EL0009

lte_power_two_exact

The relational second power is the actual square, including a zero base.

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,2,a · a)

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

Definition DAG

Pow(a,e,z) · 1

Actual proof prerequisites

pow_successor_compose · checked external prerequisitelte_power_one_exact
Original expanded first-order statement 
forall a. (exists pa_b_olte_two pa_c_olte_two. ((forall pa_i_olte_two_repeat. (exists pa_lt_olte_two_repeat_bound. pa_lt_olte_two_repeat_bound + S pa_i_olte_two_repeat = 2) -> (((exists pa_h_olte_two_repeat_decoded. pa_h_olte_two_repeat_decoded + S (a) = S ((S (pa_i_olte_two_repeat)) * pa_c_olte_two)) /\ exists pa_q_olte_two_repeat_decoded. pa_b_olte_two = pa_q_olte_two_repeat_decoded * S ((S (pa_i_olte_two_repeat)) * pa_c_olte_two) + (a)))) /\ (exists pa_u_olte_two_product pa_v_olte_two_product. ((((exists pa_h_olte_two_product_start. pa_h_olte_two_product_start + S (1) = S ((S (0)) * pa_v_olte_two_product)) /\ exists pa_q_olte_two_product_start. pa_u_olte_two_product = pa_q_olte_two_product_start * S ((S (0)) * pa_v_olte_two_product) + (1))) /\ ((((exists pa_h_olte_two_product_terminal. pa_h_olte_two_product_terminal + S (a * a) = S ((S (2)) * pa_v_olte_two_product)) /\ exists pa_q_olte_two_product_terminal. pa_u_olte_two_product = pa_q_olte_two_product_terminal * S ((S (2)) * pa_v_olte_two_product) + (a * a))) /\ forall pa_i_olte_two_product. (exists pa_lt_olte_two_product_bound. pa_lt_olte_two_product_bound + S pa_i_olte_two_product = 2) -> exists pa_p_olte_two_product pa_r_olte_two_product pa_s_olte_two_product. ((((exists pa_h_olte_two_product_factor. pa_h_olte_two_product_factor + S (pa_p_olte_two_product) = S ((S (pa_i_olte_two_product)) * pa_c_olte_two)) /\ exists pa_q_olte_two_product_factor. pa_b_olte_two = pa_q_olte_two_product_factor * S ((S (pa_i_olte_two_product)) * pa_c_olte_two) + (pa_p_olte_two_product))) /\ ((((exists pa_h_olte_two_product_partial. pa_h_olte_two_product_partial + S (pa_r_olte_two_product) = S ((S (pa_i_olte_two_product)) * pa_v_olte_two_product)) /\ exists pa_q_olte_two_product_partial. pa_u_olte_two_product = pa_q_olte_two_product_partial * S ((S (pa_i_olte_two_product)) * pa_v_olte_two_product) + (pa_r_olte_two_product))) /\ ((((exists pa_h_olte_two_product_successor. pa_h_olte_two_product_successor + S (pa_s_olte_two_product) = S ((S (S pa_i_olte_two_product)) * pa_v_olte_two_product)) /\ exists pa_q_olte_two_product_successor. pa_u_olte_two_product = pa_q_olte_two_product_successor * S ((S (S pa_i_olte_two_product)) * pa_v_olte_two_product) + (pa_s_olte_two_product))) /\ pa_s_olte_two_product = pa_r_olte_two_product * pa_p_olte_two_product))))))))

Complete tactic proof in conservative notation

All 9 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

9 script commands · 3 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.

Named ingredients (1)
lte power one exact
01Fix variables and assumptionsL1–1

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

  1. L1
    intro a
02Use earlier factsL2–8

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

  1. L2
    specialize pow_successor_compose (a)
  2. L3
    specialize pow_successor_compose (1)
  3. L4
    specialize pow_successor_compose (a)
  4. L5
    specialize pow_successor_compose ((a * a))
  5. L6
    apply pow_successor_compose
  6. L7
    specialize lte_power_one_exact (a)
  7. L8
    apply lte_power_one_exact
03Calculate and transport equalitiesL9–9

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

  1. L9
    refl

Library-wide reading audit

Original defined command ledger · 9 lines
  1. 0001intro a
  2. 0002specialize pow_successor_compose (a)
  3. 0003specialize pow_successor_compose (1)
  4. 0004specialize pow_successor_compose (a)
  5. 0005specialize pow_successor_compose ((a * a))
  6. 0006apply pow_successor_compose
  7. 0007specialize lte_power_one_exact (a)
  8. 0008apply lte_power_one_exact
  9. 0009refl