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.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

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

Exact expanded first-order arithmetic 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))

Constructive proof overview

Generated structural guide

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

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

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

Proof neighborhood

Direct dependencies

prime_nonzero Alpha theorem; checked-use authorized one_le_of_ne_zero Alpha theorem; checked-use authorized

Direct dependents

FP0007 prime_field_residue_modulus_zero FP0014 prime_field_add_zero_right FP0018 prime_field_multiply_zero_right FP001B prime_field_negate_exists FP002A prime_field_arithmetic_laws FP002D prime_field_zero_extended_inverse_exists FP0046 prime_field_inverse_table_zero FP004F prime_field_unit_trace_residue

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. The literal dependency-closed bundle is checked by original HA and the independently compiled Lean verifier. Public delivery grants no Alpha checked-use authority or Stable membership.

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.

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 exact 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