Impressum allen.thy

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

section


theory

imports axioms 

  Main
  "HOL- <<>


begin

section ‹

text‹::('a\times>'a set"  and
  e, m, b, ov, d, s and f stand for equal, meets, before, overls, durig ststarts aand finishs, r restively.\\close>

  arelations = interval +
 fixes
 e:"('a\times>'a) set" and
 m::"('a\<>"'a)"and
 b::"('a×'a) set" and
 ov::"('a×'a) set" and
 d::"('a×'a) set" an
 s::f::(a\timesa set"
 
 
 e:"(p,q) ∈
 m:"(p,q) ∈ m = p∥> b = (∃t ∧q)" and
 b:"(p,q) ∈ ov = (∃ luvt::'a.
 ov:"(p,q) \<                   (
 (k∥p ∧ p∥u ∧ u∥v) ∧ (k∥l ∧:"(,)\in=exists>k u v::'a. k∥p ∧ p∥u ∧ u∥v ∧ k∥q ∧ q∥v)" and
 s:"(p,q) ∈k u v::'a. k∥ p∥ u∥ k∥ q∥
 f:"(p,q) ∈k l u ::'a. k∥ l∥ p∥ k∥ q<>u
 d:"(p,q) ∈e-composition›
 

(** e compositions **)
subsectionRelation e is the identity relation for composition.›<> 
text ‹Relation e is the identity relation for composition.›

lemma cer:
assumes  "r \<in> {e,m,b,ov,s,f,d,m^-1,b^-1,ov^-1,s^-1,f^-1,d^-1}" 
java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 17
proof -
  { fix x y assume a:"(x,y) \<in> e O r" 
    then obtain z where "(x,z) \<in> e" and "(z,y) \<in> r" by auto
     \open(x,z \in> e<> "      java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
    with \<open>(z,y)\<in> r\<close> have "(x,y) \<in> r" by simp} note c1with havexy <     blast} notec2 = this
  
 { fix x y assume a:"(x,y) \<in>  r"
   have "(x,x) \<in> e" using e by auto
    a  "x \in>e  r"by blast} note  = java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
 
 from c1 c2 show ?thesis by auto
qed

lemma cre:
assumes  "r \<in> {e,m,b,ov,s,f,d,m^-1,b^-1,ov^-1,s^-1,f^-1,d^-1}"
shows " r O e = r"
proof -
  {fixx y assumea:"xy \<inrOe 
    then obtain z where "(x,z) \<in> r" and "(z,y) \<in> e" by auto
    from \<open>(z,y) \<in> e\<close> have "z = y" using e by auto
    with \<open>(x,z)\<in> r\<close> have "(x,y) \<in> r" by simp} note c1 = this
  
 { fix x y assume a:"(x,y) \<in>  r"
   have "(y,y) \<in> e" using e by auto
   with a have "(x,y) \<in> r O e" by blast} note c2 = this
 
 from c1 c2 show ?thesis by auto
qed

lemmas ceb = cer[of b]
lemmas cebi = cer[of"^-"
lemmas cem = cer[of m]
lemmas cemi = cer[of "m^-1"]
lemmas cee = cer[of e]
lemmas ces = cer[of s]
lemmas cesi = cer[of "s^-1"]
lemmas cef = cer[of f]
lemmas cefi = cer[of "f^-1"]
lemmas ceov = cer[of ov]
lemmas ceovi = cer[of "ov^-1"]
lemmas ced = cer[of d]
lemmas cedi = cer[of "d^-1"]
lemmas cbe = cre[of b]
lemmas cbie = cre[of "b^-1"]
lemmas cme = cre[of m]
lemmas cmie = cre[of "m^-1"]
lemmas cse = cre[of s]
lemmas csie = cre[of "s^-1"]
lemmas cfe = cre[of f]
lemmas cfie = cre[of "f^-1"]
lemmas cove = cre[of ov]
lemmas covie = cre[of "ov^-1"]
lemmas cde = cre[of d]
lemmas cdie = cre[of "d^-1"]

(*******)

(* composition with single relation *)
subsection
text java.lang.NullPointerException

method (in arelations) r_compose uses r1 r2 r3 = ((autobstst)meson)


lemmarelations b"
  by

lemma (in arelations) cbm:"b  msubseteqb"
  by (r_compose r1:b r2:m r3:b)

lemma cbov:"b O ov::)
  apply (auto simp:)   bmsubseteq b"
  using M1 M5exist_var by blast

lemma cbfi:"b O f^-1 ⊆  uto
  mma b"
  by (meson M1 M5exist_var)

lemma cbdi:"b O d^-1 ⊆ b"
  apply (auto simp: b d)
  byeso 5itv
 
lemma cbs:"b Ojava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  apply (auto simp: b s)
  by (meson M1 M5exist_var)

lemma cbsii::"O f^-1 \subseteq b"
  apply (auto simp: b s)
  by (meson M1 M5exist_varby meson

lemma lemma:" -1<s b"
  by (r_compose r1using

lemma cmm:"m O m ⊆ b"
  by (auto simp: b m)

lemma cmov:"m O ov ⊆ b"
  apply (auto simp:b m ov)
  using M1 M5exist_var by blast

lemma cmfi:"m O f^-1 ⊆ b"
  apply (r_compose r1:m r2:f r3:b)
  by (meson M1)

di\subseteq b"
  apply (auto simp add:m d b)
  using M1 by blast

lemma cms:"m O s ⊆ m"
  apply (auto simp add:m s)
  using M1 by auto

lemma cmsi:"m O s^-1 ⊆java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
  apply (auto simp add:m s)
  using M1ovwherex" and xu:"x∥d"\parallel" and ty:"t∥v" and ku:"k∥

lemma covb:"ov O b ⊆ b"
  apply (auto simp:ov b)
  using M1 M5exist_var by blast

lemma covm:"ov O m ⊆<>ul1z" using M1by blast

  using M1 by blast

lemma covs:"ov O s
proof cfiov ov"
  fix p::"'a×obtain= (,)nd ov" and yzs:"(y,z) ∈
  from xyov obtain r u v herex" and xu:"x∥v" and rt:"r∥k" and ty:"t∥v" and ku:"k∥
  from yzsreparallell1" and l1l2:"l1∥l2" using s by blast
  from uv yl1 yv have "u∥
  withl1l2ereparallelul1" and ul1l2:"ul1∥
  from ku xu xul1 l1l2 havel1ul1" using M1 by blast
  from ty yzs have "tveparallelz" using M1 by blast
  with rx rt xul1 ul1l2 zl2 tk kul1 have "(  "<>ov" by simp
  with p showlemma"^1\ f^-1"
qed

lemma cfib:"f^-1 O b ⊆
  apply (auto simp:f b)
  using M1 by blast

lemma cfim:"f^-1 O m ⊆u'" using M1 by blast
  apply (auto simp:f m)
  using M1 by auto

lemma cfiov:"f^-1 O ov ⊆p∥u'›w"x \ f^-1"usingfbyblast
proof 
    fix p::"'a×'a" assume "p ∈ f^-1 O ov" then obtain x y z where p:"p = (x,z)" and xyfi:"(x,y)∈ f^-1" and yzov:"(y,z) ∈-1 d-1 ⊆
    from xyfi yzov obtain t' r u where tpr:"<parallel:r<>y" and yu:"y∥x" and xu:"x∥
    from yzov  ry  obtain v k t u' wherep:<paralleluandparallelv" and rk:"r∥z" and zv:"z∥t" and tup:"t∥u'"
    using ov using M1 by blast
    from yu xu yup have xup:"x∥ u " using M1 by blast
    from tpr rk kt obtain r' where tprp: kpu qup d b by bla
    from kt rpt kz have rpz:"r'∥
    from tprp rpz rpt tpx xup tup< ov" using ov by blast
with<> ov" by simp
qed

lemma cfifi:"f^-1 O f^-1 ⊆
proof
  fix x::"'a× f^-1 O f^-1" then obtain p q z where x:"x = (p, q  ",) \<^- 
  from ‹p" and kl:"k∥z" and pu:"p∥u" using f by blast
  from ‹
  from zu zup pu have "  :f-1O-1 \subseteq d^-1"
 from lz kpz kplp have "l∥'a" assume "x ∈1"hen obbta p q z were x:,q)" and()\in f^-1" and "(z,q) ∈ s^-1" by auto
 with kl lpq obtain ll where "k∥q" using M5exist_var by blast
 with kp ‹v'" and zvp:"z∥(z,q): s^-1› by blast
 

  cfdi:"^1Od-\subseteq d^-1"
 
 fix x::"'a× f^-1" and "(z,q) ∈ao
 then obtain k l u where kp:"k ∥
 obtain kcdi:"d^-1 Of-1 ⊆^-1"
 from lz kpz kplp have "l∥ng M by by blast
 with kl lpq obtain ll where "k∥'a" assume "x : d^-1 O f^-1" then obtain p q z where x:"x = (p,q)" and "(p,\<in d^ and"zq) ∈
 over r fromm zu zvp upvp have "u' ∥ u " using M1 by blast
 ultimately show "x ∈obtain k' l' u' e kpz:"k:"k\<parallelz" and kplp:"k' ∥q" and qup:"q ∥u'" using f ‹
 

 ^-O s ⊆
 
 fix x::"'a×
 from ‹)" annd "(p,z) d^-1" and "(z,q) ∈
 from ‹z" and kpq:"k'∥u'" and upvp:"u'∥p:"q\q\parallel'" using s M1 by blast
 from pu zu zup have pup:"p∥u'" using M1 by blast
 moreover from lz kpz kpq have lq:"l∥
 ultimately show "x ∈in xx zzup kp kl upvp uppvpov qvp by last
 

  cfisi:"f^-1 O s^-1 ⊆ u " using M1 by blast
 
 fix x::"'a×obi qz where x"x (p,q)" nd ",z) \in> f^-1" and "(z,q) ∈
 then obtain k l u where kp:"k ∥
 obtain k' u' v' where kpz:"k' ∥ d^-1"
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
java.lang.StringIndexOutOfBoundsException: Index 178 out of bounds for length 67
 ultimatelyu' v' where kpz:"' \<parallelz" and kpq:"k' ∥parallel>u'" and upvp:"u' ∥v'" and zvp:"z ∥v'" using s ‹(z,q): s^-1› by blast
 

  cdifi:"d^-1 O f^-1 ⊆ ^-1"
 
 fix x::"'a×u" using M1 by blast
 then obtain k l u v where kp:"k ∥here"\<>uu
java.lang.StringIndexOutOfBoundsException: Range [89, 3) out of bounds for length 98
 from lz kpz kplp have "l∥l'" using M1 by blast
 with kl lpq obtain ll where "k∥ll" and "ll∥q" using M5exist_var by blast
 moreover from zu qup zup have "q ∥ u " using M1 by blast
 ultimately show "x ∈ d^-1" using x d kp uv pv by blast
 

  cdidi:"d^-1 O d^-1 ⊆ d^-1"
 
 fix x::"'a×'a" assume "x : d^-1 O d^-1" then obtain p q z where x:"x = (p,q)" and "(p,z) ∈ d^-1" and "(z,q) ∈ d^-1" by auto
 then obtain k l u v where kp:"k ∥ p" and kl:"k∥l" and lz:"l∥z" and zu:"z ∥u" and uv:"u∥v" and pv:"p∥v" using d by blast
 obtain k' l' u' v' where kpz:"k' ∥z" and kplp:"k' ∥l'" and lpq:"l' ∥q" and qup:"q ∥u'" and upvp:"u' ∥v'" and zvp:"z ∥v'" using d ‹(z,q): d^-1› by blast
 from lz kpz kplp have "l∥l'" using M1 by blast
 with kl lpq obtain ll where "k∥ll" and "ll∥q" using M5exist_var by blast
 moreover from zvp zu upvp have "u' ∥ u " using M1 by blast
 moreover with qup uv obtain uu where "q∥uu" and "uu∥v" using M5exist_var by blast
 ultimately show "x ∈ d^-1" using x d kp pv by blast
 

  cdisi:"d^-1 O s^-1 ⊆ d^-1"
 
 fix x::"'a×'a" assume "x : d^-1 O s^-1" then obtain p q z where x:"x = (p,q)" and "(p,z) ∈ d^-1" and "(z,q) ∈ s^-1" by auto
 then obtain k l u v where kp:"k ∥p" and kl:"k∥l" and lz:"l∥z" and zu:"z∥🚫<>ov
 obtain k' u' v' where kpz:"k' ∥z" and kpq:"k' ∥q" and qup:"q ∥u'" and upvp:"u' ∥v'" and zvp:"z ∥v'" using s ‹(z,q): s^-1› by blast
 from upvp zvp zu have "u'∥u" using M1 by blast
 with qup uv obtain uu where "q∥uu" and "uu∥v" using M5exist_var by blast
 moreover from kpz lz kpq have "l ∥q " using M1 by blast
 ultimately show "x ∈ d^-1" using x d kp kl pv by blast
 

  csb:"s O b ⊆ b"
  (auto simp:s b)
  M1 M5exist_var by blast

  csm:"s O m ⊆ b"
  (auto simp:s m b)
  M1 by blast

  css:"s O s ⊆ s"
 
 fix x::"'a×'a" assume "x ∈ s O s" then obtain p q z where x:"x = (p,q)" and "(p,z) ∈ s" and "(z,q) ∈ s" by auto
 from ‹(p,z) ∈ s› obtain k u v where kp:"k∥p" and kz:"k∥z" and pu:"p∥u" and uv:"u∥v" and zv:"z∥v" using s by blast
 from ‹(z,q) ∈ s› obtain k' u' v' where kpq:"k'∥q" and kpz:"k'∥z" and zup:"z∥u'" and upvp:"u'∥v'" and qvp:"q∥v'" using s by blast
 from kp kpz kz have "k'∥p" using M1 by blast
 moreover from uv zup zv have "u∥u'" using M1 by blast
 moreover with pu upvp obtain uu where "p∥uu" and "uu∥v'" using M5exist_var by blast
 ultimately show "x ∈ s" using x s kpq qvp by blast
 

  csifi:"s^-1 O f^-1 ⊆ d^-1"
 
 fix x::"'a×'a" assume "x : s^-1 O f^-1" then obtain p q z where x:"x = (p,q)" and "(p,z) ∈ s^-1" and "(z,q) ∈ f^-1" by auto
 then obtain k u v where kp:"k ∥ p" and kz:"k∥z" and zu:"z ∥u" and uv:"u∥v" and pv:"p∥v" using s by blast
 obtain k' l' u' where kpz:"k' ∥z" and kplp:"k' ∥l'" and lpq:"l' ∥q" and zup:"z∥u'" and qup:"q∥u'" using f ‹(z,q): f^-1› by blast
 from kz kpz kplp have "k∥l'" using M1 by blast
 moreover from qup zup zu have "q ∥ u " using M1 by blast
 ultimately show "x ∈ d^-1" using x d kp lpq pv uv by blast
 

  csidi:"s^-1 O d^-1 ⊆ d^-1"
 
 fix x::"'a×'a" assume "x : s^-1 O d^-1" then obtain p q z where x:"x = (p,q)" and "(p,z) ∈ s^-1" and "(z,q) ∈ d^-1" by auto
 then obtain k u v where kp:"k ∥ p" and kz:"k∥z" and zu:"z ∥u" and uv:"u∥v" and pv:"p∥v" using s by blast
 obtain k' l' u' v' where kpz:"k' ∥z" and kplp:"k' ∥l'" and lpq:"l'∥q" and qup:"q ∥u'" and upvp:"u' ∥v'" and zvp:"z∥v'" using d ‹(z,q): d^-1› by blast
 from zvp upvp zu have "u'∥u" using M1 by blast
 with qup uv obtain uu where "q∥uu" and "uu∥v" using M5exist_var by blast
 moreover from kz kpz kplp have "k ∥l' " using M1 by blast
 ultimately show "x ∈ d^-1" using x d kp lpq pv by blast
 

  cdb:"d O b ⊆ b"
  (auto simp:d b)
  M1 M5exist_var by blast

  cdm:"d O m ⊆ b"
  (auto simp:d m b)
  M1 by blast

  cfb:"f O b ⊆ b"
  (auto simp:f b)
  M1 by blast

  cfm:"f O m ⊆ m"
 
 fix x::"'a×'a" assume "x ∈ f O m" then obtain p q z where x:"x = (p,q)" and 1:"(p,z) ∈ f" and 2:"(z,q) ∈ m" by auto
 from 1 obtain u where pu:"p∥u" and zu:"z∥u" using f by auto
 with 2 have "(p,q) ∈ m" using M1 m by blast
 thus "x∈ m" using x by auto
 


(* ========= $\alpah_1$ compositions ============ *)
subsection ‹$\alpha$-composition›
text ‹We prove compositions of the form $r_1 \circ r_2 \subseteq s \cup ov \cup d$.›


lemma (in arelations) cmd:"m O d ⊆ s ∪ ov ∪ d"
proof 
  fix x::"'a×'a" assume a:"x ∈ m O d" then obtain p q z where x:"x =(p,q)" and 1:"(p,z) ∈ m" and 2:"(z,q) ∈ d" by auto
  then obtain k l u v  where pz:"p∥z" and kq:"k∥q" and kl:"k∥l" and lz:"l∥z" and zu:"z∥u" and uv:"u∥v" and qv:"q∥v" using m d by blast
  obtain k' where kpp:"k'∥p" using M3 meets_wd pz by blast
  from pz zu uv obtain zu where pzu:"p∥zu" and zuv:"zu∥v" using M5exist_var  by blast
  from kpp kq have "k'∥q ⊕ ((∃t. k'∥t ∧ t∥q) ⊕ (∃t. k∥t ∧ t∥p))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast 
  then have "(?A∧¬?B∧¬?C)∨(¬?A∧?B∧¬?C)∨(¬?A∧¬?B∧?C)"  using local.meets_atrans xor_distr_L[of ?A ?B ?C]  by blast
  thus "x ∈ s ∪ ov ∪ d"    
  proof (elim disjE)
    {assume "(?A∧\<not  thus ?thesis using x by auto}
     then have "(p,q) ∈ s" using s qv kpp pzu zuv by blast
     thus ?thesis using x by simp }
    next
    {assume "(¬?A∧?B∧¬?C)" then have "?B" by simp
     then obtain t where kpt:"k'∥t" and tq:"t∥q" by auto
     moreover from kq kl tq have "t∥l" using M1 by blast
    next
     ultimately have "(p,q) ∈\notp" by auto
     thus ?thesis using x by simp}
    
    {assume "(< thus
 >t and tp:"t\lel by auto
     with kq pzu zuv qv have "(p,q)∈d" using d by blast
     thus ?thesis using x by simp}
  qed
qed

lemma (in arelations) cmf:"m O f ⊆ s ∪ ov ∪ d"
proof
  fix x::"'a×'a" assume a:"x ∈
  then obtain k l u   where pz:"p∥z" lemma covf:"ov O f \<subseteq> s \<union> ov \<uni> d"
  obtain k' where kpp:"k'∥p" using M3 meets_wd pz by blast
  from kpp kq have "k'∥q ⊕ ((∃t. k'∥t ∧ t∥q) ⊕ (∃t. k∥  fix:"'a\<times'in> ov O f"then obtain p q z where x:"x=(p,q)" and "(p,z) ∈ ov" and "(z,q) ∈ f" by auto
  then have "(?A∧¬?B∧¬?C)∨(¬?A∧?B∧¬?C)∨(∥u\parallel>v"  zv:z<"k∥l" and lz::"c∥
java.lang.StringIndexOutOfBoundsException: Range [3, 2) out of bounds for length 45
  proof (elim disjE)
    {assume "(?A∧¬?B∧¬?C)" then have "?A" by simp
     then have "(p,  then have "(?A\<and¬¬?C \or (\n>?\>?B\and><not?) \o (<otA∧>?B∧?C))" by (inserert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
     thus ?thesis using x by simp }
    next
    {assume "(   "x n> ov ∪ d"
   hen t where:"k'∥t" and tq:"t∥q" by auto
     moreover from kq kl tq have "t∥l" using M1 by blast 
lz  "using M1 by blast
     ultimately have "(p,q) ∈ ov" then have "(p,q) ∈ s" using s kp qup uu pu by blast
     thus ?thesis using x by simp}
next
    {assume "(java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 8
     then obtain t       thenobtain twhere:k\"and tq:"t\>"byauto
     with kq pz zu qu have "(p,q)∈d" using d by blast
     thus ?thesis using x by simp}
  
qed

lemma cmovi:"m O ov^ultimately obtainl here "t lc"and lc<parallelu  _var
proof
  fix x: m   pqzwhere"=(,q)" 1"p,z) ∈ m" and 2:"(z,q) ∈ ov^-1" by auto
  then obtaink l c u v  here :"p\<>zk\parallel>l"and:"\parallel>z"andqu"q\<paralleluparallel>v" andzv:z\parallel>" nd lc:"<parallel  cu"c∥u" using  ovby 
  obtain k' where kpp:"k'∥next
java.lang.StringIndexOutOfBoundsException: Range [12, 2) out of bounds for length 56
  rom ppq have "k' q \oplus ((∃< t<parallel>q) <plus  (?B ⊕
  then have "(?A∧¬?B∧¬?C)∨(¬?A\<and      pkp have "(p)<>" using d by blast
  thus "x ∈ s ∪ ov ∪ ?thesis x by}
  java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 5
    {me(A\a>\n>?\and\not>C" then have "?A" by simp
     then have "(p,q) ∈ s" using s kpp qu cu pc by blast
     thus ?thesis using x by simp }
    next
    {assume "(¬?A∧?B∧¬?C)" then have "?B" by simp
     then obtain t where kpt:"k'∥t" and tq:"t∥
  from ‹(z,q) ∈ d› obtain k' l' u' v where  from pu zu zup have pup:"p\<parallel>u'" using M1 by blast
     ultimately have "(p,q) ∈ ov" using ovcjava.lang.StringIndexOutOfBoundsException: Range [65, 63) out of bounds for length 71
      
    next
    {assume "(\  thus "x ∈ s ∪ ov <union 
     then injava.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 74
     java.lang.StringIndexOutOfBoundsException: Range [15, 10) out of bounds for length 35
 x
  qed
qed

lemma covd:"ov O d ⊆ s ∪ ov ∪ d"

  fix       pz<and<>ujava.lang.StringIndexOutOfBoundsException: Range [85, 83) out of bounds for length 103
  from \
  fromopenz,q) ∈ d\c>obtain k' l' u' v' where kpq:"k'∥q" and kplp:"k'∥l'" and lpz:"l'∥z" and qvp:"q∥v'" and zup:"z∥v'" using  
  from uv zv zup have      thus java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
  from pu upvp obtainjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  from kp k<> <oplus( <parallel<and <arallel)⊕ (∃t. k'∥>t∥?<plus ?C)") using M2 by blast
  then have "(?A∧¬?B∧(p,z) ∈ f›z"<p>p"< java.lang.StringIndexOutOfBoundsException: Range [153, 150) out of bounds for length 182
  thus "x ns ∪ ov ∪
  proof (elim disjE)
    { assume "?A∧parallel> ((∃parallelt ∧q) ⊕t. l'<<java.lang.StringIndexOutOfBoundsException: Range [140, 139) out of bounds for length 209
      then have "(p,q) ∈¬\>C) ∨>?A∧¬> (¬¬_A uoiplimmees
     thus ?thesis using x by blast}
    next
    { assume "¬?A<      {
      then obtain t where        with  lp<n  t
      from        thus ?hesisto
      moreover from kpq tq {java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 68
      moreover from lpz lz lc       thenjava.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 75
      ultimately obtain lc where "t\      with lp lt tq pup upv qv cup have "(p,q)∈ov" using ov by blast
      then have "(p,q) ∈ ov" using ov kp kt tq puu uuvp qvp by blast
      thus ?thesis using x by auto}
    next
    { assume "¬?A <      thus ?thesis using x by 
      { assume "<>A <and> 🚫
      with puu uuvp qvp kpq then obtain t where "l'∥p" by auto
       with lpq pup upv qv have "(p,q) ∈ d" using d by blast
  qed
qed


lemma thus ?thesis using x by auto}
proof
  fix x::"'ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  from java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  from \<open>(z,q)     fix x::'a<times> assume " \<n  O s-" then obtain p q z here x"=(,"and (,)\in ov and(zq)\in s^"b 
  from uv zv zup have uu:"u\<parallel>u'" using M1 by auto
  from kp kpq have "k\<parallel>q \<oplus> ((\<exists>t. k\<parallel>t     from \<pen>(z,) \<n s^1<close> obtain k u' v'where kpz:"k\parallel>"andkpq:k<parallel and kpz:k\<arallel>" and zup"z\<parallel>>u'"  nd qvp:"\<arallel>v'" using sbyblast
  then have "(?A\<and>\<not>?B\<and>\<not>?C) \<or> ((\<not>?A\<and>?B\<and>\<not>?C) \<or> (\<not>?A\<and>\<not>?B\<and>?C))    from u qvp have "\p>v' < (<t p\<parallel>t \<and> t\<parallel>v') \<oplus> (\<exists>t. q\<parallel>t \<and> t\<parallel>u))" (is "?A \<oplus> (?B \<oplus> ?C)") using M2 by blast
   xin  s \<union> ov \<union> d"
  proof (elim disjE)
java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 66
then  (<   kpq    blast
      ? using byb}
    next
    { assume "\<not      java.lang.StringIndexOutOfBoundsException: Range [10, 11) out of bounds for length 10
 obtain t where kt:"k\<parallel>t" and tq:"t\<parallel>q" by auto
              ultimately have "(p,q)\<in> ov" using kp klz lzq pt tvp qvp ov by blast
     moreover from lpz lz   lpc:l'<el sing auto
      ultimately       ultimately obtain qt"\>"and "\parallel>"b auto
      then have "(p         ?hesis using byauto}
      thus ?thesis usingjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

    { assume "\<not>?A \<    fromo>p,q) \<in> d^-1\<close> obtain u v k l  where kp:"k\<parallel>p" and pv:"p\<parallel>v" k\parallel>"l<arallel>" :"\>u"and:u<>dby blast
  t where<>" and "t\<parallel>p" by auto
      with pu uu qup kpq have "(p,q) \<in> d" using d by blast
      thus ?thesis using x by auto}
  qed
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemma cfid then have "(?A\><>?B<><not>?C) \<> (\not>Aand?\<and>\<not>C < \>\and><>\>?)"bjava.lang.StringIndexOutOfBoundsException: Range [153, 151) out of bounds for length 186
proof
  fixjava.lang.StringIndexOutOfBoundsException: Range [8, 1) out of bounds for length 84
  from \<open>(p,        moreover from lptlpr llr ave :ll<  M1  blast
  from \<open>(z,q) \<in> d\<close> obtain k' l' u' v         from  tup have "\<>u  java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
        with kp  tpvp kll llrrvp  ave"p,r)\in>ov"usingov by blast
  from kp kpq     next
  then have "(?A\<and>\<not>?B\<and>\<not>?C) \<or> ((\<not>?A\<and>?B\<and>\      { ssume "<>? <> <? <>?"thenjava.lang.StringIndexOutOfBoundsException: Range [72, 73) out of bounds for length 72
  thus "x \<in> s \<union> ov \<union> pjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
     <nd>\<not>B\<and<not?Chjava.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
      with pup upv kp qv have "(  frompv qvpe<el<(<java.lang.StringIndexOutOfBoundsException: Range [56, 54) out of bounds for length 208
      ?
    next
    { assume "\<  then have "(<and><><>java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 184
<t"\parallel>  
      from tq kpq kplp have "t\<parallel>l'" using M1 by blast
 lpz zup obtain lpz where "t\<parallel>lpz" and "lpz\<parallel>u'" using M5exist_var by blast
      with kp pup upv kt tq qv have "(p,q)\<in>ov" using ov by blast
java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
       next
    {        thussy
            
      with puppq< d" using d by blast
      thus java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
    qed


lemma cfov:"f O ov \<subseteq> ov \<union> s \<unionjava.lang.StringIndexOutOfBoundsException: Range [54, 52) out of bounds for length 56
proof
    fixjava.lang.StringIndexOutOfBoundsException: Range [31, 28) out of bounds for length 37
    from \<open>(p,z) \<in> f\<close> obtain  k l u where lemma csim:s- <subseteq>java.lang.StringIndexOutOfBoundsException: Range [37, 35) out of bounds for length 64
    from \<open>(z,q) \<in> ov\<close> obtain k' l' c  u' v where "k'\<parallel>l'" and kpz:"k'\<parallel>z" and lpq:"l'\<parallel> q" and  zup:"z\<parallel>u'" and upv:"u'\<parallel>v" and qv:"q\<parallel>v" and lpc:"l'\<parallel>c" and cup:"c\<parallel>u'"  using  ov by blast
    from pu zu zup have pup:"p\<parallel>u'" using M1 by blast
java.lang.StringIndexOutOfBoundsException: Range [11, 8) out of bounds for length 209
java.lang.StringIndexOutOfBoundsException: Range [10, 4) out of bounds for length 186
    thus "x \<in> ov \<union> s \<union> d"
    proof (elim disjE)
      { assume "?A\<and>\<not>?B\<and>\<not>?C" then have ?A by simp
        with  lp pup upv qv have "(p,q) \<in> s" using s by blast
        thus ?thesis using x by auto}
      next
      { assume "\<not>?A\<and>?B\<and>\<not>?C" java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 22
            java.lang.StringIndexOutOfBoundsException: Range [10, 11) out of bounds for length 10
      t\parallel> last
      with lp lt tq pup upv qv cup        from pt e<parallel>"
      thus ?thesis using x by blast}
      next
      { assume "\<not>?A \<and> \<not>?B \<and> ?C" then have ?C      { ssume "\not>A \<and> \<not>?B \<and> ?C" then have ?C by simp
      then  
      with lpq pup upv qv have "(p,q) \<in> d" using d by blast
      thushus ?s y}
    qed
qed

(* ========= $\alpha_2$ composition ========== *)
 <open>We prove compositions of the form $ circ 2subseteqcupcup<close>

lemma covsi:"ov O s^-1 ⊆ ov ∪> d^-1"
proof
    fix x::"'a×'a" assume "x ∈ ov O s^-1" then obtain p q z where x:"x = (p,q)" and "(p,z) ∈ ov" and "(z,q) \  from kz kpz kplp have klp:"k∥l'" using M1 by auto
java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 212
    from ‹ obtain k' u' v' where kpz:"k'∥"'\parallel>z" an zup:"z\parallel>u'" and qvp:"q∥v'" using s by blast
    from lz kpz kpq have lq:" ∥q" using M1 by blast
 from pu qvp have "p∥oplus> ((\exists>t. p∥t ∧ t∥v') ⊕ (∃t. q∥t ∧ t∥u))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
java.lang.StringIndexOutOfBoundsException: Range [12, 9) out of bounds for length 186
java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 49
 proof (elim disjE)
 A\and🚫
 with qvp kp kl lq have "(p,q) ∈ f^-1" using f by blast
 thus ?thesis using x by auto}
 { assume <<and>?B∧¬?C" then have ?B by simp
 { assume "¬?A∧?B∧ then obtain t where pt:"p∥t" and tvp:"t∥v'" by auto
java.lang.StringIndexOutOfBoundsException: Range [22, 19) out of bounds for length 76
 moreover with pu cu have "c∥t" using M1 by blast
 ultimately hve "(,q)\in> ov" using kp kl lc cu lq qvp ov by blast
 thus ?thesis using x by auto}

 { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp
 then obtain t where qt:"q∥t" and "t∥u" by auto
 with kp kl lq pu have "(p,q) ∈ d^-1" using d by blast
 thus ?thesis using x by auto}
 qed
 


  cdim:"d^-1 O m ⊆ ov ∪ d^-1 ∪ f^-1"
 
 fix x::"'a×'a" assume "x ∈ d^-1 O m" then obtain p q z where x:"x = (p,q)" and "(p,z) ∈ d^-1" and "(z,q) ∈ m" by auto
 from ‹(p,z) ∈ d^-1› obtain k l u v where kp:"k∥p" and pv:"p∥v" and kl:"k∥l" and lz:"l∥z" and zu:"z∥u" and uv:"u∥v" using d by blast
 from ‹(z,q) ∈ m› have zq:"z∥q" using m by blast
 obtain v' where qvp:"q∥v'" using M3 meets_wd zq by blast
 from kl lz zq obtain lz where klz:"k∥lz" and lzq:"lz∥q" using M5exist_var by blast
 from pv qvp have "p∥v' ⊕ato}
 then have "(?A∧ { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp
java.lang.StringIndexOutOfBoundsException: Range [19, 16) out of bounds for length 50
 proof (elim disjE)
 ?and>\not>?B\and>¬?C" then have ?A by simp
 with qvp kp klz lzq‹
 thus ?thesis using x by auto}
 next
 >?A∧?B∧?C" then have ?B by simp
 then obtain t where pt:"p∥t" and tvp:"t∥
 from zq lzq zu have "lz∥u" using M1 by auto
 moreover from pt pv uv have "u\<parallel    v'" using M3 meets_wd zq by blast
 ultimately have (p,q\<>o
 thus ?thesis usingx by auto} }
 next then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧<<unio> f^-> d^-1"
java.lang.StringIndexOutOfBoundsException: Range [24, 21) out of bounds for length 72
 then obtain t where qt:"q∥
 with kp klz lzq pv have "(p,q) \<      { assume "\<not>?A\<and>?B\<and>\<not{
 ultimately have "(p,q)∈ ov" using lp lc cq qvp cu ov by blast
 
 

  cdiov:"d^-1 O ov ⊆ then obtain t where qt:"q∥t" and "t∥v" by auto
 
java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 142
java.lang.StringIndexOutOfBoundsException: Range [9, 10) out of bounds for length 9
 from ‹
 from lq kplp kpq have "l∥l'" using M1 by blast
 with kl lpr obtain ll where kll:"k∥
 from > ((\exists>t'. p∥t' ∧ t'∥v') ⊕ (∃t'. r∥t' ∧ t'∥v))" (is "?A ⊕ (?B ⊕
java.lang.StringIndexOutOfBoundsException: Range [109, 107) out of bounds for length 136
 thus "x ∈ f^-1 ∪
 proof (elim disjE)
 { assume "?A∧l'" using M1 by blast
 with rvp llr kp kll have "(p,r) \in f^-1"using f by blast
java.lang.StringIndexOutOfBoundsException: Range [51, 48) out of bounds for length 105
 next
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
 then obtain t' where ptp:"p∥t'" and tpvp:"t'∥v'" by auto
 moreover from lpt lpr llr have llt:"ll∥t" using M1 by blast
 moreover from ptp uv pv have utp:"u∥t'" using M1 by blast
 moreover from qu tup qup have "t∥u" using M1 by blast
 moreover with utp llt obtain tu where "ll∥tu" and "tu∥t'" using M5exist_var by blast
 with kp ptp tpvp kll llr rvp have "(p,r)∈ ov" using ov by blast
 thus ?thesis using x by auto}
 next
 { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp
 then obtain t' where rtp:"r∥t'" and "t'∥v" by auto
 with kll llr kp pv have "(p,r) ∈ d^-1" using d by blast
 thus ?thesis using x by auto}
 qed
 

  cdis:"d^-1 O s ⊆ ov ∪ f^-1 ∪ then han have "(?A\<nd\not>?B∧?C) ∨?A∧¬ (¬¬?C))" by (insert xr_dist_L[of ?A ?B ?C], ausmp:eimes
 
 fix x::"'a×'a" assume "x ∈ d^-1 O s" then obtain p q z where x:"x = (p,q)" and "(p,z) ∈ d^-1" and "(z,q) ∈ s" by auto
 from ‹(p,z)∈d^-1› obtain k l u v where kl:"k∥l" and lz:"l∥z" and kp:"k∥p" and zu:"z∥u" and uv:"u∥v" and pv:"p∥v" using d by blast
 from ‹(z,q) ∈ s› obtain l' v' where lpz:"l'∥z" and lpq:"l'∥q" and qvp:"q∥v'" using s by blast
 from lz lpz lpq have lq:"l∥q" using M1 by blast
 from pv qvp have "p∥v' ⊕ ((∃t. p∥t ∧ t∥v') ⊕ (∃t. q∥t ∧ t∥v))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 thus "x ∈ ov ∪ f^-1 ∪ d^-1"
 roof (elim disj)
java.lang.StringIndexOutOfBoundsException: Range [16, 14) out of bounds for length 68
 with kl lq qvp kp have "(p,q) ∈ f^-1" using f by blast
 thus ?thesis using x by auto}
 next
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
 then obtain t where pt:"p∥t" and tvp:"t∥v'" by auto
 from pt pv uv have "u∥t" using M1 by blast
 with lz zu obtain zu where "l∥zu" and "zu∥t" using M5exist_var by blast
 with kp pt tvp kl lq qvp have "(p,q) ∈ ov" using ov by blast
 thus ?thesis using x by auto}
 thus ?thesis using x by auto}
 { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp
 then obtain t where "q\<q <tp>uu🚫
 with kl lq kp pv have "(p,q)∈d^-1" using d by blast
 thus ?thesis using x by auto}
 
 

  csim:"s^-1 O m ⊆ ov ∪ f^-1 ∪ d^-1"
 
 fix x::"'a×'a" assume "x ∈ s^-1 O m" then obtain p q z where x:"x = (p,q)" and "(p,z) ∈ s^-1" and "(z,q) ∈ m" by auto
 from ‹(p,z)∈s^-1› obtain k u v where kp:"k∥p" and kz:"k∥z" and zu:"z∥u" and uv:"u∥v" and pv:"p∥v" using s by blast
 from ‹(z,q) ∈ m› have zq:"z∥q" using m by auto
 obtain v' where qvp:"q∥v'" using M3 meets_wd zq by blast
 from pv from \<pen(q" and qvp:"q\<parallel" av'" using f by blast
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 thus "x ∈ ov ∪ f^-1 ∪ d^-1"
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have ?A by simp
 with kp kz zq qvp have "(p,q) ∈ f^-1" using f by blast
 thus ?thesis using x by auto}
 next
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
 then obtain t where pt:"p∥t" and tvp:"t∥v'" by auto
 from pt pv uv have "u∥t" using M1 by blast
 with kp pt tvp kz zq qvp zu have "(p,q) ∈ ov" using ov by blast
 thus ?thesis using x by auto}
 next
 { assume "¬?A ∧ ¬?B ∧?C" then have ?C bybysim
 then obtain t where "q∥l'" using M1 by blast
 with kp kz zq pv have "(p,q)∈d^-1" using d by blast
 thus ?thesis using x by auto}
 qed
 
 
  csiov:"s^-1 O ov ⊆ ov ∪ f^-1 ∪ d^-1"
 
 fix x::"'a×'a" assume "x ∈ s^-1 O ov" then obtain p q z where x:"x = (p,q)" and "(p,z) \<in thus "x \<<union> m ∪ ov"
 from ‹proof (eli dsE
  ‹(z,q) ∈ ov› obtain k' l' u' v' c where kpz:"k'∥z" and zup:"z∥u'" and upvp:"u'∥v'" and kplp:"'∥q" and qvp:"q∥parallel>c" and cup:"c∥u'" using ov by blast
 from kz kpz kplp have klp:"k∥l'" using M1 by auto
  next
 >\not?B\<>\not>?C) \or> ((¬?B∧?C) \ (¬¬?C))" by (insert xor_distr_L[of ?A B ??C,auto imp:emp:elimmeets)
 thus "x then have "(p,q) ∈
 proof (elim disjE)
 { assume "?A\     next
 with kp kplp lpq qvp klp have "(p,q) ∈ f^-1" using f by blast
 thus ?thesis using x by auto}
 next
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
 then obtain t where pt:"p\parallel>t ad tv"t<parallelv
 from pt pv uv have "u∥t" using M1 by blast
 moreover from cup zup zu have cu:"c∥u" using M1 by auto
 ultimately obtain cu where "l'∥cu" and "cu∥t" using lpc M5exist_var by blast
 with kp pt tvp klp lpq qvp have "(p,q) ∈ ov" using ov by blast
 thus ?thesis using x by auto}
 next
 { assume "¬ p pta l hr ll\parallelll" and llr:"ll∥q" using M5exist_var by blast
java.lang.StringIndexOutOfBoundsException: Range [43, 40) out of bounds for length 71
 using d by blast
 thus ?thesis using x by auto}
 qed
 

  qed
 
 fix x::"'a×'a" assume "x ∈ ov^-1 O m" then obtain p q z where x:"x = (p,q)" and "(p,z) ∈ ov^-1" and "(z,q) ∈qed
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 from ‹
 obtain v' where qvp:"q∥
 rom zu zq u ae q:"c\parallel>q"using M1 by blast
 from pv qvp have "p∥v' ⊕ ((∃t. p∥t ∧ tel>z" and kplp:"k'∥l'" and lpq:"l'∥q" and zup:"z∥u'" and qvp:"q∥v'" and upvp:"u'∥v'" using ov by blast
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
  ov ∪ d^-1"
java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 68
  v ae"(q)\i> f^-1" using f by blast
 thus ?thesis using x by auto}
 next
 { assume "¬🚫
 then obtain t where ptp:"p∥ { assume "\<<>\not?B ∧
 moreover with pv uv have "u∥
  ov" using lp lc cq qvp cu ov by blast
 thus ?thesis using x by auto}
 next
 { assume "¬?A ∧
 then obtain t where qt:"q∥t" and "t∥
 with lp lc cq pv have "(p,q) ∈< ∪ ov"
 thus ?thesis using x by auto}
 qed
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

(* =========$\alpha_3$ compositions========== *)
textWe prove compositions hesubseteqcupjava.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 97
p>'" using M1 by blast
lemma covov:"ov O ov ⊆ b ∪ m ∪ ov"
proof
   atimes>'a" assume "x ∈ ov O ov" then obtain p q z where x:"x = (p,q)" and "(p,z) ∈ ov" and "(z,q)∈ ov" by auto
open(p,z) ∈ ov<>ek<java.lang.StringIndexOutOfBoundsException: Range [78, 76) out of bounds for length 252
   m<pen<n <lose obtain k' l' y u' v' where kplp:"k'∥l'" dpz<>zq" and lpy:"l'<el"
   from lz kplp kpz have llp:"l∥l'" using M1 by blast
   fromv<u'" using M1 by blast
   with pu upvp obtain uu where puu:"p∥uu" and uuv:"uu∥v'" using M5exist_var by blast
   from puu lpq have "p∥q ⊕ ((∃t'. p∥t' ∧ t'∥q) ⊕ (∃t'. l<parallel>' <and<paralleljava.lang.StringIndexOutOfBoundsException: Range [168, 167) out of bounds for length 216
    then have "(?A∧¬?B∧{ assume "A¬and¬
    thus "x ∈
    proof (elim disjE)
java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 68
< m" usingjava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
              java.lang.StringIndexOutOfBoundsException: Range [11, 8) out of bounds for length 72
      
<?A∧?B∧¬?C" then have ?B by simp
        then have "(p,q) gjava.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 37
        thus ?(* =========$\alpha_4$ 
     next
      { assume java.lang.StringIndexOutOfBoundsException: Range [0, 37) out of bounds for length 0
        then obtain t' where lptp:"l'∥t'" and "t'∥
java.lang.StringIndexOutOfBoundsException: Range [44, 39) out of bounds for length 113
        with lpq lptp  have "ll∥t'" using M1 by blast
java.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for length 106
        thus ?thesis
      qed
qed

lemma  fromkp<arallelq<plusjava.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 208
proof
   fix x::"'a×'a" assume "x ∈ ov O f^-1" then obtain p q z where x:"x = (p,q)" and "(p,z) ∈ ov" and "(z,q)∈ f^-1" by auto
   from ‹(p,z) ∈ ov› obtain k u l c v where kp:"k∥p" and pu:"p∥u" and kl:"k∥l" and lz:"l∥z" and "l∥c" and "c∥u" and uv:"u∥v" and zv"z\<parallelv
   from ‹(z,q) ∈ f^-1› obtain k' l' v' where kplp:"k'∥l'" and kpz:"k'∥z" and lpq:"l'∥q" and qvp:"q∥v'" and zvp:"z∥v'" using f by blast
   from lz kplp kpz have llp:"l∥l'" using M1 b>?A∧and>\not>C)∨(¬?A∧\<B<(nser xrditr_Lof ?A ?B?C, ato sip:lmmets
   from zv qvp zvp have qv:"q∥
   from pu lpq have "p∥q ⊕ ((∃t. p∥t ∧ t∥q) ⊕ (\<    {
    then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
    thus "x ∈ b ∪ m<unionjava.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
     "p<> gpz
)i> m" using m by auto
        thus ?thesis using x by auto}
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
      { assume "¬f
        then qjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
        thus ?thesis <java.lang.StringIndexOutOfBoundsException: Range [34, 31) out of bounds for length 60
     next
      <<>\<not>?B ∧ ?C" then have ?C by simp
        then obtain t where lptp:"l<<paralleljava.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
        lpq <<java.lang.StringIndexOutOfBoundsException: Range [86, 85) out of bounds for length 113
   t" using M1 by blast
java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 101
        thus ?thesis using x by auto}
      qed
ed


lemma csov:"s O ov <eqvjava.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 50
proof
   fix x::"'a×?A∧<nd>\not>?)"then have 
   from ‹ obtain k u v where kp:"k∥<apu:"p\parallel>u"u" and uv:"u∥v" and zv:"z∥v" using s by blast
 from ‹(z,q) ∈ ov› obtain k' l' u' v' where kpz:"k'∥z" and kplp:"k'∥l'" and lpq:"l'∥q" and zup:"z∥u'" and qvp:"q∥v'" and upvp:"u'∥ then ha(,)\in f^-1" using f qu pu kp by blast
 from kz kpz kplp have klp:"k∥?A∧not>?B🪙?C)" then hae "?C" by simp
java.lang.StringIndexOutOfBoundsException: Range [9, 7) out of bounds for length 63
 with pu upvp obtain uu where puu:"p∥uu" and uuvp:"uu∥thus ?thesis using x by simp}
 from pu lpq have "p∥q ⊕ ((∃t. p∥t ∧ t∥q) ⊕
  v (A∧?B∧?C) ∨?A∧not>?C) ∨ (¬?A🪙not>?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 thus "x ∈ b ∪ f ∪
 proof (elim disjE)
 ¬\<not?
java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 145
java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
 next
 { assume "¬?A∧?B∧¬from kp kpr have "k∥r ⊕ ((∃t. k∥t ∧ t∥r) ⊕ (∃t. k'∥then have "(?A\and>¬∧¬?C) \or>((\not>?A∧?B∧\<ot<¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 then have "(p,q) ∈ b" using b by auto
 thus ?thesis using x by auto}
 next
 { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp
 then obtain t where lpt:"l'∥t" and "t∥u" by auto
 with pu puu have "t∥uu" using M1 by blast
 with lpt kp puu uuvp klp lpq qvp have "(p,q) ∈ ov" using ov by blast
 thus ?thesis using x by auto}
 qed
 


  csfi:"s O f^-1 ⊆ b ∪ m ∪ ov"
 
 fix x::"'a×'a" assume "x ∈ s O f^-1" then obtain p q r where x:"x = (p,r)" and "(p,q) ∈ s" and "(q,r)∈ f^-1" by auto
 from ‹(p,q) ∈ s› obtain k u v where kp:"k∥p" and kq:"k∥q" and pu:"p∥u" and uv:"u∥v" and qv:"q∥v" using s by blast
 from ‹(q,r) ∈ f^-1› obtain k' l v' where kpq:"k'∥q" and kpl:"k'∥l" and lr:"l∥r" and rvp:"r∥v'" and qvp:"q∥v'" using f by blast
 from kpq kpl kq have kl:"k∥l" using M1 by blast
 from qvp qv uv have uvp:"u∥v'" using M1 by blast
 from pu lr have "p∥r ⊕ ((∃t'. p∥t' ∧ t'∥r) ⊕ (∃t'. l∥t' ∧ t'∥u))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 thus "x ∈ b ∪ m ∪ ov"
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have ?A by simp
 then have "(p,r) ∈ m" using m by auto
 thus ?thesis using x by auto}
 next
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
 then have "(p,r) ∈
 thus ?thesis using x by auto}
 next
 { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp
 then obtain t' where ltp:"l∥t'" and "t'∥u" by auto
 with kp pu uvp kl lr rvp have "(p,r) ∈ ov" using ov by blast
 thus ?thesis using x by auto}
 qed
 

(* =========$\alpha_4$ compositions========== *)
textWe prove compositions ofcirc cup <>

lemma cmmi:"m O m^-1 ⊆ f ∪ f^-1 ∪ e"
proof
   x::"'a×'a" assume a:"x ∈ m O m^-1" then obtain p q z where x:"x =(p,q)" and 1:"(p,z) ∈ m" and 2:"(z,q) ∈ m^-1" by auto
"d"<>
  obtain w<d<singjava.lang.StringIndexOutOfBoundsException: Range [74, 72) out of bounds for length 96
  java.lang.StringIndexOutOfBoundsException: Range [17, 13) out of bounds for length 208
<and>¬?B∧¬?C)∨(<<and<><>C)∨notA∧¬?B\and?)" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
  thus "x ∈f ∪ f^-1 ∪ e"
  proof (elim disjE)
    {assume "(?A∧¬?B∧¬?C)" then have "?A" by simp
     then have "p = q" using M4 kp pz qz by blast
     then have "(p,q) ∈ e" using e by auto
     thus ?thesis using x by simp }
    next
    {assume "(¬?A∧?B∧¬?C)" then have "?B" by simp
     then obtain t where kt:"k∥t" and tq:"t∥q" by auto
     then have "(p,q) ∈ f^-1" using f qz pz kp by blast
     thus ?thesis using x by simp}
    next
    {assume "(¬?A∧¬?B∧?C)" then have "?C" by simp
     then obtain t where kt:"k'∥t" and tp:"t∥p" by auto
     with kpq pz qz have "(p,q)∈f" using f by blast
     thus ?thesis using x by simp}
  qed
qed
  

lemma cfif:"f^-1 O f ⊆ e ∪ f^-1 ∪ f"
proof
  fix x::"'a×'a" assume a:"x ∈ f^-1 O f" then obtain p q z where x:"x =(p,q)" and 1:"        withkpr rung
  from 1 obtain k l u where kp:"k∥pqed
  from 2 obtain k' l' u' where kpq:"k(* =======(
 zup qup have qu:"q∥u" using M1 by auto
  from kp kpq have "k∥q ⊕t. k∥ t∥ (∃t. k'∥t ∧ t∥p))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast 
  then have "?A∧?B∧?C)∨not>?A\and>?B∧?C)∨(¬¬?C)" by (insert[of A ? C,auto:elimmeets
 e ∪ f"
  proof (elim disjE)
 \><>?B\>\<>?
     then have "p = q" using M4 kp pu qu by blast
java.lang.StringIndexOutOfBoundsException: Range [55, 54) out of bounds for length 146
     thus ?thesis using x by simp }
    next
   {assume "(¬?A∧andnotC)" then have "?B" by simp
     then obtain t where kt:"k∥ \in e🚫<¬java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 68
     then have "(p,q) ∈
     thus ?thesis using x by simp}
    
    "n?A∧¬?B∧?C)" then have "?C" by simp
     then obtain t where kt:"k'java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
pp)<>fjava.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 55
     ain<> t\>"byauto
  qed
qed

lemma cffi:"f java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 3
proof
   fix x::"'a×'a" assume "x \    ::'a\>'a"assume" \in> s^-1 O O s" then obtain p q  where:"x = (p,r)" and()\in and()<java.lang.StringIndexOutOfBoundsException: Range [128, 127) out of bounds for length 136
   from <,< ‹(q,r) ∈ f^-1\close> obtain k k' where kp:"k∥∥
 from \open∈ ‹ obtain u where pu:"p∥ "q∥u" using f M1 by blast
 from kp kpr have "k\pl>\<oplus exists>tt. k\parallel<><t. k'∥ t\<parallel)opl (?B \oplus> ?C)") using M2 by blast
 then have "(?A∧¬?B∧?C) \<r<?B\<and<?C) ∨ (¬¬?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 thus "x ∈ e ∪ f ∪ f^-1"
 (elim disjE)
 { assume "?A∧ "x \<in  s ∪
 r kp have "p = r" using M4 by auto
 thus ?thesis using x e by auto}
 next
 "<>\
 then obtain t where kt:"k∥t" and tr:"t∥r" by auto
 with ru kp pu show ?thesis using x f by blast}
 next
 { assume "¬?A ∧ ¬?B ∧>u' \oplus ((\exists>. p\<><u)" (is "?A \oplus (?B 🚫
 then obtain t where rtp:"k'∥t" and "t∥p" by auto
 with kpr ru pu show ?thesis using x f by blast}
 qed
 

(* =========$\alpha_5$ composition========== *)
text ‹We prove compositions of the form $r_1 \circ r_2 \subseteq e \cup s \cup s^{-1 proof (elim disjE)

  cssi:"s O s^-1 ⊆ e ∪ s ∪ s^-1"
 
 fix x::"'a×'a" assume "x ∈ s O s^-1" then obtain p q r where x:"x = (p,r)" and "(p,q)∈ ?thesis u using x e by auto}
 from ‹(p,q)∈s› ‹(q,r) ∈ s^-1› obtain k where kp:"k∥p" and kr:"k∥r" and kq:"k∥q" using s M1 by blast
 from ‹{ assume "\not>?A∧?B∧¬?C" then have ?B by simp
 then have "p\parallel>u' ⊕ ((∃t. p∥t ∧ t∥u') ⊕ (∃t. r∥t ∧ t∥u))" (is "?A ⊕ (?B \        with rup qp qr show ?thesis using x s by blast}
 (?A\\not>?B\and>>\not>?C) rand>?B\and>\not>?C) \or> (\not>?\<>< ?B ?C], auto simp:elimmeets)
 thus "x ∈ e ∪ s ∪ s^-1"
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have ?A by simp
 with rup kp kr have "p = r" using M4 by auto
 thus ?thesis using x e by auto}
 next
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
 then obtain t where kt:"p∥t" and tr:"t∥then obtain t where rtp:"r🚫
 with rup kp kr show ?thesis using x s by blast}
 next
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 then obtain t where rtp:"r∥t" and "t∥u" by auto
 with pu kp kr show ?thesis using x s by blast}
 qed
 

  csis:"s^-1 O s ⊆

java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 108
 om<openp ‹ s\> obtain k where kp:"k∥\parallel>r" and kq:"∥
 from ‹
 >u' \ ((∃t ∧u') ⊕t. r∥ t∥ (?B ⊕)") using M2 by blast
 then have "(?A\<and  zu obtain cz where pcz:"p∥cz" and czu:"cz∥u" using M5exist_var by blast
 thus "x ∈ uv obtain czu where pczu:"p\parallel>czu" and czuv:"czu\\ v"" using MM5exist_var by blast
 (elim disjE)
 { assume "?A∧¬?B∧¬ have "(?A\and>🚫 (¬¬?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 with rup kp kr have "p = r" using M4 by auto
 thus ?thesis using x e by auto}
 next
 { assume "\       assume "?A∧¬?B∧¬?C" then have ?A by simp
 then obtain t where kt:"p\<>u
 with rup kp kr show ?thesis using x s ththus ?thesis using x by auto}
 next
 A ∧ ¬?B ∧ ?C" then have ?C by simp
 then obtain t where rtp:"r∥t" and "t∥u" by auto
 with pu kp kr show ?thesis using x s by blast}
 qed
 

  cmim:"m^-1 O m ⊆then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬ thus "x ∈> m ∪ ov ∪s 🚫
 
 fix x::"'a×'a" assume "x \             thus ?thesis using x m by auto}
 from ‹next
  u' where pu:"p\pa>u" and rup:"r\<parallelu
 u' \<(exists
 then have "(?A∧¬?B∧ t' wheparallel>t'" and "t'∥c" by auto
 thus "x ∈ s \<ct'\parallel>czu" using M1 by auto
 proof (elim disjE)
 { assume "?A∧\             with at tq ap pczu czuv qv ‹t∥t'› have "(p,q)∈thus ?thesis using x by auto}
 with rup qp qr have "p = r" using M4 by auto
 thus ?thesis using x e by auto}
 next
  ssume"\<>< then have ?B ?B by simp
 then obtain t where kt:"p∥t" and tr:"t\ then obtain t wt where "k∥t" and "t∥p" by auto
  rup qp qr show ?thesis using x s by blast}
 next
java.lang.StringIndexOutOfBoundsException: Range [13, 6) out of bounds for length 72
 then obtain t where rtp:"r∥t" and "t∥u" by auto
 using x s by blast}
 qed
 

(* =========$\beta_1$ composition========== *)
subsection ‹
  ‹p" using M3 meets_wd pc by blast

java.lang.StringIndexOutOfBoundsException: Range [54, 53) out of bounds for length 76
 from ap khave "ahend='alert("unbekannte/s Formatierung/Symbol >");' >🪙∥ t\parallel>q) \oplus (\<>t 
 then have "(?A\and>\not>?B∧¬?C) \or> ((¬?A∧and>¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
  \open>(p,z) z) \ <>nz" using b by auto
 obtain a where ap:"a∥p" using M3 meets_wd pc by blast
 from ‹?B∧>?C then have ?A by simp
 from pc cz zu obtain cz where pcz:"p∥ with ap pcz czu qu have "(p,q) ∈ s" using s by blast
java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 39
 from ap kq have "a∥ then obtaint where a:a∥q" by auto
java.lang.StringIndexOutOfBoundsException: Range [102, 101) out of bounds for length 184
 <>o
 
java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 53
 with ap pczu czuv uv qv have "(p,q) ∈ ((\>?C)") using M2 by blast
  tthen have "(?A\and>\not>>?B∧¬?C) ∨ ((¬and>?B\and>¬?C) ∨¬¬?C))" by (insert xor_distr_L[oatosip:mees
 next
 { thus ?thesis using x by auto}
 then obtain t where at:"a∥t" and tq:"t∥q" by auto
java.lang.StringIndexOutOfBoundsException: Range [38, 12) out of bounds for length 218
 hve "(?A\<dn?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by(insset xxr_dt_[f AB?] aospeimees
java.lang.StringIndexOutOfBoundsException: Range [114, 112) out of bounds for length 190
 proof (elim disjE)
 { assume "?A∧¬?B∧¬
 thus ?thesis using x m by auto}
 next
 "not>?A\and?B∧?C" then have ?B by simp
 thus ?thesis using x b by auto}
 next
 { assume "¬?A ∧ ¬?B ∧c" by auto
 then obtain t' where "t∥t'" and "t'∥c" by auto
 with pc pczu have "t'∥o
java.lang.StringIndexOutOfBoundsException: Range [84, 83) out of bounds for length 105
 thus ?thesis using x by auto}
 qed
 }
 next
 { assume "¬?A ∧ ¬<not? ∧ ¬?B ∧
 then obtain t where "k∥p" by auto
 with kq pczu czuv uv qv have "(p,q) ∈with kq cw cu q ave (,) ∈
 thus ?thesis using x by auto}
 qed
 

  cbf:"b O f ⊆uni s ∪ d"
 
 fix x::"'a×'a" assume "x \in> b O " then obtin p q z where x:"x = (p,q)" and "(p,z) ∈ b" and "(z,q) ∈ f" by auto
 from ‹fix x::"'a×'a" assume "x ∈ b O m^-1" then obtain p q z where x:"x = (p,q)" and "(p,z) ∈ b" and "(z,q) ∈ m^-1" by auto
 obtain a where ap:"a∥p" using M3 meets_wd pc by blast
 from ‹(p,z) ∈ b› obtain c where pc:"p∥c" and cz:"c∥" using b byat
 from pc cz zu obtain cz where pcz:"p∥cz" and czu:"cz∥u" using M5exist_var by blast
 from ap kq hve "a\l⊕parallel>t \and> t∥q) ⊕ (∃t. k∥t ∧ t∥p))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
 then have "(?A∧¬?B∧¬?C) ∨ ((¬from \open>(z,q) \in> m^-1\close> have qz:"q∥z" using m by auto
 thus"x <>< m ∪ ov ∪ s ∪ d"
java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 { assume "?A∧in> b ∪ m ∪ ov ∪ s ∪ d"
 with ap pcz czu qu have "(p,q) ∈ s" using s by blast
java.lang.StringIndexOutOfBoundsException: Range [15, 8) out of bounds for length 39
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
 { assume "¬?A\<
 then obtain t where at:"a∥t" and tq:"t∥q" by auto
java.lang.StringIndexOutOfBoundsException: Range [39, 37) out of bounds for length 218
 then have "(?A∧?B\<and\?C) \< ((¬?B∧?C) ∨?A∧?B∧ ?B ?C,aut slimees)
 thus "x ∈ b ∪ m ∪ ov ∪t" and tq:"t∥
java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 26
 { assume "?A∧?B∧?C" then have ?A by simp
 thus ?thesis using x m by auto}
 next
java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 73
 thus ?thesis using x b by auto}
 next
 { assume "¬?A ∧ proof (elim disjE)
 then obtain t' where "t∥t'" and "t'∥c" by auto
 with pc pcz have "t'∥cz" using M1 by auto
 with at tq ap pcz czu qu ‹ { assume "?A∧¬?B∧\not>?C" then have ?A by simp
 thus ?thesis using x by auto}
 qed
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 next
 { assume "¬and> ¬?B ∧ ?C" then have ?C by simp
 then obtain t where "k∥t" and "t∥p" by auto
java.lang.StringIndexOutOfBoundsException: Range [15, 12) out of bounds for length 65
 thus ?thesis using x by auto}
 qed
 

 > b \< < d"
  { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp
  obtain t where "k'\parallelt" and "t∥p" by auto
 from \ with kpq pc cz hae ) \in d" using d by blast
 obtain a where ap:"a∥p" using M3 meets_wd pc by blast
 from ‹
java.lang.StringIndexOutOfBoundsException: Range [23, 15) out of bounds for length 53
 with pc wu obtain cw where pcw:"p∥cw" and cwu:"cw∥u" using M5exist_var by blast
 from ap kq have "a∥q ⊕ ((∃t. a∥t ∧ t∥q) ⊕ (∃t. k∥t ∧ t∥p))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
  "x ∈ b ∪ m ∪ ov ∪ s ∪ d"
 proof (elim disjE)
 { assume "?A∧¬
  wu have "((p,q) \in> s" using s by blast
 thus ?thesis using x by auto}
 next
 { assume "¬from \<>) ov› obtain k' l' u' v' c where kplp:"k'∥l'" and kpz:"k'∥z" and lpq:"l'∥q" and zup:"z∥u'" and upvp:"u'∥v'" and qvp:"q∥v'" and "l'∥c" and "c∥u'" using ov by blast
 then obtain t where at:"a∥t" and tq:"t∥q" by auto
java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 218
 then have "(?A∧¬?B∧ppvbainuwhere puuu:p\parallel>uu" and uuvp:"u<>'" using M5exist_var by blast
 thus "x ∈ b ∪ m ∪ ov ∪ s ∪ d"
 proof (elim disjE)
 { assume "?A∧\<ot<then have "(?A∧?B∧?C) \<r<?B∧¬or> (\not>?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
  u ?theis uing by uto}
java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 { assume "¬?A∧thus "x \in> b ∪ m ∪ ov ∪ s ∪ d"
 thus ?thesis using x b by auto}
 next
 { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp
 then obtain t' where "t∥t'" and "t'∥
  🚫
 <>< have "(p,q)∈
 thus ?thesis using x by auto}
 qed
 }
 next
 { assume "¬?A ∧ ¬}
  oin t whereere "k∥p" by auto
 withme "\<not\ ¬> ?C" then have ?C by simp
 thus ?thesis using x by auto}
 qed
 

  cbmi:"b O m^-1 ⊆ b ∪ m ∪ ov ∪ s ∪ d"
 
 fix x::"'a×'a" assume "x ∈ b O m^-1" then obtain p q z where x:"x = (p,q)" and "(p,z) ∈
 from ‹
 obtain k wherekp:"k\<arallel"using M3 meets_wd pc by blast
 from ‹(z,q) ∈ m^-1› have qz:"q∥z" using m by auto
 obtain k' where kpq:"k'∥q" using M3 meets_wd qz by blast
 from kp kpq have "k∥q ⊕ ((∃t. k∥t ∧ t∥q) ⊕ (∃t. k'∥t ∧ t∥p))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 > ov ∪ s ∪ d"
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have ?A by simp
 with kp pcpc z have "(p,q)<sg
 thus ?theis usingg x by by ay auto}
 next
 { assume "¬?B\and>\not>?C" then have ?B by simp
 then obtain t where kt:"k∥t" and tq:"t∥ "?A\<<not>?B∧ ((¬?B∧?C) ∨>?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
java.lang.StringIndexOutOfBoundsException: Range [41, 39) out of bounds for length 64
 then have "(?A∧¬?B∧¬or> ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 thus "x ∈ b ∪ m ∪ ov ∪ s ∪ d"
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have ?A by simp
 thus ?thesis using x m by auto}
 next
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
 thus ?thesis using x b by auto}
 next
 { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp
 then obtain t' where "t\<parallelnc" by auto
 with pc cz qz kt tq kp have "(p,q) ∈ ov" using ov by blast
 ssisusing x y auto}
 qed
 }
 next
 { assume "¬ \<not<
 then obtain t where "k'∥parallel>t" and tq:"t∥q" by auto
 with kpq pc cz qz have "(p,q) ∈ d" using d by blast
 thus ?thesis using x by auto}
 qed
 

  cdov:"d O ov ⊆b ∪ m ∪ ov ∪ s ∪ d"
 
 fix x::"'a×'a" assume "x \<        thus
java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 208
 from ‹
 from zup zv uv have "u∥u'" using M1 by auto
 with pu upvp obtain uu where puu:"p∥v'" using M5exist_var by blast
 parallel>q ⊕ ((∃t. l∥t ∧ t∥q) ⊕ (∃t. l'∥t ∧ t∥
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧{ assume " ?A ∧ ¬?B ∧ ?C" then have ?C by simp
 thus "x ∈ b ∪ in t' whee tp:"\parallel>t'" anpu:"t'\parallel" by auto
 proof (elim disjE)
java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
 with lp puu uuvp qvp have "(p,q) ∈ s" using s by blast
 thus ?thesis using x by auto}
 next
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
 then obtain t where lt:"l∥t" { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp
java.lang.StringIndexOutOfBoundsException: Range [21, 18) out of bounds for length 217
 then have "(?A<and\?B∧?C) ∨?A∧¬ (¬¬?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmets)
 thus "x ∈ b ∪ m ∪ ov ∪ s ∪ d"
 f (eliim disjE)
 qed
 thus ?thesis using x m by auto}
 next
 { assume "¬We prove compositions of the form $r_1 \circ r_2 \subseteq b \cup m \cup ov \cup f^{-1} \cup d^{-1}$.›
  covdi:"o O d^subsete> b ∪ m ∪ ov ∪ f^-1 ∪ d^-1"
 next
 <>A?B \and> ?C" the have ?C by simp
java.lang.StringIndexOutOfBoundsException: Range [14, 2) out of bounds for length 255
 with pu puu have "t'∥uu" using M1 by auto
 with lp puu qvp uuvp lt tq ttp have "(p,q) ∈ ov" using ov by blast
 thus ?thesis using x by auto}
 
java.lang.StringIndexOutOfBoundsException: Range [15, 13) out of bounds for length 57
 next
java.lang.StringIndexOutOfBoundsException: Range [6, 1) out of bounds for length 72
 then obtain t where "l'∥t" and "t∥p" by auto
 with lpq puu uuvp qvp have "(p,q) ∈ d" using d by blast
 thus ?thesis using x by auto}
 qed
 

  cdfi:"d O f^-1 ⊆ b ∪ m ∪ ov ∪ s ∪ d"
 
 fix x::"'a×'a" assume "x ∈ d O f^-1" then obtain p q z where x:"x = (p,q)" and "(p,z) ∈ d" and "(z,q) ∈ f^-1" by auto
 from ‹(p,z) ∈ d› obtain k l u v where kl:"k∥l" and lp:"l∥p" and kz:"k∥z" and pu:"p∥u" and uv:"u∥v" and zv:"z∥v" using d by blast
 from ‹(z,q) ∈ f^-1› obtain k' l' u' where kpz:"k'∥z" and kplp:"k'∥'" and lpq:"l'∥d zp:"z\parallel>u'" and q qup:"q🚫
 from zup zv uv have uup:"u∥ proof (elim disjE)
 from lp lpq have "l∥q ⊕ ((∃t. l∥t ∧ t∥q) ⊕ (∃t. l'∥t ∧ t∥p))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 thus "x ∈ b ∪ m ∪ ov ∪ s ∪ d"
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have ?A by simp
 with lp pu uup qup have "(p,q) ∈ s" using s by blast
 next
 next
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
 then obtain t where lt:"l∥t" and tq:"t∥q" by auto
 from pu tq have "p∥q ⊕ ((∃t'. p∥t' ∧ t'∥q) ⊕ (∃t'. t∥t' ∧ t'∥u))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 thus "x ∈ b ∪ m ∪ ov ∪ s ∪ d"
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have ?A by simp
 thus ?thesis using x m by auto}
 next
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
 thus ?thesis using x b by auto}
 next
 { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp
 then obtain t' where ttp:"t∥t'" and tpu:"t'∥u" by auto
 with lt tq lp pu uup qup have "(p,q) ∈ ov" using ov by blast
 thus ?thesis using x by auto}
 qed
 }
 next
 { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp
 then obtain t where "l'∥t" and "t∥p" by auto
 with lpq pu uup qup have "(p,q) ∈ d" using d by blast
 thus ?thesis using x by auto}
 qed
 

(* =========$\beta_2$ composition ==========*)
text ‹We prove compositions of the form $r_1 \circ r_2 \subseteq b \cup m \cup ov \cup f^{-1} \cup d^{-1}$.›

lemma covdi:"ov O d^-1 ⊆ b ∪ m ∪ ov ∪ f^-1 ∪ d^-1"
proof
  fix x::"'a×'a" assume "x ∈ ov O d^-1" then obtain p q z where "(p,z) : ov" and "(z,q) : d^-1" and x:"x = (p,q)" by auto
  from ‹(p,z) : ov› obtain k l u v c  where kp:"k∥p" and kl:"k∥l" and lz:"l∥z" and pu:"p∥u" and uv:"u∥v"  and zv:"z∥v" and lc:"l∥c" and cu:"c∥u" using ov by blast
  from ‹(z,q) : d^-1› obtain l' k' u' v'  where lpq:"l'∥q" and kplp:"k'∥l'" and kpz:"k'∥z" and qup:"q∥u'" and upvp:"u'∥v'" and zvp:"z∥v'"  using d  by blast
  from lz kpz kplp have "l∥l'" using M1 by auto
  with kl lpq obtain ll where kll:"k∥ll" and llq:"ll∥q" using M5exist_var by blast
  from pu qup  have "p∥u' ⊕ ((∃t. p∥t ∧ t∥u') ⊕ (∃t. q∥t ∧ t∥u))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
  then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
  thus "x ∈ b ∪ m ∪ ov ∪ f^-1 ∪ d^-1"
  proof (elim disjE)
      { assume "?A∧¬?B∧¬?C" then have ?A by simp
        with qup kll llq kp  have "(p,q) ∈ f^-1" using f by blast
        thus ?thesis using x  by auto} 
      next
      { assume "¬?A∧?B∧¬?C" then have ?B by simp 
        then obtain t where pt:"p∥t" and tup:"t∥u'" by auto
        from pt lpq have "p∥q ⊕ ((∃t'. p∥t' ∧ t'∥q) ⊕ (∃t'. l'∥t' ∧ t'∥t))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
        then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
        thus "x ∈ b ∪ m ∪ ov ∪ f^-1 ∪ d^-1"
        proof (elim disjE)
           { assume "?A∧¬?B∧¬?C" then have ?A by simp
             thus ?thesis using x m by auto}
           next
           { assume "¬?A∧?B∧¬?C" then have ?B by simp
             thus ?thesis using x b by auto}
           next
           { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp
             then obtain t' where lptp:"l'∥t'" and tpt:"t'∥t" by auto
             from lpq lptp llq have "ll∥t'" using M1 by auto
             with kp kll llq pt tup qup tpt  have "(p,q) ∈ ov" using ov by blast
             thus ?thesis using x by auto}
        qed
        }  
      next
      { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp
        then obtain t where "q∥t" and "t∥u" by auto
        with pu kll llq kp   have "(p,q) ∈ d^-1" using d by blast
        thus ?thesis using x  by auto}
       qed
qed

lemma cdib:"d^-1 O b ⊆ b ∪ m ∪ ov ∪ f^-1 ∪ d^-1"
proof
  fix x::"'a×'a" assume "x ∈ d^-1 O b" then obtain p q z where "(p,z) : d^-1" and "(z,q) : b" and x:"x = (p,q)" by auto
  from ‹(p,z) : d^-1› obtain k l u v  where kp:"k∥p" and kl:"k∥l" and lz:"l∥z" and pv:"p∥v" and uv:"u∥v"  and zu:"z∥u"  using d by blast
  from ‹(z,q) : b› obtain c  where  zc:"z∥c" and cq:"c∥q"  using b by blast
  with kl lz obtain lzc where klzc:"k∥lzc" and lzcq:"lzc∥q" using M5exist_var by blast
  obtain v' where qvp:"q∥v'" using M3 meets_wd cq by blast
  from pv qvp  have "p∥v' ⊕ ((∃t. p∥t ∧ t∥v') ⊕ (∃t. q∥t ∧ t∥v))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
  then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
  thus "x ∈ b ∪ m ∪ ov ∪ f^-1 ∪ d^-1"
  proof (elim disjE)
      { assume "?A∧¬?B∧¬?C" then have ?A by simp
        with qvp kp klzc lzcq  have "(p,q) ∈ f^-1" using f by blast
        thus ?thesis using x  by auto} 
      subsection\open> Convergence{sec:onconv<>
      { assume "¬?A∧?B∧¬?C" then have ?B by simp 
 <>Asensible for integral isthat be
        frompt havep<> ⊕t'. p<>t<and t'∥ (<existst' <dis<<java.lang.StringIndexOutOfBoundsException: Range [198, 196) out of bounds for length 217
        T are sequencesof numbers  functions nd
        thussetsMonotone convergence even defined generally
        roof elim)
           { assume "?A∧>?B\and>¬ ?A bysimp
             thus ?thesis using x m by auto}
           next
           { assume "\not?A<>B<><>C"then hhave ?B by simp
             thus ?thesis us b yato}
java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
ume "<>A<>\not? \and C"then have ?C by simp
             then obtain t' where ctp:"parallel tpt>" by auto
              lzcq cq ctphave "lzct'" using M1 by auto
              p vpp qvp pkzclc tpt hav "(, <> ovjava.lang.StringIndexOutOfBoundsException: Range [75, 74) out of bounds for length 85
gjava.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 42
        qed
        }  
      next
      { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp
        then obtain t where
           klzc  have (<>^  dbyblast
t thesis xby auto
       qed
qed

 csdi:"s O d^-1 ⊆ b ∪ m ∪ ov ∪ f^-1 ∪ d^-1"
theor mon_conv x<>al
x:"a\times>'a" x<>java.lang.StringIndexOutOfBoundsException: Range [48, 45) out of bounds for length 130
  from <>)java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 182
  from <open( obtain l' k' 'v where:"'\parallel>q"and kplp:k<  kpzparallel and qup:"q∥u'" and upvp:"u'∥v'" and zvp:"z∥v'"  using d  by blast
  from kp   ultimately showjava.lang.StringIndexOutOfBoundsException: Range [27, 25) out of bounds for length 72
from(<ambda.x i w)↑( )"
  then have "(?A∧ (imp:realfun_mon_conv_iff)   
  thus" \in b \union> m \union> ov \union> f^-1 \<> 
  proof (elim disjE)
{assume "Aa>\not?B<and<>C" then have ?Aby simp
        with qup kpp kplp lpq have ""p,q) ∈ f^-1" using f by blast
        thus ?esisingutojava.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
      java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
      {\n>A∧?B∧¬?C" then have ?B by simp
        then obtain t where pt:"p∥ and tupt<parallel  auto
          show? 
        then have(A<><>B<><?) \or> (\n>?\and>?B<nd>\not?) ∨?A∧?B∧" y (inser[f ?A?BC, atlmets)
        x \ m  ov ∪ f^-1 ∪ d^-1"
        proof  from assms "\A> xn \< "
           { assume "?A\<and 
              ?thesis using x m by auto}
           next
           { assume "¬?A∧ : realfun_mon_conv_iff real_mon_conv_times
             theoremreal_mon_conv_add: 
          
           {     havejava.lang.StringIndexOutOfBoundsException: Range [25, 23) out of bounds for length 67
 ':l<>t and:t\ by 
        java.lang.StringIndexOutOfBoundsException: Range [15, 13) out of bounds for length 82
              ? using byautojava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
       qed
        } 
      next
      { assume "¬l. c↑ l≤
         where "q<arallel"and "<>u
        with pu p(<-ng
?hesisx bya}
       qed
qed

lemma csib:"s^-1 O b ⊆ b ∪ m ∪ ov ∪ f^-1 ∪ d^-1"
proof
  fix x::"'a×'a" assume "x ∈ s^-1 O b" then obtain p q z where "(p,z) : s^-1" and "(z,q) : b" and x:"x = (p,q)" by auto
  from ‹(p,z) : s^-1› obtain   also notemon
  fromjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  from kz zc cq obtain zc where kzc  bound"<nd>n. c n ≤ (x::'a ==> real)"
  obtain v' where qvp:"q∥v'" using M3(*<*)proof 
  from pv qvp  have "p∥v' ⊕ ((∃t. p∥t ∧ t∥v') ⊕ (∃t. q∥t ∧ t∥v))" (is "?A ⊕ (?B ⊕ ?C)"   ix
  then have "(?A∧¬?B∧¬?C) ∨
  thus "x ∈ b ∪ m ∪ ov ∪ f^-1 ∪ d^-1"
  proof (elim disjE)
      { assume "?A∧¬?B∧¬?C" then have ?A by simp
        with qvp kp kzc zcq have "(p,q) ∈ f^-1" using f by blast
         ?thesis using x by auto}
    hence "P(OME P l) y(ule)
 "<ot>?A\and>?B\and>\<>C
        then obtain t where pt:"p∥t" and tvp:"t∥v'" by auto
        from pt cq have "p∥q ⊕ ((∃t'. p∥t' ∧ t'∥q) ⊕ (∃t'. c∥t' ∧ t'∥t))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
        then have "(?A∧¬?B∧¬java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
        thus "x ∈mo
        proof (elim disjE)
w
thus ?thesis using x m by auto}
          
          { assume "¬and?B\and>¬?C" then have ?B by simp
             thus ?thesis using x b by auto}
           nextexists>j. x \in> A j"
           {  (<>.mk_mon \subseteq>(i. A i)java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
             then obtain t'wherectpc< andtptt<  auto
             from zcq  zc<>'
             with zcqjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
             thus ?thesis using x by auto}
        qed
        }  
      next
      { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp
        then obtain t where "q∥t" and java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
 blast
        thus ?thesis using x by auto}
       qed
qed

lemma covib:"ov^-1 O b ⊆ b ∪ m ∪ ov ∪ f^-1 ∪ d^-1"
proof
  fix x::"'a×'a" assume "x ∈ ov^-1 O b" then obtain p q z where "(p,z) : ov^-1" and "(z,q) : b" and x:"x = (p,q)" by auto
  from ‹(p,z) : ov^-1› obtain k l u v c where kz:"k∥z" and kl:"k∥l" and lp:"l∥p" and zu:"z∥u" and uv:"u∥v" and pv:"p∥v" and lc:"l∥c" and cu:"c∥u" using ov by blast
  from ‹(z,q) : b› obtain w where zw:"z∥w" and wq:"w∥q" using b by blast
  from cu zu zw have cw:"c∥w" using M1 by auto
  with lc wq obtain cw where lcw:"l∥cw" and cwq:"cw∥q" using M5exist_var by blast
  obtain v' where qvp:"q∥v'" using M3 meets_wd wq by blast
  from pv qvp have "p∥v' ⊕ ((∃t. p∥t ∧ t∥v') ⊕ (∃t. q∥t ∧ t∥v))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
  then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (<not<>B∧ _L
  thus "x ∈
  proof (elim disjE)
      { assume "?A∧      {java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 72
        with qvp lp lcw cwq  have  qed
        thusjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      next
java.lang.StringIndexOutOfBoundsException: Range [17, 14) out of bounds for length 69
        then obtain t where pt:"p∥v'" by auto
        from pt wq have<><<xists. p∥t' ∧ t'∥q) ⊕ (∃t'. w∥t' ∧ t'∥t))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
        then have "and<not>?B∧¬ ((\>andB∧\?C) < <?)java.lang.StringIndexOutOfBoundsException: Range [144, 143) out of bounds for length 190
        thus "x ∈ b ∪from pv qvp have "p∥v' ⊕ ((∃t. p∥t ∧ t∥v') ⊕ (∃t. q∥t ∧ t∥v))" (is "?A ⊕ (?B ⊕A<\n>B∧¬?C) ∨ ((¬?A∧?B∧¬?C <<notjava.lang.StringIndexOutOfBoundsException: Range [102, 101) out of bounds for length 184
        proof (elim
           { assume "<and>\not>?B\\<not>?C"
             thus ?thesis java.lang.StringIndexOutOfBoundsException: Range [22, 20) out of bounds for length 39
                 { assume<><<and<>njava.lang.StringIndexOutOfBoundsException: Range [58, 57) out of bounds for length 69
           { assume "¬?A∧?B∧¬q ⊕t'. p∥ t'∥q) ⊕t'. c∥t))"java.lang.StringIndexOutOfBoundsException: Range [173, 171) out of bounds for length 217
              java.lang.StringIndexOutOfBoundsException: Range [27, 25) out of bounds for length 44
           next
           { assume "¬?A ∧ ¬ { assume "?\<><ot<and\not>C then have ?A by simp
lel<java.lang.StringIndexOutOfBoundsException: Range [84, 81) out of bounds for length 87
             moreover with wq cwq  java.lang.StringIndexOutOfBoundsException: Range [14, 11) out of bounds for length 73
             ultimately have "(p,q) ∈ ov" using ov 
             thus ?thesis using n>A ∧ ¬?B ∧ ?C" then have ?C by simp
java.lang.StringIndexOutOfBoundsException: Range [54, 52) out of bounds for length 87
        }
      next
      { assume "<not
        then obtain t wherejava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
        with pv lp lcw cwqjava.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 71
        thus is
       qed
qedjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

 cmib<><union<union <union><-
proof
  fix x::"'a×'a" assume "x ∈ m^-1 O b" then obtain p q f<open< v\parallelp" and kz:"k∥z" and pu:"p∥u" and uv:"u∥v" and zv:"z∥v" using s by blast
  from ‹(p,z) : m^-1› have zp:"z∥p" using m by auto
  from ‹(z,q) : b› obtain w where zw:"z∥w" and wq:"w∥q" using b by blast
  obtain v where pv:"p∥v" using M3 meets_wd zp by blast
  obtain v' where qvp:"q∥v"M et_dw ybat

  from pv qvp have "p∥v' ⊕ ((∃t. p∥t ∧ t∥v') ⊕ (∃t. q∥t ∧ t∥v))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
  then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
  thus "x ∈ b ∪ m ∪ ov ∪ f^-1 ∪ d^-1"
  proof (elim disjE)
      { assume "?A∧¬?B∧¬?C" then have ?A by simp
        with zp zw wq qvp have "(p,q) ∈ f^-1" using f by blast
        thus ?thesis using x by auto}
      next
      { assume "¬?A∧?B∧¬?C" then have ?B by simp
        then obtain t where pt:"p∥t" and tvp:"t∥v'" by auto
        from pt wq have "p∥q ⊕ ((∃from pu qup  have "p\\> ((\<existstt ∧u') ⊕<> t∥ A \oplus(?B ⊕ ?C)") using M2 by blast
not>?A∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simpmees
        thus "x ∈ b ∪ b ∪ ov ∪ d^-1"
        proof (elim disjE)
           { assume "?A∧¬?B∧plp f^-1" using f by blast
             thus ?thesis using x m by auto}
           next
           { assume "¬?A∧?B∧q ⊕t'. p∥ t'∥ (∃t' ∧t)

           next
           { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp
             then obtain t' where wtp:"w∥t'" and tpt:"t'∥t" by auto
             with zp zw wq pt tvp qvp have "(p,q) ∈ ov" using ov  by blast
             thus ?thesis using ot>B<nd?C" then have ?A by simp
        qed
        }
      next
      { assume "¬
        then obtain t where "q∥t" and "t∥v" by auto
        with zp zw wq pv   have "(p,q) ∈ d^-1" using d by blast
        thus ?thesis using x  by auto}
       qed
qed

(*==========$\gamma$ composition =======*)
subsection ‹
  ‹

  covovi:"ov O ov^-1 \<subseteq\ ov \<unionf d^-"
 
 fix:"'a×-1 then obtain p q z whre x:" = (p,q)" and "(p,p,z) \in ovov" and "(z, q) ∈ ov^-1" by auto
 from ‹(p,z) ∈ ov› obtain k l rom ‹p" and kz:"k∥u" and uv:"u∥v" using s by blast
 from ‹

 from kp kpq have "k∥¬ ((¬?B∧? <>(?B\?C))" by (insert xor_distr_L[of ?A ?B ?C],,auto simmp:eimmees)
  have "(?A∧not>>?∧?C) ∨?A∧¬ (¬¬?C))" by (insert xo_ditr_L_L[of ? ?B C], u ipeimes
 ¬¬
 proof (elim disjE)
 { assume "?A∧
 ve\parallelu' ⊕t'. p∥ t'∥ (∃t' ∧u))" (is "?A ⊕ ?C)") using M2 by blast
 then have "(?A∧¬t" and tvp:"t∥
 thus ?thesis
 proof (elim disjE)
 ¬and>¬?C" then have ?byimp
 with kq kp qup have "p = q" using M4 by auto
 thus ?thesis using x e by auto}
 next
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
 with kq kp qup show ?thesis using x s by blast}
 next
 { assume "\<not             
 with kq kp pu show ?thesis using x s by bl thus ?t?thsi using by ao}
 qed}
 next
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
 then obtain t where kt:"k∥t" and tq:"t∥q" by auto
 from pu qup have "p∥u' ⊕ ((∃t'. p∥t' ∧ t'∥u') ⊕ (∃t'. q∥t' ∧ t'∥u))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 thus ?thesis
 proof (elim disE)
 { assume "?A∧¬?B∧t" and "t∥v" by auto
 with qup kp kt tq show ?thesis using x f by blast}
 next
 { assume "¬
 then obtain t' where ptp:"p∥ b ∪ ov ∪ d^-
 from tq kpq kplp have "t∥f x:"a<>'from \open(p,z) : ov^-1› obtain k l u v c where kz:"k∥l" and lp:"l∥u" and uv:"u∥v" and lc:"l∥u" using ov by blast
 overwt lz c v"∥
 moreover with cu pu ptp have "c∥t'" using M1 by auto
 ultimatelyoan wee "t\parallel" and "lc∥t'" using M5exist_var by blast
 with ptp tpup kp kt tq qup show ?thesis using x ov by blast}
 next
 { assume "¬?A∧¬then have "(?A∧?B∧¬ ((¬?B∧¬ (¬¬?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 with pu kp kt tq show ?thesis using x d by blat}

 
 next
 {assume "¬?A∧¬
 then obtain wee kpt"k∥p" by auto
 from pu qphae"p\parallel' ⊕ ((∃t' ∧u') ⊕t'. q∥ '∥" s ?A\oplus (?B ⊕lst
  v (?Aan>¬?B∧?C) ∨?A∧¬ (¬¬y ierxorsL??C, uoiplm)
 thus ?te
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have ?A by simp
 with kpq kpt tp qup show ?thesis using x f by blast}
 next
 { assume "¬?A∧{ assume "¬ ¬?B ∧
 then obtain t' where "p∥t'" and "t'∥u'" by auto
 with kpq kpt tp qup show ?thesis using x d by blast}
 next
 { assume "¬?A∧¬?B∧
 then obtain t' where qtp:"q∥t'" and tpu:"t'∥
 mp kp kl hve "t\parallelusing M1 by auto
 moreoverwthlp zlz have"l\parallelc" using M1 by auto
 moreover with cpup qup qtp have "c'∥ d^-1" using d by blast
 ultimately obtain lc where "t∥lc" and "lc∥t'" using M5exist_var by blast
 with kpt tp kpq qtp tpu pu show ?thesis using x ov by blast}
 qed}
 qed
 


  cdid:"d^-1 O d ⊆ e ∪ ov ∪ ov^-1 ∪ d^-1 ∪ s^-1 ∪ f ∪
 
 fix x::"'a×' obtain v he pv:"p<rallelv¬?C) ∨?A∧¬ (\not>?<><
 from ‹p" and kl:"k\<allell
 from { assume "A\and¬?B∧?C" then have ?A by simp
 
 from next
 then have "(?A∧?B∧?C) ∨?A∧¬?C) ∨?A∧?B∧y inert xor_itrL[oof ?A ?B ?C,at sim:limees
 thus "x <> <>  s ∪ f ∪
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have kq:?A by simp
 from pv qvp have "p∥ ((∃t' ∧ t'∥ (∃t' ∧v))" (is "?A ⊕ ?C)") using M2 by blast
 then have "(?A∧?B∧?C) ∨?A∧>¬?C) ∨?A∧?B∧?C)" y (set or_dstrLof ???] ato spelmeet)
 thus hesi
 proof (elim disjE)
 { assume "?A∧¬
 with kq kp qvp have "p = q" using M4 by auto
 thus ?thesis using x e by auto}
 next
 { assume "¬?A∧
 with kq kp qvp show ?thesis using x s by blast}
 next
 { assume "¬?A∧¬?B∧?C" then have ?C by simp
 with kq kpvsw heiuin yblt
 qed}
 next
 { assume "¬?A∧?B\<andn›
 then obtain t where kt:"k∥ and tq:"t∥q" by auto
 from pv qvp have "p∥v' ⊕ ((∃ O ov^-1 ⊆ ov ∪ ov^-1 ∪ d^-1 ∪ s^-1 ∪ f^-1 "
 then have "(?A∧¬
 thus ?thesis
  (elim disjE)
 { assume "?A∧(z,q) ∈ ov^-1›'u he q'\parallel>q" and kplp:"k'∥z" and lpcp:"l'∥u'" and cpup:"c'∥
 with qvp kp kt tq show ?thesis using x f by blast}
 next
 { assume "¬¬
 then obtain t' where ptp:"p∥v'" by auto
 from tq kpq kplp have "t∥l'" using M1 by auto
 eoverovrwth ptp pv uv hae "\parallel'" using M1 by auto
 moreover with lpz zu ‹t∥l'› pu qup hve"p<rallel 
 ultimately show ?thesis using x ov kt tq kp ptp tpvp qvp by blast}
 next
 { assume "¬?B∧
 with pv kp kt tq show ?thesis using x d by blast}

 qed}
 next
 {sue \notA∧¬?B∧?C" then have ?C by auto
 then obtain t where kpt:"k'∥t" and tp:"t∥u' ⊕t'. p∥ t'∥ (∃t'. q∥ t'\arallelu))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
 from pv qvp have "p∥v' ⊕ ((∃t'. p∥ roof (elim disjE)
 then have "(?A∧?B∧?C) ∨?A∧¬ (¬¬?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 thus ?thesis
 proof (elim disjE)
 >¬¬p
 with kpq kpt tp qvp show ?thesis using x f by blast}
 next
 { assume "¬
 henotaint whr "p<>'" and "t'∥v'" by auto
 with kpq kpt tp qvp show ?thesis using x d by blast}
 next
 { assume "¬?A∧next
 then obtant hr qtp:"q∥v" by auto
 from tp kp kl have "t∥l" using M1 by auto
 moreover with qtp qvp upvp have "u'∥t'" using M1 by auto
 moreover with lz zup ‹t\      me ¬¬?C" then have ?C by auto
 ultimately show ?thesis using x ov kpt tp kpq qtp tpv pv by blast}
 from uqupae "p<>'pparall>t' 🪙 t'∥ (∃t' ∧u))" (is "?A ⊕ ?C)") using M2 by blast
 qed
 

java.lang.NullPointerException
 
 fix x::"'a×'a" assume "x ∈ ov^-1 O ov" then obtain p q z where x:"x = (p,q)" and "(p,z) ∈ ov^-1" and "(z, q) ∈ ov" by auto
 from ‹(p,z) ∈ ov^-1›
 from ‹(z,q) ∈ ov› obtain k' l' c' u' v' where kpz:"k'∥z" and kplp:"k'∥l'" and lpq:"l'∥q" and lpcp:"l'∥c'" and qvp:"q∥v'" and zup:"z∥u'" and cpup:"c'∥u'" and upvp:"u'∥v'" using ov by blast

 from lp lpq have "l∥q ⊕ ((∃t. l∥t ∧ t∥q) ⊕ (∃t. l'∥t ∧ t∥p))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 thus "x ∈ e ∪ ov ∪ ov^-1 ∪ d ∪ d^-1 ∪ s ∪ s^-1 ∪ f ∪ f^-1"
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have lq:?A by simp
 from pv qvp have "p∥v' ⊕ ((∃t'. p∥t' ∧ t'∥v') ⊕ (∃t'. q∥t' ∧ t'∥v))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 thus ?thesis
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have ?A by simp
 with lq lp qvp have "p = q" using M4 by auto
 thus ?thesis using x e by auto}
 next
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
 with lq lp qvp show ?thesis using x s by blast}
 next
 { assume "¬?A∧¬?B∧?C" then have ?C by simp
 with lq lp pv show ?thesis using x s by blast}
 qed}
 next
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
 then obtain t where lt:"l∥t" and tq:"t∥q" by auto
 from pv qvp have "p∥v' ⊕ ((∃t'. p∥t' ∧ t'∥v') ⊕ (∃t'. q∥t' ∧ t'∥v))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 thus ?thesis
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have ?A by simp
 with qvp lp lt tq show ?thesis using x f by blast}
 next
 { assume "¬?B∧?C" then have ?B by simp
 then obtain t' where ptp:"p∥t'" and tpvp:"t'∥v'" by auto
 from tq lpq lpcp have "t∥c'" using M1 by auto
 moreover with cpup zup zu have "c'∥
 moreover with ptp pv uv have "u∥
 ultimately obtain cu where "t∥cu" and "cu∥t'" using M5exist_var by blast
 with lt tq lp ptp tpvp qvp show ?thesis using x ov by blast}
 next
 { assume "¬¬
 with pv lp lt tq show ?thesis using x d by blast}

 qed}
 next
 {assume "¬?A∧¬?B∧?C" then have ?C by auto
 then btin t where lpt:"'🚫
 from pv qvp have "p∥v' ⊕ ((∃
 then have "(?A∧¬s}
 thus ?thesis
 proof (elim disjE)
 { assume "?A∧¬
 with qvp lpq lpt tp show ?thesis using x f by blast}
 next
 \?A∧¬
 v'" by auto
 with qvp lpq lpt tp show ?thesis using x d by blast}
 next
 { assume "¬?A∧¬?B∧?C" then have ?C by simp
 then obtain t' where qtp:"q∥
 from tp lp lc have "t∥c" using M1 by auto
 moreover with cu zu zup have "c∥
 moreover with qtp qvp upvp have "u'∥
 ultimately uweet<> 
 with lpt tp lpq pv qtp tpv show ?thesis using x ov by blast}
 qed}
 qed
 

(* ===========$\delta$ composition =========*)
subsection
text 


lemma cbbi:"b O b^-1 ⊆ b ∪ b^-1f pphv <v <> ((<>t' \parallelt' ∧v') ⊕t'. q∥v))" (is (?B ⊕
proof
  fix x::"'a×'a" assume "x ∈
  from ‹?C" then have ?A
  from ‹(z,q) ∈ b^-1›
  obtain k k' where kp:" ∥q" using M3 meets_wd pc qcp by fastforce
 then have "k∥t'" and "t'∥
 then have "(?A∧¬?B∧¬?A∧?B∧e ip
 thus "x ∈l" using M1 by auto
 proof (elim disjE)
 { assume "?A∧¬t∥l› obtainlzu" and "lzut'" using M5exist_var by blast
        from pc qcp have "p∥c' ⊕ ((∃t'. p∥t' ∧
        then
        thus ?thesis
        proof (elim disjE)
          {assume "(?A∧¬?B∧¬'a" assume "x ∈< ov^-1" and "(z, q) ∈ ov" by auto
            qhave
           thus ?thesis using x e  by auto}
          next
          {assume "\?A∧
           with kq kp qcp show ?thesis using x s by blast}
          next
          {assume "(¬?A∧
           with kq kp pc show ?thesis using x s by blast}
        qed}
      next
      { <><nd?<nd¬?C" then have ?B by simp
        then obtain t where kt:"k∥t" and tq:"t∥q" by auto
        from pc qcp have "p<parallel        
        then have "(?A∧¬?B∧¬?C) ∨proof (elim disjE)
        thus ?thesis
         (lidisjE)
          {assume "?A∧¬?B∧¬?C" then have ?A by simp
           with kp qcp kt tq show ?thesis using f x by blast}
          next
          {assume "¬?A∧?B∧ ereparallelt'" and tpvp:"t'∥
           then obtainzupparallelu" using M1 by auto
           from pc tq have "p∥q ⊕ ((∃t''. p∥t'' ∧ t''∥q) ⊕ (∃t''. t∥t'' ∧ t''∥c))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
           then have "(?A∧¬?B∧cu "parallel>t'" using M5exist_vart
           thus ?thesis
           proof (elim disjE)
              {assume "?A∧?B∧?C" then have ?A imp
               thus ?thesis using x m by auto}
              next
              {assume "¬?A∧?B∧¬?C" then have ?B by simp
               thus ?thesis using x b by auto}
              next
               ?A∧?C" then have ?C by simp
                then obtaing het<g" and "g∥c" by auto
                moreover with pc ptp have "g∥
                 l
           qed}
         next
          {assume "¬?A∧¬?B∧?C" then have ?C by simp
           then obtain t' where "q∥t'" and "t'∥c" by auto
           kp kt tqc sow ??hesisusing d byblast}
         qed}
      next
      { assume "¬?A∧¬?C" thn hae?B by simp
        then obtain t where kpt:"k'∥
        from  pc qcp have "p∥c' ⊕ ((∃t'. p∥t' ∧ t'∥c') ⊕ (∃t'. q∥t' ∧ t'∥c))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
        then have "(?A∧¬t'" and tpv:"t'\>"
        thus ?thesiszuu'" using M1 by au
        proof (elim disjE)
          {assume "?A∧¬?B∧¬?C" then have ?A by simp
           with qcp kpt tp kpq show ?thesis using x f by blast}
          next
          {assume "java.lang.NullPointerException
withhesis
          next
          {assume "¬?A∧¬?B∧?C" then obtain t' where qt'java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
           from qcp tp have "q∥p ⊕ ((∃t''. q∥t'' ∧ t''∥p) ⊕ (∃t''. t∥t'' ∧ t''∥c'))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
           then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
           thus ?thesis
           proof (elim disjE)
              {assume "?A∧¬?B∧¬?C" then have ?A by simp
               thus ?thesis using x m by auto}
              next
              {assume "¬?A∧?B∧(p,z)∈b› obtain c where pc:"p∥c" and "c∥z" using b by blast
               thus ?thesis using x b by auto}
              next
              { assume "¬?A∧¬?B∧?C" then obtain g where tg:"t∥g" and "g∥c'" by auto
                with qcp qt' have "g∥t'" using M1 by blast
                with qt' tpc pc kpq kpt tp tg show ?thesis using x ov by blast}
          qed}
     qed}
 qed
qed
       


lemmabib:1 O ⊆ b^-1 ∪ m^-1 ∪ e ∪ ov ∪ ov^-1 ∪ s ∪ s^-1 ∪ d ∪ d^-1 ∪ f ∪ f^-1" (is "b^-1 O b ⊆ ?R")
proof
  fix x::"'a×'a" assume "x ∈ b^-1 O b" then obtain p q z::'a where x:"x = (p,q)" and "(p,z) ∈ b^-1" and "(z,q) ∈the hav "<¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
  from ‹(p,z)∈b^-1› obtain c where zc:"z∥c" and cp:"c∥p" using b by blast
  from ‹(z,q) ∈ b› obtain c' where zcp:"z∥c'" and cpq:"c'∥q" using b by blast
  obtain u u' where pu:"p\<parallel>u" and qup:"q\<parallel>u'" using M3 meets_wd cp cpq by fastforce
  from cp cpq have "c\<parallel>q \<oplus> ((\<exists>t. c\<parallel>t \<and> t\<parallel>q) \<oplus> (\<exists>t. c'\<parallel>t \<and> t\<parallel>p))" (is "?A \<oplus> (?B \<oplus> ?C)") using M2 by blast
  then have "(?A\<and>\<not>?B\<and>\<not>?C) \<or> ((\<not>?A\<and>?B\<and>\<not>?C) \<or> (\<not>?A\<and>\<not>?B\<and>?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
  thus "x \<in>?R"
  proof (elim disjE)
      { assume "?A\<and>\<not>?B\<and>\<not>?C" then have cq:?A by simp
        from pu qup have "p\<parallel>u' \<oplus> ((\<exists>t'. p\<parallel>t' \<and> t'\<parallel>u') \<oplus> (\<exists>t'. q\<parallel>t' \<and> t'\<parallel>u))" (is "?A \<oplus> (?B \<oplus> ?C)") using M2 by blast
        then have "(?A\<and>\<not>?B\<and>\<not>?C) \<or> ((\<not>?A\<and>?B\<and>\<not>?C) \<or> (\<not>?A\<and>\<not>?B\<and>?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
        thus ?thesis
        proof (elim disjE)
          {assume "(?A\<and>\<not>?B\<and>\<not>?C)" then have "?A" by simp
           with cq cp qup have "p = q" using M4 by auto
         thesisinge yauto
          next
          {assume "\<not>?A\<and>?B\<and>\<not>?C" then have "?B" by simp
           with cq cp qup w using last
          next
          {assume "(\<not>?A\<and>\<not>?B\<and>?C)" then have "?C" by simp
           with pu cq cp show ?thesis using x s by blast}
        qed}
      next
      { assume "\<not>?A\<nd?B\<and>\<not>?C" then have ?B by simp
        then obtain t where            with kq kp qcp show hesisusingx  blast
        from pu qup have "p\<parallel>u' \<oplus> ((\<exists>t'. p\<parallel>t' \<and> t'\<parallel>u') \<oplus> (\<exists>t'. q          {ssume "(\<not>A\<and\<not?B\<and>?C)" then have "?C" by simp
        then have "(?A\<and>\<not>?B\<and>\<not>?C) \<or> ((\<not>?A\<and>?B\<and>\<not>?C) \<or> (\<not>?A\<and>\<not>?B\<and>?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
        thus ?thesis
        proof (elim disjE)
          {assume "?A\<and>\<not>?B\<and>\<not>?C" then have ?A by simp
           with qup ct tq cp show ?thesis using f x by blast}
          next
          {assume "\<not>?A\<and>?B\<and>\<not>?C"  then have ?B by simp
           then obtain t' where ptp:"p\<parallel>t'" and tpup:"t'\<parallel>u'" by auto
           from tqave \arallelq \<oplus> ((\<exists>t''. p\<parallel>t'' \<and> t''\<parallel>q) \<oplus> (\<exists>t''. t\<parallel>t'' \<and> t''\<parallel>u))" (is "?A \<oplus> (?B \<oplus> ?C)") using M2 by blast
           then have "(?A\<and>\<not>?B\<and>\<not>?C) \<or> ((\<not>?A\<and>?B\<and>\<not>?C) \<or> (\<not>?A\<and>\<not>?B\<and>?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
           thus ?thesis
           proof (elim disjE)
              {assume "?A\<and>\<not>?B\<and>\<not>?C" then have ?A by simp
               thus ?thesis using x m by auto}
              next
              {assume "\<not>?A\<and>?B\<and>\<not>?C" then have ?B by simp
          s  o
              and>\<not>?B\<and>\<not>?C) \<or> ((\<not>?A\<and>?B\<and>\<not>?C) \<or> (\<not>?A\<and>\<not>?B\<and>?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
              { assume "\<not>?A\<and>\<not>?B\<and>?C" then have ?C by simp
                then obtain g where "t\<parallel>g" and "g\<parallel>u" by auto
                moreover with pu ptp have "g\<parallel>t'" using M1 by blast
                ultimately  show ?thesis using x ov ct tq cp ptp tpup qup   by blast}
           qed}
         next
          {assume "\<not>?<>\<otB<>C" then have ?C by simp
             where\<parallel>t'" and "t'\<arallelu" by auto
           with cp  ct tq pu  show ?thesis using d x by blast}
         qed}
      next
      { assume "\<not>?A\<and>\<not>?B\<and>?C" then have ?C by simp
        then obtain t where cpt:"c'\<parallel>t" and tp:"t\<parallel>p" by auto
        from  pu qup have "p\<parallel>u' \<oplus> ((\<exists>t'. p\<parallel>t' \<and> t'\<parallel>u') \<oplus> (\<exists>t'. q\<parallel>t' \<and> t'\<parallel>u))" (is "?A \<oplus> (?B \<oplus> ?C)") using M2             pparallelp \<oplus> ((\<exists>t''. q\<parallel>t< t''\<parallel>p) \<oplus> (existst''. parallelt'' \<and> t''\<parallel>)(sA<> ?oplus ?C)") using M2 by blast
        then
        thus ?thesis
        f(misjE
          {assume "?A\<and>\<not>?B\<and>\<not>?C" then have ?A by simp
             pt cpqhow?hesissingfbylast
          next
          {assume "\<not>?A\<and>?B\<and>\<not>?C" then have ?B by simp
           with qup cpt tp cpq show thesissingxd byastjava.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
          next
          {assume "\<ot?\and>\<not>>?B\<and>C hennbtain t'wherere'"\<parallelt'"and tpc:"t'\<parallel>u" by auto
           romtp q<arallel<plus> ((\<exists>t''. q\<parallel>t'' \<and> t''\<parallel>p) \<oplus> (\<exists>t''. t\<parallel>t'' \<and> t''\<parallel>u'))" (is "?A \<oplus> (?B \<oplus> ?C)") using M2 by blast
           then have "(?A\<and>\<not>?B\<and>\<not>?C) \<or> ((\<not>?A\<and>?B\<and>\<not>?C) \<or> (\<not>?A\<and>\<not>?B\<and>?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
           thus ?thesis
           proof (elim disjE)
              {assume "?A\<and>\<not>?B\<and>\<not>?C" then have ?A by simp
               thus ?thesis using x m by auto}
              next
              {assume "\<not>?A\<and>?B\<and>\<not>?C" then have ?B by simp
               us?thesisusingsingx  uto
              next
              { assume "\<not>?A\<and>\<not>?B\<and>?C" then obtain g where tg:"t\<parallel>g" and "g\<parallel>u'" by auto
                with qup qt' have "g\<parallel>t'" using M1 by blast
                with qt' tpc pu cpq cpt tp tg show ?thesis using x ov by blast}
          qed}
     qed}
 qed
qed

lemma cddi:"d O d^-1 \<subseteq> b \<union> b^-1 \<union> m \<union> m^-1 \<union> e \<union> ov \<union> ov^-1 \<union> s \<union> s^-1 \<union> d \<union> d^-1 \<union> f \<union> f^-1" (is "d O d^-1 \<subseteq> ?R")
proof
  fix x::"'a\<times>'a" assume "x \<in> d O d^-1" then obtain p q z::'a where x:"x = (p,q)" and "(p,z) \<in> d" and "(z,q) \<in> d^-1" by auto
  from \<open>(p,z) \<in> d\<close> obtain k l u v where lp:"l\<parallel>p" and kl:"k\<parallel>l" and kz:"k\<parallel>z" and pu:"p\<parallel>u" and uv:"u\<parallel>v" and zv:"z\<parallel>v"  using d  by blast
  (*<*)
  from lp lpq have "l\<parallel>q \<oplus> ((\<exists>t. l\<parallel>t \<and> t\<parallel>q) \<oplus> (\<exists>t. l'\<parallel>t \<and> t\<parallel>p))" (isLicense
  then have "(?A\<and>\<not>?B\<and>\<not>?C) \<or> ((\<not>?A\<and>?B\<and>\<not>?C) \<or> (\<not>?A\<and>\<not>?B\<and>?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
  thus "x \<in>?R"
  proof (elim disjE)
      { assume "?A\<and>\<not>?B\<and>\<not>?C" then have lq:?A by simp
        from pu qup have "p\<parallel>u' \<oplus> ((\<exists>t'. p\<parallel>t' \<and> t'\<parallel>u') \<oplus> (\<exists>t'. q\<parallel>t' \<and> t'\<parallel>u))" (is "?A \<oplus> (?B \<oplus> ?C)") using M2 by blast
        then have "(?A\<and>?B\<and>\<not>?C) \<or> ((\not?\nd<\<not>?C) \<or> (\<not>?A\and<>B\<and>?C))" by (insert xor_distr_L[of ?A? toplimmeets
        thus ?thesis
proof( disjE)
         ?\>\<>?B\<and>>? ?" simp
           with lq lp qup have "punfolding unwrap_defby rulecfun_eqI, simp)
     using x e  by auto
          next
          \>?\nd<\<not>?C" then have "?B" by simp
           with lq lp qup show ?thesis using x s by blast} -
          next
          assume "(\<not>?A\<and>\<not>?B\<and>?C)" then have "?C" by simp
            pulqqed
   
      next
      by casesr,simp_all add: SetMem_SetInsert TR_deMorgan)
        then obtain t where         obtain t where pt<arallelt" and tp:"t\<parallel>p" by auto
        from pu qup have "p\<parallel>u' \<oplus> ((\<exists>t'. p\<parallel>t' \<and> t'\<parallel>u') \<oplus> (\<exists>t'. q\<parallel>t' \<and> t'\<parallel>u))" (is "?A \<oplus> (?B \<oplus> ?C)") using M2 by blast
        then have "(?A\<and>\<not>?Bthus?esis
        thus ?thesis
        proof (elim disjE)
          {assume "?A\<and>\<not>?B\<and>\<not>?C" then have ?A by simp
           with qup lt tq lp show ?thesis using f x by blast}
          next
          {assume "\<not>?A\<and>?B\<and>\<not>?C"  then have ?B by simp
           then obtain t' where ptp:"p\<parallel>t'" and tpup:"t'\<parallel>u'" by autolemma cddi d-<ubseteq b \<union> b^-1 \<union> m \<union> m^-1 \<union> e \<union> ov \<union> ov^-1 \<union> s \<union> s^-1 \<union> d \<union> d^-1 \<union> f \<union> f^-1" (is "d O d^-1 \<subseteq> ?R")
           from pu tq  have "p\<parallel>        from pu ve\parallel>u'\> \existstp<arallel><>t'\<parallel>u') \<oplus> (\<exists>t'. q\<parallel>t t'\<parallel>u))" ( A< (?B \<oplus> ?C)") using M2 by blast
           then have "(?A\<and>\<not>?B\<and>\<not>?C) \<or> ((\<not>?A\<and>?B\<and>\<not>?C) \<or> (\<not>?A\<and>\<not>?B\<and>?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
           thus ?thesis
       sjE)
              {assume "?A\<and>\<not>?B\<and>\<not>?C" then have 
               thus ?thesis using x m                 obtain g ere"parallelg"and "g\<parallel>u" by auto
              next
              {assume "\<not>?A\<and>?B\<and>\<not>?C" then have ?B by simp
               thus ?thesis using x b by auto}
              next
              { assume "\<not>?A\<and>\<not>?B\<and>?C" then have ?C by simp
                then obtain g where "t\<parallel>g" and
                moreover with pu ptp have "g\<parallel>t'" using M1 by blast
                ultimately  show ?thesis using x ov lt tq ptppupyst
           qed}
         next
          {assume "\<not>?A\<and>\<not>?B\<and>?C" then have ?C by simp
           then obtain t' where "q\<parallel>t'" and "t'\<parallel>u" by auto
           with lp  lt tq pu  owthesis singd yblastst
         qed}
      next
      { assume "\<not>?A\<and>\<not>?B\<and>?C" then have ?C by simp
        then obtain t where lpt:"l'\<parallel>t" and tp:"t\<parallel>p" by auto
        from  pu qup have "p\<parallel>u' \<oplus> ((\<exists>t'. p\<parallel>t' \<and> t'\<parallel>u') \<oplus> (\<exists>t'. q\<parallel>t' \<and> t'\<parallel>u))" (is "?A \<oplus> (?B \<oplus> ?C)") using M2 by blast
        then have "(?A\<and>\<not>?B\<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        thus ?thesis
        proof (elim disjE)
          {assume "?A\<and>\<not>?B\<and>\<not>?C" then have ?A by simp
           with qup lpt tp lpq show ?thesis using x f by blast}
          next
          {assume "\<not>?A\<and>?B\<and>\<not>?C" then have ?B by simp
   tp lpq show ?thesis using x d by blast}
          next
          {assume "\<not>?A\<and>\<not>?B\<and>?C" then obtain t' where qt':"q\<parallelt'" and tpc:"t'\<parallel>u" by auto
           from qup tp have "q\<parallel>p \<oplus> (  using cfid by auto
           then have "(?A\<and>\<not>?B\<and>\<not>?C) \<or> ((\<not>?A\<and>?B\<and>\<not>?C) \<or> (\<not>?A\<and>\<not>?B\<and>?C))" by (insert xor_distr_L[of ?A ?B ?C],   covd byauto
           thus ?thesis
           proof (elim disjE)
              {assume "?A\<and>\<not>?B\<and>\<not>?C" then have ?A by simp
               thus ?thesis using x m by auto}
              next
              {assume "\<not>?A\<and>?B\<and>\<not>               auto
               thus ?thesis using x b by auto}
              next
              { assume "\<not>?A\<and>\<not               csisi"^1Os-<subseteq> s-java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
                 qup qt' have "\parallel>t"using   ast
                with qt' tpc pu lpq lpt tp tg show ?thesis using x ov by blast}
          qed}
qed
 qed
qed


(* ========= inverse ========== *)
subsection java.lang.NullPointerException
text ‹


  s^-1 ∪
 using cbmi by auto


  covmi:"ov O m^-1 \<bseteqeteq
 using cmovi by auto

  covbi:"ov O b^-1 ⊆ b^-1 ∪ auto
 using cbovi by auto

  cfiovi:"f^-1 O ov^-1 ⊆ ov^-
 using covf by auto

  cfm:"f- ^1 \<>s
 using cmf by auto

  cfibi:"f^-1 O b^-1 ⊆ b^-1 ∪ m^-1 ∪ ov^-1 ∪ s^-1 ∪:"ov^ Of^1 \\⊆ s^-1 ∪d-1"
 using cbf by auto

  cdif:"d^-1 O f ⊆ ov^-1 ∪ s^-1 ∪ d^-1"
 using cfid by auto

  cdiovi:"d^-1 O ov ^-1 ⊆ ov^-1 ∪
 using covd by auto

  cdimi:"d^-1 O m^-1 ⊆ s^-1 ∪ ov^-1 ∪ d^-1 "
 using cmd by auto

  cdibi:"d^-1 O b^-1 ⊆ b^-1 ∪ m^-1 ∪
  by auto

  csd:"s O d ⊆ d"
 using cdisi by auto

  csf:"s O f ⊆ d"
 using cfisi by auto

  csovi:"s O ov^-1 ⊆ ov^-1 ∪ f ∪
 using covsi by auto

  csmi:"s O m^-1 ⊆ m^-1"
 using cmsi by auto

  csbi:"s O b^-1 ⊆ b^-1"
 using cbsi by auto

  csisi:"s^-1 O s^-1 🚫
 using css by auto

  csid:"s^-1 O d ⊆ ov^-1 ∪ f ∪ d"
 using cdis by auto

  csif:"s^-1 O f ⊆ ov^-1"
 using cfis by auto

  csiovi:"s^-1 O ov^-1 ⊆ ov^-1" 
 using covs by auto

  csimi:"s^-1 O m^-1 ⊆ m^-1"
 using cms by auto

  csibi:"s^-1 O b^-1 ⊆
 sing by aauto

  cds:"d O s ⊆ d"
 using csidi by auto

 cdsi:d O s^-1 ⊆ b^-1 ∪ m^-1 ∪ ov^-1 ∪ f ∪ d"
 using csdi by auto

  cdd:"d O d ⊆ d"
 using cdidi by auto

  cdf:"d O f ⊆ d"
 using cfidi by auto

 lemma c cbibi:"b^-1 O b^-1 ⊆
 using covdi by auto

  cdmi:"d O m^-1 ⊆ b^-1"
 using cmdi by auto

  cdbi:"d O b^-1 ⊆ b^-1"
 using cbdi by auto

  cfdi:"f O d^-1 ⊆ b^-1 ∪ m^-1 ∪ ov^-1 ∪ s^-1 ∪cb_ruecr be[c_rle]ad cbm_uls acb[cb_uls n bo[b_uls] an b [cbrues n cb[b_rls n bfcb_rls] an
 using cdfi by auto

  cfs:"f O s ⊆ d"
 using csifi by auto

  cfsi:"f O s^-1 ⊆
 using csfi by auto

  cfd:"f O d ⊆ d"
 using cdifi by auto


 <>f"
 using cfifi by auto

  cfovi:"f O ov^-1 ⊆ b^-1 ∪ m^-1 ∪ ov^-1"
 vfiy uto

  cfmi:"f O m^-1 ⊆named_theorem csi_rules declare csie[ci_rue] ndcsibcsirue] ad sb[si_ules ndciov[s_ues and csis [csi_rule] andcsd[cirues and cif[cs_rule] and
 using cmfi by auto

  cfbi:"f O b^-1 ⊆ b^-1"
 using cbfi by auto

  covifi:"ov^-1 O f^-1 ⊆ ov^-1 ∪ s^-1 ∪ d^-1"
 

  covidi:"ov^-1 O d^-1 ⊆ b^-1 ∪ m^-1 ∪ declre emcm_uls] and cmcr_ruls] adcm[r_ues]and vmcrm_ules nd sm[cr_uls]andcfm[crm_rules and cdm[rmrules]and
 using cdov by auto

  covis:"ov^-1 O s ⊆ ov^-1 ∪ f ∪ d"
 using csiov by auto

  covisi:"ov^-1 O s^-1 ⊆]ndcmi[csi_ues]] and os[cri_us] ad ci[crsi_ues] and ficsiule] an cs[rs_ruls] nd
 using csov by auto

  covid:"ov^-1 O d ⊆ ov^-1 ∪ f ∪ d"
 ingdiov b aut

  covif:"ov^-1 O f ⊆
 using cfiov by auto

 ^-1 O ov^-1 ⊆uni m^-1 ∪ ov^-1"
 using covov by auto

  covimi:"ov^-1 O m^-1 ⊆ b^-1"
 using cmov by auto

  covibi:"ov^-1 O b^-1 ⊆ b^-1"
 using cbov by auto

  cmiov:"m^-1 O ov ⊆ ov^-1 ∪ d ∪ f"
 using covim by auto

  cmifi:"m^-1 O f^-1 ⊆ m^-1"
 using cfm by auto

  cmidi:"m^-1 O d^-1 ⊆
 using cdm by auto

  cmis:"m^-1 O s ⊆ ov^-1 ∪ d ∪ f"
 using csim by auto

  cmisi:"m^-1 O s^-1 ⊆ b^-1"
 using csm by auto

  cmid:"m^-1 O d ⊆ ov^-1 ∪ d ∪ f"
 using cdim by auto

  cmif:"m^-1 O f ⊆ m^-1"
 using cfim by auto

  cmiovi:"m^-1 O ov^-1 ⊆ b^-1"
 using covm by auto

  cmimi:"m^-1 O m^-1 ⊆ b^-1"
 using cmm by auto

  cmibi:"m^-1 O b^-1 ⊆ b^-1"
 using cbm by auto

  cbim:"b^-1 O m ⊆ b^-1 ∪ m^-1 ∪ ov^-1 ∪ f ∪ d"
 using cmib by auto

  cbiov:"b^-1 O ov ⊆ b^-1 ∪ m^-1 ∪ ov^-1 ∪ f ∪ d"
 using covib by auto

  cbifi:"b^-1 O f^-1 ⊆ b^-1"
 using cfb by auto

  cbidi:"b^-1 O d^-1 ⊆ b^-1"
 using cdb by auto

  cbis:"b^-1 O s ⊆ b^-1 ∪ m^-1 ∪ ov^-1 ∪ f ∪ d"
 using csib by auto

  cbisi:"b^-1 O s^-1 ⊆ b^-1"
 using csb by auto

  cbid:"b^-1 O d ⊆ b^-1 ∪ m^-1 ∪ ov^-1 ∪ f ∪ d"
 using cdib by auto

  cbif:"b^-1 O f ⊆ b^-1"
 using cfib by auto

  cbiovi:"b^-1 O ov^-1 ⊆ b^-1"
 using covb by auto

  cbimi:"b^-1 O m^-1 ⊆ b^-1"
 using cmb by auto

  cbibi:"b^-1 O b^-1 ⊆ b^-1"
 using cbb by auto

(****)

subsection ‹Composition rules›
named_theorems ce_rules declare cem[ce_rules] and ceb[ce_rules] and ceov[ce_rules] and ces[ce_rules] and cef[ce_rules] and ced[ce_rules] and 
cemi[ce_rules] and cebi[ce_rules] and ceovi[ce_rules] and cesi[ce_rules] and cefi[ce_rules] and cedi[ce_rules]

named_theorems cm_rules declare cme[cm_rules] and cmb[cm_rules] and cmm[cm_rules] and cmov[cm_rules] and cms [cm_rules] and cmd[cm_rules] and cmf[cm_rules] and
cmbi[cm_rules] and cmmi[cm_rules] and cmovi[cm_rules] and cmsi[cm_rules] and cmdi[cm_rules] and cmfi[cm_rules]

named_theorems cb_rules declare cbe[cb_rules] and cbm[cb_rules] and cbb[cb_rules] and cbov[cb_rules] and cbs [cb_rules] and cbd[cb_rules] and cbf[cb_rules] and
cbbi[cb_rules] and cbbi[cb_rules] and cbovi[cb_rules] and cbsi[cb_rules] and cbdi[cb_rules] and cbfi[cb_rules]

named_theorems cov_rules declare cove[cov_rules] and covb[cov_rules] and covb[cov_rules] and covov[cov_rules] and covs [cov_rules] and covd[cov_rules] and covf[cov_rules] and
covbi[cov_rules] and covbi[cov_rules] and covovi[cov_rules] and covsi[cov_rules] and covdi[cov_rules] and covfi[cov_rules]

named_theorems cs_rules declare cse[cs_rules] and csb[cs_rules] and csb[cs_rules] and csov[cs_rules] and css [cs_rules] and csd[cs_rules] and csf[cs_rules] and
csbi[cs_rules] and csbi[cs_rules] and csovi[cs_rules] and cssi[cs_rules] and csdi[cs_rules] and csfi[cs_rules]

named_theorems cf_rules declare cfe[cf_rules] and cfb[cf_rules] and cfb[cf_rules] and cfov[cf_rules] and cfs [cf_rules] and cfd[cf_rules] and cff[cf_rules] and
cfbi[cf_rules] and cfbi[cf_rules] and cfovi[cf_rules] and cfsi[cf_rules] and cfdi[cf_rules] and cffi[cf_rules]

named_theorems cd_rules declare cde[cd_rules] and cdb[cd_rules] and cdb[cd_rules] and cdov[cd_rules] and cds [cd_rules] and cdd[cd_rules] and cdf[cd_rules] and
cdbi[cd_rules] and cdbi[cd_rules] and cdovi[cd_rules] and cdsi[cd_rules] and cddi[cd_rules] and cdfi[cd_rules]

named_theorems cmi_rules declare cmie[cmi_rules] and cmib[cmi_rules] and cmib[cmi_rules] and cmiov[cmi_rules] and cmis [cmi_rules] and cmid[cmi_rules] and cmif[cmi_rules] and
cmibi[cmi_rules] and cmibi[cmi_rules] and cmiovi[cmi_rules] and cmisi[cmi_rules] and cmidi[cmi_rules] and cmifi[cmi_rules]

named_theorems cbi_rules declare cbie[cbi_rules] and cbim[cbi_rules] and cbib[cbi_rules] and cbiov[cbi_rules] and cbis [cbi_rules] and cbid[cbi_rules] and cbif[cbi_rules] and
cbimi[cbi_rules] and cbibi[cbi_rules] and cbiovi[cbi_rules] and cbisi[cbi_rules] and cbidi[cbi_rules] and cbifi[cbi_rules]

named_theorems covi_rules declare covie[covi_rules] and covib[covi_rules] and covib[covi_rules] and coviov[covi_rules] and covis [covi_rules] and covid[covi_rules] and covif[covi_rules] and
covibi[covi_rules] and covibi[covi_rules] and coviovi[covi_rules] and covisi[covi_rules] and covidi[covi_rules] and covifi[covi_rules]

named_theorems csi_rules declare csie[csi_rules] and csib[csi_rules] and csib[csi_rules] and csiov[csi_rules] and csis [csi_rules] and csid[csi_rules] and csif[csi_rules] and
csibi[csi_rules] and csibi[csi_rules] and csiovi[csi_rules] and csisi[csi_rules] and csidi[csi_rules] and csifi[csi_rules]

named_theorems cfi_rules declare cfie[cfi_rules] and cfib[cfi_rules] and cfib[cfi_rules] and cfiov[cfi_rules] and cfis [cfi_rules] and cfid[cfi_rules] and cfif[cfi_rules] and
cfibi[cfi_rules] and cfibi[cfi_rules] and cfiovi[cfi_rules] and cfisi[cfi_rules] and cfidi[cfi_rules] and cfifi[cfi_rules]

named_theorems cdi_rules declare cdie[cdi_rules] and cdib[cdi_rules] and cdib[cdi_rules] and cdiov[cdi_rules] and cdis [cdi_rules] and cdid[cdi_rules] and cdif[cdi_rules] and
cdibi[cdi_rules] and cdibi[cdi_rules] and cdiovi[cdi_rules] and cdisi[cdi_rules] and cdidi[cdi_rules] and cdifi[cdi_rules]
(**)
named_theorems cre_rules declare cee[cre_rules] and cme[cre_rules] and cbe[cre_rules] and cove[cre_rules] and cse[cre_rules] and cfe[cre_rules] and cde[cre_rules] and 
cmie[cre_rules] and cbie[cre_rules] and covie[cre_rules] and csie[cre_rules] and cfie[cre_rules] and cdie[cre_rules]

named_theorems crm_rules declare cem[crm_rules] and cbm[crm_rules] and cmm[crm_rules]  and covm[crm_rules] and csm[crm_rules] and cfm[crm_rules] and cdm[crm_rules] and 
cmim[crm_rules] and cbim[crm_rules] and covim[crm_rules] and csim[crm_rules] and cfim[crm_rules] and cdim[crm_rules]

named_theorems crmi_rules declare cemi[crmi_rules] and cbmi[crmi_rules] and cmmi[crmi_rules]  and covmi[crmi_rules] and csmi[crmi_rules] and cfmi[crmi_rules] and cdmi[crmi_rules] and 
cmimi[crmi_rules] and cbimi[crmi_rules] and covimi[crmi_rules] and csimi[crmi_rules] and cfimi[crmi_rules] and cdimi[crmi_rules]

named_theorems crs_rules declare ces[crs_rules] and cbs[crs_rules] and cms[crs_rules]  and covs[crs_rules] and css[crs_rules] and cfs[crs_rules] and cds[crs_rules] and 
cmis[crs_rules] and cbis[crs_rules] and covis[crs_rules] and csis[crs_rules] and cfis[crs_rules] and cdis[crs_rules]

named_theorems crsi_rules declare cesi[crsi_rules] and cbsi[crsi_rules] and cmsi[crsi_rules]  and covsi[crsi_rules] and cssi[crsi_rules] and cfsi[crsi_rules] and cdsi[crsi_rules] and 
cmisi[crsi_rules] and cbisi[crsi_rules] and covisi[crsi_rules] and csisi[crsi_rules] and cfisi[crsi_rules] and cdisi[crsi_rules]

named_theorems crb_rules declare ceb[crb_rules] and cbb[crb_rules] and cmb[crb_rules]  and covb[crb_rules] and csb[crb_rules] and cfb[crb_rules] and cdb[crb_rules] and 
cmib[crb_rules] and cbib[crb_rules] and covib[crb_rules] and csib[crb_rules] and cfib[crb_rules] and cdib[crb_rules]

named_theorems crbi_rules declare cebi[crbi_rules] and cbbi[crbi_rules] and cmbi[crbi_rules]  and covbi[crbi_rules] and csbi[crbi_rules] and cfbi[crbi_rules] and cdbi[crbi_rules] and 
cmibi[crbi_rules] and cbibi[crbi_rules] and covibi[crbi_rules] and csibi[crbi_rules] and cfibi[crbi_rules] and cdibi[crbi_rules]

named_theorems crov_rules declare ceov[crov_rules] and cbov[crov_rules] and cmov[crov_rules]  and covov[crov_rules] and csov[crov_rules] and cfov[crov_rules] and cdov[crov_rules] and 
cmiov[crov_rules] and cbiov[crov_rules] and coviov[crov_rules] and csiov[crov_rules] and cfiov[crov_rules] and cdiov[crov_rules]

named_theorems crovi_rules declare ceovi[crovi_rules] and cbovi[crovi_rules] and cmovi[crovi_rules]  and covovi[crovi_rules] and csovi[crovi_rules] and cfovi[crovi_rules] and cdovi[crovi_rules] and 
cmiovi[crovi_rules] and cbiovi[crovi_rules] and coviovi[crovi_rules] and csiovi[crovi_rules] and cfiovi[crovi_rules] and cdiovi[crovi_rules]

named_theorems crf_rules declare cef[crf_rules] and cbf[crf_rules] and cmf[crf_rules]  and covf[crf_rules] and csf[crf_rules] and cff[crf_rules] and cdf[crf_rules] and 
cmif[crf_rules] and cbif[crf_rules] and covif[crf_rules] and csif[crf_rules] and cfif[crf_rules] and cdif[crf_rules]

named_theorems crfi_rules declare cefi[crfi_rules] and cbfi[crfi_rules] and cmfi[crfi_rules]  and covfi[crfi_rules] and csfi[crfi_rules] and cffi[crfi_rules] and cdfi[crfi_rules] and 
cmifi[crfi_rules] and cbifi[crfi_rules] and covifi[crfi_rules] and csifi[crfi_rules] and cfifi[crfi_rules] and cdifi[crfi_rules]

named_theorems crd_rules declare ced[crd_rules] and cbd[crd_rules] and cmd[crd_rules]  and covd[crd_rules] and csd[crd_rules] and cfd[crd_rules] and cdd[crd_rules] and 
cmid[crd_rules] and cbid[crd_rules] and covid[crd_rules] and csid[crd_rules] and cfid[crd_rules] and cdid[crd_rules]

named_theorems crdi_rules declare cedi[crdi_rules] and cbdi[crdi_rules] and cmdi[crdi_rules]  and covdi[crdi_rules] and csdi[crdi_rules] and cfdi[crdi_rules] and cddi[crdi_rules] and 
cmidi[crdi_rules] and cbidi[crdi_rules] and covidi[crdi_rules] and csidi[crdi_rules] and cfidi[crdi_rules] and cdidi[crdi_rules]



end