FS0009

four_square_gap_balance_right

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

Exact expanded first-order arithmetic statement

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

Constructive proof overview

Generated structural guide

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

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

FS0008 four_square_sum_expansion add_mul Stable theorem; checked-use authorized mul_add Stable theorem; checked-use authorized FS0007 four_square_additive_gap_reorder

Direct dependents

FS000A four_square_gap_balance_left FS000B four_square_absolute_square_balance

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 · 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.

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 exact 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