FP000C

prime_field_multiply_exists

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

Construct the unique canonical multiply output for every pair of residues.

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 b. (~((p) = 1) /\ forall pfa_factor_left_multiplydomain pfa_factor_right_multiplydomain. (p) = pfa_factor_left_multiplydomain * pfa_factor_right_multiplydomain -> pfa_factor_left_multiplydomain = 1 \/ pfa_factor_right_multiplydomain = 1) -> (exists pfa_gap_multiplyleft. pfa_gap_multiplyleft + S (a) = (p)) -> (exists pfa_gap_multiplyright. pfa_gap_multiplyright + S (b) = (p)) -> exists c. (((exists pfa_gap_multiplyexistsleft. pfa_gap_multiplyexistsleft + S (a) = (p)) /\ (((exists pfa_gap_multiplyexistsright. pfa_gap_multiplyexistsright + S (b) = (p)) /\ ((((exists pfa_gap_multiplyexistsresultbound. pfa_gap_multiplyexistsresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_multiplyexistsresultcongruence pfa_offset_right_multiplyexistsresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_multiplyexistsresultcongruence = (c) + (p) * pfa_offset_right_multiplyexistsresultcongruence)))))))))

Constructive proof overview

Generated structural guide

Construct the unique canonical multiply output for every pair of residues.

The unchanged tactic script uses 2 declared prerequisites and contains 22 exact native proof lines.

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

Proof neighborhood

Direct dependencies

prime_nonzero Stable theorem; checked-use authorized hensel_canonical_residue_exists Alpha theorem; checked-use authorized

Direct dependents

FP000E prime_field_multiply_exists_unique FP002C prime_field_multiply_grid_value_exists

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

22 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.

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 hrL7–15

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hensel canonical residue exists.

  1. L7
    have hr : exists c. (((exists pfa_gap_multiplyresiduebound. pfa_gap_multiplyresiduebound + S (c) = (p)) /\ ((exists pfa_offset_left_multiplyresiduecongruence pfa_offset_right_multiplyresiduecongruence. (a * b) + (p) * pfa_offset_left_multiplyresiduecongruence = (c) + (p) * pfa_offset_right_multiplyresiduecongruence))))
  2. L8
    specialize hensel_canonical_residue_exists (p)
  3. L9
    specialize hensel_canonical_residue_exists (a * b)
  4. L10
    apply hensel_canonical_residue_exists
  5. L11
    intro hz
  6. L12
    specialize prime_nonzero (p)
  7. L13
    apply prime_nonzero
  8. L14
    exact hp
  9. L15
    exact hz
03Separate the logical casesL16–16

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

  1. L16
    cases hr
04Construct an explicit witnessL17–17

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

  1. L17
    exists x
05Separate the logical casesL18–18

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

  1. L18
    split
06Use earlier factsL19–19

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

  1. L19
    exact ha
07Separate the logical casesL20–20

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

  1. L20
    split
08Use earlier factsL21–22

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

  1. L21
    exact hb
  2. L22
    exact hr_witness

Library-wide reading audit

Original exact command ledger · 22 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro b
  4. 0004intro hp
  5. 0005intro ha
  6. 0006intro hb
  7. 0007have hr : exists c. (((exists pfa_gap_multiplyresiduebound. pfa_gap_multiplyresiduebound + S (c) = (p)) /\ ((exists pfa_offset_left_multiplyresiduecongruence pfa_offset_right_multiplyresiduecongruence. (a * b) + (p) * pfa_offset_left_multiplyresiduecongruence = (c) + (p) * pfa_offset_right_multiplyresiduecongruence))))
  8. 0008specialize hensel_canonical_residue_exists (p)
  9. 0009specialize hensel_canonical_residue_exists (a * b)
  10. 0010apply hensel_canonical_residue_exists
  11. 0011intro hz
  12. 0012specialize prime_nonzero (p)
  13. 0013apply prime_nonzero
  14. 0014exact hp
  15. 0015exact hz
  16. 0016cases hr
  17. 0017exists x
  18. 0018split
  19. 0019exact ha
  20. 0020split
  21. 0021exact hb
  22. 0022exact hr_witness