FS000O · theorem body

four_square_non_two_prime_bounded_modular_seed

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

Every prime other than two has a strictly prime-bounded constructive modular four-square seed.

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. Prime(p) → ¬p = 2 → ∃ x. ∃ y. ∃ z. x · x + y · y + 1 = p · z ∧ Lt(z,p)

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

PD0010 Odd
Exact expanded first-order statement
forall p. ((~(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)) -> ~(p = 2) -> (exists fsbs_a_non_two_bounded fsbs_b_non_two_bounded fsbs_k_non_two_bounded. ((fsbs_a_non_two_bounded * fsbs_a_non_two_bounded + fsbs_b_non_two_bounded * fsbs_b_non_two_bounded + 1 = p * fsbs_k_non_two_bounded) /\ (exists fsbs_lt_gap_non_two_bounded_multiplier. fsbs_lt_gap_non_two_bounded_multiplier + S (fsbs_k_non_two_bounded) = (p))))

Proof neighborhood

Direct theorem prerequisites

prime_ne_two_is_odd · Stable closed FS000N four_square_odd_prime_bounded_modular_seed

Direct theorem dependents

FS000P four_square_prime_bounded_modular_seed

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

14 script commands · 4 reading checkpoints · 1 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.

Named ingredients (1)
four square odd prime bounded modular seed
01Fix variables and assumptionsL1–3

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

  1. L1
    intro p
  2. L2
    intro hprime
  3. L3
    intro hnot_two
02Establish hoddL4–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime ne two is odd.

  1. L4
  2. L5
    specialize prime_ne_two_is_odd p
  3. L6
    apply prime_ne_two_is_odd
  4. L7
    exact hprime
  5. L8
    exact hnot_two
03Separate the logical casesL9–9

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

  1. L9
    cases hodd
04Use earlier factsL10–14

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

  1. L10
    specialize four_square_odd_prime_bounded_modular_seed p
  2. L11
    specialize four_square_odd_prime_bounded_modular_seed x
  3. L12
    apply four_square_odd_prime_bounded_modular_seed
  4. L13
    exact hodd_witness
  5. L14
    exact hprime

Library-wide reading audit

Original defined command ledger · 14 lines
  1. 0001intro p
  2. 0002intro hprime
  3. 0003intro hnot_two
  4. 0004have hodd : Odd(p)
    Exact native replay linehave hodd : exists h. p = 2 * h + 1
  5. 0005specialize prime_ne_two_is_odd p
  6. 0006apply prime_ne_two_is_odd
  7. 0007exact hprime
  8. 0008exact hnot_two
  9. 0009cases hodd
  10. 0010specialize four_square_odd_prime_bounded_modular_seed p
  11. 0011specialize four_square_odd_prime_bounded_modular_seed x
  12. 0012apply four_square_odd_prime_bounded_modular_seed
  13. 0013exact hodd_witness
  14. 0014exact hprime