Plain-English statement
- Normalized scalar coordinate law from a vector standard Gaussian law. This cycle-178 lower_2 bridge narrows the scalar coordinate-law source field: once the paper correspondence supplies the normalized vector increment law as `stdGaussian E` and defines the scalar coordinate law as the map by a standard orthonormal coordinate, Mathlib's multivariate Gaussian API gives `gaussianReal 0 1`. The variance packaging field remains separate.
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
Read a Lean theorem left-to-right exactly as you would unpack a mathematical sentence: name → ambient types → automatically inferred structures → explicit hypotheses → conclusion → proof. Then read the proof top-to-bottom as transformations of the current goal.
- NameWhat reusable mathematical fact is being created?
- ParametersWhich symbols are arbitrary, and which structures are inferred by typeclass search?
- PropositionAfter the colon, translate the Lean expression back into a paper statement.
- Proof actionsAfter
by, ask what each tactic does to the mathematical goal—not only what syntax it uses.
Syntax used on this page
This glossary is filtered to syntax that actually occurs in the declaration above. Open a symbol only when you meet it, rather than memorizing Lean grammar in advance.
Source voice and ASTIS voice stay separate
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Lean statement
theorem selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw
{Test E : Type*} [NormedAddCommGroup E] [InnerProductSpace Real E]
[FiniteDimensional Real E] [MeasurableSpace E] [BorelSpace E]
(normalizedVectorLaw : Test → E → MeasureTheory.Measure E)
(normalizedCoordinateLaw :
Test → E → Fin (Module.finrank Real E) → MeasureTheory.Measure Real)
(testRegular : Prop)
(hNormalizedVectorLaw :
testRegular →
∀ φ x,
normalizedVectorLaw φ x = ProbabilityTheory.stdGaussian E)
(hCoordinateLawDef :
testRegular →
∀ φ x i,
normalizedCoordinateLaw φ x i =
(normalizedVectorLaw φ x).map
(fun y : E => inner Real ((stdOrthonormalBasis Real E) i) y)) :
testRegular →
∀ φ x i,
normalizedCoordinateLaw φ x i =
ProbabilityTheory.gaussianReal (0 : Real) (1 : NNReal) := by
intro htests φ x i
calc
normalizedCoordinateLaw φ x i =
(normalizedVectorLaw φ x).map
(fun y : E => inner Real ((stdOrthonormalBasis Real E) i) y) :=
hCoordinateLawDef htests φ x i
_ = (ProbabilityTheory.stdGaussian E).map
(fun y : E => inner Real ((stdOrthonormalBasis Real E) i) y) := by
rw [hNormalizedVectorLaw htests φ x]
_ = ProbabilityTheory.gaussianReal (0 : Real) (1 : NNReal) := by
let L : StrongDual Real E :=
InnerProductSpace.toDual Real E ((stdOrthonormalBasis Real E) i)
change (ProbabilityTheory.stdGaussian E).map L =
ProbabilityTheory.gaussianReal (0 : Real) (1 : NNReal)
rw [ProbabilityTheory.IsGaussian.map_eq_gaussianReal L]
rw [ProbabilityTheory.integral_strongDual_stdGaussian L]
rw [ProbabilityTheory.variance_dual_stdGaussian L]
simp [L]
/-- Source coordinate-generator integral from the normalized Brownian law.
This cycle-184 bridge narrows `hBrownianCoordinateGeneratorSourceIntegralDef`.
The source work below it is the stochastic definition of the scalar Brownian
coordinate generator as an integral under the normalized scalar-coordinate law,
plus the already local Gaussian-coordinate/variance packaging fields from the
frozen EM Brownian increment. It does not prove the scalar Taylor identity or
the normalized-remainder domination package.
-/Proof architecture
active Brownian/Ito scalar generator backend and coordinate variance leaves
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `intro` introduces quantified hypotheses into the local proof context.
- `calc` records an equality or inequality chain matching a paper calculation.
- `rw` rewrites by an established identity.
- `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.
- A formal generator display does not establish a closed operator domain.