FS0034 · theorem body

four_square_prime_from_bounded_strict_descent_and_seed

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

An actual modular prime seed with multiplier strictly below the prime needs only the bounded strict-descent step to construct a four-square 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 → Lt(y,x) → (∃ 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 → Lt(n,x) → ∃ 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

none
Exact expanded first-order statement
(forall fslb_bounded_prime_bridge fslb_bounded_multiplier_bridge. ((~(fslb_bounded_prime_bridge = 1) /\ forall frm_prime_left_fslb_bounded_prime_bridge frm_prime_right_fslb_bounded_prime_bridge. fslb_bounded_prime_bridge = frm_prime_left_fslb_bounded_prime_bridge * frm_prime_right_fslb_bounded_prime_bridge -> frm_prime_left_fslb_bounded_prime_bridge = 1 \/ frm_prime_right_fslb_bounded_prime_bridge = 1)) -> ~(fslb_bounded_multiplier_bridge = 0) -> ~(fslb_bounded_multiplier_bridge = 1) -> (exists fslb_bounded_upper_gap_bridge. fslb_bounded_upper_gap_bridge + S fslb_bounded_multiplier_bridge = fslb_bounded_prime_bridge) -> (exists fsl_a_fslb_bounded_source_bridge fsl_b_fslb_bounded_source_bridge fsl_c_fslb_bounded_source_bridge fsl_d_fslb_bounded_source_bridge. (fslb_bounded_prime_bridge * fslb_bounded_multiplier_bridge) = fsl_a_fslb_bounded_source_bridge * fsl_a_fslb_bounded_source_bridge + fsl_b_fslb_bounded_source_bridge * fsl_b_fslb_bounded_source_bridge + fsl_c_fslb_bounded_source_bridge * fsl_c_fslb_bounded_source_bridge + fsl_d_fslb_bounded_source_bridge * fsl_d_fslb_bounded_source_bridge) -> exists fslb_bounded_smaller_bridge. (~(fslb_bounded_smaller_bridge = 0) /\ ((exists fslb_bounded_lower_gap_bridge. fslb_bounded_lower_gap_bridge + S fslb_bounded_smaller_bridge = fslb_bounded_multiplier_bridge) /\ (exists fsl_a_fslb_bounded_target_bridge fsl_b_fslb_bounded_target_bridge fsl_c_fslb_bounded_target_bridge fsl_d_fslb_bounded_target_bridge. (fslb_bounded_prime_bridge * fslb_bounded_smaller_bridge) = fsl_a_fslb_bounded_target_bridge * fsl_a_fslb_bounded_target_bridge + fsl_b_fslb_bounded_target_bridge * fsl_b_fslb_bounded_target_bridge + fsl_c_fslb_bounded_target_bridge * fsl_c_fslb_bounded_target_bridge + fsl_d_fslb_bounded_target_bridge * fsl_d_fslb_bounded_target_bridge)))) -> forall p a b k. ((~(p = 1) /\ forall frm_prime_left_fslb_prime frm_prime_right_fslb_prime. p = frm_prime_left_fslb_prime * frm_prime_right_fslb_prime -> frm_prime_left_fslb_prime = 1 \/ frm_prime_right_fslb_prime = 1)) -> a * a + b * b + 1 = p * k -> (exists gap. gap + S k = p) -> (exists fsl_a_fslb_prime_result fsl_b_fslb_prime_result fsl_c_fslb_prime_result fsl_d_fslb_prime_result. (p) = fsl_a_fslb_prime_result * fsl_a_fslb_prime_result + fsl_b_fslb_prime_result * fsl_b_fslb_prime_result + fsl_c_fslb_prime_result * fsl_c_fslb_prime_result + fsl_d_fslb_prime_result * fsl_d_fslb_prime_result)

Proof neighborhood

Direct theorem prerequisites

le_refl · Stable closed FS001F four_square_descent_modular_seed_multiplier_nonzero FS003F four_square_prime_modular_seed_multiple FS0033 four_square_descent_below_prime_multiplier_bounded

Direct theorem dependents

FS0035 four_square_prime_from_bounded_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

31 script commands · 6 reading checkpoints · 0 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 below prime multiplier bounded
01Fix variables and assumptionsL1–8

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

  1. L1
    intro hstep
  2. L2
    intro p
  3. L3
    intro a
  4. L4
    intro b
  5. L5
    intro k
  6. L6
    intro hprime
  7. L7
    intro hseed
  8. L8
    intro hbelow
02Use earlier factsL9–18

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

  1. L9
    specialize four_square_descent_below_prime_multiplier_bounded k
  2. L10
    specialize four_square_descent_below_prime_multiplier_bounded p
  3. L11
    specialize four_square_descent_below_prime_multiplier_bounded k
  4. L12
    apply four_square_descent_below_prime_multiplier_bounded
  5. L13
    specialize le_refl k
  6. L14
    exact le_refl
  7. L15
    exact hprime
  8. L16
    specialize four_square_descent_modular_seed_multiplier_nonzero p
  9. L17
    specialize four_square_descent_modular_seed_multiplier_nonzero a
  10. L18
    specialize four_square_descent_modular_seed_multiplier_nonzero b
03Use earlier factsL19–19

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

  1. L19
    specialize four_square_descent_modular_seed_multiplier_nonzero k
04Fix variables and assumptionsL20–20

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

  1. L20
    intro hzero
05Use earlier factsL21–30

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

  1. L21
    apply four_square_descent_modular_seed_multiplier_nonzero
  2. L22
    exact hseed
  3. L23
    exact hzero
  4. L24
    exact hbelow
  5. L25
    specialize four_square_prime_modular_seed_multiple p
  6. L26
    specialize four_square_prime_modular_seed_multiple a
  7. L27
    specialize four_square_prime_modular_seed_multiple b
  8. L28
    specialize four_square_prime_modular_seed_multiple k
  9. L29
    apply four_square_prime_modular_seed_multiple
  10. L30
    exact hseed
06Use earlier factsL31–31

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

  1. L31
    exact hstep

Library-wide reading audit

Original defined command ledger · 31 lines
  1. 0001intro hstep
  2. 0002intro p
  3. 0003intro a
  4. 0004intro b
  5. 0005intro k
  6. 0006intro hprime
  7. 0007intro hseed
  8. 0008intro hbelow
  9. 0009specialize four_square_descent_below_prime_multiplier_bounded k
  10. 0010specialize four_square_descent_below_prime_multiplier_bounded p
  11. 0011specialize four_square_descent_below_prime_multiplier_bounded k
  12. 0012apply four_square_descent_below_prime_multiplier_bounded
  13. 0013specialize le_refl k
  14. 0014exact le_refl
  15. 0015exact hprime
  16. 0016specialize four_square_descent_modular_seed_multiplier_nonzero p
  17. 0017specialize four_square_descent_modular_seed_multiplier_nonzero a
  18. 0018specialize four_square_descent_modular_seed_multiplier_nonzero b
  19. 0019specialize four_square_descent_modular_seed_multiplier_nonzero k
  20. 0020intro hzero
  21. 0021apply four_square_descent_modular_seed_multiplier_nonzero
  22. 0022exact hseed
  23. 0023exact hzero
  24. 0024exact hbelow
  25. 0025specialize four_square_prime_modular_seed_multiple p
  26. 0026specialize four_square_prime_modular_seed_multiple a
  27. 0027specialize four_square_prime_modular_seed_multiple b
  28. 0028specialize four_square_prime_modular_seed_multiple k
  29. 0029apply four_square_prime_modular_seed_multiple
  30. 0030exact hseed
  31. 0031exact hstep