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The new lemma is

Lemma map_filter_insert m i x :
  filter P (<[i:=x]> m)
  = if decide (P (i, x)) then <[i:=x]> (filter P m) else filter P (delete i m).

We need the delete because there might be another entry with key i in m.

In addition there are the lemmas:

Lemma map_filter_insert_True m i x : (* used to be map_filter_insert *)
  P (i, x) → filter P (<[i:=x]> m) = <[i:=x]> (filter P m).
Lemma map_filter_insert_False m i x : (* used to be map_filter_insert_not_delete *)
  ¬ P (i, x) → filter P (<[i:=x]> m) = filter P (delete i m).

I use the _True and _False suffixes similar to decide_True and decide_False.

See discussion in !245 (335bc4d6, comment 72118) for the discussion that led to this MR.