PP0001

prime_field_polynomial_normalization_from_division

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

Actual finite quotient/remainder witnesses give genuine coefficientwise canonical normalization.

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 qb qc rb rc l. (forall fdp_index_pfp_division. (exists gsp_lt_gap_pfp_division_index_bound. gsp_lt_gap_pfp_division_index_bound + S fdp_index_pfp_division = l) -> exists fdp_value_pfp_division fdp_quotient_pfp_division fdp_remainder_pfp_division. (((exists ff_h_fdp_pfp_division_source. ff_h_fdp_pfp_division_source + S (fdp_value_pfp_division) = S ((S (fdp_index_pfp_division)) * c)) /\ exists ff_q_fdp_pfp_division_source. b = ff_q_fdp_pfp_division_source * S ((S (fdp_index_pfp_division)) * c) + (fdp_value_pfp_division))) /\ ((((exists ff_h_fdp_pfp_division_quotient_entry. ff_h_fdp_pfp_division_quotient_entry + S (fdp_quotient_pfp_division) = S ((S (fdp_index_pfp_division)) * qc)) /\ exists ff_q_fdp_pfp_division_quotient_entry. qb = ff_q_fdp_pfp_division_quotient_entry * S ((S (fdp_index_pfp_division)) * qc) + (fdp_quotient_pfp_division))) /\ ((((exists ff_h_fdp_pfp_division_remainder_entry. ff_h_fdp_pfp_division_remainder_entry + S (fdp_remainder_pfp_division) = S ((S (fdp_index_pfp_division)) * rc)) /\ exists ff_q_fdp_pfp_division_remainder_entry. rb = ff_q_fdp_pfp_division_remainder_entry * S ((S (fdp_index_pfp_division)) * rc) + (fdp_remainder_pfp_division))) /\ (fdp_value_pfp_division = p * fdp_quotient_pfp_division + fdp_remainder_pfp_division /\ (exists gsp_lt_gap_pfp_division_remainder_bound. gsp_lt_gap_pfp_division_remainder_bound + S fdp_remainder_pfp_division = p))))) -> (forall pfp_index_division_result. (exists pfa_gap_division_resultindex. pfa_gap_division_resultindex + S (pfp_index_division_result) = (l)) -> exists pfp_source_division_result pfp_residue_division_result. ((((exists ff_h_pfp_division_resultsource. ff_h_pfp_division_resultsource + S (pfp_source_division_result) = S ((S (pfp_index_division_result)) * c)) /\ exists ff_q_pfp_division_resultsource. b = ff_q_pfp_division_resultsource * S ((S (pfp_index_division_result)) * c) + (pfp_source_division_result))) /\ (((((exists ff_h_pfp_division_resulttarget. ff_h_pfp_division_resulttarget + S (pfp_residue_division_result) = S ((S (pfp_index_division_result)) * rc)) /\ exists ff_q_pfp_division_resulttarget. rb = ff_q_pfp_division_resulttarget * S ((S (pfp_index_division_result)) * rc) + (pfp_residue_division_result))) /\ ((((exists pfa_gap_division_resultresiduebound. pfa_gap_division_resultresiduebound + S (pfp_residue_division_result) = (p)) /\ ((exists pfa_offset_left_division_resultresiduecongruence pfa_offset_right_division_resultresiduecongruence. (pfp_source_division_result) + (p) * pfa_offset_left_division_resultresiduecongruence = (pfp_residue_division_result) + (p) * pfa_offset_right_division_resultresiduecongruence)))))))))

Constructive proof overview

Generated structural guide

Actual finite quotient/remainder witnesses give genuine coefficientwise canonical normalization.

The unchanged tactic script uses 2 declared prerequisites and contains 42 exact native proof lines.

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

Proof neighborhood

Direct dependencies

remainder_decomposition_to_mod_eq Stable theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized

Direct dependents

PP0002 prime_field_polynomial_normalization_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

42 script commands · 16 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 assumptionsL1–10

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 qb
  5. L5
    intro qc
  6. L6
    intro rb
  7. L7
    intro rc
  8. L8
    intro l
  9. L9
    intro h
  10. L10
    intro i
02Fix variables and assumptionsL11–11

Work with arbitrary variables or the premises of the current implication.

  1. L11
    intro hi
03Establish hpointL12–15

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

  1. L12
    have hpoint : ∃ a. ∃ q. ∃ r. BetaAt(b,c,i,a) ∧ (BetaAt(qb,qc,i,q) ∧ (BetaAt(rb,rc,i,r) ∧ DivRem(a,p,q,r)))Definitions: DivRemBetaAt
  2. L13
    specialize h (i)
  3. L14
    apply h
  4. L15
    exact hi
04Separate the logical casesL16–22

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

  1. L16
    cases hpoint
  2. L17
    cases hpoint_witness
  3. L18
    cases hpoint_witness_witness
  4. L19
    cases hpoint_witness_witness_witness
  5. L20
    cases hpoint_witness_witness_witness_right
  6. L21
    cases hpoint_witness_witness_witness_right_right
  7. L22
    cases hpoint_witness_witness_witness_right_right_right
05Construct an explicit witnessL23–24

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

  1. L23
    exists x
  2. L24
    exists x2
06Separate the logical casesL25–25

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

  1. L25
    split
07Use earlier factsL26–26

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

  1. L26
    exact hpoint_witness_witness_witness_left
08Separate the logical casesL27–27

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

  1. L27
    split
09Use earlier factsL28–28

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

  1. L28
    exact hpoint_witness_witness_witness_right_right_left
10Separate the logical casesL29–29

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

  1. L29
    split
11Use earlier factsL30–35

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

  1. L30
    exact hpoint_witness_witness_witness_right_right_right_right
  2. L31
    specialize remainder_decomposition_to_mod_eq (p)
  3. L32
    specialize remainder_decomposition_to_mod_eq (x)
  4. L33
    specialize remainder_decomposition_to_mod_eq (x1)
  5. L34
    specialize remainder_decomposition_to_mod_eq (x2)
  6. L35
    apply remainder_decomposition_to_mod_eq
12Calculate and transport equalitiesL36–36

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

  1. L36
    trans p*x1+x2
13Use earlier factsL37–37

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

  1. L37
    exact hpoint_witness_witness_witness_right_right_right_left
14Calculate and transport equalitiesL38–38

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

  1. L38
    congr
15Use earlier factsL39–41

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

  1. L39
    specialize mul_comm (p)
  2. L40
    specialize mul_comm (x1)
  3. L41
    apply mul_comm
16Calculate and transport equalitiesL42–42

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

  1. L42
    refl

Library-wide reading audit

Original exact command ledger · 42 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro qb
  5. 0005intro qc
  6. 0006intro rb
  7. 0007intro rc
  8. 0008intro l
  9. 0009intro h
  10. 0010intro i
  11. 0011intro hi
  12. 0012have hpoint : exists a q r. ((((exists ff_h_pfp_division_source. ff_h_pfp_division_source + S (a) = S ((S (i)) * c)) /\ exists ff_q_pfp_division_source. b = ff_q_pfp_division_source * S ((S (i)) * c) + (a))) /\ (((((exists ff_h_pfp_division_quotient. ff_h_pfp_division_quotient + S (q) = S ((S (i)) * qc)) /\ exists ff_q_pfp_division_quotient. qb = ff_q_pfp_division_quotient * S ((S (i)) * qc) + (q))) /\ (((((exists ff_h_pfp_division_remainder. ff_h_pfp_division_remainder + S (r) = S ((S (i)) * rc)) /\ exists ff_q_pfp_division_remainder. rb = ff_q_pfp_division_remainder * S ((S (i)) * rc) + (r))) /\ (((a=p*q+r) /\ ((exists pfa_gap_division_bound. pfa_gap_division_bound + S (r) = (p))))))))))
  13. 0013specialize h (i)
  14. 0014apply h
  15. 0015exact hi
  16. 0016cases hpoint
  17. 0017cases hpoint_witness
  18. 0018cases hpoint_witness_witness
  19. 0019cases hpoint_witness_witness_witness
  20. 0020cases hpoint_witness_witness_witness_right
  21. 0021cases hpoint_witness_witness_witness_right_right
  22. 0022cases hpoint_witness_witness_witness_right_right_right
  23. 0023exists x
  24. 0024exists x2
  25. 0025split
  26. 0026exact hpoint_witness_witness_witness_left
  27. 0027split
  28. 0028exact hpoint_witness_witness_witness_right_right_left
  29. 0029split
  30. 0030exact hpoint_witness_witness_witness_right_right_right_right
  31. 0031specialize remainder_decomposition_to_mod_eq (p)
  32. 0032specialize remainder_decomposition_to_mod_eq (x)
  33. 0033specialize remainder_decomposition_to_mod_eq (x1)
  34. 0034specialize remainder_decomposition_to_mod_eq (x2)
  35. 0035apply remainder_decomposition_to_mod_eq
  36. 0036trans p*x1+x2
  37. 0037exact hpoint_witness_witness_witness_right_right_right_left
  38. 0038congr
  39. 0039specialize mul_comm (p)
  40. 0040specialize mul_comm (x1)
  41. 0041apply mul_comm
  42. 0042refl