| import Mathlib | |
| import OAI.Probability.LatticeSurface.Convexity.FreeHilbert | |
| import OAI.Probability.LatticeSurface.Dynamics.ProjectionMesh | |
| import OAI.Probability.LatticeSurface.Skorokhod.SkorokhodRegularity | |
| namespace OAI | |
| namespace ProjectionComparison | |
| open Set ClusterDisplacement | |
| attribute [local instance] Classical.propDecidable | |
| variable {V E : Type*} [Fintype V] [Fintype E] | |
| noncomputable def pathIncrement (S : System V E) (w : ℝ → FreeSpace S) (N : ℕ) (i : ℕ) : V → ℝ := | |
| extendFree S (fun _ => 0) (w ((i+1:ℕ)/(N:ℝ))-w ((i:ℝ)/N)) | |
| noncomputable def pathMesh (S : System V E) (b h : V → ℝ) (hh : h∈feasible S b) | |
| (w : ℝ → FreeSpace S) (N : ℕ) (t : ℝ) : V → ℝ := | |
| recursion S b ⟨h,hh⟩ h (pathIncrement S w N) ⌊t*N⌋₊ | |
| def pathTail (S : System V E) (u₀ u₁ : ℝ → V → ℝ) (a t : ℝ) : {x // x∉S.pins} → ℝ := | |
| fun x => max (u₁ t x-u₀ t x-a) 0 | |
| end ProjectionComparison | |
| namespace ProjectionComparison | |
| open Set Filter ClusterDisplacement | |
| attribute [local instance] Classical.propDecidable | |
| variable {V E : Type*} [Fintype V] [Fintype E] | |
| omit [Fintype V] [Fintype E] in | |
| lemma coordinates_extendFree (S : System V E) (b : V → ℝ) (v : FreeSpace S) : | |
| freeCoordinates S (extendFree S b v)=v := by | |
| apply PiLp.ext | |
| intro x | |
| simp [freeCoordinates,extendFree,x.property] | |
| omit [Fintype E] in | |
| lemma pathMesh_eq_hilbert (S : System V E) (b h : V → ℝ) (hh : h∈feasible S b) | |
| (B : ConvexSkorokhod.Body (FreeSpace S)) | |
| (he : ∀v:V → ℝ,extendFree S b (ConvexSkorokhod.project B (freeCoordinates S v))=project S b ⟨h,hh⟩ v) | |
| (w : ℝ → FreeSpace S) (N : ℕ) (t : ℝ) : | |
| pathMesh S b h hh w N t= | |
| extendFree S b (ConvexSkorokhod.mesh B (freeCoordinates S h) w N t) := by | |
| have hne : (feasible S b).Nonempty := ⟨h,hh⟩ | |
| have hi : ∀i,recursion S b hne h (pathIncrement S w N) i= | |
| extendFree S b (ConvexSkorokhod.walk B (freeCoordinates S h) (ConvexSkorokhod.sampledDriver w N) i) := by | |
| intro i | |
| induction i with | |
| | zero => exact (extendFree_coordinates S b h hh.2).symm | |
| | succ i ih => | |
| rw [ProjectionComparison.recursion,←he,ConvexSkorokhod.walk,ih] | |
| apply congrArg (extendFree S b) | |
| apply congrArg (ConvexSkorokhod.project B) | |
| apply PiLp.ext | |
| intro x | |
| simp [freeCoordinates,extendFree,x.property,pathIncrement,ConvexSkorokhod.sampledDriver] | |
| exact hi _ | |
| omit [Fintype E] in | |
| lemma extendFree_lipschitz (S : System V E) (b : V → ℝ) : LipschitzWith 1 (extendFree S b) := by | |
| apply LipschitzWith.of_dist_le_mul | |
| intro u v | |
| simp only [NNReal.coe_one,one_mul,dist_eq_norm] | |
| apply (pi_norm_le_iff_of_nonneg (norm_nonneg (u-v))).mpr | |
| intro x | |
| by_cases hx : x∈S.pins | |
| · simp only [Pi.sub_apply,extendFree,dite_eq_left hx,sub_self,norm_zero] | |
| exact norm_nonneg _ | |
| · simpa only [Pi.sub_apply,extendFree,dite_eq_right hx,PiLp.sub_apply] using PiLp.norm_apply_le (u-v) ⟨x,hx⟩ | |
| omit [Fintype E] in | |
| lemma exists_projection_path (S : System V E) (b h : V → ℝ) (hh : h∈feasible S b) | |
| (c : FreeSpace S) (r : ℝ) (hr : 0 < r) (hball : Metric.closedBall c r⊆freeCarrier S b) | |
| (w : ℝ → FreeSpace S) (hw : ContinuousOn w (Icc 0 1)) | |
| (hosc : ∀t∈Icc (0:ℝ) 1,‖w t-w 0‖ ≤ r/2) : | |
| ∃u : ℝ → V → ℝ,ContinuousOn u (Icc 0 1) ∧ u 0=h ∧ | |
| (∀t∈Icc (0:ℝ) 1,u t∈feasible S b) ∧ | |
| TendstoUniformlyOn (fun n => pathMesh S b h hh w (n+1)) u atTop (Icc 0 1) := by | |
| obtain ⟨B,hB,he⟩ := exists_free_body S b ⟨h,hh⟩ | |
| have hh' : freeCoordinates S h∈B.carrier := by | |
| rw [hB];change extendFree S b (freeCoordinates S h)∈feasible S b | |
| rwa [extendFree_coordinates S b h hh.2] | |
| obtain ⟨v,hvcont,hv,hvmem,hvvar⟩ := ConvexSkorokhod.projection_mesh_regular_limit B | |
| (freeCoordinates S h) c r hh' hr (by rwa [hB]) w hw hosc | |
| let u := fun t => extendFree S b (v t) | |
| have hlim : TendstoUniformlyOn (fun n => pathMesh S b h hh w (n+1)) u atTop (Icc 0 1) := by | |
| apply Metric.tendstoUniformlyOn_iff.mpr | |
| intro ε hε | |
| filter_upwards [Metric.tendstoUniformlyOn_iff.mp hv ε hε] with n hn t ht | |
| rw [pathMesh_eq_hilbert S b h hh B he] | |
| exact ((extendFree_lipschitz S b).dist_le_mul _ _).trans_lt (by simpa using hn t ht) | |
| refine ⟨u,(extendFree_continuous S b).comp_continuousOn hvcont,?_,fun t ht => ?_,hlim⟩ | |
| · have hhlim := hlim.tendsto_at (show (0:ℝ)∈Icc (0:ℝ) 1 by constructor <;> norm_num) | |
| have hhmesh : (fun n => pathMesh S b h hh w (n+1) 0)=fun _ : ℕ => h := by | |
| funext n;simp [pathMesh,recursion] | |
| rw [hhmesh] at hhlim | |
| exact tendsto_nhds_unique hhlim tendsto_const_nhds | |
| · have hmem := hvmem t ht | |
| rwa [hB] at hmem | |
| end ProjectionComparison | |
| end OAI | |