| import Mathlib |
| import OAI.Probability.LatticeSurface.Dynamics.ProjectionMesh |
|
|
| namespace OAI |
|
|
| namespace ProjectionComparison |
|
|
| open Set Filter |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| variable {V E : Type*} [Fintype V] [Fintype E] |
|
|
| def restrictedSystem (S : System V E) (W : Set V) : |
| System W {e : E |
| left e := ⟨S.left e,e.property.1⟩ |
| right e := ⟨S.right e,e.property.2⟩ |
| cost e := S.cost e |
| pins := {x | x.val∈S.pins} |
|
|
| omit [Fintype V] [Fintype E] in |
| lemma feasible_restrict (S : System V E) (W : Set V) (b h : V → ℝ) |
| (hh : h∈feasible S b) : |
| (fun x : W => h x)∈feasible (restrictedSystem S W) (fun x => b x) := |
| ⟨fun e => hh.1 e,fun x hx => hh.2 x hx⟩ |
|
|
| end ProjectionComparison |
|
|
| namespace ProjectionComparison |
|
|
| open Set Filter |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| variable {V E : Type*} [Fintype V] [Fintype E] |
|
|
| lemma one_sided_tangent_variational (S : System V E) (b v h d : V → ℝ) |
| (hp : IsProjection S b v h) (hpin : ∀x∈S.pins,d x=0) |
| (hedge : ∀e,h (S.left e)-h (S.right e)=S.cost e → d (S.left e)-d (S.right e)≤0) : |
| 0 ≤ ∑x,(h x-v x)*d x := by |
| have hev : ∀ᶠ t in nhds (0:ℝ),0≤t → (fun x => h x+t*d x)∈feasible S b := by |
| have he : ∀ᶠ t in nhds (0:ℝ),∀e,0≤t → |
| (h (S.left e)+t*d (S.left e))-(h (S.right e)+t*d (S.right e))≤S.cost e := by |
| apply Filter.eventually_all.mpr |
| intro e |
| by_cases ha : h (S.left e)-h (S.right e)=S.cost e |
| · exact Filter.Eventually.of_forall (fun t ht => by nlinarith [mul_nonpos_of_nonneg_of_nonpos ht (hedge e ha)]) |
| · have hs : h (S.left e)-h (S.right e)<S.cost e := lt_of_le_of_ne (hp.1.1 e) ha |
| have hc : Continuous (fun t : ℝ => (h (S.left e)+t*d (S.left e))-(h (S.right e)+t*d (S.right e))) := by fun_prop |
| have he0 : (h (S.left e)+(0:ℝ)*d (S.left e))-(h (S.right e)+0*d (S.right e))<S.cost e := by simpa using hs |
| exact (hc.continuousAt.eventually (gt_mem_nhds he0)).mono fun t ht _ => ht.le |
| filter_upwards [he] with t ht hn |
| exact ⟨fun e => ht e hn,fun x hx => by simp [hpin x hx,hp.1.2 x hx]⟩ |
| obtain ⟨ε,hε,hball⟩ := Metric.eventually_nhds_iff.mp hev |
| have ht : 0<ε/2 := by linarith |
| have hv := variational S b v h hp (fun x => h x+(ε/2)*d x) |
| (hball (by simpa only [Real.dist_eq,sub_zero,abs_of_pos ht] using (show ε/2<ε by linarith)) ht.le) |
| have hid : (∑x,(h x-v x)*((h x+(ε/2)*d x)-h x))=(ε/2)*∑x,(h x-v x)*d x := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl;intro x hx;ring |
| rw [hid] at hv |
| exact nonneg_of_mul_nonneg_right hv ht |
|
|
| end ProjectionComparison |
|
|
| namespace ProjectionComparison |
|
|
| open Set Filter |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| variable {V E : Type*} [Fintype V] [Fintype E] |
|
|
| theorem projection_restrict (S : System V E) (W : Set V) (hW : S.pins⊆W) |
| (b v h : V → ℝ) (hp : IsProjection S b v h) |
| (hcut : ∀e,h (S.left e)-h (S.right e)=S.cost e → |
| S.left e∉S.pins → S.right e∉S.pins → (S.left e∈W ↔ S.right e∈W)) : |
| IsProjection (restrictedSystem S W) (fun x => b x) (fun x => v x) (fun x => h x) := by |
| refine ⟨feasible_restrict S W b h hp.1,?_⟩ |
| intro k hk |
| let d : V → ℝ := fun x => if hx : x∈W then k ⟨x,hx⟩-h x else 0 |
| have hd (x : W) : d x=k x-h x := by simp [d,x.property] |
| have hpins : ∀x∈S.pins,d x=0 := by |
| intro x hx |
| have hxW := hW hx |
| rw [show d x=k ⟨x,hxW⟩-h x by simp [d,hxW]] |
| rw [hk.2 ⟨x,hxW⟩ hx,hp.1.2 x hx,sub_self] |
| have hdir (e : E) (ha : h (S.left e)-h (S.right e)=S.cost e) : |
| d (S.left e)-d (S.right e)≤0 := by |
| by_cases hl : S.left e∈W <;> by_cases hr : S.right e∈W |
| · have hkedge := hk.1 ⟨e,hl,hr⟩ |
| change k ⟨S.left e,hl⟩-k ⟨S.right e,hr⟩≤S.cost e at hkedge |
| simp only [d,dite_eq_left hl,dite_eq_left hr] |
| linarith |
| · have hpr : S.right e∉S.pins := fun hp => hr (hW hp) |
| have hpl : S.left e∈S.pins := by |
| by_contra hn |
| exact hr ((hcut e ha hn hpr).mp hl) |
| rw [hpins _ hpl] |
| simp [d,hr] |
| · have hpl : S.left e∉S.pins := fun hp => hl (hW hp) |
| have hpr : S.right e∈S.pins := by |
| by_contra hn |
| exact hl ((hcut e ha hpl hn).mpr hr) |
| rw [hpins _ hpr] |
| simp [d,hl] |
| · simp [d,hl,hr] |
| have hv := one_sided_tangent_variational S b v h d hp hpins hdir |
| have he : (∑x,(h x-v x)*d x)=∑x : W,(h x-v x)*(k x-h x) := by |
| apply Finset.sum_congr_set W |
| · intro x hx;rw [show d x=k ⟨x,hx⟩-h x by simp [d,hx]] |
| · intro x hx;simp [d,hx] |
| rw [he] at hv |
| have hq : 0 ≤ ∑x : W,(k x-h x)^2 := Finset.sum_nonneg fun x _ => sq_nonneg _ |
| have hid : objective (fun x : W => v x) k=objective (fun x : W => v x) (fun x => h x)+ |
| 2*(∑x : W,(h x-v x)*(k x-h x))+∑x : W,(k x-h x)^2 := by |
| simp only [objective,Finset.mul_sum,←Finset.sum_add_distrib] |
| apply Finset.sum_congr rfl;intro x hx;ring |
| linarith |
|
|
| end ProjectionComparison |
|
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| end OAI |
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|
