| import OAI.Probability.ClassicalON.PinnedSigns |
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| namespace OAI |
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| universe uE uV |
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| noncomputable section |
| open scoped Classical |
| namespace ClassicalON |
| variable {V : Type uV} {E : Type uE} |
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| def bondGraph (left right : E → V) (η : E → Bool) : SimpleGraph V where |
| Adj x y := x≠y ∧ ∃ e,η e=true ∧ |
| ((left e=x ∧ right e=y) ∨ (left e=y ∧ right e=x)) |
| symm := ⟨by |
| intro x y ⟨h,e,he,hor⟩ |
| exact ⟨h.symm,e,he,hor.elim (fun ⟨a,b⟩ => Or.inr ⟨a,b⟩) (fun ⟨a,b⟩ => Or.inl ⟨a,b⟩)⟩⟩ |
| loopless := ⟨by intro x; simp⟩ |
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| theorem bondGraph_endpoint_reachable (left right : E → V) (η : E → Bool) (e : E) |
| (he : η e=true) : (bondGraph left right η).Reachable (left e) (right e) := by |
| by_cases h : left e=right e |
| · rw [h] |
| · exact SimpleGraph.Adj.reachable ⟨h,e,he,Or.inl ⟨rfl,rfl⟩⟩ |
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| theorem bondConnected_iff_reachable (left right : E → V) (η : E → Bool) (x y : V) : |
| bondConnected left right η x y ↔ (bondGraph left right η).Reachable x y := by |
| constructor |
| · intro h |
| let s : V → SignField := fun v => if (bondGraph left right η).Reachable x v then 0 else 1 |
| have hs : s∈bondSubspace left right η := by |
| intro e he |
| have hh := bondGraph_endpoint_reachable left right η e he |
| have hi : (bondGraph left right η).Reachable x (left e) ↔ |
| (bondGraph left right η).Reachable x (right e) := |
| ⟨fun hp => hp.trans hh,fun hp => hp.trans hh.symm⟩ |
| simp only [s,hi] |
| have hh := h s hs |
| by_contra hy |
| simp [s,hy] at hh |
| · rintro ⟨p⟩ s hs |
| induction p with |
| | nil => rfl |
| | @cons u v w huv p ih => |
| apply Eq.trans _ ih |
| obtain ⟨_,e,he,hor⟩ := huv |
| rcases hor with ⟨rfl,rfl⟩ | ⟨rfl,rfl⟩ |
| · exact hs e he |
| · exact (hs e he).symm |
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| end ClassicalON |
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| end |
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| end OAI |
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