FS001J · theorem body

four_square_descent_three_mod_four_primes_from_seed_and_step

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

All remaining three-modulo-four primes are represented once their modular square seeds and the decreasing quaternion step are supplied.

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. Dvd(x,y · y + z · z + 1)) → (∀ x. ∀ y. Prime(x) → ¬y = 0 → ¬y = 1 → (∃ z. ∃ n. ∃ m. ∃ k. x · y = z · z + n · n + m · m + k · k) → ∃ z. ¬z = 0 ∧ (Lt(z,y) ∧ (∃ n. ∃ m. ∃ k. ∃ i. x · z = n · n + m · m + k · k + i · i))) → ∀ x. Prime(x) → Mod4Three(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

PD0002 Lt PD0003 Dvd PD0004 Prime PD0012 Mod4Three

In local proof propositions

PD0003 Dvd PD0004 Prime
Exact expanded first-order statement
(forall fsd_seed_prime_universal. ((~(fsd_seed_prime_universal = 1) /\ forall frm_prime_left_fsd_seed_universal frm_prime_right_fsd_seed_universal. fsd_seed_prime_universal = frm_prime_left_fsd_seed_universal * frm_prime_right_fsd_seed_universal -> frm_prime_left_fsd_seed_universal = 1 \/ frm_prime_right_fsd_seed_universal = 1)) -> (exists fsd_seed_residue_universal. fsd_seed_prime_universal = 4 * fsd_seed_residue_universal + 3) -> exists fsd_seed_first_universal fsd_seed_second_universal fsd_seed_multiplier_universal. fsd_seed_first_universal * fsd_seed_first_universal + fsd_seed_second_universal * fsd_seed_second_universal + 1 = fsd_seed_prime_universal * fsd_seed_multiplier_universal) -> (forall fsd_prime_universal fsd_multiplier_universal. ((~(fsd_prime_universal = 1) /\ forall frm_prime_left_fsd_universal_prime frm_prime_right_fsd_universal_prime. fsd_prime_universal = frm_prime_left_fsd_universal_prime * frm_prime_right_fsd_universal_prime -> frm_prime_left_fsd_universal_prime = 1 \/ frm_prime_right_fsd_universal_prime = 1)) -> ~(fsd_multiplier_universal = 0) -> ~(fsd_multiplier_universal = 1) -> (exists fsl_a_fsd_universal_source fsl_b_fsd_universal_source fsl_c_fsd_universal_source fsl_d_fsd_universal_source. (fsd_prime_universal * fsd_multiplier_universal) = fsl_a_fsd_universal_source * fsl_a_fsd_universal_source + fsl_b_fsd_universal_source * fsl_b_fsd_universal_source + fsl_c_fsd_universal_source * fsl_c_fsd_universal_source + fsl_d_fsd_universal_source * fsl_d_fsd_universal_source) -> exists fsd_smaller_universal. (~(fsd_smaller_universal = 0) /\ ((exists fsd_gap_universal. fsd_gap_universal + S fsd_smaller_universal = fsd_multiplier_universal) /\ (exists fsl_a_fsd_universal_target fsl_b_fsd_universal_target fsl_c_fsd_universal_target fsl_d_fsd_universal_target. (fsd_prime_universal * fsd_smaller_universal) = fsl_a_fsd_universal_target * fsl_a_fsd_universal_target + fsl_b_fsd_universal_target * fsl_b_fsd_universal_target + fsl_c_fsd_universal_target * fsl_c_fsd_universal_target + fsl_d_fsd_universal_target * fsl_d_fsd_universal_target)))) -> (forall p. ((~(p = 1) /\ forall frm_prime_left_fsd_bad_prime frm_prime_right_fsd_bad_prime. p = frm_prime_left_fsd_bad_prime * frm_prime_right_fsd_bad_prime -> frm_prime_left_fsd_bad_prime = 1 \/ frm_prime_right_fsd_bad_prime = 1)) -> (exists t. p = 4 * t + 3) -> (exists fsl_a_fsd_bad_result fsl_b_fsd_bad_result fsl_c_fsd_bad_result fsl_d_fsd_bad_result. (p) = fsl_a_fsd_bad_result * fsl_a_fsd_bad_result + fsl_b_fsd_bad_result * fsl_b_fsd_bad_result + fsl_c_fsd_bad_result * fsl_c_fsd_bad_result + fsl_d_fsd_bad_result * fsl_d_fsd_bad_result))

Proof neighborhood

Direct theorem prerequisites

FS001I four_square_descent_prime_from_modular_seed_and_step

Direct theorem dependents

FS001K four_square_lagrange_from_modular_seeds_and_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

23 script commands · 5 reading checkpoints · 2 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 descent prime from modular seed and step
01Fix variables and assumptionsL1–5

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

  1. L1
    intro hseeds
  2. L2
    intro hstep
  3. L3
    intro p
  4. L4
    intro hprime
  5. L5
    intro hclass
02Use earlier factsL6–6

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

  1. L6
    specialize hseeds p
03Establish hseedL7–10

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hseeds.

  1. L7
    have hseed : ∃ x. ∃ y. Dvd(p,x · x + y · y + 1)Definitions: Dvd(p,x · x + y · y + 1)Original native command in the exact edition
  2. L8
    apply hseeds
  3. L9
    exact hprime
  4. L10
    exact hclass
04Separate the logical casesL11–13

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

  1. L11
    cases hseed
  2. L12
    cases hseed_witness
  3. L13
    cases hseed_witness_witness
05Establish hseed_descentL14–23

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply four square descent prime from modular seed and step.

  1. L14
    have hseed_descent : ∀ p. ∀ x. ∀ y. ∀ k. Prime(p) → x · x + y · y + 1 = p · k → ∃ z. ∃ n. ∃ m. ∃ i. p = z · z + n · n + m · m + i · iDefinitions: Prime(p)Original native command in the exact edition
  2. L15
    apply four_square_descent_prime_from_modular_seed_and_step
  3. L16
    exact hstep
  4. L17
    specialize hseed_descent p
  5. L18
    specialize hseed_descent x
  6. L19
    specialize hseed_descent x1
  7. L20
    specialize hseed_descent x2
  8. L21
    apply hseed_descent
  9. L22
    exact hprime
  10. L23
    exact hseed_witness_witness_witness

Library-wide reading audit

Original defined command ledger · 23 lines
  1. 0001intro hseeds
  2. 0002intro hstep
  3. 0003intro p
  4. 0004intro hprime
  5. 0005intro hclass
  6. 0006specialize hseeds p
  7. 0007have hseed : ∃ x. ∃ y. Dvd(p,x · x + y · y + 1)
    Exact native replay linehave hseed : exists x y k. x * x + y * y + 1 = p * k
  8. 0008apply hseeds
  9. 0009exact hprime
  10. 0010exact hclass
  11. 0011cases hseed
  12. 0012cases hseed_witness
  13. 0013cases hseed_witness_witness
  14. 0014have hseed_descent : ∀ p. ∀ x. ∀ y. ∀ k. Prime(p) → x · x + y · y + 1 = p · k → ∃ z. ∃ n. ∃ m. ∃ i. p = z · z + n · n + m · m + i · i
    Exact native replay linehave hseed_descent : forall p x y k. ((~(p = 1) /\ forall frm_prime_left_fsd_seed_local_prime frm_prime_right_fsd_seed_local_prime. p = frm_prime_left_fsd_seed_local_prime * frm_prime_right_fsd_seed_local_prime -> frm_prime_left_fsd_seed_local_prime = 1 \/ frm_prime_right_fsd_seed_local_prime = 1)) -> x * x + y * y + 1 = p * k -> (exists fsl_a_fsd_seed_local_result fsl_b_fsd_seed_local_result fsl_c_fsd_seed_local_result fsl_d_fsd_seed_local_result. (p) = fsl_a_fsd_seed_local_result * fsl_a_fsd_seed_local_result + fsl_b_fsd_seed_local_result * fsl_b_fsd_seed_local_result + fsl_c_fsd_seed_local_result * fsl_c_fsd_seed_local_result + fsl_d_fsd_seed_local_result * fsl_d_fsd_seed_local_result)
  15. 0015apply four_square_descent_prime_from_modular_seed_and_step
  16. 0016exact hstep
  17. 0017specialize hseed_descent p
  18. 0018specialize hseed_descent x
  19. 0019specialize hseed_descent x1
  20. 0020specialize hseed_descent x2
  21. 0021apply hseed_descent
  22. 0022exact hprime
  23. 0023exact hseed_witness_witness_witness