| import OAI.Probability.InvariantIsing.Core.FiniteGaussianMaximum |
| import Mathlib.Topology.Order.MonotoneConvergence |
| import Mathlib.Topology.DenseEmbedding |
|
|
| namespace OAI |
|
|
| /-! Increasing finite maxima exhaust a continuous function on a dense sequence. -/ |
| noncomputable section |
| open Filter Set |
| open scoped Topology |
| namespace InvariantIsing |
|
|
| def prefixMaximum (f : ℕ → ℝ) (n : ℕ) : ℝ := |
| Finset.univ.sup' Finset.univ_nonempty (fun i : Fin (n+1) => f i) |
|
|
| lemma le_prefixMaximum (f : ℕ → ℝ) {i n : ℕ} (h : i ≤ n) : f i ≤ prefixMaximum f n := by |
| exact Finset.le_sup' (fun j : Fin (n+1) => f j) (Finset.mem_univ ⟨i,by omega⟩) |
|
|
| lemma prefixMaximum_le (f : ℕ → ℝ) {n : ℕ} {b : ℝ} (h : ∀ i ≤ n, f i ≤ b) : |
| prefixMaximum f n ≤ b := by |
| apply Finset.sup'_le |
| intro i _ |
| exact h i (by omega) |
|
|
| lemma prefixMaximum_monotone (f : ℕ → ℝ) : Monotone (prefixMaximum f) := by |
| intro n m hnm |
| exact prefixMaximum_le f (fun i hi => le_prefixMaximum f (hi.trans hnm)) |
|
|
| lemma prefixMaximum_bddAbove {f : ℕ → ℝ} (hf : BddAbove (range f)) : |
| BddAbove (range (prefixMaximum f)) := by |
| obtain ⟨b,hb⟩ := hf |
| exact ⟨b,fun _ ⟨n,hn⟩ => hn ▸ prefixMaximum_le f (fun i _ => hb (mem_range_self i))⟩ |
|
|
| lemma prefixMaximum_tendsto {f : ℕ → ℝ} (hf : BddAbove (range f)) : |
| Tendsto (prefixMaximum f) atTop (𝓝 (⨆ i, f i)) := by |
| have hb := prefixMaximum_bddAbove hf |
| have he : (⨆ n, prefixMaximum f n) = ⨆ i, f i := by |
| apply le_antisymm |
| · exact ciSup_le (fun n => prefixMaximum_le f (fun i _ => le_ciSup hf i)) |
| · exact ciSup_le (fun i => (le_prefixMaximum f (le_refl i)).trans (le_ciSup hb i)) |
| rw [← he] |
| exact tendsto_atTop_ciSup (prefixMaximum_monotone f) hb |
|
|
| theorem dense_prefixMaximum_tendsto {S : Type*} [TopologicalSpace S] [Nonempty S] |
| (s : ℕ → S) (hs : DenseRange s) (f : S → ℝ) (hf : Continuous f) |
| (hb : BddAbove (range f)) : |
| Tendsto (prefixMaximum (f ∘ s)) atTop (𝓝 (⨆ x, f x)) := by |
| have hbs : BddAbove (range (f ∘ s)) := hb.mono (range_comp_subset_range _ _) |
| have he : (⨆ i, f (s i)) = ⨆ x, f x := by |
| apply le_antisymm |
| · exact ciSup_le (fun i => le_ciSup hb (s i)) |
| · apply ciSup_le |
| exact isClosed_property hs (isClosed_le hf continuous_const) |
| (fun i => le_ciSup hbs i) |
| simpa only [Function.comp_def,he] using prefixMaximum_tendsto hbs |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
