| import OAI.Probability.ContinuumTransition.IdealSlopes |
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| namespace OAI |
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| universe u_1 u_2 |
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| section |
| open scoped Topology |
| open Set Filter Metric |
| noncomputable section |
| namespace ContinuumTemperature |
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| lemma locallyUniform_reindex {ι : Type u_1} {κ : Type u_2} {l : Filter ι} {k : Filter κ} |
| {Q : ι → Parameter → ℝ} {q : Parameter → ℝ} |
| (hQ : TendstoLocallyUniformlyOn Q q l pressureDomain) |
| {θ : κ → ι} (hθ : Tendsto θ k l) : |
| TendstoLocallyUniformlyOn (fun n => Q (θ n)) q k pressureDomain := by |
| intro u hu x hx |
| obtain ⟨t,ht,h⟩ := hQ u hu x hx |
| exact ⟨t,ht,hθ.eventually h⟩ |
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| lemma quadraticCost_tendsto {Q : ℕ → Parameter → ℝ} {q : Parameter → ℝ} |
| (hQ : TendstoLocallyUniformlyOn Q q atTop pressureDomain) |
| (hq : ContinuousOn q pressureDomain) |
| {a d : ℕ → ℝ} {a₀ d₀ : ℝ} (ha : Tendsto a atTop (𝓝 a₀)) |
| (hd : Tendsto d atTop (𝓝 d₀)) {X : ℕ → Parameter} {x : Parameter} |
| (hx : x ∈ pressureDomain) (hX : Tendsto X atTop (𝓝 x)) : |
| Tendsto (fun n => quadraticCost (Q n) (a n) (X n) (d n)) atTop |
| (𝓝 (quadraticCost q a₀ x d₀)) := by |
| have hX1 := (continuous_fst.tendsto x).comp hX |
| have hX2 := (continuous_snd.tendsto x).comp hX |
| have hf : Tendsto (fun n => shiftedField (a n) (X n) (d n)) atTop |
| (𝓝 (shiftedField a₀ x d₀)) := |
| hX1.prodMk_nhds (hX2.add ((hX1.mul ha).mul hd)) |
| have he : ∀ᶠ n in atTop, shiftedField (a n) (X n) (d n) ∈ pressureDomain := |
| (hX.eventually (isOpen_pressureDomain.mem_nhds hx)) |
| have hval := hQ.tendsto_comp (hq _ (show shiftedField a₀ x d₀ ∈ pressureDomain from hx)) |
| (show shiftedField a₀ x d₀ ∈ pressureDomain from hx) (tendsto_nhdsWithin_iff.2 ⟨hf,he⟩) |
| exact hval.sub (((hX1.mul ha).mul (hd.pow 2)).div_const 2) |
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| lemma meanField_tendsto_of_bounded_maximizers {Q : ℕ → Parameter → ℝ} {q : Parameter → ℝ} |
| {P : ℕ → Parameter → ℝ} {p : Parameter → ℝ} {a : ℕ → ℝ} {a₀ : ℝ} |
| (hQ : TendstoLocallyUniformlyOn Q q atTop pressureDomain) (hq : ContinuousOn q pressureDomain) |
| (ha : Tendsto a atTop (𝓝 a₀)) (hP : ∀ n, IsMeanField (Q n) (a n) (P n)) |
| (hp : IsMeanField q a₀ p) {X : ℕ → Parameter} {x : Parameter} |
| (hx : x ∈ pressureDomain) (hX : Tendsto X atTop (𝓝 x)) |
| (hXS : ∀ n, X n ∈ pressureDomain) {D : ℝ} |
| (hatt : ∀ n, ∃ d ∈ Icc 0 D, quadraticCost (Q n) (a n) (X n) d = P n (X n)) : |
| Tendsto (fun n => P n (X n)) atTop (𝓝 (p x)) := by |
| choose d hd he using hatt |
| apply metric_tendsto_of_subseq |
| intro φ hφ |
| obtain ⟨r,_,ψ,hψ,hr⟩ := (isCompact_Icc : IsCompact (Icc (0:ℝ) D)).tendsto_subseq |
| (fun n => hd (φ n)) |
| let θ := φ ∘ ψ |
| have hθ := (hφ.comp hψ).tendsto_atTop |
| have hlim := quadraticCost_tendsto (locallyUniform_reindex hQ hθ) hq (ha.comp hθ) hr hx (hX.comp hθ) |
| have hle : quadraticCost q a₀ x r ≤ p x := (hp x hx).1 (mem_range_self r) |
| have hge : p x ≤ quadraticCost q a₀ x r := by |
| apply (hp x hx).2 |
| rintro _ ⟨s,rfl⟩ |
| have hs := quadraticCost_tendsto (locallyUniform_reindex hQ hθ) hq (ha.comp hθ) |
| (tendsto_const_nhds (x := s)) hx (hX.comp hθ) |
| apply le_of_tendsto_of_tendsto hs hlim |
| exact Eventually.of_forall fun n => by |
| change quadraticCost (Q (θ n)) (a (θ n)) (X (θ n)) s ≤ |
| quadraticCost (Q (θ n)) (a (θ n)) (X (θ n)) (d (θ n)) |
| rw [he] |
| exact (hP (θ n) _ (hXS (θ n))).1 (mem_range_self s) |
| have heq := le_antisymm hle hge |
| refine ⟨ψ,hψ,?_⟩ |
| rw [heq] at hlim |
| exact hlim.congr (fun n => he (θ n)) |
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| end ContinuumTemperature |
| end |
| end |
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| end OAI |
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