PP0004

prime_field_polynomial_normalization_bounded

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

The normalized table really has every coefficient strictly below the modulus.

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 d e l. (forall pfp_index_bounded_source. (exists pfa_gap_bounded_sourceindex. pfa_gap_bounded_sourceindex + S (pfp_index_bounded_source) = (l)) -> exists pfp_source_bounded_source pfp_residue_bounded_source. ((((exists ff_h_pfp_bounded_sourcesource. ff_h_pfp_bounded_sourcesource + S (pfp_source_bounded_source) = S ((S (pfp_index_bounded_source)) * c)) /\ exists ff_q_pfp_bounded_sourcesource. b = ff_q_pfp_bounded_sourcesource * S ((S (pfp_index_bounded_source)) * c) + (pfp_source_bounded_source))) /\ (((((exists ff_h_pfp_bounded_sourcetarget. ff_h_pfp_bounded_sourcetarget + S (pfp_residue_bounded_source) = S ((S (pfp_index_bounded_source)) * e)) /\ exists ff_q_pfp_bounded_sourcetarget. d = ff_q_pfp_bounded_sourcetarget * S ((S (pfp_index_bounded_source)) * e) + (pfp_residue_bounded_source))) /\ ((((exists pfa_gap_bounded_sourceresiduebound. pfa_gap_bounded_sourceresiduebound + S (pfp_residue_bounded_source) = (p)) /\ ((exists pfa_offset_left_bounded_sourceresiduecongruence pfa_offset_right_bounded_sourceresiduecongruence. (pfp_source_bounded_source) + (p) * pfa_offset_left_bounded_sourceresiduecongruence = (pfp_residue_bounded_source) + (p) * pfa_offset_right_bounded_sourceresiduecongruence))))))))) -> (forall fom_index_pfp_bounded_result. (exists fom_gap_pfp_bounded_result_index_bound. fom_gap_pfp_bounded_result_index_bound + S (fom_index_pfp_bounded_result) = l) -> exists fom_value_pfp_bounded_result. ((((exists fom_beta_height_pfp_bounded_result_entry. fom_beta_height_pfp_bounded_result_entry + S (fom_value_pfp_bounded_result) = S ((S (fom_index_pfp_bounded_result)) * e)) /\ exists fom_beta_quotient_pfp_bounded_result_entry. d = fom_beta_quotient_pfp_bounded_result_entry * S ((S (fom_index_pfp_bounded_result)) * e) + (fom_value_pfp_bounded_result))) /\ (exists fom_gap_pfp_bounded_result_value_bound. fom_gap_pfp_bounded_result_value_bound + S (fom_value_pfp_bounded_result) = p)))

Constructive proof overview

Generated structural guide

The normalized table really has every coefficient strictly below the modulus.

The unchanged tactic script uses 0 declared prerequisites and contains 22 exact native proof lines.

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

Proof neighborhood

Direct dependencies

none

Direct dependents

PP0008 prime_field_polynomial_normalization_idempotent PP002F prime_field_polynomial_normalized_horner_iff 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

22 script commands · 6 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.

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 assumptionsL1–9

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 d
  5. L5
    intro e
  6. L6
    intro l
  7. L7
    intro h
  8. L8
    intro i
  9. L9
    intro hi
02Establish hpointL10–13

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

  1. L10
    have hpoint : ∃ a. ∃ r. BetaAt(b,c,i,a) ∧ (BetaAt(d,e,i,r) ∧ CanonicalModularResidue(p,a,r))Definitions: CanonicalModularResidueBetaAt
  2. L11
    specialize h (i)
  3. L12
    apply h
  4. L13
    exact hi
03Separate the logical casesL14–18

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

  1. L14
    cases hpoint
  2. L15
    cases hpoint_witness
  3. L16
    cases hpoint_witness_witness
  4. L17
    cases hpoint_witness_witness_right
  5. L18
    cases hpoint_witness_witness_right_right
04Construct an explicit witnessL19–19

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

  1. L19
    exists x1
05Separate the logical casesL20–20

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

  1. L20
    split
06Use earlier factsL21–22

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

  1. L21
    exact hpoint_witness_witness_right_left
  2. L22
    exact hpoint_witness_witness_right_right_left

Library-wide reading audit

Original exact command ledger · 22 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro e
  6. 0006intro l
  7. 0007intro h
  8. 0008intro i
  9. 0009intro hi
  10. 0010have hpoint : exists a r. ((((exists ff_h_pfp_bounded_source_entry. ff_h_pfp_bounded_source_entry + S (a) = S ((S (i)) * c)) /\ exists ff_q_pfp_bounded_source_entry. b = ff_q_pfp_bounded_source_entry * S ((S (i)) * c) + (a))) /\ (((((exists ff_h_pfp_bounded_target_entry. ff_h_pfp_bounded_target_entry + S (r) = S ((S (i)) * e)) /\ exists ff_q_pfp_bounded_target_entry. d = ff_q_pfp_bounded_target_entry * S ((S (i)) * e) + (r))) /\ ((((exists pfa_gap_bounded_residuebound. pfa_gap_bounded_residuebound + S (r) = (p)) /\ ((exists pfa_offset_left_bounded_residuecongruence pfa_offset_right_bounded_residuecongruence. (a) + (p) * pfa_offset_left_bounded_residuecongruence = (r) + (p) * pfa_offset_right_bounded_residuecongruence))))))))
  11. 0011specialize h (i)
  12. 0012apply h
  13. 0013exact hi
  14. 0014cases hpoint
  15. 0015cases hpoint_witness
  16. 0016cases hpoint_witness_witness
  17. 0017cases hpoint_witness_witness_right
  18. 0018cases hpoint_witness_witness_right_right
  19. 0019exists x1
  20. 0020split
  21. 0021exact hpoint_witness_witness_right_left
  22. 0022exact hpoint_witness_witness_right_right_left