File size: 2,190 Bytes
f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 | import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.Density
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open Set
open scoped BigOperators
namespace CriticalPercolation.FiniteProduct
variable {E A : Type*} [Fintype E] [Fintype A]
lemma expect_zero (w : E → A → ℝ) : expect w (fun _ => 0) = 0 := by
simp [expect]
lemma expect_sum {J : Type*} [Fintype J] (w : E → A → ℝ)
(f : J → (E → A) → ℝ) :
expect w (fun ω => ∑ j, f j ω) = ∑ j, expect w (f j) := by
classical
simp only [expect, Finset.mul_sum]
exact Finset.sum_comm
lemma expect_nonneg {w : E → A → ℝ} (hw : ∀ e a, 0 ≤ w e a)
{f : (E → A) → ℝ} (hf : ∀ ω, 0 ≤ f ω) : 0 ≤ expect w f := by
rw [← expect_zero w]
exact expect_mono hw hf
noncomputable def distribution {B : Type*} [DecidableEq B]
(w : E → A → ℝ) (φ : (E → A) → B) (b : B) : ℝ :=
expect w (fun ω => if φ ω = b then 1 else 0)
lemma expect_comp {B : Type*} [Fintype B] [DecidableEq B]
(w : E → A → ℝ) (φ : (E → A) → B) (f : B → ℝ) :
expect w (fun ω => f (φ ω)) = ∑ b, distribution w φ b * f b := by
classical
simp only [distribution, ← expect_mul_const, ← expect_sum]
congr 1
funext ω
simp
lemma distribution_nonneg {B : Type*} [DecidableEq B]
{w : E → A → ℝ} (hw : ∀ e a, 0 ≤ w e a)
(φ : (E → A) → B) (b : B) : 0 ≤ distribution w φ b :=
expect_nonneg hw fun _ => by split_ifs <;> norm_num
lemma distribution_mass {B : Type*} [Fintype B] [DecidableEq B]
(w : E → A → ℝ) (hw : ∀ e, ∑ a, w e a = 1)
(φ : (E → A) → B) : ∑ b, distribution w φ b = 1 := by
simpa only [mul_one] using (expect_comp w φ (fun _ => 1)).symm.trans (expect_const w hw 1)
end CriticalPercolation.FiniteProduct
namespace CriticalPercolation.FiniteProduct
variable {E A : Type*} [Fintype E] [Fintype A]
lemma expect_const_mul (w : E → A → ℝ) (c : ℝ) (f : (E → A) → ℝ) :
expect w (fun ω => c * f ω) = c * expect w f := by
simp only [mul_comm c]
exact expect_mul_const w f c
end CriticalPercolation.FiniteProduct
end
end
end OAI
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