A fixed rotation or reflection preserves the draw
AutoSamplingTheory.TechnicalLemmas.Probability.OrthogonalHaar.orthogonalHaar_map_mul_left · theorem · Teaching coverage
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.
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.
theorem orthogonalHaar_map_mul_left (n : ℕ)
(Q : Matrix.orthogonalGroup (Fin n) ℝ) :
(orthogonalHaar n).map (fun O => Q * O) = orthogonalHaar nProof 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.
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.
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.
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.OrthogonalHaarSource assumptions versus formal assumptions
| Source | Actual Lean | Difference kind | Why it matters |
|---|---|---|---|
| The paper invokes full orthogonal Haar probability in dimension n>=3 or d>=2 and uses fixed row-reflection invariance. | All natural finite dimensions, canonical Borel full group, actual Haar constructor. The n=0 case is a singleton; no determinant component or conditional law is asserted. | generalization | Standard background is expanded and proved internally. This disclosed dimension extension does not strengthen the paper conclusions or identify the two experiments beyond their common full Haar law. |
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.
Encoder–denoiser: accepted · equivalent-after-elaboration
Detected semantic differences
- domains: The shared Haar background is validly extended beyond the source application dimensions, including the singleton case n=0. — The raw applications are in n>=3 (Section B.2) or d>=2 (Section 4.1); the candidate expressly extends only the standard Haar background to all natural dimensions. The n=0 carrier is a singleton, so this extension is valid and does not import the source's two-component statement into dimension zero.
- scopes: The candidate explicates the standard Haar background invoked by the paper rather than reproducing a verbatim source theorem. — The pinned original_text is an attributed ASTIS explication of the source's Haar notation, rather than a verbatim theorem proved in the paper. The construction/uniqueness/topology details are checked here against the current full module and pinned APIs; source attribution and source scope identify them as shared background.
- scopes: The external focused-test history is outside the evidence certified by this bounded independent source review. — The lesson contains a focused-test description. The allowed packet does not include that test file, which was not opened; this review verifies the fixed-reflection specialization mathematically through full-group membership and the genuine map law, without certifying the external test history.
A generalization is not a source correction. Proposed missing conditions require separate independent repair review. No proposed repair silently changes the original theorem.
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
AutoSamplingTheory.TechnicalLemmas.Probability.OrthogonalHaar.orthogonalHaarAutoSamplingTheory.TechnicalLemmas.Probability.OrthogonalHaar.orthogonalHaar_spec
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.