TS0006

prime_mod_four_one_bounded_minus_one_square_exists

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

Every prime congruent to one modulo four has a canonical root of minus one strictly below the prime.

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_bounded_predecessor. (exists qr_h_fts_bounded_predecessor. qr_h_fts_bounded_predecessor + S qr_x_fts_bounded_predecessor = p) /\ exists qr_u_fts_bounded_predecessor qr_v_fts_bounded_predecessor. qr_x_fts_bounded_predecessor * qr_x_fts_bounded_predecessor + p * qr_u_fts_bounded_predecessor = n + p * qr_v_fts_bounded_predecessor)

Constructive proof overview

Generated structural guide

Every prime congruent to one modulo four has a canonical root of minus one strictly below the prime.

The unchanged tactic script uses 2 declared prerequisites and contains 25 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

TS0005 prime_mod_four_one_minus_one_square_exists quadratic_residue_bounded_equiv Stable theorem; checked-use authorized

Direct dependents

TS0009 prime_mod_four_one_bounded_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

25 script commands · 6 reading checkpoints · 3 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 mod four one minus one square exists

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

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 hresidueL6–12

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime mod four one minus one square exists.

  1. L6
    have hresidue : 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. L7
    specialize prime_mod_four_one_minus_one_square_exists p
  3. L8
    specialize prime_mod_four_one_minus_one_square_exists n
  4. L9
    apply prime_mod_four_one_minus_one_square_exists
  5. L10
    exact hpredecessor
  6. L11
    exact hprime
  7. L12
    exact hfourone
03Establish hnonzeroL13–17

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply PA1.

  1. L13
    have hnonzero : ~(p = 0)
  2. L14
    intro hzero
  3. L15
    rewrite hpredecessor at hzero
  4. L16
    apply PA1
  5. L17
    exact hzero
04Establish hequivalenceL18–22

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply quadratic residue bounded equiv.

  1. L18
    have hequivalence : (QRes(p,n) → BoundedQRes(p,n)) ∧ (BoundedQRes(p,n) → QRes(p,n))Definitions: QResBoundedQRes
  2. L19
    specialize quadratic_residue_bounded_equiv p
  3. L20
    specialize quadratic_residue_bounded_equiv n
  4. L21
    apply quadratic_residue_bounded_equiv
  5. L22
    exact hnonzero
05Separate the logical casesL23–23

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

  1. L23
    cases hequivalence
06Use earlier factsL24–25

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

  1. L24
    apply hequivalence_left
  2. L25
    exact hresidue

Library-wide reading audit

Original exact command ledger · 25 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro hpredecessor
  4. 0004intro hprime
  5. 0005intro hfourone
  6. 0006have hresidue : 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
  7. 0007specialize prime_mod_four_one_minus_one_square_exists p
  8. 0008specialize prime_mod_four_one_minus_one_square_exists n
  9. 0009apply prime_mod_four_one_minus_one_square_exists
  10. 0010exact hpredecessor
  11. 0011exact hprime
  12. 0012exact hfourone
  13. 0013have hnonzero : ~(p = 0)
  14. 0014intro hzero
  15. 0015rewrite hpredecessor at hzero
  16. 0016apply PA1
  17. 0017exact hzero
  18. 0018have hequivalence : (((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 qr_x_fts_bounded_predecessor. (exists qr_h_fts_bounded_predecessor. qr_h_fts_bounded_predecessor + S qr_x_fts_bounded_predecessor = p) /\ exists qr_u_fts_bounded_predecessor qr_v_fts_bounded_predecessor. qr_x_fts_bounded_predecessor * qr_x_fts_bounded_predecessor + p * qr_u_fts_bounded_predecessor = n + p * qr_v_fts_bounded_predecessor)) /\ ((exists qr_x_fts_bounded_predecessor. (exists qr_h_fts_bounded_predecessor. qr_h_fts_bounded_predecessor + S qr_x_fts_bounded_predecessor = p) /\ exists qr_u_fts_bounded_predecessor qr_v_fts_bounded_predecessor. qr_x_fts_bounded_predecessor * qr_x_fts_bounded_predecessor + p * qr_u_fts_bounded_predecessor = n + p * qr_v_fts_bounded_predecessor) -> (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)))
  19. 0019specialize quadratic_residue_bounded_equiv p
  20. 0020specialize quadratic_residue_bounded_equiv n
  21. 0021apply quadratic_residue_bounded_equiv
  22. 0022exact hnonzero
  23. 0023cases hequivalence
  24. 0024apply hequivalence_left
  25. 0025exact hresidue