(*  Author:     Gertrud Bauer
*)

section‹Plane Graph Enumeration›

theory Plane
imports Enumerator FaceDivision RTranCl
begin


definition maxGon :: "nat → nat" where
"maxGon p ≡ p+3"

declare maxGon_def [simp]


definition duplicateEdge :: "graph → face → vertex → vertex → bool" where
 "duplicateEdge g f a b ≡
  2 ≤ directedLength f a b ∧ 2 ≤ directedLength f b a ∧ b ∈ set (neighbors g a)"

primrec containsUnacceptableEdgeSnd :: 
      "(nat → nat → bool) → nat → nat list → bool" where
 "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)"

primrec containsUnacceptableEdge :: "(nat → nat → bool) → nat list → bool" where
 "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)"

definition containsDuplicateEdge :: "graph → face → vertex → nat list → bool" where
 "containsDuplicateEdge g f v is ≡
     containsUnacceptableEdge (λi j. duplicateEdge g f (f∙v) (f∙v)) is" 

definition containsDuplicateEdge' :: "graph → face → vertex → nat list → bool" where
 "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 (f ∙v) (f ∙v)) ∧ (i0 < i1) ∧ (i1 < i2))
  ∨ (let i0 = is!0; i1 = is!1 in
    (duplicateEdge g f (f ∙v) (f ∙v)) ∧ (i0 < i1)))" 


definition generatePolygon :: "nat → vertex → face → graph → graph list" where
 "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]"

definition next_plane0 :: "nat → graph → graph list" (‹next'_plane0›) where
 "next_plane0 g ≡
     if final g then []
     else ⊔_{∈nonFinals g} ⊔_{∈vertices f} ⊔_{∈[3..<Suc(maxGon p)]} generatePolygon i v f g"


definition Seed :: "nat → graph" (‹Seed›) where
  "Seed ≡ graph(maxGon p)"

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

definition PlaneGraphs0 :: "graph set" where 
"PlaneGraphs0 ≡ ∪p. {g. Seed [next_plane0]→* g ∧ final g}"

end