FP0020

prime_field_inverse_exists_unique

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.

Proved unique inverse existence has exactly the nonzero carrier domain, not an odd-prime or supplied-inverse premise.

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. Prime(p) → Lt(a,p) → ¬a = 0 → ∃ x. FpInv(p,a,x) ∧ (∀ y. FpInv(p,a,y) → y = x)

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

Definition DAG

FpInv(p,a,b) · 3Lt(a,b) · 1Prime(p) · 1

Actual proof prerequisites

prime_field_inverse_existsprime_field_inverse_functional
Original expanded first-order statement 
forall p a. (~((p) = 1) /\ forall pfa_factor_left_inverse_unique_domain pfa_factor_right_inverse_unique_domain. (p) = pfa_factor_left_inverse_unique_domain * pfa_factor_right_inverse_unique_domain -> pfa_factor_left_inverse_unique_domain = 1 \/ pfa_factor_right_inverse_unique_domain = 1) -> (exists pfa_gap_inverse_unique_input. pfa_gap_inverse_unique_input + S (a) = (p)) -> ~(a = 0) -> exists b. (((~((a) = 0)) /\ ((((exists pfa_gap_inverse_chosenmultiplicationleft. pfa_gap_inverse_chosenmultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverse_chosenmultiplicationright. pfa_gap_inverse_chosenmultiplicationright + S (b) = (p)) /\ ((((exists pfa_gap_inverse_chosenmultiplicationresultbound. pfa_gap_inverse_chosenmultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_chosenmultiplicationresultcongruence pfa_offset_right_inverse_chosenmultiplicationresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_inverse_chosenmultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_chosenmultiplicationresultcongruence)))))))))))) /\ forall c. (((~((a) = 0)) /\ ((((exists pfa_gap_inverse_othermultiplicationleft. pfa_gap_inverse_othermultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverse_othermultiplicationright. pfa_gap_inverse_othermultiplicationright + S (c) = (p)) /\ ((((exists pfa_gap_inverse_othermultiplicationresultbound. pfa_gap_inverse_othermultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_othermultiplicationresultcongruence pfa_offset_right_inverse_othermultiplicationresultcongruence. ((a) * (c)) + (p) * pfa_offset_left_inverse_othermultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_othermultiplicationresultcongruence)))))))))))) -> c = b

Complete tactic proof in conservative notation

All 25 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

25 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.

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 existsprime field inverse functional
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 hp
  4. L4
    intro ha
  5. L5
    intro hn
02Establish hbL6–12

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime field inverse exists.

  1. L6
    have hb : ∃ b. FpInv(p,a,b)Definitions: FpInv(p,a,b)Original native command in the exact edition
  2. L7
    specialize prime_field_inverse_exists (p)
  3. L8
    specialize prime_field_inverse_exists (a)
  4. L9
    apply prime_field_inverse_exists
  5. L10
    exact hp
  6. L11
    exact ha
  7. L12
    exact hn
03Separate the logical casesL13–13

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

  1. L13
    cases hb
04Construct an explicit witnessL14–14

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

  1. L14
    exists x
05Separate the logical casesL15–15

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

  1. L15
    split
06Use earlier factsL16–16

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

  1. L16
    exact hb_witness
07Fix variables and assumptionsL17–18

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

  1. L17
    intro c
  2. L18
    intro hc
08Use earlier factsL19–25

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

  1. L19
    specialize prime_field_inverse_functional (p)
  2. L20
    specialize prime_field_inverse_functional (a)
  3. L21
    specialize prime_field_inverse_functional (c)
  4. L22
    specialize prime_field_inverse_functional (x)
  5. L23
    apply prime_field_inverse_functional
  6. L24
    exact hc
  7. L25
    exact hb_witness

Library-wide reading audit

Original defined command ledger · 25 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro hp
  4. 0004intro ha
  5. 0005intro hn
  6. 0006have hb : ∃ b. FpInv(p,a,b)
  7. 0007specialize prime_field_inverse_exists (p)
  8. 0008specialize prime_field_inverse_exists (a)
  9. 0009apply prime_field_inverse_exists
  10. 0010exact hp
  11. 0011exact ha
  12. 0012exact hn
  13. 0013cases hb
  14. 0014exists x
  15. 0015split
  16. 0016exact hb_witness
  17. 0017intro c
  18. 0018intro hc
  19. 0019specialize prime_field_inverse_functional (p)
  20. 0020specialize prime_field_inverse_functional (a)
  21. 0021specialize prime_field_inverse_functional (c)
  22. 0022specialize prime_field_inverse_functional (x)
  23. 0023apply prime_field_inverse_functional
  24. 0024exact hc
  25. 0025exact hb_witness