FP0047

prime_field_inverse_table_nonzero

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

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

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

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. ∀ v. FpInvPrefix(p,B,C,p) → Lt(a,p) → ¬a = 0 → BetaAt(B,C,a,v) → FpMul(p,a,v,1)

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

Definition DAG

FpMul(p,a,b,c) · 1FpZeroExtendedInv(p,a,b) · 1FpInvPrefix(p,b,c,l) · 1Lt(a,b) · 1BetaAt(b,c,i,x) · 1

Actual proof prerequisites

prime_field_inverse_table_lookup
Original expanded first-order 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)))))))))

Complete tactic proof in conservative notation

All 28 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

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.

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 (1)
prime field inverse table lookup
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(p,a,v)Original native command in the exact edition
  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 defined 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 : FpZeroExtendedInv(p,a,v)
  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