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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 | import OAI.Probability.MatroidProphet.Pivots.Phases
namespace OAI
namespace MatroidProphet.Pivots
open Set Finset
variable {α G Z B : Type*} [Fintype α] [LinearOrder α] [Fintype G] [Fintype Z]
[Fintype B] [LinearOrder B] {n : ℕ}
lemma lex_lt_true_iff_le_false
{B : Type u_4} [Fintype B] [LinearOrder B] (p : B ×ₗ Bool) (b : B) :
p < toLex (b, true) ↔ p ≤ toLex (b, false) := by
rw [Prod.Lex.lt_iff, Prod.Lex.le_iff]
cases (ofLex p).2 <;> simp
lemma phase_before_true
{α : Type u_1} {G : Type u_2} {Z : Type u_3} {B : Type u_4}
[Fintype α] [LinearOrder α] [Fintype G] [Fintype Z] [Fintype B] [LinearOrder B]
(F K : B → Set α) (a : G → α) (z : Z → α)
(θ : G → B) (β : Z → B) (b : B) :
blockBefore (phaseOld F K) (Sum.elim a z) (phasePlacement θ β) (toLex (b, true)) =
blockPrefix (phaseOld F K) (Sum.elim a z) (phasePlacement θ β) (toLex (b, false)) := by
unfold blockBefore blockPrefix
congr 1
ext o
exact lex_lt_true_iff_le_false _ _
lemma mem_phase_pattern (M : Matroid α) (hE : M.E = univ)
(F K : B → Set α) (hF : Monotone F) (hK : ∀ b, K b ⊆ F b)
(a : G → α) (z : Z → α) (θ : G → B) (β : Z → B)
(hz : ∀ w, z w ∈ upperPath M F a θ (β w)) (test : Fin n → α) (f : Fin n) :
f ∈ blockBirthPattern M (phaseOld F K) (Sum.elim a z) (phasePlacement θ β) test
(fun p => (ofLex p).2) ↔
∃ b, test f ∈ upperPath M F a θ b ∧
test f ∉ M.closure (lowerPath M F a θ b ∪ K b ∪ z '' {w | β w = b}) := by
classical
simp only [blockBirthPattern, Finset.mem_filter, Finset.mem_univ, true_and]
constructor
· rintro ⟨p, hp, hafter, hbefore⟩
change (ofLex p).2 = true at hp
have hp' : p = toLex ((ofLex p).1, true) := congrArg toLex (Prod.ext rfl hp)
rw [hp'] at hafter hbefore
rw [phase_endpoint M hE F K hF hK a z θ β hz] at hafter
rw [phase_before_true, phase_interior M hE F K hK a z θ β hz] at hbefore
exact ⟨(ofLex p).1, hafter, hbefore⟩
· rintro ⟨b, hafter, hbefore⟩
refine ⟨toLex (b, true), rfl, ?_, ?_⟩
· rwa [phase_endpoint M hE F K hF hK a z θ β hz]
· rwa [phase_before_true, phase_interior M hE F K hK a z θ β hz]
noncomputable def residualOfLayers (M : Matroid α) (F K : B → Set α) (a : G → α)
(z : Z → α) (θ : G → B) (β : Z → B) (test : Fin n → α)
(P : B → Finset (Fin n)) : Finset (Fin n) := by
classical
exact Finset.univ.filter fun f => ∃ b, f ∈ P b ∧
test f ∉ M.closure (lowerPath M F a θ b ∪ K b ∪ z '' {w | β w = b})
lemma first_inter_phase_eq_residual (M : Matroid α) (hE : M.E = univ)
(F K : B → Set α) (hF : Monotone F) (hK : ∀ b, K b ⊆ F b)
(a : G → α) (z : Z → α) (θ : G → B) (β : Z → B)
(hz : ∀ w, z w ∈ upperPath M F a θ (β w)) (test : Fin n → α)
(P : B → Finset (Fin n)) (first : Finset (Fin n))
(hfirst : ∀ f, f ∈ first ↔ ∃ b, f ∈ P b)
(hP : ∀ b f, f ∈ P b → test f ∈ upperPath M F a θ b ∧ test f ∉ lowerPath M F a θ b) :
first ∩ blockBirthPattern M (phaseOld F K) (Sum.elim a z) (phasePlacement θ β) test
(fun p => (ofLex p).2) = residualOfLayers M F K a z θ β test P := by
classical
ext f
rw [Finset.mem_inter, mem_phase_pattern M hE F K hF hK a z θ β hz]
simp only [residualOfLayers, Finset.mem_filter, Finset.mem_univ, true_and]
constructor
· rintro ⟨hf, c, hc, hcnot⟩
obtain ⟨b, hb⟩ := (hfirst f).1 hf
have hpb := hP b f hb
have hbc : b = c := by
rcases lt_trichotomy b c with h | h | h
· apply False.elim
apply hcnot
exact M.subset_closure _ (by simp [hE]) (Or.inl (Or.inl
(upperPath_subset_lowerPath M F a θ h hpb.1)))
· exact h
· exact (hpb.2 (upperPath_subset_lowerPath M F a θ h hc)).elim
exact ⟨b, hb, hbc.symm ▸ hcnot⟩
· rintro ⟨b, hb, hnot⟩
exact ⟨(hfirst f).2 ⟨b, hb⟩, b, (hP b f hb).1, hnot⟩
end MatroidProphet.Pivots
end OAI
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