argmax_cons — Mathlib

English

For any a, l, b, bs, zipRight(a :: l, b :: bs) equals (Some a, b) :: zipRight l bs.

Русский

Для любых a, l, b, bs, zipRight(a :: l, b :: bs) равно (Some a, b) :: zipRight l bs.

LaTeX

$$$ \operatorname{zipRight}(a :: as, b :: bs) = (\mathrm{Some}\, a, b) :: \operatorname{zipRight}(as, bs)$$$

Lean4

theorem argmax_cons (f : α → β) (a : α) (l : List α) :
    argmax f (a :: l) = Option.casesOn (argmax f l) (some a) fun c => if f a < f c then some c else some a :=
  List.reverseRecOn l rfl fun hd tl ih =>
    by
    rw [← cons_append, argmax_concat, ih, argmax_concat]
    rcases h : argmax f hd with - | m
    · simp
    dsimp
    rw [← apply_ite, ← apply_ite]
    dsimp
    split_ifs <;> try rfl
    · exact absurd (lt_trans ‹f a < f m› ‹_›) ‹_›
    · cases (‹f a < f tl›.gt_or_lt _).elim ‹_› ‹_›

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