TS0005

prime_mod_four_one_minus_one_square_exists

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

Every prime congruent to one modulo four has a constructive square root of minus one.

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 original first-admission records.

Exact expanded first-order arithmetic statement

forall p n. p = S n -> ((~(p = 1) /\ forall frm_prime_left_fts_prime frm_prime_right_fts_prime. p = frm_prime_left_fts_prime * frm_prime_right_fts_prime -> frm_prime_left_fts_prime = 1 \/ frm_prime_right_fts_prime = 1)) -> (exists fts_four_prime_one. p = 4 * fts_four_prime_one + 1) -> (exists qr_x_fts_predecessor. exists qr_u_fts_predecessor qr_v_fts_predecessor. qr_x_fts_predecessor * qr_x_fts_predecessor + p * qr_u_fts_predecessor = n + p * qr_v_fts_predecessor)

Constructive proof overview

Generated structural guide

Every prime congruent to one modulo four has a constructive square root of minus one.

The unchanged tactic script uses 2 declared prerequisites and contains 19 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

mod4_one_is_odd Stable theorem; checked-use authorized quadratic_supplement_minus_one_residue_iff_mod_four_one Alpha theorem; checked-use authorized

Direct dependents

TS0006 prime_mod_four_one_bounded_minus_one_square_exists TS0008 prime_mod_four_one_divisible_two_square_norm_exists

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

19 script commands · 5 reading checkpoints · 2 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.

01Fix variables and assumptionsL1–5

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro hpredecessor
  4. L4
    intro hprime
  5. L5
    intro hfourone
02Establish hoddL6–9

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mod4 one is odd.

  1. L6
    have hodd : exists h. p = 2 * h + 1
  2. L7
    specialize mod4_one_is_odd p
  3. L8
    apply mod4_one_is_odd
  4. L9
    exact hfourone
03Establish hlawL10–16

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply quadratic supplement minus one residue iff mod four one.

  1. L10
    have hlaw : (((exists qr_x_fts_predecessor. exists qr_u_fts_predecessor qr_v_fts_predecessor. qr_x_fts_predecessor * qr_x_fts_predecessor + p * qr_u_fts_predecessor = n + p * qr_v_fts_predecessor) -> (exists fts_four_prime_one. p = 4 * fts_four_prime_one + 1)) /\ ((exists fts_four_prime_one. p = 4 * fts_four_prime_one + 1) -> (exists qr_x_fts_predecessor. exists qr_u_fts_predecessor qr_v_fts_predecessor. qr_x_fts_predecessor * qr_x_fts_predecessor + p * qr_u_fts_predecessor = n + p * qr_v_fts_predecessor)))
  2. L11
    specialize quadratic_supplement_minus_one_residue_iff_mod_four_one p
  3. L12
    specialize quadratic_supplement_minus_one_residue_iff_mod_four_one n
  4. L13
    apply quadratic_supplement_minus_one_residue_iff_mod_four_one
  5. L14
    exact hpredecessor
  6. L15
    exact hprime
  7. L16
    exact hodd
04Separate the logical casesL17–17

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

  1. L17
    cases hlaw
05Use earlier factsL18–19

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

  1. L18
    apply hlaw_right
  2. L19
    exact hfourone

Library-wide reading audit

Original exact command ledger · 19 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro hpredecessor
  4. 0004intro hprime
  5. 0005intro hfourone
  6. 0006have hodd : exists h. p = 2 * h + 1
  7. 0007specialize mod4_one_is_odd p
  8. 0008apply mod4_one_is_odd
  9. 0009exact hfourone
  10. 0010have hlaw : (((exists qr_x_fts_predecessor. exists qr_u_fts_predecessor qr_v_fts_predecessor. qr_x_fts_predecessor * qr_x_fts_predecessor + p * qr_u_fts_predecessor = n + p * qr_v_fts_predecessor) -> (exists fts_four_prime_one. p = 4 * fts_four_prime_one + 1)) /\ ((exists fts_four_prime_one. p = 4 * fts_four_prime_one + 1) -> (exists qr_x_fts_predecessor. exists qr_u_fts_predecessor qr_v_fts_predecessor. qr_x_fts_predecessor * qr_x_fts_predecessor + p * qr_u_fts_predecessor = n + p * qr_v_fts_predecessor)))
  11. 0011specialize quadratic_supplement_minus_one_residue_iff_mod_four_one p
  12. 0012specialize quadratic_supplement_minus_one_residue_iff_mod_four_one n
  13. 0013apply quadratic_supplement_minus_one_residue_iff_mod_four_one
  14. 0014exact hpredecessor
  15. 0015exact hprime
  16. 0016exact hodd
  17. 0017cases hlaw
  18. 0018apply hlaw_right
  19. 0019exact hfourone