FP0022

prime_field_inverse_output_nonzero

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

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

Exact expanded first-order arithmetic 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)

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 21 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

FP0021 prime_field_zero_has_no_multiplicative_inverse FP000F prime_field_multiply_commutative

Direct dependents

FP0023 prime_field_inverse_symmetric

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

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.

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
    have hm : ((exists pfa_gap_inverse_swappedleft. pfa_gap_inverse_swappedleft + S (b) = (p)) /\ (((exists pfa_gap_inverse_swappedright. pfa_gap_inverse_swappedright + S (a) = (p)) /\ ((((exists pfa_gap_inverse_swappedresultbound. pfa_gap_inverse_swappedresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_swappedresultcongruence pfa_offset_right_inverse_swappedresultcongruence. ((b) * (a)) + (p) * pfa_offset_left_inverse_swappedresultcongruence = (1) + (p) * pfa_offset_right_inverse_swappedresultcongruence))))))))
  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 exact 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 : ((exists pfa_gap_inverse_swappedleft. pfa_gap_inverse_swappedleft + S (b) = (p)) /\ (((exists pfa_gap_inverse_swappedright. pfa_gap_inverse_swappedright + S (a) = (p)) /\ ((((exists pfa_gap_inverse_swappedresultbound. pfa_gap_inverse_swappedresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_inverse_swappedresultcongruence pfa_offset_right_inverse_swappedresultcongruence. ((b) * (a)) + (p) * pfa_offset_left_inverse_swappedresultcongruence = (1) + (p) * pfa_offset_right_inverse_swappedresultcongruence))))))))
  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