openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /ProductMeasure /ExpectMulReal.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.Distribution | |
| import OAI.Probability.CriticalPercolation.Modular.DensityModular | |
| namespace OAI | |
| section | |
| section | |
| open Set MeasureTheory ProbabilityTheory | |
| open Set | |
| open scoped BigOperators | |
| namespace CriticalPercolation.FiniteProduct | |
| open scoped Classical | |
| variable {E : Type*} [Fintype E] | |
| lemma expect_mul_ge_real (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1) | |
| (hpos : ∀ e b, 0 ≤ w e b) (f g : (E → Prop) → ℝ) | |
| (hf : Monotone f) (hg : Monotone g) : | |
| expect w f * expect w g ≤ expect w (fun ω => f ω * g ω) := by | |
| classical | |
| let l := Finset.univ.inf' Finset.univ_nonempty f | |
| let k := Finset.univ.inf' Finset.univ_nonempty g | |
| have hb := expect_mul_ge w hw hpos (fun ω => f ω-l) (fun ω => g ω-k) | |
| (fun ω => sub_nonneg.mpr (Finset.inf'_le _ (Finset.mem_univ ω))) | |
| (fun ω => sub_nonneg.mpr (Finset.inf'_le _ (Finset.mem_univ ω))) | |
| (fun _ _ h => sub_le_sub_right (hf h) l) (fun _ _ h => sub_le_sub_right (hg h) k) | |
| have he : (fun ω => (f ω-l)*(g ω-k)) = | |
| (fun ω => f ω*g ω - l*g ω - k*f ω + l*k) := by funext ω; ring | |
| rw [he, expect_sub, expect_sub, expect_const w hw l, expect_const w hw k, | |
| expect_add, expect_sub, expect_sub, expect_const_mul, expect_const_mul, | |
| expect_const w hw (l*k)] at hb | |
| nlinarith | |
| end CriticalPercolation.FiniteProduct | |
| end | |
| end | |
| end OAI | |