PP0003

prime_field_polynomial_normalization_entry

All decoded entries satisfy normalization, not just the initially chosen beta witnesses.

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. ∀ d. ∀ e. ∀ l. ∀ i. ∀ a. ∀ r. FpCoefficientReduction(p,b,c,d,e,l) → Lt(i,l) → BetaAt(b,c,i,a) → BetaAt(d,e,i,r) → CanonicalModularResidue(p,a,r)

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

Definition DAG

CanonicalModularResidue(m,a,r) · 2FpCoefficientReduction(p,b,c,d,e,l) · 1Lt(a,b) · 1BetaAt(b,c,i,x) · 4

Actual proof prerequisites

beta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall p b c d e l i a r. (forall pfp_index_entry_table. (exists pfa_gap_entry_tableindex. pfa_gap_entry_tableindex + S (pfp_index_entry_table) = (l)) -> exists pfp_source_entry_table pfp_residue_entry_table. ((((exists ff_h_pfp_entry_tablesource. ff_h_pfp_entry_tablesource + S (pfp_source_entry_table) = S ((S (pfp_index_entry_table)) * c)) /\ exists ff_q_pfp_entry_tablesource. b = ff_q_pfp_entry_tablesource * S ((S (pfp_index_entry_table)) * c) + (pfp_source_entry_table))) /\ (((((exists ff_h_pfp_entry_tabletarget. ff_h_pfp_entry_tabletarget + S (pfp_residue_entry_table) = S ((S (pfp_index_entry_table)) * e)) /\ exists ff_q_pfp_entry_tabletarget. d = ff_q_pfp_entry_tabletarget * S ((S (pfp_index_entry_table)) * e) + (pfp_residue_entry_table))) /\ ((((exists pfa_gap_entry_tableresiduebound. pfa_gap_entry_tableresiduebound + S (pfp_residue_entry_table) = (p)) /\ ((exists pfa_offset_left_entry_tableresiduecongruence pfa_offset_right_entry_tableresiduecongruence. (pfp_source_entry_table) + (p) * pfa_offset_left_entry_tableresiduecongruence = (pfp_residue_entry_table) + (p) * pfa_offset_right_entry_tableresiduecongruence))))))))) -> (exists pfa_gap_entry_index. pfa_gap_entry_index + S (i) = (l)) -> (((exists ff_h_pfp_entry_source. ff_h_pfp_entry_source + S (a) = S ((S (i)) * c)) /\ exists ff_q_pfp_entry_source. b = ff_q_pfp_entry_source * S ((S (i)) * c) + (a))) -> (((exists ff_h_pfp_entry_target. ff_h_pfp_entry_target + S (r) = S ((S (i)) * e)) /\ exists ff_q_pfp_entry_target. d = ff_q_pfp_entry_target * S ((S (i)) * e) + (r))) -> (((exists pfa_gap_entry_valuebound. pfa_gap_entry_valuebound + S (r) = (p)) /\ ((exists pfa_offset_left_entry_valuecongruence pfa_offset_right_entry_valuecongruence. (a) + (p) * pfa_offset_left_entry_valuecongruence = (r) + (p) * pfa_offset_right_entry_valuecongruence))))

Complete tactic proof in conservative notation

All 43 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

43 script commands · 8 reading checkpoints · 3 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 d
  5. L5
    intro e
  6. L6
    intro l
  7. L7
    intro i
  8. L8
    intro a
  9. L9
    intro r
  10. L10
    intro h
02Fix variables and assumptionsL11–13

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

  1. L11
    intro hi
  2. L12
    intro ha
  3. L13
    intro hr
03Establish hpointL14–17

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

  1. L14
    have hpoint : ∃ u. ∃ v. BetaAt(b,c,i,u) ∧ (BetaAt(d,e,i,v) ∧ CanonicalModularResidue(p,u,v))Definitions: BetaAt(b,c,i,u)BetaAt(d,e,i,v)CanonicalModularResidue(p,u,v)Original native command in the exact edition
  2. L15
    specialize h (i)
  3. L16
    apply h
  4. L17
    exact hi
04Separate the logical casesL18–21

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

  1. L18
    cases hpoint
  2. L19
    cases hpoint_witness
  3. L20
    cases hpoint_witness_witness
  4. L21
    cases hpoint_witness_witness_right
05Establish heqL22–30

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

  1. L22
    have heq : x=a
  2. L23
    specialize beta_at_unique (b)
  3. L24
    specialize beta_at_unique (c)
  4. L25
    specialize beta_at_unique (i)
  5. L26
    specialize beta_at_unique (x)
  6. L27
    specialize beta_at_unique (a)
  7. L28
    apply beta_at_unique
  8. L29
    exact hpoint_witness_witness_left
  9. L30
    exact ha
06Establish hresL31–40

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

  1. L31
    have hres : x1=r
  2. L32
    specialize beta_at_unique (d)
  3. L33
    specialize beta_at_unique (e)
  4. L34
    specialize beta_at_unique (i)
  5. L35
    specialize beta_at_unique (x1)
  6. L36
    specialize beta_at_unique (r)
  7. L37
    apply beta_at_unique
  8. L38
    exact hpoint_witness_witness_right_left
  9. L39
    exact hr
  10. L40
    rewrite heq at hpoint_witness_witness_right_right
07Calculate and transport equalitiesL41–42

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

  1. L41
    rewrite hres at hpoint_witness_witness_right_right
  2. L42
    rewrite hres at hpoint_witness_witness_right_right
08Use earlier factsL43–43

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

  1. L43
    exact hpoint_witness_witness_right_right

Library-wide reading audit

Original defined command ledger · 43 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro e
  6. 0006intro l
  7. 0007intro i
  8. 0008intro a
  9. 0009intro r
  10. 0010intro h
  11. 0011intro hi
  12. 0012intro ha
  13. 0013intro hr
  14. 0014have hpoint : ∃ u. ∃ v. BetaAt(b,c,i,u) ∧ (BetaAt(d,e,i,v) ∧ CanonicalModularResidue(p,u,v))
  15. 0015specialize h (i)
  16. 0016apply h
  17. 0017exact hi
  18. 0018cases hpoint
  19. 0019cases hpoint_witness
  20. 0020cases hpoint_witness_witness
  21. 0021cases hpoint_witness_witness_right
  22. 0022have heq : x=a
  23. 0023specialize beta_at_unique (b)
  24. 0024specialize beta_at_unique (c)
  25. 0025specialize beta_at_unique (i)
  26. 0026specialize beta_at_unique (x)
  27. 0027specialize beta_at_unique (a)
  28. 0028apply beta_at_unique
  29. 0029exact hpoint_witness_witness_left
  30. 0030exact ha
  31. 0031have hres : x1=r
  32. 0032specialize beta_at_unique (d)
  33. 0033specialize beta_at_unique (e)
  34. 0034specialize beta_at_unique (i)
  35. 0035specialize beta_at_unique (x1)
  36. 0036specialize beta_at_unique (r)
  37. 0037apply beta_at_unique
  38. 0038exact hpoint_witness_witness_right_left
  39. 0039exact hr
  40. 0040rewrite heq at hpoint_witness_witness_right_right
  41. 0041rewrite hres at hpoint_witness_witness_right_right
  42. 0042rewrite hres at hpoint_witness_witness_right_right
  43. 0043exact hpoint_witness_witness_right_right