File size: 3,055 Bytes
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 80 81 82 83 84 85 86 87 88 89 90 | import OAI.Probability.FreeIsing.Trees.BroadcastLocality
import OAI.Probability.FreeIsing.Trees.Transitivity
namespace OAI
noncomputable section
namespace Problem367
namespace RootedBroadcast
def branchRoot {d : ℕ} (c : Fin d) : TreeVertex d :=
⟨[c], List.IsChain.singleton c⟩
@[simp] theorem branchRoot_val {d : ℕ} (c : Fin d) : (branchRoot c).val = [c] := rfl
@[simp] theorem prefixFlip_root {d : ℕ} (c : Fin d) :
prefixFlip c (root d) = branchRoot c := rfl
@[simp] theorem prefixFlip_branchRoot {d : ℕ} (c : Fin d) :
prefixFlip c (branchRoot c) = root d := by
apply Subtype.ext
simp [prefixFlip, prefixFlipWord, branchRoot, root]
theorem not_prefix_flip_of_prefix {d : ℕ} (c : Fin d) (v : TreeVertex d)
(hv : [c] <+: v.val) : ¬ [c] <+: (prefixFlip c v).val := by
rcases v with ⟨w, hw⟩
rcases hv with ⟨t, ht⟩
have heq : w = c :: t := by simpa using ht.symm
subst w
simp only [prefixFlip, prefixFlipWord]
cases t with
| nil => simp
| cons b t =>
have hcb : c ≠ b := (List.isChain_cons_cons.mp hw).1
intro hprefix
have heq : [c] = [b] :=
(List.prefix_of_prefix_length_le hprefix (by simp) (by simp)).eq_of_length (by simp)
exact hcb (by simpa using heq)
theorem prefix_flip_iff_not_prefix {d : ℕ} (c : Fin d) (v : TreeVertex d) :
[c] <+: (prefixFlip c v).val ↔ ¬ [c] <+: v.val := by
constructor
· intro h
simpa only [prefixFlip_twice] using
not_prefix_flip_of_prefix c (prefixFlip c v) h
· intro h
rcases v with ⟨w, hw⟩
cases w with
| nil => simp [prefixFlip, prefixFlipWord]
| cons b t =>
have hcb : c ≠ b := by
intro heq
subst b
exact h (by simp)
simp [prefixFlip, prefixFlipWord, hcb]
def flipBranchToComplement {d : ℕ} (c : Fin d) (v : BranchVertex c) :
{w : TreeVertex d // ¬ [c] <+: w.val} :=
⟨prefixFlip c v.val, not_prefix_flip_of_prefix c v.val v.property⟩
@[simp] theorem flipBranchToComplement_val {d : ℕ} (c : Fin d) (v : BranchVertex c) :
(flipBranchToComplement c v).val = prefixFlip c v.val := rfl
def flipBranchEquivComplement {d : ℕ} (c : Fin d) :
BranchVertex c ≃ {w : TreeVertex d // ¬ [c] <+: w.val} where
toFun := flipBranchToComplement c
invFun := fun w => ⟨prefixFlip c w.val, (prefix_flip_iff_not_prefix c w.val).mpr w.property⟩
left_inv := fun v => Subtype.ext (prefixFlip_twice c v.val)
right_inv := fun w => Subtype.ext (prefixFlip_twice c w.val)
def branchRootVertex {d : ℕ} (c : Fin d) : BranchVertex c :=
⟨branchRoot c, List.prefix_refl [c]⟩
@[simp] theorem prefixTreeAut_symm_root {d : ℕ} (c : Fin d) :
(prefixTreeAut c).toEquiv.symm (root d) = branchRoot c := rfl
@[simp] theorem prefixTreeAut_symm_branchRoot {d : ℕ} (c : Fin d) :
(prefixTreeAut c).toEquiv.symm (branchRoot c) = root d :=
prefixFlip_branchRoot c
theorem treeAdj_root_branchRoot {d : ℕ} (c : Fin d) :
treeAdj (root d) (branchRoot c) := Or.inl ⟨c, rfl⟩
end RootedBroadcast
end Problem367
end
end OAI
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