| import Mathlib |
| import OAI.Probability.SKBarriers.Scalar.DyadicCDFStability |
| import OAI.Probability.SKBarriers.Scalar.ScalarGeneralTerminal |
|
|
| namespace OAI |
|
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| section |
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| noncomputable section |
| open scoped BigOperators NNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
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|
| theorem dyadicScalar_cdf_mass_bound_general {f : ℝ → ℝ} (hf : BoundedDerivs f) |
| {K : ℝ≥0} (hLip : LipschitzWith K f) (β : ℝ) {α γ : ℝ → ℝ} |
| (hα : ∀ z, α z ∈ Icc (0:ℝ) 1) (hγ : ∀ z, γ z ∈ Icc (0:ℝ) 1) |
| (n : ℕ) (s : ℝ) (t : ℝ≥0) (ht : t ≤ 1) (x : ℝ) : |
| |dyadicScalar β α n s t f x-dyadicScalar β γ n s t f x| ≤ |
| scalarTimeMassConstantK β K*dyadicCDFDistance α γ n s t := by |
| induction n generalizing f s t x with |
| | zero => |
| simp only [dyadicScalar_zero,dyadicCDFDistance] |
| have H := scalarTimeStep_mass_lipschitz_general hf hLip β ht (hγ s) (hα s) x |
| exact H.trans_eq (by dsimp [scalarTimeMassConstantK]; ring) |
| | succ n ih => |
| rw [dyadicScalar_succ β α n s t f,dyadicScalar_succ β γ n s t f,dyadicCDFDistance] |
| have hhalf : t/2 ≤ 1 := (div_le_self (show (0:ℝ≥0) ≤ t from bot_le) (by norm_num)).trans ht |
| let fα := dyadicScalar β α n (s+(t:ℝ)/2) (t/2) f |
| let fγ := dyadicScalar β γ n (s+(t:ℝ)/2) (t/2) f |
| have hfα : BoundedDerivs fα := dyadicScalar_regular hf β α _ _ _ |
| have hfγ : BoundedDerivs fγ := dyadicScalar_regular hf β γ _ _ _ |
| have hlγ : LipschitzWith K fγ := dyadicScalar_lipschitz hf hLip β hγ _ _ _ |
| have H₁ := dyadicScalar_uniform_nonexpansive hfα hfγ β hα n s (t/2) |
| (fun z => ih hf hLip (s+(t:ℝ)/2) (t/2) hhalf z) x |
| have H₂ := ih hfγ hlγ s (t/2) hhalf x |
| calc |
| _ ≤ |dyadicScalar β α n s (t/2) fα x-dyadicScalar β α n s (t/2) fγ x|+ |
| |dyadicScalar β α n s (t/2) fγ x-dyadicScalar β γ n s (t/2) fγ x| := abs_sub_le _ _ _ |
| _ ≤ _+_ := add_le_add H₁ H₂ |
| _ = _ := by ring |
|
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| theorem scalarCDFOperator_cdf_lipschitz {f : ℝ → ℝ} (hf : BoundedDerivs f) |
| {K : ℝ≥0} (hLip : LipschitzWith K f) (β : ℝ) {α γ : ℝ → ℝ} |
| (hα : ∀ z, α z ∈ Icc (0:ℝ) 1) (hαm : Monotone α) |
| (hγ : ∀ z, γ z ∈ Icc (0:ℝ) 1) (hγm : Monotone γ) |
| (s : ℝ) (t : ℝ≥0) (ht : t ≤ 1) (x : ℝ) : |
| |scalarCDFOperator β α s t f x-scalarCDFOperator β γ s t f x| ≤ |
| scalarTimeMassConstantK β K*(∫ z in s..s+t, |α z-γ z|) := by |
| have H (n : ℕ) := (dyadicScalar_cdf_mass_bound_general hf hLip β hα hγ n s t ht x).trans |
| (mul_le_mul_of_nonneg_left (dyadicCDFDistance_integral_bound hαm hγm n s t) |
| (scalarTimeMassConstantK_nonneg β K)) |
| have hv := ((dyadicScalar_general_uniform hf hLip β hα hαm s t ht).tendsto_at x).sub |
| ((dyadicScalar_general_uniform hf hLip β hγ hγm s t ht).tendsto_at x) |
| have hzero : Tendsto (fun n : ℕ => (t:ℝ)/(2:ℝ)^n) atTop (𝓝 0) := |
| tendsto_const_nhds.div_atTop (tendsto_pow_atTop_atTop_of_one_lt (by norm_num)) |
| have hr := (((hzero.mul_const ((α (s+t)-α s)+(γ (s+t)-γ s))).const_add |
| (∫ z in s..s+t, |α z-γ z|)).const_mul (scalarTimeMassConstantK β K)) |
| have Hlim := le_of_tendsto_of_tendsto hv.abs hr (Eventually.of_forall H) |
| simpa only [zero_mul,add_zero] using Hlim |
|
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| end SK.Analytic |
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| end |
| end |
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| end OAI |
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