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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 | import OAI.Probability.MatroidProphet.Pivots.Basic
import OAI.Probability.MatroidProphet.Pivots.GreedyScan
namespace OAI
namespace MatroidProphet
namespace Pivots
open Set Finset
variable {α β : Type*} [Fintype β] [DecidableEq β]
lemma kept_label_injOn
{α : Type u_1} {β : Type u_2} [Fintype β] [DecidableEq β]
(M : Matroid α) (label : β → α) (time : β → ℕ)
(hground : ∀ o, label o ∈ M.E) (htime : Function.Injective time) :
Set.InjOn label (keptOccurrences M label time : Set β) := by
classical
have hd : ∀ x y, x ∈ keptOccurrences M label time → y ∈ keptOccurrences M label time →
time x < time y → label x ≠ label y := by
intro x y _ hy hxy hlabels
apply (Finset.mem_filter.mp hy).2
rw [← hlabels]
exact M.mem_closure_of_mem ⟨x, hxy, rfl⟩ (by rintro _ ⟨z, _, rfl⟩; exact hground z)
intro x hx y hy hlabels
apply htime
rcases lt_trichotomy (time x) (time y) with hlt | heq | hgt
· exact False.elim (hd x y hx hy hlt hlabels)
· exact heq
· exact False.elim (hd y x hy hx hgt hlabels.symm)
noncomputable def occurrenceSupport (M : Matroid α) (label : β → α)
(J : Finset β) (e : α) : Finset β := by
classical
exact J.filter (fun o => label o ∈ support M (label '' (J : Set β)) e)
lemma mem_occurrenceSupport
{α : Type u_1} {β : Type u_2} [Fintype β] [DecidableEq β]
{M : Matroid α} {label : β → α} {J : Finset β} {e : α} {o : β} :
o ∈ occurrenceSupport M label J e ↔
o ∈ J ∧ label o ∈ support M (label '' (J : Set β)) e := by
classical
simp [occurrenceSupport]
lemma mem_closure_iff_occurrenceSupport (M : Matroid α) (label : β → α)
(J : Finset β) (e : α) (hI : M.Indep (label '' (J : Set β)))
(he : e ∈ M.closure (label '' (J : Set β)))
(hinj : Set.InjOn label (J : Set β)) (P : β → Prop) :
e ∈ M.closure (label '' {o | o ∈ J ∧ P o}) ↔
∀ o ∈ occurrenceSupport M label J e, P o := by
classical
rw [mem_closure_iff_support_subset hI he (by
rintro _ ⟨o, ⟨ho, _⟩, rfl⟩; exact ⟨o, ho, rfl⟩)]
constructor
· intro h o ho
obtain ⟨hoJ, hol⟩ := mem_occurrenceSupport.mp ho
obtain ⟨p, ⟨hpJ, hpP⟩, hp⟩ := h hol
exact (hinj hpJ hoJ hp) ▸ hpP
· intro h x hx
obtain ⟨o, ho, rfl⟩ := support_subset he hx
exact ⟨o, ⟨ho, h o (mem_occurrenceSupport.mpr ⟨ho, hx⟩)⟩, rfl⟩
def IsPivot (M : Matroid α) (label : β → α) (time : β → ℕ) (e : α) (o : β) : Prop :=
e ∈ M.closure (label '' {p | time p ≤ time o}) ∧
e ∉ M.closure (label '' {p | time p < time o})
lemma pivot_mem_support_and_max (M : Matroid α) (label : β → α) (time : β → ℕ)
(hground : ∀ o, label o ∈ M.E) (htime : Function.Injective time)
{e : α} {o : β} (hpivot : IsPivot M label time e o) :
o ∈ occurrenceSupport M label (keptOccurrences M label time) e ∧
∀ p ∈ occurrenceSupport M label (keptOccurrences M label time) e, time p ≤ time o := by
classical
let J := keptOccurrences M label time
have hI := keptOccurrences_indep M label time hground htime
have hinj := kept_label_injOn M label time hground htime
have hup : e ∈ M.closure (label '' {p | p ∈ J ∧ time p ≤ time o}) := by
have h := hpivot.1
have hcl := greedy_prefix_closure M label time hground (time o + 1)
simp only [Nat.lt_succ_iff] at hcl
rw [hcl] at h
exact h
have he : e ∈ M.closure (label '' (J : Set β)) := by
apply M.closure_subset_closure ?_ hup
rintro _ ⟨p, ⟨hp, _⟩, rfl⟩
exact ⟨p, hp, rfl⟩
have hmax := (mem_closure_iff_occurrenceSupport M label J e hI he hinj
(fun p => time p ≤ time o)).mp hup
have hnot : ¬ ∀ p ∈ occurrenceSupport M label J e, time p < time o := by
intro h
apply hpivot.2
rw [greedy_prefix_closure M label time hground (time o)]
exact (mem_closure_iff_occurrenceSupport M label J e hI he hinj
(fun p => time p < time o)).mpr h
push Not at hnot
obtain ⟨p, hp, hge⟩ := hnot
have heq : p = o := htime (Nat.le_antisymm (hmax p hp) hge)
exact ⟨heq ▸ hp, hmax⟩
end Pivots
end MatroidProphet
end OAI
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