feat: prove independent increments for Poisson process#4
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Complete the proof of HasIndependentIncrements for the Poisson process
constructed via Kolmogorov extension. This closes the last sorry in
PoissonProcess.lean.
Key additions:
- kolmogorovExtension_indep_increments: main independence theorem
reducing to coordinate independence under Measure.pi
- poissonProcessCumSum_succ_sub: increment equals coordinate difference
- Helper lemmas bridging iIndepFun_pi to grouped increment independence
via Finset.prod_image and set_biInter_finset_image
The proof works by showing that under the Kolmogorov extension,
increments X(t_{k+1}) - X(t_k) are measurable functions of independent
coordinates, and their joint distribution factors as a product.
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Summary
HasIndependentIncrementsfor the Poisson process Kolmogorov extension, closing the last sorry inPoissonProcess.leankolmogorovExtension_indep_incrementsreducing increment independence to coordinate independence underMeasure.pipoissonProcessCumSum_succ_sub,orderIsoOfFin_symm_succ_of_lt,kolmogorovExtension_coord_zero_aeTest plan
lake buildsucceeds (2910 jobs)PoissonProcess.lean