deterministic_comp_copy — Mathlib

English

ParallelComp is associative: (η ∥ η') ∘ (κ ∥ κ') = (η ∘ κ) ∥ (η' ∘ κ').

Русский

Параллельная компоновка ассоциативна.

LaTeX

$$$$ (\\eta \\parallelκ η') \\circκ (κ' \\parallelκ κ'') = (\\eta ∘ κ) \\parallelκ (η' ∘ κ') $$$$

Lean4

/-- For a deterministic kernel, copying then applying the kernel to the two copies is the same
as first applying the kernel then copying. -/
theorem deterministic_comp_copy {f : X → Y} (hf : Measurable f) :
    (deterministic f hf ∥ₖ deterministic f hf) ∘ₖ copy X = copy Y ∘ₖ deterministic f hf := by
  simp_rw [parallelComp_comp_copy, deterministic_prod_deterministic, copy, deterministic_comp_deterministic,
    Function.comp_def]

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