| import OAI.Probability.DilutedSpin.VertexContinuity |
|
|
| namespace OAI |
|
|
| section |
| section |
| namespace DilutedSpinGlass.ReducedTopology |
| open scoped BigOperators |
| noncomputable local instance admissibleIntervalPropDecidable (proposition : Prop) : |
| Decidable proposition := Classical.propDecidable proposition |
|
|
| / |
| coordinate form an interval. No spacing or regularity is needed here. -/ |
| lemma admissible_update_between (S : ReducedTopology) (q : S.Vertex → ℕ) (v : S.Vertex) |
| {a b t lo hi : ℕ} (hat : a≤t) (htb : t≤b) |
| (ha : Admissible S (updateCoordinate q v a) lo hi) |
| (hb : Admissible S (updateCoordinate q v b) lo hi) : |
| Admissible S (updateCoordinate q v t) lo hi := by |
| induction S generalizing lo hi with |
| | leaf => exact v.elim |
| | node k hk C ih => |
| cases v with |
| | none => |
| simp only [update_root (hk := hk),Admissible,rootSchedule] at ha hb ⊢ |
| exact ⟨ha.1.trans hat,htb.trans_lt hb.2.1, |
| fun i => admissible_mono_lower (C i) _ (Nat.succ_le_succ htb) (hb.2.2 i)⟩ |
| | some a => |
| obtain ⟨i,v⟩ := a |
| constructor |
| · simpa only [update_desc_root (hk := hk)] using ha.1 |
| constructor |
| · simpa only [update_desc_root (hk := hk)] using ha.2.1 |
| intro j |
| by_cases hj : j=i |
| · subst j |
| have haa := ha.2.2 i |
| have hbb := hb.2.2 i |
| simp only [update_desc_root (hk := hk),update_desc_same (hk := hk)] at haa hbb ⊢ |
| exact ih i _ v haa hbb |
| · have haa := ha.2.2 j |
| simpa only [update_desc_root (hk := hk),update_desc_other q i v _ j hj] using haa |
|
|
| / |
| theorem scheduled_vertex_Ico_energy_le {Ω : Type} [Fintype Ω] |
| (H d : ℕ) (S : ReducedTopology) (q : S.Vertex → ℕ) (v : S.Vertex) |
| (a b : ℕ) (hab : a≤b) |
| (ha : Admissible S (updateCoordinate q v a) d (d+H)) |
| (hb : Admissible S (updateCoordinate q v b) d (d+H)) |
| (T : KernelTower Ω H) (f : FinitePath Ω H → ℝ) (hf : ∀ x, |f x| ≤ 1) : |
| (∑ t ∈ Finset.Ico a b, |
| (PrescribedTree.treeMean (realize H d S (updateCoordinate q v t)) T f- |
| PrescribedTree.treeMean (realize H d S (updateCoordinate q v (t+1))) T f)^2) ≤ |
| (vertexArity S v:ℝ)^2 := by |
| have he := scheduled_vertex_energy_le H d S q v a (b-a) (by |
| intro t |
| apply admissible_update_between S q v (Nat.le_add_right _ _) ?_ ha hb |
| have ht := t.isLt |
| omega) T f hf |
| let E : ℕ → ℝ := fun t => |
| (PrescribedTree.treeMean (realize H d S (updateCoordinate q v t)) T f- |
| PrescribedTree.treeMean (realize H d S (updateCoordinate q v (t+1))) T f)^2 |
| have he' : (∑ t : Fin (b-a), E (a+t.val)) ≤ (vertexArity S v:ℝ)^2 := by |
| simpa only [E,Fin.val_castSucc,Fin.val_succ,Nat.add_assoc] using he |
| rw [Fin.sum_univ_eq_sum_range (fun t => E (a+t)) (b-a)] at he' |
| change (∑ t ∈ Finset.Ico a b, E t) ≤ _ |
| rwa [Finset.sum_Ico_eq_sum_range] |
|
|
|
|
| / |
| single-coordinate energy. The proof uses the actual admissible hull. -/ |
| theorem scheduled_vertex_subset_energy_le {Ω : Type} [Fintype Ω] |
| (H d : ℕ) (S : ReducedTopology) (q : S.Vertex → ℕ) (v : S.Vertex) |
| (I : Finset ℕ) |
| (ha : ∀ t ∈ I, Admissible S (updateCoordinate q v t) d (d+H) ∧ |
| Admissible S (updateCoordinate q v (t+1)) d (d+H)) |
| (T : KernelTower Ω H) (f : FinitePath Ω H → ℝ) (hf : ∀ x, |f x| ≤ 1) : |
| (∑ t ∈ I, |
| (PrescribedTree.treeMean (realize H d S (updateCoordinate q v t)) T f- |
| PrescribedTree.treeMean (realize H d S (updateCoordinate q v (t+1))) T f)^2) ≤ |
| (vertexArity S v:ℝ)^2 := by |
| classical |
| by_cases hI : I.Nonempty |
| · have hm := I.min'_mem hI |
| have hM := I.max'_mem hI |
| have hsub : I ⊆ Finset.Ico (I.min' hI) (I.max' hI+1) := by |
| intro t ht |
| exact Finset.mem_Ico.mpr ⟨I.min'_le t ht,Nat.lt_succ_of_le (I.le_max' t ht)⟩ |
| calc |
| _ ≤ ∑ t ∈ Finset.Ico (I.min' hI) (I.max' hI+1), |
| (PrescribedTree.treeMean (realize H d S (updateCoordinate q v t)) T f- |
| PrescribedTree.treeMean (realize H d S (updateCoordinate q v (t+1))) T f)^2 := |
| Finset.sum_le_sum_of_subset_of_nonneg hsub (fun _ _ _ => sq_nonneg _) |
| _ ≤ _ := scheduled_vertex_Ico_energy_le H d S q v _ _ |
| ((I.min'_le _ hM).trans (Nat.le_succ _)) (ha _ hm).1 (ha _ hM).2 T f hf |
| · have he : I=∅ := Finset.not_nonempty_iff_eq_empty.mp hI |
| simp only [he,Finset.sum_empty] |
| positivity |
|
|
| end DilutedSpinGlass.ReducedTopology |
| end |
|
|
| end |
|
|
| end OAI |
|
|
