FP001E

prime_field_inverse_exists

Every nonzero representative has an actual bounded multiplicative inverse, uniformly at every prime.

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

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

Definition DAG

FpInv(p,a,b) · 1Lt(a,b) · 1Prime(p) · 1BoundedNonzeroInverse(m,a) · 1

Actual proof prerequisites

prime_bounded_nonzero_mod_inverse · checked external prerequisiteprime_two_le · checked external prerequisite
Original expanded first-order statement 
forall p a. (~((p) = 1) /\ forall pfa_factor_left_inverse_domain pfa_factor_right_inverse_domain. (p) = pfa_factor_left_inverse_domain * pfa_factor_right_inverse_domain -> pfa_factor_left_inverse_domain = 1 \/ pfa_factor_right_inverse_domain = 1) -> (exists pfa_gap_inverse_input. pfa_gap_inverse_input + S (a) = (p)) -> ~(a = 0) -> exists b. (((~((a) = 0)) /\ ((((exists pfa_gap_inverse_existsmultiplicationleft. pfa_gap_inverse_existsmultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverse_existsmultiplicationright. pfa_gap_inverse_existsmultiplicationright + S (b) = (p)) /\ ((((exists pfa_gap_inverse_existsmultiplicationresultbound. pfa_gap_inverse_existsmultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_existsmultiplicationresultcongruence pfa_offset_right_inverse_existsmultiplicationresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_inverse_existsmultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_existsmultiplicationresultcongruence))))))))))))

Complete tactic proof in conservative notation

All 27 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

27 script commands · 12 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.

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 hiL6–12

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

  1. L6
    have hi : BoundedNonzeroInverse(p,a)Definitions: BoundedNonzeroInverse(p,a)Original native command in the exact edition
  2. L7
    specialize prime_bounded_nonzero_mod_inverse (p)
  3. L8
    specialize prime_bounded_nonzero_mod_inverse (a)
  4. L9
    apply prime_bounded_nonzero_mod_inverse
  5. L10
    exact hp
  6. L11
    exact hn
  7. L12
    exact ha
03Separate the logical casesL13–15

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

  1. L13
    cases hi
  2. L14
    cases hi_witness
  3. L15
    cases hi_witness_right
04Construct an explicit witnessL16–16

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

  1. L16
    exists x
05Separate the logical casesL17–17

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

  1. L17
    split
06Use earlier factsL18–18

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

  1. L18
    exact hn
07Separate the logical casesL19–19

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

  1. L19
    split
08Use earlier factsL20–20

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

  1. L20
    exact ha
09Separate the logical casesL21–21

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

  1. L21
    split
10Use earlier factsL22–22

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

  1. L22
    exact hi_witness_right_left
11Separate the logical casesL23–23

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

  1. L23
    split
12Use earlier factsL24–27

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

  1. L24
    specialize prime_two_le (p)
  2. L25
    apply prime_two_le
  3. L26
    exact hp
  4. L27
    exact hi_witness_right_right

Library-wide reading audit

Original defined command ledger · 27 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro hp
  4. 0004intro ha
  5. 0005intro hn
  6. 0006have hi : BoundedNonzeroInverse(p,a)
  7. 0007specialize prime_bounded_nonzero_mod_inverse (p)
  8. 0008specialize prime_bounded_nonzero_mod_inverse (a)
  9. 0009apply prime_bounded_nonzero_mod_inverse
  10. 0010exact hp
  11. 0011exact hn
  12. 0012exact ha
  13. 0013cases hi
  14. 0014cases hi_witness
  15. 0015cases hi_witness_right
  16. 0016exists x
  17. 0017split
  18. 0018exact hn
  19. 0019split
  20. 0020exact ha
  21. 0021split
  22. 0022exact hi_witness_right_left
  23. 0023split
  24. 0024specialize prime_two_le (p)
  25. 0025apply prime_two_le
  26. 0026exact hp
  27. 0027exact hi_witness_right_right