PQ0037

prime_field_polynomial_monic_normalization_entry

Each in-range output coefficient is the actual canonical product by the recorded inverse scalar.

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. ∀ k. ∀ ab. ∀ ac. ∀ bb. ∀ bc. ∀ L. ∀ i. ∀ a. ∀ r. FpMonicNormalization(p,k,ab,ac,bb,bc,L) → Lt(i,L) → BetaAt(ab,ac,i,a) → BetaAt(bb,bc,i,r) → FpMul(p,k,a,r)

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

Definition DAG

FpMul(p,a,b,c) · 1FpMonicNormalization(p,k,ab,ac,bb,bc,L) · 1Lt(a,b) · 1BetaAt(b,c,i,x) · 2

Actual proof prerequisites

prime_field_polynomial_scale_entry · checked external prerequisite
Original expanded first-order statement 
forall p k ab ac bb bc L i a r. (((~((L) = 0)) /\ (((exists pfm_leading_normalization_entry_source. ((((exists ff_h_pfp_normalization_entry_sourcesource. ff_h_pfp_normalization_entry_sourcesource + S (pfm_leading_normalization_entry_source) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_entry_sourcesource. ab = ff_q_pfp_normalization_entry_sourcesource * S ((S (0)) * ac) + (pfm_leading_normalization_entry_source))) /\ ((((~((pfm_leading_normalization_entry_source) = 0)) /\ ((((exists pfa_gap_normalization_entry_sourceinversemultiplicationleft. pfa_gap_normalization_entry_sourceinversemultiplicationleft + S (pfm_leading_normalization_entry_source) = (p)) /\ (((exists pfa_gap_normalization_entry_sourceinversemultiplicationright. pfa_gap_normalization_entry_sourceinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_entry_sourceinversemultiplicationresultbound. pfa_gap_normalization_entry_sourceinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_entry_sourceinversemultiplicationresultcongruence pfa_offset_right_normalization_entry_sourceinversemultiplicationresultcongruence. ((pfm_leading_normalization_entry_source) * (k)) + (p) * pfa_offset_left_normalization_entry_sourceinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_entry_sourceinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_entry_sourcescalescalar. pfa_gap_normalization_entry_sourcescalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_entry_sourcescale. (exists pfa_gap_normalization_entry_sourcescaleindex. pfa_gap_normalization_entry_sourcescaleindex + S (pfp_index_normalization_entry_sourcescale) = (L)) -> exists pfp_source_normalization_entry_sourcescale pfp_value_normalization_entry_sourcescale. ((((exists ff_h_pfp_normalization_entry_sourcescalesource. ff_h_pfp_normalization_entry_sourcescalesource + S (pfp_source_normalization_entry_sourcescale) = S ((S (pfp_index_normalization_entry_sourcescale)) * ac)) /\ exists ff_q_pfp_normalization_entry_sourcescalesource. ab = ff_q_pfp_normalization_entry_sourcescalesource * S ((S (pfp_index_normalization_entry_sourcescale)) * ac) + (pfp_source_normalization_entry_sourcescale))) /\ (((((exists ff_h_pfp_normalization_entry_sourcescaletarget. ff_h_pfp_normalization_entry_sourcescaletarget + S (pfp_value_normalization_entry_sourcescale) = S ((S (pfp_index_normalization_entry_sourcescale)) * bc)) /\ exists ff_q_pfp_normalization_entry_sourcescaletarget. bb = ff_q_pfp_normalization_entry_sourcescaletarget * S ((S (pfp_index_normalization_entry_sourcescale)) * bc) + (pfp_value_normalization_entry_sourcescale))) /\ ((((exists pfa_gap_normalization_entry_sourcescaleoperationleft. pfa_gap_normalization_entry_sourcescaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_entry_sourcescaleoperationright. pfa_gap_normalization_entry_sourcescaleoperationright + S (pfp_source_normalization_entry_sourcescale) = (p)) /\ ((((exists pfa_gap_normalization_entry_sourcescaleoperationresultbound. pfa_gap_normalization_entry_sourcescaleoperationresultbound + S (pfp_value_normalization_entry_sourcescale) = (p)) /\ ((exists pfa_offset_left_normalization_entry_sourcescaleoperationresultcongruence pfa_offset_right_normalization_entry_sourcescaleoperationresultcongruence. ((k) * (pfp_source_normalization_entry_sourcescale)) + (p) * pfa_offset_left_normalization_entry_sourcescaleoperationresultcongruence = (pfp_value_normalization_entry_sourcescale) + (p) * pfa_offset_right_normalization_entry_sourcescaleoperationresultcongruence)))))))))))))))))))))) -> (exists pfa_gap_normalization_entry_bound. pfa_gap_normalization_entry_bound + S (i) = (L)) -> (((exists ff_h_pfp_normalization_entry_input. ff_h_pfp_normalization_entry_input + S (a) = S ((S (i)) * ac)) /\ exists ff_q_pfp_normalization_entry_input. ab = ff_q_pfp_normalization_entry_input * S ((S (i)) * ac) + (a))) -> (((exists ff_h_pfp_normalization_entry_output. ff_h_pfp_normalization_entry_output + S (r) = S ((S (i)) * bc)) /\ exists ff_q_pfp_normalization_entry_output. bb = ff_q_pfp_normalization_entry_output * S ((S (i)) * bc) + (r))) -> (((exists pfa_gap_normalization_entry_multiplyleft. pfa_gap_normalization_entry_multiplyleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_entry_multiplyright. pfa_gap_normalization_entry_multiplyright + S (a) = (p)) /\ ((((exists pfa_gap_normalization_entry_multiplyresultbound. pfa_gap_normalization_entry_multiplyresultbound + S (r) = (p)) /\ ((exists pfa_offset_left_normalization_entry_multiplyresultcongruence pfa_offset_right_normalization_entry_multiplyresultcongruence. ((k) * (a)) + (p) * pfa_offset_left_normalization_entry_multiplyresultcongruence = (r) + (p) * pfa_offset_right_normalization_entry_multiplyresultcongruence)))))))))

Complete tactic proof in conservative notation

All 31 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

31 script commands · 5 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 assumptionsL1–10

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

  1. L1
    intro p
  2. L2
    intro k
  3. L3
    intro ab
  4. L4
    intro ac
  5. L5
    intro bb
  6. L6
    intro bc
  7. L7
    intro L
  8. L8
    intro i
  9. L9
    intro a
  10. L10
    intro r
02Fix variables and assumptionsL11–14

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

  1. L11
    intro h
  2. L12
    intro hi
  3. L13
    intro ha
  4. L14
    intro hr
03Separate the logical casesL15–16

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

  1. L15
    cases h
  2. L16
    cases h_right
04Use earlier factsL17–26

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

  1. L17
    specialize prime_field_polynomial_scale_entry (p)
  2. L18
    specialize prime_field_polynomial_scale_entry (k)
  3. L19
    specialize prime_field_polynomial_scale_entry (ab)
  4. L20
    specialize prime_field_polynomial_scale_entry (ac)
  5. L21
    specialize prime_field_polynomial_scale_entry (bb)
  6. L22
    specialize prime_field_polynomial_scale_entry (bc)
  7. L23
    specialize prime_field_polynomial_scale_entry (L)
  8. L24
    specialize prime_field_polynomial_scale_entry (i)
  9. L25
    specialize prime_field_polynomial_scale_entry (a)
  10. L26
    specialize prime_field_polynomial_scale_entry (r)
05Use earlier factsL27–31

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

  1. L27
    apply prime_field_polynomial_scale_entry
  2. L28
    exact h_right_right
  3. L29
    exact hi
  4. L30
    exact ha
  5. L31
    exact hr

Library-wide reading audit

Original defined command ledger · 31 lines
  1. 0001intro p
  2. 0002intro k
  3. 0003intro ab
  4. 0004intro ac
  5. 0005intro bb
  6. 0006intro bc
  7. 0007intro L
  8. 0008intro i
  9. 0009intro a
  10. 0010intro r
  11. 0011intro h
  12. 0012intro hi
  13. 0013intro ha
  14. 0014intro hr
  15. 0015cases h
  16. 0016cases h_right
  17. 0017specialize prime_field_polynomial_scale_entry (p)
  18. 0018specialize prime_field_polynomial_scale_entry (k)
  19. 0019specialize prime_field_polynomial_scale_entry (ab)
  20. 0020specialize prime_field_polynomial_scale_entry (ac)
  21. 0021specialize prime_field_polynomial_scale_entry (bb)
  22. 0022specialize prime_field_polynomial_scale_entry (bc)
  23. 0023specialize prime_field_polynomial_scale_entry (L)
  24. 0024specialize prime_field_polynomial_scale_entry (i)
  25. 0025specialize prime_field_polynomial_scale_entry (a)
  26. 0026specialize prime_field_polynomial_scale_entry (r)
  27. 0027apply prime_field_polynomial_scale_entry
  28. 0028exact h_right_right
  29. 0029exact hi
  30. 0030exact ha
  31. 0031exact hr