FP0029

prime_field_positive_below_modulus_not_zero

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

No positive natural below p has residue zero; the modulus boundary is sharp.

Exact expanded first-order arithmetic statement

forall p n. (exists pfa_gap_small_nonzero_bound. pfa_gap_small_nonzero_bound + S (n) = (p)) -> ~(n = 0) -> ~(((exists pfa_gap_small_nonzerobound. pfa_gap_small_nonzerobound + S (0) = (p)) /\ ((exists pfa_offset_left_small_nonzerocongruence pfa_offset_right_small_nonzerocongruence. (n) + (p) * pfa_offset_left_small_nonzerocongruence = (0) + (p) * pfa_offset_right_small_nonzerocongruence))))

Constructive proof overview

Generated structural guide

No positive natural below p has residue zero; the modulus boundary is sharp.

The unchanged tactic script uses 1 declared prerequisite and contains 15 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

FP0006 prime_field_residue_bounded_value

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. 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

15 script commands · 2 reading checkpoints · 1 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 residue bounded value
01Fix variables and assumptionsL1–5

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro hn
  4. L4
    intro hpositive
  5. L5
    intro hzero
02Establish heqL6–15

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime field residue bounded value.

  1. L6
    have heq : 0 = n
  2. L7
    specialize prime_field_residue_bounded_value (p)
  3. L8
    specialize prime_field_residue_bounded_value (n)
  4. L9
    specialize prime_field_residue_bounded_value (0)
  5. L10
    apply prime_field_residue_bounded_value
  6. L11
    exact hn
  7. L12
    exact hzero
  8. L13
    apply hpositive
  9. L14
    symm
  10. L15
    exact heq

Library-wide reading audit

Original exact command ledger · 15 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro hn
  4. 0004intro hpositive
  5. 0005intro hzero
  6. 0006have heq : 0 = n
  7. 0007specialize prime_field_residue_bounded_value (p)
  8. 0008specialize prime_field_residue_bounded_value (n)
  9. 0009specialize prime_field_residue_bounded_value (0)
  10. 0010apply prime_field_residue_bounded_value
  11. 0011exact hn
  12. 0012exact hzero
  13. 0013apply hpositive
  14. 0014symm
  15. 0015exact heq