PP0006

prime_field_polynomial_normalization_reflexive

Public research checkpoint, not admitted to Alpha or Stable. Alpha v30 remains 3222 checked-use theorems; Stable remains 432. The on-demand Alpha Lean service does not yet expose these checkpoint theorems; their independently checked literal bundles and unchanged sources are available below.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

A table already consisting of canonical coefficients normalizes to itself.

Exact expanded first-order arithmetic statement

forall p b c l. (forall fom_index_pfp_reflexive_source. (exists fom_gap_pfp_reflexive_source_index_bound. fom_gap_pfp_reflexive_source_index_bound + S (fom_index_pfp_reflexive_source) = l) -> exists fom_value_pfp_reflexive_source. ((((exists fom_beta_height_pfp_reflexive_source_entry. fom_beta_height_pfp_reflexive_source_entry + S (fom_value_pfp_reflexive_source) = S ((S (fom_index_pfp_reflexive_source)) * c)) /\ exists fom_beta_quotient_pfp_reflexive_source_entry. b = fom_beta_quotient_pfp_reflexive_source_entry * S ((S (fom_index_pfp_reflexive_source)) * c) + (fom_value_pfp_reflexive_source))) /\ (exists fom_gap_pfp_reflexive_source_value_bound. fom_gap_pfp_reflexive_source_value_bound + S (fom_value_pfp_reflexive_source) = p))) -> (forall pfp_index_reflexive_result. (exists pfa_gap_reflexive_resultindex. pfa_gap_reflexive_resultindex + S (pfp_index_reflexive_result) = (l)) -> exists pfp_source_reflexive_result pfp_residue_reflexive_result. ((((exists ff_h_pfp_reflexive_resultsource. ff_h_pfp_reflexive_resultsource + S (pfp_source_reflexive_result) = S ((S (pfp_index_reflexive_result)) * c)) /\ exists ff_q_pfp_reflexive_resultsource. b = ff_q_pfp_reflexive_resultsource * S ((S (pfp_index_reflexive_result)) * c) + (pfp_source_reflexive_result))) /\ (((((exists ff_h_pfp_reflexive_resulttarget. ff_h_pfp_reflexive_resulttarget + S (pfp_residue_reflexive_result) = S ((S (pfp_index_reflexive_result)) * c)) /\ exists ff_q_pfp_reflexive_resulttarget. b = ff_q_pfp_reflexive_resulttarget * S ((S (pfp_index_reflexive_result)) * c) + (pfp_residue_reflexive_result))) /\ ((((exists pfa_gap_reflexive_resultresiduebound. pfa_gap_reflexive_resultresiduebound + S (pfp_residue_reflexive_result) = (p)) /\ ((exists pfa_offset_left_reflexive_resultresiduecongruence pfa_offset_right_reflexive_resultresiduecongruence. (pfp_source_reflexive_result) + (p) * pfa_offset_left_reflexive_resultresiduecongruence = (pfp_residue_reflexive_result) + (p) * pfa_offset_right_reflexive_resultresiduecongruence)))))))))

Constructive proof overview

Generated structural guide

A table already consisting of canonical coefficients normalizes to itself.

The unchanged tactic script uses 1 declared prerequisite and contains 23 exact native proof lines.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

Proof neighborhood

Direct dependencies

prime_field_residue_reflexive external actual_inherited_body_freshly_checked_in_complete_bundle; no checked-use authority

Direct dependents

PP0008 prime_field_polynomial_normalization_idempotent PP0028 prime_field_polynomial_horner_residue

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. The literal dependency-closed bundle is checked by original HA and the independently compiled Lean verifier. Public delivery grants no Alpha checked-use authority or Stable membership.

Read the argument

Proof checkpoints

23 script commands · 8 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.

01Fix variables and assumptionsL1–7

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 h
  6. L6
    intro i
  7. L7
    intro hi
02Establish haL8–11

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

  1. L8
    have ha : exists a. ((((exists ff_h_pfp_reflexive_entry. ff_h_pfp_reflexive_entry + S (a) = S ((S (i)) * c)) /\ exists ff_q_pfp_reflexive_entry. b = ff_q_pfp_reflexive_entry * S ((S (i)) * c) + (a))) /\ ((exists pfa_gap_reflexive_bound. pfa_gap_reflexive_bound + S (a) = (p))))
  2. L9
    specialize h (i)
  3. L10
    apply h
  4. L11
    exact hi
03Separate the logical casesL12–13

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

  1. L12
    cases ha
  2. L13
    cases ha_witness
04Construct an explicit witnessL14–15

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

  1. L14
    exists x
  2. L15
    exists x
05Separate the logical casesL16–16

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

  1. L16
    split
06Use earlier factsL17–17

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

  1. L17
    exact ha_witness_left
07Separate the logical casesL18–18

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

  1. L18
    split
08Use earlier factsL19–23

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

  1. L19
    exact ha_witness_left
  2. L20
    specialize prime_field_residue_reflexive (p)
  3. L21
    specialize prime_field_residue_reflexive (x)
  4. L22
    apply prime_field_residue_reflexive
  5. L23
    exact ha_witness_right

Library-wide reading audit

Original exact command ledger · 23 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro l
  5. 0005intro h
  6. 0006intro i
  7. 0007intro hi
  8. 0008have ha : exists a. ((((exists ff_h_pfp_reflexive_entry. ff_h_pfp_reflexive_entry + S (a) = S ((S (i)) * c)) /\ exists ff_q_pfp_reflexive_entry. b = ff_q_pfp_reflexive_entry * S ((S (i)) * c) + (a))) /\ ((exists pfa_gap_reflexive_bound. pfa_gap_reflexive_bound + S (a) = (p))))
  9. 0009specialize h (i)
  10. 0010apply h
  11. 0011exact hi
  12. 0012cases ha
  13. 0013cases ha_witness
  14. 0014exists x
  15. 0015exists x
  16. 0016split
  17. 0017exact ha_witness_left
  18. 0018split
  19. 0019exact ha_witness_left
  20. 0020specialize prime_field_residue_reflexive (p)
  21. 0021specialize prime_field_residue_reflexive (x)
  22. 0022apply prime_field_residue_reflexive
  23. 0023exact ha_witness_right