PF001C

square_zero_root

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

An actual natural square is zero only when its root is zero.

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 a. a * a = 0 -> a = 0

Constructive proof overview

Generated structural guide

An actual natural square is zero only when its root is zero.

The unchanged tactic script uses 1 declared prerequisite and contains 10 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_eq_zero Stable theorem; checked-use authorized

Direct dependents

PF001G square_divides_square_root

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

10 script commands · 4 reading checkpoints · 1 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 a
  2. L2
    intro heq
02Establish hzeroL3–7

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul eq zero.

  1. L3
    have hzero : a = 0 \/ a = 0
  2. L4
    specialize mul_eq_zero a
  3. L5
    specialize mul_eq_zero a
  4. L6
    apply mul_eq_zero
  5. L7
    exact heq
03Separate the logical casesL8–8

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

  1. L8
    cases hzero
04Use earlier factsL9–10

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

  1. L9
    exact hzero_left
  2. L10
    exact hzero_right

Library-wide reading audit

Original exact command ledger · 10 lines
  1. 0001intro a
  2. 0002intro heq
  3. 0003have hzero : a = 0 \/ a = 0
  4. 0004specialize mul_eq_zero a
  5. 0005specialize mul_eq_zero a
  6. 0006apply mul_eq_zero
  7. 0007exact heq
  8. 0008cases hzero
  9. 0009exact hzero_left
  10. 0010exact hzero_right