| import Mathlib |
| import OAI.Probability.SKBarriers.Replicas.ReplicaFlip |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem replicaGibbsMass_le_or_or_flip {n d : ℕ} (a : Fin d) (β : ℝ) (J : Disorder n) |
| (S T P : Finset (ReplicaConfig n d)) |
| (h : ∀ s∈S,s∈T ∨ replicaFlip a s∈T ∨ s∈P) : |
| replicaGibbsMass β J S≤2*replicaGibbsMass β J T+replicaGibbsMass β J P := by |
| classical |
| have hsub : S⊆(T∪T.image (replicaFlip a))∪P := by |
| intro s hs |
| rcases h s hs with h|h|h |
| · exact Finset.mem_union_left _ (Finset.mem_union_left _ h) |
| · exact Finset.mem_union_left _ (Finset.mem_union_right _ (Finset.mem_image.mpr |
| ⟨replicaFlip a s,h,replicaFlip_involutive a s⟩)) |
| · exact Finset.mem_union_right _ h |
| have H := (replicaGibbsMass_mono β J hsub).trans |
| (replicaGibbsMass_union_le β J (T∪T.image (replicaFlip a)) P) |
| have H' := replicaGibbsMass_union_le β J T (T.image (replicaFlip a)) |
| rw [replicaGibbsMass_flip] at H' |
| linarith |
|
|
| theorem replicaGibbsMass_integral_or_or_flip {n d : ℕ} (a : Fin d) (β : ℝ) |
| (S T P : Finset (ReplicaConfig n d)) |
| (h : ∀ s∈S,s∈T ∨ replicaFlip a s∈T ∨ s∈P) : |
| (∫ J,replicaGibbsMass β J S ∂disorderLaw n)≤ |
| 2*(∫ J,replicaGibbsMass β J T ∂disorderLaw n)+(∫ J,replicaGibbsMass β J P ∂disorderLaw n) := by |
| have H := integral_mono (replicaGibbsMass_integrable β S) |
| (((replicaGibbsMass_integrable β T).const_mul 2).add (replicaGibbsMass_integrable β P)) |
| (fun J => replicaGibbsMass_le_or_or_flip a β J S T P h) |
| simpa only [Pi.add_apply,integral_add ((replicaGibbsMass_integrable β T).const_mul 2) (replicaGibbsMass_integrable β P), |
| integral_const_mul] using H |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
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| end OAI |
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|
