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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 | import OAI.Probability.MatroidProphet.Pivots.Blocks
namespace OAI
namespace MatroidProphet.Pivots
open Set Finset
variable {α A T : Type*} [Fintype α] [LinearOrder α] [Fintype A] [Fintype T] [LinearOrder T] {n : ℕ}
def blockLabel (F : T → Set α) (a : A → α) : BlockOld F ⊕ A → α :=
Sum.elim (blockOldLabel (F := F)) a
def blockPrefix (F : T → Set α) (a : A → α) (τ : A → T) (k : T) : Set α :=
blockLabel F a '' {o | blockStage F τ o ≤ k}
def blockBefore (F : T → Set α) (a : A → α) (τ : A → T) (k : T) : Set α :=
blockLabel F a '' {o | blockStage F τ o < k}
noncomputable def blockBirthPattern (M : Matroid α) (F : T → Set α)
(a : A → α) (τ : A → T) (test : Fin n → α) (mark : T → Bool) : Finset (Fin n) := by
classical
exact Finset.univ.filter fun f => ∃ k, mark k = true ∧
test f ∈ M.closure (blockPrefix F a τ k) ∧ test f ∉ M.closure (blockBefore F a τ k)
lemma pivot_block_birth (M : Matroid α) (F : T → Set α) (a : A → α)
(τ : A → T) {e : α} {o : BlockOld F ⊕ A}
(hp : IsPivot M (blockLabel F a) (blockTime F τ) e o) :
e ∈ M.closure (blockPrefix F a τ (blockStage F τ o)) ∧
e ∉ M.closure (blockBefore F a τ (blockStage F τ o)) := by
constructor
· apply M.closure_subset_closure (Set.image_mono ?_) hp.1
exact fun p h => blockTime_le_implies_stage_le F τ h
· intro h
apply hp.2
apply M.closure_subset_closure (Set.image_mono ?_) h
exact fun p h => stage_lt_implies_blockTime_lt F τ h
lemma block_birth_unique
{α : Type u_1} {A : Type u_2} {T : Type u_3}
[Fintype α] [LinearOrder α] [Fintype A] [Fintype T] [LinearOrder T]
(M : Matroid α) (F : T → Set α) (a : A → α)
(τ : A → T) {e : α} {k l : T}
(hk : e ∈ M.closure (blockPrefix F a τ k) ∧ e ∉ M.closure (blockBefore F a τ k))
(hl : e ∈ M.closure (blockPrefix F a τ l) ∧ e ∉ M.closure (blockBefore F a τ l)) :
k = l := by
rcases lt_trichotomy k l with h | rfl | h
· exact (hl.2 (M.closure_subset_closure (Set.image_mono (fun _ hx => lt_of_le_of_lt hx h)) hk.1)).elim
· rfl
· exact (hk.2 (M.closure_subset_closure (Set.image_mono (fun _ hx => lt_of_le_of_lt hx h)) hl.1)).elim
lemma blockBirthPattern_mem_freeMarkedPatterns (M : Matroid α) (F : T → Set α)
(a : A → α) (τ : A → T) (test : Fin n → α) (mark : T → Bool)
(hnonloop : ∀ f, test f ∉ M.closure ∅)
(hspan : ∀ f, test f ∈ M.closure (Set.range (blockLabel F a))) :
blockBirthPattern M F a τ test mark ∈
freeMarkedPatterns M (blockOldLabel (F := F)) a test (fun o : BlockOld F => mark (blockOldStage (F := F) o)) := by
classical
apply Finset.mem_filter.mpr
refine ⟨Finset.mem_univ _, blockTime F τ, blockTime_old_strictMono F τ,
blockTime_injective F τ, mark ∘ τ, ?_⟩
intro f
obtain ⟨o, hp⟩ := exists_pivot M (blockLabel F a) (blockTime F τ) (hnonloop f) (hspan f)
refine ⟨o, hp, ?_⟩
have hb := pivot_block_birth M F a τ hp
change Sum.elim (fun o : BlockOld F => mark (blockOldStage (F := F) o)) (mark ∘ τ) o = true ↔ _
have hm : Sum.elim (fun o : BlockOld F => mark (blockOldStage (F := F) o)) (mark ∘ τ) o =
mark (blockStage F τ o) := by cases o <;> rfl
rw [hm]
simp only [blockBirthPattern, Finset.mem_filter, Finset.mem_univ, true_and]
constructor
· intro h
exact ⟨blockStage F τ o, h, hb⟩
· rintro ⟨k, hmark, hk⟩
have heq := block_birth_unique M F a τ hb hk
simpa only [heq] using hmark
lemma blockPrefix_eq
{α : Type u_1} {A : Type u_2} {T : Type u_3}
[Fintype α] [LinearOrder α] [Fintype A] [Fintype T] [LinearOrder T]
(F : T → Set α) (hF : Monotone F)
(a : A → α) (τ : A → T) (k : T) :
blockPrefix F a τ k = F k ∪ a '' {g | τ g ≤ k} := by
ext e
constructor
· rintro ⟨o, ho, rfl⟩
cases o with
| inl o => exact Or.inl (hF ho o.property)
| inr g => exact Or.inr ⟨g, ho, rfl⟩
· rintro (he | ⟨g, hg, rfl⟩)
· exact ⟨Sum.inl ⟨toLex (k, e), he⟩, by change k ≤ k; exact le_rfl, rfl⟩
· exact ⟨Sum.inr g, hg, rfl⟩
lemma blockBefore_eq_iUnion
{α : Type u_1} {A : Type u_2} {T : Type u_3}
[Fintype α] [LinearOrder α] [Fintype A] [Fintype T] [LinearOrder T]
(F : T → Set α) (a : A → α) (τ : A → T)
(k : T) :
blockBefore F a τ k = (⋃ j : T, ⋃ (_ : j < k), F j) ∪ a '' {g | τ g < k} := by
ext e
constructor
· rintro ⟨o, ho, rfl⟩
cases o with
| inl o => exact Or.inl (mem_iUnion.mpr ⟨blockOldStage (F := F) o, mem_iUnion.mpr ⟨ho, o.property⟩⟩)
| inr g => exact Or.inr ⟨g, ho, rfl⟩
· rintro (he | ⟨g, hg, rfl⟩)
· obtain ⟨j, hj, he⟩ := mem_iUnion.mp he |>.imp fun _ h => mem_iUnion.mp h
exact ⟨Sum.inl ⟨toLex (j, e), he⟩, hj, rfl⟩
· exact ⟨Sum.inr g, hg, rfl⟩
end MatroidProphet.Pivots
end OAI
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