GI0006

gaussian_signed_square_exists

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Construct the actual natural square of every represented integer from its witnessed absolute difference.

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 p n. exists s. (((((p) * (p))) + (((n) * (n)))) = ((s) + (((((p) * (n))) + (((n) * (p)))))))

Constructive proof overview

Generated structural guide

Construct the actual natural square of every represented integer from its witnessed absolute difference.

The unchanged tactic script uses 2 declared prerequisites and contains 11 exact native proof lines.

Alpha v34 checked-use · first admitted v28 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

matrix_lattice_absolute_difference_exists Alpha theorem; checked-use authorized four_square_absolute_square_balance Alpha theorem; checked-use authorized

Direct dependents

GI0012 gaussian_signed_norm_exists GI001A gaussian_signed_norm_product

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

11 script commands · 5 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.

01Fix variables and assumptionsL1–2

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

  1. L1
    intro p
  2. L2
    intro n
02Use earlier factsL3–4

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

  1. L3
    specialize matrix_lattice_absolute_difference_exists p
  2. L4
    specialize matrix_lattice_absolute_difference_exists n
03Separate the logical casesL5–5

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

  1. L5
    cases matrix_lattice_absolute_difference_exists
04Construct an explicit witnessL6–6

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

  1. L6
    exists x * x
05Use earlier factsL7–11

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

  1. L7
    specialize four_square_absolute_square_balance p
  2. L8
    specialize four_square_absolute_square_balance n
  3. L9
    specialize four_square_absolute_square_balance x
  4. L10
    apply four_square_absolute_square_balance
  5. L11
    exact matrix_lattice_absolute_difference_exists_witness

Library-wide reading audit

Original exact command ledger · 11 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003specialize matrix_lattice_absolute_difference_exists p
  4. 0004specialize matrix_lattice_absolute_difference_exists n
  5. 0005cases matrix_lattice_absolute_difference_exists
  6. 0006exists x * x
  7. 0007specialize four_square_absolute_square_balance p
  8. 0008specialize four_square_absolute_square_balance n
  9. 0009specialize four_square_absolute_square_balance x
  10. 0010apply four_square_absolute_square_balance
  11. 0011exact matrix_lattice_absolute_difference_exists_witness