FS003Y

four_square_parity_swap_middle_coordinates

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

Swapping the middle coordinates preserves an explicitly left-associated four-square norm.

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 c d. a * a + b * b + c * c + d * d = a * a + c * c + b * b + d * d

Constructive proof overview

Generated structural guide

Swapping the middle coordinates preserves an explicitly left-associated four-square norm.

The unchanged tactic script uses 2 declared prerequisites and contains 10 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

Historical empty-context replay experiment only; that experiment persisted no certificate and granted no release authority. Current checked use follows separately sealed, independently verified proof bundles; there is no Stable promotion.

Proof neighborhood

Direct dependencies

add_assoc Stable theorem; checked-use authorized add_shuffle_middle Alpha theorem; checked-use authorized

Direct dependents

FS0040 four_square_parity_even_multiplier_halving

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

10 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.

01Fix variables and assumptionsL1–4

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro d
02Calculate and transport equalitiesL5–5

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

  1. L5
    trans (a * a + b * b) + (c * c + d * d)
03Use earlier factsL6–6

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

  1. L6
    apply add_assoc
04Calculate and transport equalitiesL7–7

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

  1. L7
    trans (a * a + c * c) + (b * b + d * d)
05Use earlier factsL8–8

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

  1. L8
    apply add_shuffle_middle
06Calculate and transport equalitiesL9–9

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

  1. L9
    symm
07Use earlier factsL10–10

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

  1. L10
    apply add_assoc

Library-wide reading audit

Original exact command ledger · 10 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005trans (a * a + b * b) + (c * c + d * d)
  6. 0006apply add_assoc
  7. 0007trans (a * a + c * c) + (b * b + d * d)
  8. 0008apply add_shuffle_middle
  9. 0009symm
  10. 0010apply add_assoc