PQ0035

prime_field_polynomial_monic_normalization_inverse

The recorded scalar is an actual inverse of every decoding of the source leading coefficient.

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. ∀ a. FpMonicNormalization(p,k,ab,ac,bb,bc,L) → BetaAt(ab,ac,0,a) → FpInv(p,a,k)

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

Definition DAG

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

Actual proof prerequisites

beta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall p k ab ac bb bc L a. (((~((L) = 0)) /\ (((exists pfm_leading_normalization_inverse_source. ((((exists ff_h_pfp_normalization_inverse_sourcesource. ff_h_pfp_normalization_inverse_sourcesource + S (pfm_leading_normalization_inverse_source) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_inverse_sourcesource. ab = ff_q_pfp_normalization_inverse_sourcesource * S ((S (0)) * ac) + (pfm_leading_normalization_inverse_source))) /\ ((((~((pfm_leading_normalization_inverse_source) = 0)) /\ ((((exists pfa_gap_normalization_inverse_sourceinversemultiplicationleft. pfa_gap_normalization_inverse_sourceinversemultiplicationleft + S (pfm_leading_normalization_inverse_source) = (p)) /\ (((exists pfa_gap_normalization_inverse_sourceinversemultiplicationright. pfa_gap_normalization_inverse_sourceinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_inverse_sourceinversemultiplicationresultbound. pfa_gap_normalization_inverse_sourceinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_inverse_sourceinversemultiplicationresultcongruence pfa_offset_right_normalization_inverse_sourceinversemultiplicationresultcongruence. ((pfm_leading_normalization_inverse_source) * (k)) + (p) * pfa_offset_left_normalization_inverse_sourceinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_inverse_sourceinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_inverse_sourcescalescalar. pfa_gap_normalization_inverse_sourcescalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_inverse_sourcescale. (exists pfa_gap_normalization_inverse_sourcescaleindex. pfa_gap_normalization_inverse_sourcescaleindex + S (pfp_index_normalization_inverse_sourcescale) = (L)) -> exists pfp_source_normalization_inverse_sourcescale pfp_value_normalization_inverse_sourcescale. ((((exists ff_h_pfp_normalization_inverse_sourcescalesource. ff_h_pfp_normalization_inverse_sourcescalesource + S (pfp_source_normalization_inverse_sourcescale) = S ((S (pfp_index_normalization_inverse_sourcescale)) * ac)) /\ exists ff_q_pfp_normalization_inverse_sourcescalesource. ab = ff_q_pfp_normalization_inverse_sourcescalesource * S ((S (pfp_index_normalization_inverse_sourcescale)) * ac) + (pfp_source_normalization_inverse_sourcescale))) /\ (((((exists ff_h_pfp_normalization_inverse_sourcescaletarget. ff_h_pfp_normalization_inverse_sourcescaletarget + S (pfp_value_normalization_inverse_sourcescale) = S ((S (pfp_index_normalization_inverse_sourcescale)) * bc)) /\ exists ff_q_pfp_normalization_inverse_sourcescaletarget. bb = ff_q_pfp_normalization_inverse_sourcescaletarget * S ((S (pfp_index_normalization_inverse_sourcescale)) * bc) + (pfp_value_normalization_inverse_sourcescale))) /\ ((((exists pfa_gap_normalization_inverse_sourcescaleoperationleft. pfa_gap_normalization_inverse_sourcescaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_inverse_sourcescaleoperationright. pfa_gap_normalization_inverse_sourcescaleoperationright + S (pfp_source_normalization_inverse_sourcescale) = (p)) /\ ((((exists pfa_gap_normalization_inverse_sourcescaleoperationresultbound. pfa_gap_normalization_inverse_sourcescaleoperationresultbound + S (pfp_value_normalization_inverse_sourcescale) = (p)) /\ ((exists pfa_offset_left_normalization_inverse_sourcescaleoperationresultcongruence pfa_offset_right_normalization_inverse_sourcescaleoperationresultcongruence. ((k) * (pfp_source_normalization_inverse_sourcescale)) + (p) * pfa_offset_left_normalization_inverse_sourcescaleoperationresultcongruence = (pfp_value_normalization_inverse_sourcescale) + (p) * pfa_offset_right_normalization_inverse_sourcescaleoperationresultcongruence)))))))))))))))))))))) -> (((exists ff_h_pfp_normalization_inverse_entry. ff_h_pfp_normalization_inverse_entry + S (a) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_inverse_entry. ab = ff_q_pfp_normalization_inverse_entry * S ((S (0)) * ac) + (a))) -> (((~((a) = 0)) /\ ((((exists pfa_gap_normalization_inverse_resultmultiplicationleft. pfa_gap_normalization_inverse_resultmultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_normalization_inverse_resultmultiplicationright. pfa_gap_normalization_inverse_resultmultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_inverse_resultmultiplicationresultbound. pfa_gap_normalization_inverse_resultmultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_inverse_resultmultiplicationresultcongruence pfa_offset_right_normalization_inverse_resultmultiplicationresultcongruence. ((a) * (k)) + (p) * pfa_offset_left_normalization_inverse_resultmultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_inverse_resultmultiplicationresultcongruence))))))))))))

Complete tactic proof in conservative notation

All 27 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

27 script commands · 5 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 assumptionsL1–10

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 a
  9. L9
    intro h
  10. L10
    intro ha
02Separate the logical casesL11–14

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

  1. L11
    cases h
  2. L12
    cases h_right
  3. L13
    cases h_right_left
  4. L14
    cases h_right_left_witness
03Establish heL15–24

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta at unique.

  1. L15
    have he : a=x
  2. L16
    specialize beta_at_unique (ab)
  3. L17
    specialize beta_at_unique (ac)
  4. L18
    specialize beta_at_unique (0)
  5. L19
    specialize beta_at_unique (a)
  6. L20
    specialize beta_at_unique (x)
  7. L21
    apply beta_at_unique
  8. L22
    exact ha
  9. L23
    exact h_right_left_witness_left
  10. L24
    rewrite he
04Calculate and transport equalitiesL25–26

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L25
    rewrite he
  2. L26
    rewrite he
05Use earlier factsL27–27

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

  1. L27
    exact h_right_left_witness_right

Library-wide reading audit

Original defined command ledger · 27 lines
  1. 0001intro p
  2. 0002intro k
  3. 0003intro ab
  4. 0004intro ac
  5. 0005intro bb
  6. 0006intro bc
  7. 0007intro L
  8. 0008intro a
  9. 0009intro h
  10. 0010intro ha
  11. 0011cases h
  12. 0012cases h_right
  13. 0013cases h_right_left
  14. 0014cases h_right_left_witness
  15. 0015have he : a=x
  16. 0016specialize beta_at_unique (ab)
  17. 0017specialize beta_at_unique (ac)
  18. 0018specialize beta_at_unique (0)
  19. 0019specialize beta_at_unique (a)
  20. 0020specialize beta_at_unique (x)
  21. 0021apply beta_at_unique
  22. 0022exact ha
  23. 0023exact h_right_left_witness_left
  24. 0024rewrite he
  25. 0025rewrite he
  26. 0026rewrite he
  27. 0027exact h_right_left_witness_right