FS002U · theorem body

four_square_euler_left_decomposition

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

All eight positive/negative Hamilton-coordinate squares split exactly into sixteen diagonal squares plus twelve paired mixed blocks.

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

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

Proof neighborhood

Direct theorem prerequisites

FS002R four_square_euler_coordinate_single_decompose FS002S four_square_euler_coordinate_triple_decompose FS002D four_square_euler_four_add_shuffle

Direct theorem dependents

FS002X four_square_euler_global_compensation

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

17 script commands · 5 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 (3)
four square euler coordinate single decomposefour square euler coordinate triple decomposefour square euler four add shuffle
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 equalitiesL9–11

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

  1. L9
    trans ((((((((a * e) * (a * e)) + (((((b * f) * (b * f)) + ((c * g) * (c * g))) + ((d * h) * (d * h)))))) · expand full local formula (1,017 characters)trans ((((((((a * e) * (a * e)) + (((((b * f) * (b * f)) + ((c * g) * (c * g))) + ((d * h) * (d * h)))))) + (((((((b * f) * (c * g)) + ((c * g) * (b * f)))) + ((((b * f) * (d * h)) + ((d * h) * (b * f))))) + ((((c * g) * (d * h)) + ((d * h) * (c * g)))))))) + (((((((((a * f) * (a * f)) + ((b * e) * (b * e))) + ((c * h) * (c * h)))) + ((d * g) * (d * g)))) + (((((((a * f) * (b * e)) + ((b * e) * (a * f)))) + ((((a * f) * (c * h)) + ((c * h) * (a * f))))) + ((((b * e) * (c * h)) + ((c * h) * (b * e))))))))) + ((((((((((a * g) * (a * g)) + ((c * e) * (c * e))) + ((d * f) * (d * f)))) + ((b * h) * (b * h)))) + (((((((a * g) * (c * e)) + ((c * e) * (a * g)))) + ((((a * g) * (d * f)) + ((d * f) * (a * g))))) + ((((c * e) * (d * f)) + ((d * f) * (c * e)))))))) + (((((((((a * h) * (a * h)) + ((b * g) * (b * g))) + ((d * e) * (d * e)))) + ((c * f) * (c * f)))) + (((((((a * h) * (b * g)) + ((b * g) * (a * h)))) + ((((a * h) * (d * e)) + ((d * e) * (a * h))))) + ((((b * g) * (d * e)) + ((d * e) * (b * g))))))))))
  2. L10
    congr
  3. L11
    congr
03Use earlier factsL12–13

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

  1. L12
    apply four_square_euler_coordinate_single_decompose
  2. L13
    apply four_square_euler_coordinate_triple_decompose
04Calculate and transport equalitiesL14–14

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

  1. L14
    congr
05Use earlier factsL15–17

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

  1. L15
    apply four_square_euler_coordinate_triple_decompose
  2. L16
    apply four_square_euler_coordinate_triple_decompose
  3. L17
    apply four_square_euler_four_add_shuffle

Library-wide reading audit

Original defined command ledger · 17 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 * e) * (a * e)) + (((((b * f) * (b * f)) + ((c * g) * (c * g))) + ((d * h) * (d * h)))))) + (((((((b * f) * (c * g)) + ((c * g) * (b * f)))) + ((((b * f) * (d * h)) + ((d * h) * (b * f))))) + ((((c * g) * (d * h)) + ((d * h) * (c * g)))))))) + (((((((((a * f) * (a * f)) + ((b * e) * (b * e))) + ((c * h) * (c * h)))) + ((d * g) * (d * g)))) + (((((((a * f) * (b * e)) + ((b * e) * (a * f)))) + ((((a * f) * (c * h)) + ((c * h) * (a * f))))) + ((((b * e) * (c * h)) + ((c * h) * (b * e))))))))) + ((((((((((a * g) * (a * g)) + ((c * e) * (c * e))) + ((d * f) * (d * f)))) + ((b * h) * (b * h)))) + (((((((a * g) * (c * e)) + ((c * e) * (a * g)))) + ((((a * g) * (d * f)) + ((d * f) * (a * g))))) + ((((c * e) * (d * f)) + ((d * f) * (c * e)))))))) + (((((((((a * h) * (a * h)) + ((b * g) * (b * g))) + ((d * e) * (d * e)))) + ((c * f) * (c * f)))) + (((((((a * h) * (b * g)) + ((b * g) * (a * h)))) + ((((a * h) * (d * e)) + ((d * e) * (a * h))))) + ((((b * g) * (d * e)) + ((d * e) * (b * g))))))))))
  10. 0010congr
  11. 0011congr
  12. 0012apply four_square_euler_coordinate_single_decompose
  13. 0013apply four_square_euler_coordinate_triple_decompose
  14. 0014congr
  15. 0015apply four_square_euler_coordinate_triple_decompose
  16. 0016apply four_square_euler_coordinate_triple_decompose
  17. 0017apply four_square_euler_four_add_shuffle