FS0009 · theorem body

four_square_gap_balance_right

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

A larger coordinate and its base have squared sum equal to the gap square plus their two ordered cross products.

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

Proof neighborhood

Direct theorem prerequisites

FS0008 four_square_sum_expansion add_mul · Stable closed mul_add · Stable closed FS0007 four_square_additive_gap_reorder

Direct theorem dependents

FS000A four_square_gap_balance_left 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

14 script commands · 8 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)
four square sum expansionfour square additive gap reorder
01Fix variables and assumptionsL1–2

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

  1. L1
    intro a
  2. L2
    intro b
02Use earlier factsL3–4

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

  1. L3
    specialize four_square_sum_expansion a
  2. L4
    specialize four_square_sum_expansion b
03Calculate and transport equalitiesL5–5

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

  1. L5
    rewrite four_square_sum_expansion
04Use earlier factsL6–8

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

  1. L6
    specialize add_mul a
  2. L7
    specialize add_mul b
  3. L8
    specialize add_mul a
05Calculate and transport equalitiesL9–9

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

  1. L9
    rewrite add_mul
06Use earlier factsL10–12

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

  1. L10
    specialize mul_add a
  2. L11
    specialize mul_add a
  3. L12
    specialize mul_add b
07Calculate and transport equalitiesL13–13

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

  1. L13
    rewrite mul_add
08Use earlier factsL14–14

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

  1. L14
    apply four_square_additive_gap_reorder

Library-wide reading audit

Original defined command ledger · 14 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003specialize four_square_sum_expansion a
  4. 0004specialize four_square_sum_expansion b
  5. 0005rewrite four_square_sum_expansion
  6. 0006specialize add_mul a
  7. 0007specialize add_mul b
  8. 0008specialize add_mul a
  9. 0009rewrite add_mul
  10. 0010specialize mul_add a
  11. 0011specialize mul_add a
  12. 0012specialize mul_add b
  13. 0013rewrite mul_add
  14. 0014apply four_square_additive_gap_reorder