FP0034

prime_field_multiply_table_exists

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

Construct every entry of the finite multiply table from primality alone.

Exact expanded first-order arithmetic statement

forall p. (~((p) = 1) /\ forall pfa_factor_left_multiplytable_domain pfa_factor_right_multiplytable_domain. (p) = pfa_factor_left_multiplytable_domain * pfa_factor_right_multiplytable_domain -> pfa_factor_left_multiplytable_domain = 1 \/ pfa_factor_right_multiplytable_domain = 1) -> exists b c. (forall pft_index_multiplytable_result. (exists pfa_gap_multiplytable_resultprefix. pfa_gap_multiplytable_resultprefix + S (pft_index_multiplytable_result) = ((p) * (p))) -> exists pft_value_multiplytable_result. (((((exists ff_h_pft_multiplytable_resultpointentry. ff_h_pft_multiplytable_resultpointentry + S (pft_value_multiplytable_result) = S ((S (pft_index_multiplytable_result)) * c)) /\ exists ff_q_pft_multiplytable_resultpointentry. b = ff_q_pft_multiplytable_resultpointentry * S ((S (pft_index_multiplytable_result)) * c) + (pft_value_multiplytable_result))) /\ ((exists pft_row_multiplytable_resultpointvalue pft_column_multiplytable_resultpointvalue. (((pft_index_multiplytable_result) = pft_row_multiplytable_resultpointvalue * (p) + pft_column_multiplytable_resultpointvalue) /\ ((((exists pfa_gap_multiplytable_resultpointvalueoperationleft. pfa_gap_multiplytable_resultpointvalueoperationleft + S (pft_row_multiplytable_resultpointvalue) = (p)) /\ (((exists pfa_gap_multiplytable_resultpointvalueoperationright. pfa_gap_multiplytable_resultpointvalueoperationright + S (pft_column_multiplytable_resultpointvalue) = (p)) /\ ((((exists pfa_gap_multiplytable_resultpointvalueoperationresultbound. pfa_gap_multiplytable_resultpointvalueoperationresultbound + S (pft_value_multiplytable_result) = (p)) /\ ((exists pfa_offset_left_multiplytable_resultpointvalueoperationresultcongruence pfa_offset_right_multiplytable_resultpointvalueoperationresultcongruence. ((pft_row_multiplytable_resultpointvalue) * (pft_column_multiplytable_resultpointvalue)) + (p) * pfa_offset_left_multiplytable_resultpointvalueoperationresultcongruence = (pft_value_multiplytable_result) + (p) * pfa_offset_right_multiplytable_resultpointvalueoperationresultcongruence))))))))))))))))

Constructive proof overview

Generated structural guide

Construct every entry of the finite multiply table from primality alone.

The unchanged tactic script uses 2 declared prerequisites and contains 12 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

FP0030 prime_field_multiply_prefix_choice FP002C prime_field_multiply_grid_value_exists

Direct dependents

FP0037 prime_field_operation_tables_exists

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

12 script commands · 4 reading checkpoints · 0 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 (2)
prime field multiply prefix choiceprime field multiply grid value exists
01Fix variables and assumptionsL1–2

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

  1. L1
    intro p
  2. L2
    intro hp
02Use earlier factsL3–5

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

  1. L3
    specialize prime_field_multiply_prefix_choice (p)
  2. L4
    specialize prime_field_multiply_prefix_choice ((p) * (p))
  3. L5
    apply prime_field_multiply_prefix_choice
03Fix variables and assumptionsL6–7

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

  1. L6
    intro i
  2. L7
    intro hi
04Use earlier factsL8–12

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

  1. L8
    specialize prime_field_multiply_grid_value_exists (p)
  2. L9
    specialize prime_field_multiply_grid_value_exists (i)
  3. L10
    apply prime_field_multiply_grid_value_exists
  4. L11
    exact hp
  5. L12
    exact hi

Library-wide reading audit

Original exact command ledger · 12 lines
  1. 0001intro p
  2. 0002intro hp
  3. 0003specialize prime_field_multiply_prefix_choice (p)
  4. 0004specialize prime_field_multiply_prefix_choice ((p) * (p))
  5. 0005apply prime_field_multiply_prefix_choice
  6. 0006intro i
  7. 0007intro hi
  8. 0008specialize prime_field_multiply_grid_value_exists (p)
  9. 0009specialize prime_field_multiply_grid_value_exists (i)
  10. 0010apply prime_field_multiply_grid_value_exists
  11. 0011exact hp
  12. 0012exact hi