FS002J · theorem body

four_square_euler_add_permute_six

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

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

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

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

Proof neighborhood

Direct theorem prerequisites

add_assoc · Stable closed add_comm · Stable closed FS0006 four_square_add_swap_right_tail

Direct theorem dependents

FS002P four_square_euler_cross_triple_expansion

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

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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]