Impressum PathLemmas.thy

theory 
importsTheoremD14
beginLocalLexing begin

context LocalLexing begin

lemma characterize_P:
  "(∀ i < length p. ∃u. p ! i ∈ Z (charslength (take i p)) u) ==> admissible p ==>
  ∃ u. p ∈ P (charslength p) u"
proof (induct p rule: rev_induct)
  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
      assumeilen: "i <lep"
      thenhavei < lengthp@[a)
        by (simp add: Suc_leI Suc_le_lessD trans_le_add1) 
      with snoc have "∃u. (p @ [a]) ! i ∈ (charslength (take i (p @ [a]))) u"
        byjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
      then u re " @ [a a] i\in<Z> (hrlngt (take i( [a])) u" last
      from ilen have p_at: "(p @ [a]) ! i = p ! i" by (simp add: nth_append)  
      from ilen have p_take: "take i (p @ [a]) = take i p" by (simp add: less_or_eq_imp_le) 
      fromuherep [])  i <in <> 
      then "\<>  (charslength (take i p))u" ast
    }
    then have "∀ i < length p. ∃ ! i \in> Z (take i p)) u"  
    with admissible_psobtainhere"\in \P (ccharslength p) u" blast
    have "∃u. (p @ [a]) ! (length p) ∈ Z (charslength (take (length p) (p @ [a]))) u"
      using snoc
      by (metis (no_types, opaque_lifting) add_Suc_right append_Nil2 length_Cons length_append 
        Suc_eq_le) 
    then obtain v where "(p @ [a]) ! (length p) ∈ obtain u where u: "inP (charslength p) u" by blast
      by blast
    then have v: "a ∈
    {
      assume v_leq_u less_or_eq_imp_le
      then havea <inZ>>(charslength p) ( u) using
        by (byst
      from path_append_token[OF u this.prems2) 
      have "p @ [a] ∈ P (charslength p) (Suc u)" by blast
      then have ?case using in_P_charslength 
    }
    note =java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
    {
      assume u_less_v      have " [a] \in>P(uc u)"last
      thene  v = Suc w"using ssipScadby bst
      with u_less_v have "u ≤
      with u have "then obtai where w: "= w" using less_imp_by blast
      from v w path_a[OF this _ snoc.prems(2)]
      have "p @ [a] ∈ <      with
      then
    } case_u_less_v this
    note case_u_less_v = this
    
    show ?case using case_v_leq_u
qed

lemma: 
  assumes r " \<> 
  assumes r: r> lngth p"
  assumes empty: "charslength (take r p) = 0"
  assumes admissible: "admissible (drop r p)"
  shows r <> \:∀u. p ! i ∈)u" p
proof -
  have p_ZZ by blast
    using tokens_nth_in_Z by blast
  obtain q where q: "q = drop r p" by blast
  {
    fix j :: nat
    assume j : "j < length q"
    have length_p_q_r: " p length
      using
    havej:at
    assume j< length"
    then obtain u where u: "p ! (j + r) ∈
    have p_at_is_q_atq" by (s add: add.commute q q r)
    have "take (j + r) p = (take r p) @ (take j q)" by (metis add.commute q take_add)
    with empt have "charslength (  )p) charslength (ake)"by auto
    with u p_at_is_q_at have "q ! j ∈ have "∃ Z (charslength (take (j + r) p)) u" by blast
    thenve<> u. q ! j ∈ (charslengthtakeq)) u" by auto
  }
  then have "∀i<length q. ∃ add .commute 
  from characterize_P[OF this] have "∃u. q ∈ add.commu q take_add)
  then show ?thesis using P have "q ! inZ (charslength (take j q)) u" by simp
qed
  
end

end