FP003D

prime_field_multiply_table_reflect

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.

Every genuine canonical multiply result occurs at its actual finite table index.

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

This checkpoint constructs prime-order fields (k=1), with genuine finite arithmetic tables, cardinality, and characteristic. Inversion is proved only for nonzero elements; the table's zero entry is a zero-to-zero convention. G091 for every prime power p^k, with an irreducible polynomial of degree k, remains open. No extension-field construction or G091 closure is claimed.

Exact theorem in conservative defined notation

∀ p. ∀ B. ∀ C. ∀ a. ∀ b. ∀ v. FpMulPrefix(p,B,C,p · p) → FpMul(p,a,b,v) → BetaAt(B,C,a · p + b,v)

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

Definition DAG

FpMul(p,a,b,c) · 2FpMulPrefix(p,b,c,l) · 1BetaAt(b,c,i,x) · 2

Actual proof prerequisites

beta_at_exists · checked external prerequisiteprime_field_multiply_table_lookupprime_field_multiply_functional
Original expanded first-order statement 
forall p B C a b v. (forall pft_index_multiplyreflect_table. (exists pfa_gap_multiplyreflect_tableprefix. pfa_gap_multiplyreflect_tableprefix + S (pft_index_multiplyreflect_table) = ((p) * (p))) -> exists pft_value_multiplyreflect_table. (((((exists ff_h_pft_multiplyreflect_tablepointentry. ff_h_pft_multiplyreflect_tablepointentry + S (pft_value_multiplyreflect_table) = S ((S (pft_index_multiplyreflect_table)) * C)) /\ exists ff_q_pft_multiplyreflect_tablepointentry. B = ff_q_pft_multiplyreflect_tablepointentry * S ((S (pft_index_multiplyreflect_table)) * C) + (pft_value_multiplyreflect_table))) /\ ((exists pft_row_multiplyreflect_tablepointvalue pft_column_multiplyreflect_tablepointvalue. (((pft_index_multiplyreflect_table) = pft_row_multiplyreflect_tablepointvalue * (p) + pft_column_multiplyreflect_tablepointvalue) /\ ((((exists pfa_gap_multiplyreflect_tablepointvalueoperationleft. pfa_gap_multiplyreflect_tablepointvalueoperationleft + S (pft_row_multiplyreflect_tablepointvalue) = (p)) /\ (((exists pfa_gap_multiplyreflect_tablepointvalueoperationright. pfa_gap_multiplyreflect_tablepointvalueoperationright + S (pft_column_multiplyreflect_tablepointvalue) = (p)) /\ ((((exists pfa_gap_multiplyreflect_tablepointvalueoperationresultbound. pfa_gap_multiplyreflect_tablepointvalueoperationresultbound + S (pft_value_multiplyreflect_table) = (p)) /\ ((exists pfa_offset_left_multiplyreflect_tablepointvalueoperationresultcongruence pfa_offset_right_multiplyreflect_tablepointvalueoperationresultcongruence. ((pft_row_multiplyreflect_tablepointvalue) * (pft_column_multiplyreflect_tablepointvalue)) + (p) * pfa_offset_left_multiplyreflect_tablepointvalueoperationresultcongruence = (pft_value_multiplyreflect_table) + (p) * pfa_offset_right_multiplyreflect_tablepointvalueoperationresultcongruence)))))))))))))))) -> (((exists pfa_gap_multiplyreflect_graphleft. pfa_gap_multiplyreflect_graphleft + S (a) = (p)) /\ (((exists pfa_gap_multiplyreflect_graphright. pfa_gap_multiplyreflect_graphright + S (b) = (p)) /\ ((((exists pfa_gap_multiplyreflect_graphresultbound. pfa_gap_multiplyreflect_graphresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_multiplyreflect_graphresultcongruence pfa_offset_right_multiplyreflect_graphresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplyreflect_graphresultcongruence = (v) + (p) * pfa_offset_right_multiplyreflect_graphresultcongruence))))))))) -> (((exists ff_h_pft_multiplyreflect_at. ff_h_pft_multiplyreflect_at + S (v) = S ((S (a*p+b)) * C)) /\ exists ff_q_pft_multiplyreflect_at. B = ff_q_pft_multiplyreflect_at * S ((S (a*p+b)) * C) + (v)))

Complete tactic proof in conservative notation

All 40 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

40 script commands · 9 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.

Named ingredients (2)
prime field multiply table lookupprime field multiply functional
01Fix variables and assumptionsL1–8

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 a
  5. L5
    intro b
  6. L6
    intro v
  7. L7
    intro htable
  8. L8
    intro hop
02Separate the logical casesL9–10

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

  1. L9
    cases hop
  2. L10
    cases hop_right
03Establish hatL11–15

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

  1. L11
    have hat : ∃ w. BetaAt(B,C,a · p + b,w)Definitions: BetaAt(B,C,a · p + b,w)Original native command in the exact edition
  2. L12
    specialize beta_at_exists (B)
  3. L13
    specialize beta_at_exists (C)
  4. L14
    specialize beta_at_exists (a*p+b)
  5. L15
    apply beta_at_exists
04Separate the logical casesL16–16

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

  1. L16
    cases hat
05Establish houtL17–26

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime field multiply table lookup.

  1. L17
    have hout : FpMul(p,a,b,x)Definitions: FpMul(p,a,b,x)Original native command in the exact edition
  2. L18
    specialize prime_field_multiply_table_lookup (p)
  3. L19
    specialize prime_field_multiply_table_lookup (B)
  4. L20
    specialize prime_field_multiply_table_lookup (C)
  5. L21
    specialize prime_field_multiply_table_lookup (a)
  6. L22
    specialize prime_field_multiply_table_lookup (b)
  7. L23
    specialize prime_field_multiply_table_lookup (x)
  8. L24
    apply prime_field_multiply_table_lookup
  9. L25
    exact htable
  10. L26
    exact hop_left
06Use earlier factsL27–28

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

  1. L27
    exact hop_right_left
  2. L28
    exact hat_witness
07Establish heqL29–38

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime field multiply functional.

  1. L29
    have heq : x = v
  2. L30
    specialize prime_field_multiply_functional (p)
  3. L31
    specialize prime_field_multiply_functional (a)
  4. L32
    specialize prime_field_multiply_functional (b)
  5. L33
    specialize prime_field_multiply_functional (x)
  6. L34
    specialize prime_field_multiply_functional (v)
  7. L35
    apply prime_field_multiply_functional
  8. L36
    exact hout
  9. L37
    exact hop
  10. L38
    rewrite heq at hat_witness
08Calculate and transport equalitiesL39–39

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

  1. L39
    rewrite heq at hat_witness
09Use earlier factsL40–40

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

  1. L40
    exact hat_witness

Library-wide reading audit

Original defined command ledger · 40 lines
  1. 0001intro p
  2. 0002intro B
  3. 0003intro C
  4. 0004intro a
  5. 0005intro b
  6. 0006intro v
  7. 0007intro htable
  8. 0008intro hop
  9. 0009cases hop
  10. 0010cases hop_right
  11. 0011have hat : ∃ w. BetaAt(B,C,a · p + b,w)
  12. 0012specialize beta_at_exists (B)
  13. 0013specialize beta_at_exists (C)
  14. 0014specialize beta_at_exists (a*p+b)
  15. 0015apply beta_at_exists
  16. 0016cases hat
  17. 0017have hout : FpMul(p,a,b,x)
  18. 0018specialize prime_field_multiply_table_lookup (p)
  19. 0019specialize prime_field_multiply_table_lookup (B)
  20. 0020specialize prime_field_multiply_table_lookup (C)
  21. 0021specialize prime_field_multiply_table_lookup (a)
  22. 0022specialize prime_field_multiply_table_lookup (b)
  23. 0023specialize prime_field_multiply_table_lookup (x)
  24. 0024apply prime_field_multiply_table_lookup
  25. 0025exact htable
  26. 0026exact hop_left
  27. 0027exact hop_right_left
  28. 0028exact hat_witness
  29. 0029have heq : x = v
  30. 0030specialize prime_field_multiply_functional (p)
  31. 0031specialize prime_field_multiply_functional (a)
  32. 0032specialize prime_field_multiply_functional (b)
  33. 0033specialize prime_field_multiply_functional (x)
  34. 0034specialize prime_field_multiply_functional (v)
  35. 0035apply prime_field_multiply_functional
  36. 0036exact hout
  37. 0037exact hop
  38. 0038rewrite heq at hat_witness
  39. 0039rewrite heq at hat_witness
  40. 0040exact hat_witness