Samplinglib mathematical lesson

Characterize the constructed probability law

AutoSamplingTheory.TechnicalLemmas.Probability.OrthogonalHaar.orthogonalHaar_spec · theorem · Teaching coverage

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Mathematical statement

For every natural finite dimension n, the constructed full orthogonal Haar law is a probability Haar measure. Every probability Haar measure on that same Borel group equals it.

\[\mu_n(O(n))=1,\quad\mu_n\text{ is Haar},\quad\forall\nu:\ (\nu(O(n))=1\land\nu\text{ is Haar})\Rightarrow\nu=\mu_n\]

Lean statement · orthogonalHaar_spec

The measure lives on the actual full orthogonal subtype with its canonical Borel space. No abstract Haar law or compactness premise is supplied by the caller.

Braces mark parameters Lean can infer; square brackets request structures such as a measurable space or probability measure. Named hypotheses are mathematical premises, not facts established by this declaration. Section parameters are described in the mathematical hypotheses above; the module link retains their exact source context.

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theorem orthogonalHaar_spec (n : ℕ) :
    IsProbabilityMeasure (orthogonalHaar n) ∧ (orthogonalHaar n).IsHaarMeasure ∧
      ∀ μ : Measure (Matrix.orthogonalGroup (Fin n) ℝ),
        IsProbabilityMeasure μ → μ.IsHaarMeasure → μ = orthogonalHaar n

Exact module and namespace context

Proof roadmap

1Prove normalization and the Haar property for this constructor
Unfold the definition. The constructor normalization set is the entire group, so its normalization theorem gives total mass one; the actual constructor instance supplies the Haar property.
2Consume the existence/uniqueness theorem
The constructed measure and any competing Haar probability both satisfy the predicate proved uniquely inhabited above. Uniqueness therefore identifies them.

Full assumptions and notation

All objects and hypotheses

n is an arbitrary natural finite dimension. The paper applications use n≥3 or d≥2; the shared background also covers n=0.
O(n) is the full real orthogonal group O^T O=I, equipped with the subspace topology of finite real matrices and its Borel sigma-algebra. Both determinant signs are allowed whenever they exist.
Haar means left invariant, finite on compact sets and positive on nonempty open sets. Probability means total mass one. Compactness and second countability are proved internally, not assumed as public premises.

Mathematical proof

1. Prove normalization and the Haar property for this constructor

Unfold the definition. The constructor normalization set is the entire group, so its normalization theorem gives total mass one; the actual constructor instance supplies the Haar property.

\[\mu_n(O(n))=1,\qquad\mu_n\text{ is Haar}\]

Corresponding Lean step

Use haarMeasure_self and the actual IsHaarMeasure instance, not a probability premise supplied by the caller.

2. Consume the existence/uniqueness theorem

The constructed measure and any competing Haar probability both satisfy the predicate proved uniquely inhabited above. Uniqueness therefore identifies them.

\[\exists!\mu:\Phi(\mu),\quad\Phi(\nu),\Phi(\mu_n)\ \Longrightarrow\nu=\mu_n\]

Corresponding Lean step

The proof genuinely calls existsUnique_orthogonalHaar n and its unique field.

Lean verification

The exact proof begins folded immediately after the mathematical argument; the statement is folded beside the claim above.

Lean proof · orthogonalHaar_spec

The adjacent exact Lean constructs the law and discharges its topology, normalization and invariance dependencies internally.

Download module (.lean)

theorem orthogonalHaar_spec (n : ℕ) :
    IsProbabilityMeasure (orthogonalHaar n) ∧ (orthogonalHaar n).IsHaarMeasure ∧
      ∀ μ : Measure (Matrix.orthogonalGroup (Fin n) ℝ),
        IsProbabilityMeasure μ → μ.IsHaarMeasure → μ = orthogonalHaar n := by
  have hprob : IsProbabilityMeasure (orthogonalHaar n) := ⟨by
    simpa only [orthogonalHaar, PositiveCompacts.coe_top] using
      (Measure.haarMeasure_self (K₀ :=
        (⊤ : PositiveCompacts (Matrix.orthogonalGroup (Fin n) ℝ))))⟩
  have hhaar : (orthogonalHaar n).IsHaarMeasure := by
    dsimp only [orthogonalHaar]
    infer_instance
  refine ⟨hprob, hhaar, ?_⟩
  intro μ hp hh
  exact (existsUnique_orthogonalHaar n).unique ⟨hp, hh⟩ ⟨hprob, hhaar⟩

Exact module and namespace context

Scope and omitted-condition boundaries

Only the common full orthogonal Haar probability and fixed left multiplication invariance. Determinant-conditioned laws, component half-mass, principal-minor sign/zero expectation, Gaussian densities, independence from an algorithm seed, adaptive posterior invariance, Krylov reduction and both main lower bounds remain open. Dimension zero is included as a singleton group; no two-components claim is made there.

Sources, dependencies, and reuse

Source and reuse

Samplinglib parents called

Mathlib API called (external library)

MeasureTheory.Measure.haarMeasure_self

Mathematical sources

Shared full Haar background — Background common to both experiments; only the fixed left-invariance part of the row-reflection argument.

Samplinglib prose is not a quotation or a source-equivalence certificate. Definitions and aliases are explained as constructions, not counted as new mathematical proofs.