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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | 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
end OAI
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