Impressum allen.thy

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

section*


theory allen

imports allen

  Mainaxioms 
  "HOL-Ei axioms


begin

section \open>Basic relations\close

text‹Basic relations›
Relations e, m, b, ov, d, s and f stand for equal, meets, before, overlaps, during, starts and finishes, respectively.›

class arelations = interval +
 fixes
  e:"a<>)set and
  m::"('a×aps durn, starts and finishes, respectively.›::"<)settimes>'a set and 
  b::"('a×'set"
  ov::"('a×'a) set" and
  d::"('a×'a) set" and
  s::"('a×nd
 ::"'<>')set
assumes
  e:"(p,q) ∈ e = (p = q)" and
  m:"(p,q) ∈q" and
  b:"(p,q) \<in t::'a. p∥ t∥
  ov:"(p,q) ∈kl  t:a 
                   k∥p ∧ p∥u ∧ u∥v) ∧ (k∥l ∧ l∥q ∧ q∥v) ∧ (l∥t ∧ t∥u))" and
  s:pq <i> s (🚫
  f:"(p,q) ∈ f = (∃ s =  (∃p ∧u ∧v ∧q ∧v)" and
  d:"(p,q) ∈ f = (∃l ∧p ∧u ∧q ∧<parallelparallel)" and
 

(** e compositions **)
subsection ‹
text ‹

lemma cer:
assumes "r in{e,m,b,ov,s,f,d,m^-1,b^-1,ov^-1,s^-1,f^-1,d^-1}" 
shows "e O r = r"
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
    from \<open>(x,z) \<in> e\<close> have "x = z" using e by auto
    with \<open>(z,y)\<in> r\<close> have "(x,y) \<in> r" by simp} note c1 = this
  
 { fix x y assume a:"(x,y) \<in>  r"
   have "(x,x) \<in> e" using e by autofrom<>x,z)\in \close have"x= zz"usingebyauto
    ahave "(,y) \in> e Or"byblast  c2
 
 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 -
  { fix x y assume a:"(x,y)    with ahave"x,y)\in eO  blast}notec2 =this
    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    assume :(,)\<>   "java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
  
 { 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 "b^-1"]
lemmas cem = cer[of m]
lemmas cemi = cer[of "m^-1"]
lemmas cee = cer[of e]
lemmas ces = cer[of s]
lemmas s^-]
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^-

(*******)

(* composition with single relation *)
subsection ‹r-composition›
text ‹, (subst (asm) r1 ), (subst (asm) r2), (subst r3)) , (meson M5exist_var)

  (in arelations) r_compose uses r1 r2 r3 = ((auto, (subst (asm) r1 ), (subst (asm) r2), (subst r3)) , (mesonlemma (in arela) cbb:"b O b ⊆


  (in arelations) cbb:"b O b ⊆O \<> 
 b r2:b r3:b)

 ns) cbm:"b O m ⊆
 by (r_compose r1:b r2:m r3:b)

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

 mma cbf cbfi:"b O f^-1 ⊆
 apply (auto simp:b f)
 by (meson M1 M5exist_var)

  cbdi:"b O d^-1 ⊆
 apply (auto simp: b d)
 by (meson M1 M5exist_var)
 
  cbs:"b O s ⊆ (msnM1 M5xistar)
 apply (auto simp: b s)
 by (meson M1 M5exist_var)

  cbsi:"b O s^-1 ⊆ b"
 apply (auto simp: b s)
 by (meson M1 M5exist_var)

  (in arelations) cmb:"m O b ⊆ b"
 by (r_compose r1:m r2:b r3:b)

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

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

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

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

java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
 apply :"m O d^-1 ⊆
 using M1 by auto

  cmsi:"m O s^-1 ⊆
 apply (auto simp add:m s)
 using M1 by blast

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

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

  covs:"ov O s ⊆ ov"
 
 fix p::"'a×'a" assume "p ∈
 from xy obtain r u v t k wrx:"r∥u" and uv:"u∥v" and rt:"r∥t" an tk:t\<>ky" and yv:"y∥u" using ov by blast
 from yzs obtain l1 l2 where yl1:"y∥l1" and l1l2:"l1∥l2" and zl2:"z∥
 from uv yl1 yv have "u∥l1" using M1 by blast
 with xu l1l2 obtain ul1 where xul1:"x∥
 from ku xu xul1 l1l2 have kul1:"k\<parallelul1" using M1 by blast
 from ty yzs have "t∥ng s M1 by blast
 with rx rt xul1 ul1l2 zl2 tk kul1 have "(x,z) ∈ ov" usi
 with p show "p ∈
 

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

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

  cfiov:"f^-1 O ov ⊆:"f^-1 O ov ⊆
 
 fix p::"'a×'a" assume "p ∈ ov O s" then x y z where p:"p = (xx,z)" a xyov:"(x,y)∈ s" by auto
 from xyfi yzov obtain t' r u where tpr:"t'∥t k whrx:"r∥u" and uv:"u∥t" and tk:"t∥y" and yv:"y∥u" using ov by blast
 from yzov ry obtain v k t u' where yup:"y∥u'" and upv:"u'∥ obtain l1 l2 where yl1:"y∥l2" and zl2:"z∥
 using ov using M1 by blast
 from yu xu yup have xup:"x∥ xu l1l2 obtain ul1 where xul1:"x∥l2" using M5exist_var by blast
 from tpr rk kt obtain r' where tprp:"t'∥hakul1:":"k∥
  rpz:"r'∥
 from tprp rpz rpt tpx xup zv upv tup have "(x,z) ∈ ov" using ov by blast
 with p show "p\<in 
 

  cfifi:"f^- O f- ⊆
 
 fix x::"'a×
 from ‹ b"
 from ‹
 from zu zup pu have "p∥
 from lz kpz kplp have "l∥
 with kl lpq obtain ll where "k∥
 with kp ‹ qup sho "x ∈ using x f by blast
 

  cfidi:"f^O ^-1\subseteq d^-1"
 
 fix x::"'a×'a" assume "x : f^-1 O d^-1" then obtain p q z where x:"x = (p,q)" and "(p,z) ∈ f^-1" and "(z,q) ∈ d^-1" by auto
 then obtain k l u where kp:"k ∥t'\>r" and ry:"r∥u" and tpx:"t'∥u" using f by blast
 obtain k' l' u' v' where kpz:"k' ∥ yup:y∥'" and upv:"u'∥k" and kz:"k∥v" and kt:"k∥
 from lz kpz kplp have "l∥l'" using M1 by blast
 with kl lpq obtain ll where "k∥
 moreover from zu zvp upvp have "u' ∥
 p d by bast
 

  cfis:"f^-1 O s ⊆ zv upv t have "(x,z) ∈
 with p show "p ∈
 fix x::"'a×'a" aqed
 from ‹ f^-1"
 from ‹'a" assume "x ∈ q)"and "(p,in> f-1" nd "("(z,q) ∈ f^-1" by auto
 from pu zu zup have pup:"p∥u'" using M1 by blast
 moreover from lz kpz kpq have lq:"l∥(p,z) ∈ f^-1› obtain k l u where kp:"k∥l" and lz:"l∥u" and zu:"z∥
 ultimately show "x ∈ ov" using x lz zup kp kl upvp upvp ov qvp by blast
 

 cfisi"^-1 s^1 ⊆
 
 fix x::"'a× f^-1 O s^- then obtain p p q z w hre x:"x = (p,q)" an"p,z)<>^
 then obtain k l u where kp:"k ∥ll" and "ll∥
 obtain k' u' v' where kpz:"k' ∥z" and kpq:"k' ∥q" and qup:"q ∥u'" and upvp:"u'∥v'" using s ‹
 from zu zvp upvp have "u'∥u" using M1 by blast
 moreover from lz kpz kpq have "l ∥q " using M1 by blast
 ultimately show "x ∈ d^- cffdi:"f- ^1 <>d'a" assume "x : f^-1 O d^-1" then obtain p q z where x:"x = (p,q)" and "(p,z) ∈ d^-1" by ut
 

 cdifidifi"d^-1^ \< dl'" using 1 bylast
 
 fix x::"'a×z) ∈-1" (,\in f^-1" by auto
 then obtain k l u v where kp:"k ∥ oreover ffrom
  ll' u' whe wherpzk' \<>zl'" and lpq:"l' ∥u'" and zup:"z∥(z,q): f^-1› by blast
 from lz kpz kplp have "l∥l'" using M1 by blast
 with kl lpq obtain ll where "k∥ll" and "ll∥
 moreover from zu qup zup have "q ∥ u " using M1 cfis:"f^-11 s \\s> ov
 ultimately show "x ∈
 

  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)" ad "(p,z) ∈ d^-1" by auto
 then obtain k l u v where kp:"k ∥ p" and kl:"k∥(z,q)∈ s› obtain k' u' v' where kpz:"k'∥q" and zup:"z∥v'" and qvp:"q<>v
 obtain k' l' u' v' where kpz:"k' ∥q" using M1 by blast
 from lz kpz kplp have "l∥ ov" using x l zupkp kl upvp uv v qvp y blast
 with kl lpq obtain ll where "k∥
 moreover from zvp zu upvp have "u' ∥
 moreover with qup uv obtain uu where "q∥'a" assume "x ∈ f^-1 O s^-1" then tain q z e :"x =(p,q)" ad "(p,z) ∈ s^-1" by auto
 ultimately show "x ∈ d^-1" using x d kp pv by blast
 

  cdisi:"d^-1 O s^-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∥u" and uv:"u∥v" and pv:"p∥v" using d by bl
 u' v' where kk' \<>zq" and qup:"q 🚫d^-1"
 from upvp zvp zu have "u'∥
 with qup uv obtain uu wee "q\parallel" 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∧¬?B∧¬?C)" then have "?A" by simp 
     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
     moreover from lz pz pzu have "l∥zu" using M1 by blast
     ultimately have "(p,q) ∈ ov" using ov kpp qv pzu zuv 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 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 ∈ m O f" then obtain p q z where x:"x =(p,q)" and 1:"(p,z) ∈ m" and 2:"(z,q) ∈ f" by auto
  then obtain k l u   where pz:"p∥z" and kq:"k∥q" and kl:"k∥l" and lz:"l∥z" and zu:"z∥u" and qu:"q∥u" using m f by blast
  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∥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∧¬?B∧¬?C)" then have "?A" by simp 
     then have "(p,q) ∈ s" using  s qu kpp pz zu 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 
     moreover from lz pz pz have "l∥z" using M1 by blast
     ultimately have "(p,q) ∈ ov" using ov kpp qu pz zu 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 kq pz zu qu  have "(p,q)∈d" using d by blast
     thus ?thesis using x by simp}
  qed
qed

lemma cmovi:"m O ov^-1 ⊆ s ∪ ov ∪ d"
proof 
  fix x::"'a×'a" assume a:"x ∈ m O ov^-1" then obtain p q z where x:"x =(p,q)" and 1:"(p,z) ∈ m" and 2:"(z,q) ∈ ov^-1" by auto
  then obtain k l c u v  where pz:"p∥z" and kq:"k∥q" and kl:"k∥l" and lz:"l∥z" and qu:"q∥u" and uv:"u∥v" and zv:"z∥v" and lc:"l∥c" and cu:"c∥u" using m ov by blast
  obtain k' where kpp:"k'∥p" using M3 meets_wd pz by blast
  from lz lc pz have pc:"p∥c" using M1 by auto
  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)" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
  thus "x ∈ s ∪ ov ∪ d"    
  proof (elim disjE)
    {assume "(?A∧¬?B∧¬?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∥q" by auto
     moreover from kq kl tq have "t∥l" using M1 by auto
     ultimately have "(p,q) ∈ ov" using ov kpp qu cu lc pc 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
     then  have "(p,q)∈d" using d kq cu qu pc by blast
     thus ?thesis using x by simp}
  qed
qed

lemma covd:"ov O d ⊆ s ∪ ov ∪ d"
proof
  fix x::"'a×'a" assume "x ∈ ov O d" then obtain p q z where x:"x=(p,q)" and "(p,z) ∈ ov" and "(z,q) ∈ d" by auto
  from ‹(p,z) ∈ ov› obtain k u v l c where kp:"k∥p" and pu:"p∥u" and uv:"u∥v" and zv:"z∥v" and lc:"l∥c" and cu:"c∥u" and kl:"k∥l" and lz:"l∥z" and cu:"c∥u" using ov by blast
  from ‹(z,q) ∈ d› obtain k' l' u' v' where kpq:"k'∥q" and kplp:"k'∥l'" and lpz:"l'∥z" and qvp:"q∥v'" and zup:"z∥u'" and upvp:"u'∥v'" using d by blast
  from uv zv zup have "u∥u'" using M1 by auto
  from pu upvp obtain uu where puu:"p∥uu" and uuvp:"uu∥v'" using ‹u∥u'› using M5exist_var 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)
  thus "x ∈ s ∪ ov ∪ d"
  proof (elim disjE)
    { assume "?A∧¬?B∧¬?C" then have ?A by simp
      then have "(p,q) ∈ s" using s kp qvp puu uuvp by blast
      thus ?thesis using x by blast}
    next
    { assume "¬?A∧?B∧¬?C" then have ?B by simp
      then obtain t where kt:"k∥t" and tq:"t∥q" by auto
      from cu pu puu have "c∥uu" using M1 by auto
      moreover from kpq tq kplp have "t∥l'" using M1 by auto
      moreover from lpz lz lc have lpc:"l'∥c" using M1 by auto
      ultimately obtain lc where "t∥lc" and "lc∥uu" using M5exist_var>" and uv:"upv" and pv:"p∥v" using d by blast
      then have "(p,q) ∈" using ov kp kt tq puu uuvp qvp by blast
java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35

    { assume "<>?A ∧ ¬?B ∧ ?C" then have ?C by simp
      then obtain t where "k'∥t" and "t∥
      withnext
      ?thesis using x by auto}
  qed
qed     then obtain t where kt"k\<parallel"<parallel>p"java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74


on
proof
 x:<>a" assume "x <" java.lang.StringIndexOutOfBoundsException: Index 132 out of bounds for length 132
parallelu" and uv:"<>and"\parallel>v" and lc:"l∥c" and cu:"c∥u" and kl:"l∥z" and cuu" using ov by blast
  from thus "x ∈ s ∪ ov ∪ d"
  from uv zv zup have uu:"u∥u'" using M1 by auto
  from kp kpq have "k∥q ⊕ ((java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 20
>?B∧)\>(<otA<and<nd\not>C <r>\n>?\><not?)
thus<  s <union 
  t obtain kptjava.lang.StringIndexOutOfBoundsException: Range [33, 31) out of bounds for length 76
    { assume "?A∧¬ moreover from lz p pz pz have "\parallelz t
java.lang.StringIndexOutOfBoundsException: Range [10, 6) out of bounds for length 61
      thus ?thesis    next
    next
    { assume "¬?A∧?B∧¬?C" then have ?B by simp
      then obtain  kt"\parallel>t"tq<parallelq  java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
      moreover fromqed
      
       obtain cwhereparallel "\>"using cu M5exist_var by blast
      then have "(p,q) <proof :"'a×'a" assume a:"x ∈O ov^-1"thenobtain     x:x ( and:(java.lang.StringIndexOutOfBoundsException: Range [104, 103) out of bounds for length 145
      thus ?thesis   pz<arallel" and kq:"k∥q" and kl:"<  lzl<  :<>" and uv:"u<  "\val\>c"and:m blast
    
    {   from lz lc pz have pc:"p∥c" using M1 by auto
      then obtain t where "frkp have "'<arallel<>t. k'∥t ∧<>(∃t. k∥t ∧ t∥p))" (is "?A ⊕\ ?C)") using M2 by blast
with pu uu qu kpq ave,q) \in d"java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
      thus using  auto
  qed
qed

lemma cfid:"f^-1 O d ⊆ s ∪ ov ∪assu "?<nd<otB<><?) java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
proof
  fix x:java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 69
  from ‹q" by auto
java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 222

 from kp kpq have "k∥q ⊕ kpp qu cu lc pc by blast
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧ thus ?thesis using x by simp}
java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 41
 proof (elim disjE)
 { assume "?A∧¬obtain t where kt:"k∥t" and tp:"t∥p" by auto
 with pup upv kp qv have "(p,q) ∈ s" using s by blast
 thus ?thesis using x by auto}
 next
 { assume "¬?A∧?B∧¬?C" then have thus ?thesis using x by by simp}
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 3
 proof
 with 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)∈ov" using ov by blast
 thus ?thesis using x by blast}
 next
 { assume "¬ \<>(<lose u'" and upvp:"u'∥d by last
 then obtain t where "k'∥t" and "t∥p" by auto
 with pup upv kpq qv have "(p,q) ∈ d" using d by blast
  ?thesis using x by auto}
 qed
 

  cfov:"f O ov ⊆ ov ∪ s ∪ d"
 
 fix x::"'a×'a" assume "x ∈ f O ov" then obtain p q z where x:"x = (p,q)" and "(p,z) ∈kpq have "k\parallelq \> ((\existst. k\>t \>t\<>q t \<and p))" (is "?A \op> (?B ⊕
 from ‹ obtain k l u where "k∥l" and kz:"k∥ and lp:"l\arallel and pu:"p\parallel>u" and zu:"z∥u" using f by blast
 from ‹<>  d"
 from pu zu zup have pup:"p∥
 from lp lpq have "l q \<oplus t. l∥ t∥ (∃\parallel>t \and> t∥p))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
 then have "(?A∧?B∧<not? ((\<notA?B∧?C) \<or ?A∧?B∧?C))" by (insert xor_distr_Lof ?A ?B ?C],auttosmp:elimmeets)tss)
 thus "x ∈ ov \<union 
 proof (elim disjE)
 assume "?A∧¬?B∧¬?C" then have ?A by simp
 lpup upv qv have "(p,q) \i> " using s by blast
 t using x by auto}
 next
  assume "¬?A∧?B∧¬?C" then have ?B by simp
  obtain t where lt:"l∥t" and tq:"t∥q" by auto
 from tq lpq lpc have "t∥c" using M1 by blast
java.lang.StringIndexOutOfBoundsException: Range [6, 1) out of bounds for length 72
 }
 next
 "not>?\n>?B ∧ ?C" then have ?C by simp
 >" and "t∥
java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 63
 
 qed
 

(* ========= $\alpha_2$ composition ========== *)
text ‹We prove compositions of the form $r_1 \circ r_2 \subseteq ov \cup f^{-1} \cup d^{-1}$.›

lemma covsi:"ov O s^-1 ⊆ ov ∪ f^-1 ∪ d^-1"
proof
"\'a"xi>ov^1 w:x  p,) "pz \>" ", <>s-1 yauto
    from ‹(p,z) ∈ ov› obtain k l c u  where kp:"k∥p" and pu:"p∥u" and kl:"k∥l" and lz:"l∥z" and lc:"l∥c" and cu:"c∥u" using ov by blast
oq<>^\'u' v where'<z  :"'\parallel>q"and:'<z : q<v   java.lang.StringIndexOutOfBoundsException: Index 202 out of bounds for length 202
    from lz kpz kpq have lq:"l∥q" using M1 by blast
 qvpp<arallel \oplus>(\exists>t.java.lang.StringIndexOutOfBoundsException: Range [104, 60) out of bounds for length 210
    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 qvp kp kl lq 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 ptp:"p∥t" and "t∥v'" by auto
        moreover with pu cu have "c∥t" using M1 by blast
        ultimately have "(p,q)∈ ov" using kp kl lc cu lq qvp  ov by blast
        thus ?thesis using x by auto}        
     next
      { 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
qed


lemma cdim:"d^-1 O m ⊆ ov ∪ d^-1 ∪ f^-1"
proof 
    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› thus" \<>  
    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" { assume "?A∧¬?B∧¬?C" then have ?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 "x ∈  ov ∪ d^-1 ∪ f^-1"
    proof (elim disjE)
      { assume "?A∧<not       have"p,q) \in> s"usings kp upuu puby 
        with qvp kp klz lzqthus thesis x last
        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 zq lzq zu have "lz∥u" using M1 by auto
        moreover from pt pv uv have "u\<      then
java.lang.StringIndexOutOfBoundsException: Range [18, 8) out of bounds for length 79
        thus ?thesis using x by auto}
     next
      { assume "¬?A ∧ <not lchave"\parallel>>c"using M1 byjava.lang.StringIndexOutOfBoundsException: Range [72, 73) out of bounds for length 72
here:q<parallelt andt<v y 
        with kp klz lzq pv have "(p,q) ∈ d^-1" using d by blast 
        thus?hesis usingx by }
      qed
qed

lemma cdiov:"d^-1 O ov ⊆ ov ∪ f^-1 ∪ d^-1"
proof
    fix x::"'a×'a" assume "x \  next
   from \<pen(and kl:"<l"and lq:"\pq  andquq<parallel  uv"\parallel>v"using  
    from ‹(q,r) ∈ ov› obtain k' l' t u' v' where lpr:"l'∥r" and kpq:"k'∥q" and kplp:"k'∥l'" and qup:"q∥u'" and "u'∥v'" and rvp:"r∥v'" and lpt:"l'∥t" and tup:"t∥u'" using ov by blast
    from lq kplp kpq have "l∥l'" using M1 by blast
    with kl lpr  obtain ll where  kll:"k∥ll" and llr:"ll∥r"  using      thenobtain "k'\<arallelt ed
    from pv rvp have "p∥v' ⊕ ((∃
   then<and\not<nd\or(<not?<>B?) \or>(\not?A<and\not?B<andC)" 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 rvp llr kp kll have "(p,r) ∈ f^-1" using f by blast
        thus ?thesis using x by auto}
      next
      { assume "¬?A∧?B∧¬
        then obtain t' where ptp:"p∥t'" and tpvp:"t'∥v'" by auto
 h llt"\parallel>t"usingbyblast
        moreover from ptp uv pv have utp:"u∥t'" using M1 by blast
moreoverfromqu quptparallel"using M1 by blast
        moreover with utp llt obtain tu where "ll∥tu" and "tu∥t'" using M5exist_var by blast
ptp  h (\    blast
        thus ?thesis using x by auto}        

 \notA\and \not>B\and C" 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
qed

lemma cdis:"d^-1 O s ⊆ ov ∪ f^-1 ∪ d^-1"
roof
  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 proof (elim disjE)
  from ‹{assume "?A\<?an>\>" thenen have ?A by simp
  from lz lpz lpq have lq:" ∥q" using M1 by blast
  have "p\parallel>v' \oplus> ((\exists>t. p∥t ∧ t∥v') ⊕ (∃t. q∥t ∧ t∥v))" (is "?A ⊕ (?B ⊕thus ?thesis using x by auto}
 (?A\\not?B\and¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto then obtain t where kt:"k\parallel>t" and tq:t\q" by uto
 thus "x ∈ ov \<      with
 proof (elim disjE)
 { assume "?A∧¬?B\      thus ?the ?thesis using x by blast}
 with kl lq qvp kp have "(p,q) ∈ f^-1" using f by blast
  ?thesis using x by auto}
 
 { assume "¬ upv kpq qv have "(p,,q) n
 then obtain t where pt:"p∥t" and tvp:"t∥?thesis using x 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) ∈qed
 union> d"
 next
 { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp
 then obtain t where "q∥t" and "t∥v" by auto
 with kl lq kp pv have "(p,q)∈d^-1" using d by blast
 thus ?thesis using x by auto}
 qed
 

 :"s^-1 O m 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) ∈
 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 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\    from lp lpq have "l∥q ⊕ ((∃t. l∥t ∧ t∥q) ⊕ (∃t. l'∥ then have "(?A∧¬?B∧¬
 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∥ rom tq lpq lpc ha have "t\c" using M1 by blas
 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
 <? 
 then obtain t where "q∥t" and "t∥v" by auto
 with kp kz zq pv have "(p,q)∈d^-1" using d by blast
 thus ?thesis using x by auto}
 qed
 

 ?thesis usingx by auto} 
 
 fix x::"'a×'a" assume "x ∈
 from \<text\r_1 \ r_2 \ ov \ f^{-1} \ d^{-1}$.\close
java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 61
java.lang.StringIndexOutOfBoundsException: Range [169, 165) out of bounds for length 202
 then have "(?A∧\<notv' \<us<
 thus "x \    then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨in> ov ∪ f^-1 ∪ d^-1"
  { assume "?A\<>n?B∧¬?C" then have ?A by simp
 { assume "?A∧¬?B∧¬?C" then have ?A by simp
 with kp kplp lpq qvp klp have "(p,q) ∈ f^-1" using f by blast
 thus ?thesis using x by auto}
 next
java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 69
java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 79
 from pt pv uv have "u∥ ththen obtain t where ptp:"p∥t" and "t∥v'" by auto
 moreover from cup zup zu have cu:"c∥u" using M1 by auto
 ultimately obtain cu where "l'∥cu" and "cu∥avep,q)<>
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 thus ?thesis using x by au
 next
java.lang.StringIndexOutOfBoundsException: Range [16, 14) out of bounds for length 72
 then obtain t where "q∥in> ov ∪ d^-1 ∪ f^-1"
 with kp klp lpq pv have "(p,q)∈d^-1" using d by blast
 thus ?thesis using x by auto}
 { assume "?A\\🪙
 

  covim:"ov^-1 O m ⊆
 
 fix x::"'a\<\<¬
 from ‹v'" by auto
 from ‹
 obtain v' where qvp:"q∥
 from zu zq cu have cq:"c∥"(p,qq)\<in v" using kp klz lzq pt tvp qvp ov by blast
 s by auto}
java.lang.StringIndexOutOfBoundsException: Range [15, 8) out of bounds for length 186
 thus "x \in> ov \unionn> f^-1 \<union 
 proof (elim disjE)
 { assume "?A∧ { assume "\not>?A ∧ ¬?B ∧ ?C" then have ?C by simp
 with lp lc cq 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 ptp:"p∥t" and "t∥v'" by auto
 moreover with pv uv have "u∥t" using M1 by blast
java.lang.StringIndexOutOfBoundsException: Range [24, 18) out of bounds for length 73
 d
 next
 { assume "¬
java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 74
 with lp lc cq pv have "(p,q) ∈fix x::"'a×'a" assume "x ∈ d^-1 O ov" then obtain p q r where x:"x = (p,r)" and "(p,q) ∈ d^-1" and "(q,r) ∈ ov" by auto
 thus ?thesis using x by auto}
 qed
 

(* =========$\alpha_3$ compositions========== *)
text ‹

  covov:"ov O ov ⊆ b \<pv rvp have "p\<parallel>v' \<oplus< ?C)") using M2 by blast
 
 fix x::"'a× ov O ov" then obtain p q z where x:"x = (p,q)and(p,z) \in> ov" and "(z,q)∈ ov" by auto
 from ‹ ov ∪ d^-1"
 from ‹
 from lz kplp kpz have llp:"l∥
 from uv zv zup have "u∥<>-
 parallel>uu" and uuv:"uu∥v'" using M5exist_var by blast
 from puu lpq have "p∥q ⊕ ((∃t'. p∥
 <><¬ ((¬?B∧?C) ∨?A∧?B∧ xor_d_distr_L[[of ?A ?B ?C?C], auuto sip:elmet)
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 E
 { assume "?A∧¬?B∧¬?C" then have ?A by simp
 then have "(p,q) ∈ m" using m by auto
java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 37
 next
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
 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 lptp:"l'∥t'" and "t'∥uu" by auto
 from kl llp lpq obtain ll where kll:"k∥ll" and llq:"ll∥q" using M5exist_var by blast
 with lpq lptp have "ll∥t'" using M1 by blast
 with kp puu uuv kll llq vp \open>t'<parallel<c have "(p,q) ∈ ov" using ov by blast
 thus ?thesis using x by auto}
 qed
 

  covfi:"ov O f^-1 ⊆ b ∪ m ∪ ov"
 
 fix x::"'a×'a" assume "x ∈qed
 from ‹
 >(,q) ∈ f^-1› obtain k' l' v' where kplp:"k'∥l'" and kpz:"k'∥z" and lpq:"l'∥>v'" and zvp:"z∥ ?C"n v?Cbyim
 from lz kplp kpz have llp:"l∥
 from zv qvp zvp have qv:"q∥v" using M1 by blast
 from pu lpq have "p∥q ⊕ ((\<exists
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 43
 mij)
 { assume "?A∧¬?B∧\<  from"\l'" and lpq:"l'∥v'" and lpc:"l'\<rallel"
 then have "(p,q) ∈ m" using m by auto
 thus ?thesis using x by auto}
 
 { then have "(?A\and<>and<t<?A∧¬∨?A∧?B∧r_[ ?A? ?C u impeets
 > b" using b by auto
 
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 { assume "¬?A ∧
 then obtain t where lptp:"l'∥t" and "t∥\t nd v:\parallel>'" by auto
 from kl lp lq obai llwhr kl:"∥
  obi wer "qparallel>t" and "t∥v" by auto
 with kp pu uv kll llr qv ‹t\<        with kp klp lpq pv have "(p,q)\<in>d^-1"
 thus ?thesis using x by auto}
java.lang.StringIndexOutOfBoundsException: Range [8, 6) out of bounds for length 9



  csov:"s O ov ⊆ b ∪ m ∪ ov"
 
 fix x::"'a×v'" using M3 meets_wd zq by blast
java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
java.lang.StringIndexOutOfBoundsException: Range [112, 111) out of bounds for length 236
 from kz kpz kplp have klp:"k∥l'" using M1 by blast
 :"u\>u'" using M1 by blast
 with pu upvp obtain uu where puu:"p∥uu" and uuvp:"uu∥v'" using M5exist_var by blast
 from pu lpq have "p∥q ⊕ ((∃t. p∥t ∧ t∥q) ⊕ (∃t. l'∥t ∧ t∥u))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
 then have "(?A∧¬?B∧\    thus "x n f^-1 ∪
 proof (elim disjE)
 proof (elim disjE)
 { assume "?A∧¬ assm"A\and>\not>?B\and>\not>?C" then have ?A by simp
java.lang.StringIndexOutOfBoundsException: Range [46, 44) out of bounds for length 66
 thus ?thesis using x by auto}
 next
 { assume "\<?A∧?B∧¬?C" then have ?B by simp
 then have "(p,q) ∈ b" using b by auto
 thus ?thesis using x by auto}
 next
 \not>?>?A \and ¬ ?C" then have ?C by simp
 then obtain t where lpt:"l'∥t" using M1 by blast
 with pu puu ultimately have "(p,q)∈
 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 \subseteq> bb \ m ∪
 
 fix x::"'a×'a" assume "x ∈ s O f^-1" then obtain p q r where x:"x = (p,r)" and "(p,q) qed
 from ‹(p,q) ∈ s› obtain k u v where kp:"k∥p" and kq:"k∥q" and pu:"p∥
 from ‹ ‹of the form $r_1 \circ r_2 \ b \cup m \cup ov$.›
 from kpq kpl kq have kl:"k∥l" using M1 by blast
 from qvp qv uv have uvp:"u🚫
java.lang.StringIndexOutOfBoundsException: Range [23, 21) out of bounds for length 136
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧ from ‹\close obtain k u l t v whre kp:"k\parallel>p" and pu:"p∥u" and kl:"k∥l" and lz:"l∥z" and "l∥t" and "t∥u" and uv:"u∥v" and zv:"z∥from  (z,q) nov🚫and kpz:"k'\parallel" and lpq:"l'∥parallel>y" and "y∥u'" and zup:"z∥u'" and upvp:"u'∥v'" and qvp:"q∥v'" using ov by blast
 thus "x ∈ b ∪ uv zv zup have "u🚫l\>\> t'\parallel>uu))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
 proof (elim disjE)
 ?∧?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∧{ assume "?A ¬?B∧¬?C" then have ?A by simp
 then then have "(p,q) n m by auto
 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'∥next
  { assume "🚫
  by auto}
 qed
 

(* =========$\alpha_4$ compositions========== *)
text ‹

  cmmi:"m O m^-1 ⊆ f ∪ f^-1 ∪ e"
 
 fix x::"'a×'a" assume a:"x ∈ m O m^-1" then obtain from kl llp lpq obtain ll where kll:"k∥ll" and llq:"ll∥q" using M5exist_var by blast
 then have p with kp puu uuv kll llq qvp ‹ using x by auto}
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
  kkpq have "k q \oplus> ((∃t. k∥t \z>v" using ov by blast
 B🪙not>?B\and>?C)" by (inseertor_dsr_of ?A? ?] aut sp:limes)
 thus "x ∈v" using M1 by blast
 proof (elim disjE)
  "(?A∧¬?B∧¬?C)" then have "?A" by simp
 then have "p = q" using M4 kp pz qz by blast
 then have "(p,q) ∈ \> ov"
 thus ?thesis using x by simp }
 next
 {assume "(¬?A∧?B∧ proof (elim disjE)
 then obtain t where kt:"k∥t" and tq:"t∥q" by auto
 then have "(p,q) \(p,q) \in> f^-1" usingf qz pz kp by blast
 thus ?thesve "(p,q) \<n
 next
 {assume "(¬?A∧ next
 then obtain t where kt:"k'∥t" and tp:"t∥p" by auto
 with kpq pz qz have "(p,q)∈" using f by blast
 thus ?thesis using x by simp}
 ed
 
 

  cfif:"f^-1O f \subseteq> e ∪ f^-1 ∪ f"
 
 fix x::"'a×'a" assume a:"x ∈ f^-1 O f" then obtain p q z where x:"x =(p,q)" and 1:"(p,z) ∈ assume "\not>?A \\<nd 
 from 1 obtain k l u where kp:"k∥p" and kl:"k∥l" and lz:"l∥l'\parallel>t" and "t\>u" by auto
 from 2 obtain k' l' u' where kpq:"k'∥q" and kplp:"k'∥l'" and lpz:"l'∥z" and zup:"z∥u'" and qup:"q∥u'" using f by rom kl llp lpq obtain ll where kll:"k\parallel>ll" and llr:"ll\parallel>q" using M5exist_var by blast
 from zu zup qup have qu:"q∥u" using M1 by auto
 from kp kpq have "k∥ witith lpq lptp have "ll∥
 then with kp pu uv kll llr qv ‹t∥
 thus "x ∈ e \<unionq
 proof (elim disjE)
 {assume "(?A∧\<notsubseteq>> b \ ∪ m ∪ ov"
 then have "p = q" using M4 kp pu qu by blast
 then have "(p,q) ∈ e" using e by auto
 thus ?thesis using x by simp }
 next
 {assume "(¬?B\a<C  "?B" by simp
java.lang.StringIndexOutOfBoundsException: Range [126, 125) out of bounds for length 184
 have"pq <>-1
 thus ?thesis using x by simp}
 next
 {assume "(¬\<>aen ve "??C" by simp
 then obtain t where kt:"k'∥t" an from uv zv zup have uup:"u∥u'" using M1 by blast
 with kpq pu qu have "(p,q)∈f" using f by blast
java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
 qed
 
 then have"(\and¬¬ ((¬?B∧\<>?\<nd\<ot?
  cffi:"f O f^-1 ⊆ e ∪ f^-1"
 
 fix x::"'a×'a" assume "x ∈ f O f^-1" then obtain p q r where x:"x = (p,r)" and "(p,q)∈ { assume "?A∧?B∧>?C" then have ?A by simp
 open>(p,q)∈f› ‹(q,r) ∈ f^-1› obtain k k' where kp:"k∥p" and kpr:" thus ?thesis using x by auto}
 from ‹
java.lang.StringIndexOutOfBoundsException: Range [16, 14) out of bounds for length 208
 ?<>B< <not>?C) \or> (¬?A∧ b" using b by auto
 thus "x ∈ ‹ the form $r_1 \ r_2 \subseteq f \ f^{-1}\cup e$.\close>
 proof (elim disjE)
 { assume "?A∧
 with pu ru kp have "p = r" using M4 by autojava.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 143
 thus ?thesis using x e by auto}
 next
 then have pz:"p\parallel>z" an qz:"q\parallel>z" sing m by auto
 then obtain t where kt:"k∥k k' where kp:"k\parallel>p" and kpq:"k'\parallel>q" using M3 meets_wd qz pz by blast
 with ru kp pu show ?thesis using x f by blast}
 next
 { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp
 then obtain t where rtp:"k'∥t" and "t∥p" by auto from kp kpq have "k∥q ⊕ ((∃t. k∥t ∧ t∥q) ⊕ (∃t then have "(?A\(not>?A\>?B\and>\<not?(\<>?<>C
  pu show ?thesis usinx f by blast}
 qed


(* =========$\alpha_5$ composition========== *)
text <  from zu  ((∃t ∧q) ⊕java.lang.StringIndexOutOfBoundsException: Range [116, 113) out of bounds for length 208
(¬¬(<not<¬?A∧?B∧ xor_distr_L? B?]  simp)
lemma cssi:"s O s^-1 ⊆ e ∪ s ∪ thus "x ∈ f^-1 ∪
proof
   fix x    {assume (?A\and\not<andnotC)" then have "?A" by simp
   from ‹
<,)\in>\close<open>(q,r) <in> s^-1› obtain u u' where pu:"p∥u" and rup:"r∥u'" using s by blast
   then have "p∥
   then have "(?A∧¬?B∧\<    ?B \<>?
   thus "x\> union 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∧
        then obtain t wherenext
        with{assume "🚫
      next
      { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp     with kpq pu qu have "(p,q\inf"using f by blast
   then obtain t where rtp:"r\parallel>"and "t\parallelu  java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
        with pu kp kr show ?java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    qed
qed

lemma csis:"s^-1 O s ⊆ e ∪
proof
   fixx:"<times  x < thenr xx "p,q)\>s^-1" "q,r) \in>s" by auto
   from ‹(p,q)∈open>(p,)\<in>f›" and kpr:"k'r" using f by blast
   <o>(p,q)\>f›(q,r) ∈ f^-1›u" andu" and ru:"r∥
   then have "p∥u' ⊕ ((∃<parallel> ⊕((\<<>t \and t\parallel>r) ⊕ (∃t ∧>p))" (is "?A \<plus<¬or> ((\not>?A∧>🚫?A∧?B∧
   then have "(?A∧¬?B∧\    lim
   thus>e ∪ s^-1"
   proof (elim disjE)
      { assume "        withpu ujava.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 51
        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∥u'" by auto
        with rup kp kr  show ?thesis using x s by blast}
      next
      { assume "¬?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
qed

lemma cmim:"m^-1 O m ⊆ s ∪ s^-1 ∪ e"
proof
   fix x::"'a×'a" assume "x ∈ m^-1 O m" then obtain p q r where x:"x = (p,r)" and "(p,q)∈m^-1" and "(q,r) ∈m" by auto
   from ‹(p,q)∈m^-1› ‹(q,r) ∈ m› have qp:"q∥p" and qr:"q∥r" using m by auto
   obtain u u'  where pu:"p\<parallel>u" and  rup:"r\<parallel>u'" using M3 meets_wd qp      { assume "\not>A<and>?B\<and>\<not>?C" then have ?B by simp 
   then have "p\<parallel<oplus> <existstparallel> \<and> t\<parallel>u') \<oplus> (\<exists>t. r\<parallel>t \<and> t\parallel>)"<> <plus ?C)") using M2 by blast
   then have "(?A\<and>\<
   thus "x \<in>  s \<union> s^-1 \<union> e"
m)
      { assume "?A\<and>\<not>?B\<and>\<not>?C" then have ?A by simp
        with rup qp qr have "p = r" using M4 by auto
        thusjava.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 39
      next
       <java.lang.StringIndexOutOfBoundsException: Range [23, 21) out of bounds for length 69
        then obtain t where kt:"p\<parallel>t" and tr:"t\<parallel>   then<java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 202
thjava.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 56
      next
      { assume "\<not>?A \<and> \<not>?B \<and> ?C"   then ave "?and><B<d<C<r>((\<not>?A\<d<and<<<Aand\not>?B\<and>?C))" by (insert xor_distr_L[of ?Aajava.lang.StringIndexOutOfBoundsException: Range [172, 171) out of bounds for length 185
        p>" and "t\<parallel>u" by auto
        with pu qp qr show ?thesis using x s  by blast}
    qed
qed

(* =========$\beta_1$ composition========== *)
subsection ‹$\beta$-composition›
text ‹

  cbd:"b O d ⊆ b ∪ m ∪
 
 fix x::"'a×'a" assume "x ∈ b O d" then obtain p q z where x:"xproof
 m ‹(p,z) ∈ b› obtain c where pc:"p∥from \open>(p,q∈s^-1›(q,r) ∈<closep" and kr:"k<kq" using s M1 by blast
  obtain a where ap:"a∥p" using M3 meets_wd pc by blast
  from \have "p<parallel⊕t. p∥ t∥ (∃t ∧u))" (is "?A ⊕ ?Cu2last
 from pc czjava.lang.StringIndexOutOfBoundsException: Range [18, 16) out of bounds for length 104
  with<au\parallelvrt
  from ap kqproof java.lang.StringIndexOutOfBoundsException: Range [16, 14) out of bounds for length 21
  then<<ot?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨?A∧>?B∧java.lang.StringIndexOutOfBoundsException: Range [162, 160) out of bounds for length 184
  thuss
  proof       next
{java.lang.StringIndexOutOfBoundsException: Range [16, 14) out of bounds for length 68
        with ap pczu czuv uv qv have "(p,q) ∈ s" parallel>t" and tr:"t\<parallel'" by auto
java.lang.StringIndexOutOfBoundsException: Range [22, 20) out of bounds for length 39
      next
      { assume "🚫
        then obtain t where at:"a∥
        from pc tq have "p∥q ⊕
        java.lang.StringIndexOutOfBoundsException: Range [14, 12) out of bounds for length 190
x b <union  < d"
        proof (elim disjE)
           { assume "?A∧¬?B∧
java.lang.StringIndexOutOfBoundsException: Range [29, 25) out of bounds for length 44
           
              obtain ue<arallel>'" using M3 meets_wd qp qr by fastforce
             thus ?thesis using x b by auto}
           next
           { assume "¬   then have "p∥oplus> ((\<>t. p∥t ∧ t∥u') ⊕ (∃t. r∥t ∧ t∥u))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
             then obtainwhere "t\<arallelel
             with pc pczu have "<java.lang.StringIndexOutOfBoundsException: Range [47, 44) out of bounds for length 66
java.lang.StringIndexOutOfBoundsException: Range [28, 26) out of bounds for length 107
             java.lang.StringIndexOutOfBoundsException: Range [27, 25) out of bounds for length 42
        qed
        }  
      next
      { assume "\{ asume \not>?A\and>?B∧¬?C" ve
       java.lang.StringIndexOutOfBoundsException: Range [30, 28) out of bounds for length 71
        with kq pczu czuv uv qv have        withjava.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 56
        thus      { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp
       qed
qed

lemma cbf:"b O f ⊆ b ∪ with pu p qr show ?thesis sin xx s bbyblast
proof
  fix x::"'a×
  from ‹$\beta$-composition›
  obtain a where ap:"a∥
  from ‹lemma cbd:"b O d \subseteq> b > b \union> m \union> ov ∪ s ∪ d"
  from pc cz zu obtain cz where pcz:" ∥cz" and czu:"cz∥u" using M5exist_var by blast
 froma ve "a\parallelq \<oplus ((∃t. a\<t ∧<<oplus>(\<exists.k∥t ∧ t∥p))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
java.lang.StringIndexOutOfBoundsException: Range [76, 75) out of bounds for length 184
 thus "x ∈ b ∪ m ∪ ov ∪from \<<in> b\close> obtain c where pc:"p∥c" and cz:"c∥
 proof (elim disjE)
 { assume "?A∧¬\<not"
java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 65
 thus ?thesis using x by auto}
 next
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
  whre at"a\<t" and tq:"t∥
 from pc tq have "p\<parallel  🪙 ((¬?B∧>?C) ∨>?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 then have "(?A∧¬thus "x \in> b ∪ m ∪ ov ∪ s ∪ d"
 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∥t'" and "t'∥c" by auto
 with pc pcz have "t'∥cz" using M1 by auto
 with at tq ap pcz czu qu ‹t∥t'› 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 "k∥t" and "t∥p" by auto
 with kq pcz czu qu have "(p,q) ∈ d" using d by blast
 thus ?thesis using x by auto}
 qed
 

  cbovi:"b O ov^-1 ⊆ b ∪ m ∪ ov ∪ s ∪ d"
 
 fix x::"'a×'a" assume "x ∈ b O ov^-1" then obtain p q z where x:"x = (p,q)" and "(p,z) ∈ b" and "(z,q) ∈ ov^-1" by auto
 from ‹(p,z) ∈ b› obtain c where pc:"p∥c" and cz:"c∥z" using b by auto
 btain a where ap:"a∥p" using M3 meets_wd pc by blast
 from \  proof (elof (elim disjE)
 from cz lz lw have "c∥w" using M1 by auto
 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 ⊕
 <<?A\<<\ (\?A∧?B∧f ?A ?B ?C], uto siip:elimmeeets
 thus "x ∈ b ∪ m ∪ ov ∪ s ∪ d"
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have ?A by simp
 with ap qu pcw cwu have "(p,q) ∈ s" using s by blast
java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
 next
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
 then obtain t where at:"a∥t" and tq:"t∥q" by auto
 from pc tq have "p∥from pc tq have "p∥q ⊕ ((∃t'. p∥t' ∧ t'∥q) ⊕ (∃t'. t∥t' ∧ t'∥c))" (is "?A ⊕ (?B ⊕ tthen haand>🚫 (inser or_disrLo ? ? ?C, at sm:elmmeets)
 then have "(?A∧<not?\or> ((¬?A∧?B∧¬?C) ∨ (¬and>¬?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∧ { assume "\t∧¬
 thus ?thesis using x b by auto}
 next
 { assume "¬?A ∧ ?C" then have ?C by simp
 then obtain t' where "t∥t'" and "t'∥
 with pc pcw have "t'∥czu" using M1 by auto
 with at tq ap pcw cwu qu ‹in>ov" using ov by blast
 thus ?thesis using x by auto}
 qed
 
 next
 { assume "\<>A ?C" then have ?C by simp
 then obtain t where "k∥t" and "t∥
 thkq pcw cwu uhve "pq) \< d" using d by blast
 thus ?thesis using x by auto}
 qed
 

  cbmi:"b O m^-1 ⊆ b ∪ m ∪ ov 🚫< ay au
 
java.lang.StringIndexOutOfBoundsException: Range [9, 6) out of bounds for length 139
 from ‹z b by uo
 obtain k where kp:"k∥ "parallel>q \>((∃t. a\<lel<
 \> < \
 thus "x \in b 🚫
 from kp kpq have "k∥q ⊕ ((∃proof (elim disjE)
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
java.lang.StringIndexOutOfBoundsException: Range [17, 15) out of bounds for length 64
 proof (elim disjE)
 { assume "?A∧¬?B∧ thus ?thesis using x by auto}
 with kp pc cz qz have "(p,q) \      next
 thus ?thesis using x by auto}
 next
 { assume "¬?A∧?B∧¬parallel>q ⊕ ((∃t'. p∥t' ∧ t'∥q) ⊕ (∃t'. t∥¬>¬∨?A∧¬ (¬¬?C))" by (insert xor_distr_L[of ?A B ?C], ato siimp:elimeetts)
 then obtain t where kt:"k∥q" by auto
 from pc tq have "p∥ proof (elim disjE)
 then have "(?A∧¬¬
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
java.lang.StringIndexOutOfBoundsException: Range [9, 7) out of bounds for length 26
 
 thus ?thesis using x m by auto}
 next
 { assume "¬?A\        
 thus ?thesis using x b by auto}
 next
java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 72
  with kq pcz czu qu have "(p,q) ∈ d" using d by blast
 with pc cz qz kt tq kp have "(p,q) ∈ ov" using ov by blast
 
 qed
 }
 lemma cbovi:"b O ov^-1 ⊆union> m \union> ov ∪ s ∪
java.lang.StringIndexOutOfBoundsException: Range [10, 8) out of bounds for length 72
 re"'\<>
java.lang.StringIndexOutOfBoundsException: Range [50, 47) out of bounds for length 63
 thus ?thesis using x by auto}
 qed
  from cz lz lw have "c∥w" using M1 by auto

  cdov:"d O ov \<  thus
 
 fix x::"'a× with ap qu pcw u have "(p,q) \<  
 from ‹
 m \open>(z,q) ∈
 from zup zv uv have "u∥parallel>q ⊕ ((∃t'. p∥t' ∧ t'∥q) ⊕ (∃t'. t∥t' ∧ t'∥c))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
 with pu uvp btin u uu here puu:"p\<":""uu<arallelv
 from lp lpq have "l∥no?B\and>¬?C" then have ?A by simp
 ¬¬or> ((\not>?A∧?C) \<(<
 thus "x ∈ b ∪ m ∪ ov ∪ s ∪ d"
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have ?A by simp
 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" and tq:"t∥q" by auto
 from pu tq have "p∥q ⊕ ((∃t'. p∥t' ∧ t'∥q) ⊕ (∃ thu ?hi ixmbao}
 then have "(?A\<           next
java.lang.StringIndexOutOfBoundsException: Range [25, 23) out of bounds for length 69
 proof (elim disjE)
java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 thus ?thesis using x m by auto}
 next
 { assume "¬?A∧?B∧\             wi pc pcw have "t' w" using M1 by auto
 thus ?thesis using x b by auto}
 next
 { assume "¬?A ∧ with at tq ap pcw cwu qu \opent∥t'›ov" using ov by blast
 then obtain t' where ttp:"t∥t'" and "t'∥
 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}
 qed
 
  hen obtan t wherewhere \>t" ad "t∥
 >?A ∧?B \<and 
 then obtain t where "l'∥t" and "t∥p" by auto
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 thus ?thesis using x by auto}
 qed
 

  cdfi:"d O f^-1 ⊆ b ∪re kp:"k\<>p 
 
java.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 64
 from ‹ppc cz qz haveve "(p,q) \in> s" using s by blast
 from \<thussi using x by auto}
 from zup zv uv have uup:"u∥
 from lp lpq have "l∥q ⊕ ((∃>?A∧<><
java.lang.StringIndexOutOfBoundsException: Range [103, 102) out of bounds for length 185
 <>bunion> ov ∪ s ∪ d"
java.lang.StringIndexOutOfBoundsException: Range [61, 59) out of bounds for length 190
 { assume "?A∧¬?B∧pt'" and "t'∥
 with thus ?the uing xy
 thus ?thesis using x by auto}
 next
 { assume "¬?A∧?B∧\<not ?A ∧>?B \and> ?C" then have ?C by simp
 then obtain t where lt:"l\<elt
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 then have "(?A∧¬?B∧¬?C) ∨
  "x ∈ b ∪ m ∪ ov ∪ s ∪ d"
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have ?A by 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
 thus ?thesis using x m by auto}
 next
 { assume "¬uu" and uuvp:"uu∥
java.lang.StringIndexOutOfBoundsException: Range [35, 33) out of bounds for length 209
 next
 🚫
java.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 87
 with lt tq lp pu uup qup have "(p,q) ∈ ov" usingmp
 thus ?thesis using x by auto}
 qed
java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
 next
java.lang.StringIndexOutOfBoundsException: Range [17, 14) out of bounds for length 72
 then obtain t where "l'∥t" and "t∥from pu tq have "p∥q ⊕ ((∃t'. p∥t' ∧ t'∥q) ⊕ (∃\and>¬¬ ((¬?B∧?C) ∨?A∧?B∧mmets
 with lpq pu uup qup have "(p,q) ∈proof (eli dis
 thus ?thesis using x by auto}
 
 

(* =========$\beta_2$ composition ==========*)
text ‹

 :"v O d^-1 <seteqq
 
 fix x::"'a×{ assume "\<not? ∧ ¬<d"
 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"
  qed
 from lz kpz kplp have "l∥l'" using M1 by auto
 with kl lpq obtain ll where kll:"k∥
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 3
 then have "(?A∧¬q" and zup<pu'" using f by blast
 thus "x ∈ b ∪ m ∪ ov ∪ f^-1 ∪ d^-1"
 
 { assume "?A∧
 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}
 
 { assume "¬
 then obtain t' where lptp:"l'∥\parallel>v" by auto
 from lpq lptp llq have "ll∥t'" using M1 by auto
 with kp kll llq pt tup qup tpt have "(p,q) \<        with pv kp kzc zcq   have "(p,q) \<in> d^-1" using d byblastnot>?A\and>\<not??C))" by (insertxor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 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}
 
 

  cdib:"d^-1 O b ⊆ b ∪ m ∪ ov ∪ f^-1 ∪ d^-1"
 
 fix x::"'a×'a" assume "x ∈ { assume "¬?A∧?B∧¬?C" then have ?B by simp
 from ‹t" and tvp:"t∥
 from ‹ "p\parallelq \oplus> ((\<>t'
 with kl lz obtain lzc where klzc:"k∥lzc" and lzcq:"lzc∥then have "(?A\<>\?C) ∨\not>?A\<>?¬or> (\not>?A∧¬?B∧C)" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 obtain v' where qvp:"q∥v'" using M3 meets_wd cq by blast
java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 209
 then have "(?A\\and>\<ot?) \or> (\not>?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 thus "x ∈ b ∪ m ∪ ov ∪ f^-1 ∪ d^-1"
  disjE)
 { assume "?A∧¬"?A\and<B∧ 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}
 next
 "\not?A\and>?B\>\not?C" the have ?B by simp
 then obtain t where pt:"p∥t" and tvp:"t∥v'" by auto
 from pt cq have "p∥ ((∃t' ∧ (∃t' ∧ t'∥ (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
 then have "(?A∧¬thus ?th?thesis using x b by auto}
 thus "x ∈ b ∪
 proof (elim disjE)
 A\and ?B\><?"
 thus ?thesis using x m by autparallel>t'" and tpt:"t'\parallel>t" by auto
 next
  { assume "¬?A∧?B∧¬?C" then have ?B by simp

 next
 { assume "\<ot?
 obtaini t' wheree ctp"c\parallel>t'" and tpt:"t'∥t" by auto
 from lzcq cq ctp have "lzc∥t'" using M1 by auto
 with pt tvp qvp kp klzc lzcq 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∥v" by auto
 with pv kp klzc lzcq have "(p,q) ∈ ??thesis u using x by auto}
 thus ?thesis using x by auto}
 qed
 

  csdi:"s O d^-1 ⊆
 :"m^-1 O b \subseteq b \> m \> ov \> f^-1 \union> d^-1"
 fix x::"'a×'a" assume "x ∈ s O d^-1" then obtain p q z where "(p,z) : s" and "(z,q) : d^-1" and x:"x = (p,q)" by auto
 rom \>(p,z) : s\close> obtaink u v where kp:"k🚫' using Mmesw qb ls
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 from kp kz kpz have kpp:"k'∥p" using M1 by auto
 ∥u' \opluse>t. p∥ t∥ (∃t. q\parallel>t ∧u))" (is ?A<> 
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?¬¬?B∧mp:limmees
 thus "x ∈ m ∪ f^-1 ∪
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have ?A by simp
 with qup kpp kp lpq have "(p,q) ∈
 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∥ ((∃t' ∧q) ⊕t'. l'∥ t'∥)" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 thus "x ∈ b ∪ m ∪ ov ∪ f^-1 ∪ d^-1"
 proof (elim disjE)
 ?B\\a>¬
 thus ?thesis using x m by auto}
 next
 { assume "¬?A∧?B∧¬
 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
 with pt tup qup kpp kplp lpq 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 kpp kplp lpq have "(p,q) ∈ d^-1" using d by blast
 thus ?thesis using x by auto}
 qed
 

 union> m ∪> ^-1 ∪1
 
 fix x::"'a×'a" assume "x ∈ s^-1 O b" then obtain p q z wh x::"'a\times>'a" assume "x ∈ ov O ov^" the obtain z where x:" p,q)" and "(z) \<> 
 (p,z) : s^-1› obtain k u v where kp:"k∥z" and zu:"z∥v" and pv:"p∥
 from ‹(z,q) : b› obtain c where zc:"z∥c" and cq:"c∥q" using b by blast
 from kz zc cq obtain zc where kzc:"k∥zc" and zcq:"zc∥q" using M5exist_var 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∧¬C) r¬?A∧¬∧], ao ip:lmms
 thus "x ∈ b ∪ m \<union  \<tB¬ ((¬?B∧?C) ∨?A∧?B∧xor_st_[fA?B ], utosm:lmet)
 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
 thus ?thesis using x by auto}
 next
 { assume "¬?A∧?B∧ from pu qup have "p "p∥ ((∃t' ∧u') ⊕t'. q∥ t'∥ (?B ⊕
 then obtain t where pt:"p∥v'" by auto
 from pt cq have "p∥q ⊕ ((∃t'. p∥
 then have "(?A∧
 thus "x ∈ b ∪ m ∪ ov ∪ f^-1 ∪ d^-1"
 proof (elim { assume "?A∧?B∧A s
 { assume "?A∧
 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 ctp:"c∥
 from zcq cq ctp have "zc∥t'" using M1 by auto
 with zcq pt tvp qvp kzc kp ctp tpt have "(p,q) ∈ ov" using ov by blast
  esuinx auo
 
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
 next
 { assume "¬?A ∧ ¬?B ∧jE)
 then obtain t where "q∥
 with pv kp kzc zcq have "(p,q) ∈ d^-1" using d by blast
 thus ?thesis using x by auto}
 qed
 

  covib:"ov^-1 O b ⊆ m ∪ f^-1 ∪-"
 
 fixx"a\timesa" 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
 ‹z" and kl:"k∥p" and zu:"z∥v" and pv:"p∥c" and cu:"c∥
 from ‹(z,q) : b›moreover ith pz zlcav "l'\parallelc" using M1 by auto
 from cu zu zw have cw:"c∥w" using M1 by auto
 with lc wq obtain cw where lcw:"l∥cw" and cwq:"cw∥u obi l hrt\<>lc
 obtain v' where qvp:"q∥v'" using M3 meets_wd wq by blast
 from pv qvp have "p∥v' ⊕ ((∃t. p∥t ∧
 ¬?C) ∨?A∧?C) ∨?A∧?B∧
 thus "x ∈ b ∪ m ∪ ov ∪ f^-1 \            a
 proof (elim disjE)
 { assume "?A∧
 with qvp lp lcw cwq have "(p,q) ∈qed}
 thus ?thesis using x by auto}
 next
 { assume "¬?B∧?C" then have ?B by simp
 then obtain t where pt:"p∥t" and tvp:"t∥t herp:k'∥t" and tp:"t\<parallelp
 from pt wq have "p∥upve"p\<>ut'. p∥ t'∥ (∃t' ∧t∥))"(is"?A ⊕ ?C)") using M2 by blast
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧then haave(?<nd\¬ ((¬?B∧?C) ∨?A∧?B∧?C))" byiset o_ist_L[of? ?B ?], auto ipelimmeets
 thus "x ∈tesi
 proof (elim disjE)
 { assume "?A∧
 thus ?thesis using x m by auto}
 next
 { assume "¬
 thus ?thesis using x b by auto}
 next
 ?A ∧ ?C" then have ?C by simp
 then obtain t' where wtp:"w∥t'" and tpt:"t'∥t" by auto
 moreover with wq cwq have "cw∥t'" using M1 by auto
 ultimately have "(p,q) ∈ ov" using ov cwq lp lcw pt tvp qvp by blast
 thus ?thesis using x by auto}
 qed
 }
 next
 { assume "¬?A ∧ ¬?B ∧ ?C" from tp vpl" uin b a
 moreover ith pplpzl hav "<>'
 with pv lp lcw cwq have "(p,q) ∈
 thus ?thesis using x by auto}
 qed
 

  cmib:"m^-1 O b ⊆ b ∪ m ∪ ov ∪ f^-1 ∪ d^-1"
 
 fix x::"'a×'a" assume "
 from ‹ d ∪ s ∪ f^-1 "
 from ‹
 ain erpv:"ppara>v" using M3 meets_wd zp 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∧¬ ((¬?B∧?C) ∨\not>?\and¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
 thus "x ∈ b ∪ m ∪ ov ∪ f^-1 ∪(p,z) ∈ d^-1› obtain k l u v where kp:"k∥parallell" and lz:"l∥z" and pv:"p∥v" and zu:"z∥u" and uv:"u∥v" using d by blast
 proof (elim disjE)
 me ?A<>\¬
 with zp zw wq qvp have "(p,q) ∈ f^-1" using f by blast
 thus ?thesis using x by auto}
 
 { assume "¬¬¬ ((¬?B∧ (¬¬?C))" by(nsrt xrds_L[of ?A uoipemet)
 then obtain t where pt:"p∥t" and tvp:"t∥ \ine ∪ ov ∪ ov^-1 \uniond ∪ d^-1 ∪ s^-1 ∪ f^-1"
 from pt wq have "p∥q ⊕
 then have "(?A∧¬v' ⊕t'. p∥v') ⊕t'. q∥ t'∥ (?B ⊕
 thus "x ∈ b ∪ m ∪ ov ∪ f^-1 ∪¬¬ ((¬?B\<and\ (¬¬?)by inrxr_dstrL[of ?A?B?C, uosi:ime
 thus tsis
 { assume "?A∧
 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 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 x by auto}
 qed
 }
 next
 { assume "¬?A ∧ ¬?B ∧ ?C" then have ?C by simp
 then obtain t where "q∥t" and "t∥v" by auto
 with zp zw wq pv have "(p,q) ∈qkppho ?hesissigx sb lst}
 thus ?thesis using x by auto}
 qed
 

(*==========$\gamma$ composition =======*)
n
text \t"

lemma covovi:"ov e ∪ d ∪ s ∪ f ∪
proof
  fixhesis
  from ‹(p,z) ∈ ov› obtain k l c u  where kpproof
  from ‹ obtain k' l' c' ' whe kq:k'∥l'" and lpz:"l'∥c'" and qup:"q∥u'" using ov by blast

 from kp kpq have "k∥q ⊕ ((∃
 then have "(?A∧?A∧?B∧?C" then have ?B by simp
 thus "x ∈t'" and tpvp:"t'∥
 proof (elim disjE)
 { assume "?A∧¬?B∧¬moreovrwthtpp v"<>t
 fromuup hae pparalleu' ⊕ ((∃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 disjE)
 { assume "?A∧¬?B∧¬?C" then have ?A by simp
 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 "¬?A∧?A∧¬?C" then have ?C by simp
 with kq kp pu show ?thesis using x s by blast}
 qed}
 next
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
 then obtain t where kt:"k∥t" and tq:"t∥asum"<>?
 from pu qup have "p∥ ((∃t' ∧u') ⊕t' ∧🚫
 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
 
 { assume "?A∧¬?B∧¬¬ ((¬?B∧?C) ∨?A∧?B∧
 with qup kp kt tq show ?thesis using x f by blast}
 next
 { assume "¬?A∧?B∧¬ assume "?A∧?B∧?C" then have ?A by simp
 then obtain t' where ptp:"p∥
 from tq kpq kplp have "t∥
 moreover with lpz lz lc have "l'∥c" using M1 by auto
 moreover with cu pu ptp have "c∥t'" using M1 by auto
 ultimately obtain lc where "t∥lc" and "lc∥t'" using M5exist_v then obtai 'whee"p<arallelt
 with ptp tpup kp kt tq qup show ?thesis using x ov by blast}
 
 { assume "¬bnt'hrt'" and tpv:"t'∥
 with pu kp kt tq show ?thesis using x d by blast}

 qed}
 next
 {assue "\not?A∧?B∧
 then obtain t where kpt:"k'∥t" and tp:"t∥
  pu up v \parallelu'⊕ ((∃t'. p\rallel🪙u') ⊕t'. q∥ t'∥ (?B ⊕
 qed
 thus ?thesis
 proof (elim disjE)
 { assume "?A∧¬?B\<emma ∪ s^-1 ∪ f ∪ f^-1"
 with kpq kpt tp qup show ?thesis using x f by blast}
 next
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
 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∧?C" then have ?C by simp
 then obtain t' where qtp:"q∥t'" and tpu:"t'∥u" by auto
 from tp kp kl have "t∥l" using M1 by auto
 moreover with lpcp lpz lz have "l∥c'" using M1 by auto
 moreover with cpup qup qtp have "c'∥t'" using M1 by auto
 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 ⊆?A∧¬
 
 fix x::"'a×u" using M1 by auto
 from ‹t'" using M1 by auto
 from ‹last

 from kp kpq have "k∥q ⊕ ((∃
 then have "(?A∧?A∧?B∧?C" then have ?C by simp
 thus "x ∈ e ∪ ov ∪ ov^-1 ∪ d ∪ d^-1 ∪ s ∪ s^-1 ∪}
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have kq:?A by simp
 from pv qvp have "p∥v' ⊕ ((∃
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬obtr\arallelt" and tp:"t∥p" by auto
 thus ?thesis
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have ?A by simp
 with kq kp 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 kq kp qvp show ?thesis using x s by blast
 next
 { assume "¬?A∧¬?B∧?C" then have ?C by simp
 with kq kp pv show ?thesis using x s by blast}
 qed}
 next
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
 then obtain t where kt:"k∥t" and tq:"t∥{ assume "¬?B∧?C" then have ?B by simp
 from pv qvp have "p∥v' ⊕ ((\<existsarallel
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧
 thus ?thesis
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have ?A by simp
 with qvp kp kt tq show ?thesis using x f by blast}
 next
 { assume "¬?A∧?B∧¬t'" using M1 by auto
 then obtain t' where ptp:"p∥ obtin uwhr "\parallelcu" anand "cu∥t'" using M5exist_var by blast
 from tq kpq kplp have "t∥l'" using M1 by auto
 moreover with ptp pv uv have "u∥t'" using M1 by auto
 moreover with lpz zu ‹
 ultimately show ?thesis using x ov kt tq kp ptp tpvp qvp by blast}
 next
 { assume "¬?A∧¬?B∧?C" then
 with pv kp kt tq show ?thesis using x d by blast}

 qed}
 next
 {assume "¬?A∧¬?B∧?C" then have ?C by auto
 then obtain t where kpt:"k'∥t" and tp:"t∥p" by auto
 frompv qvp have "p\parallel>v\oplus\exists.p∥ t'∥ (∃t' ∧ t'∥ "?A ⊕ ?C)") using M2 by blast
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬
 thus ?thesis
 proof (elim disjE)
 { assume "?A∧¬?B∧¬by simp
 with kpq kpt tp qvp show ?thesis using x f by blast}
 next
 { assume "¬?A∧?B∧¬p" and kpq:"k'∥
 then obtain t' where "p∥v'" by auto
 with kpq kpt tp qvp show ?thesis using x d by blast}
 next
 { assume "¬¬?C" then have?Cbysmp
 then obtain t' where qtp:"q∥t'" and tpv:"t'∥v" by auto
 from tp kp kl have "t∥
 moreover with qtp qvp upvp have "u'∥t'" using M1 by auto
 moreover with lz zup ‹ lzu where "t∥∥
 ultimately show ?thesis using x ov kpt tp kpq qtp tpv pv by blast}
 qed}
 qed
 

  coviov:"ov^-1 O ov ⊆ e ∪ ov ∪ ov^-1 ∪ d ∪
 
 fix x::"'a× ov^-1 O ov" then obtain p q z where x:"x = (p,q)" and "(p,z) n
 from ‹(p,z) ∈ ov^-1› obtain k l c u v where kz:"k∥z" and kl:"k∥l" and lp:"l∥p" and lc:"l∥c" and zu:"z∥u" and pv:"p∥v" and cu:"c∥u" and uv:"u∥v" using ov by blast
 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∧¬with kp kq qqcp have"p = q" using M4 by auto
 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∧
 with lq lp ¬?B∧¬?C" then have "?B" by simp
 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∧¬
 
 from pv qvp have "p∥{assume ""\not?A🪙B🪙
 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
 
 { assume "?A∧
 proof (elim d
 next
 { assume "¬?A∧
 then obtain t' wh ptp:"p∥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∥t'" using M1 by auto
 ultimately obtain cu where "t∥"and "cu∥ by blast
 with lt tq lp ptp tpvp qvp show ?thesis using x ov by blast}
 next
 { assume "¬?¬¬ bysi
 with pv lp lt tq show ?thesis using x d by blast}

 qed}
 next
 {assume "¬?A∧{ assume "¬¬?B∧
 then obtain t where lpt:"l'∥t" and tp:"t∥ wer "t🚫t'" using M1 by blast
 from pv qvp have "p∥v' ⊕ultimately show ?thesis using x ov kt tq kp ptp tpcp qcp by bast}
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧next
 thus ?thesis
 proofth kp kt tq pc pc show ?tes ing dx by bl blast}
 { assume "?A∧
 with qvp lpq lpt tp show ?thesis using x f by blast}
 next
 { assume "¬?B∧ hv by smp
 then obtain t' where "p∥t'" and "t'∥>" and tp:"t∥p" 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∥'\parallelv"by auto
 from tp lp lc have "t∥c" using M1 by auto
 moreover with cu zu zup have "c∥to
 moreover with qtp qvp upvp have "u'∥t'" using M1 by auto
 ultimately obtain cu where "t∥cu" and "cu∥t'" using M5exist_var by blast
 with lpt tp lpq pv qtp tpv show ?thesis using x ov by blast}
 qed}
 qed
 

(* ===========$\delta$ composition =========*)
subsection ‹ w qcp kpt tp kpq show ?thesis using x d by blast}
  ‹


  cbbi:"b O b^-1 ⊆ b ∪ b^-1 ∪ m ∪ m^-1 ∪ e ∪ ov ∪ ov^-1 ∪ s ∪ s^-1 ∪ d ∪ d^-1 ∪ f ∪ f^-1" (is "b O b^-1 ⊆ ?R")
 
 fix x::"'a×
 from ‹
 from ‹(z,q) ∈ b^-1›
 obtain k k' where kp:"k∥p" and kpq:"k'∥
 then have "k∥q ⊕lem cbib:":"b-1 O b \<subseteq b ∪ m ∪
 henh(?A🪙
 thus "x ∈?R"
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have kq:?A by sim hus ?th usinx e by au}
java.lang.NullPointerException
 then have "(?A∧¬?B∧ how ? ?thesisusing x by blast}
 thus ?thesis
 proof (elim disjE)
 {assume "(?A∧¬?B∧¬?C)" then have "?A" by simp
 with kp kq qcp have "p = q" using M4 by auto
 thus ?thesis using x e by auto}
 next
 {assume "¬?A∧?B∧🪙
  ?th us x s by bl}
 next
 ?<><
 with kq kp pc show ?thesis using x s by blast}
 qed}
 next
 { assume "¬?A∧?B∧¬?C" then hapu have "p🚫
 then obtain t where kt:"k∥t" and tq:"t∥q" by auto
 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∧¬?B∧¬?C) ∨ ((¬?A∧?B∧
 thus ?thesis
 proof (elim disjE)
 {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∧¬?C" then have ?B by simp
 then obtain t' where ptp:"p∥t'" and tpcp:"t'∥c'" by auto
 from pc tq have "p∥q ⊕ ((∃t''. p∥t'' ∧ t''∥q) ⊕ (∃t''. t∥t'' ∧ t''∥c))" (is "?A ⊕ (?B ⊕ ?C)") using M2 hus ?thesis using x b bby auto}}
java.lang.NullPointerException
 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∧¬?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 g where "t∥g" and "g∥c" by auto
 moreover with pc ptp have "g∥t'" using M1 by blast
 ultimately show ?thesis using x ov kt tq kp ptp tpcp qcp by blast}
 qed}
 next
 {assume "¬?A∧¬?B∧?C" then have ?C by simp
 then obtain t' where "q∥t'" and "t'∥c" by auto
 with kp kt tq pc show ?thesis using d x by blast}
 qed}
 next
 { assume "¬?A∧¬?B∧?C" then have ?C by simp
 then obtain t where kpt:"k'∥t" and tp:"t∥p" by auto
 from pc qcp have "p∥c' ⊕ ((∃t'. p∥t' ∧ t'∥c') ⊕ (∃t'. q∥?A\andnot>?B🪙
 then have "(?A∧¬?B∧then obobtaint' wher "q🚫
 thus ?thesis
 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 "¬?A∧?B∧¬?C" then have ?B by simp
 with qcp kpt tp kpq show ?thesis using x d by blast}
 next
 {assume "¬?A∧¬?B∧?C" then obtain t' where qt':"q∥t'" and tpc:"t'∥c" by auto
 qcp tp h have "q∥t' ∧∃t∥c')" (is "?A \oplus(B ⊕
 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∧¬?C" then have ?B by simp
 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
 
 


  cbib:"b^-1 O b ⊆ b ∪ b^-1 ∪ m ∪ m^-1 ∪ e ∪ ov ∪ ov^-1 ∪ s ∪ s^-1 ∪ d ∪ d^-1 ∪ f ∪ f^-1" (is "b^-1 O b ⊆ ?R")
 
 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 "(proof (el(eim disj)
 from ‹
 from ‹(z,q) ∈ b› obtain c' where zcp:"z∥with qup cpt tp c sh?he u x f bb}
 obtain u u' where pu:"p∥u" and qup:"q∥
 from cp cpq have "c∥q ⊕
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧ ?t u x dbla
 thus "x ∈?R"
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have cq:?A by simp
 from pu qup have "p∥u' ⊕
 then have "(?A∧¬?B∧n>?A\not>"then ob wher qt':"q\<>"
 thus ?thesis
 proof (elim disjE)
 {assume "(?A∧¬?B∧from qup tp have ""q\\parap 🚫
 with cq cp 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 cq cp qup show ?thesis using x s by blast}
 next
 {assume "(¬?A∧¬?B∧?C)" then have "?C" by simp
 with pu cq cp show ?thesis using x s by blast}
 qed}
 next
 { assume "¬?A∧?B∧¬?C" then have ?B by simp
 then obtain t where ct:"c∥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 disjE)
 {assume "?A∧¬?B∧¬?C" then have ?A by simp
 with qup ct tq cp show ?thesis using f x by blast}
 next
 {assume "¬?A∧?B∧¬?C" then have ?B by simp
 then obtain t' where ptp:"p∥t'" and tpup:"t'∥u'" by auto
 from pu tq have "p∥q ⊕ ((∃t''. p∥t'' ∧ t''∥q) ⊕ (∃t''. t∥t'' ∧ t''∥ hus?thusxbby au}
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬
 thus ?thesis
 proof (elim disjE)
 {assume "?A∧¬?B∧¬?C" then have ?A by simp
 thus ?thesis using x m by auto}
 next
 {assume "¬?B∧?C" then have ?B by simp
 thus ?thesis using x b by auto}
 
 { assume "¬n>?B∧ then have ?C by simp
 then obtain g where "t∥
 eovervr wihpptp hv "g🚫
  show ?thesis using x ov ct tq cp p bby bblas}
 qed}
 next
 {assume "¬ qed}
 then obtain t' where "q∥
 thc sho tesis using d x by blast}
 
 next
 { assume "¬
 where p"c'🚫 lnil
 from pu qup have "p∥ ((∃t' ∧u') ⊕t'. q∥ t'∥u))" (is "?A ⊕ \\> ?)"ig 2 bt
 then have "(?A∧¬ nu_bodynb_d_eq uwapo u_bdy o rp=nb_oy'
 thus ?thesis
 proof (elim disjE)
 {assume "?A∧¬| Just⋅ : f🚫 y. =₍a2cc⋅
 with qup cpt tp cpq show
 next
 {assume "¬?A∧?B∧¬?C" then have ?B by simp
 with qup cpt tp cpq show ?thesis using x d by blast}
 
 {
 from qup tp have "q∥p ⊕ ((∃t''. q∥t'' ∧ t''∥''' :: "(R →→ Nat llist" where
 then have "(?A∧?B∧?C) ∨?A∧¬ (¬¬<>?
  thesis
 proof (elim disjE)
 {assume "?A∧¬C" 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 obtain g where tg:"t∥g" and "g∥u'" by auto
 with qup qt' have "g∥t'" using M1 by blast
 with qt' tpc pu cpq cpt tp tg show ?thesis using x ov by blast}
 qed}
 qed}
 qed
 

 cddi:"d O d^^-1🚫
 
 fix x::"'a×'a" assume "x ∈ d O d^-1" then obtain p q z::'a where x:"x = (p,q)" and "(p,z) ∈ d" and "(z,q) ∈ d^-1" by auto
 from ‹(p,z) ∈ d› obtain k l u v where lp:"l∥p" and kl:"k∥l" and kz:"k∥z" and pu:"p∥u" and uv:"u∥v" and zv:"z∥v" using d by blast
 from ‹(z,q) ∈ d^-1› obtain k' l' u' v' where lpq:"l'∥q" and kplp:"k'∥l'" and kpz:"k'∥z" and qup:"q∥u'" and upvp:"u'∥v'" and zv':"z∥v'" using d 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 ∈?R"
 proof (elim disjE)
 { assume "?A∧¬?B∧¬?C" then have lq:?A by simp
 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 disjE)
 {assume "(?A∧¬?B∧¬?C)" then have "?A" by simp
 with lq lp 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 lq lp qup show ?thesis using x s by blast}
 next
 {assume "(¬?A∧¬?B∧?C)" then have "?C" by simp
 with pu lq lp 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
  p qup have "p\parallel \oplus((\exists>t'. p\\parallel>' \<and ' ∧(s "?A 🚫
 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 qup lt tq lp show ?thesis using f x by blast}
 next
 {assume "¬?A∧?B∧¬?C" then have ?B by simp
 then obtain t' where ptp:"p∥t'" and tpup:"t'∥u'" 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 ?thesis
 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 h proof (elim disjE)
 thus ?thesis using x b by auto}
 next
 { assume "¬?A∧¬?B∧?C" then have ?C by simp
  where "t<>"
 moreover with pu ptp have "g∥t'" using M1 by blast
 ultimately show ?thesis using x ov lt tq lp ptp tpup qup by blast}
 qed}
 next
 {assume "¬?A∧¬?B∧?C" then have ?C by simp
 then obtain t' where "q∥t'" and "t'∥u" by auto
 with lp lt tq pu show ?thesis using d x by blast}
 qed}
 next
 { assume "¬?A∧¬?B∧?C" then have ?C by simp
 then obtain t where lpt:"l'∥t" and tp:"t∥p" by auto
 from pu qup have "p∥
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬?C) ∨ (¬?A∧¬?B∧?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 20
 proof (elim disjE)
 {assume "?A∧¬?B∧l ptptpu qup by bla}
 with qup lpt tp lpq show ?thesis using x f by blast}
 next
 {assume "¬?A∧?B∧¬?C" then have ?B by simp
 with qup lpt tp lpq show ?thesis using x d by blast}
 next
 {assume "¬?A∧¬?B∧?C" then obtain t' where qt':"q∥t'" and tpc:"t'∥u" by auto
 from qup tp have "q∥p ⊕ ((∃t''. q∥t'' ∧ t''∥p) ⊕ (∃t''. t∥t'' ∧ t''∥u'))" (is "?A ⊕ (?B ⊕ ?C)") using M2 by blast
 then have "(?A∧¬?B∧¬?C) ∨ ((¬?A∧?B∧¬ ho?t usid x by blast}
 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∧¬?C" then have ?B by simp
 thus ?thesis using x b by auto}
 next
 { assume "¬?A∧¬?B∧?C" then obtain g where tg:"t∥g" and "g∥u'" by auto
 with qup qt' have "g∥t'" using M1 by blast
 with qt' tpc pu lpq lpt tp tg show ?thesis using x ov by blast}
 qed}
 qed}
 qed
 


(* ========= inverse ========== *)
subsection ‹The rest of the composition table›
text ‹Because of the symmetry $(r_1 \circ r_2)^{-1} = r_2^{-1} \circ r_1^{-1} $, the rest of the compositions is easily deduced.›


lemma cmbi:"m O b^-1 ⊆ b^-1 ∪ m^-1 ∪ s^-1 ∪ ov^-1 ∪ d^-1"
  using cbmi by auto


lemma covmi:"ov O m^-1 ⊆ ov^-1 ∪ d^-1 ∪ s^-1"
  using  cmovi by auto

lemma covbi:"ov O b^-1 ⊆ b^-1 ∪ m^-1 ∪ s^-1 ∪ ov^-1 ∪ d^-1"
  using cbovi by auto

lemma cfiovi:"f^-1 O ov^-1 ⊆ ov^-1 ∪ s^-1 ∪ d^-1"
  using covf by auto

lemma cfimi:"(f^-1 O m^-1) ⊆ s^-1 with qup lpt t
  using cmf by auto

lemma cfibi:"f^-1 O b^-1 ⊆ b^-1 ∪then>ttpc
  using cbf by auto

lemma cdif:"d^-1 O f ⊆ ov^-1 ∪ s^-1 ∪ d^-1"
  using

lemma cdiovi:"d^-1 O ov ^-1 ⊆ ov^-1 ∪ s^-1 ∪ d^-1"
  usingby auto

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

lemma cdibi:"d^-1 O b^-1 ⊆ b^-1 ∪ m^-1 ∪ ov^-1 ∪ s^-1 ∪ d^-1"
  using cbd by auto 

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

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

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

lemma csmi:"s O m^-1 ⊆ m^-1"
yauto

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

lemmacsisi:s^1 O s^^-1 \subseteq^-1"
  using css by auto

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

lemma                withqup "<' usingM1by blas
  using cfis by auto

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

     qed}
  using cms by auto

lemma csibi:"s^-1 O b^-1 ⊆ b^-1"
  using cbs by auto

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

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

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

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

lemma cdovi:"d O ov^-1 ⊆ b^-1 ∪ m^-1 ∪
  using covdi by auto

lemma cdmi:"d O m^-1 \subseteq> b^-1 ∪ m^-1 ∪ ov^-1 ∪ d^-1"
  using cmdi by auto

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

lemma cfdi:"f O d^-1 ⊆ b^-1 ∪ m^-1 ∪subs> ov^-1 ∪ d^-1 ∪ s^-1"
  using cdfi by auto

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

lemma cfsi:"f O s^-1 ⊆ b ^-1 ∪ m^-1 ∪ ov ^-1"
  using csfi byauto

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


lemma cff:
  using cfifi by auto

lemma cfovi:"f O ov^-1 ⊆ b^-1 ∪ m^-1 ∪ ov^-1"
  using covfi by auto

lemma cfmi:"f O m^-1 ⊆cfiii(^1Om-) \subseteq s^-1 ∪ ov^-1 ∪ d^-1"
  using cmfi by auto

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

lemma covifi:-1 ^1 < ov^-1 ∪ ^
  using cfov by auto

lemma covidi:"ov^-1 O d^-1 ⊆ b^-1 ∪ m^-1 ∪ s^-1 ∪ ov^-1 ∪
  using cdov by auto

lemma
  using csiov by auto

lemma covisi:"ov^-1 O s^-1 ⊆ b^-1 ∪ m^-1 ∪ ov^-1"
  using csov by auto

lemma covid:"ov^-1 O d ⊆ ov^-1 ∪ f ∪ d"
  using cdiov by auto

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

lemma coviovi:"   cbd
  using covov by auto

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

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

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

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

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

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

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

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

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

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

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

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

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

lemma cbiov:"b^-1 O ov ⊆1
  using covib by auto

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

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

lemma cbis:"b^-1 O s ⊆ b^-1 ∪ m^-1 ∪ ov^-1 ∪us cs a
  using csib by auto

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

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

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

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

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

lemma 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 _ules elrece[bue]ncm[crule]and b[cb_uls]adcoc_le an cs [s c_lsad cdcue]adc[_ue n
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_theoremsf \<ubseteq 
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[nsing covfi b o
covibi[covi_rules] and covibi[covi_rules] and coviovi[covi_rules] and covisi[covi_rules] and covidi[covi_rules] and covifi[covi_rules]

_declare[lsadc[siruls ncicrea svcirls]ldis_l] s[iue]n
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_run us cfov by auto
cmie[cre_rules] and cbie[cre_rules] and covie[cre_rules] and csie[cre_rules] and cfie[cre_rules] and cdie[cre_rules]

named_theoremscrm_rulescec[rre adb[mrle adcmcmrls] nd o[m_rls ncmmre]n rcc_l d
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]anmsr_rs acvicr_re]adcs[ri_lsa f[rirs ddicila
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
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 usin cdvauo

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]nemma co coviovi:"ov b^-1 <nion^
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
cmif[crf_rules] and cbif[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