FS000L

four_square_odd_prime_half_positive

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

The half h of an odd prime p=2h+1 is constructively at least 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 h. p = 2 * h + 1 -> ((~(p = 1) /\ forall frm_prime_left_fsbs_prime frm_prime_right_fsbs_prime. p = frm_prime_left_fsbs_prime * frm_prime_right_fsbs_prime -> frm_prime_left_fsbs_prime = 1 \/ frm_prime_right_fsbs_prime = 1)) -> (exists fsbs_le_gap_positive_half. fsbs_le_gap_positive_half + (1) = (h))

Constructive proof overview

Generated structural guide

The half h of an odd prime p=2h+1 is constructively at least one.

The unchanged tactic script uses 1 declared prerequisite 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

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

nonzero_is_succ Stable theorem; checked-use authorized

Direct dependents

FS000M four_square_odd_prime_half_seed_norm_strict

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 · 9 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 assumptionsL1–4

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

  1. L1
    intro p
  2. L2
    intro h
  3. L3
    intro hodd
  4. L4
    intro hprime
02Establish hnonzeroL5–6

Establish this local claim before using it. It is not an additional assumption.

  1. L5
    have hnonzero : ~(h = 0)
  2. L6
    intro hzero
03Separate the logical casesL7–7

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

  1. L7
    cases hprime
04Use earlier factsL8–8

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

  1. L8
    apply hprime_left
05Calculate and transport equalitiesL9–11

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

  1. L9
    rewrite hodd
  2. L10
    rewrite hzero
  3. L11
    norm_num
06Establish hsuccessorL12–15

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

  1. L12
    have hsuccessor : exists t. h = S t
  2. L13
    specialize nonzero_is_succ h
  3. L14
    apply nonzero_is_succ
  4. L15
    exact hnonzero
07Separate the logical casesL16–16

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

  1. L16
    cases hsuccessor
08Construct an explicit witnessL17–17

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

  1. L17
    exists x
09Calculate and transport equalitiesL18–19

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

  1. L18
    rewrite hsuccessor_witness
  2. L19
    simp

Library-wide reading audit

Original exact command ledger · 19 lines
  1. 0001intro p
  2. 0002intro h
  3. 0003intro hodd
  4. 0004intro hprime
  5. 0005have hnonzero : ~(h = 0)
  6. 0006intro hzero
  7. 0007cases hprime
  8. 0008apply hprime_left
  9. 0009rewrite hodd
  10. 0010rewrite hzero
  11. 0011norm_num
  12. 0012have hsuccessor : exists t. h = S t
  13. 0013specialize nonzero_is_succ h
  14. 0014apply nonzero_is_succ
  15. 0015exact hnonzero
  16. 0016cases hsuccessor
  17. 0017exists x
  18. 0018rewrite hsuccessor_witness
  19. 0019simp