TS003H

two_square_norm_zero_iff_coordinates_zero

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

Zero is an explicit boundary: a two-square norm vanishes exactly when both natural coordinates vanish.

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 b. ((a * a + b * b = 0 -> (a = 0 /\ b = 0)) /\ ((a = 0 /\ b = 0) -> a * a + b * b = 0))

Constructive proof overview

Generated structural guide

Zero is an explicit boundary: a two-square norm vanishes exactly when both natural coordinates vanish.

The unchanged tactic script uses 3 declared prerequisites and contains 18 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

add_eq_zero_left Stable theorem; checked-use authorized add_eq_zero_right Stable theorem; checked-use authorized TS003G two_square_self_square_zero_reflects

Direct dependents

none

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

18 script commands · 8 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.

Named ingredients (1)
two square self square zero reflects
01Fix variables and assumptionsL1–2

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

  1. L1
    intro a
  2. L2
    intro b
02Separate the logical casesL3–3

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

  1. L3
    split
03Fix variables and assumptionsL4–4

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

  1. L4
    intro hzero
04Separate the logical casesL5–5

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

  1. L5
    split
05Use earlier factsL6–15

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

  1. L6
    apply two_square_self_square_zero_reflects
  2. L7
    specialize add_eq_zero_left (a * a)
  3. L8
    specialize add_eq_zero_left (b * b)
  4. L9
    apply add_eq_zero_left
  5. L10
    exact hzero
  6. L11
    apply two_square_self_square_zero_reflects
  7. L12
    specialize add_eq_zero_right (a * a)
  8. L13
    specialize add_eq_zero_right (b * b)
  9. L14
    apply add_eq_zero_right
  10. L15
    exact hzero
06Fix variables and assumptionsL16–16

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

  1. L16
    intro hcoordinates
07Separate the logical casesL17–17

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

  1. L17
    cases hcoordinates
08Calculate and transport equalitiesL18–18

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L18
    simp [hcoordinates_left, hcoordinates_right]

Library-wide reading audit

Original exact command ledger · 18 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003split
  4. 0004intro hzero
  5. 0005split
  6. 0006apply two_square_self_square_zero_reflects
  7. 0007specialize add_eq_zero_left (a * a)
  8. 0008specialize add_eq_zero_left (b * b)
  9. 0009apply add_eq_zero_left
  10. 0010exact hzero
  11. 0011apply two_square_self_square_zero_reflects
  12. 0012specialize add_eq_zero_right (a * a)
  13. 0013specialize add_eq_zero_right (b * b)
  14. 0014apply add_eq_zero_right
  15. 0015exact hzero
  16. 0016intro hcoordinates
  17. 0017cases hcoordinates
  18. 0018simp [hcoordinates_left, hcoordinates_right]