FP0029

prime_field_positive_below_modulus_not_zero

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

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. ∀ n. Lt(n,p) → ¬n = 0 → ¬CanonicalModularResidue(p,n,0)

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

Definition DAG

CanonicalModularResidue(m,a,r) · 1Lt(a,b) · 1

Actual proof prerequisites

prime_field_residue_bounded_value
Original expanded first-order 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))))

Complete tactic proof in conservative notation

All 15 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

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.

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 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 defined 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