openAI-math-dataset / lean /OAI /Probability /SKBarriers /Interpolation /RestrictedConcentration.lean
| import Mathlib | |
| import OAI.Probability.SKBarriers.Interpolation.RestrictedReplicas | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open scoped BigOperators | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| namespace SK.Analytic | |
| theorem restrictedLogPartition_subGaussian {n d : ℕ} (hn : 0 < n) (hd : 0 < d) | |
| (β : ℝ) {S : Finset (ReplicaConfig n d)} (hS : S.Nonempty) : | |
| HasSubgaussianMGF (fun J : Disorder n => restrictedLogPartition β S J- | |
| ∫ K, restrictedLogPartition β S K ∂disorderLaw n) | |
| (logPartitionProxy (β*(d:ℝ)) n) (disorderLaw n) := by | |
| let inst : SmoothFamily.Model n := restrictedModel hd S hS | |
| have H := @SmoothFamily.logPartition_subGaussian n inst hn (β*(d:ℝ)) | |
| change HasSubgaussianMGF (fun J => restrictedLogPartition ((β*(d:ℝ))/(d:ℝ)) S J- | |
| ∫ K, restrictedLogPartition ((β*(d:ℝ))/(d:ℝ)) S K ∂disorderLaw n) | |
| (logPartitionProxy (β*(d:ℝ)) n) (disorderLaw n) at H | |
| have hd' : (d:ℝ) ≠ 0 := by exact_mod_cast hd.ne' | |
| simpa only [mul_div_cancel_right₀ _ hd'] using H | |
| theorem restrictedLogPartition_integrable {n d : ℕ} (hn : 0 < n) (hd : 0 < d) | |
| (β : ℝ) {S : Finset (ReplicaConfig n d)} (hS : S.Nonempty) : | |
| Integrable (restrictedLogPartition β S) (disorderLaw n) := by | |
| let inst : SmoothFamily.Model n := restrictedModel hd S hS | |
| have H := @SmoothFamily.logPartition_integrable n inst hn (β*(d:ℝ)) | |
| change Integrable (restrictedLogPartition ((β*(d:ℝ))/(d:ℝ)) S) (disorderLaw n) at H | |
| have hd' : (d:ℝ) ≠ 0 := by exact_mod_cast hd.ne' | |
| simpa only [mul_div_cancel_right₀ _ hd'] using H | |
| theorem restrictedLogPartition_upper_tail {n d : ℕ} (hn : 0 < n) (hd : 0 < d) | |
| (β u : ℝ) {S : Finset (ReplicaConfig n d)} (hS : S.Nonempty) (hu : 0 ≤ u) : | |
| (disorderLaw n).real {J | u ≤ restrictedLogPartition β S J- | |
| ∫ K, restrictedLogPartition β S K ∂disorderLaw n} ≤ | |
| Real.exp (-u^2/(2*(logPartitionProxy (β*(d:ℝ)) n:ℝ))) := | |
| (restrictedLogPartition_subGaussian hn hd β hS).measure_ge_le hu | |
| def replicaGibbsMass {n d : ℕ} (β : ℝ) (J : Disorder n) (S : Finset (ReplicaConfig n d)) : ℝ := | |
| ∑ s ∈ S, ∏ a, gibbs β J (s a) | |
| theorem replicaGibbsMass_nonneg {n d : ℕ} (β : ℝ) (J : Disorder n) (S : Finset (ReplicaConfig n d)) : | |
| 0 ≤ replicaGibbsMass β J S := | |
| Finset.sum_nonneg (fun s _ => Finset.prod_nonneg (fun a _ => (gibbs_pos β J (s a)).le)) | |
| theorem replicaGibbsMass_le_one {n d : ℕ} (β : ℝ) (J : Disorder n) (S : Finset (ReplicaConfig n d)) : | |
| replicaGibbsMass β J S ≤ 1 := by | |
| have H : replicaGibbsMass β J S ≤ ∑ s : ReplicaConfig n d, ∏ a, gibbs β J (s a) := | |
| Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ S) | |
| (fun s _ _ => Finset.prod_nonneg (fun a _ => (gibbs_pos β J (s a)).le)) | |
| rw [← Fintype.prod_sum] at H | |
| simpa only [gibbs_sum,Finset.prod_const_one] using H | |
| theorem replicaGibbsMass_eq_ratio {n d : ℕ} (β : ℝ) (J : Disorder n) (S : Finset (ReplicaConfig n d)) : | |
| replicaGibbsMass β J S = restrictedPartition β S J/(partition β J)^d := by | |
| unfold replicaGibbsMass restrictedPartition | |
| rw [Finset.sum_div] | |
| apply Finset.sum_congr rfl | |
| intro s _ | |
| simp only [gibbs,Finset.prod_div_distrib,Finset.prod_const,Finset.card_univ,Fintype.card_fin] | |
| congr 1 | |
| rw [replicaEnergy,Finset.mul_sum,Real.exp_sum] | |
| theorem replicaGibbsMass_eq_exp {n d : ℕ} (β : ℝ) (J : Disorder n) {S : Finset (ReplicaConfig n d)} | |
| (hS : S.Nonempty) : replicaGibbsMass β J S = | |
| Real.exp (restrictedLogPartition β S J-(d:ℝ)*logPartition β J) := by | |
| rw [replicaGibbsMass_eq_ratio,Real.exp_sub,Real.exp_nat_mul,restrictedLogPartition,logPartition, | |
| Real.exp_log (restrictedPartition_pos β hS J),Real.exp_log (partition_pos β J)] | |
| theorem continuous_replicaGibbsMass {n d : ℕ} (β : ℝ) (S : Finset (ReplicaConfig n d)) : | |
| Continuous (fun J : Disorder n => replicaGibbsMass β J S) := by | |
| apply continuous_finsetSum | |
| intro s _ | |
| exact continuous_finsetProd _ (fun a _ => continuous_gibbs β (s a)) | |
| theorem replicaGibbsMass_integrable {n d : ℕ} (β : ℝ) (S : Finset (ReplicaConfig n d)) : | |
| Integrable (fun J : Disorder n => replicaGibbsMass β J S) (disorderLaw n) := by | |
| apply (integrable_const (1:ℝ)).mono' (continuous_replicaGibbsMass β S).aestronglyMeasurable | |
| filter_upwards [] with J | |
| simpa only [Real.norm_eq_abs,abs_of_nonneg (replicaGibbsMass_nonneg β J S)] using | |
| replicaGibbsMass_le_one β J S | |
| end SK.Analytic | |
| end | |
| end | |
| end | |
| end OAI | |