FP0040

prime_field_inverse_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.

Every decoded inverse table lookup has its exact unary meaning, including the inverse-at-zero convention.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

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) → BetaAt(B,C,a,v) → FpZeroExtendedInv(p,a,v)

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

Definition DAG

FpZeroExtendedInv(p,a,b) · 2FpInvPrefix(p,b,c,l) · 1Lt(a,b) · 1BetaAt(b,c,i,x) · 2

Actual proof prerequisites

beta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall p B C a v. (forall pft_index_inverseunary_lookup_table. (exists pfa_gap_inverseunary_lookup_tableprefix. pfa_gap_inverseunary_lookup_tableprefix + S (pft_index_inverseunary_lookup_table) = (p)) -> exists pft_value_inverseunary_lookup_table. (((((exists ff_h_pft_inverseunary_lookup_tablepointentry. ff_h_pft_inverseunary_lookup_tablepointentry + S (pft_value_inverseunary_lookup_table) = S ((S (pft_index_inverseunary_lookup_table)) * C)) /\ exists ff_q_pft_inverseunary_lookup_tablepointentry. B = ff_q_pft_inverseunary_lookup_tablepointentry * S ((S (pft_index_inverseunary_lookup_table)) * C) + (pft_value_inverseunary_lookup_table))) /\ ((((exists pfa_gap_inverseunary_lookup_tablepointvalueinput. pfa_gap_inverseunary_lookup_tablepointvalueinput + S (pft_index_inverseunary_lookup_table) = (p)) /\ (((exists pfa_gap_inverseunary_lookup_tablepointvalueoutput. pfa_gap_inverseunary_lookup_tablepointvalueoutput + S (pft_value_inverseunary_lookup_table) = (p)) /\ ((((pft_index_inverseunary_lookup_table) = 0 /\ (pft_value_inverseunary_lookup_table) = 0) \/ (((~((pft_index_inverseunary_lookup_table) = 0)) /\ ((((exists pfa_gap_inverseunary_lookup_tablepointvaluenonzeromultiplicationleft. pfa_gap_inverseunary_lookup_tablepointvaluenonzeromultiplicationleft + S (pft_index_inverseunary_lookup_table) = (p)) /\ (((exists pfa_gap_inverseunary_lookup_tablepointvaluenonzeromultiplicationright. pfa_gap_inverseunary_lookup_tablepointvaluenonzeromultiplicationright + S (pft_value_inverseunary_lookup_table) = (p)) /\ ((((exists pfa_gap_inverseunary_lookup_tablepointvaluenonzeromultiplicationresultbound. pfa_gap_inverseunary_lookup_tablepointvaluenonzeromultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverseunary_lookup_tablepointvaluenonzeromultiplicationresultcongruence pfa_offset_right_inverseunary_lookup_tablepointvaluenonzeromultiplicationresultcongruence. ((pft_index_inverseunary_lookup_table) * (pft_value_inverseunary_lookup_table)) + (p) * pfa_offset_left_inverseunary_lookup_tablepointvaluenonzeromultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverseunary_lookup_tablepointvaluenonzeromultiplicationresultcongruence)))))))))))))))))))))) -> (exists pfa_gap_inverseunary_lookup_bound. pfa_gap_inverseunary_lookup_bound + S (a) = (p)) -> (((exists ff_h_pft_inverseunary_lookup_at. ff_h_pft_inverseunary_lookup_at + S (v) = S ((S (a)) * C)) /\ exists ff_q_pft_inverseunary_lookup_at. B = ff_q_pft_inverseunary_lookup_at * S ((S (a)) * C) + (v))) -> (((exists pfa_gap_inverseunary_lookup_graphinput. pfa_gap_inverseunary_lookup_graphinput + S (a) = (p)) /\ (((exists pfa_gap_inverseunary_lookup_graphoutput. pfa_gap_inverseunary_lookup_graphoutput + S (v) = (p)) /\ ((((a) = 0 /\ (v) = 0) \/ (((~((a) = 0)) /\ ((((exists pfa_gap_inverseunary_lookup_graphnonzeromultiplicationleft. pfa_gap_inverseunary_lookup_graphnonzeromultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverseunary_lookup_graphnonzeromultiplicationright. pfa_gap_inverseunary_lookup_graphnonzeromultiplicationright + S (v) = (p)) /\ ((((exists pfa_gap_inverseunary_lookup_graphnonzeromultiplicationresultbound. pfa_gap_inverseunary_lookup_graphnonzeromultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverseunary_lookup_graphnonzeromultiplicationresultcongruence pfa_offset_right_inverseunary_lookup_graphnonzeromultiplicationresultcongruence. ((a) * (v)) + (p) * pfa_offset_left_inverseunary_lookup_graphnonzeromultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverseunary_lookup_graphnonzeromultiplicationresultcongruence))))))))))))))))))

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 · 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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–8

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 hat
02Establish hpointL9–12

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

  1. L9
    have hpoint : ∃ w. BetaAt(B,C,a,w) ∧ FpZeroExtendedInv(p,a,w)Definitions: BetaAt(B,C,a,w)FpZeroExtendedInv(p,a,w)Original native command in the exact edition
  2. L10
    specialize htable (a)
  3. L11
    apply htable
  4. L12
    exact ha
03Separate the logical casesL13–14

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

  1. L13
    cases hpoint
  2. L14
    cases hpoint_witness
04Establish heqL15–24

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

  1. L15
    have heq : x = v
  2. L16
    specialize beta_at_unique (B)
  3. L17
    specialize beta_at_unique (C)
  4. L18
    specialize beta_at_unique (a)
  5. L19
    specialize beta_at_unique (x)
  6. L20
    specialize beta_at_unique (v)
  7. L21
    apply beta_at_unique
  8. L22
    exact hpoint_witness_left
  9. L23
    exact hat
  10. L24
    rewrite heq at hpoint_witness_right
05Calculate and transport equalitiesL25–27

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

  1. L25
    rewrite heq at hpoint_witness_right
  2. L26
    rewrite heq at hpoint_witness_right
  3. L27
    rewrite heq at hpoint_witness_right
06Use earlier factsL28–28

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

  1. L28
    exact hpoint_witness_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 hat
  9. 0009have hpoint : ∃ w. BetaAt(B,C,a,w) ∧ FpZeroExtendedInv(p,a,w)
  10. 0010specialize htable (a)
  11. 0011apply htable
  12. 0012exact ha
  13. 0013cases hpoint
  14. 0014cases hpoint_witness
  15. 0015have heq : x = v
  16. 0016specialize beta_at_unique (B)
  17. 0017specialize beta_at_unique (C)
  18. 0018specialize beta_at_unique (a)
  19. 0019specialize beta_at_unique (x)
  20. 0020specialize beta_at_unique (v)
  21. 0021apply beta_at_unique
  22. 0022exact hpoint_witness_left
  23. 0023exact hat
  24. 0024rewrite heq at hpoint_witness_right
  25. 0025rewrite heq at hpoint_witness_right
  26. 0026rewrite heq at hpoint_witness_right
  27. 0027rewrite heq at hpoint_witness_right
  28. 0028exact hpoint_witness_right