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Probability: kernels, pushforward and sequential composition of laws - #21
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`Distribution.bindK μ f`, the mixture of the kernel `f` over the law `μ` (the composition of conditional distributions, CR18 §3.6.1), with the lemmas the probabilistic automata of the next layer use: linearity, composition of mixtures, the point kernel, pushforwards as point kernels and through mixtures, restriction through mixtures, weight and nonnegativity, and independent products as mixtures (`prod_eq_bindK`, `pi_eq_bindK_update`). Probability laws get `ProbDist.map` with `map_map`, and `ProbDist.bind`, defined from `bindK` (the one bind). Part of #14. Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>
…or steps
CR18 §3.6.1 defines a conditional distribution p_{Y|X}; it does not define its
mixture over a law, so the module doc and THEORY.md now say only that. The root
doc of Probability is wrapped to 100 characters, and the steps of Kernel.lean
are lemmas (#18).
Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>
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The restriction and pushforward facts that do not involve kernels (`support_restrict_subset`, `restrict_restrict`, `restrict_congr`, `restrict_false`, `single_restrict`, `fTransform_single`, `restrict_fTransform`) move to Distribution.lean, next to `restrict` and `fTransform`. Kernel.lean no longer opens `Classical`: `bindK_ite` states its condition under `open scoped Classical in`, and the four proofs that decide equalities or events start with `classical`. Two `show` steps become direct terms. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01E7GPy1oUx164NSZsxJT3N9
Each induction on a law says what both sides are at a sum of two laws and at a point mass; the product lemmas comment each step (the factor split off, the terms that survive the sum, the support cases). Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01E7GPy1oUx164NSZsxJT3N9
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Adds
Distribution.bindK, the mixture of a kernel over a law (CR18 §3.6.1), with the lemmas #22 uses, and the probability lawsProbDist.mapandProbDist.bindbuilt on it. Seven restriction and pushforward lemmas that the kernel lemmas need are added toDistribution.lean.Part of #14.