TS001Z · theorem body

every_natural_square_is_sum_of_two_squares

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

Every natural square has the direct representation n²+0².

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 n. exists ftsc_first_natural_square ftsc_second_natural_square. (n * n) = ftsc_first_natural_square * ftsc_first_natural_square + ftsc_second_natural_square * ftsc_second_natural_square

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 n. exists ftsc_first_natural_square ftsc_second_natural_square. (n * n) = ftsc_first_natural_square * ftsc_first_natural_square + ftsc_second_natural_square * ftsc_second_natural_square

Proof neighborhood

Direct theorem prerequisites

none

Direct theorem dependents

TS002R represented_factor_product_times_square_is_two_square TS002S beta_grouped_prime_square_factor_product_is_two_square TS003M two_square_representation_preserved_by_square_factor

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

4 script commands · 3 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.

01Fix variables and assumptionsL1–1

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

  1. L1
    intro n
02Construct an explicit witnessL2–3

Supply the displayed value, then prove that it has the required property.

  1. L2
    exists n
  2. L3
    exists 0
03Calculate and transport equalitiesL4–4

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

  1. L4
    simp

Library-wide reading audit

Original defined command ledger · 4 lines
  1. 0001intro n
  2. 0002exists n
  3. 0003exists 0
  4. 0004simp