openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /ProductMeasure /ExpectClosedFlip.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.Delta | |
| namespace OAI | |
| section | |
| section | |
| open Set MeasureTheory ProbabilityTheory | |
| open Set | |
| open scoped BigOperators | |
| open scoped Classical | |
| namespace CriticalPercolation.FiniteProduct | |
| open scoped Classical | |
| variable {E : Type*} [Fintype E] | |
| lemma expect_closed_flip (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1) | |
| (e : E) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) | |
| (h0 : w e False = 1-p) (h1 : w e True = p) (f : Set E → ℝ) : | |
| expect w (fun ω => if ¬ω e then p*f (Function.update ω e True) else 0) = | |
| expect w (fun ω => if ω e then (1-p)*f ω else 0) := by | |
| have he0 : w e False ≠ 0 := by rw [h0]; linarith [hp.2] | |
| have he1 : w e True ≠ 0 := by rw [h1]; exact hp.1.ne' | |
| have he := expect_pin_transport w hw e False True he0 he1 f | |
| have hc : expect w (fun ω => if ¬ω e then p*f (Function.update ω e True) else 0) = | |
| w e False * expect (pin w e False) (fun ω => p*f (Function.update ω e True)) := by | |
| convert expect_prop_mask_pin w e False (fun ω => p*f (Function.update ω e True)) using 1 | |
| congr 1 | |
| funext ω | |
| by_cases hb : ω e <;> simp [hb] | |
| have ho : expect w (fun ω => if ω e then (1-p)*f ω else 0) = | |
| w e True * expect (pin w e True) (fun ω => (1-p)*f ω) := by | |
| convert expect_prop_mask_pin w e True (fun ω => (1-p)*f ω) using 1 | |
| congr 1 | |
| funext ω | |
| by_cases hb : ω e <;> simp [hb] | |
| erw [hc,ho,expect_const_mul,expect_const_mul,h0,h1,he] | |
| ring | |
| lemma expect_closed_flip_decidable [d : DecidableEq E] | |
| (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1) | |
| (e : E) {p : ℝ} (hp : p ∈ Set.Ioo 0 1) | |
| (h0 : w e False = 1-p) (h1 : w e True = p) (f : Set E → ℝ) : | |
| expect w (fun ω => if ¬ω e then p*f (Function.update ω e True) else 0) = | |
| expect w (fun ω => if ω e then (1-p)*f ω else 0) := by | |
| have hup (ω : Set E) : | |
| @Function.update E (fun _ => Prop) (fun a b => Classical.propDecidable (a=b)) ω e True = | |
| Function.update ω e True := by | |
| have hd : (fun a b => Classical.propDecidable (a=b)) = d := Subsingleton.elim _ _ | |
| exact congrArg (fun d' : DecidableEq E => | |
| @Function.update E (fun _ => Prop) d' ω e True) hd | |
| have h := expect_closed_flip w hw e hp h0 h1 f | |
| convert h using 1 | |
| congr 1 | |
| funext ω | |
| by_cases hc : ω e | |
| · simp only [hc,not_true_eq_false,ite_false] | |
| · simp only [hc,not_false_eq_true,ite_true] | |
| exact congrArg (fun σ : Set E => p*f σ) (hup ω).symm | |
| end CriticalPercolation.FiniteProduct | |
| end | |
| end | |
| end OAI | |