FP0018

prime_field_multiply_zero_right

Public research checkpoint, not admitted to Alpha or Stable. Alpha v30 remains 3222 checked-use theorems; Stable remains 432. The on-demand Alpha Lean service does not yet expose these checkpoint theorems; their independently checked literal bundles and unchanged sources are available below.

Multiplication by the actual zero representative produces zero.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

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,a,0,0)

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

Definition DAG

FpMul(p,a,b,c) · 1Lt(a,b) · 2Prime(p) · 1

Actual proof prerequisites

prime_field_zero_below_primeprime_field_mod_of_equal
Original expanded first-order 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)))))))))

Complete tactic proof in conservative notation

All 19 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

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.

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 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
  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 defined command ledger · 19 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro hp
  4. 0004intro ha
  5. 0005have hz : Lt(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