| import OAI.Probability.ContinuumTransition.ExactAssignments |
|
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| namespace OAI |
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| section |
| open scoped BigOperators |
| open MeasureTheory Set |
| namespace ContinuumTemperature |
|
|
| abbrev ConfigFamily := (ι : Type) → [Fintype ι] → (ι → Space) → ℝ |
|
|
| def RelabelInvariant (F : ConfigFamily) : Prop := |
| ∀ (ι κ : Type) [Fintype ι] [Fintype κ] (e : ι ≃ κ) (x : κ → Space), |
| F ι (x ∘ e) = F κ x |
|
|
| noncomputable def familyIntegral (F : ConfigFamily) (A : Set Space) |
| (ι : Type) [Fintype ι] : ℝ := |
| ∫ x in {x : ι → Space | ∀ i, x i ∈ A}, F ι x |
|
|
| noncomputable def familyTerm (F : ConfigFamily) (A : Set Space) (n : ℕ) : ℝ := |
| familyIntegral F A (Fin n)/(n.factorial : ℝ) |
|
|
| noncomputable def grandIntegral (F : ConfigFamily) (A : Set Space) : ℝ := |
| ∑' n : ℕ, familyTerm F A n |
|
|
| theorem familyIntegral_relabel (F : ConfigFamily) (hF : RelabelInvariant F) |
| (A : Set Space) {ι κ : Type} [Fintype ι] [Fintype κ] (e : ι ≃ κ) : |
| familyIntegral F A ι = familyIntegral F A κ := by |
| let p : (ι → Space) ≃ᵐ (κ → Space) := MeasurableEquiv.piCongrLeft (fun _ : κ => Space) e |
| have hp : MeasurePreserving p volume volume := by |
| simpa only [volume_pi] using measurePreserving_piCongrLeft (fun _ : κ => volume) e |
| have hpx (x : ι → Space) : p x = x ∘ e.symm := by |
| funext i |
| simpa only [p, Equiv.apply_symm_apply, Function.comp_apply] using |
| (MeasurableEquiv.piCongrLeft_apply_apply e (β := fun _ : κ => Space) x (e.symm i)) |
| have heq := hp.setIntegral_preimage_emb p.measurableEmbedding (F κ) |
| {x : κ → Space | ∀ i, x i ∈ A} |
| have hpre : p ⁻¹' {x : κ → Space | ∀ i, x i ∈ A} = |
| {x : ι → Space | ∀ i, x i ∈ A} := by |
| ext x |
| simp only [mem_preimage, mem_ofPred_eq, hpx, Function.comp_apply] |
| constructor |
| · intro hh i |
| simpa only [Equiv.symm_apply_apply] using hh (e i) |
| · intro hh i |
| exact hh (e.symm i) |
| rw [hpre] at heq |
| simpa only [familyIntegral, hpx, hF κ ι e.symm] using heq |
|
|
| theorem familyIntegral_eq_Fin (F : ConfigFamily) (hF : RelabelInvariant F) |
| (A : Set Space) (ι : Type) [Fintype ι] : |
| familyIntegral F A ι = familyIntegral F A (Fin (Fintype.card ι)) := |
| familyIntegral_relabel F hF A (Fintype.equivFin ι) |
|
|
| theorem integral_binary_bound {ι : Type} [Fintype ι] [DecidableEq ι] |
| {A B : Set Space} (hA : MeasurableSet A) (hB : MeasurableSet B) (hd : Disjoint A B) |
| {F : (ι → Space) → ℝ} (a b : ℕ → ℝ) |
| (hF : IntegrableOn F {x : ι → Space | ∀ i, x i ∈ A ∪ B}) |
| (hbound : ∀ s : Finset ι, (∫ x in binaryRegion A B s, F x) ≤ |
| a s.card*b (Fintype.card ι-s.card)) : |
| (∫ x in {x : ι → Space | ∀ i, x i ∈ A ∪ B}, F x) ≤ |
| ∑ k ∈ Finset.range (Fintype.card ι+1), |
| ((Fintype.card ι).choose k : ℝ)*a k*b (Fintype.card ι-k) := by |
| rw [show volume = Measure.pi (fun _ : ι => (volume : Measure Space)) by exact volume_pi] |
| rw [integral_binary_assignments_eq volume hA hB hd (by simpa only [← volume_pi] using hF)] |
| calc |
| _ ≤ ∑ s : Finset ι, a s.card*b (Fintype.card ι-s.card) := by |
| apply Finset.sum_le_sum |
| intro s _ |
| simpa only [← volume_pi] using hbound s |
| _ = _ := by |
| have hh := Finset.sum_powerset_apply_card (fun k => a k*b (Fintype.card ι-k)) |
| (x := (Finset.univ : Finset ι)) |
| simpa only [Finset.powerset_univ, Finset.card_univ, nsmul_eq_mul, mul_assoc] using hh |
|
|
| theorem factorial_binomial_sum (a b : ℕ → ℝ) (N : ℕ) : |
| (∑ k ∈ Finset.range (N+1), (N.choose k : ℝ)*a k*b (N-k))/(N.factorial : ℝ) = |
| ∑ k ∈ Finset.range (N+1), (a k/(k.factorial : ℝ))*(b (N-k)/((N-k).factorial : ℝ)) := by |
| rw [Finset.sum_div] |
| apply Finset.sum_congr rfl |
| intro k hk |
| rw [Nat.cast_choose ℝ (by have := Finset.mem_range.mp hk; omega)] |
| have hN : (N.factorial : ℝ) ≠ 0 := by exact_mod_cast Nat.factorial_ne_zero N |
| field_simp |
|
|
| theorem familyTerm_nonneg (F : ConfigFamily) (A : Set Space) |
| (hF : ∀ (ι : Type) [Fintype ι] (x : ι → Space), 0 ≤ F ι x) (n : ℕ) : |
| 0 ≤ familyTerm F A n := by |
| apply div_nonneg (integral_nonneg (hF (Fin n))) |
| positivity |
|
|
| theorem integrableOn_family_of_bound (F : ConfigFamily) {A : Set Space} |
| (hA : volume A ≠ ⊤) {ι : Type} [Fintype ι] |
| (hFm : Measurable (F ι)) (hF0 : ∀ x, 0 ≤ F ι x) |
| {C : ℝ} (hbound : ∀ x, F ι x ≤ C) : |
| IntegrableOn (F ι) {x : ι → Space | ∀ i, x i ∈ A} := by |
| have hvol : volume {x : ι → Space | ∀ i, x i ∈ A} ≠ ⊤ := by |
| rw [volume_generalConfigurations] |
| exact ENNReal.pow_ne_top hA |
| let : IsFiniteMeasure (volume.restrict {x : ι → Space | ∀ i, x i ∈ A}) := ⟨by |
| simpa only [Measure.restrict_apply_univ] using lt_top_iff_ne_top.mpr hvol⟩ |
| apply (integrable_const C).mono' hFm.aestronglyMeasurable |
| filter_upwards [] with x |
| rw [Real.norm_eq_abs, abs_of_nonneg (hF0 x)] |
| exact hbound x |
|
|
| theorem familyTerm_le_exponential (F : ConfigFamily) {A : Set Space} |
| (hA : volume A ≠ ⊤) {d : ℝ} |
| (hFm : ∀ (ι : Type) [Fintype ι], Measurable (F ι)) |
| (hF0 : ∀ (ι : Type) [Fintype ι] (x : ι → Space), 0 ≤ F ι x) |
| (hbound : ∀ (ι : Type) [Fintype ι] (x : ι → Space), |
| F ι x ≤ Real.exp (d*Fintype.card ι)) (n : ℕ) : |
| familyTerm F A n ≤ (volume.real A*Real.exp d)^n/(n.factorial : ℝ) := by |
| have hvol : volume {x : Fin n → Space | ∀ i, x i ∈ A} ≠ ⊤ := by |
| rw [volume_generalConfigurations] |
| exact ENNReal.pow_ne_top hA |
| let : IsFiniteMeasure (volume.restrict {x : Fin n → Space | ∀ i, x i ∈ A}) := ⟨by |
| simpa only [Measure.restrict_apply_univ] using lt_top_iff_ne_top.mpr hvol⟩ |
| have hh := integral_mono (integrableOn_family_of_bound F hA (hFm (Fin n)) |
| (hF0 (Fin n)) (hbound (Fin n))) (integrable_const _) (hbound (Fin n)) |
| have hn : (0 : ℝ) ≤ n.factorial := Nat.cast_nonneg _ |
| apply (div_le_div_of_nonneg_right hh hn).trans_eq |
| rw [setIntegral_const, Measure.real, volume_generalConfigurations, ENNReal.toReal_pow, |
| Fintype.card_fin, smul_eq_mul, mul_comm d (n : ℝ), Real.exp_nat_mul, mul_pow] |
| rfl |
|
|
| theorem summable_familyTerm_of_bound (F : ConfigFamily) {A : Set Space} |
| (hA : volume A ≠ ⊤) {d : ℝ} |
| (hFm : ∀ (ι : Type) [Fintype ι], Measurable (F ι)) |
| (hF0 : ∀ (ι : Type) [Fintype ι] (x : ι → Space), 0 ≤ F ι x) |
| (hbound : ∀ (ι : Type) [Fintype ι] (x : ι → Space), |
| F ι x ≤ Real.exp (d*Fintype.card ι)) : Summable (familyTerm F A) := |
| Summable.of_nonneg_of_le (familyTerm_nonneg F A hF0) |
| (familyTerm_le_exponential F hA hFm hF0 hbound) |
| (Real.summable_pow_div_factorial _) |
|
|
| theorem tsum_le_convolution {f a b : ℕ → ℝ} (hf : Summable f) |
| (ha : Summable a) (hb : Summable b) |
| (hle : ∀ n, f n ≤ ∑ k ∈ Finset.range (n+1), a k*b (n-k)) : |
| (∑' n, f n) ≤ (∑' n, a n)*(∑' n, b n) := by |
| have hab : Summable (fun p : ℕ × ℕ => a p.1*b p.2) := |
| summable_mul_of_summable_norm (summable_norm_iff.mpr ha) (summable_norm_iff.mpr hb) |
| rw [ha.tsum_mul_tsum_eq_tsum_sum_range hb hab] |
| exact hf.tsum_le_tsum hle (summable_sum_mul_range_of_summable_mul hab) |
|
|
| theorem grandIntegral_insertion_bound (F G : ConfigFamily) |
| (hG : RelabelInvariant G) {A B : Set Space} |
| (hA : MeasurableSet A) (hB : MeasurableSet B) (hd : Disjoint A B) |
| (hAfin : volume A ≠ ⊤) |
| (hFi : ∀ n, IntegrableOn (F (Fin n)) {x | ∀ i, x i ∈ A ∪ B}) |
| (hGi : ∀ (ι : Type) [Fintype ι], IntegrableOn (G ι) {x | ∀ i, x i ∈ B}) |
| (hFs : Summable (familyTerm F (A ∪ B))) (hGs : Summable (familyTerm G B)) |
| {d : ℝ} |
| (hbound : ∀ (n : ℕ) (s : Finset (Fin n)) (x : Fin n → Space), |
| x ∈ binaryRegion A B s → F (Fin n) x ≤ |
| Real.exp (d*s.card) * G {i : Fin n // i ∉ s} (fun i => x i)) : |
| grandIntegral F (A ∪ B) ≤ |
| Real.exp (volume.real A*Real.exp d)*grandIntegral G B := by |
| let a (n : ℕ) := (volume.real A*Real.exp d)^n/(n.factorial : ℝ) |
| have ha : HasSum a (Real.exp (volume.real A*Real.exp d)) := |
| by |
| simpa only [a, Real.exp_eq_exp_ℝ] using |
| NormedSpace.expSeries_div_hasSum_exp (volume.real A*Real.exp d) |
| change (∑' n, familyTerm F (A ∪ B) n) ≤ |
| Real.exp (volume.real A*Real.exp d)*(∑' n, familyTerm G B n) |
| rw [← ha.tsum_eq] |
| apply tsum_le_convolution hFs ha.summable hGs |
| intro n |
| have hb := integral_binary_bound hA hB hd |
| (fun k => (volume.real A)^k*Real.exp (d*k)) |
| (fun k => familyIntegral G B (Fin k)) (hFi n) ?_ |
| swap |
| · intro s |
| have hv : volume {x : {i : Fin n // i ∈ s} → Space | ∀ i, x i ∈ A} ≠ ⊤ := by |
| rw [volume_generalConfigurations] |
| exact ENNReal.pow_ne_top hAfin |
| let : IsFiniteMeasure (volume.restrict |
| {x : {i : Fin n // i ∈ s} → Space | ∀ i, x i ∈ A}) := ⟨by |
| simpa only [Measure.restrict_apply_univ] using lt_top_iff_ne_top.mpr hv⟩ |
| have hprod := integrableOn_binary_product volume A B s |
| (integrable_const (Real.exp (d*s.card))) (by simpa only [← volume_pi] using hGi {i : Fin n // i ∉ s}) |
| have hi := setIntegral_mono_on |
| ((hFi n).mono_set (binaryRegion_subset_union A B s)) |
| (by simpa only [← volume_pi] using hprod) |
| (measurableSet_binaryRegion hA hB s) (hbound n s) |
| rw [show volume = Measure.pi (fun _ : Fin n => (volume : Measure Space)) by exact volume_pi, |
| setIntegral_binary_product volume A B s (fun _ => Real.exp (d*s.card)) |
| (G {i : Fin n // i ∉ s})] at hi |
| simp only [← volume_pi] at hi |
| have heg := familyIntegral_eq_Fin G hG B {i : Fin n // i ∉ s} |
| have hgc : familyIntegral G B {i : Fin n // i ∉ s} = |
| familyIntegral G B (Fin (n-s.card)) := by |
| have hc : Fintype.card {i : Fin n // i ∉ s} = n-s.card := by |
| rw [Fintype.card_subtype_compl (fun i : Fin n => i ∈ s), |
| Fintype.card_fin, Fintype.card_coe] |
| rw [hc] at heg |
| exact heg |
| rw [setIntegral_const, Measure.real, volume_generalConfigurations, ENNReal.toReal_pow, |
| Fintype.card_coe, smul_eq_mul] at hi |
| exact hi.trans_eq (by |
| change volume.real A ^ s.card * Real.exp (d*s.card) * |
| familyIntegral G B {i : Fin n // i ∉ s} = _ |
| rw [hgc, Fintype.card_fin]) |
| have hf : (0 : ℝ) ≤ n.factorial := Nat.cast_nonneg _ |
| have hh := div_le_div_of_nonneg_right hb hf |
| rw [Fintype.card_fin, factorial_binomial_sum |
| (fun k => (volume.real A)^k*Real.exp (d*k)) |
| (fun k => familyIntegral G B (Fin k)) n] at hh |
| apply hh.trans_eq |
| apply Finset.sum_congr rfl |
| intro k _ |
| dsimp only [a, familyTerm] |
| rw [mul_comm d (k : ℝ), Real.exp_nat_mul, mul_pow] |
|
|
| end ContinuumTemperature |
| end |
|
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| end OAI |
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|
