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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 61 62 | 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
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