FS002D

four_square_euler_four_add_shuffle

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

Eight additive terms are transposed as four independent magnitude/cross-term pairs.

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 e f g h. ((a + b) + (c + d)) + ((e + f) + (g + h)) = ((a + c) + (e + g)) + ((b + d) + (f + h))

Constructive proof overview

Generated structural guide

Eight additive terms are transposed as four independent magnitude/cross-term pairs.

The unchanged tactic script uses 1 declared prerequisite and contains 13 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_shuffle_middle Alpha theorem; checked-use authorized

Direct dependents

FS000X four_square_conjugate_left_decomposition FS002E four_square_euler_balance_aggregate FS002U four_square_euler_left_decomposition FS004N four_square_signed_conjugate_negative_blocks

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

13 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 assumptionsL1–8

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
  5. L5
    intro e
  6. L6
    intro f
  7. L7
    intro g
  8. L8
    intro h
02Calculate and transport equalitiesL9–10

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

  1. L9
    trans ((a + c) + (b + d)) + ((e + g) + (f + h))
  2. L10
    congr
03Use earlier factsL11–13

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

  1. L11
    apply add_shuffle_middle
  2. L12
    apply add_shuffle_middle
  3. L13
    apply add_shuffle_middle

Library-wide reading audit

Original exact command ledger · 13 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro e
  6. 0006intro f
  7. 0007intro g
  8. 0008intro h
  9. 0009trans ((a + c) + (b + d)) + ((e + g) + (f + h))
  10. 0010congr
  11. 0011apply add_shuffle_middle
  12. 0012apply add_shuffle_middle
  13. 0013apply add_shuffle_middle