TS003G

two_square_self_square_zero_reflects

Dependency-curried candidate body; not Alpha-enrolled; no checked-use authority

A natural square is zero only when its coordinate 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

A natural square is zero only when its coordinate 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

TS003H two_square_norm_zero_iff_coordinates_zero

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This dependency-curried candidate body does not grant checked theorem use or 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 hzero
02Establish hsplitL3–7

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

  1. L3
    have hsplit : 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 hzero
03Separate the logical casesL8–8

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

  1. L8
    cases hsplit
04Use earlier factsL9–10

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

  1. L9
    exact hsplit_left
  2. L10
    exact hsplit_right

Library-wide reading audit

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