| import OAI.Probability.InvariantIsing.Core.DenseFiniteMaximum |
|
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| namespace OAI |
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| /-! Decreasing finite minima exhaust continuous functions on a dense sequence. -/ |
| noncomputable section |
| open Filter Set |
| open scoped Topology |
| namespace InvariantIsing |
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|
| def prefixMinimum (f : ℕ → ℝ) (n : ℕ) : ℝ := |
| Finset.univ.inf' Finset.univ_nonempty (fun i : Fin (n+1) => f i) |
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| lemma prefixMinimum_le (f : ℕ → ℝ) {i n : ℕ} (h : i ≤ n) : prefixMinimum f n ≤ f i := |
| Finset.inf'_le (fun j : Fin (n+1) => f j) (Finset.mem_univ (⟨i,by omega⟩ : Fin (n+1))) |
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| lemma le_prefixMinimum (f : ℕ → ℝ) {n : ℕ} {b : ℝ} (h : ∀ i ≤ n, b ≤ f i) : |
| b ≤ prefixMinimum f n := by |
| apply Finset.le_inf' |
| intro i _ |
| exact h i (by omega) |
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| lemma prefixMinimum_antitone (f : ℕ → ℝ) : Antitone (prefixMinimum f) := by |
| intro n m hnm |
| exact le_prefixMinimum f (fun i hi => prefixMinimum_le f (hi.trans hnm)) |
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| lemma prefixMinimum_bddBelow {f : ℕ → ℝ} (hf : BddBelow (range f)) : |
| BddBelow (range (prefixMinimum f)) := by |
| obtain ⟨b,hb⟩ := hf |
| exact ⟨b,fun _ ⟨n,hn⟩ => hn ▸ le_prefixMinimum f (fun i _ => hb (mem_range_self i))⟩ |
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| lemma prefixMinimum_tendsto {f : ℕ → ℝ} (hf : BddBelow (range f)) : |
| Tendsto (prefixMinimum f) atTop (𝓝 (⨅ i, f i)) := by |
| have hb := prefixMinimum_bddBelow hf |
| have he : (⨅ n, prefixMinimum f n) = ⨅ i, f i := by |
| apply le_antisymm |
| · exact le_ciInf (fun i => (ciInf_le hb i).trans (prefixMinimum_le f (le_refl i))) |
| · exact le_ciInf (fun n => le_prefixMinimum f (fun i _ => ciInf_le hf i)) |
| rw [← he] |
| exact tendsto_atTop_ciInf (prefixMinimum_antitone f) hb |
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| theorem dense_prefixMinimum_tendsto {S : Type*} [TopologicalSpace S] [Nonempty S] |
| (s : ℕ → S) (hs : DenseRange s) (f : S → ℝ) (hf : Continuous f) |
| (hb : BddBelow (range f)) : |
| Tendsto (prefixMinimum (f ∘ s)) atTop (𝓝 (⨅ x, f x)) := by |
| have hbs : BddBelow (range (f ∘ s)) := hb.mono (range_comp_subset_range _ _) |
| have he : (⨅ i, f (s i)) = ⨅ x, f x := by |
| apply le_antisymm |
| · apply le_ciInf |
| exact isClosed_property hs (isClosed_le continuous_const hf) |
| (fun i => ciInf_le hbs i) |
| · exact le_ciInf (fun i => ciInf_le hb (s i)) |
| simpa only [Function.comp_def,he] using prefixMinimum_tendsto hbs |
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| end InvariantIsing |
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| end |
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| end OAI |
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