norm_terminal_sub_eq_process_sub

Plain-English statement

Two elementary Ito integrals built on different dyadic grids have exactly the same L2 distance as their integrands.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

Mathematical statement

||I_T(eta)-I_T(xi)||_2 = ||eta-xi||_L2 for heterogeneous dyadic elementary processes.

Intuition

Both grids are refined to one common level; coefficient replication leaves the step process unchanged and Brownian increments telescope inside every coarse block.

Conditions

two dyadic elementary adapted processes
Brownian motion relative to their filtration
a common terminal horizon

Why these conditions cannot be dropped

dyadic nesting supplies an exact common grid
filtration monotonicity preserves adapted coefficients
the Brownian isometry controls the terminal sums

Proof route

refine both grids to their maximum level
prove pointwise process preservation
telescope each block of Brownian increments
apply the same-grid distance isometry

Lean interface notes

refinement changes neither processToLp nor terminalToLp
this theorem makes terminal completion independent of heterogeneous approximation grids

Lean learning studio · mathematics → formal proof

Read the mathematics first, then descend into Lean

You do not need to know Lean before opening this panel. The page keeps the paper-level theorem, rigorous proof obligations, exact Lean declaration, and proof dependencies as separate layers so a first-time reader can move down one layer at a time.

BeginnerWhy this theorem exists → intuition → statement → one hand calculation. Hide proof-engineering detail.

RigorousExpose assumptions, hidden measure/limit/domain issues, proof route, and rigorous references.

Lean learnerOpen the exact declaration, proof tree/network, syntax glossary, and line-by-line explanation.

Proof architecture

Start with the tree when learning: prerequisites sit below the theorem and downstream results sit above it. Switch to the network when you want to understand where this declaration lives in the local formal library. Every mapped node is clickable.

Loading source-derived dependency evidence…

Graph rule: only dependencies found by the ASTIS source scan are drawn. Missing tactic indirection is treated as an under-approximation; the site never invents an edge just to make a prettier graph.

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

When Samplinglib shows a short quotation from Chewi, it is labeled as a source excerpt and linked to the canonical book page. Intuition, expanded proof steps, hidden regularity assumptions, and Lean explanations are ASTIS-authored commentary. A quotation never substitutes for a formal proof, and an ASTIS explanation is never attributed to the textbook author.

Lean statement

theorem norm_terminal_sub_eq_process_sub
    (eta xi : DyadicElementaryProcess filtration T)
    (hB : IsBrownianMotionWithFiltration B filtration mu) :
    ‖terminalToLp eta hB - terminalToLp xi hB‖ =
      ‖processToLp eta hB - processToLp xi hB‖ := by
  let _ : IsProbabilityMeasure mu := hB.isProbabilityMeasure
  let eta' := commonRefinementLeft eta xi
  let xi' := commonRefinementRight eta xi
  calc
    ‖terminalToLp eta hB - terminalToLp xi hB‖ =
        ‖terminalToLp eta' hB - terminalToLp xi' hB‖ := by
      rw [commonRefinementLeft_terminalToLp_eq eta xi hB,
        commonRefinementRight_terminalToLp_eq eta xi hB]
    _ = ‖elementaryProcessToLp eta'.process hB T -
        elementaryProcessToLp xi'.process hB T‖ :=
      norm_elementaryItoTerminalToLp_sub eta'.process xi'.process
        (commonRefinement_times_eq eta xi) hB T
    _ = ‖processToLp eta' hB - processToLp xi' hB‖ := by
      rw [elementaryProcessToLp_eq_processToLp eta' hB,
        elementaryProcessToLp_eq_processToLp xi' hB]
    _ = ‖processToLp eta hB - processToLp xi hB‖ := by
      rw [commonRefinementLeft_processToLp_eq eta xi hB,
        commonRefinementRight_processToLp_eq eta xi hB]

end DyadicElementaryRefinement
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheory

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:443published source at 7bcd37294df1