PP0009

prime_field_polynomial_repeat_coefficients

A genuinely repeated canonical value forms a bounded coefficient table, including length zero.

Alpha v34 checked-use · first admitted v31 · 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.

Length is representation length, not polynomial degree. Leading zeros and the empty zero polynomial are allowed; the canonical argument guard x<p also applies to the empty case. Evaluation is defined by actual field-operation steps, not an assumed residue invariant. Polynomial division, gcd, irreducibles and general prime-power extension fields remain open; this does not close G091.

Exact theorem in conservative defined notation

∀ p. ∀ b. ∀ c. ∀ a. ∀ l. Lt(a,p) → Repeat(b,c,a,l) → BetaPrefixInto(b,c,l,p)

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

Definition DAG

BetaPrefixInto(b,c,l,B) · 1Lt(a,b) · 1Repeat(b,c,a,l) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall p b c a l. (exists pfa_gap_constant_bound. pfa_gap_constant_bound + S (a) = (p)) -> (forall pfp_repeat_index_constant_repeat. (exists pfa_gap_constant_repeatindex. pfa_gap_constant_repeatindex + S (pfp_repeat_index_constant_repeat) = (l)) -> (((exists ff_h_pfp_constant_repeatentry. ff_h_pfp_constant_repeatentry + S (a) = S ((S (pfp_repeat_index_constant_repeat)) * c)) /\ exists ff_q_pfp_constant_repeatentry. b = ff_q_pfp_constant_repeatentry * S ((S (pfp_repeat_index_constant_repeat)) * c) + (a)))) -> (forall fom_index_pfp_constant_result. (exists fom_gap_pfp_constant_result_index_bound. fom_gap_pfp_constant_result_index_bound + S (fom_index_pfp_constant_result) = l) -> exists fom_value_pfp_constant_result. ((((exists fom_beta_height_pfp_constant_result_entry. fom_beta_height_pfp_constant_result_entry + S (fom_value_pfp_constant_result) = S ((S (fom_index_pfp_constant_result)) * c)) /\ exists fom_beta_quotient_pfp_constant_result_entry. b = fom_beta_quotient_pfp_constant_result_entry * S ((S (fom_index_pfp_constant_result)) * c) + (fom_value_pfp_constant_result))) /\ (exists fom_gap_pfp_constant_result_value_bound. fom_gap_pfp_constant_result_value_bound + S (fom_value_pfp_constant_result) = p)))

Complete tactic proof in conservative notation

All 15 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

15 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.

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 assumptionsL1–9

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

  1. L1
    intro p
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro a
  5. L5
    intro l
  6. L6
    intro ha
  7. L7
    intro hr
  8. L8
    intro i
  9. L9
    intro hi
02Construct an explicit witnessL10–10

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

  1. L10
    exists a
03Separate the logical casesL11–11

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

  1. L11
    split
04Use earlier factsL12–15

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

  1. L12
    specialize hr (i)
  2. L13
    apply hr
  3. L14
    exact hi
  4. L15
    exact ha

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro a
  5. 0005intro l
  6. 0006intro ha
  7. 0007intro hr
  8. 0008intro i
  9. 0009intro hi
  10. 0010exists a
  11. 0011split
  12. 0012specialize hr (i)
  13. 0013apply hr
  14. 0014exact hi
  15. 0015exact ha