FS005F · theorem body

four_square_signed_sum_two_decomposition

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

The square of two natural addends separates its diagonal squares from its symmetric cross correction.

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 x y. (x + y) * (x + y) = (x * x + y * y) + (x * y + y * x)

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 x y. (x + y) * (x + y) = (x * x + y * y) + (x * y + y * x)

Proof neighborhood

Direct theorem prerequisites

FS0008 four_square_sum_expansion add_assoc · Stable closed add_comm · Stable closed FS0006 four_square_add_swap_right_tail

Direct theorem dependents

FS005G four_square_signed_sum_four_decomposition FS005H four_square_signed_pair_block_decomposition

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

20 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 (2)
four square sum expansionfour square add swap right tail
01Fix variables and assumptionsL1–2

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

  1. L1
    intro x
  2. L2
    intro y
02Calculate and transport equalitiesL3–3

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

  1. L3
    trans (x * x + x * y) + (y * x + y * y)
03Use earlier factsL4–4

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

  1. L4
    apply four_square_sum_expansion
04Calculate and transport equalitiesL5–13

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

  1. L5
    trans ((x * x) + ((x * y) + ((y * x) + (y * y))))
  2. L6
    simp [add_assoc]
  3. L7
    trans ((x * x) + ((y * y) + ((x * y) + (y * x))))
  4. L8
    congr
  5. L9
    refl
  6. L10
    trans ((y * y) + ((x * y) + (y * x)))
  7. L11
    trans ((x * y) + ((y * y) + (y * x)))
  8. L12
    congr
  9. L13
    refl
05Use earlier factsL14–15

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

  1. L14
    apply add_comm
  2. L15
    apply four_square_add_swap_right_tail
06Calculate and transport equalitiesL16–20

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

  1. L16
    congr
  2. L17
    refl
  3. L18
    refl
  4. L19
    symm
  5. L20
    simp [add_assoc]

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro x
  2. 0002intro y
  3. 0003trans (x * x + x * y) + (y * x + y * y)
  4. 0004apply four_square_sum_expansion
  5. 0005trans ((x * x) + ((x * y) + ((y * x) + (y * y))))
  6. 0006simp [add_assoc]
  7. 0007trans ((x * x) + ((y * y) + ((x * y) + (y * x))))
  8. 0008congr
  9. 0009refl
  10. 0010trans ((y * y) + ((x * y) + (y * x)))
  11. 0011trans ((x * y) + ((y * y) + (y * x)))
  12. 0012congr
  13. 0013refl
  14. 0014apply add_comm
  15. 0015apply four_square_add_swap_right_tail
  16. 0016congr
  17. 0017refl
  18. 0018refl
  19. 0019symm
  20. 0020simp [add_assoc]