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.

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

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

Exact expanded first-order arithmetic 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

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 28 exact native proof lines.

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

Proof neighborhood

Direct dependencies

FP000C prime_field_multiply_exists FP000D prime_field_multiply_functional

Direct dependents

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. The literal dependency-closed bundle is checked by original HA and the independently compiled Lean verifier. Public delivery grants no Alpha checked-use authority or Stable membership.

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.

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 : exists c. (((exists pfa_gap_multiplychoiceleft. pfa_gap_multiplychoiceleft + S (a) = (p)) /\ (((exists pfa_gap_multiplychoiceright. pfa_gap_multiplychoiceright + S (b) = (p)) /\ ((((exists pfa_gap_multiplychoiceresultbound. pfa_gap_multiplychoiceresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_multiplychoiceresultcongruence pfa_offset_right_multiplychoiceresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplychoiceresultcongruence = (c) + (p) * pfa_offset_right_multiplychoiceresultcongruence)))))))))
  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 exact 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 : exists c. (((exists pfa_gap_multiplychoiceleft. pfa_gap_multiplychoiceleft + S (a) = (p)) /\ (((exists pfa_gap_multiplychoiceright. pfa_gap_multiplychoiceright + S (b) = (p)) /\ ((((exists pfa_gap_multiplychoiceresultbound. pfa_gap_multiplychoiceresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_multiplychoiceresultcongruence pfa_offset_right_multiplychoiceresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplychoiceresultcongruence = (c) + (p) * pfa_offset_right_multiplychoiceresultcongruence)))))))))
  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