FS000P · theorem body

four_square_prime_bounded_modular_seed

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

Every prime, including two, has actual witnesses a²+b²+1=p·k with the constructive strict bound k<p.

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) → ∃ 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

none
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)) -> (exists fsbs_a_prime_bounded fsbs_b_prime_bounded fsbs_k_prime_bounded. ((fsbs_a_prime_bounded * fsbs_a_prime_bounded + fsbs_b_prime_bounded * fsbs_b_prime_bounded + 1 = p * fsbs_k_prime_bounded) /\ (exists fsbs_lt_gap_prime_bounded_multiplier. fsbs_lt_gap_prime_bounded_multiplier + S (fsbs_k_prime_bounded) = (p))))

Proof neighborhood

Direct theorem prerequisites

eq_decidable · Stable closed FS000O four_square_non_two_prime_bounded_modular_seed

Direct theorem dependents

FS0035 four_square_prime_from_bounded_strict_descent

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

20 script commands · 10 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 non two prime bounded modular seed
01Fix variables and assumptionsL1–2

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

  1. L1
    intro p
  2. L2
    intro hprime
02Establish hcaseL3–6

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

  1. L3
    have hcase : p = 2 \/ ~(p = 2)
  2. L4
    specialize eq_decidable p
  3. L5
    specialize eq_decidable 2
  4. L6
    exact eq_decidable
03Separate the logical casesL7–7

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

  1. L7
    cases hcase
04Calculate and transport equalitiesL8–9

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

  1. L8
    rewrite hcase_left
  2. L9
    rewrite hcase_left
05Construct an explicit witnessL10–12

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

  1. L10
    exists 1
  2. L11
    exists 0
  3. L12
    exists 1
06Separate the logical casesL13–13

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

  1. L13
    split
07Calculate and transport equalitiesL14–14

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

  1. L14
    norm_num
08Construct an explicit witnessL15–15

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

  1. L15
    exists 0
09Calculate and transport equalitiesL16–16

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

  1. L16
    norm_num
10Use earlier factsL17–20

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

  1. L17
    specialize four_square_non_two_prime_bounded_modular_seed p
  2. L18
    apply four_square_non_two_prime_bounded_modular_seed
  3. L19
    exact hprime
  4. L20
    exact hcase_right

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro p
  2. 0002intro hprime
  3. 0003have hcase : p = 2 \/ ~(p = 2)
  4. 0004specialize eq_decidable p
  5. 0005specialize eq_decidable 2
  6. 0006exact eq_decidable
  7. 0007cases hcase
  8. 0008rewrite hcase_left
  9. 0009rewrite hcase_left
  10. 0010exists 1
  11. 0011exists 0
  12. 0012exists 1
  13. 0013split
  14. 0014norm_num
  15. 0015exists 0
  16. 0016norm_num
  17. 0017specialize four_square_non_two_prime_bounded_modular_seed p
  18. 0018apply four_square_non_two_prime_bounded_modular_seed
  19. 0019exact hprime
  20. 0020exact hcase_right