sigma_keys_lookup — Mathlib

English

The sigma-keys lookup identity equates the sigma-encoded keys with the entries.

Русский

Идентичность sigma-ключей lookup совпадает с записями.

LaTeX

$$$$ \mathrm{keys}.\sigma (\lambda i . (\mathrm{lookup}\ i\ s).toFinset) = \langle s.\mathrm{entries}, s.\mathrm{nodup\_entries} \rangle $$$$

Lean4

@[simp]
theorem sigma_keys_lookup (s : Finmap β) :
    s.keys.sigma (fun i => (s.lookup i).toFinset) = ⟨s.entries, s.nodup_entries⟩ :=
  by
  ext x
  have : x ∈ s.entries → x.1 ∈ s.keys := Multiset.mem_map_of_mem _
  simpa [lookup_eq_some_iff]

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