| import Mathlib | |
| import OAI.Analysis.HardSphere.Operators.FreeTransition | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped ENNReal Topology BigOperators | |
| open scoped BigOperators ComplexConjugate | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| open scoped ENNReal NNReal Topology BigOperators | |
| open MeasureTheory ProbabilityTheory Real Complex | |
| open scoped ENNReal ComplexConjugate | |
| namespace HardSphere | |
| open Wiener | |
| open scoped ComplexConjugate | |
| lemma avoidance_reflection_norm (N : ℕ) (L : Length) (a t : ℝ) (f : FreePathL2 N L t) : | |
| ‖(2:ℂ) • avoidanceMultiplier N L a t f-f‖≤‖f‖ := by | |
| apply Lp.norm_le_norm_of_ae_le | |
| filter_upwards [Lp.coeFn_sub ((2:ℂ) • avoidanceMultiplier N L a t f) f, | |
| Lp.coeFn_smul (2:ℂ) (avoidanceMultiplier N L a t f), | |
| avoidanceMultiplier_ae N L a t f] with ω hsub hsmul hf | |
| rw [hsub,Pi.sub_apply,hsmul,Pi.smul_apply,hf] | |
| by_cases h : ω∈staysAllowed N L a | |
| · simp only [Set.indicator_of_mem h,smul_eq_mul] | |
| convert le_refl ‖f ω‖ using 1 | |
| congr 1 | |
| ring | |
| · simp only [Set.indicator_of_notMem h,smul_zero,zero_sub,norm_neg,le_refl] | |
| lemma free_deficit_le_twice_killed (N : ℕ) (L : Length) (a t : ℝ) (f : ScalarL2 N L) : | |
| ‖f‖^2-(inner ℂ f (freeTransition N L t f)).re ≤ | |
| 2*(‖f‖^2-(inner ℂ f (killedTransition N L a t f)).re) := by | |
| have h := re_inner_le_norm (𝕜 := ℂ) (endpointPull N L t 0 f) | |
| ((2:ℂ) • avoidanceMultiplier N L a t (endpointPull N L t 1 f)-endpointPull N L t 1 f) | |
| have hb := avoidance_reflection_norm N L a t (endpointPull N L t 1 f) | |
| have hn0 := (endpointPull N L t 0).norm_map f | |
| have hn1 := (endpointPull N L t 1).norm_map f | |
| have hi0 : inner ℂ (endpointPull N L t 0 f) (endpointPull N L t 1 f) = | |
| inner ℂ f (freeTransition N L t f) := by | |
| rw [freeTransition,ContinuousLinearMap.comp_apply,ContinuousLinearMap.adjoint_inner_right] | |
| rfl | |
| have hi1 : inner ℂ (endpointPull N L t 0 f) | |
| (avoidanceMultiplier N L a t (endpointPull N L t 1 f)) = | |
| inner ℂ f (killedTransition N L a t f) := by | |
| rw [killedTransition,ContinuousLinearMap.comp_apply,ContinuousLinearMap.comp_apply, | |
| ContinuousLinearMap.adjoint_inner_right] | |
| rfl | |
| rw [inner_sub_right,inner_smul_right,hi0,hi1] at h | |
| change (2 * inner ℂ f (killedTransition N L a t f)-inner ℂ f (freeTransition N L t f)).re≤_ at h | |
| rw [Complex.sub_re,Complex.mul_re] at h | |
| norm_num at h | |
| simp only [LinearIsometry.norm_map] at hb | |
| have hmul := mul_le_mul_of_nonneg_left hb (norm_nonneg f) | |
| nlinarith | |
| end HardSphere | |
| end | |
| end OAI | |