PX0009

prime_field_polynomial_power_index_bound

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

A coefficient indexed by its power is genuinely inside the highest-degree-first prefix.

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 expanded first-order arithmetic statement

forall i k L. i+S k=L -> (exists pfa_gap_power_index_bound. pfa_gap_power_index_bound + S (i) = (L))

Constructive proof overview

Generated structural guide

A coefficient indexed by its power is genuinely inside the highest-degree-first prefix.

The unchanged tactic script uses 1 declared prerequisite and contains 8 exact native proof lines.

Alpha v34 checked-use · first admitted v33 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

add_comm Alpha theorem; checked-use authorized

Direct dependents

PX000E prime_field_polynomial_power_coefficient_transport PX001A prime_field_polynomial_left_pad_power_coefficient PX0021 prime_field_polynomial_zero_power_coefficient

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

8 script commands · 4 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.

01Fix variables and assumptionsL1–4

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

  1. L1
    intro i
  2. L2
    intro k
  3. L3
    intro L
  4. L4
    intro h
02Construct an explicit witnessL5–5

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

  1. L5
    exists k
03Calculate and transport equalitiesL6–7

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

  1. L6
    trans i+S k
  2. L7
    simp [add_comm]
04Use earlier factsL8–8

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

  1. L8
    exact h

Library-wide reading audit

Original exact command ledger · 8 lines
  1. 0001intro i
  2. 0002intro k
  3. 0003intro L
  4. 0004intro h
  5. 0005exists k
  6. 0006trans i+S k
  7. 0007simp [add_comm]
  8. 0008exact h