FP000A

prime_field_add_exists_unique

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

Existence and uniqueness of canonical prime-field add are both proved.

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. (~((p) = 1) /\ forall pfa_factor_left_addunique_domain pfa_factor_right_addunique_domain. (p) = pfa_factor_left_addunique_domain * pfa_factor_right_addunique_domain -> pfa_factor_left_addunique_domain = 1 \/ pfa_factor_right_addunique_domain = 1) -> (exists pfa_gap_addunique_left. pfa_gap_addunique_left + S (a) = (p)) -> (exists pfa_gap_addunique_right. pfa_gap_addunique_right + S (b) = (p)) -> exists c. (((exists pfa_gap_addchosenleft. pfa_gap_addchosenleft + S (a) = (p)) /\ (((exists pfa_gap_addchosenright. pfa_gap_addchosenright + S (b) = (p)) /\ ((((exists pfa_gap_addchosenresultbound. pfa_gap_addchosenresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_addchosenresultcongruence pfa_offset_right_addchosenresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addchosenresultcongruence = (c) + (p) * pfa_offset_right_addchosenresultcongruence))))))))) /\ forall d. (((exists pfa_gap_addcomparisonleft. pfa_gap_addcomparisonleft + S (a) = (p)) /\ (((exists pfa_gap_addcomparisonright. pfa_gap_addcomparisonright + S (b) = (p)) /\ ((((exists pfa_gap_addcomparisonresultbound. pfa_gap_addcomparisonresultbound + S (d) = (p)) /\ ((exists pfa_offset_left_addcomparisonresultcongruence pfa_offset_right_addcomparisonresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addcomparisonresultcongruence = (d) + (p) * pfa_offset_right_addcomparisonresultcongruence))))))))) -> d = c

Constructive proof overview

Generated structural guide

Existence and uniqueness of canonical prime-field add are both proved.

The unchanged tactic script uses 2 declared prerequisites and contains 28 exact native proof lines.

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

Proof neighborhood

Direct dependencies

FP0008 prime_field_add_exists FP0009 prime_field_add_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

28 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 add existsprime field add functional
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 hp
  5. L5
    intro ha
  6. L6
    intro hb
02Establish hcL7–14

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

  1. L7
    have hc : exists c. (((exists pfa_gap_addchoiceleft. pfa_gap_addchoiceleft + S (a) = (p)) /\ (((exists pfa_gap_addchoiceright. pfa_gap_addchoiceright + S (b) = (p)) /\ ((((exists pfa_gap_addchoiceresultbound. pfa_gap_addchoiceresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_addchoiceresultcongruence pfa_offset_right_addchoiceresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addchoiceresultcongruence = (c) + (p) * pfa_offset_right_addchoiceresultcongruence)))))))))
  2. L8
    specialize prime_field_add_exists (p)
  3. L9
    specialize prime_field_add_exists (a)
  4. L10
    specialize prime_field_add_exists (b)
  5. L11
    apply prime_field_add_exists
  6. L12
    exact hp
  7. L13
    exact ha
  8. L14
    exact hb
03Separate the logical casesL15–15

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

  1. L15
    cases hc
04Construct an explicit witnessL16–16

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

  1. L16
    exists x
05Separate the logical casesL17–17

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

  1. L17
    split
06Use earlier factsL18–18

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

  1. L18
    exact hc_witness
07Fix variables and assumptionsL19–20

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

  1. L19
    intro d
  2. L20
    intro hd
08Use earlier factsL21–28

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

  1. L21
    specialize prime_field_add_functional (p)
  2. L22
    specialize prime_field_add_functional (a)
  3. L23
    specialize prime_field_add_functional (b)
  4. L24
    specialize prime_field_add_functional (d)
  5. L25
    specialize prime_field_add_functional (x)
  6. L26
    apply prime_field_add_functional
  7. L27
    exact hd
  8. L28
    exact hc_witness

Library-wide reading audit

Original exact command ledger · 28 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro b
  4. 0004intro hp
  5. 0005intro ha
  6. 0006intro hb
  7. 0007have hc : exists c. (((exists pfa_gap_addchoiceleft. pfa_gap_addchoiceleft + S (a) = (p)) /\ (((exists pfa_gap_addchoiceright. pfa_gap_addchoiceright + S (b) = (p)) /\ ((((exists pfa_gap_addchoiceresultbound. pfa_gap_addchoiceresultbound + S (c) = (p)) /\ ((exists pfa_offset_left_addchoiceresultcongruence pfa_offset_right_addchoiceresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_addchoiceresultcongruence = (c) + (p) * pfa_offset_right_addchoiceresultcongruence)))))))))
  8. 0008specialize prime_field_add_exists (p)
  9. 0009specialize prime_field_add_exists (a)
  10. 0010specialize prime_field_add_exists (b)
  11. 0011apply prime_field_add_exists
  12. 0012exact hp
  13. 0013exact ha
  14. 0014exact hb
  15. 0015cases hc
  16. 0016exists x
  17. 0017split
  18. 0018exact hc_witness
  19. 0019intro d
  20. 0020intro hd
  21. 0021specialize prime_field_add_functional (p)
  22. 0022specialize prime_field_add_functional (a)
  23. 0023specialize prime_field_add_functional (b)
  24. 0024specialize prime_field_add_functional (d)
  25. 0025specialize prime_field_add_functional (x)
  26. 0026apply prime_field_add_functional
  27. 0027exact hd
  28. 0028exact hc_witness