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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 | 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 // S.left e∈W ∧ S.right e∈W} where
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
end OAI
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