| import OAI.Probability.ClassicalON.ProductFKG |
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| namespace OAI |
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| universe uE uV uX |
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| noncomputable section |
| open MeasureTheory Set |
| open scoped BigOperators |
| namespace ClassicalON |
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| section |
| variable {X : Type uX} {V : Type uV} {E : Type uE} [TopologicalSpace X] [CompactSpace X] [MeasurableSpace X] |
| [BorelSpace X] [Fintype E] (μ : Measure X) [IsProbabilityMeasure μ] |
| (T : E → X → ℝ) (left right : E → V) (b : E → ℝ) |
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| def amplitudeCoupling (a : V → ℝ) (e : E) : ℝ := b e*a (left e)*a (right e) |
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| def amplitudePartition (a : V → ℝ) : ℝ := |
| ∫ x,Real.exp (edgeHamiltonian T (amplitudeCoupling left right b a) x) ∂μ |
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| def amplitudeSlope (a d : V → ℝ) (t : ℝ) (e : E) : ℝ := |
| b e*(d (left e)*(a (right e)+t*d (right e))+(a (left e)+t*d (left e))*d (right e)) |
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| omit [CompactSpace X] [MeasurableSpace X] [BorelSpace X] in |
| theorem continuous_amplitudeHamiltonian (hT : ∀ e,Continuous (T e)) : |
| Continuous (fun p : (V → ℝ)×X => edgeHamiltonian T |
| (amplitudeCoupling left right b p.1) p.2) := by |
| unfold edgeHamiltonian amplitudeCoupling |
| fun_prop |
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| theorem amplitudePartition_pos (hT : ∀ e,Continuous (T e)) (a : V → ℝ) : |
| 0<amplitudePartition μ T left right b a := by |
| exact compact_integral_pos (continuous_edgeHamiltonian T hT _).rexp (fun x => Real.exp_pos _) |
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| theorem continuous_amplitudePartition [Fintype V] (hT : ∀ e,Continuous (T e)) : |
| Continuous (amplitudePartition μ T left right b) := by |
| exact compact_parametric_integral _ (continuous_amplitudeHamiltonian T left right b hT).rexp |
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| omit [TopologicalSpace X] [CompactSpace X] [MeasurableSpace X] [BorelSpace X] in |
| theorem amplitudeHamiltonian_direction_derivative (a d : V → ℝ) (t : ℝ) (x : X) : |
| HasDerivAt (fun u => edgeHamiltonian T |
| (amplitudeCoupling left right b (fun v => a v+u*d v)) x) |
| (edgeHamiltonian T (amplitudeSlope left right b a d t) x) t := by |
| unfold edgeHamiltonian |
| apply HasDerivAt.fun_sum |
| intro e _ |
| have hl := ((hasDerivAt_id t).mul_const (d (left e))).const_add (a (left e)) |
| have hr := ((hasDerivAt_id t).mul_const (d (right e))).const_add (a (right e)) |
| exact (((hl.const_mul (b e)).mul hr).mul_const (T e x)).congr_deriv (by |
| simp only [id_eq,amplitudeSlope]; ring) |
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| theorem log_amplitudePartition_direction_derivative (hT : ∀ e,Continuous (T e)) |
| (a d : V → ℝ) (t : ℝ) : |
| HasDerivAt (fun u => Real.log (amplitudePartition μ T left right b (fun v => a v+u*d v))) |
| (∑ e,amplitudeSlope left right b a d t e* |
| edgeMean μ T (amplitudeCoupling left right b (fun v => a v+t*d v)) (T e)) t := by |
| let H := fun u x => edgeHamiltonian T |
| (amplitudeCoupling left right b (fun v => a v+u*d v)) x |
| let A := fun u x => edgeHamiltonian T (amplitudeSlope left right b a d u) x |
| have hc : Continuous (fun p : ℝ×X => H p.1 p.2) := by |
| unfold H edgeHamiltonian amplitudeCoupling; fun_prop |
| have ha : Continuous (fun p : ℝ×X => A p.1 p.2) := by |
| unfold A edgeHamiltonian amplitudeSlope; fun_prop |
| have hI := hasDerivAt_integral_compact μ (fun u x => Real.exp (H u x)) |
| (fun u x => Real.exp (H u x)*A u x) hc.rexp (hc.rexp.mul ha) (fun u x => |
| (amplitudeHamiltonian_direction_derivative T left right b a d u x).exp) t |
| have hz := amplitudePartition_pos μ T left right b hT (fun v => a v+t*d v) |
| apply (hI.log hz.ne').congr_deriv |
| change exponentialMean μ (H t) (A t)=_ |
| unfold A edgeHamiltonian |
| rw [exponentialMean_sum μ (continuous_edgeHamiltonian T hT _) _ (fun e => by fun_prop)] |
| simp only [exponentialMean_const_mul,edgeMean] |
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| omit [TopologicalSpace X] [CompactSpace X] [MeasurableSpace X] [BorelSpace X] [Fintype E] in |
| theorem amplitudeCoupling_mono (hb : ∀ e,0≤b e) {a c : V → ℝ} |
| (ha : ∀ v,0≤a v) (hac : ∀ v,a v≤c v) (e : E) : |
| amplitudeCoupling left right b a e≤amplitudeCoupling left right b c e := by |
| unfold amplitudeCoupling |
| exact mul_le_mul (mul_le_mul_of_nonneg_left (hac _) (hb e)) (hac _) |
| (ha _) (mul_nonneg (hb e) ((ha _).trans (hac _))) |
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| omit [TopologicalSpace X] [CompactSpace X] [MeasurableSpace X] [BorelSpace X] [Fintype E] in |
| theorem amplitudeSlope_nonneg (hb : ∀ e,0≤b e) {a d : V → ℝ} |
| (ha : ∀ v,0≤a v) (hd : ∀ v,0≤d v) {t : ℝ} (ht : 0≤t) (e : E) : |
| 0≤amplitudeSlope left right b a d t e := by |
| unfold amplitudeSlope |
| exact mul_nonneg (hb e) (add_nonneg |
| (mul_nonneg (hd _) (add_nonneg (ha _) (mul_nonneg ht (hd _)))) |
| (mul_nonneg (add_nonneg (ha _) (mul_nonneg ht (hd _))) (hd _))) |
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| omit [TopologicalSpace X] [CompactSpace X] [MeasurableSpace X] [BorelSpace X] [Fintype E] in |
| theorem amplitudeSlope_mono (hb : ∀ e,0≤b e) {a c d : V → ℝ} |
| (hd : ∀ v,0≤d v) (hac : ∀ v,a v≤c v) (t : ℝ) (e : E) : |
| amplitudeSlope left right b a d t e≤amplitudeSlope left right b c d t e := by |
| unfold amplitudeSlope |
| apply mul_le_mul_of_nonneg_left _ (hb e) |
| exact add_le_add |
| (mul_le_mul_of_nonneg_left (add_le_add (hac _) le_rfl) (hd _)) |
| (mul_le_mul_of_nonneg_right (add_le_add (hac _) le_rfl) (hd _)) |
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| end |
| end ClassicalON |
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| end |
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| end OAI |
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