TS003H · theorem body

two_square_norm_zero_iff_coordinates_zero

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

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.

Statement with defined notation

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

Proof neighborhood

Direct theorem prerequisites

add_eq_zero_left · Stable closed add_eq_zero_right · Stable closed TS003G two_square_self_square_zero_reflects

Direct theorem dependents

none

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

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.

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

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 defined 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]