FP0003

prime_field_residue_reflexive

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 bounded representative represents itself, including zero.

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. ∀ a. Lt(a,p) → CanonicalModularResidue(p,a,a)

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

Definition DAG

CanonicalModularResidue(m,a,r) · 1Lt(a,b) · 1

Actual proof prerequisites

mod_eq_refl · checked external prerequisite
Original expanded first-order statement 
forall p a. (exists pfa_gap_reflexive_bound. pfa_gap_reflexive_bound + S (a) = (p)) -> (((exists pfa_gap_reflexivebound. pfa_gap_reflexivebound + S (a) = (p)) /\ ((exists pfa_offset_left_reflexivecongruence pfa_offset_right_reflexivecongruence. (a) + (p) * pfa_offset_left_reflexivecongruence = (a) + (p) * pfa_offset_right_reflexivecongruence))))

Complete tactic proof in conservative notation

All 8 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

8 script commands · 3 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 assumptionsL1–3

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro ha
02Separate the logical casesL4–4

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

  1. L4
    split
03Use earlier factsL5–8

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

  1. L5
    exact ha
  2. L6
    specialize mod_eq_refl (p)
  3. L7
    specialize mod_eq_refl (a)
  4. L8
    apply mod_eq_refl

Library-wide reading audit

Original defined command ledger · 8 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro ha
  4. 0004split
  5. 0005exact ha
  6. 0006specialize mod_eq_refl (p)
  7. 0007specialize mod_eq_refl (a)
  8. 0008apply mod_eq_refl