openAI-math-dataset / lean /OAI /Probability /SKBarriers /Gaussian /LipschitzChainRepresentation.lean
| import Mathlib | |
| import OAI.Probability.SKBarriers.Parisi.CDFPartitionRepresentation | |
| import OAI.Probability.SKBarriers.Gaussian.LipschitzObservable | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open scoped NNReal Topology | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| namespace SK.Analytic | |
| theorem scalarWindowDerivative_lipschitz_bound {g : ℝ → ℝ} {L : ℝ≥0} | |
| (hg : LipschitzWith L g) {h : ℝ} (hh : 0<h) (x : ℝ) : | |
| |scalarWindowDerivative g h x| ≤ L := by | |
| have H := hg.dist_le_mul (x+h) x | |
| simp only [Real.dist_eq,add_sub_cancel_left,abs_of_pos hh] at H | |
| rw [scalarWindowDerivative,abs_div,abs_of_pos hh,div_le_iff₀ hh] | |
| exact H | |
| theorem scalarWindow_lipschitz {g : ℝ → ℝ} {L : ℝ≥0} | |
| (hg : LipschitzWith L g) {h : ℝ} (hh : 0<h) : LipschitzWith L (scalarWindow g h) := by | |
| apply lipschitzWith_of_nnnorm_deriv_le | |
| (fun x => (scalarWindow_hasDerivAt hg.continuous h x).differentiableAt) | |
| intro x | |
| rw [(scalarWindow_hasDerivAt hg.continuous h x).deriv] | |
| exact_mod_cast scalarWindowDerivative_lipschitz_bound hg hh x | |
| theorem scalarCDFAverage_eq_chainAverage_lipschitz {f g : ℝ → ℝ} | |
| (hf : BoundedDerivs f) {K B L : ℝ≥0} (hfK : LipschitzWith K f) | |
| (hgL : LipschitzWith L g) (hB : ∀ z, |g z| ≤ B) | |
| (β : ℝ) {α : ℝ → ℝ} (hα : ∀ z, α z∈Icc (0:ℝ) 1) (hmono : Monotone α) | |
| (l : List (ℝ × ℝ≥0)) (hm : ∀ p∈l, p.1∈Icc (0:ℝ) 1) | |
| (s : ℝ) (t : ℝ≥0) (ht : t≤1) (hdur : chainDuration l=t) | |
| (hmodel : TimeChainModels α s l) (x : ℝ) : | |
| scalarCDFAverage β α s t f g x=scalarTimeChainAverage β l f g x := by | |
| apply sub_eq_zero.mp | |
| apply abs_eq_zero.mp | |
| apply le_antisymm ?_ (abs_nonneg _) | |
| apply le_of_forall_pos_le_add | |
| intro ε hε | |
| simp only [zero_add] | |
| let h := ε/(2*((L:ℝ)+1)) | |
| have hL : 0<(L:ℝ)+1 := by positivity | |
| have hh : 0<h := div_pos hε (mul_pos (by norm_num) hL) | |
| have hsmall : (L:ℝ)*h ≤ ε/2 := by | |
| calc | |
| _ ≤ ((L:ℝ)+1)*h := mul_le_mul_of_nonneg_right (by linarith) hh.le | |
| _ = ε/2 := by dsimp [h]; field_simp | |
| have hd (z) : |g z-scalarWindow g h z| ≤ (L:ℝ)*h := by | |
| rw [abs_sub_comm] | |
| exact scalarWindow_approx hgL hh z | |
| have H₁ := scalarCDFAverage_observable_distance β hα hmono hf hfK hgL | |
| (scalarWindow_lipschitz hgL hh) hB (scalarWindow_bound hgL.continuous hB hh) | |
| (mul_nonneg L.coe_nonneg hh.le) hd s t ht x | |
| have H₂ := scalarHierarchyAverage_observable_distance hf | |
| ⟨hgL.continuous,B,B.coe_nonneg,hB⟩ | |
| ⟨(scalarWindow_lipschitz hgL hh).continuous,B,B.coe_nonneg,scalarWindow_bound hgL.continuous hB hh⟩ | |
| (mul_nonneg L.coe_nonneg hh.le) hd l.length (fun i => (l.get i).1) | |
| (fun i => β*Real.sqrt ((l.get i).2:ℝ)) x | |
| change |scalarTimeChainAverage β l f g x-scalarTimeChainAverage β l f (scalarWindow g h) x|≤_ at H₂ | |
| have HE := scalarCDFAverage_eq_chainAverage hf | |
| (scalarWindow_hasDerivAt hgL.continuous h) (scalarWindowDerivative_continuous hgL.continuous h) | |
| hfK (scalarWindow_bound hgL.continuous hB hh) (scalarWindowDerivative_lipschitz_bound hgL hh) | |
| β hα hmono l hm s t ht hdur hmodel x | |
| rw [HE] at H₁ | |
| have H := abs_sub_le (scalarCDFAverage β α s t f g x) | |
| (scalarTimeChainAverage β l f (scalarWindow g h) x) (scalarTimeChainAverage β l f g x) | |
| rw [abs_sub_comm (scalarTimeChainAverage β l f (scalarWindow g h) x)] at H | |
| linarith | |
| theorem scalarCDFAverage_eq_partition_lipschitz {f g : ℝ → ℝ} (hf : BoundedDerivs f) | |
| {K B L : ℝ≥0} (hK : LipschitzWith K f) (hgL : LipschitzWith L g) (hB : ∀ y, |g y|≤B) | |
| (β : ℝ) {α : ℝ → ℝ} (hα : ∀ z, α z∈Icc (0:ℝ) 1) (hαm : Monotone α) | |
| (n : ℕ) (q : Fin (n+1) → ℝ) (hq : Monotone q) (m : Fin n → ℝ) | |
| (hm : ∀ i, m i∈Icc (0:ℝ) 1) | |
| (hmodel : ∀ i : Fin n, ∀ z∈Ico (q i.castSucc) (q i.succ), α z=m i) | |
| (t : ℝ≥0) (ht : t≤1) (hd : (t:ℝ)=q (Fin.last n)-q 0) (x : ℝ) : | |
| scalarCDFAverage β α (q 0) t f g x=scalarHierarchyAverage n m (timeGridCoefficients n β q) f g x := by | |
| have hdur : chainDuration (partitionTimeChain n q hq m)=t := by | |
| apply NNReal.coe_injective | |
| exact (partitionTimeChain_duration n q hq m).trans hd.symm | |
| have hml (p) (hp : p∈partitionTimeChain n q hq m) : p.1∈Icc (0:ℝ) 1 := by | |
| obtain ⟨i,rfl⟩ := List.mem_ofFn.mp hp | |
| exact hm i | |
| have H := scalarCDFAverage_eq_chainAverage_lipschitz hf hK hgL hB β hα hαm | |
| (partitionTimeChain n q hq m) hml (q 0) t ht hdur (partitionTimeChain_models n q hq m hmodel) x | |
| simp only [partitionTimeChain,scalarTimeChainAverage_ofFn,NNReal.coe_mk] at H | |
| exact H | |
| end SK.Analytic | |
| end | |
| end | |
| end OAI | |