File size: 3,589 Bytes
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 | import OAI.Probability.MatroidProphet.Algorithm.Feasible
namespace OAI
namespace MatroidProphet.MainAlgorithm
open Set Finset
variable {n : ℕ}
lemma candidate_congr (M : Matroid (Fin n)) (d : MainMasks n)
(s s' : Fin n → Option ℤ) (h : ∀ e ∈ d.H, s e = s' e) (e : Fin n) (w : Option ℤ) :
candidate M d s e w ↔ candidate M d s' e w := by
have heq : {f | f ∈ d.H ∧ higher f (s f) e w} = {f | f ∈ d.H ∧ higher f (s' f) e w} := by
ext f
by_cases hf : f ∈ d.H
· simp only [Set.mem_ofPred_eq, hf, true_and, h f hf]
· simp [hf]
simp only [candidate, heq]
lemma listed_congr (M : Matroid (Fin n)) (d : MainMasks n)
(s s' : Fin n → Option ℤ)
(hH : ∀ e ∈ d.H, s e = s' e) (hD : ∀ e ∈ d.D, s e = s' e) :
listed M d s = listed M d s' := by
classical
unfold listed
apply Finset.biUnion_congr rfl
intro e he
have hc := candidate_congr M d s s' hH e (s e)
rw [hD e he] at hc
simp only [hD e he, hc]
lemma groups_congr (M : Matroid (Fin n)) (d : MainMasks n)
(s s' : Fin n → Option ℤ)
(hH : ∀ e ∈ d.H, s e = s' e) (hD : ∀ e ∈ d.D, s e = s' e) :
groups M d s = groups M d s' := by
rw [groups, groups, listed_congr M d s s' hH hD]
lemma groupMask_congr (M : Matroid (Fin n)) (d : MainMasks n)
(s s' : Fin n → Option ℤ) (mask : Finset (Fin n))
(hH : ∀ e ∈ d.H, s e = s' e) (hD : ∀ e ∈ d.D, s e = s' e)
(hmask : ∀ e ∈ mask, s e = s' e) :
groupMask M d s mask = groupMask M d s' mask := by
funext j
ext e
by_cases he : e ∈ mask
· have hc := candidate_congr M d s s' hH e (s e)
rw [hmask e he] at hc
simp only [groupMask, Set.mem_ofPred_eq, he, true_and,
hmask e he, hc, groups_congr M d s s' hH hD]
· simp [groupMask, he]
lemma preprocessing_congr (M : Matroid (Fin n)) (hE : M.E = Set.univ) (d : MainMasks n)
(s s' : Fin n → Option ℤ) (h : ∀ e ∈ d.H ∪ d.D ∪ d.C, s e = s' e) :
finalPath M hE d s = finalPath M hE d s' ∧
birth M hE d s = birth M hE d s' ∧ assign M hE d s = assign M hE d s' := by
classical
have hH : ∀ e ∈ d.H, s e = s' e := fun e he => h e (mem_union_left _ (mem_union_left _ he))
have hD : ∀ e ∈ d.D, s e = s' e := fun e he => h e (mem_union_left _ (mem_union_right _ he))
have hC : ∀ e ∈ d.C, s e = s' e := fun e he => h e (mem_union_right _ he)
have hgd := groupMask_congr M d s s' d.D hH hD hD
have hgc := groupMask_congr M d s s' d.C hH hD hC
have hgs := groups_congr M d s s' hH hD
have hf : finalPath M hE d s = finalPath M hE d s' := by
simp only [finalPath, hgd, hgc, hgs]
have hb : birth M hE d s = birth M hE d s' := by
funext e i
simp only [birth, hgd, hgc, hgs]
refine ⟨hf, hb, ?_⟩
funext e w
have hel (i : ℤ) : eligible M hE d s e i ↔ eligible M hE d s' e i := by
simp only [eligible, hf, hb, hgs, candidate_congr M d s s' hH]
cases w <;> simp only [assign, hel, hb]
lemma observed_preprocessing (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(r : Seed (mainSeedBits n)) (w : Weights n) :
let s := fun e => roundedLevel weightBase (observed (hidden M hE) r w e)
let s' := fun e => roundedLevel weightBase (w e)
finalPath M hE (mainMasks r) s = finalPath M hE (mainMasks r) s' ∧
birth M hE (mainMasks r) s = birth M hE (mainMasks r) s' ∧
assign M hE (mainMasks r) s = assign M hE (mainMasks r) s' := by
apply preprocessing_congr
intro e he
unfold observed
have hm : e ∈ (hidden M hE).mask r := he
rw [ite_eq_left hm]
end MatroidProphet.MainAlgorithm
end OAI
|