# Big operators on a multiset in ordered rings #

This file contains the results concerning the interaction of multiset big operators with ordered rings.

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@[simp]
theorem CanonicallyOrderedAdd.multiset_prod_pos {R : Type u_1} [CommSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R] [NoZeroDivisors R] [Nontrivial R] {m : Multiset R} :
0 < m.prod ↔ ∀ x ∈ m, 0 < x
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theorem Multiset.le_prod_of_submultiplicative_on_pred_of_nonneg {α : Type u_1} {β : Type u_2} [CommMonoid α] [CommMonoidWithZero β] [PartialOrder β] [PosMulMono β] (f : α → β) (p : α → Prop) (h0 : ∀ (a : α), 0 ≤ f a) (h_one : f 1 ≤ 1) (h_mul : ∀ (a b : α), p a → p b → f (a * b) ≤ f a * f b) (hp_mul : ∀ (a b : α), p a → p b → p (a * b)) (s : Multiset α) (hps : ∀ a ∈ s, p a) :
f s.prod ≤ (map f s).prod
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theorem Multiset.le_prod_of_submultiplicative_of_nonneg {α : Type u_1} {β : Type u_2} [CommMonoid α] [CommMonoidWithZero β] [PartialOrder β] [PosMulMono β] (f : α → β) (h0 : ∀ (a : α), 0 ≤ f a) (h_one : f 1 ≤ 1) (h_mul : ∀ (a b : α), f (a * b) ≤ f a * f b) (s : Multiset α) :
f s.prod ≤ (map f s).prod