FP004A

prime_field_enumeration_value

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Every decoded entry of the identity enumeration equals its actual index.

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

Exact expanded first-order arithmetic 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

Constructive proof overview

Generated structural guide

Every decoded entry of the identity enumeration equals its actual index.

The unchanged tactic script uses 1 declared prerequisite and contains 18 exact native proof lines.

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

Proof neighborhood

Direct dependencies

beta_at_unique Stable theorem; checked-use authorized

Direct dependents

FP004B prime_field_enumeration_is_bijection

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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