FP0007

prime_field_residue_modulus_zero

The modulus itself has canonical residue zero, the characteristic-p boundary.

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. Prime(p) → CanonicalModularResidue(p,p,0)

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

Definition DAG

CanonicalModularResidue(m,a,r) · 1Prime(p) · 1

Actual proof prerequisites

prime_field_zero_below_primezero_add · checked external prerequisite
Original expanded first-order statement 
forall p. (~((p) = 1) /\ forall pfa_factor_left_modulus_domain pfa_factor_right_modulus_domain. (p) = pfa_factor_left_modulus_domain * pfa_factor_right_modulus_domain -> pfa_factor_left_modulus_domain = 1 \/ pfa_factor_right_modulus_domain = 1) -> (((exists pfa_gap_modulus_zerobound. pfa_gap_modulus_zerobound + S (0) = (p)) /\ ((exists pfa_offset_left_modulus_zerocongruence pfa_offset_right_modulus_zerocongruence. (p) + (p) * pfa_offset_left_modulus_zerocongruence = (0) + (p) * pfa_offset_right_modulus_zerocongruence))))

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 · 5 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.

Named ingredients (1)
prime field zero below prime
01Fix variables and assumptionsL1–2

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

  1. L1
    intro p
  2. L2
    intro hp
02Separate the logical casesL3–3

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

  1. L3
    split
03Use earlier factsL4–6

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

  1. L4
    specialize prime_field_zero_below_prime (p)
  2. L5
    apply prime_field_zero_below_prime
  3. L6
    exact hp
04Construct an explicit witnessL7–8

Supply the displayed value, then prove that it has the required property.

  1. L7
    exists 0
  2. L8
    exists 1
05Calculate and transport equalitiesL9–9

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

  1. L9
    simp [zero_add]

Library-wide reading audit

Original defined command ledger · 9 lines
  1. 0001intro p
  2. 0002intro hp
  3. 0003split
  4. 0004specialize prime_field_zero_below_prime (p)
  5. 0005apply prime_field_zero_below_prime
  6. 0006exact hp
  7. 0007exists 0
  8. 0008exists 1
  9. 0009simp [zero_add]