FS000B · theorem body

four_square_absolute_square_balance

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

Either constructive absolute-difference branch transports coordinate squares to its magnitude square and cross 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 left right magnitude. (left = right + magnitude \/ right = left + magnitude) -> left * left + right * right = magnitude * magnitude + (left * right + right * left)

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 left right magnitude. (left = right + magnitude \/ right = left + magnitude) -> left * left + right * right = magnitude * magnitude + (left * right + right * left)

Proof neighborhood

Direct theorem prerequisites

FS0009 four_square_gap_balance_right FS000A four_square_gap_balance_left

Direct theorem dependents

FS000C signed_balance_square_transport FS000F quaternion_coordinate_square_transport FS0011 four_square_conjugate_coordinate_square_transport

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

15 script commands · 6 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 gap balance rightfour square gap balance left
01Fix variables and assumptionsL1–4

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

  1. L1
    intro left
  2. L2
    intro right
  3. L3
    intro magnitude
  4. L4
    intro hdifference
02Separate the logical casesL5–5

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L5
    cases hdifference
03Calculate and transport equalitiesL6–9

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

  1. L6
    rewrite hdifference_left
  2. L7
    rewrite hdifference_left
  3. L8
    rewrite hdifference_left
  4. L9
    rewrite hdifference_left
04Use earlier factsL10–10

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

  1. L10
    apply four_square_gap_balance_right
05Calculate and transport equalitiesL11–14

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

  1. L11
    rewrite hdifference_right
  2. L12
    rewrite hdifference_right
  3. L13
    rewrite hdifference_right
  4. L14
    rewrite hdifference_right
06Use earlier factsL15–15

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

  1. L15
    apply four_square_gap_balance_left

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro left
  2. 0002intro right
  3. 0003intro magnitude
  4. 0004intro hdifference
  5. 0005cases hdifference
  6. 0006rewrite hdifference_left
  7. 0007rewrite hdifference_left
  8. 0008rewrite hdifference_left
  9. 0009rewrite hdifference_left
  10. 0010apply four_square_gap_balance_right
  11. 0011rewrite hdifference_right
  12. 0012rewrite hdifference_right
  13. 0013rewrite hdifference_right
  14. 0014rewrite hdifference_right
  15. 0015apply four_square_gap_balance_left