Graphs

Graph: Graph theory


Require Export Rel.
Fixpoint sum (ns : list nat) : nat :=
  match ns with
  | [] ⇒ 0
  | n::ns ⇒ n + sum ns
  end.
Fixpoint sum_map {X : Type} (f : X → nat) (xs : list X) : nat :=
  match xs with
  | [] ⇒ 0
  | x::xs ⇒ f x + sum_map f xs
  end.
Fixpoint remove_first {X : Type} (eq_dec : ∀ (x y : X), {x=y} + {x≠y}) (y:X) (xs : list X) : list X :=
  match xs with
  | [] ⇒ []
  | (x::xs) ⇒ if eq_dec y x then xs else x::remove_first eq_dec y xs
  end.
Inductive setlike {X:Type} : list X → Prop :=
  | setlike_nil : setlike []
  | setlike_cons : ∀ (v:X) (vs:list X),
      setlike vs → ¬In v vs → setlike (v::vs).
Module ExplicitType.
  (* adapted from https://coq.inria.fr/V8.2pl1/contribs/GraphBasics.html *)
  Parameter X : Type.
  Axiom eq_dec_V : ∀ (x y : X), {x = y} + {x ≠ y}.
  Lemma eq_dec_E : ∀ (e1 e2 : X * X), {e1 = e2} + {e1 ≠ e2}.
  Proof.
    intros.
    destruct e1 as [src1 tgt1]. destruct e2 as [src2 tgt2].
    destruct (eq_dec_V src1 src2) as [Heqsrc | Hneqsrc];
    destruct (eq_dec_V tgt1 tgt2) as [Heqtgt | Hneqtgt].
    - left. rewrite Heqsrc. rewrite Heqtgt. reflexivity.
    - right. intros Hcontra. inversion Hcontra. contradiction.
    - right. intros Hcontra. inversion Hcontra. contradiction.
    - right. intros Hcontra. inversion Hcontra. contradiction.
  Qed.
  Inductive graph : list X → list (X * X) → Type :=
  | g_empty : graph [] []
  | g_vertex :
      ∀ (V : list X) (E : list (X * X)) (g : graph V E) (v : X),
        ¬ In v V → graph (v::V) E
  | g_arc :
      ∀ (V : list X) (E : list (X * X)) (g : graph V E) (src tgt : X),
      In src V →
      In tgt V →
      ¬ In (src,tgt) E → graph V ((src,tgt)::E).
  Lemma vertex_setlike : ∀ {V : list X} {E : list (X * X)} (g:graph V E), setlike V.
  Proof.
    (* FILL IN HERE *) Admitted.
  Fixpoint in_degree {V : list X} {E : list (X * X)} (v:X) (g:graph V E) : nat :=
    match g with
    | g_empty ⇒ 0
    | g_vertex V' E' g' _ _ ⇒ in_degree v g'
    | g_arc V' E' g' _ tgt _ _ _ ⇒
      if eq_dec_V v tgt
      then 1 + in_degree v g'
      else in_degree v g'
    end.
  Fixpoint out_degree {V : list X} {E : list (X * X)} (v:X) (g:graph V E) : nat (* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
  Definition degree {V : list X} {E : list (X * X)} (v:X) (g:graph V E) : nat :=
    in_degree v g + out_degree v g.
  Fixpoint sum_vertex_degree {V : list X} {E : list (X * X)} (g:graph V E) : nat :=
    match g with
    | g_empty ⇒ 0
    | g_vertex V' E' g' _ _ ⇒ sum_vertex_degree g'
    | g_arc V' E' g' _ _ _ _ _ ⇒ 2 + sum_vertex_degree g'
    end.
  Lemma not_in__degree_O :
    ∀ {V : list X} {E : list (X * X)} (v:X) (g:graph V E),
      ¬In v V → degree v g = 0.
  Proof.
    (* FILL IN HERE *) Admitted.
  Lemma sum_vertex_degree__sum_map_degree :
    ∀ {V : list X} {E : list (X * X)} (g:graph V E),
      sum_vertex_degree g = sum_map (fun v ⇒ in_degree v g) V + sum_map (fun v ⇒ out_degree v g) V.
  Proof.
    (* FILL IN HERE *) Admitted.
  Fixpoint num_edges {V : list X} {E : list (X * X)} (g:graph V E) : nat :=
    match g with
    | g_empty ⇒ 0
    | g_vertex V' E' g' _ _ ⇒ num_edges g'
    | g_arc V' E' g' _ _ _ _ _ ⇒ 1 + num_edges g'
    end.
  Lemma num_edges__length : ∀ {V : list X} {E : list (X * X)} (g:graph V E),
        num_edges g = length E.
  Proof.
    (* FILL IN HERE *) Admitted.
  Lemma inductive_handshake : ∀ {V : list X} {E : list (X * X)} (g:graph V E),
      sum_vertex_degree g = 2 * num_edges g.
  Proof.
    (* FILL IN HERE *) Admitted.
  Lemma euler_handshake : ∀ {V : list X} {E : list (X * X)} (g:graph V E),
      sum (map (fun v ⇒ degree v g) V) = 2 * length E.
  Proof.
    (* FILL IN HERE *) Admitted.
End ExplicitType.