File size: 5,209 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 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 | import OAI.Probability.MatroidProphet.Density.Guarded
import Mathlib.Data.Nat.Find
namespace OAI
namespace MatroidProphet
open Set
variable {α : Type*} [Fintype α]
attribute [local instance] Classical.propDecidable
def pathHorizon (h : ℕ) : ℕ := 3 * (h + 1)
lemma activation_add_horizon (h : ℕ) : activation h + (pathHorizon h : ℤ) = 0 := by
dsimp [activation, pathHorizon]
omega
lemma exists_nominal_member (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (e : α) :
∃ j : ℕ, e ∈ nominalPath M hE κ D C h (activation h + j) := by
refine ⟨pathHorizon h, ?_⟩
rw [activation_add_horizon, nominalPath_nonnegative M hE κ D C h (by omega)]
trivial
noncomputable def nominalBirth (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (e : α) : ℤ :=
activation h + Nat.find (exists_nominal_member M hE κ D C h e)
lemma nominalBirth_mem (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (e : α) :
e ∈ nominalPath M hE κ D C h (nominalBirth M hE κ D C h e) :=
Nat.find_spec (exists_nominal_member M hE κ D C h e)
lemma nominalBirth_bounds (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (e : α) :
activation h ≤ nominalBirth M hE κ D C h e ∧ nominalBirth M hE κ D C h e ≤ 0 := by
have hmem : e ∈ nominalPath M hE κ D C h (activation h + (pathHorizon h : ℤ)) := by
rw [activation_add_horizon, nominalPath_nonnegative M hE κ D C h (by omega)]
trivial
have hle := Nat.find_min' (exists_nominal_member M hE κ D C h e) hmem
have hi : (Nat.find (exists_nominal_member M hE κ D C h e) : ℤ) ≤ pathHorizon h := by
exact_mod_cast hle
have hh := activation_add_horizon h
dsimp only [nominalBirth]
omega
lemma nominalBirth_not_activation_succ (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (e : α) :
nominalBirth M hE κ D C h e ≠ activation h + 1 := by
intro heq
have hm := nominalBirth_mem M hE κ D C h e
rw [heq, nominalPath_baseline_gap] at hm
have hm0 : e ∈ nominalPath M hE κ D C h (activation h + (0 : ℕ)) := by simpa using hm
have hj := Nat.find_min' (exists_nominal_member M hE κ D C h e) hm0
have hj0 : Nat.find (exists_nominal_member M hE κ D C h e) = 0 := Nat.eq_zero_of_le_zero hj
dsimp only [nominalBirth] at heq
rw [hj0] at heq
omega
lemma nominalBirth_range (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (e : α) :
nominalBirth M hE κ D C h e = activation h ∨
(activation h + 2 ≤ nominalBirth M hE κ D C h e ∧ nominalBirth M hE κ D C h e ≤ 0) := by
have hb := nominalBirth_bounds M hE κ D C h e
have hg := nominalBirth_not_activation_succ M hE κ D C h e
omega
lemma not_mem_nominal_before_birth (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (e : α) (he : e ∉ M.closure ∅)
{k : ℤ} (hk : k < nominalBirth M hE κ D C h e) :
e ∉ nominalPath M hE κ D C h k := by
by_cases hka : k < activation h
· rwa [nominalPath_early M hE κ D C h hka]
· intro hm
have heq : activation h + ((k - activation h).toNat : ℤ) = k := by
rw [Int.toNat_of_nonneg (by omega)]
omega
have hm' : e ∈ nominalPath M hE κ D C h
(activation h + ((k - activation h).toNat : ℤ)) := by rwa [heq]
have hf := Nat.find_min' (exists_nominal_member M hE κ D C h e) hm'
have hfi : (Nat.find (exists_nominal_member M hE κ D C h e) : ℤ) ≤
((k - activation h).toNat : ℤ) := by exact_mod_cast hf
dsimp only [nominalBirth] at hk
omega
lemma mem_nominal_iff_birth_le (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (D C : ℕ → Set α) (h : ℕ) (e : α) (he : e ∉ M.closure ∅) (k : ℤ) :
e ∈ nominalPath M hE κ D C h k ↔ nominalBirth M hE κ D C h e ≤ k := by
constructor
· intro hm
by_contra hn
exact not_mem_nominal_before_birth M hE κ D C h e he (by omega) hm
· intro hk
exact nominalPath_mono M hE κ D C h hk (nominalBirth_mem M hE κ D C h e)
lemma nominalPath_generators (M : Matroid α) (hE : M.E = univ)
(κ : ℕ) (hκ : 0 < κ) (D C : ℕ → Set α) (h : ℕ) :
∃ Jtotal : Set α, Jtotal ⊆ D h ∧ κ * Jtotal.ncard ≤ (D h).ncard ∧
∃ J : ℕ → Set α, Monotone J ∧ (∀ j ≤ pathHorizon h, J j ⊆ Jtotal) ∧
(∀ j ≤ pathHorizon h,
M.closure (guardedPath M hE κ D C h (activation h + j) ∪ J j) =
nominalPath M hE κ D C h (activation h + j)) := by
have hmono : Monotone (fun j : ℕ => guardedPath M hE κ D C h (activation h + j)) := by
intro i j hij
apply guardedPath_mono M hE κ D C h
have hi : (i : ℤ) ≤ j := by exact_mod_cast hij
omega
obtain ⟨Jtotal, hJD, hcard, J, hJmono, hJJ, hgen⟩ :=
densityPath_generators M hE κ hκ (D h)
(fun j : ℕ => guardedPath M hE κ D C h (activation h + j)) hmono (pathHorizon h)
refine ⟨Jtotal, hJD, hcard, J, hJmono, hJJ, ?_⟩
intro j hj
rw [nominalPath, densityEnabled, ite_eq_left (by omega)]
exact hgen j hj
end MatroidProphet
end OAI
|