English
A specialized version of updating on the left branch: (update f (inl i) x) (inl j) equals updating the left-indexed family: (update (λ j, f (inl j)) i x) (inl j).
Русский
Узко-formулированное обновление слева: (update f (inl i) x) (inl j) = (update (λ j, f (inl j)) i x) (inl j).
LaTeX
$$$$ (\\operatorname{update} f (\\mathrm{inl} i) x)(\\mathrm{inl} j) = (\\operatorname{update} (\\lambda j \\mapsto f(\\mathrm{inl} j)) i\\; x)(\\mathrm{inl} j). $$$$
Lean4
@[simp]
theorem update_inl_apply_inl' {γ : α ⊕ β → Type*} [DecidableEq α] [DecidableEq (α ⊕ β)] {f : (i : α ⊕ β) → γ i} {i : α}
{x : γ (.inl i)} (j : α) : update f (.inl i) x (Sum.inl j) = update (fun j ↦ f (.inl j)) i x j :=
Function.update_apply_of_injective f Sum.inl_injective i x j
