FP001C

prime_field_negate_functional

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Actual additive inverses are unique among canonical representatives.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

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.

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; 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. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply 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