EL0007

lte_power_zero_exact

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

Alpha v34 checked-use · first admitted v29 · 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.

All displayed hypotheses are required. Powers and positive differences are actual existential outputs. The proof constructs second-order correction identities and iterates the prime step; no binomial expansion or LTE oracle is assumed. The 2-adic variants remain separate open targets.

Exact theorem in conservative defined notation

∀ a. Pow(a,0,1)

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

Definition DAG

Pow(a,e,z) · 2

Actual proof prerequisites

pow_exists · checked external prerequisitepow_zero · checked external prerequisite
Original expanded first-order 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))))))))

Complete tactic proof in conservative notation

All 16 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

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.

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 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(a,0,x)Original native command in the exact edition
  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 defined command ledger · 16 lines
  1. 0001intro a
  2. 0002have hexists : ∃ x. Pow(a,0,x)
  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