Theory Slicing.AuxLemmas

section ‹Auxiliary lemmas›

theory AuxLemmas imports Main begin

abbreviation "arbitrary == undefined"

text ‹Lemmas about left- and rightmost elements in lists›

lemma leftmost_element_property:
  assumes "∃x ∈ set xs. P x"
  obtains zs x' ys where "xs = zs@x'#ys" and "P x'" and "∀z ∈ set zs. ¬ P z"
proof(atomize_elim)
  from ‹∃x ∈ set xs. P x› 
  show "∃zs x' ys. xs = zs @ x' # ys ∧ P x' ∧ (∀z∈set zs. ¬ P z)"
  proof(induct xs)
    case Nil thus ?case by simp
  next
    case (Cons x' xs')
    note IH = ‹∃a∈set xs'. P a
      ⟹ ∃zs x' ys. xs' = zs@x'#ys ∧ P x' ∧ (∀z∈set zs. ¬ P z)›
    show ?case
    proof (cases "P x'")
      case True
      then have "(∃ys. x' # xs' = [] @ x' # ys) ∧ P x' ∧ (∀x∈set []. ¬ P x)" by simp
      then show ?thesis by blast
    next
      case False
      with ‹∃y∈set (x'#xs'). P y› have "∃y∈set xs'. P y" by simp
      from IH[OF this] obtain y ys zs where "xs' = zs@y#ys"
        and "P y" and "∀z∈set zs. ¬ P z" by blast
      from ‹∀z∈set zs. ¬ P z› False have "∀z∈set (x'#zs). ¬ P z" by simp
      with ‹xs' = zs@y#ys› ‹P y› show ?thesis by (metis Cons_eq_append_conv)
    qed
  qed
qed



lemma rightmost_element_property:
  assumes "∃x ∈ set xs. P x"
  obtains ys x' zs where "xs = ys@x'#zs" and "P x'" and "∀z ∈ set zs. ¬ P z"
proof(atomize_elim)
  from ‹∃x ∈ set xs. P x›
  show "∃ys x' zs. xs = ys @ x' # zs ∧ P x' ∧ (∀z∈set zs. ¬ P z)"
  proof(induct xs)
    case Nil thus ?case by simp
  next
    case (Cons x' xs')
    note IH = ‹∃a∈set xs'. P a
      ⟹ ∃ys x' zs. xs' = ys @ x' # zs ∧ P x' ∧ (∀z∈set zs. ¬ P z)›
    show ?case
    proof(cases "∃y∈set xs'. P y")
      case True
      from IH[OF this] obtain y ys zs where "xs' = ys @ y # zs"
        and "P y" and "∀z∈set zs. ¬ P z" by blast
      thus ?thesis by (metis Cons_eq_append_conv)
    next
      case False
      with ‹∃y∈set (x'#xs'). P y› have "P x'" by simp
      with False show ?thesis by (metis eq_Nil_appendI)
    qed
  qed
qed


text ‹Lemma concerning maps and ‹@››

lemma map_append_append_maps:
  assumes map:"map f xs = ys@zs"
  obtains xs' xs'' where "map f xs' = ys" and "map f xs'' = zs" and "xs=xs'@xs''"
by (metis append_eq_conv_conj append_take_drop_id assms drop_map take_map that)


text ‹Lemma concerning splitting of @{term list}s›

lemma  path_split_general:
assumes all:"∀zs. xs ≠ ys@zs"
obtains j zs where "xs = (take j ys)@zs" and "j < length ys"
  and "∀k > j. ∀zs'. xs ≠ (take k ys)@zs'"
proof(atomize_elim)
  from ‹∀zs. xs ≠ ys@zs›
  show "∃j zs. xs = take j ys @ zs ∧ j < length ys ∧ 
               (∀k>j. ∀zs'. xs ≠ take k ys @ zs')"
  proof(induct ys arbitrary:xs)
    case Nil thus ?case by auto
  next
    case (Cons y' ys')
    note IH = ‹⋀xs. ∀zs. xs ≠ ys' @ zs ⟹
      ∃j zs. xs = take j ys' @ zs ∧ j < length ys' ∧ 
      (∀k. j < k ⟶ (∀zs'. xs ≠ take k ys' @ zs'))›
    show ?case
    proof(cases xs)
      case Nil thus ?thesis by simp
    next
      case (Cons x' xs')
      with ‹∀zs. xs ≠ (y' # ys') @ zs› have "x' ≠ y' ∨ (∀zs. xs' ≠ ys' @ zs)"
        by simp
      show ?thesis
      proof(cases "x' = y'")
        case True
        with ‹x' ≠ y' ∨ (∀zs. xs' ≠ ys' @ zs)› have "∀zs. xs' ≠ ys' @ zs" by simp
        from IH[OF this] have "∃j zs. xs' = take j ys' @ zs ∧ j < length ys' ∧
          (∀k. j < k ⟶ (∀zs'. xs' ≠ take k ys' @ zs'))" .
        then obtain j zs where "xs' = take j ys' @ zs"
          and "j < length ys'"
          and all_sub:"∀k. j < k ⟶ (∀zs'. xs' ≠ take k ys' @ zs')"
          by blast
        from ‹xs' = take j ys' @ zs› True
          have "(x'#xs') = take (Suc j) (y' # ys') @ zs"
          by simp
        from all_sub True have all_imp:"∀k. j < k ⟶ 
          (∀zs'. (x'#xs') ≠ take (Suc k) (y' # ys') @ zs')"
          by auto
        { fix l assume "(Suc j) < l"
          then obtain k where [simp]:"l = Suc k" by(cases l) auto
          with ‹(Suc j) < l› have "j < k" by simp
          with all_imp 
          have "∀zs'. (x'#xs') ≠ take (Suc k) (y' # ys') @ zs'"
            by simp
          hence "∀zs'. (x'#xs') ≠ take l (y' # ys') @ zs'"
            by simp }
        with ‹(x'#xs') = take (Suc j) (y' # ys') @ zs› ‹j < length ys'› Cons
        show ?thesis by (metis Suc_length_conv less_Suc_eq_0_disj)
      next
        case False
        with Cons have "∀i zs'. i > 0 ⟶ xs ≠ take i (y' # ys') @ zs'"
          by auto(case_tac i,auto)
        moreover
        have "∃zs. xs = take 0 (y' # ys') @ zs" by simp
        ultimately show ?thesis by(rule_tac x="0" in exI,auto)
      qed
    qed
  qed
qed

end