FP001B

prime_field_negate_exists

Construct a bounded additive inverse by actual signed floor division; the zero input is included.

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)

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

Definition DAG

FpAdd(p,a,b,c) · 1Lt(a,b) · 2Prime(p) · 1

Actual proof prerequisites

signed_integer_floor_exists · checked external prerequisiteprime_nonzero · checked external prerequisiteprime_field_zero_below_primeadd_assoc · checked external prerequisiteadd_comm · checked external prerequisite
Original expanded first-order statement 
forall p a. (~((p) = 1) /\ forall pfa_factor_left_negate_domain pfa_factor_right_negate_domain. (p) = pfa_factor_left_negate_domain * pfa_factor_right_negate_domain -> pfa_factor_left_negate_domain = 1 \/ pfa_factor_right_negate_domain = 1) -> (exists pfa_gap_negate_input. pfa_gap_negate_input + S (a) = (p)) -> exists b. (((exists pfa_gap_negate_existsadditionleft. pfa_gap_negate_existsadditionleft + S (a) = (p)) /\ (((exists pfa_gap_negate_existsadditionright. pfa_gap_negate_existsadditionright + S (b) = (p)) /\ ((((exists pfa_gap_negate_existsadditionresultbound. pfa_gap_negate_existsadditionresultbound + S (0) = (p)) /\ ((exists pfa_offset_left_negate_existsadditionresultcongruence pfa_offset_right_negate_existsadditionresultcongruence. ((a) + (b)) + (p) * pfa_offset_left_negate_existsadditionresultcongruence = (0) + (p) * pfa_offset_right_negate_existsadditionresultcongruence)))))))))

Complete tactic proof in conservative notation

All 33 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

33 script commands · 13 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 (1)
prime field zero below prime
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 hfL5–14

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply signed integer floor exists.

  1. L5
    have hf : ∃ q. ∃ n. ∃ r. 0 + p · n = a + p · q + r ∧ Lt(r,p)Definitions: Lt(r,p)Original native command in the exact edition
  2. L6
    specialize signed_integer_floor_exists (0)
  3. L7
    specialize signed_integer_floor_exists (a)
  4. L8
    specialize signed_integer_floor_exists (p)
  5. L9
    apply signed_integer_floor_exists
  6. L10
    intro hz
  7. L11
    specialize prime_nonzero (p)
  8. L12
    apply prime_nonzero
  9. L13
    exact hp
  10. L14
    exact hz
03Separate the logical casesL15–18

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

  1. L15
    cases hf
  2. L16
    cases hf_witness
  3. L17
    cases hf_witness_witness
  4. L18
    cases hf_witness_witness_witness
04Construct an explicit witnessL19–19

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

  1. L19
    exists x2
05Separate the logical casesL20–20

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

  1. L20
    split
06Use earlier factsL21–21

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

  1. L21
    exact ha
07Separate the logical casesL22–22

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

  1. L22
    split
08Use earlier factsL23–23

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

  1. L23
    exact hf_witness_witness_witness_right
09Separate the logical casesL24–24

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

  1. L24
    split
10Use earlier factsL25–27

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

  1. L25
    specialize prime_field_zero_below_prime (p)
  2. L26
    apply prime_field_zero_below_prime
  3. L27
    exact hp
11Construct an explicit witnessL28–29

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

  1. L28
    exists x
  2. L29
    exists x1
12Calculate and transport equalitiesL30–32

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L30
    trans (a + p * x) + x2
  2. L31
    simp [add_assoc, add_comm]
  3. L32
    symm
13Use earlier factsL33–33

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

  1. L33
    exact hf_witness_witness_witness_left

Library-wide reading audit

Original defined command ledger · 33 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro hp
  4. 0004intro ha
  5. 0005have hf : ∃ q. ∃ n. ∃ r. 0 + p · n = a + p · q + r ∧ Lt(r,p)
  6. 0006specialize signed_integer_floor_exists (0)
  7. 0007specialize signed_integer_floor_exists (a)
  8. 0008specialize signed_integer_floor_exists (p)
  9. 0009apply signed_integer_floor_exists
  10. 0010intro hz
  11. 0011specialize prime_nonzero (p)
  12. 0012apply prime_nonzero
  13. 0013exact hp
  14. 0014exact hz
  15. 0015cases hf
  16. 0016cases hf_witness
  17. 0017cases hf_witness_witness
  18. 0018cases hf_witness_witness_witness
  19. 0019exists x2
  20. 0020split
  21. 0021exact ha
  22. 0022split
  23. 0023exact hf_witness_witness_witness_right
  24. 0024split
  25. 0025specialize prime_field_zero_below_prime (p)
  26. 0026apply prime_field_zero_below_prime
  27. 0027exact hp
  28. 0028exists x
  29. 0029exists x1
  30. 0030trans (a + p * x) + x2
  31. 0031simp [add_assoc, add_comm]
  32. 0032symm
  33. 0033exact hf_witness_witness_witness_left