FS004M · theorem body

four_square_prime_modular_seed

dependency-curried kernel-checked candidate body; not enrolled in Alpha or Stable

Every prime, including the exceptional prime two, admits explicit constructive witnesses a²+b²+1=p·k without any supplementary hypothesis.

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. Dvd(p,x · x + y · y + 1)

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

PD0003 Dvd PD0004 Prime

In local proof propositions

none
Exact expanded first-order statement
forall p. ((~(p = 1) /\ forall frm_prime_left_fsri_prime frm_prime_right_fsri_prime. p = frm_prime_left_fsri_prime * frm_prime_right_fsri_prime -> frm_prime_left_fsri_prime = 1 \/ frm_prime_right_fsri_prime = 1)) -> exists a b k. a * a + b * b + 1 = p * k

Proof neighborhood

Direct theorem prerequisites

eq_decidable · Stable closed FS004L four_square_non_two_prime_modular_seed

Direct theorem dependents

FS0031 four_square_prime_from_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

16 script commands · 7 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 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 equalitiesL8–8

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

  1. L8
    rewrite hcase_left
05Construct an explicit witnessL9–11

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

  1. L9
    exists 1
  2. L10
    exists 0
  3. L11
    exists 1
06Calculate and transport equalitiesL12–12

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

  1. L12
    norm_num
07Use earlier factsL13–16

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

  1. L13
    specialize four_square_non_two_prime_modular_seed p
  2. L14
    apply four_square_non_two_prime_modular_seed
  3. L15
    exact hprime
  4. L16
    exact hcase_right

Library-wide reading audit

Original defined command ledger · 16 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. 0009exists 1
  10. 0010exists 0
  11. 0011exists 1
  12. 0012norm_num
  13. 0013specialize four_square_non_two_prime_modular_seed p
  14. 0014apply four_square_non_two_prime_modular_seed
  15. 0015exact hprime
  16. 0016exact hcase_right