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b317718 | 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 | import OAI.Probability.FreeIsing.Trees.Parents
namespace OAI
namespace Problem367
private lemma mem_inducedEdges_8911 {d : ℕ} (D : Finset (TreeVertex d))
(u v : TreeVertex d) :
(u, v) ∈ inducedEdges D ↔
(u ∈ D ∧ v ∈ D) ∧ treeAdj u v ∧ u.1.length < v.1.length := by
classical
rw [inducedEdges, Finset.mem_filter]
exact and_congr Finset.mem_product Iff.rfl
lemma inducedEdges_snd_injOn {d : ℕ} (D : Finset (TreeVertex d)) :
Set.InjOn Prod.snd (↑(inducedEdges D) : Set (TreeVertex d × TreeVertex d)) := by
rintro ⟨a, b⟩ hab ⟨c, e⟩ hce heq
have hab' : (a ∈ D ∧ b ∈ D) ∧ treeAdj a b ∧ a.1.length < b.1.length := by
exact (mem_inducedEdges_8911 D a b).mp hab
have hce' : (c ∈ D ∧ e ∈ D) ∧ treeAdj c e ∧ c.1.length < e.1.length := by
exact (mem_inducedEdges_8911 D c e).mp hce
change b = e at heq
subst e
exact Prod.ext (treeAdj_lower_unique hab'.2.1 hce'.2.1 hab'.2.2 hce'.2.2) rfl
lemma ConnectedFinset.inducedEdges_image_snd {d : ℕ}
{D : Finset (TreeVertex d)} (hD : ConnectedFinset D)
{r : TreeVertex d} (hr : r ∈ D)
(hmin : ∀ w ∈ D, r.1.length ≤ w.1.length) :
(inducedEdges D).image Prod.snd = D.erase r := by
classical
ext v
constructor
· intro hv
obtain ⟨⟨u, w⟩, huw, hwv⟩ := Finset.mem_image.mp hv
have huw' : (u ∈ D ∧ w ∈ D) ∧ treeAdj u w ∧ u.1.length < w.1.length := by
exact (mem_inducedEdges_8911 D u w).mp huw
change w = v at hwv
subst w
apply Finset.mem_erase.mpr
refine ⟨?_, huw'.1.2⟩
intro heq
subst v
have := hmin u huw'.1.1
omega
· intro hv
obtain ⟨hne, hv⟩ := Finset.mem_erase.mp hv
apply Finset.mem_image.mpr
refine ⟨(parentVertex v, v), ?_, rfl⟩
have hp := hD.parent_mem_of_ne_min hr hmin hv hne
have hnil := ConnectedFinset.ne_nil_of_ne_min hmin hv hne
have hpos : 0 < v.1.length := List.length_pos_iff.mpr hnil
have hadj := treeAdj_parentVertex hnil
apply (mem_inducedEdges_8911 D _ _).mpr
refine ⟨⟨hp, hv⟩, hadj, ?_⟩
rw [parentVertex_length]
omega
lemma ConnectedFinset.inducedEdges_card_add_one {d : ℕ}
{D : Finset (TreeVertex d)} (hD : ConnectedFinset D) :
(inducedEdges D).card + 1 = D.card := by
classical
obtain ⟨r, hr, hmin⟩ := D.exists_min_image (fun v => v.1.length) hD.1
have himage := hD.inducedEdges_image_snd hr hmin
have hcard := congrArg Finset.card himage
rw [Finset.card_image_of_injOn (inducedEdges_snd_injOn D), Finset.card_erase_of_mem hr] at hcard
have hpos : 0 < D.card := Finset.card_pos.mpr hD.1
omega
lemma ConnectedFinset.inducedEdges_card {d : ℕ}
{D : Finset (TreeVertex d)} (hD : ConnectedFinset D) :
(inducedEdges D).card = D.card - 1 := by
have := hD.inducedEdges_card_add_one
omega
end Problem367
end OAI
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