TS003G · theorem body

two_square_self_square_zero_reflects

dependency-curried kernel-checked candidate body; not enrolled in Alpha or Stable

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.

Statement with defined notation

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

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

Proof neighborhood

Direct theorem prerequisites

mul_eq_zero · Stable closed

Direct theorem dependents

TS003H two_square_norm_zero_iff_coordinates_zero

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

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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