Plain-English statement
- The same mean replacement bound survives a common downstream Markov kernel. In an adjacent adaptive hybrid, the downstream kernel represents all later calls after the one conditional rule being replaced.
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 abs_real_comp_commonSuffix_sub_le_integral_eventBound
{A B C : Type*} [MeasurableSpace A] [MeasurableSpace B] [MeasurableSpace C]
(μ : Measure A) [IsProbabilityMeasure μ]
(K L : Kernel A B) [IsMarkovKernel K] [IsMarkovKernel L]
(S : Kernel B C) [IsMarkovKernel S]
(ε : A → ℝ) (hε : Integrable ε μ)
(hbound : ∀ᵐ x ∂μ, ∀ s, MeasurableSet s →
|(K x).real s - (L x).real s| ≤ ε x) :
∀ t, MeasurableSet t →
|((S ∘ₖ K) ∘ₘ μ).real t - ((S ∘ₖ L) ∘ₘ μ).real t| ≤
∫ x, ε x ∂μ := by
have hKL := abs_real_comp_sub_le_integral_eventBound μ K L ε hε hbound
have hS := KernelTotalVariation.abs_real_comp_sub_le
(K ∘ₘ μ) (L ∘ₘ μ) S hKL
simpa only [← Measure.comp_assoc] using hS
/-- Finite adjacent-hybrid errors telescope. Applying this to the output laws
of hybrids `0, ..., n` turns the one-slot bounds into their accumulated error.
The result is deliberately stated for real sequences so it does not invent a
second total-variation definition. -/Open AutoSamplingTheory/TechnicalLemmas/Probability/KernelHybridTelescope.lean:59published source at ed709a7f320d
Proof architecture
absorb every later adaptive call into one common Markov suffix while preserving the mean one-slot error
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `have` creates a named intermediate mathematical fact.
- `simpa` closes the goal after a controlled simplification of a typed result.
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.
- Integrability is an input or proved output; a displayed integral alone does not supply it.