PQ0032

prime_field_polynomial_monic_represented_degree

A monic prefix of annotated length S d has represented degree d, also for d=0.

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. ∀ d. FpMonic(p,b,c,L) → L = S d → FpRepresentedDegree(p,b,c,L,d)

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

Definition DAG

FpRepresentedDegree(p,b,c,L,d) · 1FpMonic(p,b,c,L) · 1

Actual proof prerequisites

succ_ne_zero · checked external prerequisite
Original expanded first-order statement 
forall p b c L d. (((~((L) = 0)) /\ (((forall fom_index_pfp_monic_degree_sourcecoefficients. (exists fom_gap_pfp_monic_degree_sourcecoefficients_index_bound. fom_gap_pfp_monic_degree_sourcecoefficients_index_bound + S (fom_index_pfp_monic_degree_sourcecoefficients) = L) -> exists fom_value_pfp_monic_degree_sourcecoefficients. ((((exists fom_beta_height_pfp_monic_degree_sourcecoefficients_entry. fom_beta_height_pfp_monic_degree_sourcecoefficients_entry + S (fom_value_pfp_monic_degree_sourcecoefficients) = S ((S (fom_index_pfp_monic_degree_sourcecoefficients)) * c)) /\ exists fom_beta_quotient_pfp_monic_degree_sourcecoefficients_entry. b = fom_beta_quotient_pfp_monic_degree_sourcecoefficients_entry * S ((S (fom_index_pfp_monic_degree_sourcecoefficients)) * c) + (fom_value_pfp_monic_degree_sourcecoefficients))) /\ (exists fom_gap_pfp_monic_degree_sourcecoefficients_value_bound. fom_gap_pfp_monic_degree_sourcecoefficients_value_bound + S (fom_value_pfp_monic_degree_sourcecoefficients) = p))) /\ ((((exists ff_h_pfp_monic_degree_sourceleading. ff_h_pfp_monic_degree_sourceleading + S (1) = S ((S (0)) * c)) /\ exists ff_q_pfp_monic_degree_sourceleading. b = ff_q_pfp_monic_degree_sourceleading * S ((S (0)) * c) + (1)))))))) -> L=S d -> ((((L)=S (d)) /\ (((forall fom_index_pfp_monic_degree_resultcoefficients. (exists fom_gap_pfp_monic_degree_resultcoefficients_index_bound. fom_gap_pfp_monic_degree_resultcoefficients_index_bound + S (fom_index_pfp_monic_degree_resultcoefficients) = L) -> exists fom_value_pfp_monic_degree_resultcoefficients. ((((exists fom_beta_height_pfp_monic_degree_resultcoefficients_entry. fom_beta_height_pfp_monic_degree_resultcoefficients_entry + S (fom_value_pfp_monic_degree_resultcoefficients) = S ((S (fom_index_pfp_monic_degree_resultcoefficients)) * c)) /\ exists fom_beta_quotient_pfp_monic_degree_resultcoefficients_entry. b = fom_beta_quotient_pfp_monic_degree_resultcoefficients_entry * S ((S (fom_index_pfp_monic_degree_resultcoefficients)) * c) + (fom_value_pfp_monic_degree_resultcoefficients))) /\ (exists fom_gap_pfp_monic_degree_resultcoefficients_value_bound. fom_gap_pfp_monic_degree_resultcoefficients_value_bound + S (fom_value_pfp_monic_degree_resultcoefficients) = p))) /\ ((exists pfd_leading_monic_degree_result. ((((exists ff_h_pfp_monic_degree_resultentry. ff_h_pfp_monic_degree_resultentry + S (pfd_leading_monic_degree_result) = S ((S (0)) * c)) /\ exists ff_q_pfp_monic_degree_resultentry. b = ff_q_pfp_monic_degree_resultentry * S ((S (0)) * c) + (pfd_leading_monic_degree_result))) /\ ((~(pfd_leading_monic_degree_result=0))))))))))

Complete tactic proof in conservative notation

All 20 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

20 script commands · 10 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 d
  6. L6
    intro h
  7. L7
    intro hlen
02Separate the logical casesL8–10

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

  1. L8
    cases h
  2. L9
    cases h_right
  3. L10
    split
03Use earlier factsL11–11

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

  1. L11
    exact hlen
04Separate the logical casesL12–12

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

  1. L12
    split
05Use earlier factsL13–13

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

  1. L13
    exact h_right_left
06Construct an explicit witnessL14–14

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

  1. L14
    exists 1
07Separate the logical casesL15–15

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

  1. L15
    split
08Use earlier factsL16–16

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

  1. L16
    exact h_right_right
09Fix variables and assumptionsL17–17

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

  1. L17
    intro hz
10Use earlier factsL18–20

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

  1. L18
    specialize succ_ne_zero (0)
  2. L19
    apply succ_ne_zero
  3. L20
    exact hz

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro L
  5. 0005intro d
  6. 0006intro h
  7. 0007intro hlen
  8. 0008cases h
  9. 0009cases h_right
  10. 0010split
  11. 0011exact hlen
  12. 0012split
  13. 0013exact h_right_left
  14. 0014exists 1
  15. 0015split
  16. 0016exact h_right_right
  17. 0017intro hz
  18. 0018specialize succ_ne_zero (0)
  19. 0019apply succ_ne_zero
  20. 0020exact hz