PP0001

prime_field_polynomial_normalization_from_division

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

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. ∀ qb. ∀ qc. ∀ rb. ∀ rc. ∀ l. DivisionPrefix(p,b,c,qb,qc,rb,rc,l) → FpCoefficientReduction(p,b,c,rb,rc,l)

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

Definition DAG

FpCoefficientReduction(p,b,c,d,e,l) · 1DivRem(n,d,q,r) · 1BetaAt(b,c,i,x) · 3DivisionPrefix(m,b,c,qb,qc,rb,rc,l) · 1

Actual proof prerequisites

remainder_decomposition_to_mod_eq · checked external prerequisitemul_comm · checked external prerequisite
Original expanded first-order 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)))))))))

Complete tactic proof in conservative notation

All 42 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

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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: BetaAt(b,c,i,a)BetaAt(qb,qc,i,q)BetaAt(rb,rc,i,r)DivRem(a,p,q,r)Original native command in the exact edition
  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 defined 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 : ∃ a. ∃ q. ∃ r. BetaAt(b,c,i,a) ∧ (BetaAt(qb,qc,i,q) ∧ (BetaAt(rb,rc,i,r) ∧ DivRem(a,p,q,r)))
  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