FP001C

prime_field_negate_functional

Actual additive inverses are unique among canonical representatives.

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. ∀ c. FpAdd(p,a,b,0) → FpAdd(p,a,c,0) → b = c

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

Definition DAG

FpAdd(p,a,b,c) · 2

Actual proof prerequisites

prime_field_add_cancel_left
Original expanded first-order 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

Complete tactic proof in conservative notation

All 14 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

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.

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 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 defined 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