FS002J

four_square_euler_add_permute_six

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

Six abstract additive entries are paired by a bounded explicit adjacent-swap proof.

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 u0 u1 u2 u3 u4 u5. ((((((u0) + (u1)) + (u2))) + ((((u3) + (u4)) + (u5))))) = ((((((u0) + (u3))) + (((u1) + (u4)))) + (((u2) + (u5)))))

Constructive proof overview

Generated structural guide

Six abstract additive entries are paired by a bounded explicit adjacent-swap proof.

The unchanged tactic script uses 3 declared prerequisites and contains 28 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_comm Stable theorem; checked-use authorized FS0006 four_square_add_swap_right_tail

Direct dependents

FS002P four_square_euler_cross_triple_expansion

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

28 script commands · 6 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.

Named ingredients (1)
four square add swap right tail
01Fix variables and assumptionsL1–6

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

  1. L1
    intro u0
  2. L2
    intro u1
  3. L3
    intro u2
  4. L4
    intro u3
  5. L5
    intro u4
  6. L6
    intro u5
02Calculate and transport equalitiesL7–15

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

  1. L7
    trans ((u0) + ((u1) + ((u2) + ((u3) + ((u4) + (u5))))))
  2. L8
    simp [add_assoc]
  3. L9
    trans ((u0) + ((u3) + ((u1) + ((u4) + ((u2) + (u5))))))
  4. L10
    congr
  5. L11
    refl
  6. L12
    trans ((u3) + ((u1) + ((u2) + ((u4) + (u5)))))
  7. L13
    trans ((u1) + ((u3) + ((u2) + ((u4) + (u5)))))
  8. L14
    congr
  9. L15
    refl
03Use earlier factsL16–17

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

  1. L16
    apply four_square_add_swap_right_tail
  2. L17
    apply four_square_add_swap_right_tail
04Calculate and transport equalitiesL18–22

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

  1. L18
    congr
  2. L19
    refl
  3. L20
    congr
  4. L21
    refl
  5. L22
    trans ((u4) + ((u2) + (u5)))
05Use earlier factsL23–23

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

  1. L23
    apply four_square_add_swap_right_tail
06Calculate and transport equalitiesL24–28

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

  1. L24
    congr
  2. L25
    refl
  3. L26
    refl
  4. L27
    symm
  5. L28
    simp [add_assoc]

Library-wide reading audit

Original exact command ledger · 28 lines
  1. 0001intro u0
  2. 0002intro u1
  3. 0003intro u2
  4. 0004intro u3
  5. 0005intro u4
  6. 0006intro u5
  7. 0007trans ((u0) + ((u1) + ((u2) + ((u3) + ((u4) + (u5))))))
  8. 0008simp [add_assoc]
  9. 0009trans ((u0) + ((u3) + ((u1) + ((u4) + ((u2) + (u5))))))
  10. 0010congr
  11. 0011refl
  12. 0012trans ((u3) + ((u1) + ((u2) + ((u4) + (u5)))))
  13. 0013trans ((u1) + ((u3) + ((u2) + ((u4) + (u5)))))
  14. 0014congr
  15. 0015refl
  16. 0016apply four_square_add_swap_right_tail
  17. 0017apply four_square_add_swap_right_tail
  18. 0018congr
  19. 0019refl
  20. 0020congr
  21. 0021refl
  22. 0022trans ((u4) + ((u2) + (u5)))
  23. 0023apply four_square_add_swap_right_tail
  24. 0024congr
  25. 0025refl
  26. 0026refl
  27. 0027symm
  28. 0028simp [add_assoc]