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9b6e014 | 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 68 69 70 71 72 73 | import OAI.Probability.ContinuumTransition.FiniteProduct
namespace OAI
universe u_1
section
open scoped BigOperators
open MeasureTheory Set
namespace ContinuumTemperature
open Classical
theorem familyTerm_gibbs_empty_ne_zero (U : Space → Space → ℝ) (β h : ℝ) {n : ℕ} (hn : n ≠ 0) :
familyTerm (gibbsFamily U β h) ∅ n = 0 := by
have hp : 0 < n := Nat.pos_of_ne_zero hn
have he : {x : Fin n → Space | ∀ i, x i ∈ (∅ : Set Space)} = ∅ := by
ext x
simp only [mem_ofPred_eq, mem_empty_iff_false, iff_false, not_forall]
exact ⟨⟨0, hp⟩, not_false⟩
simp only [familyTerm, familyIntegral, he, setIntegral_empty, zero_div]
theorem grandFinite_gibbs_blocks {κ : Type u_1} {U : Space → Space → ℝ}
(hU : ∀ x y, U x y = U y x) {β h : ℝ} (hβ : 0 ≤ β)
(A : κ → Set Space) (s : Finset κ)
(hA : ∀ i ∈ s, MeasurableSet (A i))
(hd : (s : Set κ).Pairwise (fun i j => Disjoint (A i) (A j)))
(hcross : ∀ i ∈ s, ∀ j ∈ s, i ≠ j → ∀ x ∈ A i, ∀ y ∈ A j, 0 ≤ U x y)
(hFi : ∀ (ι : Type) [Fintype ι], IntegrableOn (gibbsFamily U β h ι)
{x | ∀ i, x i ∈ ⋃ j ∈ s, A j})
(hFA : ∀ i ∈ s, GrandFinite (gibbsFamily U β h) (A i)) :
GrandFinite (gibbsFamily U β h) (⋃ i ∈ s, A i) := by
induction s using Finset.induction_on with
| empty =>
simp only [Finset.notMem_empty, iUnion_of_empty, iUnion_empty] at hFi ⊢
refine ⟨hFi, ?_⟩
apply summable_of_ne_finset_zero (s := {0})
intro n hn
exact familyTerm_gibbs_empty_ne_zero U β h (by simpa only [Finset.mem_singleton] using hn)
| @insert i s hi ih =>
let B := ⋃ j ∈ s, A j
have hB : MeasurableSet B := MeasurableSet.biUnion s.finite_toSet.countable
(fun j hj => hA j (Finset.mem_insert_of_mem hj))
have hAi := hA i (Finset.mem_insert_self _ _)
have hAB : Disjoint (A i) B := by
apply Set.disjoint_iUnion₂_right.mpr
intro j hj
exact hd (Finset.mem_insert_self _ _) (Finset.mem_insert_of_mem hj) (fun hh => hi (hh ▸ hj))
have hieq : (⋃ j ∈ insert i s, A j) = A i ∪ B := by simp only [Finset.mem_insert, iUnion_iUnion_eq_or_left]; rfl
have hFiB (ι : Type) [Fintype ι] : IntegrableOn (gibbsFamily U β h ι)
{x | ∀ a, x a ∈ B} := by
apply (hFi ι).mono_set
intro x hx a
rw [hieq]
exact Or.inr (hx a)
have hFB := ih (fun j hj => hA j (Finset.mem_insert_of_mem hj))
(hd.mono (Finset.subset_insert _ _))
(fun j hj k hk => hcross j (Finset.mem_insert_of_mem hj) k (Finset.mem_insert_of_mem hk))
hFiB (fun j hj => hFA j (Finset.mem_insert_of_mem hj))
rw [hieq] at hFi ⊢
apply GrandFinite.union_product (gibbsFamily_relabel U β h) (gibbsFamily_relabel U β h)
hAi hB hAB (hFA i (Finset.mem_insert_self _ _)) hFB hFi (gibbsFamily_nonneg U β h)
intro n r x hx
apply gibbsFamily_binary_le hU hβ ?_ r hx
intro a ha b hb
obtain ⟨j, hj, hjb⟩ := mem_iUnion₂.mp hb
exact hcross i (Finset.mem_insert_self _ _) j (Finset.mem_insert_of_mem hj)
(fun hh => hi (hh ▸ hj)) a ha b hjb
end ContinuumTemperature
end
end OAI
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