Theory Plane

(*  ID:         $Id: Plane.thy,v 1.2 2006/01/09 06:56:13 nipkow Exp $
    Author:     Gertrud Bauer
*)

header{* Plane Graph Enumeration *}

theory Plane
imports Enumerator FaceDivision RTranCl
begin


constdefs
 maxGon :: "nat => nat"
"maxGon p ≡ p+3"

declare maxGon_def [simp]


constdefs duplicateEdge :: "graph => face => vertex => vertex => bool"
 "duplicateEdge g f a b ≡ 
  2 ≤ directedLength f a b ∧ 2 ≤ directedLength f b a ∧ b mem (neighbors g a)"

consts containsUnacceptableEdgeSnd :: 
      "(nat => nat => bool) => nat => nat list => bool"
primrec "containsUnacceptableEdgeSnd N v [] = False"
 "containsUnacceptableEdgeSnd N v (w#ws) = 
     (case ws of [] => False
         | (w'#ws') => if v < w ∧ w < w' ∧ N w w' then True
                      else containsUnacceptableEdgeSnd N w ws)"

consts containsUnacceptableEdge :: "(nat => nat => bool) => nat list => bool"
primrec "containsUnacceptableEdge N [] = False"
 "containsUnacceptableEdge N (v#vs) = 
     (case vs of [] => False
           | (w#ws) => if v < w ∧ N v w then True
                      else containsUnacceptableEdgeSnd N v vs)"

constdefs containsDuplicateEdge :: "graph => face => vertex => nat list => bool"
 "containsDuplicateEdge g f v is ≡ 
     containsUnacceptableEdge (λi j. duplicateEdge g f (fi•v) (fj•v)) is" 

constdefs containsDuplicateEdge' :: "graph => face => vertex => nat list => bool"
 "containsDuplicateEdge' g f v is ≡ 
  2 ≤ |is| ∧ 
  ((∃k < |is| - 2. let i0 = is!k; i1 = is!(k+1); i2 = is!(k+2) in
    (duplicateEdge g f (fi1 •v) (fi2 •v)) ∧ (i0 < i1) ∧ (i1 < i2))
  ∨ (let i0 = is!0; i1 = is!1 in
    (duplicateEdge g f (fi0 •v) (fi1 •v)) ∧ (i0 < i1)))" 


constdefs generatePolygon :: "nat => vertex => face => graph => graph list"
 "generatePolygon n v f g ≡ 
     let enumeration = enumerator n |vertices f|;
     enumeration = [is ∈ enumeration. ¬ containsDuplicateEdge g f v is];
     vertexLists = [indexToVertexList f v is. is ∈ enumeration] in
     [subdivFace g f vs. vs ∈ vertexLists]"

constdefs next_plane0 :: "nat => graph => graph list" ("next'_plane0_")
 "next_plane0p g ≡
     if final g then [] 
     else \<Squnion>f∈nonFinals g \<Squnion>v∈vertices f \<Squnion>i∈[3..maxGon p] generatePolygon i v f g"


constdefs Seed :: "nat => graph" ("Seed_")
  "Seedp ≡ graph(maxGon p)"

lemma Seed_not_final[iff]: "¬ final (Seed p)"
by(simp add:Seed_def graph_def finalGraph_def nonFinals_def)

constdefs
 PlaneGraphs0 :: "graph set" 
"PlaneGraphs0 ≡ \<Union>p. {g. Seedp [next_plane0p]->* g ∧ final g}"

end