FS005H

four_square_signed_pair_block_decomposition

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

Exact expanded first-order arithmetic 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))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 15 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

FS005F four_square_signed_sum_two_decomposition add_shuffle_middle Alpha theorem; checked-use authorized

Direct dependents

FS000X four_square_conjugate_left_decomposition

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

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.

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 exact 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