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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 | import OAI.Probability.Ising.Basic
namespace OAI
open scoped BigOperators Classical
namespace BufferedIsing
lemma fkWeight_nonneg {n m : ℕ} (G : FiniteGraph n m)
(I J F : Finset (Fin n)) (ω : Finset (Fin m)) :
0 ≤ fkWeight G I J F ω := by
classical
unfold fkWeight
apply mul_nonneg (pow_nonneg (by norm_num) _)
apply Finset.prod_nonneg
intro e _
split_ifs
· apply sub_nonneg.mpr
apply Real.exp_le_one_iff.mpr
nlinarith [G.coupling_nonneg e]
· exact (Real.exp_pos _).le
lemma fkWeight_empty_pos {n m : ℕ} (G : FiniteGraph n m)
(I J F : Finset (Fin n)) : 0 < fkWeight G I J F ∅ := by
classical
unfold fkWeight
simp only [Finset.notMem_empty, ↓reduceIte]
exact mul_pos (pow_pos (by norm_num) _) (Finset.prod_pos fun _ _ => Real.exp_pos _)
lemma comparison_empty_preserves_I {n m : ℕ} (G : FiniteGraph n m)
(I J F : Finset (Fin n)) (hIJ : Disjoint I J) (x y : KeptVertex F)
(hxy : (comparisonSetoid G I J F ∅).r x y) :
x.val ∈ I ↔ y.val ∈ I := by
induction hxy with
| rel x y h =>
rcases h with h | h | ⟨e, he, _⟩
· exact iff_of_true h.1 h.2
· exact iff_of_false
(fun hx => Finset.disjoint_left.mp hIJ hx h.1)
(fun hy => Finset.disjoint_left.mp hIJ hy h.2)
· exact (Finset.notMem_empty e he).elim
| refl => rfl
| symm _ _ _ ih => exact ih.symm
| trans _ _ _ _ _ ih ih' => exact ih.trans ih'
lemma qZero_nonneg_lt_one {n m : ℕ} (G : FiniteGraph n m)
(I J F : Finset (Fin n)) (hIJ : Disjoint I J)
(i j : KeptVertex F) (hi : i.val ∈ I) (hj : j.val ∈ J) :
0 ≤ qZero G I J F i j ∧ qZero G I J F i j < 1 := by
classical
have he : (∅ : Finset (Fin m)) ∈ (keptEdges G F).powerset :=
Finset.mem_powerset.mpr (Finset.empty_subset _)
have hden : 0 < ∑ ω ∈ (keptEdges G F).powerset, fkWeight G I J F ω :=
lt_of_lt_of_le (fkWeight_empty_pos G I J F)
(Finset.single_le_sum (fun ω _ => fkWeight_nonneg G I J F ω) he)
have hno : ¬(comparisonSetoid G I J F ∅).r i j := by
intro hconn
exact Finset.disjoint_left.mp hIJ
((comparison_empty_preserves_I G I J F hIJ i j hconn).mp hi) hj
unfold qZero
constructor
· apply div_nonneg _ hden.le
apply Finset.sum_nonneg
intro ω _
split_ifs
· exact fkWeight_nonneg G I J F ω
· exact le_rfl
· apply (div_lt_one hden).mpr
apply Finset.sum_lt_sum
· intro ω _
split_ifs
· exact le_rfl
· exact fkWeight_nonneg G I J F ω
· exact ⟨∅, he, by simpa only [hno, ite_false] using fkWeight_empty_pos G I J F⟩
lemma comparisonRel_terminal_swap {n m : ℕ} (G : FiniteGraph n m)
(I J F : Finset (Fin n)) (ω : Finset (Fin m)) :
comparisonRel G I J F ω = comparisonRel G J I F ω := by
funext x y
apply propext
simp only [comparisonRel]
exact or_left_comm
/-- The zero-field terminal connection probability is symmetric under exchange of the terminal sets. -/
lemma qZero_terminal_swap {n m : ℕ} (G : FiniteGraph n m)
(I J F : Finset (Fin n)) (i j : KeptVertex F) :
qZero G I J F i j = qZero G J I F j i := by
classical
have hs : ∀ ω, comparisonSetoid G I J F ω = comparisonSetoid G J I F ω := by
intro ω
unfold comparisonSetoid
rw [comparisonRel_terminal_swap G I J F ω]
have hw : ∀ ω, fkWeight G I J F ω = fkWeight G J I F ω := by
intro ω
unfold fkWeight clusterCount
rw [hs ω]
have hr : ∀ ω, (comparisonSetoid G I J F ω).r i j ↔
(comparisonSetoid G J I F ω).r j i := by
intro ω
change Relation.EqvGen (comparisonRel G I J F ω) i j ↔
Relation.EqvGen (comparisonRel G J I F ω) j i
rw [comparisonRel_terminal_swap G I J F ω]
exact ⟨fun h => Relation.EqvGen.symm _ _ h, fun h => Relation.EqvGen.symm _ _ h⟩
unfold qZero
simp only [hw, hr]
lemma qZero_independent_representatives {n m : ℕ} (G : FiniteGraph n m)
(I J F : Finset (Fin n)) (i i' j j' : KeptVertex F)
(hi : i.val ∈ I) (hi' : i'.val ∈ I) (hj : j.val ∈ J) (hj' : j'.val ∈ J) :
qZero G I J F i j = qZero G I J F i' j' := by
classical
unfold qZero
congr 1
apply Finset.sum_congr rfl
intro ω _
have hconn : (comparisonSetoid G I J F ω).r i j ↔
(comparisonSetoid G I J F ω).r i' j' := by
have hii : (comparisonSetoid G I J F ω).r i i' :=
Relation.EqvGen.rel _ _ (Or.inl ⟨hi, hi'⟩)
have hjj : (comparisonSetoid G I J F ω).r j j' :=
Relation.EqvGen.rel _ _ (Or.inr (Or.inl ⟨hj, hj'⟩))
have hsym : ∀ {u v}, (comparisonSetoid G I J F ω).r u v →
(comparisonSetoid G I J F ω).r v u := (comparisonSetoid G I J F ω).iseqv.symm
have htrans : ∀ {u v z}, (comparisonSetoid G I J F ω).r u v →
(comparisonSetoid G I J F ω).r v z → (comparisonSetoid G I J F ω).r u z :=
(comparisonSetoid G I J F ω).iseqv.trans
exact ⟨fun h => htrans (hsym hii) (htrans h hjj),
fun h => htrans hii (htrans h (hsym hjj))⟩
simp only [hconn]
end BufferedIsing
end OAI
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