FP001F

prime_field_inverse_functional

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

Multiplicative inverses are unique as bounded natural representatives.

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 c. (((~((a) = 0)) /\ ((((exists pfa_gap_inverse_firstmultiplicationleft. pfa_gap_inverse_firstmultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverse_firstmultiplicationright. pfa_gap_inverse_firstmultiplicationright + S (b) = (p)) /\ ((((exists pfa_gap_inverse_firstmultiplicationresultbound. pfa_gap_inverse_firstmultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_firstmultiplicationresultcongruence pfa_offset_right_inverse_firstmultiplicationresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_inverse_firstmultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_firstmultiplicationresultcongruence)))))))))))) -> (((~((a) = 0)) /\ ((((exists pfa_gap_inverse_secondmultiplicationleft. pfa_gap_inverse_secondmultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverse_secondmultiplicationright. pfa_gap_inverse_secondmultiplicationright + S (c) = (p)) /\ ((((exists pfa_gap_inverse_secondmultiplicationresultbound. pfa_gap_inverse_secondmultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_secondmultiplicationresultcongruence pfa_offset_right_inverse_secondmultiplicationresultcongruence. ((a) * (c)) + (p) * pfa_offset_left_inverse_secondmultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_secondmultiplicationresultcongruence)))))))))))) -> b = c

Constructive proof overview

Generated structural guide

Multiplicative inverses are unique as bounded natural representatives.

The unchanged tactic script uses 1 declared prerequisite and contains 23 exact native proof lines.

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

Proof neighborhood

Direct dependencies

bounded_mod_inverse_unique Stable theorem; checked-use authorized

Direct dependents

FP0020 prime_field_inverse_exists_unique FP002E prime_field_zero_extended_inverse_functional

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

23 script commands · 3 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 assumptionsL1–6

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 c
  5. L5
    intro hb
  6. L6
    intro hc
02Separate the logical casesL7–14

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

  1. L7
    cases hb
  2. L8
    cases hb_right
  3. L9
    cases hb_right_right
  4. L10
    cases hb_right_right_right
  5. L11
    cases hc
  6. L12
    cases hc_right
  7. L13
    cases hc_right_right
  8. L14
    cases hc_right_right_right
03Use earlier factsL15–23

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

  1. L15
    specialize bounded_mod_inverse_unique (p)
  2. L16
    specialize bounded_mod_inverse_unique (a)
  3. L17
    specialize bounded_mod_inverse_unique (b)
  4. L18
    specialize bounded_mod_inverse_unique (c)
  5. L19
    apply bounded_mod_inverse_unique
  6. L20
    exact hb_right_right_left
  7. L21
    exact hc_right_right_left
  8. L22
    exact hb_right_right_right_right
  9. L23
    exact hc_right_right_right_right

Library-wide reading audit

Original exact command ledger · 23 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro b
  4. 0004intro c
  5. 0005intro hb
  6. 0006intro hc
  7. 0007cases hb
  8. 0008cases hb_right
  9. 0009cases hb_right_right
  10. 0010cases hb_right_right_right
  11. 0011cases hc
  12. 0012cases hc_right
  13. 0013cases hc_right_right
  14. 0014cases hc_right_right_right
  15. 0015specialize bounded_mod_inverse_unique (p)
  16. 0016specialize bounded_mod_inverse_unique (a)
  17. 0017specialize bounded_mod_inverse_unique (b)
  18. 0018specialize bounded_mod_inverse_unique (c)
  19. 0019apply bounded_mod_inverse_unique
  20. 0020exact hb_right_right_left
  21. 0021exact hc_right_right_left
  22. 0022exact hb_right_right_right_right
  23. 0023exact hc_right_right_right_right