FP003E

prime_field_negate_table_lookup

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

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. FpNegPrefix(p,B,C,p) → Lt(a,p) → BetaAt(B,C,a,v) → FpAdd(p,a,v,0)

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

Definition DAG

FpAdd(p,a,b,c) · 2FpNegPrefix(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_negateunary_lookup_table. (exists pfa_gap_negateunary_lookup_tableprefix. pfa_gap_negateunary_lookup_tableprefix + S (pft_index_negateunary_lookup_table) = (p)) -> exists pft_value_negateunary_lookup_table. (((((exists ff_h_pft_negateunary_lookup_tablepointentry. ff_h_pft_negateunary_lookup_tablepointentry + S (pft_value_negateunary_lookup_table) = S ((S (pft_index_negateunary_lookup_table)) * C)) /\ exists ff_q_pft_negateunary_lookup_tablepointentry. B = ff_q_pft_negateunary_lookup_tablepointentry * S ((S (pft_index_negateunary_lookup_table)) * C) + (pft_value_negateunary_lookup_table))) /\ ((((exists pfa_gap_negateunary_lookup_tablepointvalueadditionleft. pfa_gap_negateunary_lookup_tablepointvalueadditionleft + S (pft_index_negateunary_lookup_table) = (p)) /\ (((exists pfa_gap_negateunary_lookup_tablepointvalueadditionright. pfa_gap_negateunary_lookup_tablepointvalueadditionright + S (pft_value_negateunary_lookup_table) = (p)) /\ ((((exists pfa_gap_negateunary_lookup_tablepointvalueadditionresultbound. pfa_gap_negateunary_lookup_tablepointvalueadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negateunary_lookup_tablepointvalueadditionresultcongruence pfa_offset_right_negateunary_lookup_tablepointvalueadditionresultcongruence. ((pft_index_negateunary_lookup_table) + (pft_value_negateunary_lookup_table)) + (p) * pfa_offset_left_negateunary_lookup_tablepointvalueadditionresultcongruence = (0) + (p) * pfa_offset_right_negateunary_lookup_tablepointvalueadditionresultcongruence))))))))))))) -> (exists pfa_gap_negateunary_lookup_bound. pfa_gap_negateunary_lookup_bound + S (a) = (p)) -> (((exists ff_h_pft_negateunary_lookup_at. ff_h_pft_negateunary_lookup_at + S (v) = S ((S (a)) * C)) /\ exists ff_q_pft_negateunary_lookup_at. B = ff_q_pft_negateunary_lookup_at * S ((S (a)) * C) + (v))) -> (((exists pfa_gap_negateunary_lookup_graphadditionleft. pfa_gap_negateunary_lookup_graphadditionleft + S (a) = (p)) /\ (((exists pfa_gap_negateunary_lookup_graphadditionright. pfa_gap_negateunary_lookup_graphadditionright + S (v) = (p)) /\ ((((exists pfa_gap_negateunary_lookup_graphadditionresultbound. pfa_gap_negateunary_lookup_graphadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negateunary_lookup_graphadditionresultcongruence pfa_offset_right_negateunary_lookup_graphadditionresultcongruence. ((a) + (v)) + (p) * pfa_offset_left_negateunary_lookup_graphadditionresultcongruence = (0) + (p) * pfa_offset_right_negateunary_lookup_graphadditionresultcongruence)))))))))

Complete tactic proof in conservative notation

All 26 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

26 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) ∧ FpAdd(p,a,w,0)Definitions: BetaAt(B,C,a,w)FpAdd(p,a,w,0)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 equalitiesL25–25

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
06Use earlier factsL26–26

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

  1. L26
    exact hpoint_witness_right

Library-wide reading audit

Original defined command ledger · 26 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) ∧ FpAdd(p,a,w,0)
  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. 0026exact hpoint_witness_right