PP0002

prime_field_polynomial_normalization_exists

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

Exact expanded first-order arithmetic 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)))))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 28 exact native proof lines.

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

beta_division_prefix_exists Alpha theorem; checked-use authorized PP0001 prime_field_polynomial_normalization_from_division

Direct dependents

PP000D prime_field_polynomial_add_exists PP0015 prime_field_polynomial_scale_exists PP0022 prime_field_polynomial_horner_exists PP0031 prime_field_polynomial_reduce_and_evaluate_exists

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

Named ingredients (1)
prime field polynomial normalization from division

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

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
  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 exact command ledger · 28 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro l
  5. 0005intro hp
  6. 0006have hd : exists qb qc rb rc. (forall fdp_index_pfp_exists_division. (exists gsp_lt_gap_pfp_exists_division_index_bound. gsp_lt_gap_pfp_exists_division_index_bound + S fdp_index_pfp_exists_division = l) -> exists fdp_value_pfp_exists_division fdp_quotient_pfp_exists_division fdp_remainder_pfp_exists_division. (((exists ff_h_fdp_pfp_exists_division_source. ff_h_fdp_pfp_exists_division_source + S (fdp_value_pfp_exists_division) = S ((S (fdp_index_pfp_exists_division)) * c)) /\ exists ff_q_fdp_pfp_exists_division_source. b = ff_q_fdp_pfp_exists_division_source * S ((S (fdp_index_pfp_exists_division)) * c) + (fdp_value_pfp_exists_division))) /\ ((((exists ff_h_fdp_pfp_exists_division_quotient_entry. ff_h_fdp_pfp_exists_division_quotient_entry + S (fdp_quotient_pfp_exists_division) = S ((S (fdp_index_pfp_exists_division)) * qc)) /\ exists ff_q_fdp_pfp_exists_division_quotient_entry. qb = ff_q_fdp_pfp_exists_division_quotient_entry * S ((S (fdp_index_pfp_exists_division)) * qc) + (fdp_quotient_pfp_exists_division))) /\ ((((exists ff_h_fdp_pfp_exists_division_remainder_entry. ff_h_fdp_pfp_exists_division_remainder_entry + S (fdp_remainder_pfp_exists_division) = S ((S (fdp_index_pfp_exists_division)) * rc)) /\ exists ff_q_fdp_pfp_exists_division_remainder_entry. rb = ff_q_fdp_pfp_exists_division_remainder_entry * S ((S (fdp_index_pfp_exists_division)) * rc) + (fdp_remainder_pfp_exists_division))) /\ (fdp_value_pfp_exists_division = p * fdp_quotient_pfp_exists_division + fdp_remainder_pfp_exists_division /\ (exists gsp_lt_gap_pfp_exists_division_remainder_bound. gsp_lt_gap_pfp_exists_division_remainder_bound + S fdp_remainder_pfp_exists_division = p)))))
  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