FS001W

four_square_descent_double_square_four_sum

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

Exact expanded first-order arithmetic statement

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

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 3 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

FS000E four_square_product_square four_mul_eq_double_double Stable theorem; checked-use authorized two_mul_eq_add_self Alpha theorem; checked-use authorized

Direct dependents

FS001X four_square_descent_odd_half_norm_strict

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 · 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.

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 exact 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