FS003H · theorem body

four_square_prime_case_reduction

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

Constructive prime trichotomy reduces representation of every prime exactly to the still-open three-modulo-four prime case.

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

(∀ x. Prime(x) → Mod4Three(x) → ∃ y. ∃ z. ∃ n. ∃ m. x = y · y + z · z + n · n + m · m) → ∀ x. Prime(x) → ∃ y. ∃ z. ∃ n. ∃ m. x = y · y + z · z + n · n + m · m

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

PD0004 Prime PD0012 Mod4Three

In local proof propositions

PD0011 Mod4One PD0012 Mod4Three
Exact expanded first-order statement
(forall fsl_three_prime_universal. ((~(fsl_three_prime_universal = 1) /\ forall frm_prime_left_fsl_three_universal frm_prime_right_fsl_three_universal. fsl_three_prime_universal = frm_prime_left_fsl_three_universal * frm_prime_right_fsl_three_universal -> frm_prime_left_fsl_three_universal = 1 \/ frm_prime_right_fsl_three_universal = 1)) -> (exists fsl_three_residue_universal. fsl_three_prime_universal = 4 * fsl_three_residue_universal + 3) -> (exists fsl_a_three_universal fsl_b_three_universal fsl_c_three_universal fsl_d_three_universal. (fsl_three_prime_universal) = fsl_a_three_universal * fsl_a_three_universal + fsl_b_three_universal * fsl_b_three_universal + fsl_c_three_universal * fsl_c_three_universal + fsl_d_three_universal * fsl_d_three_universal)) -> (forall fsl_prime_universal. ((~(fsl_prime_universal = 1) /\ forall frm_prime_left_fsl_all_universal frm_prime_right_fsl_all_universal. fsl_prime_universal = frm_prime_left_fsl_all_universal * frm_prime_right_fsl_all_universal -> frm_prime_left_fsl_all_universal = 1 \/ frm_prime_right_fsl_all_universal = 1)) -> (exists fsl_a_all_universal fsl_b_all_universal fsl_c_all_universal fsl_d_all_universal. (fsl_prime_universal) = fsl_a_all_universal * fsl_a_all_universal + fsl_b_all_universal * fsl_b_all_universal + fsl_c_all_universal * fsl_c_all_universal + fsl_d_all_universal * fsl_d_all_universal))

Proof neighborhood

Direct theorem prerequisites

prime_mod_four_good_or_three · Alpha closed FS003E four_square_prime_two_or_one_mod_four

Direct theorem dependents

FS003K four_square_lagrange_from_three_mod_four_primes

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 · 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 prime two or one mod four
01Fix variables and assumptionsL1–3

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

  1. L1
    intro hthree
  2. L2
    intro p
  3. L3
    intro hprime
02Establish hclassesL4–7

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime mod four good or three.

  1. L4
    have hclasses : p = 2 ∨ Mod4One(p) ∨ Mod4Three(p)Definitions: Mod4One(p)Mod4Three(p)Original native command in the exact edition
  2. L5
    specialize prime_mod_four_good_or_three p
  3. L6
    apply prime_mod_four_good_or_three
  4. L7
    exact hprime
03Separate the logical casesL8–8

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

  1. L8
    cases hclasses
04Use earlier factsL9–16

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

  1. L9
    specialize four_square_prime_two_or_one_mod_four p
  2. L10
    apply four_square_prime_two_or_one_mod_four
  3. L11
    exact hprime
  4. L12
    exact hclasses_left
  5. L13
    specialize hthree p
  6. L14
    apply hthree
  7. L15
    exact hprime
  8. L16
    exact hclasses_right

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro hthree
  2. 0002intro p
  3. 0003intro hprime
  4. 0004have hclasses : p = 2 ∨ Mod4One(p) ∨ Mod4Three(p)
    Exact native replay linehave hclasses : ((p = 2 \/ exists k. p = 4 * k + 1) \/ exists k. p = 4 * k + 3)
  5. 0005specialize prime_mod_four_good_or_three p
  6. 0006apply prime_mod_four_good_or_three
  7. 0007exact hprime
  8. 0008cases hclasses
  9. 0009specialize four_square_prime_two_or_one_mod_four p
  10. 0010apply four_square_prime_two_or_one_mod_four
  11. 0011exact hprime
  12. 0012exact hclasses_left
  13. 0013specialize hthree p
  14. 0014apply hthree
  15. 0015exact hprime
  16. 0016exact hclasses_right