| import Mathlib | |
| import OAI.Probability.SKBarriers.Interpolation.RestrictedConcentration | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open scoped BigOperators | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| namespace SK.Analytic | |
| theorem replicaGibbsMass_tail {n d : ℕ} (β : ℝ) (J : Disorder n) | |
| (S : Finset (ReplicaConfig n d)) : | |
| replicaGibbsMass β J (Finset.univ.filter (fun s : ReplicaConfig n (d+1) => Fin.tail s∈S))= | |
| replicaGibbsMass β J S := by | |
| classical | |
| unfold replicaGibbsMass | |
| rw [Finset.sum_filter] | |
| calc | |
| _ = ∑ p : Config n × ReplicaConfig n d,if p.2∈S then gibbs β J p.1*∏ a,gibbs β J (p.2 a) else 0 := by | |
| apply Fintype.sum_equiv (Fin.consEquiv (fun _ : Fin (d+1) => Config n)).symm | |
| intro s | |
| simp only [Fin.consEquiv_symm_apply,Fin.prod_univ_succ,Fin.tail] | |
| _ = _ := by | |
| rw [Fintype.sum_prod_type] | |
| simp_rw [show ∀ (x : Config n) (s : ReplicaConfig n d), | |
| (if s∈S then gibbs β J x*∏ a,gibbs β J (s a) else 0)= | |
| gibbs β J x*(if s∈S then ∏ a,gibbs β J (s a) else 0) by intros; split_ifs <;> simp] | |
| simp_rw [← Finset.mul_sum] | |
| rw [← Finset.sum_mul,gibbs_sum,one_mul,← Finset.sum_filter] | |
| simp | |
| end SK.Analytic | |
| end | |
| end | |
| end OAI | |