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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 | 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
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