FP0006

prime_field_residue_bounded_value

No two distinct representatives below p denote the same residue class.

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. ∀ a. ∀ r. Lt(a,p) → CanonicalModularResidue(p,a,r) → r = a

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

Definition DAG

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

Actual proof prerequisites

binary_canonical_residue_functional · checked external prerequisiteprime_field_residue_reflexive
Original expanded first-order statement 
forall p a r. (exists pfa_gap_bounded_value. pfa_gap_bounded_value + S (a) = (p)) -> (((exists pfa_gap_bounded_resultbound. pfa_gap_bounded_resultbound + S (r) = (p)) /\ ((exists pfa_offset_left_bounded_resultcongruence pfa_offset_right_bounded_resultcongruence. (a) + (p) * pfa_offset_left_bounded_resultcongruence = (r) + (p) * pfa_offset_right_bounded_resultcongruence)))) -> r = a

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 · 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 residue reflexive
01Fix variables and assumptionsL1–5

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

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro r
  4. L4
    intro ha
  5. L5
    intro hr
02Use earlier factsL6–15

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

  1. L6
    specialize binary_canonical_residue_functional (p)
  2. L7
    specialize binary_canonical_residue_functional (a)
  3. L8
    specialize binary_canonical_residue_functional (r)
  4. L9
    specialize binary_canonical_residue_functional (a)
  5. L10
    apply binary_canonical_residue_functional
  6. L11
    exact hr
  7. L12
    specialize prime_field_residue_reflexive (p)
  8. L13
    specialize prime_field_residue_reflexive (a)
  9. L14
    apply prime_field_residue_reflexive
  10. L15
    exact ha

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro r
  4. 0004intro ha
  5. 0005intro hr
  6. 0006specialize binary_canonical_residue_functional (p)
  7. 0007specialize binary_canonical_residue_functional (a)
  8. 0008specialize binary_canonical_residue_functional (r)
  9. 0009specialize binary_canonical_residue_functional (a)
  10. 0010apply binary_canonical_residue_functional
  11. 0011exact hr
  12. 0012specialize prime_field_residue_reflexive (p)
  13. 0013specialize prime_field_residue_reflexive (a)
  14. 0014apply prime_field_residue_reflexive
  15. 0015exact ha