FS003Z · theorem body

four_square_parity_swap_outer_coordinates

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

The second crossed coordinate partition preserves the full natural four-square norm.

historical independently replay-verified empty-context experiment; the experiment itself persisted no certificate and granted no release authority; current checked use follows separately sealed proof bundles; no Stable promotion.

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

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

Proof neighborhood

Direct theorem prerequisites

add_assoc · Stable closed add_comm · Stable closed add_shuffle_middle · Alpha closed

Direct theorem dependents

FS0040 four_square_parity_even_multiplier_halving

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

14 script commands · 9 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.

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 equalitiesL7–9

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

  1. L7
    trans (a * a + b * b) + (d * d + c * c)
  2. L8
    congr
  3. L9
    refl
05Use earlier factsL10–10

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

  1. L10
    apply add_comm
06Calculate and transport equalitiesL11–11

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

  1. L11
    trans (a * a + d * d) + (b * b + c * c)
07Use earlier factsL12–12

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

  1. L12
    apply add_shuffle_middle
08Calculate and transport equalitiesL13–13

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

  1. L13
    symm
09Use earlier factsL14–14

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

  1. L14
    apply add_assoc

Library-wide reading audit

Original defined command ledger · 14 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 + b * b) + (d * d + c * c)
  8. 0008congr
  9. 0009refl
  10. 0010apply add_comm
  11. 0011trans (a * a + d * d) + (b * b + c * c)
  12. 0012apply add_shuffle_middle
  13. 0013symm
  14. 0014apply add_assoc