TS0022

two_square_scaled_norm_identity

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

Scaling a two-square norm by z² scales both coordinates by z.

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 a b z. (a * a + b * b) * (z * z) = (a * z) * (a * z) + (b * z) * (b * z)

Constructive proof overview

Generated structural guide

Scaling a two-square norm by z² scales both coordinates by z.

The unchanged tactic script uses 2 declared prerequisites and contains 7 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

add_mul Stable theorem; checked-use authorized mul_shuffle_four Alpha theorem; checked-use authorized

Direct dependents

TS0024 prime_divisible_two_square_norm_unit_coordinate_yields_negative_one_root

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

7 script commands · 3 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.

01Fix variables and assumptionsL1–3

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro z
02Calculate and transport equalitiesL4–5

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

  1. L4
    simp [add_mul]
  2. L5
    congr
03Use earlier factsL6–7

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

  1. L6
    apply mul_shuffle_four
  2. L7
    apply mul_shuffle_four

Library-wide reading audit

Original exact command ledger · 7 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro z
  4. 0004simp [add_mul]
  5. 0005congr
  6. 0006apply mul_shuffle_four
  7. 0007apply mul_shuffle_four