FP0028

prime_field_residue_multiply

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

The natural-number residue map preserves actual canonical multiply.

Exact expanded first-order arithmetic statement

forall p a b x y z. (((exists pfa_gap_multiplynatural_leftbound. pfa_gap_multiplynatural_leftbound + S (x) = (p)) /\ ((exists pfa_offset_left_multiplynatural_leftcongruence pfa_offset_right_multiplynatural_leftcongruence. (a) + (p) * pfa_offset_left_multiplynatural_leftcongruence = (x) + (p) * pfa_offset_right_multiplynatural_leftcongruence)))) -> (((exists pfa_gap_multiplynatural_rightbound. pfa_gap_multiplynatural_rightbound + S (y) = (p)) /\ ((exists pfa_offset_left_multiplynatural_rightcongruence pfa_offset_right_multiplynatural_rightcongruence. (b) + (p) * pfa_offset_left_multiplynatural_rightcongruence = (y) + (p) * pfa_offset_right_multiplynatural_rightcongruence)))) -> (((exists pfa_gap_multiplynatural_resultleft. pfa_gap_multiplynatural_resultleft + S (x) = (p)) /\ (((exists pfa_gap_multiplynatural_resultright. pfa_gap_multiplynatural_resultright + S (y) = (p)) /\ ((((exists pfa_gap_multiplynatural_resultresultbound. pfa_gap_multiplynatural_resultresultbound + S (z) = (p)) /\ ((exists pfa_offset_left_multiplynatural_resultresultcongruence pfa_offset_right_multiplynatural_resultresultcongruence. ((x) * (y)) + (p) * pfa_offset_left_multiplynatural_resultresultcongruence = (z) + (p) * pfa_offset_right_multiplynatural_resultresultcongruence))))))))) -> (((exists pfa_gap_multiplynatural_homomorphismbound. pfa_gap_multiplynatural_homomorphismbound + S (z) = (p)) /\ ((exists pfa_offset_left_multiplynatural_homomorphismcongruence pfa_offset_right_multiplynatural_homomorphismcongruence. (a * b) + (p) * pfa_offset_left_multiplynatural_homomorphismcongruence = (z) + (p) * pfa_offset_right_multiplynatural_homomorphismcongruence))))

Constructive proof overview

Generated structural guide

The natural-number residue map preserves actual canonical multiply.

The unchanged tactic script uses 2 declared prerequisites and contains 30 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

mod_eq_trans Alpha theorem; checked-use authorized mod_eq_mul Alpha theorem; checked-use authorized

Direct dependents

none

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

30 script commands · 4 reading checkpoints · 0 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 assumptionsL1–9

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 x
  5. L5
    intro y
  6. L6
    intro z
  7. L7
    intro ha
  8. L8
    intro hb
  9. L9
    intro hz
02Separate the logical casesL10–15

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

  1. L10
    cases ha
  2. L11
    cases hb
  3. L12
    cases hz
  4. L13
    cases hz_right
  5. L14
    cases hz_right_right
  6. L15
    split
03Use earlier factsL16–25

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

  1. L16
    exact hz_right_right_left
  2. L17
    specialize mod_eq_trans (p)
  3. L18
    specialize mod_eq_trans (a * b)
  4. L19
    specialize mod_eq_trans (x * y)
  5. L20
    specialize mod_eq_trans (z)
  6. L21
    apply mod_eq_trans
  7. L22
    specialize mod_eq_mul (p)
  8. L23
    specialize mod_eq_mul (a)
  9. L24
    specialize mod_eq_mul (x)
  10. L25
    specialize mod_eq_mul (b)
04Use earlier factsL26–30

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

  1. L26
    specialize mod_eq_mul (y)
  2. L27
    apply mod_eq_mul
  3. L28
    exact ha_right
  4. L29
    exact hb_right
  5. L30
    exact hz_right_right_right

Library-wide reading audit

Original exact command ledger · 30 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro b
  4. 0004intro x
  5. 0005intro y
  6. 0006intro z
  7. 0007intro ha
  8. 0008intro hb
  9. 0009intro hz
  10. 0010cases ha
  11. 0011cases hb
  12. 0012cases hz
  13. 0013cases hz_right
  14. 0014cases hz_right_right
  15. 0015split
  16. 0016exact hz_right_right_left
  17. 0017specialize mod_eq_trans (p)
  18. 0018specialize mod_eq_trans (a * b)
  19. 0019specialize mod_eq_trans (x * y)
  20. 0020specialize mod_eq_trans (z)
  21. 0021apply mod_eq_trans
  22. 0022specialize mod_eq_mul (p)
  23. 0023specialize mod_eq_mul (a)
  24. 0024specialize mod_eq_mul (x)
  25. 0025specialize mod_eq_mul (b)
  26. 0026specialize mod_eq_mul (y)
  27. 0027apply mod_eq_mul
  28. 0028exact ha_right
  29. 0029exact hb_right
  30. 0030exact hz_right_right_right