Samplinglib mathematical lesson

A fixed rotation or reflection preserves the draw

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

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Mathematical statement

For every natural finite dimension n and every fixed Q in the full real orthogonal group O(n), the pushforward of the constructed Haar probability by O → Q O equals that same probability. This includes a fixed single-row reflection.

\[(O\mapsto QO)_{\#}\mu_n=\mu_n\qquad(Q\in O(n))\]

Lean statement · orthogonalHaar_map_mul_left

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_map_mul_left (n : ℕ)
    (Q : Matrix.orthogonalGroup (Fin n) ℝ) :
    (orthogonalHaar n).map (fun O => Q * O) = orthogonalHaar n

Exact module and namespace context

Proof roadmap

1Apply the Haar invariance of this actual law
Install the Haar property from the characterization of the constructed law. A fixed Q acts by continuous left multiplication in the orthogonal group; the Haar pushforward theorem gives the same measure. Q need not have determinant +1.

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.
Q is any fixed real orthogonal n by n matrix.

Mathematical proof

1. Apply the Haar invariance of this actual law

Install the Haar property from the characterization of the constructed law. A fixed Q acts by continuous left multiplication in the orthogonal group; the Haar pushforward theorem gives the same measure. Q need not have determinant +1.

\[\mu_n\text{ left invariant}\ \Longrightarrow\ (O\mapsto QO)_{\#}\mu_n=\mu_n\]

Corresponding Lean step

orthogonalHaar_spec supplies the actual Haar instance; map_mul_left_eq_self supplies the pushforward equality. The focused test constructs a diagonal single-row reflection with determinant -1 and checks it is measurable and measure preserving.

Lean verification

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

Lean proof · orthogonalHaar_map_mul_left

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

Download module (.lean)

theorem orthogonalHaar_map_mul_left (n : ℕ)
    (Q : Matrix.orthogonalGroup (Fin n) ℝ) :
    (orthogonalHaar n).map (fun O => Q * O) = orthogonalHaar n := by
  let := (orthogonalHaar_spec n).2.1
  exact map_mul_left_eq_self (orthogonalHaar n) Q

end AutoSamplingTheory.TechnicalLemmas.Probability.OrthogonalHaar

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.map_mul_left_eq_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.