FP004A

prime_field_enumeration_value

Every decoded entry of the identity enumeration equals its actual 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. ∀ i. ∀ a. IdentityMatrixSelector(b,c,p) → Lt(i,p) → BetaAt(b,c,i,a) → a = i

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

Definition DAG

IdentityMatrixSelector(b,c,l) · 1Lt(a,b) · 1BetaAt(b,c,i,x) · 1

Actual proof prerequisites

beta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall p b c i a. (forall pff_enumeration_index_enumeration_value_source. (exists pfa_gap_enumeration_value_sourcebound. pfa_gap_enumeration_value_sourcebound + S (pff_enumeration_index_enumeration_value_source) = (p)) -> (((exists ff_h_pft_enumeration_value_sourceentry. ff_h_pft_enumeration_value_sourceentry + S (pff_enumeration_index_enumeration_value_source) = S ((S (pff_enumeration_index_enumeration_value_source)) * c)) /\ exists ff_q_pft_enumeration_value_sourceentry. b = ff_q_pft_enumeration_value_sourceentry * S ((S (pff_enumeration_index_enumeration_value_source)) * c) + (pff_enumeration_index_enumeration_value_source)))) -> (exists pfa_gap_enumeration_value_bound. pfa_gap_enumeration_value_bound + S (i) = (p)) -> (((exists ff_h_pft_enumeration_value_at. ff_h_pft_enumeration_value_at + S (a) = S ((S (i)) * c)) /\ exists ff_q_pft_enumeration_value_at. b = ff_q_pft_enumeration_value_at * S ((S (i)) * c) + (a))) -> a = i

Complete tactic proof in conservative notation

All 18 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

18 script commands · 2 reading checkpoints · 0 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 i
  5. L5
    intro a
  6. L6
    intro henum
  7. L7
    intro hi
  8. L8
    intro hat
02Use earlier factsL9–18

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

  1. L9
    specialize beta_at_unique (b)
  2. L10
    specialize beta_at_unique (c)
  3. L11
    specialize beta_at_unique (i)
  4. L12
    specialize beta_at_unique (a)
  5. L13
    specialize beta_at_unique (i)
  6. L14
    apply beta_at_unique
  7. L15
    exact hat
  8. L16
    specialize henum (i)
  9. L17
    apply henum
  10. L18
    exact hi

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro i
  5. 0005intro a
  6. 0006intro henum
  7. 0007intro hi
  8. 0008intro hat
  9. 0009specialize beta_at_unique (b)
  10. 0010specialize beta_at_unique (c)
  11. 0011specialize beta_at_unique (i)
  12. 0012specialize beta_at_unique (a)
  13. 0013specialize beta_at_unique (i)
  14. 0014apply beta_at_unique
  15. 0015exact hat
  16. 0016specialize henum (i)
  17. 0017apply henum
  18. 0018exact hi