FP001B

prime_field_negate_exists

Public research checkpoint, not admitted to Alpha or Stable. Alpha v30 remains 3222 checked-use theorems; Stable remains 432. The on-demand Alpha Lean service does not yet expose these checkpoint theorems; their independently checked literal bundles and unchanged sources are available below.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

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

Exact expanded first-order arithmetic 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)))))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 5 declared prerequisites and contains 33 exact native proof lines.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

Proof neighborhood

Direct dependencies

signed_integer_floor_exists Alpha theorem; checked-use authorized prime_nonzero Alpha theorem; checked-use authorized FP0002 prime_field_zero_below_prime add_assoc Alpha theorem; checked-use authorized add_comm Alpha theorem; checked-use authorized

Direct dependents

FP001D prime_field_negate_exists_unique FP0035 prime_field_negate_table_exists

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. The literal dependency-closed bundle is checked by original HA and the independently compiled Lean verifier. Public delivery grants no Alpha checked-use authority or Stable membership.

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.

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 : exists q n r. (0 + p * n = (a + p * q) + r /\ (exists pfa_gap_negate_floor. pfa_gap_negate_floor + S (r) = (p)))
  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 exact command ledger · 33 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro hp
  4. 0004intro ha
  5. 0005have hf : exists q n r. (0 + p * n = (a + p * q) + r /\ (exists pfa_gap_negate_floor. pfa_gap_negate_floor + S (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