FP000D

prime_field_multiply_functional

The actual bounded multiply graph is functional without any field-law premise.

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. ∀ c. ∀ d. FpMul(p,a,b,c) → FpMul(p,a,b,d) → c = d

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

Definition DAG

FpMul(p,a,b,c) · 2

Actual proof prerequisites

binary_canonical_residue_functional · checked external prerequisite
Original expanded first-order statement 
forall p a b c d. (((exists pfa_gap_multiplyfirstleft. pfa_gap_multiplyfirstleft + S (a) = (p)) /\ (((exists pfa_gap_multiplyfirstright. pfa_gap_multiplyfirstright + S (b) = (p)) /\ ((((exists pfa_gap_multiplyfirstresultbound. pfa_gap_multiplyfirstresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_multiplyfirstresultcongruence pfa_offset_right_multiplyfirstresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplyfirstresultcongruence = (c) + (p) * pfa_offset_right_multiplyfirstresultcongruence))))))))) -> (((exists pfa_gap_multiplysecondleft. pfa_gap_multiplysecondleft + S (a) = (p)) /\ (((exists pfa_gap_multiplysecondright. pfa_gap_multiplysecondright + S (b) = (p)) /\ ((((exists pfa_gap_multiplysecondresultbound. pfa_gap_multiplysecondresultbound + S (d) = (p)) /\ ((exists pfa_offset_left_multiplysecondresultcongruence pfa_offset_right_multiplysecondresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplysecondresultcongruence = (d) + (p) * pfa_offset_right_multiplysecondresultcongruence))))))))) -> c = d

Complete tactic proof in conservative notation

All 18 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

18 script commands · 3 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 assumptionsL1–7

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 c
  5. L5
    intro d
  6. L6
    intro hc
  7. L7
    intro hd
02Separate the logical casesL8–11

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

  1. L8
    cases hc
  2. L9
    cases hc_right
  3. L10
    cases hd
  4. L11
    cases hd_right
03Use earlier factsL12–18

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

  1. L12
    specialize binary_canonical_residue_functional (p)
  2. L13
    specialize binary_canonical_residue_functional (a * b)
  3. L14
    specialize binary_canonical_residue_functional (c)
  4. L15
    specialize binary_canonical_residue_functional (d)
  5. L16
    apply binary_canonical_residue_functional
  6. L17
    exact hc_right_right
  7. L18
    exact hd_right_right

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro b
  4. 0004intro c
  5. 0005intro d
  6. 0006intro hc
  7. 0007intro hd
  8. 0008cases hc
  9. 0009cases hc_right
  10. 0010cases hd
  11. 0011cases hd_right
  12. 0012specialize binary_canonical_residue_functional (p)
  13. 0013specialize binary_canonical_residue_functional (a * b)
  14. 0014specialize binary_canonical_residue_functional (c)
  15. 0015specialize binary_canonical_residue_functional (d)
  16. 0016apply binary_canonical_residue_functional
  17. 0017exact hc_right_right
  18. 0018exact hd_right_right