| import Mathlib |
| import OAI.Probability.SKGap.Terminal.TerminalMatrixMeasurable |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open MeasureTheory ProbabilityTheory Real Set Filter |
| open scoped Topology |
|
|
| lemma terminalMatrixData_small_eventually {j ε : ℝ} (hε : 0 < ε) : |
| ∀ᶠ n : ℕ in atTop,terminalEntryScale n ≤ 1 ∧ |
| terminalEntryScale n*(1+terminalRowBound j/3) ≤ 2*ε/3 := by |
| have ht := terminalEntryScale_tendsto |
| have ht' := ht.mul_const (1+terminalRowBound j/3) |
| exact (ht.eventually (ge_mem_nhds (by norm_num : (0:ℝ) < 1))).and |
| (by simpa only [zero_mul] using ht'.eventually (ge_mem_nhds (by simpa only [zero_mul] using (show (0:ℝ) < 2*ε/3 by positivity) : (0:ℝ)*(1+terminalRowBound j/3) < 2*ε/3))) |
|
|
| lemma terminalMatrixData_actual_bad {β α ε l : ℝ} (hβ : 0 < β) (hα : 0 ≤ α) |
| (hε : 0 < ε) (hl : 0 ≤ l) (hedge : 2*sqrt (β^2*α) ≤ ε/6) : |
| ∀ᶠ n : ℕ in atTop,(disorderLaw β n).real {g | ¬ terminalMatrixData (terminalMatrixBound (β^2)) α ε g} ≤ |
| terminalRawTail (β^2) α ε n l := by |
| have hn : ∀ᶠ n : ℕ in atTop,1 ≤ n := eventually_ge_atTop 1 |
| filter_upwards [terminalMatrixData_small_eventually (j:=β^2) hε,hn] with n hn hnpos |
| let : NeZero n := ⟨by omega⟩ |
| have hm := (terminalMatrixData_measurable (n:=n) (terminalMatrixBound (β^2)) α ε).compl |
| change MeasurableSet {g : Disorder n | ¬terminalMatrixData (terminalMatrixBound (β^2)) α ε g} at hm |
| rw [← (goe_actual_disorder_hasLaw β n).measureReal_eq hm] |
| apply (measureReal_mono (μ:=gaussianCoordinates (MatrixCoordinates (Fin n))) ?_ (measure_ne_top _ _)).trans |
| (terminalRaw_bad_bound (sq_pos_of_pos hβ) hα hε hl hedge) |
| intro g hg hr |
| exact hg (terminalMatrixData_of_raw (sq_pos_of_pos hβ) hε.le hn.1 hn.2 g hr) |
|
|
| lemma terminalMatrixData_actual_whp {β α ε l : ℝ} (hβ : 0 < β) (hα : 0 ≤ α) |
| (hε : 0 < ε) (hl : 0 ≤ l) (hedge : 2*sqrt (β^2*α) ≤ ε/6) |
| (h₅ : 0 < ε^2/(36*π^2*(β^2))-(l*α+log (1+exp (-l)))) |
| (h₆ : 0 < ε/(4*(β^2))-(l*α+log (1+exp (-l))+α*log 2)) : |
| Tendsto (fun n : ℕ=>(disorderLaw β n).real {g | terminalMatrixData (terminalMatrixBound (β^2)) α ε g}) atTop (𝓝 1) := by |
| have hb : Tendsto (fun n : ℕ=>(disorderLaw β n).real {g | ¬terminalMatrixData (terminalMatrixBound (β^2)) α ε g}) atTop (𝓝 0) := |
| squeeze_zero' (Eventually.of_forall (fun _=>measureReal_nonneg)) |
| (terminalMatrixData_actual_bad hβ hα hε hl hedge) (terminalRawTail_tendsto (sq_pos_of_pos hβ) h₅ h₆) |
| have he (n : ℕ) : (disorderLaw β n).real {g | terminalMatrixData (terminalMatrixBound (β^2)) α ε g}= |
| 1-(disorderLaw β n).real {g | ¬terminalMatrixData (terminalMatrixBound (β^2)) α ε g} := by |
| have hm := terminalMatrixData_measurable (n:=n) (terminalMatrixBound (β^2)) α ε |
| have h := measureReal_compl hm (μ:=disorderLaw β n) |
| simp only [probReal_univ] at h |
| change (disorderLaw β n).real {g | ¬terminalMatrixData (terminalMatrixBound (β^2)) α ε g}=1- _ at h |
| linarith only [h] |
| simp_rw [he] |
| simpa only [sub_zero] using tendsto_const_nhds.sub hb |
| end SKGap |
|
|
| end |
| end |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Filter Real |
| open scoped Topology |
|
|
| lemma tendsto_cosh_real_atTop : Tendsto cosh atTop atTop := by |
| have he : Tendsto (fun x : ℝ=>exp x/2) atTop atTop := by |
| simpa only [div_eq_mul_inv,mul_comm,one_mul,id_eq] using tendsto_exp_atTop.const_mul_atTop (by norm_num : (0:ℝ) < 1/2) |
| apply tendsto_atTop_mono _ he |
| intro x |
| rw [cosh_eq] |
| have hp := exp_pos (-x) |
| linarith only [hp] |
|
|
| lemma terminalTime_exists {M α ε : ℝ} (_hM : 0 ≤ M) (hα : 0 < α) |
| (hε : 400001*ε < 1/4) : |
| ∃ T : ℝ,0 < T ∧ M^2 ≤ (T/4)^2*α/2 ∧ |
| 400001*(ε+M*(2*sqrt (1/cosh (T/4)^2)+1/cosh (T/4)^2)) ≤ 1/4 ∧ |
| exp (-T/8)*(exp 1-1) ≤ α/4 := by |
| have hT4 : Tendsto (fun T : ℝ=>T/4) atTop atTop := by |
| simpa only [div_eq_mul_inv,mul_comm,one_mul,id_eq] using tendsto_id.const_mul_atTop (by norm_num : (0:ℝ) < 1/4) |
| have hs : Tendsto (fun T : ℝ=>1/cosh (T/4)^2) atTop (𝓝 0) := by |
| have h := tendsto_inv_atTop_zero.comp (tendsto_cosh_real_atTop.comp hT4) |
| simpa only [Function.comp_apply,one_div,inv_pow,zero_pow (by decide : (2:ℕ) ≠ 0)] using h.pow 2 |
| have hsroot := hs.sqrt |
| have hsfull : Tendsto (fun T : ℝ=>400001*(ε+M*(2*sqrt (1/cosh (T/4)^2)+1/cosh (T/4)^2))) atTop (𝓝 (400001*ε)) := by |
| simpa only [sqrt_zero,mul_zero,add_zero] using (((hsroot.const_mul 2).add hs).const_mul M |>.const_add ε |>.const_mul 400001) |
| have he : Tendsto (fun T : ℝ=>exp (-T/8)*(exp 1-1)) atTop (𝓝 0) := by |
| have hT8 : Tendsto (fun T : ℝ=>T/8) atTop atTop := by |
| simpa only [div_eq_mul_inv,mul_comm,one_mul,id_eq] using tendsto_id.const_mul_atTop (by norm_num : (0:ℝ) < 1/8) |
| simpa only [Function.comp_apply,neg_div,zero_mul] using (tendsto_exp_neg_atTop_nhds_zero.comp hT8).mul_const (exp 1-1) |
| have ht : ∀ᶠ T : ℝ in atTop,4*sqrt (2*M^2/α)+1 ≤ T := eventually_ge_atTop _ |
| obtain ⟨T,⟨htT,htnum⟩,htp⟩ := (ht.and (hsfull.eventually (ge_mem_nhds hε))).and |
| (he.eventually (ge_mem_nhds (by positivity : (0:ℝ) < α/4))) |>.exists |
| refine ⟨T,?_,?_,htnum,htp⟩ |
| · have hr := sqrt_nonneg (2*M^2/α) |
| linarith only [htT,hr] |
| · have hr := sqrt_nonneg (2*M^2/α) |
| have hsquare := sq_sqrt (by positivity : 0 ≤ 2*M^2/α) |
| have hbound : 2*M^2/α ≤ (T/4)^2 := by nlinarith only [hr,hsquare,htT] |
| have hmul := (div_le_iff₀ hα).mp hbound |
| nlinarith only [hmul] |
| end SKGap |
|
|
| end |
| end |
|
|
| end OAI |
|
|
