FS000A · theorem body

four_square_gap_balance_left

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

The opposite coordinate orientation has the same gap-square and ordered cross-term 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 a b. a * a + (a + b) * (a + b) = b * b + (a * (a + b) + (a + b) * a)

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

Proof neighborhood

Direct theorem prerequisites

FS0009 four_square_gap_balance_right add_comm · Stable closed

Direct theorem dependents

FS000B four_square_absolute_square_balance

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

9 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 (1)
four square gap balance right
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 + b) * (a + b) + a * a
03Use earlier factsL4–4

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

  1. L4
    apply add_comm
04Calculate and transport equalitiesL5–5

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

  1. L5
    trans b * b + ((a + b) * a + a * (a + b))
05Use earlier factsL6–6

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

  1. L6
    apply four_square_gap_balance_right
06Calculate and transport equalitiesL7–8

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

  1. L7
    congr
  2. L8
    refl
07Use earlier factsL9–9

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

  1. L9
    apply add_comm

Library-wide reading audit

Original defined command ledger · 9 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003trans (a + b) * (a + b) + a * a
  4. 0004apply add_comm
  5. 0005trans b * b + ((a + b) * a + a * (a + b))
  6. 0006apply four_square_gap_balance_right
  7. 0007congr
  8. 0008refl
  9. 0009apply add_comm