FS005H · theorem body

four_square_signed_pair_block_decomposition

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

A pair-versus-pair signed coordinate square decomposes into four diagonal squares and two symmetric cross pairs.

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

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

Proof neighborhood

Direct theorem prerequisites

FS005F four_square_signed_sum_two_decomposition add_shuffle_middle · Alpha closed

Direct theorem dependents

FS000X four_square_conjugate_left_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

15 script commands · 3 reading checkpoints · 2 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 signed sum two decomposition
01Fix variables and assumptionsL1–4

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

  1. L1
    intro x
  2. L2
    intro y
  3. L3
    intro z
  4. L4
    intro w
02Establish hfirstL5–8

Establish this local claim before using it. It is not an additional assumption.

  1. L5
    have hfirst : (x + y) * (x + y) = (x * x + y * y) + (x * y + y * x)
  2. L6
    specialize four_square_signed_sum_two_decomposition x
  3. L7
    specialize four_square_signed_sum_two_decomposition y
  4. L8
    exact four_square_signed_sum_two_decomposition
03Establish hsecondL9–15

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply add shuffle middle.

  1. L9
    have hsecond : (z + w) * (z + w) = (z * z + w * w) + (z * w + w * z)
  2. L10
    specialize four_square_signed_sum_two_decomposition z
  3. L11
    specialize four_square_signed_sum_two_decomposition w
  4. L12
    exact four_square_signed_sum_two_decomposition
  5. L13
    rewrite hfirst
  6. L14
    rewrite hsecond
  7. L15
    apply add_shuffle_middle

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro x
  2. 0002intro y
  3. 0003intro z
  4. 0004intro w
  5. 0005have hfirst : (x + y) * (x + y) = (x * x + y * y) + (x * y + y * x)
  6. 0006specialize four_square_signed_sum_two_decomposition x
  7. 0007specialize four_square_signed_sum_two_decomposition y
  8. 0008exact four_square_signed_sum_two_decomposition
  9. 0009have hsecond : (z + w) * (z + w) = (z * z + w * w) + (z * w + w * z)
  10. 0010specialize four_square_signed_sum_two_decomposition z
  11. 0011specialize four_square_signed_sum_two_decomposition w
  12. 0012exact four_square_signed_sum_two_decomposition
  13. 0013rewrite hfirst
  14. 0014rewrite hsecond
  15. 0015apply add_shuffle_middle