| import Mathlib |
| import OAI.Probability.LatticeSurface.Dynamics.ProjectionFaces |
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|
| namespace OAI |
|
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| namespace ProjectionComparison |
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|
| open Set |
|
|
| variable {V E : Type*} [Fintype V] [Fintype E] |
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| omit [Fintype V] [Fintype E] in |
| lemma active_values_constant (S : System V E) (h d : V → ℝ) |
| (he : ∀ e,h (S.left e)-h (S.right e)=S.cost e → d (S.left e)=d (S.right e)) |
| (x y : V) (hxy : (activeGraph S h).Reachable x y) : d x=d y := by |
| obtain ⟨p⟩ := hxy |
| induction p with |
| | nil => rfl |
| | @cons x z y hxz p ih => |
| rcases hxz with ⟨_,e,heh,hor⟩ |
| have hek := he e heh |
| have hxz : d x=d z := by |
| rcases hor with ⟨rfl,rfl⟩ | ⟨rfl,rfl⟩ |
| · exact hek |
| · exact hek.symm |
| exact hxz.trans ih |
|
|
| omit [Fintype V] in |
| lemma active_eventually_subset (S : System V E) (H : ℝ → V → ℝ) (t : ℝ) |
| (hc : ∀ x,ContinuousAt (fun s => H s x) t) : |
| ∀ᶠ s in nhds t,∀ e,H s (S.left e)-H s (S.right e)=S.cost e → H t (S.left e)-H t (S.right e)=S.cost e := by |
| apply Filter.eventually_all.mpr |
| intro e |
| by_cases he : H t (S.left e)-H t (S.right e)=S.cost e |
| · exact Filter.Eventually.of_forall (fun _ _ => he) |
| · have hne := ((hc (S.left e)).sub (hc (S.right e))).eventually (isOpen_ne.mem_nhds he) |
| filter_upwards [hne] with s hs |
| exact fun h => (hs h).elim |
|
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| end ProjectionComparison |
|
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| namespace ProjectionComparison |
|
|
| open Set |
|
|
| variable {V E : Type*} [Fintype V] [Fintype E] |
|
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| theorem projection_derivative (S : System V E) (B U H : ℝ → V → ℝ) (dB dU dH : V → ℝ) (t : ℝ) |
| (hp : ∀ᶠ s in nhds t,IsProjection S (B s) (U s) (H s)) |
| (hb : ∀ x,HasDerivAt (fun s => B s x) (dB x) t) |
| (hu : ∀ x,HasDerivAt (fun s => U s x) (dU x) t) |
| (hh : ∀ x,HasDerivAt (fun s => H s x) (dH x) t) : |
| (∀ x p,p∈S.pins → (activeGraph S (H t)).Reachable x p → dH x=dB p) ∧ |
| (∀ x,(∀ p∈S.pins,¬(activeGraph S (H t)).Reachable x p) → |
| dH x=(∑ y∈activeComponent S (H t) x,dU y)/(activeComponent S (H t) x).card) := by |
| classical |
| have hed (e : E) (he : H t (S.left e)-H t (S.right e)=S.cost e) : dH (S.left e)=dH (S.right e) := by |
| have hmax : IsLocalMax (fun s => H s (S.left e)-H s (S.right e)) t := by |
| filter_upwards [hp] with s hs |
| rw [he] |
| exact hs.1.1 e |
| exact sub_eq_zero.mp (hmax.hasDerivAt_eq_zero ((hh (S.left e)).sub (hh (S.right e)))) |
| have hpin (p : V) (hp' : p∈S.pins) : dH p=dB p := by |
| have heq : (fun s => H s p)=ᶠ[nhds t](fun s => B s p) := hp.mono (fun s hs => hs.1.2 p hp') |
| exact ((hh p).congr_of_eventuallyEq heq.symm).unique (hb p) |
| constructor |
| · intro x p hp' hxp |
| exact (active_values_constant S (H t) dH hed x p hxp).trans (hpin p hp') |
| · intro x hground |
| let w : V → ℝ := fun y => if (activeGraph S (H t)).Reachable x y then 1 else 0 |
| have hw : ∀ p∈S.pins,w p=0 := by intro p hp';exact ite_eq_right (hground p hp') |
| have hedge (e : E) (he : H t (S.left e)-H t (S.right e)=S.cost e) : w (S.left e)=w (S.right e) := by |
| have hr := active_edge_reachable S (H t) e he |
| have hi : (activeGraph S (H t)).Reachable x (S.left e) ↔ (activeGraph S (H t)).Reachable x (S.right e) := |
| ⟨fun hx => hx.trans hr,fun hx => hx.trans hr.symm⟩ |
| simp only [w,hi] |
| have heq : (fun s => ∑ y,(H s y-U s y)*w y)=ᶠ[nhds t](fun _ => (0:ℝ)) := by |
| filter_upwards [hp,active_eventually_subset S H t (fun x => (hh x).continuousAt)] with s hs ha |
| exact tangent_balance S (B s) (U s) (H s) w hs hw (fun e he => hedge e (ha e he)) |
| have hd : HasDerivAt (fun s => ∑ y,(H s y-U s y)*w y) (∑ y,(dH y-dU y)*w y) t := by |
| apply HasDerivAt.fun_sum |
| intro y hy |
| exact ((hh y).sub (hu y)).mul_const (w y) |
| have hzero : (∑ y,(dH y-dU y)*w y)=0 := (hd.congr_of_eventuallyEq heq.symm).unique (hasDerivAt_const t 0) |
| have hbal : (∑ y∈activeComponent S (H t) x,(dH y-dU y))=0 := by |
| calc |
| _ = ∑ y∈activeComponent S (H t) x,(dH y-dU y)*w y := by |
| apply Finset.sum_congr rfl |
| intro y hy |
| simp only [w,ite_eq_left ((mem_activeComponent S (H t) x y).mp hy),mul_one] |
| _ = ∑ y,(dH y-dU y)*w y := by |
| apply Finset.sum_subset (Finset.subset_univ _) |
| intro y hy hn |
| have hn' : ¬(activeGraph S (H t)).Reachable x y := by simpa only [mem_activeComponent] using hn |
| simp only [w,ite_eq_right hn',mul_zero] |
| _ = 0 := hzero |
| have hconst : (∑ y∈activeComponent S (H t) x,dH y)=(activeComponent S (H t) x).card*dH x := by |
| trans ∑ _y∈activeComponent S (H t) x,dH x |
| · apply Finset.sum_congr rfl |
| intro y hy |
| exact (active_values_constant S (H t) dH hed x y ((mem_activeComponent S (H t) x y).mp hy)).symm |
| · simp only [Finset.sum_const,nsmul_eq_mul] |
| have hx : x∈activeComponent S (H t) x := (mem_activeComponent S (H t) x x).mpr (.refl x) |
| have hcard : (0:ℝ)<(activeComponent S (H t) x).card := Nat.cast_pos.mpr (Finset.card_pos.mpr ⟨x,hx⟩) |
| apply (eq_div_iff hcard.ne').mpr |
| rw [Finset.sum_sub_distrib,hconst] at hbal |
| linarith |
|
|
| end ProjectionComparison |
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| end OAI |
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