TS0000 — even_square_is_four_multiple

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 z k. z = 2 * k -> exists q. z * z = 4 * q + 0

Constructive proof overview

Generated structural guide

The square of an even natural has an explicit multiple-of-four witness.

The unchanged tactic script uses 2 declared prerequisites and contains 28 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

mul_assoc Stable theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized

Direct dependents

TS0002 square_mod_four_zero_or_one

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

28 script commands · 11 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 assumptionsL1–3

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

L1
intro z
L2
intro k
L3
intro heven

02Construct an explicit witnessL4–4

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

L4
exists k * k

03Calculate and transport equalitiesL5–11