FP001F

prime_field_inverse_functional

Multiplicative inverses are unique as bounded natural representatives.

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

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

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. ∀ c. FpInv(p,a,b) → FpInv(p,a,c) → b = c

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

Definition DAG

FpInv(p,a,b) · 2

Actual proof prerequisites

bounded_mod_inverse_unique · checked external prerequisite
Original expanded first-order 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

Complete tactic proof in conservative notation

All 23 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

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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