FS001E · theorem body

four_square_descent_strict_step_from_centered_quaternion

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

Every explicit centered nonzero quaternion certificate constructs a genuinely represented strictly smaller prime multiplier.

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

∀ p. ∀ k. ∀ r. ∀ a. ∀ b. ∀ c. ∀ d. ∀ e. ∀ f. ∀ g. ∀ h. ∀ u. ∀ v. ∀ w. ∀ x. ¬k = 0 → ¬r = 0 → Lt(r,k) → p · k = a · a + b · b + c · c + d · d → k · r = e · e + f · f + g · g + h · h → (a · e = b · f + c · g + d · h + k · u ∨ b · f + c · g + d · h = a · e + k · u) ∧ ((a · f + b · e + c · h = d · g + k · v ∨ d · g = a · f + b · e + c · h + k · v) ∧ ((a · g + c · e + d · f = b · h + k · w ∨ b · h = a · g + c · e + d · f + k · w) ∧ (a · h + b · g + d · e = c · f + k · x ∨ c · f = a · h + b · g + d · e + k · x))) → ∃ y. ¬y = 0 ∧ (Lt(y,k) ∧ (∃ z. ∃ n. ∃ m. ∃ i. p · y = z · z + n · n + m · m + i · i))

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

In local proof propositions

none
Exact expanded first-order statement
forall p k r a b c d e f g h u v w x. ~(k = 0) -> ~(r = 0) -> (exists fsd_center_gap. fsd_center_gap + S r = k) -> p * k = (a * a + b * b + c * c + d * d) -> k * r = (e * e + f * f + g * g + h * h) -> ((((a * e) = (b * f + c * g + d * h) + (k * u)) \/ ((b * f + c * g + d * h) = (a * e) + (k * u))) /\ ((((a * f + b * e + c * h) = (d * g) + (k * v)) \/ ((d * g) = (a * f + b * e + c * h) + (k * v))) /\ ((((a * g + c * e + d * f) = (b * h) + (k * w)) \/ ((b * h) = (a * g + c * e + d * f) + (k * w))) /\ (((a * h + b * g + d * e) = (c * f) + (k * x)) \/ ((c * f) = (a * h + b * g + d * e) + (k * x)))))) -> (exists r. (~(r = 0) /\ ((exists fsd_strict_gap. fsd_strict_gap + S r = k) /\ (exists fsl_a_fsd_strict_result fsl_b_fsd_strict_result fsl_c_fsd_strict_result fsl_d_fsd_strict_result. (p * r) = fsl_a_fsd_strict_result * fsl_a_fsd_strict_result + fsl_b_fsd_strict_result * fsl_b_fsd_strict_result + fsl_c_fsd_strict_result * fsl_c_fsd_strict_result + fsl_d_fsd_strict_result * fsl_d_fsd_strict_result))))

Proof neighborhood

Direct theorem prerequisites

FS001D four_square_descent_quaternion_quotient

Direct theorem dependents

none

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

35 script commands · 10 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 quaternion quotient
01Fix variables and assumptionsL1–10

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

  1. L1
    intro p
  2. L2
    intro k
  3. L3
    intro r
  4. L4
    intro a
  5. L5
    intro b
  6. L6
    intro c
  7. L7
    intro d
  8. L8
    intro e
  9. L9
    intro f
  10. L10
    intro g
02Fix variables and assumptionsL11–20

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

  1. L11
    intro h
  2. L12
    intro u
  3. L13
    intro v
  4. L14
    intro w
  5. L15
    intro x
  6. L16
    intro hk
  7. L17
    intro hr
  8. L18
    intro hstrict
  9. L19
    intro hfirst
  10. L20
    intro hsecond
03Fix variables and assumptionsL21–21

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

  1. L21
    intro hbalances
04Construct an explicit witnessL22–22

Supply the displayed value, then prove that it has the required property.

  1. L22
    exists r
05Separate the logical casesL23–23

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

  1. L23
    split
06Use earlier factsL24–24

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

  1. L24
    exact hr
07Separate the logical casesL25–25

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

  1. L25
    split
08Use earlier factsL26–26

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

  1. L26
    exact hstrict
09Construct an explicit witnessL27–30

Supply the displayed value, then prove that it has the required property.

  1. L27
    exists u
  2. L28
    exists v
  3. L29
    exists w
  4. L30
    exists x
10Use earlier factsL31–35

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

  1. L31
    apply four_square_descent_quaternion_quotient
  2. L32
    exact hk
  3. L33
    exact hfirst
  4. L34
    exact hsecond
  5. L35
    exact hbalances

Library-wide reading audit

Original defined command ledger · 35 lines
  1. 0001intro p
  2. 0002intro k
  3. 0003intro r
  4. 0004intro a
  5. 0005intro b
  6. 0006intro c
  7. 0007intro d
  8. 0008intro e
  9. 0009intro f
  10. 0010intro g
  11. 0011intro h
  12. 0012intro u
  13. 0013intro v
  14. 0014intro w
  15. 0015intro x
  16. 0016intro hk
  17. 0017intro hr
  18. 0018intro hstrict
  19. 0019intro hfirst
  20. 0020intro hsecond
  21. 0021intro hbalances
  22. 0022exists r
  23. 0023split
  24. 0024exact hr
  25. 0025split
  26. 0026exact hstrict
  27. 0027exists u
  28. 0028exists v
  29. 0029exists w
  30. 0030exists x
  31. 0031apply four_square_descent_quaternion_quotient
  32. 0032exact hk
  33. 0033exact hfirst
  34. 0034exact hsecond
  35. 0035exact hbalances