FP001E

prime_field_inverse_exists

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.

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

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

Exact expanded first-order arithmetic 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))))))))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 27 exact native proof lines.

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

Proof neighborhood

Direct dependencies

prime_bounded_nonzero_mod_inverse Alpha theorem; checked-use authorized prime_two_le Alpha theorem; checked-use authorized

Direct dependents

FP0020 prime_field_inverse_exists_unique FP0024 prime_field_nonzero_coprime FP002D prime_field_zero_extended_inverse_exists

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. The literal dependency-closed bundle is checked by original HA and the independently compiled Lean verifier. Public delivery grants no Alpha checked-use authority or Stable membership.

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.

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 : exists b. ~(b = 0) /\ ((exists pfa_gap_inverse_bound. pfa_gap_inverse_bound + S (b) = (p)) /\ (exists pfa_offset_left_inverse_mod pfa_offset_right_inverse_mod. (a*b) + (p) * pfa_offset_left_inverse_mod = (1) + (p) * pfa_offset_right_inverse_mod))
  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 exact command ledger · 27 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro hp
  4. 0004intro ha
  5. 0005intro hn
  6. 0006have hi : exists b. ~(b = 0) /\ ((exists pfa_gap_inverse_bound. pfa_gap_inverse_bound + S (b) = (p)) /\ (exists pfa_offset_left_inverse_mod pfa_offset_right_inverse_mod. (a*b) + (p) * pfa_offset_left_inverse_mod = (1) + (p) * pfa_offset_right_inverse_mod))
  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