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c4cdbbd | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 | import OAI.Probability.InvariantIsing.Core.DenseFiniteMaximum
namespace OAI
/-! Decreasing finite minima exhaust continuous functions on a dense sequence. -/
noncomputable section
open Filter Set
open scoped Topology
namespace InvariantIsing
def prefixMinimum (f : ℕ → ℝ) (n : ℕ) : ℝ :=
Finset.univ.inf' Finset.univ_nonempty (fun i : Fin (n+1) => f i)
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)))
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)
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))
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))⟩
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
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
end InvariantIsing
end
end OAI
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