FP003A

prime_field_add_table_reflect

Every genuine canonical add result occurs at its actual finite table index.

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. ∀ b. ∀ v. FpAddPrefix(p,B,C,p · p) → FpAdd(p,a,b,v) → BetaAt(B,C,a · p + b,v)

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

Definition DAG

FpAdd(p,a,b,c) · 2FpAddPrefix(p,b,c,l) · 1BetaAt(b,c,i,x) · 2

Actual proof prerequisites

beta_at_exists · checked external prerequisiteprime_field_add_table_lookupprime_field_add_functional
Original expanded first-order statement 
forall p B C a b v. (forall pft_index_addreflect_table. (exists pfa_gap_addreflect_tableprefix. pfa_gap_addreflect_tableprefix + S (pft_index_addreflect_table) = ((p) * (p))) -> exists pft_value_addreflect_table. (((((exists ff_h_pft_addreflect_tablepointentry. ff_h_pft_addreflect_tablepointentry + S (pft_value_addreflect_table) = S ((S (pft_index_addreflect_table)) * C)) /\ exists ff_q_pft_addreflect_tablepointentry. B = ff_q_pft_addreflect_tablepointentry * S ((S (pft_index_addreflect_table)) * C) + (pft_value_addreflect_table))) /\ ((exists pft_row_addreflect_tablepointvalue pft_column_addreflect_tablepointvalue. (((pft_index_addreflect_table) = pft_row_addreflect_tablepointvalue * (p) + pft_column_addreflect_tablepointvalue) /\ ((((exists pfa_gap_addreflect_tablepointvalueoperationleft. pfa_gap_addreflect_tablepointvalueoperationleft + S (pft_row_addreflect_tablepointvalue) = (p)) /\ (((exists pfa_gap_addreflect_tablepointvalueoperationright. pfa_gap_addreflect_tablepointvalueoperationright + S (pft_column_addreflect_tablepointvalue) = (p)) /\ ((((exists pfa_gap_addreflect_tablepointvalueoperationresultbound. pfa_gap_addreflect_tablepointvalueoperationresultbound + S (pft_value_addreflect_table) = (p)) /\ ((exists pfa_offset_left_addreflect_tablepointvalueoperationresultcongruence pfa_offset_right_addreflect_tablepointvalueoperationresultcongruence. ((pft_row_addreflect_tablepointvalue) + (pft_column_addreflect_tablepointvalue)) + (p) * pfa_offset_left_addreflect_tablepointvalueoperationresultcongruence = (pft_value_addreflect_table) + (p) * pfa_offset_right_addreflect_tablepointvalueoperationresultcongruence)))))))))))))))) -> (((exists pfa_gap_addreflect_graphleft. pfa_gap_addreflect_graphleft + S (a) = (p)) /\ (((exists pfa_gap_addreflect_graphright. pfa_gap_addreflect_graphright + S (b) = (p)) /\ ((((exists pfa_gap_addreflect_graphresultbound. pfa_gap_addreflect_graphresultbound + S (v) = (p)) /\ ((exists pfa_offset_left_addreflect_graphresultcongruence pfa_offset_right_addreflect_graphresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addreflect_graphresultcongruence = (v) + (p) * pfa_offset_right_addreflect_graphresultcongruence))))))))) -> (((exists ff_h_pft_addreflect_at. ff_h_pft_addreflect_at + S (v) = S ((S (a*p+b)) * C)) /\ exists ff_q_pft_addreflect_at. B = ff_q_pft_addreflect_at * S ((S (a*p+b)) * C) + (v)))

Complete tactic proof in conservative notation

All 40 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

40 script commands · 9 reading checkpoints · 3 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 (2)
prime field add table lookupprime field add functional
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 b
  6. L6
    intro v
  7. L7
    intro htable
  8. L8
    intro hop
02Separate the logical casesL9–10

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

  1. L9
    cases hop
  2. L10
    cases hop_right
03Establish hatL11–15

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

  1. L11
    have hat : ∃ w. BetaAt(B,C,a · p + b,w)Definitions: BetaAt(B,C,a · p + b,w)Original native command in the exact edition
  2. L12
    specialize beta_at_exists (B)
  3. L13
    specialize beta_at_exists (C)
  4. L14
    specialize beta_at_exists (a*p+b)
  5. L15
    apply beta_at_exists
04Separate the logical casesL16–16

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

  1. L16
    cases hat
05Establish houtL17–26

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

  1. L17
    have hout : FpAdd(p,a,b,x)Definitions: FpAdd(p,a,b,x)Original native command in the exact edition
  2. L18
    specialize prime_field_add_table_lookup (p)
  3. L19
    specialize prime_field_add_table_lookup (B)
  4. L20
    specialize prime_field_add_table_lookup (C)
  5. L21
    specialize prime_field_add_table_lookup (a)
  6. L22
    specialize prime_field_add_table_lookup (b)
  7. L23
    specialize prime_field_add_table_lookup (x)
  8. L24
    apply prime_field_add_table_lookup
  9. L25
    exact htable
  10. L26
    exact hop_left
06Use earlier factsL27–28

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

  1. L27
    exact hop_right_left
  2. L28
    exact hat_witness
07Establish heqL29–38

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

  1. L29
    have heq : x = v
  2. L30
    specialize prime_field_add_functional (p)
  3. L31
    specialize prime_field_add_functional (a)
  4. L32
    specialize prime_field_add_functional (b)
  5. L33
    specialize prime_field_add_functional (x)
  6. L34
    specialize prime_field_add_functional (v)
  7. L35
    apply prime_field_add_functional
  8. L36
    exact hout
  9. L37
    exact hop
  10. L38
    rewrite heq at hat_witness
08Calculate and transport equalitiesL39–39

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

  1. L39
    rewrite heq at hat_witness
09Use earlier factsL40–40

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

  1. L40
    exact hat_witness

Library-wide reading audit

Original defined command ledger · 40 lines
  1. 0001intro p
  2. 0002intro B
  3. 0003intro C
  4. 0004intro a
  5. 0005intro b
  6. 0006intro v
  7. 0007intro htable
  8. 0008intro hop
  9. 0009cases hop
  10. 0010cases hop_right
  11. 0011have hat : ∃ w. BetaAt(B,C,a · p + b,w)
  12. 0012specialize beta_at_exists (B)
  13. 0013specialize beta_at_exists (C)
  14. 0014specialize beta_at_exists (a*p+b)
  15. 0015apply beta_at_exists
  16. 0016cases hat
  17. 0017have hout : FpAdd(p,a,b,x)
  18. 0018specialize prime_field_add_table_lookup (p)
  19. 0019specialize prime_field_add_table_lookup (B)
  20. 0020specialize prime_field_add_table_lookup (C)
  21. 0021specialize prime_field_add_table_lookup (a)
  22. 0022specialize prime_field_add_table_lookup (b)
  23. 0023specialize prime_field_add_table_lookup (x)
  24. 0024apply prime_field_add_table_lookup
  25. 0025exact htable
  26. 0026exact hop_left
  27. 0027exact hop_right_left
  28. 0028exact hat_witness
  29. 0029have heq : x = v
  30. 0030specialize prime_field_add_functional (p)
  31. 0031specialize prime_field_add_functional (a)
  32. 0032specialize prime_field_add_functional (b)
  33. 0033specialize prime_field_add_functional (x)
  34. 0034specialize prime_field_add_functional (v)
  35. 0035apply prime_field_add_functional
  36. 0036exact hout
  37. 0037exact hop
  38. 0038rewrite heq at hat_witness
  39. 0039rewrite heq at hat_witness
  40. 0040exact hat_witness