FP0019

prime_field_multiply_zero_left

Zero is absorbing on both sides of multiplication.

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. Prime(p) → Lt(a,p) → FpMul(p,0,a,0)

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

Definition DAG

FpMul(p,a,b,c) · 1Lt(a,b) · 1Prime(p) · 1

Actual proof prerequisites

prime_field_multiply_commutativeprime_field_multiply_zero_right
Original expanded first-order statement 
forall p a. (~((p) = 1) /\ forall pfa_factor_left_zero_multiply_domain pfa_factor_right_zero_multiply_domain. (p) = pfa_factor_left_zero_multiply_domain * pfa_factor_right_zero_multiply_domain -> pfa_factor_left_zero_multiply_domain = 1 \/ pfa_factor_right_zero_multiply_domain = 1) -> (exists pfa_gap_zero_multiply_bound. pfa_gap_zero_multiply_bound + S (a) = (p)) -> (((exists pfa_gap_zero_multiplyleft. pfa_gap_zero_multiplyleft + S (0) = (p)) /\ (((exists pfa_gap_zero_multiplyright. pfa_gap_zero_multiplyright + S (a) = (p)) /\ ((((exists pfa_gap_zero_multiplyresultbound. pfa_gap_zero_multiplyresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_zero_multiplyresultcongruence pfa_offset_right_zero_multiplyresultcongruence. ((0) * (a)) + (p) * pfa_offset_left_zero_multiplyresultcongruence = (0) + (p) * pfa_offset_right_zero_multiplyresultcongruence)))))))))

Complete tactic proof in conservative notation

All 14 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

14 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 (2)
prime field multiply commutativeprime field multiply zero right
01Fix variables and assumptionsL1–4

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

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro hp
  4. L4
    intro ha
02Use earlier factsL5–14

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

  1. L5
    specialize prime_field_multiply_commutative (p)
  2. L6
    specialize prime_field_multiply_commutative (a)
  3. L7
    specialize prime_field_multiply_commutative (0)
  4. L8
    specialize prime_field_multiply_commutative (0)
  5. L9
    apply prime_field_multiply_commutative
  6. L10
    specialize prime_field_multiply_zero_right (p)
  7. L11
    specialize prime_field_multiply_zero_right (a)
  8. L12
    apply prime_field_multiply_zero_right
  9. L13
    exact hp
  10. L14
    exact ha

Library-wide reading audit

Original defined command ledger · 14 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro hp
  4. 0004intro ha
  5. 0005specialize prime_field_multiply_commutative (p)
  6. 0006specialize prime_field_multiply_commutative (a)
  7. 0007specialize prime_field_multiply_commutative (0)
  8. 0008specialize prime_field_multiply_commutative (0)
  9. 0009apply prime_field_multiply_commutative
  10. 0010specialize prime_field_multiply_zero_right (p)
  11. 0011specialize prime_field_multiply_zero_right (a)
  12. 0012apply prime_field_multiply_zero_right
  13. 0013exact hp
  14. 0014exact ha