FS001H · theorem body

four_square_descent_prime_from_strict_step

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

Every nonzero represented prime multiple descends all the way to a representation of the prime under the precise strict-step hypothesis.

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. Prime(x) → ¬y = 0 → (∃ z. ∃ n. ∃ m. ∃ k. x · y = z · z + n · n + m · m + k · k) → ∃ z. ∃ n. ∃ m. ∃ k. x = z · z + n · n + m · m + k · k

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 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 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)) -> ~(k = 0) -> (exists fsl_a_fsd_multiple fsl_b_fsd_multiple fsl_c_fsd_multiple fsl_d_fsd_multiple. (p * k) = fsl_a_fsd_multiple * fsl_a_fsd_multiple + fsl_b_fsd_multiple * fsl_b_fsd_multiple + fsl_c_fsd_multiple * fsl_c_fsd_multiple + fsl_d_fsd_multiple * fsl_d_fsd_multiple) -> (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

le_refl · Stable closed FS001G four_square_descent_strict_multiplier_bounded

Direct theorem dependents

FS001I four_square_descent_prime_from_modular_seed_and_step

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

16 script commands · 2 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 (1)
four square descent strict multiplier bounded
01Fix variables and assumptionsL1–6

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

  1. L1
    intro hstep
  2. L2
    intro p
  3. L3
    intro k
  4. L4
    intro hprime
  5. L5
    intro hnonzero
  6. L6
    intro hrepresented
02Use earlier factsL7–16

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

  1. L7
    specialize four_square_descent_strict_multiplier_bounded k
  2. L8
    specialize four_square_descent_strict_multiplier_bounded p
  3. L9
    specialize four_square_descent_strict_multiplier_bounded k
  4. L10
    apply four_square_descent_strict_multiplier_bounded
  5. L11
    specialize le_refl k
  6. L12
    exact le_refl
  7. L13
    exact hprime
  8. L14
    exact hnonzero
  9. L15
    exact hrepresented
  10. L16
    exact hstep

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro hstep
  2. 0002intro p
  3. 0003intro k
  4. 0004intro hprime
  5. 0005intro hnonzero
  6. 0006intro hrepresented
  7. 0007specialize four_square_descent_strict_multiplier_bounded k
  8. 0008specialize four_square_descent_strict_multiplier_bounded p
  9. 0009specialize four_square_descent_strict_multiplier_bounded k
  10. 0010apply four_square_descent_strict_multiplier_bounded
  11. 0011specialize le_refl k
  12. 0012exact le_refl
  13. 0013exact hprime
  14. 0014exact hnonzero
  15. 0015exact hrepresented
  16. 0016exact hstep