FP0018

prime_field_multiply_zero_right

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

Multiplication by the actual zero representative produces zero.

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_multiply_zero_domain pfa_factor_right_multiply_zero_domain. (p) = pfa_factor_left_multiply_zero_domain * pfa_factor_right_multiply_zero_domain -> pfa_factor_left_multiply_zero_domain = 1 \/ pfa_factor_right_multiply_zero_domain = 1) -> (exists pfa_gap_multiply_zero_bound. pfa_gap_multiply_zero_bound + S (a) = (p)) -> (((exists pfa_gap_multiply_zeroleft. pfa_gap_multiply_zeroleft + S (a) = (p)) /\ (((exists pfa_gap_multiply_zeroright. pfa_gap_multiply_zeroright + S (0) = (p)) /\ ((((exists pfa_gap_multiply_zeroresultbound. pfa_gap_multiply_zeroresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_multiply_zeroresultcongruence pfa_offset_right_multiply_zeroresultcongruence. ((a) * (0)) + (p) * pfa_offset_left_multiply_zeroresultcongruence = (0) + (p) * pfa_offset_right_multiply_zeroresultcongruence)))))))))

Constructive proof overview

Generated structural guide

Multiplication by the actual zero representative produces zero.

The unchanged tactic script uses 2 declared prerequisites and contains 19 exact native proof lines.

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

Proof neighborhood

Direct dependencies

FP0002 prime_field_zero_below_prime FP0001 prime_field_mod_of_equal

Direct dependents

FP0019 prime_field_multiply_zero_left FP0026 prime_field_no_zero_divisors 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

19 script commands · 8 reading checkpoints · 1 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 zero below primeprime field mod of equal
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
02Establish hzL5–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime field zero below prime.

  1. L5
    have hz : exists pfa_gap_multiply_zero_canonical. pfa_gap_multiply_zero_canonical + S (0) = (p)
  2. L6
    specialize prime_field_zero_below_prime (p)
  3. L7
    apply prime_field_zero_below_prime
  4. L8
    exact hp
03Separate the logical casesL9–9

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

  1. L9
    split
04Use earlier factsL10–10

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

  1. L10
    exact ha
05Separate the logical casesL11–11

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

  1. L11
    split
06Use earlier factsL12–12

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

  1. L12
    exact hz
07Separate the logical casesL13–13

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

  1. L13
    split
08Use earlier factsL14–19

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

  1. L14
    exact hz
  2. L15
    specialize prime_field_mod_of_equal (p)
  3. L16
    specialize prime_field_mod_of_equal (a * 0)
  4. L17
    specialize prime_field_mod_of_equal (0)
  5. L18
    apply prime_field_mod_of_equal
  6. L19
    apply PA5

Library-wide reading audit

Original exact command ledger · 19 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro hp
  4. 0004intro ha
  5. 0005have hz : exists pfa_gap_multiply_zero_canonical. pfa_gap_multiply_zero_canonical + S (0) = (p)
  6. 0006specialize prime_field_zero_below_prime (p)
  7. 0007apply prime_field_zero_below_prime
  8. 0008exact hp
  9. 0009split
  10. 0010exact ha
  11. 0011split
  12. 0012exact hz
  13. 0013split
  14. 0014exact hz
  15. 0015specialize prime_field_mod_of_equal (p)
  16. 0016specialize prime_field_mod_of_equal (a * 0)
  17. 0017specialize prime_field_mod_of_equal (0)
  18. 0018apply prime_field_mod_of_equal
  19. 0019apply PA5