FP0024

prime_field_nonzero_coprime

Every nonzero canonical element is genuinely coprime to its prime modulus.

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 → Coprime(a,p)

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

Definition DAG

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

Actual proof prerequisites

prime_field_inverse_existsmod_inverse_implies_coprime · checked external prerequisite
Original expanded first-order statement 
forall p a. (~((p) = 1) /\ forall pfa_factor_left_coprime_domain pfa_factor_right_coprime_domain. (p) = pfa_factor_left_coprime_domain * pfa_factor_right_coprime_domain -> pfa_factor_left_coprime_domain = 1 \/ pfa_factor_right_coprime_domain = 1) -> (exists pfa_gap_coprime_bound. pfa_gap_coprime_bound + S (a) = (p)) -> ~(a = 0) -> (forall pfa_divisor_coprime_result. (exists pfa_left_factor_coprime_result. (a) = pfa_divisor_coprime_result * pfa_left_factor_coprime_result) -> (exists pfa_right_factor_coprime_result. (p) = pfa_divisor_coprime_result * pfa_right_factor_coprime_result) -> pfa_divisor_coprime_result = 1)

Complete tactic proof in conservative notation

All 22 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

22 script commands · 4 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 (1)
prime field inverse exists
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 field inverse exists.

  1. L6
    have hi : ∃ 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 casesL13–17

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
  4. L16
    cases hi_witness_right_right
  5. L17
    cases hi_witness_right_right_right
04Use earlier factsL18–22

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

  1. L18
    specialize mod_inverse_implies_coprime (a)
  2. L19
    specialize mod_inverse_implies_coprime (p)
  3. L20
    specialize mod_inverse_implies_coprime (x)
  4. L21
    apply mod_inverse_implies_coprime
  5. L22
    exact hi_witness_right_right_right_right

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro hp
  4. 0004intro ha
  5. 0005intro hn
  6. 0006have hi : ∃ 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 hi
  14. 0014cases hi_witness
  15. 0015cases hi_witness_right
  16. 0016cases hi_witness_right_right
  17. 0017cases hi_witness_right_right_right
  18. 0018specialize mod_inverse_implies_coprime (a)
  19. 0019specialize mod_inverse_implies_coprime (p)
  20. 0020specialize mod_inverse_implies_coprime (x)
  21. 0021apply mod_inverse_implies_coprime
  22. 0022exact hi_witness_right_right_right_right