FS001A

four_square_descent_square_factor_norm

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Four coordinates each divisible by k have a norm exactly divisible by the natural square k².

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.

Exact expanded first-order arithmetic statement

forall k u v w x. ((k * u) * (k * u) + (k * v) * (k * v) + (k * w) * (k * w) + (k * x) * (k * x)) = (k * k) * (u * u + v * v + w * w + x * x)

Constructive proof overview

Generated structural guide

Four coordinates each divisible by k have a norm exactly divisible by the natural square k².

The unchanged tactic script uses 2 declared prerequisites and contains 15 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Proof neighborhood

Direct dependencies

FS000E four_square_product_square mul_add Stable theorem; checked-use authorized

Direct dependents

FS001C four_square_descent_scaled_norm_quotient FS0022 four_square_descent_divisible_coordinates_prime_factor

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

15 script commands · 4 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.

Named ingredients (1)
four square product square
01Fix variables and assumptionsL1–5

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

  1. L1
    intro k
  2. L2
    intro u
  3. L3
    intro v
  4. L4
    intro w
  5. L5
    intro x
02Calculate and transport equalitiesL6–9

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

  1. L6
    trans (k * k) * (u * u) + (k * k) * (v * v) + (k * k) * (w * w) + (k * k) * (x * x)
  2. L7
    congr
  3. L8
    congr
  4. L9
    congr
03Use earlier factsL10–13

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

  1. L10
    apply four_square_product_square
  2. L11
    apply four_square_product_square
  3. L12
    apply four_square_product_square
  4. L13
    apply four_square_product_square
04Calculate and transport equalitiesL14–15

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

  1. L14
    symm
  2. L15
    simp [mul_add]

Library-wide reading audit

Original exact command ledger · 15 lines
  1. 0001intro k
  2. 0002intro u
  3. 0003intro v
  4. 0004intro w
  5. 0005intro x
  6. 0006trans (k * k) * (u * u) + (k * k) * (v * v) + (k * k) * (w * w) + (k * k) * (x * x)
  7. 0007congr
  8. 0008congr
  9. 0009congr
  10. 0010apply four_square_product_square
  11. 0011apply four_square_product_square
  12. 0012apply four_square_product_square
  13. 0013apply four_square_product_square
  14. 0014symm
  15. 0015simp [mul_add]