FP001D

prime_field_negate_exists_unique

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

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

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. (~((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

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 23 exact native proof lines.

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

FP001B prime_field_negate_exists FP001C prime_field_negate_functional

Direct dependents

FP002A prime_field_arithmetic_laws

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

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.

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 : exists b. (((exists pfa_gap_negate_choiceadditionleft. pfa_gap_negate_choiceadditionleft + S (a) = (p)) /\ (((exists pfa_gap_negate_choiceadditionright. pfa_gap_negate_choiceadditionright + S (b) = (p)) /\ ((((exists pfa_gap_negate_choiceadditionresultbound. pfa_gap_negate_choiceadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_choiceadditionresultcongruence pfa_offset_right_negate_choiceadditionresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_negate_choiceadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_choiceadditionresultcongruence)))))))))
  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 exact command ledger · 23 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro hp
  4. 0004intro ha
  5. 0005have hb : exists b. (((exists pfa_gap_negate_choiceadditionleft. pfa_gap_negate_choiceadditionleft + S (a) = (p)) /\ (((exists pfa_gap_negate_choiceadditionright. pfa_gap_negate_choiceadditionright + S (b) = (p)) /\ ((((exists pfa_gap_negate_choiceadditionresultbound. pfa_gap_negate_choiceadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_choiceadditionresultcongruence pfa_offset_right_negate_choiceadditionresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_negate_choiceadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_choiceadditionresultcongruence)))))))))
  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