FS001A · theorem body

four_square_descent_square_factor_norm

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

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)

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

none

In local proof propositions

none
Exact expanded first-order 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)

Proof neighborhood

Direct theorem prerequisites

FS000E four_square_product_square mul_add · Stable closed

Direct theorem dependents

FS001C four_square_descent_scaled_norm_quotient FS0022 four_square_descent_divisible_coordinates_prime_factor

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

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.

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 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 defined 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]