| import OAI.Probability.ContinuumTransition.KacEnergy | |
| namespace OAI | |
| section | |
| open scoped BigOperators | |
| open MeasureTheory Set | |
| namespace ContinuumTemperature | |
| theorem familyTerm_product_bound (F G H : ConfigFamily) | |
| (hG : RelabelInvariant G) (hH : RelabelInvariant H) {A B : Set Space} | |
| (hA : MeasurableSet A) (hB : MeasurableSet B) (hd : Disjoint A B) | |
| (hGF : GrandFinite G A) (hHF : GrandFinite H B) (n : ℕ) | |
| (hFi : IntegrableOn (F (Fin n)) {x | ∀ i, x i ∈ A ∪ B}) | |
| (hbound : ∀ (s : Finset (Fin n)) (x : Fin n → Space), | |
| x ∈ binaryRegion A B s → F (Fin n) x ≤ | |
| G {i : Fin n // i ∈ s} (fun i => x i)*H {i : Fin n // i ∉ s} (fun i => x i)) : | |
| familyTerm F (A ∪ B) n ≤ ∑ k ∈ Finset.range (n+1), familyTerm G A k*familyTerm H B (n-k) := by | |
| have hb := integral_binary_bound hA hB hd | |
| (fun k => familyIntegral G A (Fin k)) (fun k => familyIntegral H B (Fin k)) | |
| hFi ?_ | |
| swap | |
| · intro s | |
| have hprod := integrableOn_binary_product volume A B s | |
| (by simpa only [← volume_pi] using hGF.integrable {i : Fin n // i ∈ s}) | |
| (by simpa only [← volume_pi] using hHF.integrable {i : Fin n // i ∉ s}) | |
| have hi := setIntegral_mono_on | |
| (hFi.mono_set (binaryRegion_subset_union A B s)) | |
| (by simpa only [← volume_pi] using hprod) | |
| (measurableSet_binaryRegion hA hB s) (hbound s) | |
| rw [show volume = Measure.pi (fun _ : Fin n => (volume : Measure Space)) by exact volume_pi, | |
| setIntegral_binary_product volume A B s (G {i : Fin n // i ∈ s}) | |
| (H {i : Fin n // i ∉ s})] at hi | |
| simp only [← volume_pi] at hi | |
| have heg := familyIntegral_eq_Fin G hG A {i : Fin n // i ∈ s} | |
| have heh := familyIntegral_eq_Fin H hH B {i : Fin n // i ∉ s} | |
| have hc : Fintype.card {i : Fin n // i ∈ s} = s.card := Fintype.card_coe s | |
| rw [hc] at heg | |
| have hcc : Fintype.card {i : Fin n // i ∉ s} = n-s.card := by | |
| have hh := Fintype.card_subtype_compl (fun i : Fin n => i ∈ s) | |
| simpa only [Fintype.card_fin, hc] using hh | |
| rw [hcc] at heh | |
| change (∫ x in binaryRegion A B s, F (Fin n) x) ≤ _ | |
| calc | |
| _ ≤ familyIntegral G A {i : Fin n // i ∈ s}* | |
| familyIntegral H B {i : Fin n // i ∉ s} := hi | |
| _ = _ := by rw [heg, heh, Fintype.card_fin] | |
| have hh := div_le_div_of_nonneg_right hb (Nat.cast_nonneg (n.factorial) : (0 : ℝ) ≤ n.factorial) | |
| rw [Fintype.card_fin, factorial_binomial_sum | |
| (fun k => familyIntegral G A (Fin k)) (fun k => familyIntegral H B (Fin k)) n] at hh | |
| exact hh | |
| theorem GrandFinite.union_product {F G H : ConfigFamily} | |
| (hG : RelabelInvariant G) (hH : RelabelInvariant H) {A B : Set Space} | |
| (hA : MeasurableSet A) (hB : MeasurableSet B) (hd : Disjoint A B) | |
| (hGF : GrandFinite G A) (hHF : GrandFinite H B) | |
| (hFi : ∀ (ι : Type) [Fintype ι], IntegrableOn (F ι) {x | ∀ i, x i ∈ A ∪ B}) | |
| (h0 : ∀ (ι : Type) [Fintype ι] (x : ι → Space), 0 ≤ F ι x) | |
| (hbound : ∀ (n : ℕ) (s : Finset (Fin n)) (x : Fin n → Space), | |
| x ∈ binaryRegion A B s → F (Fin n) x ≤ | |
| G {i : Fin n // i ∈ s} (fun i => x i)*H {i : Fin n // i ∉ s} (fun i => x i)) : | |
| GrandFinite F (A ∪ B) := by | |
| refine ⟨hFi, ?_⟩ | |
| have hab : Summable (fun p : ℕ × ℕ => familyTerm G A p.1*familyTerm H B p.2) := | |
| summable_mul_of_summable_norm (summable_norm_iff.mpr hGF.summable) | |
| (summable_norm_iff.mpr hHF.summable) | |
| exact Summable.of_nonneg_of_le (familyTerm_nonneg F (A ∪ B) h0) | |
| (fun n => familyTerm_product_bound F G H hG hH hA hB hd hGF hHF n (hFi (Fin n)) (hbound n)) | |
| (summable_sum_mul_range_of_summable_mul hab) | |
| end ContinuumTemperature | |
| end | |
| end OAI | |