FP0023

prime_field_inverse_symmetric

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.

Inversion is symmetric on the nonzero elements, with both domains proved.

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

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. Prime(p) → FpInv(p,a,b) → FpInv(p,b,a)

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

Definition DAG

FpInv(p,a,b) · 2Prime(p) · 1

Actual proof prerequisites

prime_field_inverse_output_nonzeroprime_field_multiply_commutative
Original expanded first-order statement 
forall p a b. (~((p) = 1) /\ forall pfa_factor_left_inverse_symmetric_domain pfa_factor_right_inverse_symmetric_domain. (p) = pfa_factor_left_inverse_symmetric_domain * pfa_factor_right_inverse_symmetric_domain -> pfa_factor_left_inverse_symmetric_domain = 1 \/ pfa_factor_right_inverse_symmetric_domain = 1) -> (((~((a) = 0)) /\ ((((exists pfa_gap_inverse_symmetric_sourcemultiplicationleft. pfa_gap_inverse_symmetric_sourcemultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverse_symmetric_sourcemultiplicationright. pfa_gap_inverse_symmetric_sourcemultiplicationright + S (b) = (p)) /\ ((((exists pfa_gap_inverse_symmetric_sourcemultiplicationresultbound. pfa_gap_inverse_symmetric_sourcemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_symmetric_sourcemultiplicationresultcongruence pfa_offset_right_inverse_symmetric_sourcemultiplicationresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_inverse_symmetric_sourcemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_symmetric_sourcemultiplicationresultcongruence)))))))))))) -> (((~((b) = 0)) /\ ((((exists pfa_gap_inverse_symmetric_targetmultiplicationleft. pfa_gap_inverse_symmetric_targetmultiplicationleft + S (b) = (p)) /\ (((exists pfa_gap_inverse_symmetric_targetmultiplicationright. pfa_gap_inverse_symmetric_targetmultiplicationright + S (a) = (p)) /\ ((((exists pfa_gap_inverse_symmetric_targetmultiplicationresultbound. pfa_gap_inverse_symmetric_targetmultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_symmetric_targetmultiplicationresultcongruence pfa_offset_right_inverse_symmetric_targetmultiplicationresultcongruence. ((b) * (a)) + (p) * pfa_offset_left_inverse_symmetric_targetmultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_symmetric_targetmultiplicationresultcongruence))))))))))))

Complete tactic proof in conservative notation

All 21 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

21 script commands · 6 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.

Named ingredients (2)
prime field inverse output nonzeroprime field multiply commutative
01Fix variables and assumptionsL1–5

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 hi
02Separate the logical casesL6–6

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

  1. L6
    split
03Fix variables and assumptionsL7–7

Work with arbitrary variables or the premises of the current implication.

  1. L7
    intro hz
04Use earlier factsL8–14

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

  1. L8
    specialize prime_field_inverse_output_nonzero (p)
  2. L9
    specialize prime_field_inverse_output_nonzero (a)
  3. L10
    specialize prime_field_inverse_output_nonzero (b)
  4. L11
    apply prime_field_inverse_output_nonzero
  5. L12
    exact hp
  6. L13
    exact hi
  7. L14
    exact hz
05Separate the logical casesL15–15

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

  1. L15
    cases hi
06Use earlier factsL16–21

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

  1. L16
    specialize prime_field_multiply_commutative (p)
  2. L17
    specialize prime_field_multiply_commutative (a)
  3. L18
    specialize prime_field_multiply_commutative (b)
  4. L19
    specialize prime_field_multiply_commutative (1)
  5. L20
    apply prime_field_multiply_commutative
  6. L21
    exact hi_right

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro b
  4. 0004intro hp
  5. 0005intro hi
  6. 0006split
  7. 0007intro hz
  8. 0008specialize prime_field_inverse_output_nonzero (p)
  9. 0009specialize prime_field_inverse_output_nonzero (a)
  10. 0010specialize prime_field_inverse_output_nonzero (b)
  11. 0011apply prime_field_inverse_output_nonzero
  12. 0012exact hp
  13. 0013exact hi
  14. 0014exact hz
  15. 0015cases hi
  16. 0016specialize prime_field_multiply_commutative (p)
  17. 0017specialize prime_field_multiply_commutative (a)
  18. 0018specialize prime_field_multiply_commutative (b)
  19. 0019specialize prime_field_multiply_commutative (1)
  20. 0020apply prime_field_multiply_commutative
  21. 0021exact hi_right