English
If f : α → β is submultiplicative on a predicate p with suitable closure properties, and s is a nonempty multiset of elements satisfying p, then f(s.prod) ≤ ∏_{a∈s} f(a).
Русский
Если f удовлетворяет условию субумножения на предикат p и множество s непусто и все элементы удовлетворяют p, тогда f(s.prod) ≤ ∏_{a∈s} f(a).
LaTeX
$$$ f : α \to β,\ p,\ h\_mul,\ hp\_mul,\ s : Multiset α,\ hs\_nonempty : s ≠ ∅,\ hs : ∀ a, a ∈ s → p a \Rightarrow f(s.prod) ≤ (s.map f).prod $$$
Lean4
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : α → β) (p : α → Prop) (h_one : f 1 = 1) (hp_one : p 1)
(h_mul : ∀ a b, p a → p b → f (a * b) ≤ f a * f b) (hp_mul : ∀ a b, p a → p b → p (a * b)) (s : Multiset α)
(hps : ∀ a, a ∈ s → p a) : f s.prod ≤ (s.map f).prod :=
by
revert s
refine Multiset.induction ?_ ?_
· simp [le_of_eq h_one]
intro a s hs hpsa
have hps : ∀ x, x ∈ s → p x := fun x hx => hpsa x (mem_cons_of_mem hx)
have hp_prod : p s.prod := prod_induction p s hp_mul hp_one hps
rw [prod_cons, map_cons, prod_cons]
exact (h_mul a s.prod (hpsa a (mem_cons_self a s)) hp_prod).trans (mul_le_mul_left' (hs hps) _)
