FP0019

prime_field_multiply_zero_left

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Zero is absorbing on both sides of multiplication.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

Exact expanded first-order arithmetic 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)))))))))

Constructive proof overview

Generated structural guide

Zero is absorbing on both sides of multiplication.

The unchanged tactic script uses 2 declared prerequisites and contains 14 exact native proof lines.

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

FP000F prime_field_multiply_commutative FP0018 prime_field_multiply_zero_right

Direct dependents

FP0021 prime_field_zero_has_no_multiplicative_inverse FP002A prime_field_arithmetic_laws

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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