FP003C

prime_field_multiply_table_lookup

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

Every decoded multiply table lookup has exactly the proved canonical arithmetic meaning.

Exact expanded first-order arithmetic statement

forall p B C a b v. (forall pft_index_multiplylookup_table. (exists pfa_gap_multiplylookup_tableprefix. pfa_gap_multiplylookup_tableprefix + S (pft_index_multiplylookup_table) = ((p) * (p))) -> exists pft_value_multiplylookup_table. (((((exists ff_h_pft_multiplylookup_tablepointentry. ff_h_pft_multiplylookup_tablepointentry + S (pft_value_multiplylookup_table) = S ((S (pft_index_multiplylookup_table)) * C)) /\ exists ff_q_pft_multiplylookup_tablepointentry. B = ff_q_pft_multiplylookup_tablepointentry * S ((S (pft_index_multiplylookup_table)) * C) + (pft_value_multiplylookup_table))) /\ ((exists pft_row_multiplylookup_tablepointvalue pft_column_multiplylookup_tablepointvalue. (((pft_index_multiplylookup_table) = pft_row_multiplylookup_tablepointvalue * (p) + pft_column_multiplylookup_tablepointvalue) /\ ((((exists pfa_gap_multiplylookup_tablepointvalueoperationleft. pfa_gap_multiplylookup_tablepointvalueoperationleft + S (pft_row_multiplylookup_tablepointvalue) = (p)) /\ (((exists pfa_gap_multiplylookup_tablepointvalueoperationright. pfa_gap_multiplylookup_tablepointvalueoperationright + S (pft_column_multiplylookup_tablepointvalue) = (p)) /\ ((((exists pfa_gap_multiplylookup_tablepointvalueoperationresultbound. pfa_gap_multiplylookup_tablepointvalueoperationresultbound + S (pft_value_multiplylookup_table) = (p)) /\ ((exists pfa_offset_left_multiplylookup_tablepointvalueoperationresultcongruence pfa_offset_right_multiplylookup_tablepointvalueoperationresultcongruence. ((pft_row_multiplylookup_tablepointvalue) * (pft_column_multiplylookup_tablepointvalue)) + (p) * pfa_offset_left_multiplylookup_tablepointvalueoperationresultcongruence = (pft_value_multiplylookup_table) + (p) * pfa_offset_right_multiplylookup_tablepointvalueoperationresultcongruence)))))))))))))))) -> (exists pfa_gap_multiplylookup_a. pfa_gap_multiplylookup_a + S (a) = (p)) -> (exists pfa_gap_multiplylookup_b. pfa_gap_multiplylookup_b + S (b) = (p)) -> (((exists ff_h_pft_multiplylookup_at. ff_h_pft_multiplylookup_at + S (v) = S ((S (a*p+b)) * C)) /\ exists ff_q_pft_multiplylookup_at. B = ff_q_pft_multiplylookup_at * S ((S (a*p+b)) * C) + (v))) -> (((exists pfa_gap_multiplylookup_graphleft. pfa_gap_multiplylookup_graphleft + S (a) = (p)) /\ (((exists pfa_gap_multiplylookup_graphright. pfa_gap_multiplylookup_graphright + S (b) = (p)) /\ ((((exists pfa_gap_multiplylookup_graphresultbound. pfa_gap_multiplylookup_graphresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_multiplylookup_graphresultcongruence pfa_offset_right_multiplylookup_graphresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplylookup_graphresultcongruence = (v) + (p) * pfa_offset_right_multiplylookup_graphresultcongruence)))))))))

Constructive proof overview

Generated structural guide

Every decoded multiply table lookup has exactly the proved canonical arithmetic meaning.

The unchanged tactic script uses 3 declared prerequisites and contains 39 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

matrix_recursive_flattened_index_bound Alpha theorem; checked-use authorized beta_at_unique Alpha theorem; checked-use authorized FP003B prime_field_multiply_grid_value_lookup

Direct dependents

FP003D prime_field_multiply_table_reflect FP0044 prime_field_multiply_table_commutative FP0045 prime_field_multiply_table_associative FP0048 prime_field_left_table_distributive FP0049 prime_field_right_table_distributive

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

39 script commands · 6 reading checkpoints · 2 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.

Named ingredients (1)
prime field multiply grid value lookup

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 a
  5. L5
    intro b
  6. L6
    intro v
  7. L7
    intro htable
  8. L8
    intro ha
  9. L9
    intro hb
  10. L10
    intro hat
02Establish hpointL11–19

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

  1. L11
    have hpoint : ∃ w. BetaAt(B,C,a · p + b,w) ∧ FpMulGridValue(p,a · p + b,w)Definitions: FpMulGridValueBetaAt
  2. L12
    specialize htable (a*p+b)
  3. L13
    apply htable
  4. L14
    specialize matrix_recursive_flattened_index_bound (p)
  5. L15
    specialize matrix_recursive_flattened_index_bound (a)
  6. L16
    specialize matrix_recursive_flattened_index_bound (b)
  7. L17
    apply matrix_recursive_flattened_index_bound
  8. L18
    exact ha
  9. L19
    exact hb
03Separate the logical casesL20–21

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

  1. L20
    cases hpoint
  2. L21
    cases hpoint_witness
04Establish heqL22–31

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 = v
  2. L23
    specialize beta_at_unique (B)
  3. L24
    specialize beta_at_unique (C)
  4. L25
    specialize beta_at_unique (a*p+b)
  5. L26
    specialize beta_at_unique (x)
  6. L27
    specialize beta_at_unique (v)
  7. L28
    apply beta_at_unique
  8. L29
    exact hpoint_witness_left
  9. L30
    exact hat
  10. L31
    rewrite heq at hpoint_witness_right
05Calculate and transport equalitiesL32–32

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

  1. L32
    rewrite heq at hpoint_witness_right
06Use earlier factsL33–39

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

  1. L33
    specialize prime_field_multiply_grid_value_lookup (p)
  2. L34
    specialize prime_field_multiply_grid_value_lookup (a)
  3. L35
    specialize prime_field_multiply_grid_value_lookup (b)
  4. L36
    specialize prime_field_multiply_grid_value_lookup (v)
  5. L37
    apply prime_field_multiply_grid_value_lookup
  6. L38
    exact hb
  7. L39
    exact hpoint_witness_right

Library-wide reading audit

Original exact command ledger · 39 lines
  1. 0001intro p
  2. 0002intro B
  3. 0003intro C
  4. 0004intro a
  5. 0005intro b
  6. 0006intro v
  7. 0007intro htable
  8. 0008intro ha
  9. 0009intro hb
  10. 0010intro hat
  11. 0011have hpoint : exists w. (((((exists ff_h_pft_multiplylookup_pointentry. ff_h_pft_multiplylookup_pointentry + S (w) = S ((S (a*p+b)) * C)) /\ exists ff_q_pft_multiplylookup_pointentry. B = ff_q_pft_multiplylookup_pointentry * S ((S (a*p+b)) * C) + (w))) /\ ((exists pft_row_multiplylookup_pointvalue pft_column_multiplylookup_pointvalue. (((a*p+b) = pft_row_multiplylookup_pointvalue * (p) + pft_column_multiplylookup_pointvalue) /\ ((((exists pfa_gap_multiplylookup_pointvalueoperationleft. pfa_gap_multiplylookup_pointvalueoperationleft + S (pft_row_multiplylookup_pointvalue) = (p)) /\ (((exists pfa_gap_multiplylookup_pointvalueoperationright. pfa_gap_multiplylookup_pointvalueoperationright + S (pft_column_multiplylookup_pointvalue) = (p)) /\ ((((exists pfa_gap_multiplylookup_pointvalueoperationresultbound. pfa_gap_multiplylookup_pointvalueoperationresultbound + S (w) = (p)) /\ ((exists pfa_offset_left_multiplylookup_pointvalueoperationresultcongruence pfa_offset_right_multiplylookup_pointvalueoperationresultcongruence. ((pft_row_multiplylookup_pointvalue) * (pft_column_multiplylookup_pointvalue)) + (p) * pfa_offset_left_multiplylookup_pointvalueoperationresultcongruence = (w) + (p) * pfa_offset_right_multiplylookup_pointvalueoperationresultcongruence)))))))))))))))
  12. 0012specialize htable (a*p+b)
  13. 0013apply htable
  14. 0014specialize matrix_recursive_flattened_index_bound (p)
  15. 0015specialize matrix_recursive_flattened_index_bound (a)
  16. 0016specialize matrix_recursive_flattened_index_bound (b)
  17. 0017apply matrix_recursive_flattened_index_bound
  18. 0018exact ha
  19. 0019exact hb
  20. 0020cases hpoint
  21. 0021cases hpoint_witness
  22. 0022have heq : x = v
  23. 0023specialize beta_at_unique (B)
  24. 0024specialize beta_at_unique (C)
  25. 0025specialize beta_at_unique (a*p+b)
  26. 0026specialize beta_at_unique (x)
  27. 0027specialize beta_at_unique (v)
  28. 0028apply beta_at_unique
  29. 0029exact hpoint_witness_left
  30. 0030exact hat
  31. 0031rewrite heq at hpoint_witness_right
  32. 0032rewrite heq at hpoint_witness_right
  33. 0033specialize prime_field_multiply_grid_value_lookup (p)
  34. 0034specialize prime_field_multiply_grid_value_lookup (a)
  35. 0035specialize prime_field_multiply_grid_value_lookup (b)
  36. 0036specialize prime_field_multiply_grid_value_lookup (v)
  37. 0037apply prime_field_multiply_grid_value_lookup
  38. 0038exact hb
  39. 0039exact hpoint_witness_right