openAI-math-dataset / lean /OAI /Probability /LatticeSurface /Dynamics /ProjectionComponentLimitSupport.lean
| import Mathlib | |
| import OAI.Probability.LatticeSurface.Dynamics.ProjectionComponentMesh | |
| import OAI.Probability.LatticeSurface.Dynamics.ProjectionLimitSupport | |
| namespace OAI | |
| namespace ProjectionComparison | |
| open Set Filter ClusterDisplacement | |
| attribute [local instance] Classical.propDecidable | |
| variable {V E : Type*} [Fintype V] | |
| lemma pathMesh_grid (S : System V E) (b h : V → ℝ) (hh : h∈feasible S b) | |
| (w : ℝ → FreeSpace S) (N : ℕ) (hN : 0<N) (i : ℕ) : | |
| pathMesh S b h hh w N ((i:ℝ)/N)=recursion S b ⟨h,hh⟩ h (pathIncrement S w N) i := by | |
| have hN0 : (N:ℝ)≠0 := Nat.cast_ne_zero.mpr (Nat.ne_of_gt hN) | |
| simp [pathMesh,div_mul_cancel₀ (i:ℝ) hN0] | |
| lemma pathMesh_cut_eventually (S : System V E) (W : Set V) | |
| (b h : V → ℝ) (hh : h∈feasible S b) (w : ℝ → FreeSpace S) (u : ℝ → V → ℝ) | |
| (hlim : TendstoUniformlyOn (fun n => pathMesh S b h hh w (n+1)) u atTop (Icc 0 1)) | |
| (δ : ℝ) (hδ : 0<δ) | |
| (hcut : ∀t∈Icc (0:ℝ) 1,∀e,S.cost e-(u t (S.left e)-u t (S.right e))≤δ → | |
| S.left e∉S.pins → S.right e∉S.pins → (S.left e∈W ↔ S.right e∈W)) : | |
| ∀ᶠ n in atTop,∀i<n+1,∀e, | |
| recursion S b ⟨h,hh⟩ h (pathIncrement S w (n+1)) (i+1) (S.left e)- | |
| recursion S b ⟨h,hh⟩ h (pathIncrement S w (n+1)) (i+1) (S.right e)=S.cost e → | |
| S.left e∉S.pins → S.right e∉S.pins → (S.left e∈W ↔ S.right e∈W) := by | |
| filter_upwards [Metric.tendstoUniformlyOn_iff.mp hlim (δ/4) (by linarith)] with n hn i hi e he hl hr | |
| have hN : (0:ℝ)<((n+1:ℕ):ℝ) := by positivity | |
| have ht : ((i+1:ℕ):ℝ)/((n+1:ℕ):ℝ)∈Icc (0:ℝ) 1 := by | |
| constructor | |
| · positivity | |
| · apply (div_le_one hN).mpr | |
| exact_mod_cast (Nat.succ_le_of_lt hi) | |
| have hb (x : V) : |recursion S b ⟨h,hh⟩ h (pathIncrement S w (n+1)) (i+1) x-u (((i+1:ℕ):ℝ)/((n+1:ℕ):ℝ)) x|<δ/4 := by | |
| have hhx := (dist_le_pi_dist _ _ x).trans_lt (hn _ ht) | |
| rw [pathMesh_grid S b h hh w (n+1) (Nat.succ_pos n)] at hhx | |
| simpa only [Real.dist_eq,abs_sub_comm] using hhx | |
| apply hcut _ ht e _ hl hr | |
| have hbl := abs_lt.mp (hb (S.left e)) | |
| have hbr := abs_lt.mp (hb (S.right e)) | |
| linarith | |
| end ProjectionComparison | |
| end OAI | |