| 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 |
|
|
