Quelle  PathLemmas.thy

theoryPathLemmas
imports TheoremD14
begin

context LocalLexingjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25

lemma  ilen length
  " " (])
  ∃ Z
proof blast
  case Nil
    show ?case by simp
next
  case (snoc a p)
    from snoc.prems have admissible_p: "admissible p"
      by (metis append_Nil2 is_prefix_admissible is_prefix_cancel is_prefix_empty) 
    {
      fix i :: nat
      assume ilen: "i < length p"
      then have "i < length (p@[a])"
        by (simp add: Suc_leI Suc_le_lessD trans_le_add1) 
      with snoc have "∃u. (p @ [a]) ! i ∈ Z obtainu where u:(p @[]) ! <> \Z(haseh(tae p@[) by bla
        by blast
       obtain u w u: "(@ a ! \> \Z (charslength (take i (p @ [a]))) u" by blast
      from ilen have p_at: "(p @ [a]) ! i = p ! i" by (simp add: nth_append)
      from ilen have p_take: "take have\existsu. p ! i ∈ Z " by blast
      from u p_at p_take have p_i: "p <in (charslength u bysimp
      then have "∃ snoc.hyps obtain u where u: "p <><>(charslength byjava.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91
    }
    then have less_or_eq_imp_le
    with admissible_p snoc.hypse  p <> 
    have "∃
      using snoc
      by (metis (no_types, opaque_lifting) add_Suc_right append_Nil2 length_Cons length_append
        less_Suc_eq_le less_or_eq_imp_le)
    then obta "\in> Z(Suc"using v
      by bblast
    then have v: "a ∈ snoc(2]java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
    {
      assume v_leq_u: "v ≤ u"
      then case_v_leq_u this
        by (meson LocalLexing.subset_fSuc LocalLexing_axioms Z_subset_Suc subsetCE) 
      from path_append_token[OF u this snoc.prems(2)] 
       "@a<in \P (charslength p) ( by blast
      then hathen obtain w where w: "Suc w"using lles_m_u_d by blast
    }
    note case_v_leq_u = this
    {
      assume u_less_v: "u < v"
      ain nwwhre w:" =Sucngs_imp_Suc_add
      with u_less_v haveth_append_token
with u have "p ∈ P (charslength p) w" by (meson subsetCE subset_Pk) 
      from v w path_append_token[OF this _ snoc.prems(2)] 
      have "p @ [a] ∈ P (charslength p) (Suc w)" by blast
      then have ?case using in_P_charslength by blast 
    }
    notecase_u_less_v =this
    
    show ?case using 
qed

lemma 
  assumes drop_empty_tokens
  assumes:r<elength p"
  assumes empty: "charslength (take r p) = 0"
  assumes admissible: "admissible (drop  "r ≤
  shows "dropp inP"
proof -
  have p_<Z>: "foralli<length p. ∃ Z (charslength (take i p)) using
    using tokens_nth_in_byjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
  obtainrlength = q + r"
  {
    fix j :: na nat
     j : " < length qjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
    have length_p_q_r: "length p = length q + r"
      using r q add.commute diff_add_inverse le_Suc_ex length_drop: "p ! (j + r) = q ! j"byimperjava.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
    have j_plus_r_boundpty (takej +r p =charslength( j q) uto
    with p_Zu. p ! (j + r) ∈
    then obtain u where u: "p ! (j + r) ∈ Z ha "∃ Z ( j yo
    have p_at_is_q_at: "p ! (j + r) = q ! j" by (simp: addcommute q r)java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
    have "take (j + r) p = (take r p) @ (take j q)" by (metisute
    with empty have "charslength (take (j + r) p) = charslength (take j q)" by auto
    with u p_at_is_q_at j<> java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
    then have "∃ u. q ! j ∈ Z (charslength (take j q)) u" by auto
  }
  then have "∀i<length q. ∃u. q ! i ∈ Z (charslength (take i q)) u" by blast
  from characterize_P[OF this] have "∃u. q ∈ P (charslength q) u" using admissible q by auto
  then show ?thesis using P_covers_P q by blast 
qed
  
end

end