FS003H

four_square_prime_case_reduction

Dependency-curried candidate body; not Alpha-enrolled; no checked-use authority

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.

Exact expanded first-order arithmetic 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))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 16 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

prime_mod_four_good_or_three Alpha theorem; checked-use authorized FS003E four_square_prime_two_or_one_mod_four

Direct dependents

FS003K four_square_lagrange_from_three_mod_four_primes

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This dependency-curried candidate body does not grant checked theorem use or Stable membership.

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.

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 \/ exists k. p = 4 * k + 1) \/ exists k. p = 4 * k + 3)
  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 exact command ledger · 16 lines
  1. 0001intro hthree
  2. 0002intro p
  3. 0003intro hprime
  4. 0004have 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