FP001C

prime_field_negate_functional

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

Actual additive inverses are unique among canonical representatives.

Exact expanded first-order arithmetic statement

forall p a b c. (((exists pfa_gap_negate_firstadditionleft. pfa_gap_negate_firstadditionleft + S (a) = (p)) /\ (((exists pfa_gap_negate_firstadditionright. pfa_gap_negate_firstadditionright + S (b) = (p)) /\ ((((exists pfa_gap_negate_firstadditionresultbound. pfa_gap_negate_firstadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_firstadditionresultcongruence pfa_offset_right_negate_firstadditionresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_negate_firstadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_firstadditionresultcongruence))))))))) -> (((exists pfa_gap_negate_secondadditionleft. pfa_gap_negate_secondadditionleft + S (a) = (p)) /\ (((exists pfa_gap_negate_secondadditionright. pfa_gap_negate_secondadditionright + S (c) = (p)) /\ ((((exists pfa_gap_negate_secondadditionresultbound. pfa_gap_negate_secondadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_secondadditionresultcongruence pfa_offset_right_negate_secondadditionresultcongruence. ((a) + (c)) + (p) * pfa_offset_left_negate_secondadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_secondadditionresultcongruence))))))))) -> b = c

Constructive proof overview

Generated structural guide

Actual additive inverses are unique among canonical representatives.

The unchanged tactic script uses 1 declared prerequisite and contains 14 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

FP001A prime_field_add_cancel_left

Direct dependents

FP001D prime_field_negate_exists_unique FP003F prime_field_negate_table_reflect

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

14 script commands · 2 reading checkpoints · 0 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 (1)
prime field add cancel left
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 c
  5. L5
    intro hb
  6. L6
    intro hc
02Use earlier factsL7–14

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

  1. L7
    specialize prime_field_add_cancel_left (p)
  2. L8
    specialize prime_field_add_cancel_left (a)
  3. L9
    specialize prime_field_add_cancel_left (b)
  4. L10
    specialize prime_field_add_cancel_left (c)
  5. L11
    specialize prime_field_add_cancel_left (0)
  6. L12
    apply prime_field_add_cancel_left
  7. L13
    exact hb
  8. L14
    exact hc

Library-wide reading audit

Original exact command ledger · 14 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro b
  4. 0004intro c
  5. 0005intro hb
  6. 0006intro hc
  7. 0007specialize prime_field_add_cancel_left (p)
  8. 0008specialize prime_field_add_cancel_left (a)
  9. 0009specialize prime_field_add_cancel_left (b)
  10. 0010specialize prime_field_add_cancel_left (c)
  11. 0011specialize prime_field_add_cancel_left (0)
  12. 0012apply prime_field_add_cancel_left
  13. 0013exact hb
  14. 0014exact hc