Big operators on a finset in ordered rings

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

theorem Finset.prod_nonneg {ι : Type u_1} {R : Type u_2} [CommMonoidWithZero R] [PartialOrder R] [ZeroLEOneClass R] [PosMulMono R] {f : ι → R} {s : Finset ι} (h0 : ∀ i ∈ s, 0 ≤ f i) : 0 ≤ ∏ i ∈ s, f i

theorem Finset.prod_le_prod {ι : Type u_1} {R : Type u_2} [CommMonoidWithZero R] [PartialOrder R] [ZeroLEOneClass R] [PosMulMono R] {f : ι → R} {g : ι → R} {s : Finset ι} (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, f i ≤ g i) : ∏ i ∈ s, f i ≤ ∏ i ∈ s, g i

If all f i, i ∈ s, are nonnegative and each f i is less than or equal to g i, then the product of f i is less than or equal to the product of g i. See also Finset.prod_le_prod' for the case of an ordered commutative multiplicative monoid.

theorem GCongr.prod_le_prod {ι : Type u_1} {R : Type u_2} [CommMonoidWithZero R] [PartialOrder R] [ZeroLEOneClass R] [PosMulMono R] {f : ι → R} {g : ι → R} {s : Finset ι} (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, f i ≤ g i) : s.prod f ≤ s.prod g

If all f i, i ∈ s, are nonnegative and each f i is less than or equal to g i, then the product of f i is less than or equal to the product of g i.

This is a variant (beta-reduced) version of the standard lemma Finset.prod_le_prod, convenient for the gcongr tactic.

theorem Finset.prod_le_one {ι : Type u_1} {R : Type u_2} [CommMonoidWithZero R] [PartialOrder R] [ZeroLEOneClass R] [PosMulMono R] {f : ι → R} {s : Finset ι} (h0 : ∀ i ∈ s, 0 ≤ f i) (h1 : ∀ i ∈ s, f i ≤ 1) : ∏ i ∈ s, f i ≤ 1

If each f i, i ∈ s belongs to [0, 1], then their product is less than or equal to one. See also Finset.prod_le_one' for the case of an ordered commutative multiplicative monoid.

theorem Finset.prod_pos {ι : Type u_1} {R : Type u_2} [CommMonoidWithZero R] [PartialOrder R] [ZeroLEOneClass R] [PosMulStrictMono R] [Nontrivial R] {f : ι → R} {s : Finset ι} (h0 : ∀ i ∈ s, 0 < f i) : 0 < ∏ i ∈ s, f i

theorem Finset.prod_lt_prod {ι : Type u_1} {R : Type u_2} [CommMonoidWithZero R] [PartialOrder R] [ZeroLEOneClass R] [PosMulStrictMono R] [Nontrivial R] {f : ι → R} {g : ι → R} {s : Finset ι} (hf : ∀ i ∈ s, 0 < f i) (hfg : ∀ i ∈ s, f i ≤ g i) (hlt : ∃ i ∈ s, f i < g i) : ∏ i ∈ s, f i < ∏ i ∈ s, g i

theorem Finset.prod_lt_prod_of_nonempty {ι : Type u_1} {R : Type u_2} [CommMonoidWithZero R] [PartialOrder R] [ZeroLEOneClass R] [PosMulStrictMono R] [Nontrivial R] {f : ι → R} {g : ι → R} {s : Finset ι} (hf : ∀ i ∈ s, 0 < f i) (hfg : ∀ i ∈ s, f i < g i) (h_ne : s.Nonempty) : ∏ i ∈ s, f i < ∏ i ∈ s, g i

theorem Finset.prod_add_prod_le {ι : Type u_1} {R : Type u_2} [OrderedCommSemiring R] {s : Finset ι} {i : ι} {f : ι → R} {g : ι → R} {h : ι → R} (hi : i ∈ s) (h2i : g i + h i ≤ f i) (hgf : ∀ j ∈ s, j ≠ i → g j ≤ f j) (hhf : ∀ j ∈ s, j ≠ i → h j ≤ f j) (hg : ∀ i ∈ s, 0 ≤ g i) (hh : ∀ i ∈ s, 0 ≤ h i) : ∏ i ∈ s, g i + ∏ i ∈ s, h i ≤ ∏ i ∈ s, f i

If g, h ≤ f and g i + h i ≤ f i, then the product of f over s is at least the sum of the products of g and h. This is the version for OrderedCommSemiring.

theorem Finset.sum_mul_sq_le_sq_mul_sq {ι : Type u_1} {R : Type u_2} [LinearOrderedCommSemiring R] [ExistsAddOfLE R] (s : Finset ι) (f : ι → R) (g : ι → R) : (∑ i ∈ s, f i * g i) ^ 2 ≤ (∑ i ∈ s, f i ^ 2) * ∑ i ∈ s, g i ^ 2

Cauchy-Schwarz inequality for finsets.

theorem Finset.abs_prod {ι : Type u_1} {R : Type u_2} [LinearOrderedCommRing R] (s : Finset ι) (f : ι → R) : |∏ x ∈ s, f x| = ∏ x ∈ s, |f x|

@[simp]

theorem CanonicallyOrderedCommSemiring.prod_pos {ι : Type u_1} {R : Type u_2} [CanonicallyOrderedCommSemiring R] {f : ι → R} {s : Finset ι} [Nontrivial R] : 0 < ∏ i ∈ s, f i ↔ ∀ i ∈ s, 0 < f i

Note that the name is to match CanonicallyOrderedCommSemiring.mul_pos.

theorem Finset.prod_add_prod_le' {ι : Type u_1} {R : Type u_2} [CanonicallyOrderedCommSemiring R] {f : ι → R} {g : ι → R} {h : ι → R} {s : Finset ι} {i : ι} (hi : i ∈ s) (h2i : g i + h i ≤ f i) (hgf : ∀ j ∈ s, j ≠ i → g j ≤ f j) (hhf : ∀ j ∈ s, j ≠ i → h j ≤ f j) : ∏ i ∈ s, g i + ∏ i ∈ s, h i ≤ ∏ i ∈ s, f i

If g, h ≤ f and g i + h i ≤ f i, then the product of f over s is at least the sum of the products of g and h. This is the version for CanonicallyOrderedCommSemiring.

theorem AbsoluteValue.sum_le {ι : Type u_1} {R : Type u_2} {S : Type u_3} [Semiring R] [OrderedSemiring S] (abv : AbsoluteValue R S) (s : Finset ι) (f : ι → R) : abv (∑ i ∈ s, f i) ≤ ∑ i ∈ s, abv (f i)

theorem IsAbsoluteValue.abv_sum {ι : Type u_1} {R : Type u_2} {S : Type u_3} [Semiring R] [OrderedSemiring S] (abv : R → S) [IsAbsoluteValue abv] (f : ι → R) (s : Finset ι) : abv (∑ i ∈ s, f i) ≤ ∑ i ∈ s, abv (f i)

@[deprecated IsAbsoluteValue.abv_sum]

theorem abv_sum_le_sum_abv {ι : Type u_1} {R : Type u_2} {S : Type u_3} [Semiring R] [OrderedSemiring S] (abv : R → S) [IsAbsoluteValue abv] (f : ι → R) (s : Finset ι) : abv (∑ i ∈ s, f i) ≤ ∑ i ∈ s, abv (f i)

Alias of IsAbsoluteValue.abv_sum.

theorem AbsoluteValue.map_prod {ι : Type u_1} {R : Type u_2} {S : Type u_3} [CommSemiring R] [Nontrivial R] [LinearOrderedCommRing S] (abv : AbsoluteValue R S) (f : ι → R) (s : Finset ι) : abv (∏ i ∈ s, f i) = ∏ i ∈ s, abv (f i)

theorem IsAbsoluteValue.map_prod {ι : Type u_1} {R : Type u_2} {S : Type u_3} [CommSemiring R] [Nontrivial R] [LinearOrderedCommRing S] (abv : R → S) [IsAbsoluteValue abv] (f : ι → R) (s : Finset ι) : abv (∏ i ∈ s, f i) = ∏ i ∈ s, abv (f i)

def Mathlib.Meta.Positivity.evalFinsetProd : Mathlib.Meta.Positivity.PositivityExt

The positivity extension which proves that ∏ i ∈ s, f i is nonnegative if f is, and positive if each f i is.

TODO: The following example does not work

example (s : Finset ℕ) (f : ℕ → ℤ) (hf : ∀ n, 0 ≤ f n) : 0 ≤ s.prod f := by positivity

because compareHyp can't look for assumptions behind binders.