FP0047

prime_field_inverse_table_nonzero

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

Every nonzero inverse-table entry multiplies its input to the actual one representative.

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 B C a v. (forall pft_index_inverse_nonzero_table. (exists pfa_gap_inverse_nonzero_tableprefix. pfa_gap_inverse_nonzero_tableprefix + S (pft_index_inverse_nonzero_table) = (p)) -> exists pft_value_inverse_nonzero_table. (((((exists ff_h_pft_inverse_nonzero_tablepointentry. ff_h_pft_inverse_nonzero_tablepointentry + S (pft_value_inverse_nonzero_table) = S ((S (pft_index_inverse_nonzero_table)) * C)) /\ exists ff_q_pft_inverse_nonzero_tablepointentry. B = ff_q_pft_inverse_nonzero_tablepointentry * S ((S (pft_index_inverse_nonzero_table)) * C) + (pft_value_inverse_nonzero_table))) /\ ((((exists pfa_gap_inverse_nonzero_tablepointvalueinput. pfa_gap_inverse_nonzero_tablepointvalueinput + S (pft_index_inverse_nonzero_table) = (p)) /\ (((exists pfa_gap_inverse_nonzero_tablepointvalueoutput. pfa_gap_inverse_nonzero_tablepointvalueoutput + S (pft_value_inverse_nonzero_table) = (p)) /\ ((((pft_index_inverse_nonzero_table) = 0 /\ (pft_value_inverse_nonzero_table) = 0) \/ (((~((pft_index_inverse_nonzero_table) = 0)) /\ ((((exists pfa_gap_inverse_nonzero_tablepointvaluenonzeromultiplicationleft. pfa_gap_inverse_nonzero_tablepointvaluenonzeromultiplicationleft + S (pft_index_inverse_nonzero_table) = (p)) /\ (((exists pfa_gap_inverse_nonzero_tablepointvaluenonzeromultiplicationright. pfa_gap_inverse_nonzero_tablepointvaluenonzeromultiplicationright + S (pft_value_inverse_nonzero_table) = (p)) /\ ((((exists pfa_gap_inverse_nonzero_tablepointvaluenonzeromultiplicationresultbound. pfa_gap_inverse_nonzero_tablepointvaluenonzeromultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_nonzero_tablepointvaluenonzeromultiplicationresultcongruence pfa_offset_right_inverse_nonzero_tablepointvaluenonzeromultiplicationresultcongruence. ((pft_index_inverse_nonzero_table) * (pft_value_inverse_nonzero_table)) + (p) * pfa_offset_left_inverse_nonzero_tablepointvaluenonzeromultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_nonzero_tablepointvaluenonzeromultiplicationresultcongruence)))))))))))))))))))))) -> (exists pfa_gap_inverse_nonzero_bound. pfa_gap_inverse_nonzero_bound + S (a) = (p)) -> ~(a = 0) -> (((exists ff_h_pft_inverse_nonzero_entry. ff_h_pft_inverse_nonzero_entry + S (v) = S ((S (a)) * C)) /\ exists ff_q_pft_inverse_nonzero_entry. B = ff_q_pft_inverse_nonzero_entry * S ((S (a)) * C) + (v))) -> (((exists pfa_gap_inverse_nonzero_productleft. pfa_gap_inverse_nonzero_productleft + S (a) = (p)) /\ (((exists pfa_gap_inverse_nonzero_productright. pfa_gap_inverse_nonzero_productright + S (v) = (p)) /\ ((((exists pfa_gap_inverse_nonzero_productresultbound. pfa_gap_inverse_nonzero_productresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_nonzero_productresultcongruence pfa_offset_right_inverse_nonzero_productresultcongruence. ((a) * (v)) + (p) * pfa_offset_left_inverse_nonzero_productresultcongruence = (1) + (p) * pfa_offset_right_inverse_nonzero_productresultcongruence)))))))))

Constructive proof overview

Generated structural guide

Every nonzero inverse-table entry multiplies its input to the actual one representative.

The unchanged tactic script uses 1 declared prerequisite and contains 28 exact native proof lines.

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

Proof neighborhood

Direct dependencies

FP0040 prime_field_inverse_table_lookup

Direct dependents

none

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

28 script commands · 6 reading checkpoints · 1 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 inverse table 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 assumptionsL1–9

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 v
  6. L6
    intro htable
  7. L7
    intro ha
  8. L8
    intro hn
  9. L9
    intro hat
02Establish hiL10–19

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

  1. L10
    have hi : FpZeroExtendedInv(p,a,v)Definitions: FpZeroExtendedInv
  2. L11
    specialize prime_field_inverse_table_lookup (p)
  3. L12
    specialize prime_field_inverse_table_lookup (B)
  4. L13
    specialize prime_field_inverse_table_lookup (C)
  5. L14
    specialize prime_field_inverse_table_lookup (a)
  6. L15
    specialize prime_field_inverse_table_lookup (v)
  7. L16
    apply prime_field_inverse_table_lookup
  8. L17
    exact htable
  9. L18
    exact ha
  10. L19
    exact hat
03Separate the logical casesL20–24

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

  1. L20
    cases hi
  2. L21
    cases hi_right
  3. L22
    cases hi_right_right
  4. L23
    cases hi_right_right_left
  5. L24
    exfalso
04Use earlier factsL25–26

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

  1. L25
    apply hn
  2. L26
    exact hi_right_right_left_left
05Separate the logical casesL27–27

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

  1. L27
    cases hi_right_right_right
06Use earlier factsL28–28

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

  1. L28
    exact hi_right_right_right_right

Library-wide reading audit

Original exact command ledger · 28 lines
  1. 0001intro p
  2. 0002intro B
  3. 0003intro C
  4. 0004intro a
  5. 0005intro v
  6. 0006intro htable
  7. 0007intro ha
  8. 0008intro hn
  9. 0009intro hat
  10. 0010have hi : ((exists pfa_gap_inverse_nonzero_valueinput. pfa_gap_inverse_nonzero_valueinput + S (a) = (p)) /\ (((exists pfa_gap_inverse_nonzero_valueoutput. pfa_gap_inverse_nonzero_valueoutput + S (v) = (p)) /\ ((((a) = 0 /\ (v) = 0) \/ (((~((a) = 0)) /\ ((((exists pfa_gap_inverse_nonzero_valuenonzeromultiplicationleft. pfa_gap_inverse_nonzero_valuenonzeromultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverse_nonzero_valuenonzeromultiplicationright. pfa_gap_inverse_nonzero_valuenonzeromultiplicationright + S (v) = (p)) /\ ((((exists pfa_gap_inverse_nonzero_valuenonzeromultiplicationresultbound. pfa_gap_inverse_nonzero_valuenonzeromultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_nonzero_valuenonzeromultiplicationresultcongruence pfa_offset_right_inverse_nonzero_valuenonzeromultiplicationresultcongruence. ((a) * (v)) + (p) * pfa_offset_left_inverse_nonzero_valuenonzeromultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_nonzero_valuenonzeromultiplicationresultcongruence)))))))))))))))))
  11. 0011specialize prime_field_inverse_table_lookup (p)
  12. 0012specialize prime_field_inverse_table_lookup (B)
  13. 0013specialize prime_field_inverse_table_lookup (C)
  14. 0014specialize prime_field_inverse_table_lookup (a)
  15. 0015specialize prime_field_inverse_table_lookup (v)
  16. 0016apply prime_field_inverse_table_lookup
  17. 0017exact htable
  18. 0018exact ha
  19. 0019exact hat
  20. 0020cases hi
  21. 0021cases hi_right
  22. 0022cases hi_right_right
  23. 0023cases hi_right_right_left
  24. 0024exfalso
  25. 0025apply hn
  26. 0026exact hi_right_right_left_left
  27. 0027cases hi_right_right_right
  28. 0028exact hi_right_right_right_right