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e320e82 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 | import OAI.Combinatorics.GraphThreshold.ProbabilityTreeFacts
namespace OAI
/-! Full-path interpretation of supports and structural disjointness. -/
noncomputable section
namespace LeanBlast.SecondKahnKalai.ProbabilityTree
variable {X : Type} [DecidableEq X]
theorem pathUnion_subset_support {d : ℕ} (T : ProbabilityTree X d) (p : T.Path) :
T.pathUnion p ⊆ T.support := by
induction T with
| leaf => exact Finset.empty_subset _
| @node d B inst law positive label child ih =>
rcases p with ⟨b, p⟩
exact Finset.union_subset (label_subset_support law positive label child b)
((ih b p).trans (child_support_subset law positive label child b))
theorem mem_pathUnion_iff {d : ℕ} (T : ProbabilityTree X d) (p : T.Path) (x : X) :
x ∈ T.pathUnion p ↔ ∃ S ∈ T.pathLabels p, x ∈ S := by
induction T with
| leaf => simp [pathUnion, pathLabels]
| @node d B inst law positive label child ih =>
rcases p with ⟨b, p⟩
simp [pathUnion, pathLabels, ih b p]
theorem pathLabel_subset_pathUnion {d : ℕ} (T : ProbabilityTree X d)
(p : T.Path) {S : Finset X} (hS : S ∈ T.pathLabels p) : S ⊆ T.pathUnion p := by
intro x hx
exact (mem_pathUnion_iff T p x).2 ⟨S, hS, hx⟩
theorem pathLabel_subset_support {d : ℕ} (T : ProbabilityTree X d)
(p : T.Path) {S : Finset X} (hS : S ∈ T.pathLabels p) : S ⊆ T.support :=
(pathLabel_subset_pathUnion T p hS).trans (pathUnion_subset_support T p)
/-- Every arc coordinate occurs on an actual full path, since all branches continue. -/
theorem mem_support_iff_exists_path {d : ℕ} (T : ProbabilityTree X d) (x : X) :
x ∈ T.support ↔ ∃ p : T.Path, x ∈ T.pathUnion p := by
induction T with
| leaf => simp [support, pathUnion]
| @node d B inst law positive label child ih =>
constructor
· intro hx
obtain ⟨b, _, hxb⟩ := Finset.mem_biUnion.mp hx
rcases Finset.mem_union.mp hxb with hlabel | hchild
· exact ⟨⟨b, choosePath (child b)⟩, Finset.mem_union_left _ hlabel⟩
· obtain ⟨p, hp⟩ := (ih b).1 hchild
exact ⟨⟨b, p⟩, Finset.mem_union_right _ hp⟩
· rintro ⟨⟨b, p⟩, hx⟩
apply Finset.mem_biUnion.mpr
refine ⟨b, Finset.mem_univ _, ?_⟩
rcases Finset.mem_union.mp hx with hlabel | hchild
· exact Finset.mem_union_left _ hlabel
· exact Finset.mem_union_right _ ((ih b).2 ⟨p, hchild⟩)
theorem support_eq_pathUnion_biUnion {d : ℕ} (T : ProbabilityTree X d) :
T.support = Finset.univ.biUnion T.pathUnion := by
ext x
simpa only [Finset.mem_biUnion, Finset.mem_univ, true_and] using
mem_support_iff_exists_path T x
theorem isDisjoint_iff_pathDisjoint {d : ℕ} (T : ProbabilityTree X d) :
T.IsDisjoint ↔ T.PathDisjoint := by
induction T with
| leaf => simp [IsDisjoint, PathDisjoint, pathLabels]
| @node d B inst law positive label child ih =>
constructor
· intro hT p
rcases p with ⟨b, p⟩
apply List.pairwise_cons.mpr
constructor
· intro S hS
exact (hT.1 b).mono_right (pathLabel_subset_support (child b) p hS)
· exact (ih b).1 (hT.2 b) p
· intro hT
constructor
· intro b
apply Finset.disjoint_left.mpr
intro x hxLabel hxSupport
obtain ⟨p, hp⟩ := (mem_support_iff_exists_path (child b) x).1 hxSupport
obtain ⟨S, hS, hxS⟩ := (mem_pathUnion_iff (child b) p x).1 hp
have hpw := hT ⟨b, p⟩
have hdis := (List.pairwise_cons.mp hpw).1 S hS
exact Finset.disjoint_left.mp hdis hxLabel hxS
· intro b
apply (ih b).2
intro p
exact (List.pairwise_cons.mp (hT ⟨b, p⟩)).2
theorem incoming_disjoint_subtree_of_pathDisjoint {d : ℕ} {B : Type} [Fintype B]
{law : FiniteLaw B} {positive : law.IsPositive} {label : B → Finset X}
{child : B → ProbabilityTree X d}
(hT : (node B law positive label child).PathDisjoint) (b : B) :
Disjoint (label b) (child b).support :=
((isDisjoint_iff_pathDisjoint _).2 hT).1 b
/-- The locality assertion applies even when disjointness is given in the literal path form. -/
theorem child_dependsOutside_of_pathDisjoint {d : ℕ} {B : Type} [Fintype B]
{law : FiniteLaw B} {positive : law.IsPositive} {label : B → Finset X}
{child : B → ProbabilityTree X d}
(hT : (node B law positive label child).PathDisjoint) (b : B)
{β : Type*} {f : Finset X → β} (hf : DependsOnlyOn (child b).support f) :
DependsOutside (label b) f :=
hf.dependsOutside (incoming_disjoint_subtree_of_pathDisjoint hT b)
theorem coveredBy_mono {d : ℕ} {T : ProbabilityTree X d} {W V : Finset X}
(h : T.CoveredBy W) (hWV : W ⊆ V) : T.CoveredBy V := by
obtain ⟨p, hp⟩ := h
exact ⟨p, hp.trans hWV⟩
theorem coveredBy_univ [Fintype X] {d : ℕ} (T : ProbabilityTree X d) :
T.CoveredBy Finset.univ :=
⟨choosePath T, Finset.subset_univ _⟩
end LeanBlast.SecondKahnKalai.ProbabilityTree
end
end OAI
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