Theory Girth_Independence

theory Girth_Independence imports Connectivity

section ‹ Girth and Independence ›
text ‹ We translate and extend on a number of definitions and lemmas on girth and independence
from Noschinski's ugraph representation cite"noschinski_2012". ›

context sgraph

definition girth :: "enat" where
  "girth  INF p cycles. enat (walk_length p)"

lemma girth_acyclic: "cycles = {}  girth = "
  unfolding girth_def using top_enat_def by simp

lemma girth_lte: "c  cycles  girth  walk_length c"
  using girth_def INF_lower by auto

lemma girth_obtains: 
  assumes "girth  top"
  obtains c where "c  cycles" and "walk_length c = girth"
  using enat_in_INF girth_def assms by (metis (full_types) the_enat.simps)

lemma girthI:
  assumes "c'  cycles"
  assumes " c . c  cycles  walk_length c'  walk_length c"
  shows "girth = walk_length c'"
proof (rule ccontr)
  assume "girth  walk_length c'"
  then have "girth < walk_length c'"
    using  assms girth_lte by fastforce
  then obtain c where "c  cycles" and "walk_length c < walk_length c'"
    using girth_def by (metis enat_ord_simps(2) girth_obtains infinity_ilessE top_enat_def) 
  thus False using assms(2) less_imp_le_nat le_antisym
    by fastforce

lemma (in fin_sgraph) girth_min_alt: 
  assumes "cycles  {}"
  shows "girth = Min ((λ c . enat (walk_length c)) ` cycles)" (is "girth = Min ?A")
  unfolding girth_def using finite_cycles assms Min_Inf
  by (metis (full_types) INF_le_SUP bot_enat_def ccInf_empty ccSup_empty enat_ord_code(5) finite_imageI top_enat_def zero_enat_def) 

definition is_independent_set :: "'a set  bool" where
"is_independent_set vs  vs  V  (all_edges vs)  E = {}"

text ‹ A More mathematical way of thinking about it ›
lemma is_independent_alt: "is_independent_set vs  vs  V  ( v  vs.  u  vs. ¬ vert_adj v u)"
  unfolding is_independent_set_def
proof (auto)
  fix v u assume ss: "vs  V" and inter: "all_edges vs  E = {}" and vin: "v  vs" and uin: "u  vs" and adj: "vert_adj v u"
  then have inE: "{v, u}  E" using vert_adj_def by simp
  then have imp: "{v, u}  all_edges vs" using vin uin e_in_all_edges_ss vin uin
    by (simp add: ss)
  then show False
    using inE inter by blast 
  fix x assume "vs  V" "vvs. uvs. ¬ vert_adj v u"  "x  all_edges vs" "x  E"
  then have " u v. {u, v}  vs  {u, v}  E" by (simp add: vert_adj_def)
  then have " x . x  vs  card x = 2  x  E" by (metis card_2_iff) 
  then show False using all_edges_def
    by (metis (mono_tags, lifting) x  E x  all_edges vs mem_Collect_eq) 

lemma singleton_independent_set: "v  V  is_independent_set {v}"
  by (metis empty_subsetI insert_absorb2 insert_subset is_independent_alt
      singletonD singleton_not_edge vert_adj_def) 

definition independent_sets :: "'a set set" where
  "independent_sets  {vs. is_independent_set vs}"

definition independence_number :: "enat"  where
   "independence_number  SUP vs  independent_sets. enat (card vs)"

abbreviation "α  independence_number"

lemma independent_sets_mono:
  "vs  independent_sets   us  vs  us  independent_sets "
  using Int_mono[OF all_edges_mono, of us vs "E" "E"]
  unfolding independent_sets_def is_independent_set_def by auto

lemma le_independence_iff:
  assumes "0 < k"
  shows "k  α  k  card ` independent_sets " (is "?L  ?R")
  assume ?L
  then obtain vs where "vs  independent_sets " and klt: "k  card vs"
    using assms unfolding independence_number_def enat_le_Sup_iff by auto
  obtain us where "us  vs" and "k = card us"
    using card_Ex_subset  klt by auto
  have "us  independent_sets "  by (auto intro: independent_sets_mono)
  then show ?R using k = card us by auto
qed (auto intro: SUP_upper simp: independence_number_def)

lemma zero_less_independence:
  assumes "V  {}"
  shows "0 < α"
proof -
  from assms obtain a where "a  V" by auto
  then have "0 < enat (card {a})" "{a}  independent_sets"
    using independent_sets_def is_independent_set_def all_edges_def singleton_independent_set by simp_all
  then show ?thesis unfolding independence_number_def less_SUP_iff ..


context fin_sgraph
lemma fin_independent_sets: "finite (independent_sets)" 
  unfolding independent_sets_def is_independent_set_def using finV by auto

lemma independence_le_card:
  shows "α   card V"
proof -
  { fix x assume "x  independent_sets "
    then have "x  V" by (auto simp: independent_sets_def is_independent_set_def) }
  with finV show ?thesis unfolding independence_number_def
    by (intro SUP_least) (auto intro: card_mono)

lemma independence_fin: "α  "
  using independence_le_card by (cases "α") auto

lemma independence_max_alt: "V  {}  α = Max ((λ vs . enat (card vs)) ` independent_sets)"
  unfolding independence_number_def using Sup_enat_def zero_less_independence
  by (metis i0_less independence_fin independence_number_def)

lemma independent_sets_ne: 
  assumes "V  {}"
  shows "independent_sets  {}"
proof - 
  from assms obtain a where "a  V" by auto
  then have "{a}  independent_sets" using independent_sets_def singleton_independent_set by simp
  thus ?thesis by blast

lemma independence_obtains: 
  assumes "V  {}"
  obtains vs where "is_independent_set vs" and "card vs = α"
proof -
  have "α = Max ((λ vs . enat (card vs)) ` independent_sets)" using independence_max_alt assms by simp
  then obtain vs where "vs  independent_sets" and "enat (card vs) = α" 
    using obtains_MIN[of "independent_sets" "λ vs . enat (card vs)"] assms fin_independent_sets independent_sets_ne
    by (metis (no_types, lifting) Max_in finite_imageI imageE image_is_empty) 
  thus ?thesis using independent_sets_def that by simp