FS000L · theorem body

four_square_odd_prime_half_positive

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

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

historical independently replay-verified empty-context experiment; the experiment itself persisted no certificate and granted no release authority; current checked use follows separately sealed proof bundles; no Stable promotion.

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.

Statement with defined notation

∀ p. ∀ h. p = 2 · h + 1 → Prime(p) → Lt(0,h)

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

PD0002 Lt PD0004 Prime

In local proof propositions

none
Exact expanded first-order 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))

Proof neighborhood

Direct theorem prerequisites

nonzero_is_succ · Stable closed

Direct theorem dependents

FS000M four_square_odd_prime_half_seed_norm_strict

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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