FP0028

prime_field_residue_multiply

The natural-number residue map preserves actual canonical multiply.

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. ∀ b. ∀ x. ∀ y. ∀ z. CanonicalModularResidue(p,a,x) → CanonicalModularResidue(p,b,y) → FpMul(p,x,y,z) → CanonicalModularResidue(p,a · b,z)

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

Definition DAG

CanonicalModularResidue(m,a,r) · 3FpMul(p,a,b,c) · 1

Actual proof prerequisites

mod_eq_trans · checked external prerequisitemod_eq_mul · checked external prerequisite
Original expanded first-order statement 
forall p a b x y z. (((exists pfa_gap_multiplynatural_leftbound. pfa_gap_multiplynatural_leftbound + S (x) = (p)) /\ ((exists pfa_offset_left_multiplynatural_leftcongruence pfa_offset_right_multiplynatural_leftcongruence. (a) + (p) * pfa_offset_left_multiplynatural_leftcongruence = (x) + (p) * pfa_offset_right_multiplynatural_leftcongruence)))) -> (((exists pfa_gap_multiplynatural_rightbound. pfa_gap_multiplynatural_rightbound + S (y) = (p)) /\ ((exists pfa_offset_left_multiplynatural_rightcongruence pfa_offset_right_multiplynatural_rightcongruence. (b) + (p) * pfa_offset_left_multiplynatural_rightcongruence = (y) + (p) * pfa_offset_right_multiplynatural_rightcongruence)))) -> (((exists pfa_gap_multiplynatural_resultleft. pfa_gap_multiplynatural_resultleft + S (x) = (p)) /\ (((exists pfa_gap_multiplynatural_resultright. pfa_gap_multiplynatural_resultright + S (y) = (p)) /\ ((((exists pfa_gap_multiplynatural_resultresultbound. pfa_gap_multiplynatural_resultresultbound + S (z) = (p)) /\ ((exists pfa_offset_left_multiplynatural_resultresultcongruence pfa_offset_right_multiplynatural_resultresultcongruence. ((x) * (y)) + (p) * pfa_offset_left_multiplynatural_resultresultcongruence = (z) + (p) * pfa_offset_right_multiplynatural_resultresultcongruence))))))))) -> (((exists pfa_gap_multiplynatural_homomorphismbound. pfa_gap_multiplynatural_homomorphismbound + S (z) = (p)) /\ ((exists pfa_offset_left_multiplynatural_homomorphismcongruence pfa_offset_right_multiplynatural_homomorphismcongruence. (a * b) + (p) * pfa_offset_left_multiplynatural_homomorphismcongruence = (z) + (p) * pfa_offset_right_multiplynatural_homomorphismcongruence))))

Complete tactic proof in conservative notation

All 30 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

30 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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–9

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

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro x
  5. L5
    intro y
  6. L6
    intro z
  7. L7
    intro ha
  8. L8
    intro hb
  9. L9
    intro hz
02Separate the logical casesL10–15

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L10
    cases ha
  2. L11
    cases hb
  3. L12
    cases hz
  4. L13
    cases hz_right
  5. L14
    cases hz_right_right
  6. L15
    split
03Use earlier factsL16–25

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

  1. L16
    exact hz_right_right_left
  2. L17
    specialize mod_eq_trans (p)
  3. L18
    specialize mod_eq_trans (a * b)
  4. L19
    specialize mod_eq_trans (x * y)
  5. L20
    specialize mod_eq_trans (z)
  6. L21
    apply mod_eq_trans
  7. L22
    specialize mod_eq_mul (p)
  8. L23
    specialize mod_eq_mul (a)
  9. L24
    specialize mod_eq_mul (x)
  10. L25
    specialize mod_eq_mul (b)
04Use earlier factsL26–30

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

  1. L26
    specialize mod_eq_mul (y)
  2. L27
    apply mod_eq_mul
  3. L28
    exact ha_right
  4. L29
    exact hb_right
  5. L30
    exact hz_right_right_right

Library-wide reading audit

Original defined command ledger · 30 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro b
  4. 0004intro x
  5. 0005intro y
  6. 0006intro z
  7. 0007intro ha
  8. 0008intro hb
  9. 0009intro hz
  10. 0010cases ha
  11. 0011cases hb
  12. 0012cases hz
  13. 0013cases hz_right
  14. 0014cases hz_right_right
  15. 0015split
  16. 0016exact hz_right_right_left
  17. 0017specialize mod_eq_trans (p)
  18. 0018specialize mod_eq_trans (a * b)
  19. 0019specialize mod_eq_trans (x * y)
  20. 0020specialize mod_eq_trans (z)
  21. 0021apply mod_eq_trans
  22. 0022specialize mod_eq_mul (p)
  23. 0023specialize mod_eq_mul (a)
  24. 0024specialize mod_eq_mul (x)
  25. 0025specialize mod_eq_mul (b)
  26. 0026specialize mod_eq_mul (y)
  27. 0027apply mod_eq_mul
  28. 0028exact ha_right
  29. 0029exact hb_right
  30. 0030exact hz_right_right_right