FS0035 · theorem body

four_square_prime_from_bounded_strict_descent

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

The checked unconditional bounded modular seed discharges every seed premise, so the exact below-prime strict step alone represents every 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. 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

PD0002 Lt PD0004 Prime

In local proof propositions

PD0002 Lt PD0004 Prime
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. ((~(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)) -> (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

FS000P four_square_prime_bounded_modular_seed FS0034 four_square_prime_from_bounded_strict_descent_and_seed

Direct theorem dependents

FS0036 four_square_lagrange_from_bounded_strict_descent FS003M four_square_prime_from_odd_signed_quaternion

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

22 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 (2)
four square prime bounded modular seedfour square prime from bounded strict descent and seed
01Fix variables and assumptionsL1–3

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

  1. L1
    intro hstep
  2. L2
    intro p
  3. L3
    intro hprime
02Establish hseedL4–7

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

  1. L4
    have hseed : ∃ a. ∃ b. ∃ k. a · a + b · b + 1 = p · k ∧ Lt(k,p)Definitions: Lt(k,p)Original native command in the exact edition
  2. L5
    specialize four_square_prime_bounded_modular_seed p
  3. L6
    apply four_square_prime_bounded_modular_seed
  4. L7
    exact hprime
03Separate the logical casesL8–11

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

  1. L8
    cases hseed
  2. L9
    cases hseed_witness
  3. L10
    cases hseed_witness_witness
  4. L11
    cases hseed_witness_witness_witness
04Establish hdescentL12–21

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

  1. L12
    have hdescent : ∀ q. ∀ a. ∀ b. ∀ k. Prime(q) → a · a + b · b + 1 = q · k → Lt(k,q) → ∃ x. ∃ y. ∃ z. ∃ n. q = x · x + y · y + z · z + n · nDefinitions: Prime(q)Lt(k,q)Original native command in the exact edition
  2. L13
    apply four_square_prime_from_bounded_strict_descent_and_seed
  3. L14
    exact hstep
  4. L15
    specialize hdescent p
  5. L16
    specialize hdescent x
  6. L17
    specialize hdescent x1
  7. L18
    specialize hdescent x2
  8. L19
    apply hdescent
  9. L20
    exact hprime
  10. L21
    exact hseed_witness_witness_witness_left
05Use earlier factsL22–22

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

  1. L22
    exact hseed_witness_witness_witness_right

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro hstep
  2. 0002intro p
  3. 0003intro hprime
  4. 0004have hseed : ∃ a. ∃ b. ∃ k. a · a + b · b + 1 = p · k ∧ Lt(k,p)
    Exact native replay linehave hseed : exists a b k. ((a * a + b * b + 1 = p * k) /\ (exists gap. gap + S k = p))
  5. 0005specialize four_square_prime_bounded_modular_seed p
  6. 0006apply four_square_prime_bounded_modular_seed
  7. 0007exact hprime
  8. 0008cases hseed
  9. 0009cases hseed_witness
  10. 0010cases hseed_witness_witness
  11. 0011cases hseed_witness_witness_witness
  12. 0012have hdescent : ∀ q. ∀ a. ∀ b. ∀ k. Prime(q) → a · a + b · b + 1 = q · k → Lt(k,q) → ∃ x. ∃ y. ∃ z. ∃ n. q = x · x + y · y + z · z + n · n
    Exact native replay linehave hdescent : forall q a b k. ((~(q = 1) /\ forall frm_prime_left_fslb_bounded_local_prime frm_prime_right_fslb_bounded_local_prime. q = frm_prime_left_fslb_bounded_local_prime * frm_prime_right_fslb_bounded_local_prime -> frm_prime_left_fslb_bounded_local_prime = 1 \/ frm_prime_right_fslb_bounded_local_prime = 1)) -> a * a + b * b + 1 = q * k -> (exists gap. gap + S k = q) -> (exists fsl_a_fslb_bounded_local_result fsl_b_fslb_bounded_local_result fsl_c_fslb_bounded_local_result fsl_d_fslb_bounded_local_result. (q) = fsl_a_fslb_bounded_local_result * fsl_a_fslb_bounded_local_result + fsl_b_fslb_bounded_local_result * fsl_b_fslb_bounded_local_result + fsl_c_fslb_bounded_local_result * fsl_c_fslb_bounded_local_result + fsl_d_fslb_bounded_local_result * fsl_d_fslb_bounded_local_result)
  13. 0013apply four_square_prime_from_bounded_strict_descent_and_seed
  14. 0014exact hstep
  15. 0015specialize hdescent p
  16. 0016specialize hdescent x
  17. 0017specialize hdescent x1
  18. 0018specialize hdescent x2
  19. 0019apply hdescent
  20. 0020exact hprime
  21. 0021exact hseed_witness_witness_witness_left
  22. 0022exact hseed_witness_witness_witness_right