Plain-English statement
- Canonical inclusion of elementary adapted processes into the progressive `L2` domain used for general Ito integration.
Read the mathematics first, then descend into Lean
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Proof architecture
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How to read the exact Lean declaration
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- ParametersWhich symbols are arbitrary, and which structures are inferred by typeclass search?
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Syntax used on this page
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Lean statement
noncomputable def toProgressiveL2 (eta : ElementaryAdaptedProcess filtration n)
(mu : Measure Omega) [IsFiniteMeasure mu] (T : ℝ≥0) :
ProgressiveL2Integrand filtration mu T where
process := eta.value
progressive := value_stronglyProgressive eta
memLp := value_memLp_two eta mu T
@[simp] theorem toProgressiveL2_process
(eta : ElementaryAdaptedProcess filtration n)
(mu : Measure Omega) [IsFiniteMeasure mu] (T : ℝ≥0) :
(toProgressiveL2 eta mu T).process = eta.value :=
rflOpen AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoEmbedding.lean:125published source at 7bcd37294df1
Proof architecture
compare elementary approximants and general progressive integrands in one product-space L2 domain
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `simp` normalizes through registered definitional and theorem rewrites.
Why the statement has this shape
The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.
Hidden assumptions and non-claims
- Measurability is represented explicitly or must be supplied by a dependency.
- Density statements retain normalization and absolute-continuity prerequisites.