TS001V

two_square_sum_square_blocks

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

The square of a sum separates into its diagonal square block and repeated cross-product block.

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. (a + b) * (a + b) = (a * a + b * b) + (a * b + a * b)

Constructive proof overview

Generated structural guide

The square of a sum separates into its diagonal square block and repeated cross-product block.

The unchanged tactic script uses 4 declared prerequisites and contains 14 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

TS001Q two_square_sum_square_expands add_comm Stable theorem; checked-use authorized TS001P two_square_add_swap_nested add_assoc Stable theorem; checked-use authorized

Direct dependents

TS001W brahmagupta_fibonacci_two_square_identity

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

14 script commands · 7 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 (2)
two square sum square expandstwo square add swap nested
01Fix variables and assumptionsL1–2

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

  1. L1
    intro a
  2. L2
    intro b
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 a * a + (a * b + (a * b + b * b))
03Use earlier factsL4–4

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

  1. L4
    apply two_square_sum_square_expands
04Calculate and transport equalitiesL5–10

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

  1. L5
    trans a * a + (b * b + (a * b + a * b))
  2. L6
    congr
  3. L7
    refl
  4. L8
    trans a * b + (b * b + a * b)
  5. L9
    congr
  6. L10
    refl
05Use earlier factsL11–12

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

  1. L11
    apply add_comm
  2. L12
    apply two_square_add_swap_nested
06Calculate and transport equalitiesL13–13

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

  1. L13
    symm
07Use earlier factsL14–14

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

  1. L14
    apply add_assoc

Library-wide reading audit

Original exact command ledger · 14 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003trans a * a + (a * b + (a * b + b * b))
  4. 0004apply two_square_sum_square_expands
  5. 0005trans a * a + (b * b + (a * b + a * b))
  6. 0006congr
  7. 0007refl
  8. 0008trans a * b + (b * b + a * b)
  9. 0009congr
  10. 0010refl
  11. 0011apply add_comm
  12. 0012apply two_square_add_swap_nested
  13. 0013symm
  14. 0014apply add_assoc