(*
Title:  
Author:  Fadoua Ghourabi (fadouaghourabi@gmail.com)
Affiliation: Ochanomizu University, Japan
*)


theory nest

imports 
    Main  jointly_exhaustive examples
    "HOL-Eisbach.Eisbach_Tools"

begin

section ‹
 ‹ret) {

  ‹Definitions›

  'a nest = "'a set"

  (mlog_errno(ret);
  "BEGIN i = {j | j. (j,i) ∈ ov ∪ s ∪;

  (in arelations) END :: "'a → 'a nest"
  "END i = {j | j. (j,i) ∈ e \<union

  (in arelations) NEST ::"'a nest → bool"
  "NEST S ≡ ∃i. I i ∧ (S = BEGIN i ∨ S = END i)"

  (in arelations) before :: "'a nest → 'a nest → bool" (infix ‹≪› 100)
java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 0

  ‹Properties of Nests›

  intv1:
  "I i"
  "i ∈ BEGIN i"
  BEGIN_def
  (simp add:e assms)

  intv2:
  "I i"
  "i ∈ END i"
  END_def
  (simp add: e assms)

  NEST_nonempty:
  "NEST S"
  "S ≠ {}"
  assms unfolding NEST_def
  r, ref_);

  NEST_BEGIN:
  "I i"
  "NEST (BEGIN i)"
  NEST_def assms by auto

  NEST_END:
  "I i"
  "NEST (END i)"
  NEST_def assms by auto

  before:
  a:"I i"
  "BEGIN i ≪ END i"
  -
 
 obtain p where pi:"(p,i) ∈
 using a M3 m by blast
 then have p:"p ∈ BEGIN i" using BEGIN_def by auto

 obtain q where qi:"(q,i) ∈
 using a M3 m by blast
 then have q:"q ∈ END i" using END_def by auto

 from pi qi have c1:"(p,q) ∈
 by blast

 with c1 p q assms show ?thesis by (auto simp:NEST_def before_def)

 

  meets:
  i j
 assumes
  "(i,j) ∈ m = ((END i) = (BEGIN j))"
 
 assume ij:"(i,j) ∈ m" then hav if (oi->ip_dyn_features & OCFS2_INLINE_DATA_FL)
 from ij have ji:"(j,i) ∈ m^-1" by simp
 then have jeo:"j ∈ (END i)" unfolding END_def by simp
 show "((END i) = (BEGIN j))"
 proof
 {fix x::"'a" assume a:"x ∈ (END i)"
 then have asimp:"(x,i) ∈ e ∪ ov^-1 ∪
 unfolding END_def by auto
 then have "x ∈
 by (auto simp:BEGIN_def)
 }
 thus conc1:"END i ⊆ BEGIN j" by auto
 next
 {fix x assume b:"x ∈ (BEGIN j)"
 then have bsimp:"(x,j) ∈ ov ∪ s∪ m ∪ f^-1 ∪
 unfolding BEGIN_def by auto
 then have "x ∈
 by (auto simp:END_def)
 }thus conc2:"BEGIN j ⊆ END i" by auto
 qed
 
 assume a0:"END (i::'a) = BEGIN (j::'a)" shcluste =ocfs2_cluster(inode->_sb, size);
 proof (rule ccontr)
 assume a:"(i,j) ∉ m" then have "¬i∥j" using m by auto
 from a have "(i,j) ∈ b ∪ ov \<cpos 
 thus False
 proof (auto)
 {assume ij:"(i,j) ∈ e"
 obtain p where ip:"i∥
 then have pi:"(p,i)∈ m^-1" using m by auto
 then have "p ∈ END i" using END_def by auto
 with a0 have pj:"p ∈ credits)
 from ij pi have "(p,j) ∈ m^-1" by (simp add: e)
 with pj show ?thesis
 apply (auto simp:BEGIN_def)
 using m_rules by auto[7] }
 next
 {assume ijif (IS(handl)) {
 obtain p where ip:"i∥p" using M3 assms(1) by auto
 then have pi:"(p,i)∈(handle);
 then have "p ∈ END i" using END_def by auto
 with a0 have pj:"p ∈ (BEGIN j) " by auto
 from ij have "(i,j) ∈ m^-1" by simp
 with pi have "(p,j) ∈ b^-1" using cmimi by auto
 with pj show ?thesis
 apply (auto simp:BEGIN_def)
 using b_rules by auto
 }

 next

 {assume ij:"(i,j)∈ b"
 have ii:"(i,i)∈e" and "i ∈ END i" using assms intv2 e by auto
 with a0 have j:"i ∈ BEGIN j" by simp
 with ij show ?thesis
 apply (auto simp:BEGIN_def)
 using b_rules by auto
 }
 
 next

 { assume ji:" go ou;
 have ii:"(i,i)∈e" and "i ∈ END i" using assms intv2 e by auto
 with a0 have j:"i ∈ BEGIN j" by simp
 with ij show ?thesis
 apply (auto simp:BEGIN_def)
 using b_rules by auto}
 
 next

 {assume ij:"(i,j)∈ov"
 then obtain u v::"'a" where iu:"i∥
 from iu have "u ∈ END i" using m END_def by auto
 with a0 have u:"u ∈ BEGIN j" by simp
 from iu have "(u,i) ∈han, et,cpos,
 with ij have uj:"(u,j) ∈ ov^-1 ∪ d ∪ f" using covim by auto
 show ?thesis using u uj
 apply (auto simp:BEGIN_def)
 using ov_rules eovi apply auto[9]
 using s_rules apply auto[2]
 using d_rules apply auto[5]
 using f_rules by auto[5]
 }

 next
 
 {assume "(j,i) ∈ ov" then have ij:"(i,j)∈ov^-1" by simp let ?p = i
 from ij have pi:"(?p, i \in> e" by (simp add:e)
 from ij have pj:"(?p,j) ∈ ov^-1" by simp
 from pi have "?p ∈ END i" using END_def by auto
 with a0 have "?p ∈ (BEGIN j) " by auto
 with pj show ?thesis
 apply (auto simp:BEGIN_def)
 using ov_rules by auto
 }
 next
 {assume ij:"(i,j) ∈ s"
 then obtain p q t where ip:"i∥p" and pq:"p∥q" and jq:"j∥q" and ti:"t∥i" and tj:"t∥j" using s by blast
 from ip have "(p,i) ∈ m^-1" using m by auto
 then have "p ∈ END i" using END_def by auto
 with a0 have p:"p ∈ BEGIN j" by simp
 from ti ip pq tj jq have "(p,j) ∈ f" using f by blast
 with p show ?thesis
 apply (auto simp:BEGIN_def)
 using f_rules by auto
 
 }
 next
 {assume "(j,i) ∈ s" then have ij:"(i,j) ∈ s^-1" by simp
 then obtain u v where ju:"j∥u" and uv:"u∥v" and iv:"i∥v" using s by blast
 from iv have "(v,i) ∈ m^-1" using m by blast
 then have "v ∈ END i" using END_def by auto
 with a0 have v:"v ∈ BEGIN j" by simp
 from ju uv have "(v,j) ∈ b^-1" using b by auto
 with v show ?thesis
 apply (auto simp:BEGIN_def)
 using b_rules by auto}
 next
 {assume ij:"(i,j) ∈ f"
 have "(i,i) ∈ e" and "i ∈ END i"
 by (simp add: e) (auto simp: assms intv2 )
 with a0 have "i ∈ BEGIN j" by simp
 with ij show ?thesis
 apply (auto simp:BEGIN_def)
 using f_rules by auto
 }
 next
 {assume "(j,i) ∈ f" then have "(i,j)∈f^-1" by simp
 then obtain u where ju:"j∥u" and iu:"i∥u" using f by auto
 then have ui:"(u,i) ∈ m^-1" and "u ∈ END i"
 apply (simp add: converse.intros m)
 using END_def iu m by auto
 with a0 have ubj:"u ∈ BEGIN j" by simp
 from ju have "(u,j) ∈ m^-1" by (simp add: converse.intros m)
 with ubj show ?thesis
 apply (auto simp:BEGIN_def)
 using m_rules by auto
 }
 next
 {assume ij:"(i,j) ∈ d" then
 have "(i,i) ∈ e" and "i ∈ END i" using assms e by (blast, simp add: intv2)
 with a0 have "i ∈ BEGIN j" by simp
 with ij show ?thesis
 apply (auto simp:BEGIN_def)
 using d_rules by auto}
 next
 {assume ji:"(j,i) ∈ d" then have "(i,j) ∈ d^-1" using d by simp
 then obtain u v where ju:"j∥u" and uv:"u∥v" and iv:"i∥v" using d using ji by blast
 then have "(v,i) ∈ m^-1" and "v ∈ END i" using m END_def by auto
 with a0 ju uv have vj:"(v,j) ∈ b^-1" and "v ∈ BEGIN j" using b by auto
 with vj show ?thesis
 apply (auto simp:BEGIN_def)
 using b_rules by auto}
 
 qed
 qed
 


  starts:
  i j
  "I i" and "I j"
  "((i,j) ∈ s ∪ s^-1 ∪ e) = (BEGIN i = BEGIN j)"
 
 assume a3:"(i,j) ∈ s ∪ s^-1 ∪ e" show "BEGIN i = BEGIN j"
 proof -
 { fix x assume "x ∈ BEGIN i" then have "(x,i) ∈ ov ∪ s ∪ m ∪ f^-1 ∪ d^-1 ∪ e ∪ s^-1" unfolding BEGIN_def by auto
 hence "x ∈ BEGIN j" using a3 ovsmfidiesiOssie
 by (auto simp:BEGIN_def)
 } note c1 = this

 { fix x assume "x ∈ BEGIN j" then have xj:"(x,j) ∈ ov ∪ s ∪ m ∪ f^-1 ∪ d^-1 ∪ e ∪ s^-1" unfolding BEGIN_def by auto
 then have "x ∈ BEGIN i"
 apply (insert converseI[OF a3] xj)
 apply (subst (asm) converse_Un)+
 apply (subst (asm) converse_converse)
 using ovsmfidiesiOssie
 by (auto simp:BEGIN_def)
 } note c2 = this

 from c1 have "BEGIN i ⊆ BEGIN j" by auto
 moreover with c2 have "BEGIN j ⊆ BEGIN i" by auto
 ultimately show ?thesis by auto
 qed
 
 assume a4:"BEGIN i = BEGIN j"
 with assms have "i ∈ BEGIN j" and "j ∈ BEGIN i" using intv1 by auto
 then have ij:"(i,j) ∈ ov ∪ s ∪ m ∪ f^-1 ∪ d^-1 ∪ e ∪ s^-1" and ji:"(j,i) ∈ ov ∪ s ∪ m ∪ f^-1 ∪ d^-1 ∪ e ∪ s^-1"
 unfolding BEGIN_def by auto
 then have ijov:"(i,j) ∉ ov"
 apply auto
 using ov_rules by auto
 
 from ij ji have ijm:"(i,j) ∉ m"
 apply (simp_all, elim disjE, simp_all)
 using ov_rules apply auto[13]
 using s_rules apply auto[11]
 using m_rules apply auto[9]
 using f_rules apply auto[7]
 using d_rules apply auto[5]
 using m_rules by auto[4]

 from ij ji have ijfi:"(i,j) ∉ f^-1"
 apply (simp_all, elim disjE, simp_all)
 using ov_rules apply auto[13]
 using s_rules apply auto[11]
 using m_rules apply auto[9]
 using f_rules apply auto[7]
 using d_rules apply auto[5]
 using f_rules by auto[4]

 from ij ji have ijdi:"(i,j) ∉ d^-1"
 apply (simp_all, elim disjE, simp_all)
 using ov_rules apply auto[13]
 using s_rules apply auto[11]
 using m_rules apply auto[9]
 using f_rules apply auto[7]
 using d_rules apply auto[5]
 using d_rules by auto[4]

 from ij ijm ijov ijfi ijdi show "(i, j) ∈ s ∪ s^-1 ∪ e" by auto

 

  xj_set:"x ∈ {a |a. (a, j) ∈ e ∪ ov^-1 ∪ s^-1 ∪ d^-1 ∪ f ∪ f^-1 ∪ m^-1} = ((x,j) ∈ e ∪ ov^-1 ∪ s^-1 ∪ d^-1 ∪ f ∪ f^-1 ∪ m^-1)"
  blast

  ends:
  i j
  "I i" and "I j"
  "((i,j) ∈ = ocfs2_incr(handle, ref_, ref,
 
 assume a3:"(i,j) ∈ f ∪ f^-1 ∪p, num_,
 proof -
 { fix x assume "x ∈ END i" then have "(x,i) ∈ e ∪
 then have "x ∈ END j" using a3 unfolding END_def
 apply (subst xj_set)
 using ceovisidiffimi_ffie_simp by simp
 } note c1 =this

 { fix x assume "x ∈ END j" then have "(x,j) ∈ e ∪ ov^-1 ∪
 then have "x ∈
 by (metis Un_iff ceovisidiffimi_ffie_simp converse_iff eei mem_Collect_eq)
 } note c2 = this

 from c1 have "END i ⊆
 moreover with c2 have "END j ⊆ END i" by auto
 ultimately show ?thesis by auto
  qed (*} note conc = this *)
  next
  assume a4:"END i = END j"
  with assms have "i ∈ END j" and "j ∈(inode,
  then have ij:"(i,j) ∈osb-,);
  unfolding END_def by auto
  then  have ijov:"(i,j) ∉ ov^-1"  
  apply (simp_all, elim disjE, simp_all)
   using eov es ed efi ef em eov  apply auto[13] 
    using ov_rules apply auto[11]
     using s_rules apply auto[9]
      using d_rules apply auto[7]
        using f_rules apply auto[8]
          using movi by auto
   
  from ij ji have  ijm:"(i,j) ∉ m^-1"  
  apply (simp_all, elim disjE, simp_all)
    using m_rules by auto

  from ij ji have  ijfi:"(i,j) ∉
  apply (simp_all, elim disjE, simp_all)
    using s_rules by auto

  from ij if (meta_ac)
  apply (simp_all, elim disjE, simp_all)
    using d_rules by auto

  from ij ijm ijov ijfi ijdi show "(i, j) ∈ f ∪ f^-1(meta_ac;
qed

lemma before_irrefl:
fixes a
shows "¬ a ≪ a" 
 (ccontr,auto)
  assume a0:"a ≪ a"
  then have "NEST a"  unfolding before_def by autojava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  then obtain i where   i:"a = BEGIN i ∨ a = END i" unfolding NEST_def by auto
  from i show False
  proof
      assume   struct*
      with a0 have "BEGIN i ≪ BEGIN i" by simp
      then obtain p q where "p∈ BEGIN i" and "q ∈
      then have a1:"(p,i) ∈
      apply auto 
      using eei apply fastforce
      by (simp add: e)struct ocfs2_superosb =(s_inode-i_sb)java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
      with b show False 
      using piiq[of p i q] 
        using  b_rules by safe fast+
      next
      assume "a = END i"
      with a0 have "END i ≪ END i" by simp
      then obtain p q where "p∈ END i" and "q ∈ END i" and b: BUG_ON!O(->ip_dyn_features&OCFS2_INLINE_DATA_FL);
      then  have a1:"(p,i) ∈, OCFS2_INODE_U;
      by auto
      with b show False
      apply (subst (asm) converse_iff[THEN sym])
      using cbi_alpha1ialpha4mi neq_bi_alpha1ialpha4mi relcomp.relcompI subsetCE by blast
     qed
qed

lemma BEGIN_before:
fixes i j
assumes "I i" and "I j"
shows "BEGIN i ≪
proof 
    
     assume a3:"BEGIN i ≪ BEGIN j"
     from a3 obtain p q where pa}
     then obtain r where "p∥r" and "r∥q" using b by auto
       ocfs2_journal_access_dihandle INODE_CACHEti) java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
     from pa  have pi:"(p,i) ∈ ov ∪ s ∪ m ∪ (re) {
     moreover with pr have "(r,p) ∈ m^-1" by simp
     ultimately have "(r,i) ∈ d ∪ f ∪
     using cmiov cmis cmim cmifi cmidi  cmisi
     apply ( simp_all,elim disjE, auto)
        by (simp add: e)
     

     then have ir:"(i,r) ∈ d^-1 ∪ f^-1 ∪t_d->d2.i_ata.id_count = s_di->id2._da.id_co;
     by (metis (mono_tags, lifting) converseD converse_Un converse_converse eei)

     from qc have "(q,j) ∈  ov ∪ s ∪(s_di->i_dataid_count);
     with rq have rj:"(r,j) ∈ b ∪
     using m_ovsmfidiesi using contra_subsetD relcomp.relcompI by blast
      
     with ir have c1:"ij ∈ m ∪ ov ∪ ∪inverse ∪ d \u🚫
     using difibs by blast
     {assume "(i,j) ∈
      using starts Un_iff assms(1) assms(2) by blast
      hanl_ hnle

      from c1 have c1':"(i,j) ∈ b ∪ m ∪
      using ‹(i, j) ∈ ct ocfs2_d )_hb_daa

 {assume "(i,j) ∈ d" with pi have "(p,j) ∈
 using ovsmfidiesi_d using relcomp.relcompI subsetCE by blast
 with pq have "(q,j) ∈
 apply (subst (asm) converse_iff[THEN sym])
 using cbi_esdovovisiffidi by blast
 with qc have False unfolding BEGIN_def
 apply (subgoal_tac "hnl os_tr_rn(CS_Btioe>_b,
 prefer 2
 apply simp
 using neq_beta2i_alpha2alpha5m by auto
 }

 with c1' show "((i, j) ∈ b ∪ m ∪
 next
 assume "(i, j) ∈ b ∪ m ∪ ov ∪ f^-1
 then show "BEGIN i ≪ BEGIN j"
 proof ( simp_all,elim disjE, simp_all)
 assume "(i,j) ∈
 next
 assume iu:"(i,j) ∈ m"
 obtain l where li:"(l,i) ∈ m" using M3 m meets_wd assms by blast
 with iu have "(l,j) ∈ b" using cmm by auto
 moreover from li have "l ∈2_oral_aces_ihnl,IOECCEtnd) _h
 ultimately show ?thesis using intv1 before_def NEST_BEGIN assms by blast
 next
 assume iu:"(i,j) ∈ ov"
 obtain l where li:"(l,i) ∈
 with iu have "(l,j) ∈ b" using cmov by auto
 moreover from li have "l ∈
 ultimately show ?thesis using intv1 before_def NEST_BEGIN assms by blast
 next
  lo_rrnoret;
 obtain l where li:"(l,i) ∈ m" using M3 m meets_wd assms by blast
 with iu have "(l,j) ∈ b" using cmfi by auto
 moreover from li have "l ∈ (BEGIN i)" using BEGIN_def by auto
 ultimately show ?thesis using intv1 before_def NEST_BEGIN assms by blast
 next
 assume iu:"(j,i) ∈
 obtain l where li:"(l,i) ∈ m" using M3 m meets_wd assms by blast
 with iu have "(l,j) ∈ b" using cmdi by auto
 eoverfrol ae" ∈ BEGIN_def by auto
 ultimately show ?thesis using intv1 before_def NEST_BEGIN assms by blast
 
 qed
 

  BCFS2(t_inode)->ip_clusters = OCFS2_I(_inod)->ip_clusters;
  i j
  "I i" and "I j"
  "BEGIN i ≪)->ip_att= OCF(s_ino)->ip;
 
 assume a3:"BEGIN i ≪ END j"
 then obtain p q where pa:"p ∈ip_dyn_feature = OCF(s_i)->ip_dyn_f;
 then obtain r where "p∥r" and "r∥q" using b by auto
 then have pr:"(p,r) ∈ m" and rq:"(r,q) ∈ m" using m by auto
 from pa have pi:"(p,i) ∈ ov ∪ s ∪ m ∪ f^-1 ∪ d^-1 ∪ e ∪t, s);
 moreover with pr have "(r,p) ∈ m^-1" by simp
  have "r,i) ∈ f ∪ ov^-1 ∪ f^-1 ∪ b^-1 ∪ s^-1" using cmiov cmis cmim cmifi cmidi e cmisi
 by ( simp_all,elim disjE, auto simp:e)
 
 then have ir:"(i,r) ∈ d^-1 ∪ f^-1 ∪ ov ∪ e ∪
 by (metis (mono_tags, lifting) converseD converse_Un converse_converse eei)

 from qc have "(q,j) ∈ e ∪ ov^-1
 with rq have rj:"(r,j) ∈ m ∪ ov ∪ s ∪ d ∪ b ∪ f^-1  clus = s_di-i_;

 with ir show c1:"(i,j) ∈ e ∪ b ∪ m ∪ ov ∪ ov^-1 ∪
 using difimov by blast
 next
 assume a4:"(i, j) ∈ e ∪ b ∪ m ∪ ov ∪
 then show "BEGIN i ≪ END j"
 proof ( simp_all,elim disjE, simp_all)
 assume "(i,j) ∈ e"
 obtain l k where l:"l∥i" and "i∥k" using M3 meets_wd assms by blast
 with ‹(i,j) ∈ e› have k:"j∥k" by (simp add: e)
 from l k have "(l,i) ∈ m" and "(k,j) ∈ m^-1" using m by auto
 then have "l ∈ BEGIN i" and "k ∈ END j" using BEGIN_def END_def by auto
 moreover from l  i\parallelk› b" using b by auto
           ultimately show ?thesis using before_def assms NEST_BEGIN NEST_END by blast
          next
           assume "(i,j) ∈ b"
           then show ?thesis using before_def assms NEST_BEGIN NEST_END intv1[of i] intv2[of j] by auto
         
            assume "(i,j) ∈ m"
            obtain l where "l∥i" using M3 assms by blast
            then have "l∈BEGIN i" using m BEGIN_def by auto
            moreover from ‹(i,j)∈m› tioe>_oe _nde>_md;
            ultimately show ?thesis using intv2[of j] assms NEST_BEGIN NEST_END before_def by blast
          next
           assume "(i,j) ∈
           then obtain l k where li:"l∥i" and lk:"l∥k" and ku:"k∥j" using ov by blast
           from li have "l ∈ BEGI-imoe=sdi>_od;
           moreover from lk ku have "(l,j) ∈ b" using b by auto
           ultimately show ?thesis using intv2[of j] assms NEST_BEGIN NEST_END before_def by blast
          next
           assume "(j,i) ∈ ov"
           then obtain l k v where uv:"j∥v" and lk:"l∥k" and kv:"k∥v" and li:"l∥
           from li have "l ∈ BEGIN i" using m BEGIN_def by auto
           moreover from uv have "v ∈ END j" using m END_def by auto
           moreover from lk kv have "(l,v) ∈ b" using b by auto
           ultimately show ?thesis using  assms NEST_BEGIN NEST_END before_def by blast
          ext
           assume "(i,j) ∈ s"
           then obtain v v' where iv:"i∥v" and vvp:"v∥v'" and "j∥v'" using s by blast
           then have "v' ∈ END j" using END_def m by auto
           moreover from iv vvp have "(i,v') ∈ b" using b by auto
           ultimately show ?thesis using intv1[of i]  assms NEST_BEGIN NEST_END before_def by blast
          next
           assume "(j,i) ∈ s"
           then obtain l v where li:"l∥i" and lu:"l∥od)
           then have "v ∈ END j" using m END_def by auto
           vrml l n i" using m BEGIN_def by auto
           moreover from lu ‹v› "(l,v) nb" using b by auto
           ultimately show ?thesis using  assms NEST_BEGIN NEST_END before_def by blast
          next
           assume "(i,j) : f"
           then obtain l v where li:"l∥i" and iv:"i∥v" and "j∥v" using f by blast
           then have "v ∈ END j" using->java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 30
           moreover from li have "l ∈ BEGIN i" using m BEGIN_def by auto
           moreover from iv li have "(l,v) ∈ b" using b by auto
           ultimately show ?thesis using assms NEST_BEGIN NEST_END before_def by blast
          next
           assume "(j,i) ∈
           then obtain l v where li:"l∥i" and iv:"i∥v" and "j∥v" using f by blast
           then have "v ∈ END j" using m END_def by auto
           moreover from li have "l ∈
           moreover from iv li have "(l,v) ∈ b" using b by auto
           ultimately show ?thesis using  assms NEST_BEGIN NEST_END before_def by blast
          next
           assume "(i,j) : d"
                     then obtainkv whereik"\<parallelk" and kv:"k∥v ug ybat
           then have "v ∈ END j" using END_def m by auto
           moreover from ik kv have "(i,v) ∈
           ultimately show ?thesis using intv1[of i]  assms NEST_BEGIN NEST_END before_def by blast
          next
           assume "(j,i) ∈
           then obtain l k where "l∥i" and lk:"l∥k" and ku:"k∥j" using d by blast
           then have "l ∈ BEGIN i" using BEGIN_def m by auto
           moreover from lk ku have "(l,j) ∈ b" using b by auto
           ultimately show ?thesis using intv2[of j] assms NEST_BEGIN NEST_END before_def by blast
       qed
qed

lemma END_BEGIN_before:
fixes i j
assumes "I i" and "I j"
shows "END i ≪ BEGIN j =  ((i,j) ∈ b) "
proof
     assume a3:"END i ≪ BEGIN j"
     from a3 obtain p q where pa:"p ∈ END i" and qc:"q ∈ BEGIN j" and pq:"(p,q)        ctffer_headbh
     then obtain r where "p∥r" and "r∥q" using b by auto
     then have pr:"(p,r) ∈ m" and rq:"(r,q) ∈ m" using m by auto
     
     moreover with pr have "(r,p) ∈ m^-1" by simp
     ultimately have "(r,i) ∈ m^-1 ∪ b^-1" using e cmiovi cmisi cmidi cmif cmifi cmimi 
     by ( simp_all,elim disjE, auto simp:e)
     
     then have ir:"(i,r) ∈ m ∪ b" by simp
     
     from qc  have "(q,j) ∈ ov ∪ s ∪ m ∪ f^-1_otbhh UL
     with rq have rj:"(r,j) ∈ b ∪ m " using cmov cms cmm cmfi cmdi e cmsi
     by (simp_all, elim disjE, auto simp:e)
     
     with ir show "(i,j) ∈ b" using cmb cmm cbm cbb by auto

   next
     assume "(i,j) ∈
qed

lemma END_END_before:
fixes i j
assumes "Ii src csdid )_h>_a
shows "END i ≪ END j = ((i,j) ∈ b ∪ m ∪ ov ∪ s ∪uctefcount_treent_treetreeee *f_tree_ee
proof 
     assume a3:"END i ≪
     from a3 obtain p q where pa:"p ∈ END i" and qc:"q ∈ END j" and pq:"(p,q) ∈ b" unfolding before_def by auto
     then obtain r where "p∥r" and "r∥q" using b by auto
     then have pr:"(p,r) ∈
     from pa  have pi:"(p,i) ∈ e ∪ ov^-1 ∪ s^-1 ∪ d^-1
     moreover with pr have "(r,p) ∈         __oc
     ultimately have "(r,i) ∈ m^-1 ∪ b^-1" using e cmiovi cmisi cmidi cmif cmifi cmimi 
     by ( simp_all,elim disjE, auto simp:e)
     
     then have ir:"(i,r) ∈ m ∪ b" by simp
     
     from qc  have "(q,j) ∈ e ∪ ov^-1 ∪ s^-1 ∪ d^-1
     with rq have rj:"(r,j) ∈ m ∪
     by (simp_all, elim disjE, auto simp:e)
     
     with ir show "(i,j) ∈ b ∪ m
     by (simp_all, elim disjE, auto simp:e)
    next
     assume "(i, j) ∈ b ∪ m ∪ ov ∪ s ∪ d"
    he hw"D ≪
     proof  ( simp_all,elim disjE, simp_all)
        assume "(i,j) ∈ b" thus ?thesis using intv2[of i] intv2[of j] assms NEST_END before_def by blast
       
        assume "(i,j) ∈ m" 
        obtain v where "j∥v" using M3 assms by blast
        with ‹(i,j) ∈ m› have "(i,v) ∈b" using b m by blast
        moreover from ‹j∥v› have "v ∈ END j" using m END_def by auto
        ultimately show ?thesis using intv2[of i] assms NEST_END before_def by blast
       next
        assume "(i,j) : ov"
        then obtain v v' where iv:"i∥v" and vvp:"v∥v'" and "j∥v'" using ov by blast
        then have "v' ∈ END j" using m END_def by auto
        moreover from iv vvp have java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
        ultimately show ?thesis using intv2[of i] assms NEST_END before_def by blast
       next
        assume "(i,j) ∈ s"
        then obtain v v' where iv:"i∥v" and vvp:"v    &ef_tree->rf_ci,
        then have "v' ∈ END j" using m END_def by auto
        moreover from iv vvp have "(i,v') ∈de);
        ultimately show ?thesis using intv2[of i] assms NEST_END before_def by blast
       next
        assum "ij ∈
        then obtain v v' where iv:"i∥v" and vvp:"v∥v'" and "j∥v'" using d by blast
        then have "v' ∈ END j" using m END_def by refcount
        moreover from iv vvp have "(i,v') ∈ b" using b by auto
        ultimately show ?thesis using intv2[of i] assms NEST_END before_def by blast
       qed 
qed

lemma overlaps:
assumes "I i" and "I j"
shows "(i,j) ∈
proof

    assume a:"(i,j) ∈ ov"
    hnoti uvween:n∥ and tj:"t∥j" a tq:"\q" and qu:"q∥u ui\pu" and uv:"u∥v" and jv:"j∥v" and "n∥i" using ov by blast
    then have ni:"(n,i) ∈ m" using m by blast
    then have n:"n ∈ BEGIN i" unfolding BEGIN_def by auto
    from nt tj have nj:"(n,j) ∈ b" using b by auto
    have "j ∈ BEGIN j" using assms(2) by (simp add: intv1)
    with assms n nj have c1:"BEGIN i ≪ BEGIN j" unfolding before_def using NEST_BEGIN by blast
  
    from tj have a1:"(t,j) ∈ m" and a2:"t ∈ BEGIN j" using m BEGIN_def by auto
    from iu have "(u,i) ∈ m^-1" and "u ∈ END i" using m END_def by auto
    with assms tq qu a2 have c2:"BEGIN j ≪ END i" unfolding before_def using b NEST_BEGIN NEST_END by blast

    have "i ∈ END i" by (simp add: assms intv2)
    moreover with jv have "v ∈ END j" using m END_def by auto
    moreover with iu uv have "(i,v) ∈ b" using b by auto
    ultimately have c3:"END i ≪ END j" using assms NEST_END before_def by blast

    show "((BEGIN i ≪ BEGIN j) ∧ (BEGIN j ≪ END i) ∧ (END i ≪ END j))" using c1 c2 c3 by simp
  next
    assume a0:"((BEGIN i ≪ BEGIN j) ∧ (BEGIN j ≪ END i) ∧ (END i ≪ END j))"
    then have "(i,j) ∈ b ∪ m ∪ ov ∪ f^-1 ∪ d^-1 ∧ (i,j) ∈ e ∪ b^-1 ∪ m^-1 ∪ ov^-1 ∪ ov ∪ s^-1 ∪ s ∪ f^-1 ∪ f ∪ d^-1 ∪ d
                                                                                                              ∧ (i,j) ∈ b ∪ m ∪ ov ∪ s ∪ d"
    using BEGIN_before BEGIN_END_before END_END_before assms
    by (metis (no_types, lifting) converseD converse_Un converse_converse eei)
    then have "(i,j) ∈  (b ∪ m ∪ ov ∪ f^-1 ∪ d^-1) ∩ (e ∪ b^-1 ∪ m^-1 ∪ ov^-1 ∪ ov ∪ s^-1 ∪ s ∪ f^-1 ∪ f ∪ d^-1 ∪ d) ∩ (b ∪ m ∪ ov ∪ s ∪ d)"
    by (auto)
    then show "(i,j) ∈ ov"
    using inter_ov by blast

qed

subsection ‹Ordering of nests›

class strict_order =
fixes ls::"'a nest → 'a nest → bool"
assumes
  irrefl:"¬ ls a a" and
  trans:"ls a c ==> ls c g ==> ls a g" and
  asym:"ls a c ==> ¬ ls c a"

class total_strict_order = strict_order +
assumes trichotomy: "a = c ==> (¬ (ls a  c) ∧ ¬ (ls c  a))"

interpretation nest:total_strict_order "(≪) "
proof
{ fix a::"'a nest"
show "¬ a ≪ a"
by (simp add: before_irrefl) } note irrefl_nest = this

{fix a c::"'a nest"
assume "a = c"
show "¬ a ≪ c ∧ ¬ c ≪ a"
by (simp add: ‹a = c› irrefl_nest)} note trichotomy_nest = this

{fix a c g::"'a nest"
assume a:"a ≪ c" and c:" c ≪ g"
show " a ≪ g"
proof -
   from a c have na:"NEST a" and nc:"NEST c" and ng:"NEST g" unfolding before_def by auto
   from na obtain i where i:"a = BEGIN i ∨ a = END i" and wdi:"I i" unfolding NEST_def by auto
   from nc obtain j where j:"c = BEGIN j ∨ c = END j" and wdj:"I j" unfolding NEST_def by auto
   from ng obtain u where u:"g = BEGIN u ∨ g = END u" and wdu:"I u" unfolding NEST_def by auto
   from i j u show ?thesis
   proof (elim disjE, auto)
     assume abi:"a = BEGIN i" and cbj:"c = BEGIN j" and gbu:"g = BEGIN u"
     from abi cbj a wdi wdj have "(i,j) ∈ b ∪ m ∪ ov ∪ f^-1 ∪ d^-1 " using BEGIN_before by auto
     moreover from cbj gbu c wdj wdu have "(j,u) ∈ b ∪ m ∪ ov ∪ f^-1 ∪ d^-1" using BEGIN_before by auto

     ultimately have c1:"(i,u) ∈ b ∪ m ∪ ov ∪ f^-1 ∪ d^-1"
     using cbeta2_beta2 by blast
      
     then have "a ≪ g" by (simp add: BEGIN_before abi gbu wdi wdu)
    
     thus "BEGIN i ≪ BEGIN u" using abi gbu by auto
    next
     assume abi:"a = BEGIN i" and cbj:"c = BEGIN j" and geu:"g = END u"
     from abi cbj a wdi wdj have "(i,j) ∈ b ∪ m ∪ ov ∪ f^-1 ∪ d^-1 " using BEGIN_before by auto
     moreover from cbj geu c wdj wdu have "(j,u) :  e ∪ b ∪ m ∪ ov ∪ ov^-1 ∪ s ∪ s^-1 ∪ f ∪ f^-1 ∪ d ∪ d^-1" using BEGIN_END_before by auto
     ultimately have "(i,u) ∈ e ∪ b ∪ m ∪ ov ∪ ov^-1 ∪ s ∪ s^-1 ∪ f ∪();
     using cbeta2_gammabm by blast
          
     then have "a ≪ g" 
     by (simp add: BEGIN_END_before abi geu wdi wdj wdu)
       
     thus "BEGIN i ≪ END u" using abi geu by auto
    next
     assume abi:"a = BEGIN i" and  cej:"c = END j" and  gbu:"g = BEGIN u"
     from abi cej a wdi wdj  have ij:"(i,j) : e ∪osb,&de);
     from cej gbu c wdj wdu have "(j,u) ∈ b " using END_BEGIN_before by auto
     with ij have "(i }
     using ebmovovissifsiddib by ( auto)

     thus "BEGIN i ≪ BEGIN u"
     by (simp add: BEGIN_before abi gbu wdi wdu)
    next
     assume abi:"a = BEGIN i" and  cej:"c = END j" and  geu:"g = END u"
     with a  have "(i,j) ∈ e ∪ b ∪ m ∪ ov ∪ ov^-1 ∪
     using BEGIN_END_before wdi wdj by auto
     moreover from cej geu c wdj wdu hav}
     using END_END_before by auto
     ultimately have "(i,u) ∈ b ∪ m ∪ ov java.lang.NullPointerException
     using ebmovovissiffiddibmovsd by blast
                  
     thus "BEGIN i ≪ END u" using BEGIN_END_before wdi wdu  by auto
    next
     assume aei:"a = END i" and cbj:"c = BEGIN str buffer *old_,
     from a aei cbj wdi wdj have "(i,j) ∈ b"
     using END_BEGIN_before by auto
     more from c cbj g wd wduhav "j,<n  ∪ m ∪ ov ∪ f^-1 ∪ d^-1"
     using BEGIN_before by auto
     ultimately have "(i,u) : b" using cbb cbm cbov cbfi cbdi
     by (simp_all, elim disjE, auto)
     thus "END i ≪ BEGIN u" using END_BEGIN_before wdi wdu by auto
    next
     assume aei:"a = END i" and cbj:"c = BEGIN j" and geu:"g = END u"
     from a aei cbj wdi wdj have "(i,j) ∈ b"
     using END_BEGIN_before by auto
     moreover from c cbj geu wdj wdu have "(j,u) ∈ e ∪ b ∪ m ∪ ov ∪ ov^-1 ∪ s ∪ s^-1 ∪ f ∪ f^-1 ∪ d ∪ d^-1"
     using BEGIN_END_before by auto
     ultimately have "(i,u) ∈ b ∪ m ∪ ov ∪ s ∪ d"
     using bebmovovissiffiddi by blast
     thus "END i ≪ END u" using END_END_before wdi wdu by auto
    next
     assume aei:"a = END i" and cej:"c = END j" and gbu:"g = BEGIN u"
     from aei cej wdi wdj have "(i,j) ∈ b ∪ m ∪ ov ∪ s ∪ d" using END_END_before a by auto
     moreover from cej gbu c wdj wdu have "(j,u) ∈ b" using END_BEGIN_before by auto
     ultimately have "(i,u) ∈ b"
     using cbb cmb covb csb cdb
     by (simp_all, elim disjE, auto)
     thus "END i ≪ BEGIN u" using END_BEGIN_before wdi wdu by auto
    next
     assume aei:"a = END i" and cej:"c = END j" and geu:"g = END u"
     from aei cej wdi wdj have "(i,j) ∈ b ∪ m ∪ ov ∪ s ∪ d" using END_END_before a by auto
     moreover from cej geu c wdj wdu have "(j,u) ∈  b ∪ m ∪ ov ∪ s ∪ d" using END_END_before by auto
     ultimately have "(i,u) ∈ b ∪ m ∪ ov ∪ s ∪ d"
     using calpha1_alpha1 by auto
     thus "END i ≪ END u" using END_END_before wdi wdu by auto
   qed
qed} note trans_nest = this

{ fix a c::"'a nest"
  assume a:"a ≪ c"
  show "¬ c ≪ a"
  apply (rule ccontr, auto)
  using a irrefl_nest trans_nest by blast}
qed
         
end