FS001W · theorem body

four_square_descent_double_square_four_sum

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

The square of the doubled odd half is exactly four copies of the half-square.

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 h. (2 * h) * (2 * h) = (h * h + h * h) + (h * h + h * h)

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 h. (2 * h) * (2 * h) = (h * h + h * h) + (h * h + h * h)

Proof neighborhood

Direct theorem prerequisites

FS000E four_square_product_square four_mul_eq_double_double · Stable closed two_mul_eq_add_self · Alpha closed

Direct theorem dependents

FS001X four_square_descent_odd_half_norm_strict

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 · 9 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 product square
01Fix variables and assumptionsL1–1

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

  1. L1
    intro h
02Calculate and transport equalitiesL2–2

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

  1. L2
    trans (2 * 2) * (h * h)
03Use earlier factsL3–3

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

  1. L3
    apply four_square_product_square
04Calculate and transport equalitiesL4–8

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

  1. L4
    trans 4 * (h * h)
  2. L5
    congr
  3. L6
    norm_num
  4. L7
    refl
  5. L8
    trans 2 * (2 * (h * h))
05Use earlier factsL9–9

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

  1. L9
    apply four_mul_eq_double_double
06Calculate and transport equalitiesL10–10

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

  1. L10
    trans (2 * (h * h)) + (2 * (h * h))
07Use earlier factsL11–11

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

  1. L11
    apply two_mul_eq_add_self
08Calculate and transport equalitiesL12–12

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

  1. L12
    congr
09Use earlier factsL13–14

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

  1. L13
    apply two_mul_eq_add_self
  2. L14
    apply two_mul_eq_add_self

Library-wide reading audit

Original defined command ledger · 14 lines
  1. 0001intro h
  2. 0002trans (2 * 2) * (h * h)
  3. 0003apply four_square_product_square
  4. 0004trans 4 * (h * h)
  5. 0005congr
  6. 0006norm_num
  7. 0007refl
  8. 0008trans 2 * (2 * (h * h))
  9. 0009apply four_mul_eq_double_double
  10. 0010trans (2 * (h * h)) + (2 * (h * h))
  11. 0011apply two_mul_eq_add_self
  12. 0012congr
  13. 0013apply two_mul_eq_add_self
  14. 0014apply two_mul_eq_add_self