FP0035

prime_field_negate_table_exists

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

Exact expanded first-order arithmetic statement

forall p. (~((p) = 1) /\ forall pfa_factor_left_negatetable_domain pfa_factor_right_negatetable_domain. (p) = pfa_factor_left_negatetable_domain * pfa_factor_right_negatetable_domain -> pfa_factor_left_negatetable_domain = 1 \/ pfa_factor_right_negatetable_domain = 1) -> exists b c. (forall pft_index_negatetable_result. (exists pfa_gap_negatetable_resultprefix. pfa_gap_negatetable_resultprefix + S (pft_index_negatetable_result) = (p)) -> exists pft_value_negatetable_result. (((((exists ff_h_pft_negatetable_resultpointentry. ff_h_pft_negatetable_resultpointentry + S (pft_value_negatetable_result) = S ((S (pft_index_negatetable_result)) * c)) /\ exists ff_q_pft_negatetable_resultpointentry. b = ff_q_pft_negatetable_resultpointentry * S ((S (pft_index_negatetable_result)) * c) + (pft_value_negatetable_result))) /\ ((((exists pfa_gap_negatetable_resultpointvalueadditionleft. pfa_gap_negatetable_resultpointvalueadditionleft + S (pft_index_negatetable_result) = (p)) /\ (((exists pfa_gap_negatetable_resultpointvalueadditionright. pfa_gap_negatetable_resultpointvalueadditionright + S (pft_value_negatetable_result) = (p)) /\ ((((exists pfa_gap_negatetable_resultpointvalueadditionresultbound. pfa_gap_negatetable_resultpointvalueadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negatetable_resultpointvalueadditionresultcongruence pfa_offset_right_negatetable_resultpointvalueadditionresultcongruence. ((pft_index_negatetable_result) + (pft_value_negatetable_result)) + (p) * pfa_offset_left_negatetable_resultpointvalueadditionresultcongruence = (0) + (p) * pfa_offset_right_negatetable_resultpointvalueadditionresultcongruence)))))))))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 12 exact native proof lines.

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

FP0031 prime_field_negate_prefix_choice FP001B prime_field_negate_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. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply 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 negate prefix choiceprime field negate 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_negate_prefix_choice (p)
  2. L4
    specialize prime_field_negate_prefix_choice (p)
  3. L5
    apply prime_field_negate_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_negate_exists (p)
  2. L9
    specialize prime_field_negate_exists (i)
  3. L10
    apply prime_field_negate_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_negate_prefix_choice (p)
  4. 0004specialize prime_field_negate_prefix_choice (p)
  5. 0005apply prime_field_negate_prefix_choice
  6. 0006intro i
  7. 0007intro hi
  8. 0008specialize prime_field_negate_exists (p)
  9. 0009specialize prime_field_negate_exists (i)
  10. 0010apply prime_field_negate_exists
  11. 0011exact hp
  12. 0012exact hi