| import Mathlib |
| import OAI.Probability.SKGap.Model |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open MeasureTheory ProbabilityTheory Real Set Filter |
| open scoped BigOperators ENNReal Topology |
|
|
| lemma subadditive_chain_finite {E : Type*} [AddCommGroup E] (p : E→ℝ) |
| (hp : ∀ x y,p (x+y) ≤ p x+p y) (u : ℕ→E) (r : ℕ→ℝ) |
| (hr : ∀ k,p (u (k+1)-u k) ≤ r k) (n : ℕ) : |
| p (u n) ≤ p (u 0)+∑ k∈Finset.range n,r k := by |
| induction n with |
| | zero => simp |
| | succ n hn => |
| have hh := hp (u n) (u (n+1)-u n) |
| rw [add_sub_cancel] at hh |
| rw [Finset.sum_range_succ] |
| linarith only [hh,hn,hr n] |
|
|
| lemma subadditive_chain_limit {E : Type*} [AddCommGroup E] [TopologicalSpace E] |
| (p : E→ℝ) (hc : Continuous p) (hp : ∀ x y,p (x+y) ≤ p x+p y) |
| (u : ℕ→E) {x : E} (hu : Tendsto u atTop (𝓝 x)) (r : ℕ→ℝ) |
| (hr0 : ∀ k,0 ≤ r k) (hrs : Summable r) (hr : ∀ k,p (u (k+1)-u k) ≤ r k) : |
| p x ≤ p (u 0)+∑' k,r k := by |
| apply le_of_tendsto (hc.tendsto x |>.comp hu) |
| exact Eventually.of_forall (fun n=>(subadditive_chain_finite p hp u r hr n).trans |
| (add_le_add_right (hrs.sum_le_tsum (Finset.range n) (fun i _=>hr0 i)) _)) |
|
|
| lemma arithmetic_half_geometric_hasSum : |
| HasSum (fun k : ℕ=>((k:ℝ)+1)*(1/2:ℝ)^k) 4 := by |
| have h₁ := hasSum_coe_mul_geometric_of_norm_lt_one (by norm_num : ‖(1/2:ℝ)‖ < 1) |
| have h₂ := hasSum_geometric_of_lt_one (by norm_num : (0:ℝ) ≤ 1/2) (by norm_num : (1/2:ℝ)<1) |
| have h : HasSum (fun k : ℕ => ((k:ℝ)+1)*(1/2:ℝ)^k) |
| (1/2/(1-1/2)^2+(1-1/2)⁻¹) := by |
| convert h₁.add h₂ using 1 |
| ext k |
| ring |
| norm_num at h |
| exact h |
|
|
| theorem continuous_chaining_probability |
| {Ω T E : Type*} [MeasurableSpace Ω] (μ : Measure Ω) |
| [TopologicalSpace T] [AddCommGroup E] [TopologicalSpace E] |
| (p : E→ℝ) (hc : Continuous p) (hp : ∀ x y,p (x+y) ≤ p x+p y) |
| (F : T→Ω→E) (hF : ∀ ω,Continuous (fun t=>F t ω)) |
| (S : Set T) (ι : ℕ→Type*) [∀ k,Fintype (ι k)] |
| (v : ∀ k,ι k→T) (edges : ∀ k,Finset (ι k×ι (k+1))) |
| (hcover : ∀ t∈S,∃ a : ∀ k,ι k, |
| (∀ k,(a k,a (k+1))∈edges k) ∧ Tendsto (fun k=>v k (a k)) atTop (𝓝 t)) |
| (r : ℕ→ℝ) (hr0 : ∀ k,0 ≤ r k) (hrs : Summable r) |
| (b : ℝ) (q₀ : ℝ≥0∞) (q : ℕ→ℝ≥0∞) |
| (hb : ∀ i,μ {ω | b < p (F (v 0 i) ω)} ≤ q₀) |
| (hi : ∀ k, ∀ e∈edges k,μ {ω | r k < p (F (v (k+1) e.2) ω-F (v k e.1) ω)} ≤ q k) : |
| μ {ω | ∃ t∈S,b+(∑' k,r k) < p (F t ω)} ≤ |
| (Fintype.card (ι 0):ℝ≥0∞)*q₀+∑' k,((edges k).card:ℝ≥0∞)*q k := by |
| classical |
| let B : Set Ω := ⋃ i : ι 0,{ω | b < p (F (v 0 i) ω)} |
| let I : ℕ→Set Ω := fun k=>⋃ e∈edges k,{ω | r k < p (F (v (k+1) e.2) ω-F (v k e.1) ω)} |
| have hs : {ω | ∃ t∈S,b+(∑' k,r k) < p (F t ω)} ⊆ B ∪ ⋃ k,I k := by |
| intro ω hω |
| by_contra hh |
| have hh' := not_or.mp hh |
| obtain ⟨t,ht,htp⟩ := hω |
| obtain ⟨a,ha,hat⟩ := hcover t ht |
| have hbase : p (F (v 0 (a 0)) ω) ≤ b := by |
| by_contra hn |
| exact hh'.1 (mem_iUnion.mpr ⟨a 0,not_le.mp hn⟩) |
| have hinc : ∀ k,p (F (v (k+1) (a (k+1))) ω-F (v k (a k)) ω) ≤ r k := by |
| intro k |
| by_contra hn |
| apply hh'.2 |
| exact mem_iUnion.mpr ⟨k,mem_iUnion₂.mpr ⟨(a k,a (k+1)),ha k,not_le.mp hn⟩⟩ |
| have hl := subadditive_chain_limit p hc hp (fun k=>F (v k (a k)) ω) |
| ((hF ω).tendsto t |>.comp hat) r hr0 hrs hinc |
| linarith only [hl,hbase,htp] |
| apply (measure_mono hs).trans |
| apply (measure_union_le _ _).trans |
| apply add_le_add |
| · exact (measure_iUnion_fintype_le μ _).trans ((Finset.sum_le_sum (fun i _=>hb i)).trans_eq (by simp)) |
| · apply (measure_iUnion_le I).trans |
| apply ENNReal.tsum_le_tsum |
| intro k |
| apply (measure_biUnion_finset_le (edges k) _).trans |
| exact (Finset.sum_le_sum (fun e he=>hi k e he)).trans_eq (by simp) |
| end SKGap |
| end |
| end |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open MeasureTheory ProbabilityTheory Real Set Filter |
| open scoped BigOperators ENNReal Topology |
|
|
| lemma exponential_series_bound {c : ℝ} (hc : log 2 ≤ c) : |
| (∑' k : ℕ,ENNReal.ofReal (exp (-c*((k:ℝ)+1)))) ≤ ENNReal.ofReal (2*exp (-c)) := by |
| have hc0 : 0 < c := lt_of_lt_of_le (log_pos (by norm_num : (1:ℝ)<2)) hc |
| have hr : exp (-c) < 1 := by rw [exp_lt_one_iff];linarith |
| have hr2 : exp (-c) ≤ 1/2 := by |
| rw [← exp_log (by norm_num : (0:ℝ)<(1/2))] |
| apply exp_le_exp.mpr |
| rw [log_div (by norm_num) (by norm_num),log_one] |
| linarith only [hc] |
| have he (k : ℕ) : exp (-c*((k:ℝ)+1))=exp (-c)*exp (-c)^k := by |
| rw [← exp_nat_mul,← exp_add] |
| congr 1;ring |
| have hs : HasSum (fun k : ℕ=>exp (-c*((k:ℝ)+1))) (exp (-c)*(1-exp (-c))⁻¹) := by |
| simpa only [he] using (hasSum_geometric_of_lt_one (exp_pos (-c)).le hr).mul_left (exp (-c)) |
| rw [← ENNReal.ofReal_tsum_of_nonneg (fun _=>(exp_pos _).le) hs.summable,hs.tsum_eq] |
| apply ENNReal.ofReal_le_ofReal |
| rw [← div_eq_mul_inv] |
| apply (div_le_iff₀ (by linarith only [hr] : 0 < 1-exp (-c))).mpr |
| nlinarith only [mul_le_mul_of_nonneg_left hr2 (exp_pos (-c)).le] |
|
|
| lemma chaining_entropy_tail_product {n : ℕ} (_hn : 1 ≤ n) |
| {A B c : ℝ} (hB : 0 ≤ B) (hc : 0 < c) (hA : B+c ≤ A) |
| (k : ℕ) {m q : ℝ≥0∞} |
| (hm : m ≤ ENNReal.ofReal (exp (B*((k:ℝ)+1)*(n:ℝ)))) |
| (hq : q ≤ ENNReal.ofReal (exp (-A*((k:ℝ)+1)^2*(n:ℝ)))) : |
| m*q ≤ ENNReal.ofReal (exp (-c*(n:ℝ)*((k:ℝ)+1))) := by |
| apply (mul_le_mul' hm hq).trans |
| rw [← ENNReal.ofReal_mul (exp_pos _).le,← exp_add] |
| apply ENNReal.ofReal_le_ofReal |
| apply exp_le_exp.mpr |
| have hkn : 1 ≤ (k:ℝ)+1 := by |
| have hk : (0:ℝ) ≤ k := by positivity |
| linarith only [hk] |
| have hnn : 0 ≤ (n:ℝ) := by positivity |
| have hA0 : 0 ≤ A := by linarith |
| have hs : (k:ℝ)+1 ≤ ((k:ℝ)+1)^2 := by nlinarith |
| have hh := mul_le_mul_of_nonneg_left hs hA0 |
| have hh' := mul_le_mul_of_nonneg_right hA (show 0 ≤ (k:ℝ)+1 by positivity) |
| have hh'' := mul_le_mul_of_nonneg_right (show c*((k:ℝ)+1) ≤ A*((k:ℝ)+1)^2-B*((k:ℝ)+1) by linarith only [hh,hh']) hnn |
| linarith only [hh''] |
|
|
| lemma chaining_entropy_series {n : ℕ} (hn : 1 ≤ n) |
| {A B c : ℝ} (hB : 0 ≤ B) (hc : log 2 ≤ c) (hA : B+c ≤ A) |
| (m q : ℕ→ℝ≥0∞) |
| (hm : ∀ k,m k ≤ ENNReal.ofReal (exp (B*((k:ℝ)+1)*(n:ℝ)))) |
| (hq : ∀ k,q k ≤ ENNReal.ofReal (exp (-A*((k:ℝ)+1)^2*(n:ℝ)))) : |
| (∑' k,m k*q k) ≤ ENNReal.ofReal (2*exp (-c*(n:ℝ))) := by |
| have hc0 : 0 < c := lt_of_lt_of_le (log_pos (by norm_num : (1:ℝ)<2)) hc |
| have hn' : (1:ℝ) ≤ n := by exact_mod_cast hn |
| apply (ENNReal.tsum_le_tsum (fun k=>chaining_entropy_tail_product hn hB hc0 hA k (hm k) (hq k))).trans |
| have hcn : log 2 ≤ c*(n:ℝ) := hc.trans (by nlinarith only [hn',hc0]) |
| simpa only [neg_mul] using exponential_series_bound hcn |
| end SKGap |
| end |
| end |
|
|
| end OAI |
|
|
