FS0056 · theorem body

four_square_signed_divisible_norm_product_representation

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

Any four individually k-divisible natural norm-product coordinates yield an explicit four-square representation of the exact quotient p·r.

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. ∀ m0. ∀ m1. ∀ m2. ∀ m3. ¬k = 0 → p · k · (k · r) = m0 · m0 + m1 · m1 + m2 · m2 + m3 · m3 → Dvd(k,m0) → Dvd(k,m1) → Dvd(k,m2) → Dvd(k,m3) → ∃ x. ∃ y. ∃ z. ∃ n. p · r = x · x + y · y + z · z + n · n

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

PD0003 Dvd

In local proof propositions

none
Exact expanded first-order statement
forall p k r m0 m1 m2 m3. ~(k = 0) -> (p * k) * (k * r) = (m0 * m0 + m1 * m1 + m2 * m2 + m3 * m3) -> (exists ftcn_factor_fsso_multiple_0. (m0) = (k) * ftcn_factor_fsso_multiple_0) -> (exists ftcn_factor_fsso_multiple_1. (m1) = (k) * ftcn_factor_fsso_multiple_1) -> (exists ftcn_factor_fsso_multiple_2. (m2) = (k) * ftcn_factor_fsso_multiple_2) -> (exists ftcn_factor_fsso_multiple_3. (m3) = (k) * ftcn_factor_fsso_multiple_3) -> (exists fsl_a_fsso_quotient fsl_b_fsso_quotient fsl_c_fsso_quotient fsl_d_fsso_quotient. (p * r) = fsl_a_fsso_quotient * fsl_a_fsso_quotient + fsl_b_fsso_quotient * fsl_b_fsso_quotient + fsl_c_fsso_quotient * fsl_c_fsso_quotient + fsl_d_fsso_quotient * fsl_d_fsso_quotient)

Proof neighborhood

Direct theorem prerequisites

FS001C four_square_descent_scaled_norm_quotient

Direct theorem dependents

FS0057 four_square_signed_absolute_block_representation

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

39 script commands · 7 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 scaled norm 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 m0
  5. L5
    intro m1
  6. L6
    intro m2
  7. L7
    intro m3
  8. L8
    intro hnonzero
  9. L9
    intro hnorm
  10. L10
    intro hdiv0
02Fix variables and assumptionsL11–13

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

  1. L11
    intro hdiv1
  2. L12
    intro hdiv2
  3. L13
    intro hdiv3
03Separate the logical casesL14–17

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

  1. L14
    cases hdiv0
  2. L15
    cases hdiv1
  3. L16
    cases hdiv2
  4. L17
    cases hdiv3
04Construct an explicit witnessL18–21

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

  1. L18
    exists x
  2. L19
    exists x1
  3. L20
    exists x2
  4. L21
    exists x3
05Use earlier factsL22–30

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

  1. L22
    specialize four_square_descent_scaled_norm_quotient p
  2. L23
    specialize four_square_descent_scaled_norm_quotient k
  3. L24
    specialize four_square_descent_scaled_norm_quotient r
  4. L25
    specialize four_square_descent_scaled_norm_quotient x
  5. L26
    specialize four_square_descent_scaled_norm_quotient x1
  6. L27
    specialize four_square_descent_scaled_norm_quotient x2
  7. L28
    specialize four_square_descent_scaled_norm_quotient x3
  8. L29
    apply four_square_descent_scaled_norm_quotient
  9. L30
    exact hnonzero
06Calculate and transport equalitiesL31–38

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L31
    rewrite hdiv0_witness at hnorm
  2. L32
    rewrite hdiv0_witness at hnorm
  3. L33
    rewrite hdiv1_witness at hnorm
  4. L34
    rewrite hdiv1_witness at hnorm
  5. L35
    rewrite hdiv2_witness at hnorm
  6. L36
    rewrite hdiv2_witness at hnorm
  7. L37
    rewrite hdiv3_witness at hnorm
  8. L38
    rewrite hdiv3_witness at hnorm
07Use earlier factsL39–39

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

  1. L39
    exact hnorm

Library-wide reading audit

Original defined command ledger · 39 lines
  1. 0001intro p
  2. 0002intro k
  3. 0003intro r
  4. 0004intro m0
  5. 0005intro m1
  6. 0006intro m2
  7. 0007intro m3
  8. 0008intro hnonzero
  9. 0009intro hnorm
  10. 0010intro hdiv0
  11. 0011intro hdiv1
  12. 0012intro hdiv2
  13. 0013intro hdiv3
  14. 0014cases hdiv0
  15. 0015cases hdiv1
  16. 0016cases hdiv2
  17. 0017cases hdiv3
  18. 0018exists x
  19. 0019exists x1
  20. 0020exists x2
  21. 0021exists x3
  22. 0022specialize four_square_descent_scaled_norm_quotient p
  23. 0023specialize four_square_descent_scaled_norm_quotient k
  24. 0024specialize four_square_descent_scaled_norm_quotient r
  25. 0025specialize four_square_descent_scaled_norm_quotient x
  26. 0026specialize four_square_descent_scaled_norm_quotient x1
  27. 0027specialize four_square_descent_scaled_norm_quotient x2
  28. 0028specialize four_square_descent_scaled_norm_quotient x3
  29. 0029apply four_square_descent_scaled_norm_quotient
  30. 0030exact hnonzero
  31. 0031rewrite hdiv0_witness at hnorm
  32. 0032rewrite hdiv0_witness at hnorm
  33. 0033rewrite hdiv1_witness at hnorm
  34. 0034rewrite hdiv1_witness at hnorm
  35. 0035rewrite hdiv2_witness at hnorm
  36. 0036rewrite hdiv2_witness at hnorm
  37. 0037rewrite hdiv3_witness at hnorm
  38. 0038rewrite hdiv3_witness at hnorm
  39. 0039exact hnorm