Characterize the constructed probability law
AutoSamplingTheory.TechnicalLemmas.Probability.OrthogonalHaar.orthogonalHaar_spec · theorem · Teaching coverage
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.
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.
theorem orthogonalHaar_spec (n : ℕ) :
IsProbabilityMeasure (orthogonalHaar n) ∧ (orthogonalHaar n).IsHaarMeasure ∧
∀ μ : Measure (Matrix.orthogonalGroup (Fin n) ℝ),
IsProbabilityMeasure μ → μ.IsHaarMeasure → μ = orthogonalHaar nProof 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.
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.
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.
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_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⟩Source 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.
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.existsUnique_orthogonalHaarAutoSamplingTheory.TechnicalLemmas.Probability.OrthogonalHaar.orthogonalHaar
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.