File size: 2,758 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 | import OAI.Probability.MatroidProphet.Pivots.OccurrenceSupport
namespace OAI
namespace MatroidProphet
namespace Pivots
open Set Finset
variable {α β : Type*} [Fintype β] [DecidableEq β]
lemma exists_pivot
{α : Type u_1} {β : Type u_2} [Fintype β] [DecidableEq β]
(M : Matroid α) (label : β → α) (time : β → ℕ) {e : α}
(hnonloop : e ∉ M.closure ∅) (hspan : e ∈ M.closure (Set.range label)) :
∃ o, IsPivot M label time e o := by
classical
let P := fun k => e ∈ M.closure (label '' {o | time o ≤ k})
have hex : ∃ k, P k := by
refine ⟨Finset.univ.sup time, M.closure_subset_closure ?_ hspan⟩
rintro _ ⟨o, rfl⟩
exact ⟨o, by
change time o ≤ Finset.univ.sup time
exact Finset.le_sup (Finset.mem_univ o), rfl⟩
let k := Nat.find hex
have hk : P k := Nat.find_spec hex
have hnot : e ∉ M.closure (label '' {o | time o < k}) := by
intro hn
by_cases hk0 : k = 0
· exact hnonloop (by simpa [hk0] using hn)
have hprev : P (k - 1) := by
apply M.closure_subset_closure ?_ hn
rintro _ ⟨o, ho, rfl⟩
exact ⟨o, by
change time o ≤ k - 1
change time o < k at ho
omega, rfl⟩
have hmin : k ≤ k - 1 := Nat.find_min' hex hprev
omega
have hat : ∃ o, time o = k := by
by_contra h
apply hnot
apply M.closure_subset_closure ?_ hk
rintro _ ⟨o, ho, rfl⟩
refine ⟨o, ?_, rfl⟩
have hne : time o ≠ k := fun heq => h ⟨o, heq⟩
change time o < k
change time o ≤ k at ho
omega
obtain ⟨o, ho⟩ := hat
exact ⟨o, by simpa [IsPivot, ho] using And.intro hk hnot⟩
lemma pivot_unique
{α : Type u_1} {β : Type u_2} [Fintype β] [DecidableEq β]
(M : Matroid α) (label : β → α) (time : β → ℕ)
(htime : Function.Injective time) {e : α} {o p : β}
(ho : IsPivot M label time e o) (hp : IsPivot M label time e p) : o = p := by
apply htime
apply le_antisymm
· by_contra h
apply ho.2
apply M.closure_subset_closure ?_ hp.1
rintro _ ⟨a, ha, rfl⟩
exact ⟨a, lt_of_le_of_lt ha (Nat.lt_of_not_ge h), rfl⟩
· by_contra h
apply hp.2
apply M.closure_subset_closure ?_ ho.1
rintro _ ⟨a, ha, rfl⟩
exact ⟨a, lt_of_le_of_lt ha (Nat.lt_of_not_ge h), rfl⟩
lemma existsUnique_pivot (M : Matroid α) (label : β → α) (time : β → ℕ)
(htime : Function.Injective time) {e : α}
(hnonloop : e ∉ M.closure ∅) (hspan : e ∈ M.closure (Set.range label)) :
∃! o, IsPivot M label time e o := by
obtain ⟨o, ho⟩ := exists_pivot M label time hnonloop hspan
exact ⟨o, ho, fun p hp => pivot_unique M label time htime hp ho⟩
end Pivots
end MatroidProphet
end OAI
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