FP0002

prime_field_zero_below_prime

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.

Zero is a canonical representative at every prime, including two.

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. Prime(p) → Lt(0,p)

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

Definition DAG

Lt(a,b) · 1Prime(p) · 1

Actual proof prerequisites

prime_nonzero · checked external prerequisiteone_le_of_ne_zero · checked external prerequisite
Original expanded first-order statement 
forall p. (~((p) = 1) /\ forall pfa_factor_left_zero_domain pfa_factor_right_zero_domain. (p) = pfa_factor_left_zero_domain * pfa_factor_right_zero_domain -> pfa_factor_left_zero_domain = 1 \/ pfa_factor_right_zero_domain = 1) -> (exists pfa_gap_zero_bound. pfa_gap_zero_bound + S (0) = (p))

Complete tactic proof in conservative notation

All 9 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

9 script commands · 4 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 assumptionsL1–2

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

  1. L1
    intro p
  2. L2
    intro hp
02Use earlier factsL3–4

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

  1. L3
    specialize one_le_of_ne_zero (p)
  2. L4
    apply one_le_of_ne_zero
03Fix variables and assumptionsL5–5

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

  1. L5
    intro hz
04Use earlier factsL6–9

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

  1. L6
    specialize prime_nonzero (p)
  2. L7
    apply prime_nonzero
  3. L8
    exact hp
  4. L9
    exact hz

Library-wide reading audit

Original defined command ledger · 9 lines
  1. 0001intro p
  2. 0002intro hp
  3. 0003specialize one_le_of_ne_zero (p)
  4. 0004apply one_le_of_ne_zero
  5. 0005intro hz
  6. 0006specialize prime_nonzero (p)
  7. 0007apply prime_nonzero
  8. 0008exact hp
  9. 0009exact hz