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diff --git a/backend/Interference_adjacency.v b/backend/Interference_adjacency.v
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-Require Import InterfGraphMapImp.
-Require Import Graph_Facts.
-Require Import FSets.
-Require Import SetsFacts.
-Require Import Edges.
-
-Import Edge RegFacts Props RegRegProps.
-
-(* Some properties about the interference adjacency
-   and the same about preference adjacency *)
-
-(* x is not in its own interference neighborhood *)
-Lemma not_in_interf_self : forall x g,
-~VertexSet.In x (interference_adj x g).
-
-Proof.
-intros x g. rewrite in_interf. intro H.
-elim (In_graph_edge_diff_ext _ _ H). auto.
-Qed.
-
-(* x is not in its own preference neighborhood *)
-Lemma not_in_pref_self : forall x g,
-~VertexSet.In x (preference_adj x g).
-
-Proof.
-intros x g. rewrite in_pref. intro H. destruct H.
-elim (In_graph_edge_diff_ext _ _ H). auto.
-Qed.
-
-(* If x is an interference neighbor of y in g
-   then y is an interference neighbor of x in g *)
-Lemma interf_adj_comm : forall x y g,
-VertexSet.In x (interference_adj y g) -> VertexSet.In y (interference_adj x g).
-
-Proof.
-intros x y g H. rewrite in_interf. rewrite edge_comm. rewrite <-in_interf. auto.
-Qed.
-
-(* If x is a preference neighbor of y in g
-    then y is a preference neighbor of x in g *)
-Lemma pref_adj_comm : forall x y g,
-VertexSet.In x (preference_adj y g) -> VertexSet.In y (preference_adj x g).
-
-Proof.
-intros x y g H. 
-rewrite in_pref in H. destruct H.  rewrite edge_comm in H. 
-rewrite in_pref. exists x0. assumption.
-Qed.
-
-(* If x is an interference neighbor of any vertex of g then x is in g *)
-Lemma in_interf_in : forall x r g,
-VertexSet.In x (interference_adj r g) -> In_graph x g.
-
-Proof.
-intros x r g H. rewrite in_interf in H.
-apply (proj1 (In_graph_edge_in_ext _ _ H)).
-Qed.
-
-(* If x is a preferenec neighbor of any vertex then x is in g *)
-Lemma in_pref_in : forall x r g,
-VertexSet.In x (preference_adj r g) -> In_graph x g.
-
-Proof.
-intros x r g H. rewrite in_pref in H. destruct H.
-apply (proj1 (In_graph_edge_in_ext _ _ H)).
-Qed.