FP001D

prime_field_negate_exists_unique

Every element, including zero, has a unique actual additive inverse.

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. Prime(p) → Lt(a,p) → ∃ x. FpAdd(p,a,x,0) ∧ (∀ y. FpAdd(p,a,y,0) → y = x)

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

Definition DAG

FpAdd(p,a,b,c) · 3Lt(a,b) · 1Prime(p) · 1

Actual proof prerequisites

prime_field_negate_existsprime_field_negate_functional
Original expanded first-order statement 
forall p a. (~((p) = 1) /\ forall pfa_factor_left_negate_unique_domain pfa_factor_right_negate_unique_domain. (p) = pfa_factor_left_negate_unique_domain * pfa_factor_right_negate_unique_domain -> pfa_factor_left_negate_unique_domain = 1 \/ pfa_factor_right_negate_unique_domain = 1) -> (exists pfa_gap_negate_unique_input. pfa_gap_negate_unique_input + S (a) = (p)) -> exists b. (((exists pfa_gap_negate_chosenadditionleft. pfa_gap_negate_chosenadditionleft + S (a) = (p)) /\ (((exists pfa_gap_negate_chosenadditionright. pfa_gap_negate_chosenadditionright + S (b) = (p)) /\ ((((exists pfa_gap_negate_chosenadditionresultbound. pfa_gap_negate_chosenadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_chosenadditionresultcongruence pfa_offset_right_negate_chosenadditionresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_negate_chosenadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_chosenadditionresultcongruence))))))))) /\ forall c. (((exists pfa_gap_negate_otheradditionleft. pfa_gap_negate_otheradditionleft + S (a) = (p)) /\ (((exists pfa_gap_negate_otheradditionright. pfa_gap_negate_otheradditionright + S (c) = (p)) /\ ((((exists pfa_gap_negate_otheradditionresultbound. pfa_gap_negate_otheradditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_otheradditionresultcongruence pfa_offset_right_negate_otheradditionresultcongruence. ((a) + (c)) + (p) * pfa_offset_left_negate_otheradditionresultcongruence = (0) + (p) * pfa_offset_right_negate_otheradditionresultcongruence))))))))) -> c = b

Complete tactic proof in conservative notation

All 23 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

23 script commands · 8 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 negate existsprime field negate functional
01Fix variables and assumptionsL1–4

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro hp
  4. L4
    intro ha
02Establish hbL5–10

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

  1. L5
    have hb : ∃ b. FpAdd(p,a,b,0)Definitions: FpAdd(p,a,b,0)Original native command in the exact edition
  2. L6
    specialize prime_field_negate_exists (p)
  3. L7
    specialize prime_field_negate_exists (a)
  4. L8
    apply prime_field_negate_exists
  5. L9
    exact hp
  6. L10
    exact ha
03Separate the logical casesL11–11

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

  1. L11
    cases hb
04Construct an explicit witnessL12–12

Supply the displayed value, then prove that it has the required property.

  1. L12
    exists x
05Separate the logical casesL13–13

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

  1. L13
    split
06Use earlier factsL14–14

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

  1. L14
    exact hb_witness
07Fix variables and assumptionsL15–16

Work with arbitrary variables or the premises of the current implication.

  1. L15
    intro c
  2. L16
    intro hc
08Use earlier factsL17–23

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

  1. L17
    specialize prime_field_negate_functional (p)
  2. L18
    specialize prime_field_negate_functional (a)
  3. L19
    specialize prime_field_negate_functional (c)
  4. L20
    specialize prime_field_negate_functional (x)
  5. L21
    apply prime_field_negate_functional
  6. L22
    exact hc
  7. L23
    exact hb_witness

Library-wide reading audit

Original defined command ledger · 23 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro hp
  4. 0004intro ha
  5. 0005have hb : ∃ b. FpAdd(p,a,b,0)
  6. 0006specialize prime_field_negate_exists (p)
  7. 0007specialize prime_field_negate_exists (a)
  8. 0008apply prime_field_negate_exists
  9. 0009exact hp
  10. 0010exact ha
  11. 0011cases hb
  12. 0012exists x
  13. 0013split
  14. 0014exact hb_witness
  15. 0015intro c
  16. 0016intro hc
  17. 0017specialize prime_field_negate_functional (p)
  18. 0018specialize prime_field_negate_functional (a)
  19. 0019specialize prime_field_negate_functional (c)
  20. 0020specialize prime_field_negate_functional (x)
  21. 0021apply prime_field_negate_functional
  22. 0022exact hc
  23. 0023exact hb_witness