| import Mathlib |
| import OAI.Combinatorics.IndependentSets.Geometry.RationalLaw |
|
|
| namespace OAI |
|
|
| namespace LargeIndependentSets.RationalLaw |
| open scoped BigOperators Classical |
| variable {I : Type*} [Fintype I] |
|
|
| noncomputable def uniform (I : Type*) [Fintype I] [Nonempty I] : RationalLaw I where |
| weight _ := 1 / Fintype.card I |
| nonneg _ := by positivity |
| total := by |
| simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul] |
| have h : (Fintype.card I : ℚ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero |
| field_simp |
|
|
| noncomputable def bind (p : RationalLaw I) {J : I → Type*} [∀ i, Fintype (J i)] |
| (q : ∀ i, RationalLaw (J i)) : RationalLaw (Sigma J) where |
| weight o := p.weight o.1 * (q o.1).weight o.2 |
| nonneg o := mul_nonneg (p.nonneg _) ((q _).nonneg _) |
| total := by |
| simp only [Fintype.sum_sigma, ← Finset.mul_sum, fun i => (q i).total, mul_one] |
| exact p.total |
|
|
| noncomputable def pi [DecidableEq I] {J : I → Type*} [∀ i, Fintype (J i)] |
| (q : ∀ i, RationalLaw (J i)) : RationalLaw (∀ i, J i) where |
| weight x := ∏ i, (q i).weight (x i) |
| nonneg x := Finset.prod_nonneg (fun i _ => (q i).nonneg _) |
| total := by |
| rw [← Fintype.prod_sum] |
| simp only [fun i => (q i).total, Finset.prod_const_one] |
|
|
| lemma expect_uniform (f : I → ℝ) [Nonempty I] : |
| (∑ i, ((uniform I).weight i : ℝ) * f i) = 𝔼 i, f i := by |
| simp only [uniform, Rat.cast_div, Rat.cast_one, Rat.cast_natCast, |
| Finset.expect_eq_sum_div_card, Finset.card_univ, ← Finset.mul_sum] |
| ring |
|
|
| lemma expect_bind (p : RationalLaw I) {J : I → Type*} [∀ i, Fintype (J i)] |
| (q : ∀ i, RationalLaw (J i)) (f : Sigma J → ℝ) : |
| (∑ o, ((p.bind q).weight o : ℝ) * f o) = |
| ∑ i, (p.weight i : ℝ) * ∑ j, ((q i).weight j : ℝ) * f ⟨i,j⟩ := by |
| simp only [bind, Fintype.sum_sigma, Rat.cast_mul, mul_assoc, Finset.mul_sum] |
|
|
| end LargeIndependentSets.RationalLaw |
|
|
| end OAI |
|
|
