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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 | import OAI.Probability.MatroidProphet.Density.Path
namespace OAI
namespace MatroidProphet
open Set
variable {α : Type*} [Fintype α]
noncomputable def densityEnabled (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D : Set α) (a k : ℤ) (P : Set α) : Set α :=
if a ≤ k then densityExpansion M hE κ D P else P
lemma densityEnabled_extensive (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D : Set α) (a k : ℤ) (P : Set α) :
P ⊆ densityEnabled M hE κ D a k P := by
by_cases hak : a ≤ k
· simpa only [densityEnabled, ite_eq_left hak] using densityExpansion_extensive M hE κ D P
· simp only [densityEnabled, ite_eq_right hak, Subset.rfl]
lemma densityEnabled_flat (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D : Set α) (a k : ℤ) {P : Set α} (hP : M.IsFlat P) :
M.IsFlat (densityEnabled M hE κ D a k P) := by
by_cases hak : a ≤ k
· simpa only [densityEnabled, ite_eq_left hak] using densityExpansion_flat M hE κ D P
· simpa only [densityEnabled, ite_eq_right hak] using hP
lemma densityEnabled_mono (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D : Set α) (a : ℤ) {k k' : ℤ} {P P' : Set α}
(hkk' : k ≤ k') (hPP' : P ⊆ P') :
densityEnabled M hE κ D a k P ⊆ densityEnabled M hE κ D a k' P' := by
by_cases hak : a ≤ k
· have hak' := hak.trans hkk'
simpa only [densityEnabled, ite_eq_left hak, ite_eq_left hak'] using
densityExpansion_mono M hE κ D hPP'
· by_cases hak' : a ≤ k'
· simpa only [densityEnabled, ite_eq_right hak, ite_eq_left hak'] using
hPP'.trans (densityExpansion_extensive M hE κ D P')
· simpa only [densityEnabled, ite_eq_right hak, ite_eq_right hak'] using hPP'
lemma densityEnabled_univ (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D : Set α) (a k : ℤ) : densityEnabled M hE κ D a k univ = univ :=
Set.eq_univ_of_univ_subset (densityEnabled_extensive M hE κ D a k univ)
def activation (h : ℕ) : ℤ := -3 * ((h : ℤ) + 1)
noncomputable def guardedPath (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) : ℕ → ℤ → Set α
| 0, k => if k < 0 then M.closure ∅ else univ
| h + 1, k =>
M.closure (densityEnabled M hE κ (D h) (activation h) k
(guardedPath M hE κ D C h k) ∪
(C h ∩ densityEnabled M hE κ (D h) (activation h) (k + 1)
(guardedPath M hE κ D C h (k + 1))))
noncomputable def nominalPath (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (k : ℤ) : Set α :=
densityEnabled M hE κ (D h) (activation h) k (guardedPath M hE κ D C h k)
lemma guardedPath_succ (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (k : ℤ) :
guardedPath M hE κ D C (h+1) k =
M.closure (nominalPath M hE κ D C h k ∪ (C h ∩ nominalPath M hE κ D C h (k+1))) := rfl
lemma guardedPath_flat (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (k : ℤ) :
M.IsFlat (guardedPath M hE κ D C h k) := by
cases h with
| zero =>
by_cases hk : k < 0
· simpa only [guardedPath, ite_eq_left hk] using Matroid.isFlat_closure (M := M) ∅
· simpa only [guardedPath, ite_eq_right hk, ← hE] using M.ground_isFlat
| succ h => exact Matroid.isFlat_closure (M := M) _
lemma nominalPath_flat (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (k : ℤ) :
M.IsFlat (nominalPath M hE κ D C h k) :=
densityEnabled_flat M hE κ (D h) (activation h) k (guardedPath_flat M hE κ D C h k)
lemma guardedPath_mono (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) : Monotone (guardedPath M hE κ D C h) := by
induction h with
| zero =>
intro i j hij
by_cases hi : i < 0
· by_cases hj : j < 0
· simp only [guardedPath, ite_eq_left hi, ite_eq_left hj, Subset.rfl]
· simp only [guardedPath, ite_eq_left hi, ite_eq_right hj, subset_univ]
· have hj : ¬ j < 0 := by omega
simp only [guardedPath, ite_eq_right hi, ite_eq_right hj, Subset.rfl]
| succ h ih =>
intro i j hij
apply M.closure_subset_closure
exact union_subset_union (densityEnabled_mono M hE κ (D h) (activation h) hij (ih hij))
(inter_subset_inter_right (C h)
(densityEnabled_mono M hE κ (D h) (activation h) (by omega) (ih (by omega))))
lemma nominalPath_mono (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) : Monotone (nominalPath M hE κ D C h) := by
intro i j hij
exact densityEnabled_mono M hE κ (D h) (activation h) hij
(guardedPath_mono M hE κ D C h hij)
lemma guarded_subset_nominal (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (k : ℤ) :
guardedPath M hE κ D C h k ⊆ nominalPath M hE κ D C h k :=
densityEnabled_extensive M hE κ (D h) (activation h) k _
lemma nominal_subset_guarded (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (k : ℤ) :
nominalPath M hE κ D C h k ⊆ guardedPath M hE κ D C (h+1) k :=
subset_union_left.trans (M.subset_closure _ (by simp [hE]))
lemma guarded_subset_next_nominal (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (k : ℤ) :
guardedPath M hE κ D C (h+1) k ⊆ nominalPath M hE κ D C h (k+1) := by
have hs : nominalPath M hE κ D C h k ∪ (C h ∩ nominalPath M hE κ D C h (k+1)) ⊆
nominalPath M hE κ D C h (k+1) :=
union_subset (nominalPath_mono M hE κ D C h (by omega)) inter_subset_right
exact (M.closure_subset_closure hs).trans_eq (nominalPath_flat M hE κ D C h (k+1)).closure
lemma guardedPath_nonnegative (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) {k : ℤ} (hk : 0 ≤ k) :
guardedPath M hE κ D C h k = univ := by
induction h generalizing k with
| zero => simp [guardedPath, not_lt_of_ge hk]
| succ h ih =>
rw [guardedPath, ih hk, ih (by omega), densityEnabled_univ, densityEnabled_univ]
simp [hE]
lemma nominalPath_nonnegative (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) {k : ℤ} (hk : 0 ≤ k) :
nominalPath M hE κ D C h k = univ := by
rw [nominalPath, guardedPath_nonnegative M hE κ D C h hk, densityEnabled_univ]
lemma guardedPath_early (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) {k : ℤ} (hk : k < -3 * (h : ℤ) - 1) :
guardedPath M hE κ D C h k = M.closure ∅ := by
induction h generalizing k with
| zero =>
have hk' : k < 0 := by omega
simp [guardedPath, hk']
| succ h ih =>
have hka : ¬ activation h ≤ k := by dsimp [activation]; push_cast at hk; omega
have hk'a : ¬ activation h ≤ k+1 := by dsimp [activation]; push_cast at hk; omega
have hkold : k < -3 * (h : ℤ) - 1 := by push_cast at hk; omega
have hk'old : k+1 < -3 * (h : ℤ) - 1 := by omega
rw [guardedPath, densityEnabled, densityEnabled, ite_eq_right hka, ite_eq_right hk'a,
ih hkold, ih hk'old]
rw [union_eq_self_of_subset_right inter_subset_right, M.closure_closure]
lemma nominalPath_early (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) {k : ℤ} (hk : k < activation h) :
nominalPath M hE κ D C h k = M.closure ∅ := by
rw [nominalPath, densityEnabled, ite_eq_right (not_le_of_gt hk)]
apply guardedPath_early
dsimp [activation] at hk
omega
lemma nominalPath_baseline (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) :
nominalPath M hE κ D C h (activation h) =
densityExpansion M hE κ (D h) (M.closure ∅) := by
rw [nominalPath, densityEnabled, ite_eq_left le_rfl, guardedPath_early]
dsimp [activation]
omega
lemma nominalPath_baseline_gap (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) :
nominalPath M hE κ D C h (activation h + 1) =
nominalPath M hE κ D C h (activation h) := by
rw [nominalPath_baseline, nominalPath, densityEnabled, ite_eq_left (by omega), guardedPath_early]
dsimp [activation]
omega
end MatroidProphet
end OAI
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