero_witness) + (a)))) /\ (exists pa_u_olte_zero_witness_product pa_v_olte_zero_witness_product. ((((exists pa_h_olte_zero_witness_product_start. pa_h_olte_zero_witness_product_start + S (1) = S ((S (0)) * pa_v_olte_zero_witness_product)) /\ exists pa_q_olte_zero_witness_product_start. pa_u_olte_zero_witness_product = pa_q_olte_zero_witness_product_start * S ((S (0)) * pa_v_olte_zero_witness_product) + (1))) /\ ((((exists pa_h_olte_zero_witness_product_terminal. pa_h_olte_zero_witness_product_terminal + S (x) = S ((S (0)) * pa_v_olte_zero_witness_product)) /\ exists pa_q_olte_zero_witness_product_terminal. pa_u_olte_zero_witness_product = pa_q_olte_zero_witness_product_terminal * S ((S (0)) * pa_v_olte_zero_witness_product) + (x))) /\ forall pa_i_olte_zero_witness_product. (exists pa_lt_olte_zero_witness_product_bound. pa_lt_olte_zero_witness_product_bound + S pa_i_olte_zero_witness_product = 0) -> exists pa_p_olte_zero_witness_product pa_r_olte_zero_witness_product pa_s_olte_zero_witness_product. ((((exists pa_h_olte_zero_witness_product_factor. pa_h_olte_zero_witness_product_factor + S (pa_p_olte_zero_witness_product) = S ((S (pa_i_olte_zero_witness_product)) * pa_c_olte_zero_witness)) /\ exists pa_q_olte_zero_witness_product_factor. pa_b_olte_zero_witness = pa_q_olte_zero_witness_product_factor * S ((S (pa_i_olte_zero_witness_product)) * pa_c_olte_zero_witness) + (pa_p_olte_zero_witness_product))) /\ ((((exists pa_h_olte_zero_witness_product_partial. pa_h_olte_zero_witness_product_partial + S (pa_r_olte_zero_witness_product) = S ((S (pa_i_olte_zero_witness_product)) * pa_v_olte_zero_witness_product)) /\ exists pa_q_olte_zero_witness_product_partial. pa_u_olte_zero_witness_product = pa_q_olte_zero_witness_product_partial * S ((S (pa_i_olte_zero_witness_product)) * pa_v_olte_zero_witness_product) + (pa_r_olte_zero_witness_product))) /\ ((((exists pa_h_olte_zero_witness_product_successor. pa_h_olte_zero_witness_product_successor + S (pa_s_olte_zero_witness_product) = S ((S (S pa_i_olte_zero_witness_product)) * pa_v_olte_zero_witness_product)) /\ exists pa_q_olte_zero_witness_product_successor. pa_u_olte_zero_witness_product = pa_q_olte_zero_witness_product_successor * S ((S (S pa_i_olte_zero_witness_product)) * pa_v_olte_zero_witness_product) + (pa_s_olte_zero_witness_product))) /\ pa_s_olte_zero_witness_product = pa_r_olte_zero_witness_product * pa_p_olte_zero_witness_product))))))))
  3. 0003specialize pow_exists (a)
  4. 0004specialize pow_exists (0)
  5. 0005apply pow_exists
  6. 0006cases hexists
  7. 0007have hx : x = 1
  8. 0008specialize pow_zero (a)
  9. 0009specialize pow_zero (0)
  10. 0010specialize pow_zero (x)
  11. 0011apply pow_zero
  12. 0012refl
  13. 0013exact hexists_witness
  14. 0014rewrite hx at hexists_witness
  15. 0015rewrite hx at hexists_witness
  16. 0016exact hexists_witness