FP0022

prime_field_inverse_output_nonzero

An actual inverse is itself nonzero, so reciprocal inversion has the correct domain.

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. ∀ b. Prime(p) → FpInv(p,a,b) → ¬b = 0

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

Definition DAG

FpMul(p,a,b,c) · 1FpInv(p,a,b) · 1Prime(p) · 1

Actual proof prerequisites

prime_field_zero_has_no_multiplicative_inverseprime_field_multiply_commutative
Original expanded first-order statement 
forall p a b. (~((p) = 1) /\ forall pfa_factor_left_inverse_output_domain pfa_factor_right_inverse_output_domain. (p) = pfa_factor_left_inverse_output_domain * pfa_factor_right_inverse_output_domain -> pfa_factor_left_inverse_output_domain = 1 \/ pfa_factor_right_inverse_output_domain = 1) -> (((~((a) = 0)) /\ ((((exists pfa_gap_inverse_outputmultiplicationleft. pfa_gap_inverse_outputmultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_inverse_outputmultiplicationright. pfa_gap_inverse_outputmultiplicationright + S (b) = (p)) /\ ((((exists pfa_gap_inverse_outputmultiplicationresultbound. pfa_gap_inverse_outputmultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_outputmultiplicationresultcongruence pfa_offset_right_inverse_outputmultiplicationresultcongruence. ((a) * (b)) + (p) * pfa_offset_left_inverse_outputmultiplicationresultcongruence = (1) + (p) * pfa_offset_right_inverse_outputmultiplicationresultcongruence)))))))))))) -> ~(b = 0)

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 · 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 (2)
prime field zero has no multiplicative inverseprime field multiply commutative
01Fix variables and assumptionsL1–6

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
  6. L6
    intro hz
02Separate the logical casesL7–7

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

  1. L7
    cases hi
03Use earlier factsL8–11

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

  1. L8
    specialize prime_field_zero_has_no_multiplicative_inverse (p)
  2. L9
    specialize prime_field_zero_has_no_multiplicative_inverse (a)
  3. L10
    apply prime_field_zero_has_no_multiplicative_inverse
  4. L11
    exact hp
04Establish hmL12–21

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

  1. L12
  2. L13
    specialize prime_field_multiply_commutative (p)
  3. L14
    specialize prime_field_multiply_commutative (a)
  4. L15
    specialize prime_field_multiply_commutative (b)
  5. L16
    specialize prime_field_multiply_commutative (1)
  6. L17
    apply prime_field_multiply_commutative
  7. L18
    exact hi_right
  8. L19
    rewrite hz at hm
  9. L20
    rewrite hz at hm
  10. L21
    exact hm

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. 0006intro hz
  7. 0007cases hi
  8. 0008specialize prime_field_zero_has_no_multiplicative_inverse (p)
  9. 0009specialize prime_field_zero_has_no_multiplicative_inverse (a)
  10. 0010apply prime_field_zero_has_no_multiplicative_inverse
  11. 0011exact hp
  12. 0012have hm : FpMul(p,b,a,1)
  13. 0013specialize prime_field_multiply_commutative (p)
  14. 0014specialize prime_field_multiply_commutative (a)
  15. 0015specialize prime_field_multiply_commutative (b)
  16. 0016specialize prime_field_multiply_commutative (1)
  17. 0017apply prime_field_multiply_commutative
  18. 0018exact hi_right
  19. 0019rewrite hz at hm
  20. 0020rewrite hz at hm
  21. 0021exact hm