FS002K · theorem body

four_square_euler_add_permute_nine

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

A three-by-three square expansion separates its diagonal and its six ordered mixed products.

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 u6 u7 u8. (((((((u0) + (u1)) + (u2))) + ((((u3) + (u4)) + (u5)))) + ((((u6) + (u7)) + (u8))))) = ((((((u0) + (u4)) + (u8))) + ((((((u3) + (u1))) + (((u6) + (u2)))) + (((u7) + (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 u6 u7 u8. (((((((u0) + (u1)) + (u2))) + ((((u3) + (u4)) + (u5)))) + ((((u6) + (u7)) + (u8))))) = ((((((u0) + (u4)) + (u8))) + ((((((u3) + (u1))) + (((u6) + (u2)))) + (((u7) + (u5)))))))

Proof neighborhood

Direct theorem prerequisites

add_assoc · Stable closed add_comm · Stable closed FS0006 four_square_add_swap_right_tail

Direct theorem dependents

FS002O four_square_euler_three_square_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

77 script commands · 14 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 assumptionsL1–9

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
  7. L7
    intro u6
  8. L8
    intro u7
  9. L9
    intro u8
02Calculate and transport equalitiesL10–19

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

  1. L10
    trans ((u0) + ((u1) + ((u2) + ((u3) + ((u4) + ((u5) + ((u6) + ((u7) + (u8)))))))))
  2. L11
    simp [add_assoc]
  3. L12
    trans ((u0) + ((u4) + ((u8) + ((u3) + ((u1) + ((u6) + ((u2) + ((u7) + (u5)))))))))
  4. L13
    congr
  5. L14
    refl
  6. L15
    trans ((u4) + ((u1) + ((u2) + ((u3) + ((u5) + ((u6) + ((u7) + (u8))))))))
  7. L16
    trans ((u1) + ((u4) + ((u2) + ((u3) + ((u5) + ((u6) + ((u7) + (u8))))))))
  8. L17
    congr
  9. L18
    refl
  10. L19
    trans ((u2) + ((u4) + ((u3) + ((u5) + ((u6) + ((u7) + (u8)))))))
03Calculate and transport equalitiesL20–21

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

  1. L20
    congr
  2. L21
    refl
04Use earlier factsL22–24

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

  1. L22
    apply four_square_add_swap_right_tail
  2. L23
    apply four_square_add_swap_right_tail
  3. L24
    apply four_square_add_swap_right_tail
05Calculate and transport equalitiesL25–34

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

  1. L25
    congr
  2. L26
    refl
  3. L27
    trans ((u8) + ((u1) + ((u2) + ((u3) + ((u5) + ((u6) + (u7)))))))
  4. L28
    trans ((u1) + ((u8) + ((u2) + ((u3) + ((u5) + ((u6) + (u7)))))))
  5. L29
    congr
  6. L30
    refl
  7. L31
    trans ((u2) + ((u8) + ((u3) + ((u5) + ((u6) + (u7))))))
  8. L32
    congr
  9. L33
    refl
  10. L34
    trans ((u3) + ((u8) + ((u5) + ((u6) + (u7)))))
06Calculate and transport equalitiesL35–42

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

  1. L35
    congr
  2. L36
    refl
  3. L37
    trans ((u5) + ((u8) + ((u6) + (u7))))
  4. L38
    congr
  5. L39
    refl
  6. L40
    trans ((u6) + ((u8) + (u7)))
  7. L41
    congr
  8. L42
    refl
07Use earlier factsL43–48

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

  1. L43
    apply add_comm
  2. L44
    apply four_square_add_swap_right_tail
  3. L45
    apply four_square_add_swap_right_tail
  4. L46
    apply four_square_add_swap_right_tail
  5. L47
    apply four_square_add_swap_right_tail
  6. L48
    apply four_square_add_swap_right_tail
08Calculate and transport equalitiesL49–54

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

  1. L49
    congr
  2. L50
    refl
  3. L51
    trans ((u3) + ((u1) + ((u2) + ((u5) + ((u6) + (u7))))))
  4. L52
    trans ((u1) + ((u3) + ((u2) + ((u5) + ((u6) + (u7))))))
  5. L53
    congr
  6. L54
    refl
09Use earlier factsL55–56

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

  1. L55
    apply four_square_add_swap_right_tail
  2. L56
    apply four_square_add_swap_right_tail
10Calculate and transport equalitiesL57–64

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

  1. L57
    congr
  2. L58
    refl
  3. L59
    congr
  4. L60
    refl
  5. L61
    trans ((u6) + ((u2) + ((u5) + (u7))))
  6. L62
    trans ((u2) + ((u6) + ((u5) + (u7))))
  7. L63
    congr
  8. L64
    refl
11Use earlier factsL65–66

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

  1. L65
    apply four_square_add_swap_right_tail
  2. L66
    apply four_square_add_swap_right_tail
12Calculate and transport equalitiesL67–71

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

  1. L67
    congr
  2. L68
    refl
  3. L69
    congr
  4. L70
    refl
  5. L71
    trans ((u7) + (u5))
13Use earlier factsL72–72

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

  1. L72
    apply add_comm
14Calculate and transport equalitiesL73–77

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

  1. L73
    congr
  2. L74
    refl
  3. L75
    refl
  4. L76
    symm
  5. L77
    simp [add_assoc]

Library-wide reading audit

Original defined command ledger · 77 lines
  1. 0001intro u0
  2. 0002intro u1
  3. 0003intro u2
  4. 0004intro u3
  5. 0005intro u4
  6. 0006intro u5
  7. 0007intro u6
  8. 0008intro u7
  9. 0009intro u8
  10. 0010trans ((u0) + ((u1) + ((u2) + ((u3) + ((u4) + ((u5) + ((u6) + ((u7) + (u8)))))))))
  11. 0011simp [add_assoc]
  12. 0012trans ((u0) + ((u4) + ((u8) + ((u3) + ((u1) + ((u6) + ((u2) + ((u7) + (u5)))))))))
  13. 0013congr
  14. 0014refl
  15. 0015trans ((u4) + ((u1) + ((u2) + ((u3) + ((u5) + ((u6) + ((u7) + (u8))))))))
  16. 0016trans ((u1) + ((u4) + ((u2) + ((u3) + ((u5) + ((u6) + ((u7) + (u8))))))))
  17. 0017congr
  18. 0018refl
  19. 0019trans ((u2) + ((u4) + ((u3) + ((u5) + ((u6) + ((u7) + (u8)))))))
  20. 0020congr
  21. 0021refl
  22. 0022apply four_square_add_swap_right_tail
  23. 0023apply four_square_add_swap_right_tail
  24. 0024apply four_square_add_swap_right_tail
  25. 0025congr
  26. 0026refl
  27. 0027trans ((u8) + ((u1) + ((u2) + ((u3) + ((u5) + ((u6) + (u7)))))))
  28. 0028trans ((u1) + ((u8) + ((u2) + ((u3) + ((u5) + ((u6) + (u7)))))))
  29. 0029congr
  30. 0030refl
  31. 0031trans ((u2) + ((u8) + ((u3) + ((u5) + ((u6) + (u7))))))
  32. 0032congr
  33. 0033refl
  34. 0034trans ((u3) + ((u8) + ((u5) + ((u6) + (u7)))))
  35. 0035congr
  36. 0036refl
  37. 0037trans ((u5) + ((u8) + ((u6) + (u7))))
  38. 0038congr
  39. 0039refl
  40. 0040trans ((u6) + ((u8) + (u7)))
  41. 0041congr
  42. 0042refl
  43. 0043apply add_comm
  44. 0044apply four_square_add_swap_right_tail
  45. 0045apply four_square_add_swap_right_tail
  46. 0046apply four_square_add_swap_right_tail
  47. 0047apply four_square_add_swap_right_tail
  48. 0048apply four_square_add_swap_right_tail
  49. 0049congr
  50. 0050refl
  51. 0051trans ((u3) + ((u1) + ((u2) + ((u5) + ((u6) + (u7))))))
  52. 0052trans ((u1) + ((u3) + ((u2) + ((u5) + ((u6) + (u7))))))
  53. 0053congr
  54. 0054refl
  55. 0055apply four_square_add_swap_right_tail
  56. 0056apply four_square_add_swap_right_tail
  57. 0057congr
  58. 0058refl
  59. 0059congr
  60. 0060refl
  61. 0061trans ((u6) + ((u2) + ((u5) + (u7))))
  62. 0062trans ((u2) + ((u6) + ((u5) + (u7))))
  63. 0063congr
  64. 0064refl
  65. 0065apply four_square_add_swap_right_tail
  66. 0066apply four_square_add_swap_right_tail
  67. 0067congr
  68. 0068refl
  69. 0069congr
  70. 0070refl
  71. 0071trans ((u7) + (u5))
  72. 0072apply add_comm
  73. 0073congr
  74. 0074refl
  75. 0075refl
  76. 0076symm
  77. 0077simp [add_assoc]