TS001V · theorem body

two_square_sum_square_blocks

Alpha v34 checked-use · independently kernel and Lean verified; 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.

Statement with defined notation

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

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

Proof neighborhood

Direct theorem prerequisites

TS001Q two_square_sum_square_expands add_comm · Stable closed TS001P two_square_add_swap_nested add_assoc · Stable closed

Direct theorem dependents

TS001W brahmagupta_fibonacci_two_square_identity

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

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.

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)
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 defined 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