FS003P · theorem body

four_square_lagrange

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

Lagrange's complete four-square theorem: every natural number has an unconditionally constructive, independently kernel-checked representation as a sum of four natural squares.

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 fsl_a_fslf_natural fsl_b_fslf_natural fsl_c_fslf_natural fsl_d_fslf_natural. (n) = fsl_a_fslf_natural * fsl_a_fslf_natural + fsl_b_fslf_natural * fsl_b_fslf_natural + fsl_c_fslf_natural * fsl_c_fslf_natural + fsl_d_fslf_natural * fsl_d_fslf_natural)

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 fsl_a_fslf_natural fsl_b_fslf_natural fsl_c_fslf_natural fsl_d_fslf_natural. (n) = fsl_a_fslf_natural * fsl_a_fslf_natural + fsl_b_fslf_natural * fsl_b_fslf_natural + fsl_c_fslf_natural * fsl_c_fslf_natural + fsl_d_fslf_natural * fsl_d_fslf_natural)

Proof neighborhood

Direct theorem prerequisites

Direct theorem dependents

none

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

2 script commands · 1 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)
01Use earlier factsL1–2

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

  1. L1
    apply four_square_lagrange_from_all_primes
  2. L2
    exact four_square_prime_representation

Library-wide reading audit

Original defined command ledger · 2 lines
  1. 0001apply four_square_lagrange_from_all_primes
  2. 0002exact four_square_prime_representation