English
A diagrammatic equality expresses the posterior as a parallel composition with copies, matching the id on the other side.
Русский
Диагональная ekравенство выражает постериор через параллельное сочетание с копиями, соответствующее стороне с id.
LaTeX
$$$(Kernel.id \\parallel κ†μ) \\circ κ \\circ μ = (κ \\parallel Kernel.id) \\circ Copy \\circ μ$$$
Lean4
/-- The main property of the posterior, as equality of the following diagrams:
```
-- id -- κ
μ -- κ -| = μ -|
-- κ†μ -- id
``` -/
theorem parallelProd_posterior_comp_copy_comp :
(Kernel.id ∥ₖ κ†μ) ∘ₘ Kernel.copy 𝓧 ∘ₘ κ ∘ₘ μ = (κ ∥ₖ Kernel.id) ∘ₘ Kernel.copy Ω ∘ₘ μ := by
calc
(Kernel.id ∥ₖ κ†μ) ∘ₘ Kernel.copy 𝓧 ∘ₘ κ ∘ₘ μ
_ = (κ ∘ₘ μ) ⊗ₘ κ†μ := by rw [← Measure.compProd_eq_parallelComp_comp_copy_comp]
_ = Kernel.swap _ _ ∘ₘ (μ ⊗ₘ κ) := by rw [compProd_posterior_eq_swap_comp]
_ = Kernel.swap _ _ ∘ₘ (Kernel.id ∥ₖ κ) ∘ₘ Kernel.copy Ω ∘ₘ μ := by
rw [Measure.compProd_eq_parallelComp_comp_copy_comp]
_ = (κ ∥ₖ Kernel.id) ∘ₘ Kernel.copy Ω ∘ₘ μ := by
rw [Measure.comp_assoc, Kernel.swap_parallelComp, Measure.comp_assoc, Kernel.comp_assoc, Kernel.swap_copy,
Measure.comp_assoc]
