FS001I · theorem body

four_square_descent_prime_from_modular_seed_and_step

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

A concrete modular square seed and the explicit strict multiplier step suffice for an actual representation of the prime.

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. ∀ 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. ∀ y. ∀ z. ∀ n. Prime(x) → y · y + z · z + 1 = x · n → ∃ m. ∃ k. ∃ i. ∃ j. x = m · m + k · k + i · i + j · j

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

PD0004 Prime
Exact expanded first-order statement
(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 x y k. ((~(p = 1) /\ forall frm_prime_left_fsd_p frm_prime_right_fsd_p. p = frm_prime_left_fsd_p * frm_prime_right_fsd_p -> frm_prime_left_fsd_p = 1 \/ frm_prime_right_fsd_p = 1)) -> x * x + y * y + 1 = p * k -> (exists fsl_a_fsd_prime_result fsl_b_fsd_prime_result fsl_c_fsd_prime_result fsl_d_fsd_prime_result. (p) = fsl_a_fsd_prime_result * fsl_a_fsd_prime_result + fsl_b_fsd_prime_result * fsl_b_fsd_prime_result + fsl_c_fsd_prime_result * fsl_c_fsd_prime_result + fsl_d_fsd_prime_result * fsl_d_fsd_prime_result)

Proof neighborhood

Direct theorem prerequisites

FS001F four_square_descent_modular_seed_multiplier_nonzero FS003F four_square_prime_modular_seed_multiple FS001H four_square_descent_prime_from_strict_step

Direct theorem dependents

FS001J four_square_descent_three_mod_four_primes_from_seed_and_step 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

28 script commands · 5 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 (3)
four square descent modular seed multiplier nonzerofour square prime modular seed multiplefour square descent prime from strict step
01Fix variables and assumptionsL1–7

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

  1. L1
    intro hstep
  2. L2
    intro p
  3. L3
    intro x
  4. L4
    intro y
  5. L5
    intro k
  6. L6
    intro hprime
  7. L7
    intro hseed
02Establish hdescentL8–17

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

  1. L8
    have hdescent : ∀ p. ∀ k. Prime(p) → ¬k = 0 → (∃ x. ∃ y. ∃ z. ∃ n. p · k = x · x + y · y + z · z + n · n) → ∃ x. ∃ y. ∃ z. ∃ n. p = x · x + y · y + z · z + n · nDefinitions: Prime(p)Original native command in the exact edition
  2. L9
    apply four_square_descent_prime_from_strict_step
  3. L10
    exact hstep
  4. L11
    specialize hdescent p
  5. L12
    specialize hdescent k
  6. L13
    apply hdescent
  7. L14
    exact hprime
  8. L15
    specialize four_square_descent_modular_seed_multiplier_nonzero p
  9. L16
    specialize four_square_descent_modular_seed_multiplier_nonzero x
  10. L17
    specialize four_square_descent_modular_seed_multiplier_nonzero y
03Use earlier factsL18–18

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

  1. L18
    specialize four_square_descent_modular_seed_multiplier_nonzero k
04Fix variables and assumptionsL19–19

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

  1. L19
    intro hzero
05Use earlier factsL20–28

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

  1. L20
    apply four_square_descent_modular_seed_multiplier_nonzero
  2. L21
    exact hseed
  3. L22
    exact hzero
  4. L23
    specialize four_square_prime_modular_seed_multiple p
  5. L24
    specialize four_square_prime_modular_seed_multiple x
  6. L25
    specialize four_square_prime_modular_seed_multiple y
  7. L26
    specialize four_square_prime_modular_seed_multiple k
  8. L27
    apply four_square_prime_modular_seed_multiple
  9. L28
    exact hseed

Library-wide reading audit

Original defined command ledger · 28 lines
  1. 0001intro hstep
  2. 0002intro p
  3. 0003intro x
  4. 0004intro y
  5. 0005intro k
  6. 0006intro hprime
  7. 0007intro hseed
  8. 0008have hdescent : ∀ p. ∀ k. Prime(p) → ¬k = 0 → (∃ x. ∃ y. ∃ z. ∃ n. p · k = x · x + y · y + z · z + n · n) → ∃ x. ∃ y. ∃ z. ∃ n. p = x · x + y · y + z · z + n · n
    Exact native replay linehave hdescent : forall p k. ((~(p = 1) /\ forall frm_prime_left_fsd_local_prime frm_prime_right_fsd_local_prime. p = frm_prime_left_fsd_local_prime * frm_prime_right_fsd_local_prime -> frm_prime_left_fsd_local_prime = 1 \/ frm_prime_right_fsd_local_prime = 1)) -> ~(k = 0) -> (exists fsl_a_fsd_local_multiple fsl_b_fsd_local_multiple fsl_c_fsd_local_multiple fsl_d_fsd_local_multiple. (p * k) = fsl_a_fsd_local_multiple * fsl_a_fsd_local_multiple + fsl_b_fsd_local_multiple * fsl_b_fsd_local_multiple + fsl_c_fsd_local_multiple * fsl_c_fsd_local_multiple + fsl_d_fsd_local_multiple * fsl_d_fsd_local_multiple) -> (exists fsl_a_fsd_local_result fsl_b_fsd_local_result fsl_c_fsd_local_result fsl_d_fsd_local_result. (p) = fsl_a_fsd_local_result * fsl_a_fsd_local_result + fsl_b_fsd_local_result * fsl_b_fsd_local_result + fsl_c_fsd_local_result * fsl_c_fsd_local_result + fsl_d_fsd_local_result * fsl_d_fsd_local_result)
  9. 0009apply four_square_descent_prime_from_strict_step
  10. 0010exact hstep
  11. 0011specialize hdescent p
  12. 0012specialize hdescent k
  13. 0013apply hdescent
  14. 0014exact hprime
  15. 0015specialize four_square_descent_modular_seed_multiplier_nonzero p
  16. 0016specialize four_square_descent_modular_seed_multiplier_nonzero x
  17. 0017specialize four_square_descent_modular_seed_multiplier_nonzero y
  18. 0018specialize four_square_descent_modular_seed_multiplier_nonzero k
  19. 0019intro hzero
  20. 0020apply four_square_descent_modular_seed_multiplier_nonzero
  21. 0021exact hseed
  22. 0022exact hzero
  23. 0023specialize four_square_prime_modular_seed_multiple p
  24. 0024specialize four_square_prime_modular_seed_multiple x
  25. 0025specialize four_square_prime_modular_seed_multiple y
  26. 0026specialize four_square_prime_modular_seed_multiple k
  27. 0027apply four_square_prime_modular_seed_multiple
  28. 0028exact hseed