FP0007

prime_field_residue_modulus_zero

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

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

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. (~((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))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 9 exact native proof lines.

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

Proof neighborhood

Direct dependencies

FP0002 prime_field_zero_below_prime zero_add Stable theorem; checked-use authorized

Direct dependents

FP0056 prime_field_characteristic_exact

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

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.

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 exact 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]