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auto-task(exists_boost_mul_rotation): prove restricted Lorentz elements are a boost times a rotation#1350

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auto-task(exists_boost_mul_rotation): prove restricted Lorentz elements are a boost times a rotation#1350
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Summary

Completes the TODO in Physlib/Relativity/LorentzGroup/Restricted/Basic.lean:

TODO "Prove that every member of the restricted Lorentz group is combination of a boost and a rotation."

The homeomorphism LorentzGroup.toBoostRotation in
Physlib/Relativity/LorentzGroup/Restricted/FromBoostRotation.lean already
provides the boost–rotation decomposition; its file docstring states that the
inverse "writes elements of the restricted Lorentz group as a product of a
generalized boost and a rotation". This PR turns that into an explicit,
user-facing existence theorem and removes the now-satisfied TODO note.

Changes

  • Added LorentzGroup.exists_boost_mul_rotation to FromBoostRotation.lean:
    for every Λ : LorentzGroup.restricted d there exist a velocity
    v : Lorentz.Velocity d and a rotation R : Rotations d with
    (Λ : LorentzGroup d) = generalizedBoost 0 v * (R : LorentzGroup d).
    The proof uses the existing toVelocity/toRotation maps and
    mul_inv_cancel_left.
  • Removed the corresponding TODO "..." marker from
    Restricted/Basic.lean.

How to review

  1. Read the new theorem exists_boost_mul_rotation; confirm its statement
    says every restricted Lorentz element equals a generalized boost times a
    rotation, and that it builds with no sorry.
  2. Confirm the TODO "Prove that every member of the restricted Lorentz group is combination of a boost and a rotation." note is gone from
    Restricted/Basic.lean.
  3. lake build (project builds green; the theorem depends only on
    propext, Classical.choice, Quot.sound — no new axioms).

…ts are a boost times a rotation

Co-authored-by: Claude <noreply@anthropic.com>
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github-actions Bot commented Jul 1, 2026

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Thank you for this PR, which will now be reviewed. If submitting to ./Physlib or ./QuantumInfo, please see our review guidelines if you are not familiar with the process. You should expect a back and forth with a reviewer before your PR is merged. See also that link for how to add appropriate labels to your PR. The PR will also go through a number of automated checks. You can learn more about these here, including how to run them locally.

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