FS002U

four_square_euler_left_decomposition

Alpha v34 independently verified · alpha_closed; checked-use authorized; 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.

Exact expanded first-order arithmetic 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))))))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 3 declared prerequisites and contains 17 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

Proof neighborhood

Direct dependencies

FS002R four_square_euler_coordinate_single_decompose FS002S four_square_euler_coordinate_triple_decompose FS002D four_square_euler_four_add_shuffle

Direct dependents

FS002X four_square_euler_global_compensation

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

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.

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 exact 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