piecewise_piecewise_of_subset_left

English

Piecewise over a singleton equals the update by the singleton’s value: piecewise {i} f g = update g i (f i).

Русский

Piecewise над единичным множеством равно обновлению: piecewise {i} f g = update g i (f i).

LaTeX

$$$\\text{piecewise}\\{i\\} \\; f \\; g = \\text{update } g\\ i\\ (f\\ i).$$$

Lean4

theorem piecewise_piecewise_of_subset_left {s t : Finset ι} [∀ i, Decidable (i ∈ s)] [∀ i, Decidable (i ∈ t)]
    (h : s ⊆ t) (f₁ f₂ g : ∀ a, π a) : s.piecewise (t.piecewise f₁ f₂) g = s.piecewise f₁ g :=
  s.piecewise_congr (fun _i hi => piecewise_eq_of_mem _ _ _ (h hi)) fun _ _ => rfl

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