| import Mathlib |
| import OAI.Probability.LatticeSurface.Dynamics.ProjectionPath |
|
|
| namespace OAI |
|
|
| namespace ProjectionComparison |
|
|
| open Set |
|
|
| variable {V E : Type*} |
|
|
| def faceGraph (S : System V E) (F : Finset E) : SimpleGraph V where |
| Adj x y := x≠y ∧ ∃ e,e∈F ∧ ((S.left e=x ∧ S.right e=y) ∨ (S.left e=y ∧ S.right e=x)) |
| symm := ⟨by rintro x y ⟨hne,e,he,hor⟩;exact ⟨hne.symm,e,he,hor.symm⟩⟩ |
| loopless := ⟨by intro x hx;exact hx.1 rfl⟩ |
|
|
| noncomputable def faceComponent [Fintype V] (S : System V E) (F : Finset E) (x : V) : Finset V := |
| by classical exact {y | (faceGraph S F).Reachable x y}.toFinset |
|
|
| noncomputable def faceAverage [Fintype V] (S : System V E) (F : Finset E) (d : V → ℝ) (x : V) : ℝ := by |
| classical exact if ∀ p∈S.pins,¬(faceGraph S F).Reachable x p then |
| (∑y∈faceComponent S F x,d y)/(faceComponent S F x).card else 0 |
|
|
| end ProjectionComparison |
|
|
| namespace ProjectionComparison |
|
|
| open Set MeasureTheory |
|
|
| variable {V E : Type*} [Fintype E] |
|
|
| noncomputable def activePattern (S : System V E) (h : V → ℝ) : Finset E := |
| by classical exact Finset.univ.filter (fun e => h (S.left e)-h (S.right e)=S.cost e) |
|
|
| lemma mem_activePattern (S : System V E) (h : V → ℝ) (e : E) : |
| e∈activePattern S h ↔ h (S.left e)-h (S.right e)=S.cost e := by classical simp [activePattern] |
|
|
| lemma faceGraph_pattern (S : System V E) (h : V → ℝ) : |
| faceGraph S (activePattern S h)=activeGraph S h := by |
| ext x y |
| simp only [faceGraph,activeGraph,mem_activePattern] |
|
|
| lemma faceComponent_pattern [Fintype V] (S : System V E) (h : V → ℝ) (x : V) : |
| faceComponent S (activePattern S h) x=activeComponent S h x := by |
| classical |
| simp only [faceComponent,activeComponent,faceGraph_pattern] |
|
|
| lemma measurableSet_activePattern (S : System V E) (H : ℝ → V → ℝ) |
| (hc : ∀ x,Continuous (fun t => H t x)) (F : Finset E) : |
| MeasurableSet {t | activePattern S (H t)=F} := by |
| classical |
| have hset : {t | activePattern S (H t)=F} = ⋂ e : E,{t | (H t (S.left e)-H t (S.right e)=S.cost e) ↔ e∈F} := by |
| ext t |
| simp only [mem_ofPred_eq,mem_iInter] |
| exact ⟨fun he e => by rw [← mem_activePattern,he],fun he => Finset.ext fun e => by rw [mem_activePattern];exact he e⟩ |
| rw [hset] |
| apply MeasurableSet.iInter |
| intro e |
| have hm : MeasurableSet {t | H t (S.left e)-H t (S.right e)=S.cost e} := |
| ((hc (S.left e)).sub (hc (S.right e))).measurable (measurableSet_singleton (S.cost e)) |
| by_cases he : e∈F |
| · simpa only [he,iff_true] using hm |
| · simpa only [he,iff_false,Set.compl_ofPred] using hm.compl |
|
|
| noncomputable def patternWeight (S : System V E) (H : ℝ → V → ℝ) (F : Finset E) (t : ℝ) : ℝ := |
| by classical exact if activePattern S (H t)=F then 1 else 0 |
|
|
| lemma measurable_patternWeight (S : System V E) (H : ℝ → V → ℝ) |
| (hc : ∀ x,Continuous (fun t => H t x)) (F : Finset E) : |
| Measurable (patternWeight S H F) := by |
| classical |
| exact Measurable.ite (measurableSet_activePattern S H hc F) measurable_const measurable_const |
|
|
| lemma intervalIntegrable_patternWeight (S : System V E) (H : ℝ → V → ℝ) |
| (hc : ∀ x,Continuous (fun t => H t x)) (F : Finset E) : |
| IntervalIntegrable (patternWeight S H F) volume 0 1 := by |
| have hi : IntervalIntegrable (fun _ : ℝ => (1:ℝ)) volume 0 1 := intervalIntegrable_const |
| apply hi.mono_fun (measurable_patternWeight S H hc F).aestronglyMeasurable |
| apply Filter.Eventually.of_forall |
| intro t |
| classical |
| simp only [patternWeight] |
| split_ifs <;> norm_num |
|
|
| lemma sum_patternWeight (S : System V E) (H : ℝ → V → ℝ) (t : ℝ) : |
| (∑ F,patternWeight S H F t)=1 := by |
| classical |
| simp [patternWeight,eq_comm] |
|
|
| lemma sum_patternWeight_mul (S : System V E) (H : ℝ → V → ℝ) (t : ℝ) (a : Finset E → ℝ) : |
| (∑ F,patternWeight S H F t*a F)=a (activePattern S (H t)) := by |
| classical |
| simp [patternWeight,eq_comm] |
|
|
| end ProjectionComparison |
|
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| end OAI |
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|
