PQ0034

prime_field_polynomial_monic_constant

The entire represented degree-zero monic prefix is the constant one, not an empty prefix.

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. FpMonic(p,b,c,1) → Repeat(b,c,1,1)

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

Definition DAG

FpMonic(p,b,c,L) · 1Repeat(b,c,a,l) · 1

Actual proof prerequisites

le_zero · checked external prerequisitele_of_succ_le_succ · checked external prerequisite
Original expanded first-order statement 
forall p b c. (((~((1) = 0)) /\ (((forall fom_index_pfp_monic_constant_sourcecoefficients. (exists fom_gap_pfp_monic_constant_sourcecoefficients_index_bound. fom_gap_pfp_monic_constant_sourcecoefficients_index_bound + S (fom_index_pfp_monic_constant_sourcecoefficients) = 1) -> exists fom_value_pfp_monic_constant_sourcecoefficients. ((((exists fom_beta_height_pfp_monic_constant_sourcecoefficients_entry. fom_beta_height_pfp_monic_constant_sourcecoefficients_entry + S (fom_value_pfp_monic_constant_sourcecoefficients) = S ((S (fom_index_pfp_monic_constant_sourcecoefficients)) * c)) /\ exists fom_beta_quotient_pfp_monic_constant_sourcecoefficients_entry. b = fom_beta_quotient_pfp_monic_constant_sourcecoefficients_entry * S ((S (fom_index_pfp_monic_constant_sourcecoefficients)) * c) + (fom_value_pfp_monic_constant_sourcecoefficients))) /\ (exists fom_gap_pfp_monic_constant_sourcecoefficients_value_bound. fom_gap_pfp_monic_constant_sourcecoefficients_value_bound + S (fom_value_pfp_monic_constant_sourcecoefficients) = p))) /\ ((((exists ff_h_pfp_monic_constant_sourceleading. ff_h_pfp_monic_constant_sourceleading + S (1) = S ((S (0)) * c)) /\ exists ff_q_pfp_monic_constant_sourceleading. b = ff_q_pfp_monic_constant_sourceleading * S ((S (0)) * c) + (1)))))))) -> (forall pfp_repeat_index_monic_constant_value. (exists pfa_gap_monic_constant_valueindex. pfa_gap_monic_constant_valueindex + S (pfp_repeat_index_monic_constant_value) = (1)) -> (((exists ff_h_pfp_monic_constant_valueentry. ff_h_pfp_monic_constant_valueentry + S (1) = S ((S (pfp_repeat_index_monic_constant_value)) * c)) /\ exists ff_q_pfp_monic_constant_valueentry. b = ff_q_pfp_monic_constant_valueentry * S ((S (pfp_repeat_index_monic_constant_value)) * c) + (1))))

Complete tactic proof in conservative notation

All 18 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

18 script commands · 4 reading checkpoints · 1 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 assumptionsL1–4

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 h
02Separate the logical casesL5–6

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

  1. L5
    cases h
  2. L6
    cases h_right
03Fix variables and assumptionsL7–8

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

  1. L7
    intro i
  2. L8
    intro hi
04Establish hi0L9–18

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply le zero.

  1. L9
    have hi0 : i=0
  2. L10
    specialize le_zero (i)
  3. L11
    apply le_zero
  4. L12
    specialize le_of_succ_le_succ (i)
  5. L13
    specialize le_of_succ_le_succ (0)
  6. L14
    apply le_of_succ_le_succ
  7. L15
    exact hi
  8. L16
    rewrite hi0
  9. L17
    rewrite hi0
  10. L18
    exact h_right_right

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro h
  5. 0005cases h
  6. 0006cases h_right
  7. 0007intro i
  8. 0008intro hi
  9. 0009have hi0 : i=0
  10. 0010specialize le_zero (i)
  11. 0011apply le_zero
  12. 0012specialize le_of_succ_le_succ (i)
  13. 0013specialize le_of_succ_le_succ (0)
  14. 0014apply le_of_succ_le_succ
  15. 0015exact hi
  16. 0016rewrite hi0
  17. 0017rewrite hi0
  18. 0018exact h_right_right