TS0023

negative_one_norm_multiple_yields_predecessor_residue

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

If a successor divides r²+1, its predecessor is constructively a quadratic residue.

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 r. p = S n -> (exists ftcn_factor_ftsc_root_norm. (r * r + 1) = (p) * ftcn_factor_ftsc_root_norm) -> (exists qr_x_ftsc_predecessor. exists qr_u_ftsc_predecessor qr_v_ftsc_predecessor. qr_x_ftsc_predecessor * qr_x_ftsc_predecessor + p * qr_u_ftsc_predecessor = n + p * qr_v_ftsc_predecessor)

Constructive proof overview

Generated structural guide

If a successor divides r²+1, its predecessor is constructively a quadratic residue.

The unchanged tactic script uses 5 declared prerequisites and contains 14 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

mul_one Stable theorem; checked-use authorized add_assoc Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized zero_add Stable theorem; checked-use authorized add_succ_left Stable theorem; checked-use authorized

Direct dependents

TS0024 prime_divisible_two_square_norm_unit_coordinate_yields_negative_one_root

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

14 script commands · 5 reading checkpoints · 0 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 r
  4. L4
    intro hpredecessor
  5. L5
    intro hmultiple
02Separate the logical casesL6–6

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

  1. L6
    cases hmultiple
03Construct an explicit witnessL7–9

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

  1. L7
    exists r
  2. L8
    exists 1
  3. L9
    exists x
04Calculate and transport equalitiesL10–13

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

  1. L10
    trans (r * r + 1) + n
  2. L11
    rewrite hpredecessor
  3. L12
    simp [mul_one, add_assoc, add_comm, zero_add, add_succ_left]
  4. L13
    rewrite hmultiple_witness
05Use earlier factsL14–14

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

  1. L14
    apply add_comm

Library-wide reading audit

Original exact command ledger · 14 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro r
  4. 0004intro hpredecessor
  5. 0005intro hmultiple
  6. 0006cases hmultiple
  7. 0007exists r
  8. 0008exists 1
  9. 0009exists x
  10. 0010trans (r * r + 1) + n
  11. 0011rewrite hpredecessor
  12. 0012simp [mul_one, add_assoc, add_comm, zero_add, add_succ_left]
  13. 0013rewrite hmultiple_witness
  14. 0014apply add_comm