PQ0041

prime_field_polynomial_monic_normalization_fixed

An already monic prefix normalizes by the actual scalar one using its original beta codes.

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. Prime(p) → FpMonic(p,b,c,L) → FpMonicNormalization(p,1,b,c,b,c,L)

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

Definition DAG

FpMonic(p,b,c,L) · 1FpMonicNormalization(p,k,ab,ac,bb,bc,L) · 1Prime(p) · 1

Actual proof prerequisites

succ_ne_zero · checked external prerequisiteprime_field_multiply_one_left · checked external prerequisiteprime_two_le · checked external prerequisiteprime_field_polynomial_scale_one · checked external prerequisite
Original expanded first-order statement 
forall p b c L. (~((p) = 1) /\ forall pfa_factor_left_normalization_fixed_prime pfa_factor_right_normalization_fixed_prime. (p) = pfa_factor_left_normalization_fixed_prime * pfa_factor_right_normalization_fixed_prime -> pfa_factor_left_normalization_fixed_prime = 1 \/ pfa_factor_right_normalization_fixed_prime = 1) -> (((~((L) = 0)) /\ (((forall fom_index_pfp_normalization_fixed_inputcoefficients. (exists fom_gap_pfp_normalization_fixed_inputcoefficients_index_bound. fom_gap_pfp_normalization_fixed_inputcoefficients_index_bound + S (fom_index_pfp_normalization_fixed_inputcoefficients) = L) -> exists fom_value_pfp_normalization_fixed_inputcoefficients. ((((exists fom_beta_height_pfp_normalization_fixed_inputcoefficients_entry. fom_beta_height_pfp_normalization_fixed_inputcoefficients_entry + S (fom_value_pfp_normalization_fixed_inputcoefficients) = S ((S (fom_index_pfp_normalization_fixed_inputcoefficients)) * c)) /\ exists fom_beta_quotient_pfp_normalization_fixed_inputcoefficients_entry. b = fom_beta_quotient_pfp_normalization_fixed_inputcoefficients_entry * S ((S (fom_index_pfp_normalization_fixed_inputcoefficients)) * c) + (fom_value_pfp_normalization_fixed_inputcoefficients))) /\ (exists fom_gap_pfp_normalization_fixed_inputcoefficients_value_bound. fom_gap_pfp_normalization_fixed_inputcoefficients_value_bound + S (fom_value_pfp_normalization_fixed_inputcoefficients) = p))) /\ ((((exists ff_h_pfp_normalization_fixed_inputleading. ff_h_pfp_normalization_fixed_inputleading + S (1) = S ((S (0)) * c)) /\ exists ff_q_pfp_normalization_fixed_inputleading. b = ff_q_pfp_normalization_fixed_inputleading * S ((S (0)) * c) + (1)))))))) -> (((~((L) = 0)) /\ (((exists pfm_leading_normalization_fixed_result. ((((exists ff_h_pfp_normalization_fixed_resultsource. ff_h_pfp_normalization_fixed_resultsource + S (pfm_leading_normalization_fixed_result) = S ((S (0)) * c)) /\ exists ff_q_pfp_normalization_fixed_resultsource. b = ff_q_pfp_normalization_fixed_resultsource * S ((S (0)) * c) + (pfm_leading_normalization_fixed_result))) /\ ((((~((pfm_leading_normalization_fixed_result) = 0)) /\ ((((exists pfa_gap_normalization_fixed_resultinversemultiplicationleft. pfa_gap_normalization_fixed_resultinversemultiplicationleft + S (pfm_leading_normalization_fixed_result) = (p)) /\ (((exists pfa_gap_normalization_fixed_resultinversemultiplicationright. pfa_gap_normalization_fixed_resultinversemultiplicationright + S (1) = (p)) /\ ((((exists pfa_gap_normalization_fixed_resultinversemultiplicationresultbound. pfa_gap_normalization_fixed_resultinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_fixed_resultinversemultiplicationresultcongruence pfa_offset_right_normalization_fixed_resultinversemultiplicationresultcongruence. ((pfm_leading_normalization_fixed_result) * (1)) + (p) * pfa_offset_left_normalization_fixed_resultinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_fixed_resultinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_fixed_resultscalescalar. pfa_gap_normalization_fixed_resultscalescalar + S (1) = (p)) /\ ((forall pfp_index_normalization_fixed_resultscale. (exists pfa_gap_normalization_fixed_resultscaleindex. pfa_gap_normalization_fixed_resultscaleindex + S (pfp_index_normalization_fixed_resultscale) = (L)) -> exists pfp_source_normalization_fixed_resultscale pfp_value_normalization_fixed_resultscale. ((((exists ff_h_pfp_normalization_fixed_resultscalesource. ff_h_pfp_normalization_fixed_resultscalesource + S (pfp_source_normalization_fixed_resultscale) = S ((S (pfp_index_normalization_fixed_resultscale)) * c)) /\ exists ff_q_pfp_normalization_fixed_resultscalesource. b = ff_q_pfp_normalization_fixed_resultscalesource * S ((S (pfp_index_normalization_fixed_resultscale)) * c) + (pfp_source_normalization_fixed_resultscale))) /\ (((((exists ff_h_pfp_normalization_fixed_resultscaletarget. ff_h_pfp_normalization_fixed_resultscaletarget + S (pfp_value_normalization_fixed_resultscale) = S ((S (pfp_index_normalization_fixed_resultscale)) * c)) /\ exists ff_q_pfp_normalization_fixed_resultscaletarget. b = ff_q_pfp_normalization_fixed_resultscaletarget * S ((S (pfp_index_normalization_fixed_resultscale)) * c) + (pfp_value_normalization_fixed_resultscale))) /\ ((((exists pfa_gap_normalization_fixed_resultscaleoperationleft. pfa_gap_normalization_fixed_resultscaleoperationleft + S (1) = (p)) /\ (((exists pfa_gap_normalization_fixed_resultscaleoperationright. pfa_gap_normalization_fixed_resultscaleoperationright + S (pfp_source_normalization_fixed_resultscale) = (p)) /\ ((((exists pfa_gap_normalization_fixed_resultscaleoperationresultbound. pfa_gap_normalization_fixed_resultscaleoperationresultbound + S (pfp_value_normalization_fixed_resultscale) = (p)) /\ ((exists pfa_offset_left_normalization_fixed_resultscaleoperationresultcongruence pfa_offset_right_normalization_fixed_resultscaleoperationresultcongruence. ((1) * (pfp_source_normalization_fixed_resultscale)) + (p) * pfa_offset_left_normalization_fixed_resultscaleoperationresultcongruence = (pfp_value_normalization_fixed_resultscale) + (p) * pfa_offset_right_normalization_fixed_resultscaleoperationresultcongruence))))))))))))))))))))))

Complete tactic proof in conservative notation

All 33 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

33 script commands · 11 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 assumptionsL1–6

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 hp
  6. L6
    intro h
02Separate the logical casesL7–9

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

  1. L7
    cases h
  2. L8
    cases h_right
  3. L9
    split
03Use earlier factsL10–10

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

  1. L10
    exact h_left
04Separate the logical casesL11–11

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

  1. L11
    split
05Construct an explicit witnessL12–12

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

  1. L12
    exists 1
06Separate the logical casesL13–13

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

  1. L13
    split
07Use earlier factsL14–14

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

  1. L14
    exact h_right_right
08Separate the logical casesL15–15

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

  1. L15
    split
09Fix variables and assumptionsL16–16

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

  1. L16
    intro hz
10Use earlier factsL17–26

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

  1. L17
    specialize succ_ne_zero (0)
  2. L18
    apply succ_ne_zero
  3. L19
    exact hz
  4. L20
    specialize prime_field_multiply_one_left (p)
  5. L21
    specialize prime_field_multiply_one_left (1)
  6. L22
    apply prime_field_multiply_one_left
  7. L23
    exact hp
  8. L24
    specialize prime_two_le (p)
  9. L25
    apply prime_two_le
  10. L26
    exact hp
11Use earlier factsL27–33

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

  1. L27
    specialize prime_field_polynomial_scale_one (p)
  2. L28
    specialize prime_field_polynomial_scale_one (b)
  3. L29
    specialize prime_field_polynomial_scale_one (c)
  4. L30
    specialize prime_field_polynomial_scale_one (L)
  5. L31
    apply prime_field_polynomial_scale_one
  6. L32
    exact hp
  7. L33
    exact h_right_left

Library-wide reading audit

Original defined command ledger · 33 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro L
  5. 0005intro hp
  6. 0006intro h
  7. 0007cases h
  8. 0008cases h_right
  9. 0009split
  10. 0010exact h_left
  11. 0011split
  12. 0012exists 1
  13. 0013split
  14. 0014exact h_right_right
  15. 0015split
  16. 0016intro hz
  17. 0017specialize succ_ne_zero (0)
  18. 0018apply succ_ne_zero
  19. 0019exact hz
  20. 0020specialize prime_field_multiply_one_left (p)
  21. 0021specialize prime_field_multiply_one_left (1)
  22. 0022apply prime_field_multiply_one_left
  23. 0023exact hp
  24. 0024specialize prime_two_le (p)
  25. 0025apply prime_two_le
  26. 0026exact hp
  27. 0027specialize prime_field_polynomial_scale_one (p)
  28. 0028specialize prime_field_polynomial_scale_one (b)
  29. 0029specialize prime_field_polynomial_scale_one (c)
  30. 0030specialize prime_field_polynomial_scale_one (L)
  31. 0031apply prime_field_polynomial_scale_one
  32. 0032exact hp
  33. 0033exact h_right_left