PP0002

prime_field_polynomial_normalization_exists

Every natural coefficient table has an actual canonical reduction at every nonzero modulus, including empty tables.

Alpha v34 checked-use · first admitted v31 · 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.

Length is representation length, not polynomial degree. Leading zeros and the empty zero polynomial are allowed; the canonical argument guard x<p also applies to the empty case. Evaluation is defined by actual field-operation steps, not an assumed residue invariant. Polynomial division, gcd, irreducibles and general prime-power extension fields remain open; this does not close G091.

Exact theorem in conservative defined notation

∀ p. ∀ b. ∀ c. ∀ l. ¬p = 0 → ∃ x. ∃ y. FpCoefficientReduction(p,b,c,x,y,l)

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

Definition DAG

FpCoefficientReduction(p,b,c,d,e,l) · 1DivisionPrefix(m,b,c,qb,qc,rb,rc,l) · 1

Actual proof prerequisites

beta_division_prefix_exists · checked external prerequisiteprime_field_polynomial_normalization_from_division
Original expanded first-order statement 
forall p b c l. ~(p=0) -> exists d e. (forall pfp_index_exists. (exists pfa_gap_existsindex. pfa_gap_existsindex + S (pfp_index_exists) = (l)) -> exists pfp_source_exists pfp_residue_exists. ((((exists ff_h_pfp_existssource. ff_h_pfp_existssource + S (pfp_source_exists) = S ((S (pfp_index_exists)) * c)) /\ exists ff_q_pfp_existssource. b = ff_q_pfp_existssource * S ((S (pfp_index_exists)) * c) + (pfp_source_exists))) /\ (((((exists ff_h_pfp_existstarget. ff_h_pfp_existstarget + S (pfp_residue_exists) = S ((S (pfp_index_exists)) * e)) /\ exists ff_q_pfp_existstarget. d = ff_q_pfp_existstarget * S ((S (pfp_index_exists)) * e) + (pfp_residue_exists))) /\ ((((exists pfa_gap_existsresiduebound. pfa_gap_existsresiduebound + S (pfp_residue_exists) = (p)) /\ ((exists pfa_offset_left_existsresiduecongruence pfa_offset_right_existsresiduecongruence. (pfp_source_exists) + (p) * pfa_offset_left_existsresiduecongruence = (pfp_residue_exists) + (p) * pfa_offset_right_existsresiduecongruence)))))))))

Complete tactic proof in conservative notation

All 28 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

28 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.

Named ingredients (1)
prime field polynomial normalization from division
01Fix variables and assumptionsL1–5

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
02Establish hdL6–12

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

  1. L6
    have hd : ∃ qb. ∃ qc. ∃ rb. ∃ rc. DivisionPrefix(p,b,c,qb,qc,rb,rc,l)Definitions: DivisionPrefix(p,b,c,qb,qc,rb,rc,l)Original native command in the exact edition
  2. L7
    specialize beta_division_prefix_exists (p)
  3. L8
    specialize beta_division_prefix_exists (b)
  4. L9
    specialize beta_division_prefix_exists (c)
  5. L10
    specialize beta_division_prefix_exists (l)
  6. L11
    apply beta_division_prefix_exists
  7. L12
    exact hp
03Separate the logical casesL13–16

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

  1. L13
    cases hd
  2. L14
    cases hd_witness
  3. L15
    cases hd_witness_witness
  4. L16
    cases hd_witness_witness_witness
04Construct an explicit witnessL17–18

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

  1. L17
    exists x2
  2. L18
    exists x3
05Use earlier factsL19–28

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

  1. L19
    specialize prime_field_polynomial_normalization_from_division (p)
  2. L20
    specialize prime_field_polynomial_normalization_from_division (b)
  3. L21
    specialize prime_field_polynomial_normalization_from_division (c)
  4. L22
    specialize prime_field_polynomial_normalization_from_division (x)
  5. L23
    specialize prime_field_polynomial_normalization_from_division (x1)
  6. L24
    specialize prime_field_polynomial_normalization_from_division (x2)
  7. L25
    specialize prime_field_polynomial_normalization_from_division (x3)
  8. L26
    specialize prime_field_polynomial_normalization_from_division (l)
  9. L27
    apply prime_field_polynomial_normalization_from_division
  10. L28
    exact hd_witness_witness_witness_witness

Library-wide reading audit

Original defined command ledger · 28 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro l
  5. 0005intro hp
  6. 0006have hd : ∃ qb. ∃ qc. ∃ rb. ∃ rc. DivisionPrefix(p,b,c,qb,qc,rb,rc,l)
  7. 0007specialize beta_division_prefix_exists (p)
  8. 0008specialize beta_division_prefix_exists (b)
  9. 0009specialize beta_division_prefix_exists (c)
  10. 0010specialize beta_division_prefix_exists (l)
  11. 0011apply beta_division_prefix_exists
  12. 0012exact hp
  13. 0013cases hd
  14. 0014cases hd_witness
  15. 0015cases hd_witness_witness
  16. 0016cases hd_witness_witness_witness
  17. 0017exists x2
  18. 0018exists x3
  19. 0019specialize prime_field_polynomial_normalization_from_division (p)
  20. 0020specialize prime_field_polynomial_normalization_from_division (b)
  21. 0021specialize prime_field_polynomial_normalization_from_division (c)
  22. 0022specialize prime_field_polynomial_normalization_from_division (x)
  23. 0023specialize prime_field_polynomial_normalization_from_division (x1)
  24. 0024specialize prime_field_polynomial_normalization_from_division (x2)
  25. 0025specialize prime_field_polynomial_normalization_from_division (x3)
  26. 0026specialize prime_field_polynomial_normalization_from_division (l)
  27. 0027apply prime_field_polynomial_normalization_from_division
  28. 0028exact hd_witness_witness_witness_witness