FP000E

prime_field_multiply_exists_unique

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.

Existence and uniqueness of canonical prime-field multiply are both proved.

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. ∀ b. Prime(p) → Lt(a,p) → Lt(b,p) → ∃ x. FpMul(p,a,b,x) ∧ (∀ y. FpMul(p,a,b,y) → y = x)

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

Definition DAG

FpMul(p,a,b,c) · 3Lt(a,b) · 2Prime(p) · 1

Actual proof prerequisites

prime_field_multiply_existsprime_field_multiply_functional
Original expanded first-order statement 
forall p a b. (~((p) = 1) /\ forall pfa_factor_left_multiplyunique_domain pfa_factor_right_multiplyunique_domain. (p) = pfa_factor_left_multiplyunique_domain * pfa_factor_right_multiplyunique_domain -> pfa_factor_left_multiplyunique_domain = 1 \/ pfa_factor_right_multiplyunique_domain = 1) -> (exists pfa_gap_multiplyunique_left. pfa_gap_multiplyunique_left + S (a) = (p)) -> (exists pfa_gap_multiplyunique_right. pfa_gap_multiplyunique_right + S (b) = (p)) -> exists c. (((exists pfa_gap_multiplychosenleft. pfa_gap_multiplychosenleft + S (a) = (p)) /\ (((exists pfa_gap_multiplychosenright. pfa_gap_multiplychosenright + S (b) = (p)) /\ ((((exists pfa_gap_multiplychosenresultbound. pfa_gap_multiplychosenresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_multiplychosenresultcongruence pfa_offset_right_multiplychosenresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplychosenresultcongruence = (c) + (p) * pfa_offset_right_multiplychosenresultcongruence))))))))) /\ forall d. (((exists pfa_gap_multiplycomparisonleft. pfa_gap_multiplycomparisonleft + S (a) = (p)) /\ (((exists pfa_gap_multiplycomparisonright. pfa_gap_multiplycomparisonright + S (b) = (p)) /\ ((((exists pfa_gap_multiplycomparisonresultbound. pfa_gap_multiplycomparisonresultbound + S (d) = (p)) /\ ((exists pfa_offset_left_multiplycomparisonresultcongruence pfa_offset_right_multiplycomparisonresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplycomparisonresultcongruence = (d) + (p) * pfa_offset_right_multiplycomparisonresultcongruence))))))))) -> d = c

Complete tactic proof in conservative notation

All 28 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

28 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 multiply existsprime field multiply functional
01Fix variables and assumptionsL1–6

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 hp
  5. L5
    intro ha
  6. L6
    intro hb
02Establish hcL7–14

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

  1. L7
    have hc : ∃ c. FpMul(p,a,b,c)Definitions: FpMul(p,a,b,c)Original native command in the exact edition
  2. L8
    specialize prime_field_multiply_exists (p)
  3. L9
    specialize prime_field_multiply_exists (a)
  4. L10
    specialize prime_field_multiply_exists (b)
  5. L11
    apply prime_field_multiply_exists
  6. L12
    exact hp
  7. L13
    exact ha
  8. L14
    exact hb
03Separate the logical casesL15–15

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

  1. L15
    cases hc
04Construct an explicit witnessL16–16

Supply the displayed value, then prove that it has the required property.

  1. L16
    exists x
05Separate the logical casesL17–17

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

  1. L17
    split
06Use earlier factsL18–18

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

  1. L18
    exact hc_witness
07Fix variables and assumptionsL19–20

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

  1. L19
    intro d
  2. L20
    intro hd
08Use earlier factsL21–28

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

  1. L21
    specialize prime_field_multiply_functional (p)
  2. L22
    specialize prime_field_multiply_functional (a)
  3. L23
    specialize prime_field_multiply_functional (b)
  4. L24
    specialize prime_field_multiply_functional (d)
  5. L25
    specialize prime_field_multiply_functional (x)
  6. L26
    apply prime_field_multiply_functional
  7. L27
    exact hd
  8. L28
    exact hc_witness

Library-wide reading audit

Original defined command ledger · 28 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro b
  4. 0004intro hp
  5. 0005intro ha
  6. 0006intro hb
  7. 0007have hc : ∃ c. FpMul(p,a,b,c)
  8. 0008specialize prime_field_multiply_exists (p)
  9. 0009specialize prime_field_multiply_exists (a)
  10. 0010specialize prime_field_multiply_exists (b)
  11. 0011apply prime_field_multiply_exists
  12. 0012exact hp
  13. 0013exact ha
  14. 0014exact hb
  15. 0015cases hc
  16. 0016exists x
  17. 0017split
  18. 0018exact hc_witness
  19. 0019intro d
  20. 0020intro hd
  21. 0021specialize prime_field_multiply_functional (p)
  22. 0022specialize prime_field_multiply_functional (a)
  23. 0023specialize prime_field_multiply_functional (b)
  24. 0024specialize prime_field_multiply_functional (d)
  25. 0025specialize prime_field_multiply_functional (x)
  26. 0026apply prime_field_multiply_functional
  27. 0027exact hd
  28. 0028exact hc_witness