| import OAI.Probability.GammaPassage.SpatialPositiveChunk |
|
|
| namespace OAI |
|
|
| namespace GammaFPP.CrossingBridge |
| open PlanarFPP Set |
|
|
| lemma trace_append (p : Vertex) (ds es : List Step) : |
| trace p (ds++es)=trace p ds++trace (p+displacement ds) es := by |
| induction ds generalizing p with |
| | nil => simp [trace] |
| | cons d ds ih => simp [trace,ih,add_assoc] |
|
|
| /-- A contiguous split of actual traversals is a split into actual paths. -/ |
| lemma trace_split (p : Vertex) (ds : List Step) (xs ys : List Traversal) |
| (h : trace p ds=xs++ys) : |
| ∃ pre rest : List Step, ds=pre++rest ∧ trace p pre=xs ∧ |
| trace (p+displacement pre) rest=ys := by |
| induction xs generalizing p ds with |
| | nil => exact ⟨[],ds,rfl,rfl,by simpa using h⟩ |
| | cons x xs ih => |
| cases ds with |
| | nil => simp [trace] at h |
| | cons d ds => |
| obtain ⟨hx,ht⟩ := List.cons.inj h |
| obtain ⟨pre,rest,he,hpre,hrest⟩ := ih (p+d.delta) ds ht |
| refine ⟨d::pre,rest,by simp [he],?_,?_⟩ |
| · simp only [trace,hpre,hx] |
| · simpa only [displacement_cons,add_assoc] using hrest |
|
|
| lemma trace_edges (p : Vertex) (ds : List Step) : |
| ((trace p ds).map traversalEdge).toFinset=pathEdges p ds := by |
| induction ds generalizing p with |
| | nil => rfl |
| | cons d ds ih => simp [trace,traversalEdge,pathEdges,ih] |
|
|
| lemma simple_append_edges (p : Vertex) (ds es : List Step) |
| (h : ((trace p (ds++es)).map traversalEdge).Nodup) : |
| ((trace p ds).map traversalEdge).Nodup ∧ |
| ((trace (p+displacement ds) es).map traversalEdge).Nodup ∧ |
| Disjoint (pathEdges p ds) (pathEdges (p+displacement ds) es) := by |
| rw [trace_append,List.map_append,List.nodup_append] at h |
| refine ⟨h.1,h.2.1,?_⟩ |
| rw [← trace_edges,← trace_edges] |
| exact Finset.disjoint_left.mpr (fun e he hf => |
| h.2.2 e (List.mem_toFinset.mp he) e (List.mem_toFinset.mp hf) rfl) |
|
|
| /-- Source wi:chunk-defect. Construct actual labelled negative-prefix witnesses |
| inside the original box, with disjoint physical edges and the basic bound. -/ |
| theorem spatial_negative_prefix_witnesses (w : Edge → ℝ) (hw : ∀ e, 0 ≤ w e) |
| (b m₀ am ap K η : ℝ) (ha : am ≤ ap) (hη : 0 ≤ η) |
| (k m : ℕ) (ih : ℤ×ℤ) |
| (hscale : max |m₀-((am+ap)/2)*b| |(am+ap)/2| ≤ η*(2 : ℝ)^k) |
| (p : Vertex) (ds : List Step) |
| (hin : PathIn p ds {z | z-spatialTranslate b k m ih ∈ endpointBox b 4 4 k m}) |
| (hsimple : ((trace p ds).map traversalEdge).Nodup) : |
| ∃ ws : List GammaFPP.BK.StrictPathWitness, |
| ws.Pairwise (fun q r => Disjoint q.edges r.edges) ∧ |
| (∀ q ∈ ws, q ∈ spatialWitnessFamily b m₀ am ap K η k m ih ∧ |
| q.Good w ∧ q.edges ⊆ pathEdges p ds) ∧ |
| -η*(2 : ℝ)^k-2*η*(2 : ℝ)^k*ws.length ≤ defect w b m₀ ((am+ap)/2) p ds := by |
| have hηs : 0 ≤ η*(2 : ℝ)^k := mul_nonneg hη (by positivity) |
| induction hn : ds.length using Nat.strong_induction_on generalizing p ds with |
| | h n rec => |
| by_cases hlo : -η*(2 : ℝ)^k ≤ defect w b m₀ ((am+ap)/2) p ds |
| · exact ⟨[],by simp,by simp,by simpa using hlo⟩ |
| · have hstep : ∀ t ∈ trace p ds, -(η*(2 : ℝ)^k) ≤ stepDefect w b m₀ ((am+ap)/2) t := by |
| intro t _ |
| have ht := hw (traversalEdge t) |
| have hs := (support_step_bound b m₀ ((am+ap)/2) t.2).trans hscale |
| unfold stepDefect |
| linarith |
| obtain ⟨xs,ys,he,hne,hxslo,hxshi⟩ := first_drop (stepDefect w b m₀ ((am+ap)/2)) |
| (η*(2 : ℝ)^k) (trace p ds) hstep 0 (by linarith) |
| (by rw [trace_total_defect]; linarith) |
| obtain ⟨pre,rest,heq,hpre,hrest⟩ := trace_split p ds xs ys he |
| have hpreNe : pre ≠ [] := by intro hz; subst pre; simp [trace] at hpre; exact hne hpre |
| have hlen : rest.length < n := by |
| have hh := List.length_pos_iff.mpr hpreNe |
| rw [heq,List.length_append] at hn |
| omega |
| have hsimp := simple_append_edges p pre rest (by rw [← heq]; exact hsimple) |
| have hin' : PathIn p (pre++rest) {z | z-spatialTranslate b k m ih ∈ endpointBox b 4 4 k m} := by rw [← heq]; exact hin |
| have hprelo : -2*(η*(2 : ℝ)^k) ≤ defect w b m₀ ((am+ap)/2) p pre := by |
| rw [← trace_total_defect,hpre]; simpa using hxslo |
| have hprehi : defect w b m₀ ((am+ap)/2) p pre < -η*(2 : ℝ)^k := by |
| rw [← trace_total_defect,hpre]; simpa [neg_mul] using hxshi |
| obtain ⟨q,hq,hqs,hqd,hqg⟩ := negative_midpoint_spatial_witness w b m₀ am ap K η ha k m ih p pre |
| (pathIn_append_left p pre rest _ hin') hprehi |
| have hqe : q.edges=pathEdges p pre := by change pathEdges q.start q.steps=_; rw [hqs,hqd] |
| obtain ⟨ws,hpair,hws,hbound⟩ := rec rest.length hlen (p+displacement pre) rest |
| (pathIn_append_right p pre rest _ hin') hsimp.2.1 rfl |
| refine ⟨q::ws,List.pairwise_cons.mpr ⟨?_,hpair⟩,?_,?_⟩ |
| · intro r hr |
| rw [hqe] |
| exact hsimp.2.2.mono_right (hws r hr).2.2 |
| · intro r hr |
| rw [heq,pathEdges_append] |
| rcases List.mem_cons.mp hr with rfl | hr |
| · exact ⟨hq,hqg,by rw [hqe]; exact Finset.subset_union_left⟩ |
| · exact ⟨(hws r hr).1,(hws r hr).2.1,(hws r hr).2.2.trans Finset.subset_union_right⟩ |
| · rw [heq,defect_append] |
| simp only [List.length_cons,Nat.cast_add,Nat.cast_one] |
| nlinarith |
|
|
| end GammaFPP.CrossingBridge |
|
|
| end OAI |
|
|
