FS0044

four_square_two_half_ranges_overflow_odd

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

Two inclusive odd half-ranges together have strictly more entries than the modulus.

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 h. p = 2 * h + 1 -> (exists fsri_gap_overflow. fsri_gap_overflow + S (p) = (S h + S h))

Constructive proof overview

Generated structural guide

Two inclusive odd half-ranges together have strictly more entries than the modulus.

The unchanged tactic script uses 5 declared prerequisites and contains 6 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

Historical empty-context replay experiment only; that experiment persisted no certificate and granted no release authority. Current checked use follows separately sealed, independently verified proof bundles; there is no Stable promotion.

Proof neighborhood

Direct dependencies

mul_succ_left Stable theorem; checked-use authorized mul_zero_left Stable theorem; checked-use authorized zero_add Stable theorem; checked-use authorized add_succ_left Stable theorem; checked-use authorized add_assoc Stable theorem; checked-use authorized

Direct dependents

FS000K four_square_odd_prime_half_coordinate_seed FS004K four_square_odd_prime_modular_seed

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

6 script commands · 4 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 assumptionsL1–3

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

  1. L1
    intro p
  2. L2
    intro h
  3. L3
    intro hodd
02Calculate and transport equalitiesL4–4

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

  1. L4
    rewrite hodd
03Construct an explicit witnessL5–5

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

  1. L5
    exists 0
04Calculate and transport equalitiesL6–6

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

  1. L6
    simp [mul_succ_left, mul_zero_left, zero_add, add_succ_left, add_assoc]

Library-wide reading audit

Original exact command ledger · 6 lines
  1. 0001intro p
  2. 0002intro h
  3. 0003intro hodd
  4. 0004rewrite hodd
  5. 0005exists 0
  6. 0006simp [mul_succ_left, mul_zero_left, zero_add, add_succ_left, add_assoc]