PQ0031

prime_field_polynomial_monic_leading_value

Every actual decoding of a monic leading coefficient is canonical one.

Alpha v34 checked-use · first admitted v32 · 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.

All coefficients retain the established highest-degree-first order. Trimming handles empty and all-zero prefixes; monic normalization requires a nonzero leading coefficient. Synthetic division has a nonempty input of length S n and a quotient of length n, unique in decoded values. Its coefficient recurrence, actual evaluation remainder and positive-degree drop are checked. General polynomial Euclidean division, gcd/Bezout, an arbitrary-convolution factor theorem, irreducible-polynomial existence and the full G091 prime-power-field endpoint remain open. These exact theorems are first admitted to Alpha v32; Stable remains unchanged.

Exact theorem in conservative defined notation

∀ p. ∀ b. ∀ c. ∀ L. ∀ a. FpMonic(p,b,c,L) → BetaAt(b,c,0,a) → a = 1

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

Definition DAG

FpMonic(p,b,c,L) · 1BetaAt(b,c,i,x) · 1

Actual proof prerequisites

beta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall p b c L a. (((~((L) = 0)) /\ (((forall fom_index_pfp_monic_value_sourcecoefficients. (exists fom_gap_pfp_monic_value_sourcecoefficients_index_bound. fom_gap_pfp_monic_value_sourcecoefficients_index_bound + S (fom_index_pfp_monic_value_sourcecoefficients) = L) -> exists fom_value_pfp_monic_value_sourcecoefficients. ((((exists fom_beta_height_pfp_monic_value_sourcecoefficients_entry. fom_beta_height_pfp_monic_value_sourcecoefficients_entry + S (fom_value_pfp_monic_value_sourcecoefficients) = S ((S (fom_index_pfp_monic_value_sourcecoefficients)) * c)) /\ exists fom_beta_quotient_pfp_monic_value_sourcecoefficients_entry. b = fom_beta_quotient_pfp_monic_value_sourcecoefficients_entry * S ((S (fom_index_pfp_monic_value_sourcecoefficients)) * c) + (fom_value_pfp_monic_value_sourcecoefficients))) /\ (exists fom_gap_pfp_monic_value_sourcecoefficients_value_bound. fom_gap_pfp_monic_value_sourcecoefficients_value_bound + S (fom_value_pfp_monic_value_sourcecoefficients) = p))) /\ ((((exists ff_h_pfp_monic_value_sourceleading. ff_h_pfp_monic_value_sourceleading + S (1) = S ((S (0)) * c)) /\ exists ff_q_pfp_monic_value_sourceleading. b = ff_q_pfp_monic_value_sourceleading * S ((S (0)) * c) + (1)))))))) -> (((exists ff_h_pfp_monic_value_entry. ff_h_pfp_monic_value_entry + S (a) = S ((S (0)) * c)) /\ exists ff_q_pfp_monic_value_entry. b = ff_q_pfp_monic_value_entry * S ((S (0)) * c) + (a))) -> a=1

Complete tactic proof in conservative notation

All 17 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

17 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.

01Fix variables and assumptionsL1–7

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 L
  5. L5
    intro a
  6. L6
    intro h
  7. L7
    intro ha
02Separate the logical casesL8–9

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

  1. L8
    cases h
  2. L9
    cases h_right
03Use earlier factsL10–17

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

  1. L10
    specialize beta_at_unique (b)
  2. L11
    specialize beta_at_unique (c)
  3. L12
    specialize beta_at_unique (0)
  4. L13
    specialize beta_at_unique (a)
  5. L14
    specialize beta_at_unique (1)
  6. L15
    apply beta_at_unique
  7. L16
    exact ha
  8. L17
    exact h_right_right

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro L
  5. 0005intro a
  6. 0006intro h
  7. 0007intro ha
  8. 0008cases h
  9. 0009cases h_right
  10. 0010specialize beta_at_unique (b)
  11. 0011specialize beta_at_unique (c)
  12. 0012specialize beta_at_unique (0)
  13. 0013specialize beta_at_unique (a)
  14. 0014specialize beta_at_unique (1)
  15. 0015apply beta_at_unique
  16. 0016exact ha
  17. 0017exact h_right_right