(*  Title:      HOL/Imperative_HOL/ex/Imperative_Quicksort.thy

    Author:     Lukas Bulwahn, TU Muenchen

*)

header {* An imperative implementation of Quicksort on arrays *}

theory Imperative_Quicksort

imports Imperative_HOL Subarray Multiset Efficient_Nat

begin

text {* We prove QuickSort correct in the Relational Calculus. *}

definition swap :: "nat array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> unit Heap"

where

  "swap arr i j =

     do {

       x \<leftarrow> Array.nth arr i;

       y \<leftarrow> Array.nth arr j;

       Array.upd i y arr;

       Array.upd j x arr;

       return ()

     }"

lemma crel_swapI [crel_intros]:

  assumes "i < Array.length h a" "j < Array.length h a"

    "x = get_array a h ! i" "y = get_array a h ! j"

    "h' = Array.update a j x (Array.update a i y h)"

  shows "crel (swap a i j) h h' r"

  unfolding swap_def using assms by (auto intro!: crel_intros)

lemma swap_permutes:

  assumes "crel (swap a i j) h h' rs"

  shows "multiset_of (get_array a h') 

  = multiset_of (get_array a h)"

  using assms

  unfolding swap_def

  by (auto simp add: Array.length_def multiset_of_swap dest: sym [of _ "h'"] elim!: crel_bindE crel_nthE crel_returnE crel_updE)

function part1 :: "nat array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat Heap"

where

  "part1 a left right p = (

     if (right \<le> left) then return right

     else do {

       v \<leftarrow> Array.nth a left;

       (if (v \<le> p) then (part1 a (left + 1) right p)

                    else (do { swap a left right;

  part1 a left (right - 1) p }))

     })"

by pat_completeness auto

termination

by (relation "measure (\<lambda>(_,l,r,_). r - l )") auto

declare part1.simps[simp del]

lemma part_permutes:

  assumes "crel (part1 a l r p) h h' rs"

  shows "multiset_of (get_array a h') 

  = multiset_of (get_array a h)"

  using assms

proof (induct a l r p arbitrary: h h' rs rule:part1.induct)

  case (1 a l r p h h' rs)

  thus ?case

    unfolding part1.simps [of a l r p]

    by (elim crel_bindE crel_ifE crel_returnE crel_nthE) (auto simp add: swap_permutes)

qed

lemma part_returns_index_in_bounds:

  assumes "crel (part1 a l r p) h h' rs"

  assumes "l \<le> r"

  shows "l \<le> rs \<and> rs \<le> r"

using assms

proof (induct a l r p arbitrary: h h' rs rule:part1.induct)

  case (1 a l r p h h' rs)

  note cr = `crel (part1 a l r p) h h' rs`

  show ?case

  proof (cases "r \<le> l")

    case True (* Terminating case *)

    with cr `l \<le> r` show ?thesis

      unfolding part1.simps[of a l r p]

      by (elim crel_bindE crel_ifE crel_returnE crel_nthE) auto

  next

    case False (* recursive case *)

    note rec_condition = this

    let ?v = "get_array a h ! l"

    show ?thesis

    proof (cases "?v \<le> p")

      case True

      with cr False

      have rec1: "crel (part1 a (l + 1) r p) h h' rs"

        unfolding part1.simps[of a l r p]

        by (elim crel_bindE crel_nthE crel_ifE crel_returnE) auto

      from rec_condition have "l + 1 \<le> r" by arith

      from 1(1)[OF rec_condition True rec1 `l + 1 \<le> r`]

      show ?thesis by simp

    next

      case False

      with rec_condition cr

      obtain h1 where swp: "crel (swap a l r) h h1 ()"

        and rec2: "crel (part1 a l (r - 1) p) h1 h' rs"

        unfolding part1.simps[of a l r p]

        by (elim crel_bindE crel_nthE crel_ifE crel_returnE) auto

      from rec_condition have "l \<le> r - 1" by arith

      from 1(2) [OF rec_condition False rec2 `l \<le> r - 1`] show ?thesis by fastsimp

    qed

  qed

qed

lemma part_length_remains:

  assumes "crel (part1 a l r p) h h' rs"

  shows "Array.length h a = Array.length h' a"

using assms

proof (induct a l r p arbitrary: h h' rs rule:part1.induct)

  case (1 a l r p h h' rs)

  note cr = `crel (part1 a l r p) h h' rs`

  show ?case

  proof (cases "r \<le> l")

    case True (* Terminating case *)

    with cr show ?thesis

      unfolding part1.simps[of a l r p]

      by (elim crel_bindE crel_ifE crel_returnE crel_nthE) auto

  next

    case False (* recursive case *)

    with cr 1 show ?thesis

      unfolding part1.simps [of a l r p] swap_def

      by (auto elim!: crel_bindE crel_ifE crel_nthE crel_returnE crel_updE) fastsimp

  qed

qed

lemma part_outer_remains:

  assumes "crel (part1 a l r p) h h' rs"

  shows "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array (a::nat array) h ! i = get_array a h' ! i"

  using assms

proof (induct a l r p arbitrary: h h' rs rule:part1.induct)

  case (1 a l r p h h' rs)

  note cr = `crel (part1 a l r p) h h' rs`

  show ?case

  proof (cases "r \<le> l")

    case True (* Terminating case *)

    with cr show ?thesis

      unfolding part1.simps[of a l r p]

      by (elim crel_bindE crel_ifE crel_returnE crel_nthE) auto

  next

    case False (* recursive case *)

    note rec_condition = this

    let ?v = "get_array a h ! l"

    show ?thesis

    proof (cases "?v \<le> p")

      case True

      with cr False

      have rec1: "crel (part1 a (l + 1) r p) h h' rs"

        unfolding part1.simps[of a l r p]

        by (elim crel_bindE crel_nthE crel_ifE crel_returnE) auto

      from 1(1)[OF rec_condition True rec1]

      show ?thesis by fastsimp

    next

      case False

      with rec_condition cr

      obtain h1 where swp: "crel (swap a l r) h h1 ()"

        and rec2: "crel (part1 a l (r - 1) p) h1 h' rs"

        unfolding part1.simps[of a l r p]

        by (elim crel_bindE crel_nthE crel_ifE crel_returnE) auto

      from swp rec_condition have

        "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array a h ! i = get_array a h1 ! i"

        unfolding swap_def

        by (elim crel_bindE crel_nthE crel_updE crel_returnE) auto

      with 1(2) [OF rec_condition False rec2] show ?thesis by fastsimp

    qed

  qed

qed

lemma part_partitions:

  assumes "crel (part1 a l r p) h h' rs"

  shows "(\<forall>i. l \<le> i \<and> i < rs \<longrightarrow> get_array (a::nat array) h' ! i \<le> p)

  \<and> (\<forall>i. rs < i \<and> i \<le> r \<longrightarrow> get_array a h' ! i \<ge> p)"

  using assms

proof (induct a l r p arbitrary: h h' rs rule:part1.induct)

  case (1 a l r p h h' rs)

  note cr = `crel (part1 a l r p) h h' rs`

  show ?case

  proof (cases "r \<le> l")

    case True (* Terminating case *)

    with cr have "rs = r"

      unfolding part1.simps[of a l r p]

      by (elim crel_bindE crel_ifE crel_returnE crel_nthE) auto

    with True

    show ?thesis by auto

  next

    case False (* recursive case *)

    note lr = this

    let ?v = "get_array a h ! l"

    show ?thesis

    proof (cases "?v \<le> p")

      case True

      with lr cr

      have rec1: "crel (part1 a (l + 1) r p) h h' rs"

        unfolding part1.simps[of a l r p]

        by (elim crel_bindE crel_nthE crel_ifE crel_returnE) auto

      from True part_outer_remains[OF rec1] have a_l: "get_array a h' ! l \<le> p"

        by fastsimp

      have "\<forall>i. (l \<le> i = (l = i \<or> Suc l \<le> i))" by arith

      with 1(1)[OF False True rec1] a_l show ?thesis

        by auto

    next

      case False

      with lr cr

      obtain h1 where swp: "crel (swap a l r) h h1 ()"

        and rec2: "crel (part1 a l (r - 1) p) h1 h' rs"

        unfolding part1.simps[of a l r p]

        by (elim crel_bindE crel_nthE crel_ifE crel_returnE) auto

      from swp False have "get_array a h1 ! r \<ge> p"

        unfolding swap_def

        by (auto simp add: Array.length_def elim!: crel_bindE crel_nthE crel_updE crel_returnE)

      with part_outer_remains [OF rec2] lr have a_r: "get_array a h' ! r \<ge> p"

        by fastsimp

      have "\<forall>i. (i \<le> r = (i = r \<or> i \<le> r - 1))" by arith

      with 1(2)[OF lr False rec2] a_r show ?thesis

        by auto

    qed

  qed

qed

fun partition :: "nat array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat Heap"

where

  "partition a left right = do {

     pivot \<leftarrow> Array.nth a right;

     middle \<leftarrow> part1 a left (right - 1) pivot;

     v \<leftarrow> Array.nth a middle;

     m \<leftarrow> return (if (v \<le> pivot) then (middle + 1) else middle);

     swap a m right;

     return m

   }"

declare partition.simps[simp del]

lemma partition_permutes:

  assumes "crel (partition a l r) h h' rs"

  shows "multiset_of (get_array a h') 

  = multiset_of (get_array a h)"

proof -

    from assms part_permutes swap_permutes show ?thesis

      unfolding partition.simps

      by (elim crel_bindE crel_returnE crel_nthE crel_ifE crel_updE) auto

qed

lemma partition_length_remains:

  assumes "crel (partition a l r) h h' rs"

  shows "Array.length h a = Array.length h' a"

proof -

  from assms part_length_remains show ?thesis

    unfolding partition.simps swap_def

    by (elim crel_bindE crel_returnE crel_nthE crel_ifE crel_updE) auto

qed

lemma partition_outer_remains:

  assumes "crel (partition a l r) h h' rs"

  assumes "l < r"

  shows "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array (a::nat array) h ! i = get_array a h' ! i"

proof -

  from assms part_outer_remains part_returns_index_in_bounds show ?thesis

    unfolding partition.simps swap_def

    by (elim crel_bindE crel_returnE crel_nthE crel_ifE crel_updE) fastsimp

qed

lemma partition_returns_index_in_bounds:

  assumes crel: "crel (partition a l r) h h' rs"

  assumes "l < r"

  shows "l \<le> rs \<and> rs \<le> r"

proof -

  from crel obtain middle h'' p where part: "crel (part1 a l (r - 1) p) h h'' middle"

    and rs_equals: "rs = (if get_array a h'' ! middle \<le> get_array a h ! r then middle + 1

         else middle)"

    unfolding partition.simps

    by (elim crel_bindE crel_returnE crel_nthE crel_ifE crel_updE) simp

  from `l < r` have "l \<le> r - 1" by arith

  from part_returns_index_in_bounds[OF part this] rs_equals `l < r` show ?thesis by auto

qed

lemma partition_partitions:

  assumes crel: "crel (partition a l r) h h' rs"

  assumes "l < r"

  shows "(\<forall>i. l \<le> i \<and> i < rs \<longrightarrow> get_array (a::nat array) h' ! i \<le> get_array a h' ! rs) \<and>

  (\<forall>i. rs < i \<and> i \<le> r \<longrightarrow> get_array a h' ! rs \<le> get_array a h' ! i)"

proof -

  let ?pivot = "get_array a h ! r" 

  from crel obtain middle h1 where part: "crel (part1 a l (r - 1) ?pivot) h h1 middle"

    and swap: "crel (swap a rs r) h1 h' ()"

    and rs_equals: "rs = (if get_array a h1 ! middle \<le> ?pivot then middle + 1

         else middle)"

    unfolding partition.simps

    by (elim crel_bindE crel_returnE crel_nthE crel_ifE crel_updE) simp

  from swap have h'_def: "h' = Array.update a r (get_array a h1 ! rs)

    (Array.update a rs (get_array a h1 ! r) h1)"

    unfolding swap_def

    by (elim crel_bindE crel_returnE crel_nthE crel_updE) simp

  from swap have in_bounds: "r < Array.length h1 a \<and> rs < Array.length h1 a"

    unfolding swap_def

    by (elim crel_bindE crel_returnE crel_nthE crel_updE) simp

  from swap have swap_length_remains: "Array.length h1 a = Array.length h' a"

    unfolding swap_def by (elim crel_bindE crel_returnE crel_nthE crel_updE) auto

  from `l < r` have "l \<le> r - 1" by simp

  note middle_in_bounds = part_returns_index_in_bounds[OF part this]

  from part_outer_remains[OF part] `l < r`

  have "get_array a h ! r = get_array a h1 ! r"

    by fastsimp

  with swap

  have right_remains: "get_array a h ! r = get_array a h' ! rs"

    unfolding swap_def

    by (auto simp add: Array.length_def elim!: crel_bindE crel_returnE crel_nthE crel_updE) (cases "r = rs", auto)

  from part_partitions [OF part]

  show ?thesis

  proof (cases "get_array a h1 ! middle \<le> ?pivot")

    case True

    with rs_equals have rs_equals: "rs = middle + 1" by simp

    { 

      fix i

      assume i_is_left: "l \<le> i \<and> i < rs"

      with swap_length_remains in_bounds middle_in_bounds rs_equals `l < r`

      have i_props: "i < Array.length h' a" "i \<noteq> r" "i \<noteq> rs" by auto

      from i_is_left rs_equals have "l \<le> i \<and> i < middle \<or> i = middle" by arith

      with part_partitions[OF part] right_remains True

      have "get_array a h1 ! i \<le> get_array a h' ! rs" by fastsimp

      with i_props h'_def in_bounds have "get_array a h' ! i \<le> get_array a h' ! rs"

        unfolding Array.update_def Array.length_def by simp

    }

    moreover

    {

      fix i

      assume "rs < i \<and> i \<le> r"

      hence "(rs < i \<and> i \<le> r - 1) \<or> (rs < i \<and> i = r)" by arith

      hence "get_array a h' ! rs \<le> get_array a h' ! i"

      proof

        assume i_is: "rs < i \<and> i \<le> r - 1"

        with swap_length_remains in_bounds middle_in_bounds rs_equals

        have i_props: "i < Array.length h' a" "i \<noteq> r" "i \<noteq> rs" by auto

        from part_partitions[OF part] rs_equals right_remains i_is

        have "get_array a h' ! rs \<le> get_array a h1 ! i"

          by fastsimp

        with i_props h'_def show ?thesis by fastsimp

      next

        assume i_is: "rs < i \<and> i = r"

        with rs_equals have "Suc middle \<noteq> r" by arith

        with middle_in_bounds `l < r` have "Suc middle \<le> r - 1" by arith

        with part_partitions[OF part] right_remains 

        have "get_array a h' ! rs \<le> get_array a h1 ! (Suc middle)"

          by fastsimp

        with i_is True rs_equals right_remains h'_def

        show ?thesis using in_bounds

          unfolding Array.update_def Array.length_def

          by auto

      qed

    }

    ultimately show ?thesis by auto

  next

    case False

    with rs_equals have rs_equals: "middle = rs" by simp

    { 

      fix i

      assume i_is_left: "l \<le> i \<and> i < rs"

      with swap_length_remains in_bounds middle_in_bounds rs_equals

      have i_props: "i < Array.length h' a" "i \<noteq> r" "i \<noteq> rs" by auto

      from part_partitions[OF part] rs_equals right_remains i_is_left

      have "get_array a h1 ! i \<le> get_array a h' ! rs" by fastsimp

      with i_props h'_def have "get_array a h' ! i \<le> get_array a h' ! rs"

        unfolding Array.update_def by simp

    }

    moreover

    {

      fix i

      assume "rs < i \<and> i \<le> r"

      hence "(rs < i \<and> i \<le> r - 1) \<or> i = r" by arith

      hence "get_array a h' ! rs \<le> get_array a h' ! i"

      proof

        assume i_is: "rs < i \<and> i \<le> r - 1"

        with swap_length_remains in_bounds middle_in_bounds rs_equals

        have i_props: "i < Array.length h' a" "i \<noteq> r" "i \<noteq> rs" by auto

        from part_partitions[OF part] rs_equals right_remains i_is

        have "get_array a h' ! rs \<le> get_array a h1 ! i"

          by fastsimp

        with i_props h'_def show ?thesis by fastsimp

      next

        assume i_is: "i = r"

        from i_is False rs_equals right_remains h'_def

        show ?thesis using in_bounds

          unfolding Array.update_def Array.length_def

          by auto

      qed

    }

    ultimately

    show ?thesis by auto

  qed

qed

function quicksort :: "nat array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> unit Heap"

where

  "quicksort arr left right =

     (if (right > left)  then

        do {

          pivotNewIndex \<leftarrow> partition arr left right;

          pivotNewIndex \<leftarrow> assert (\<lambda>x. left \<le> x \<and> x \<le> right) pivotNewIndex;

          quicksort arr left (pivotNewIndex - 1);

          quicksort arr (pivotNewIndex + 1) right

        }

     else return ())"

by pat_completeness auto

(* For termination, we must show that the pivotNewIndex is between left and right *) 

termination

by (relation "measure (\<lambda>(a, l, r). (r - l))") auto

declare quicksort.simps[simp del]

lemma quicksort_permutes:

  assumes "crel (quicksort a l r) h h' rs"

  shows "multiset_of (get_array a h') 

  = multiset_of (get_array a h)"

  using assms

proof (induct a l r arbitrary: h h' rs rule: quicksort.induct)

  case (1 a l r h h' rs)

  with partition_permutes show ?case

    unfolding quicksort.simps [of a l r]

    by (elim crel_ifE crel_bindE crel_assertE crel_returnE) auto

qed

lemma length_remains:

  assumes "crel (quicksort a l r) h h' rs"

  shows "Array.length h a = Array.length h' a"

using assms

proof (induct a l r arbitrary: h h' rs rule: quicksort.induct)

  case (1 a l r h h' rs)

  with partition_length_remains show ?case

    unfolding quicksort.simps [of a l r]

    by (elim crel_ifE crel_bindE crel_assertE crel_returnE) auto

qed

lemma quicksort_outer_remains:

  assumes "crel (quicksort a l r) h h' rs"

   shows "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array (a::nat array) h ! i = get_array a h' ! i"

  using assms

proof (induct a l r arbitrary: h h' rs rule: quicksort.induct)

  case (1 a l r h h' rs)

  note cr = `crel (quicksort a l r) h h' rs`

  thus ?case

  proof (cases "r > l")

    case False

    with cr have "h' = h"

      unfolding quicksort.simps [of a l r]

      by (elim crel_ifE crel_returnE) auto

    thus ?thesis by simp

  next

  case True

   { 

      fix h1 h2 p ret1 ret2 i

      assume part: "crel (partition a l r) h h1 p"

      assume qs1: "crel (quicksort a l (p - 1)) h1 h2 ret1"

      assume qs2: "crel (quicksort a (p + 1) r) h2 h' ret2"

      assume pivot: "l \<le> p \<and> p \<le> r"

      assume i_outer: "i < l \<or> r < i"

      from  partition_outer_remains [OF part True] i_outer

      have "get_array a h !i = get_array a h1 ! i" by fastsimp

      moreover

      with 1(1) [OF True pivot qs1] pivot i_outer

      have "get_array a h1 ! i = get_array a h2 ! i" by auto

      moreover

      with qs2 1(2) [of p h2 h' ret2] True pivot i_outer

      have "get_array a h2 ! i = get_array a h' ! i" by auto

      ultimately have "get_array a h ! i= get_array a h' ! i" by simp

    }

    with cr show ?thesis

      unfolding quicksort.simps [of a l r]

      by (elim crel_ifE crel_bindE crel_assertE crel_returnE) auto

  qed

qed

lemma quicksort_is_skip:

  assumes "crel (quicksort a l r) h h' rs"

  shows "r \<le> l \<longrightarrow> h = h'"

  using assms

  unfolding quicksort.simps [of a l r]

  by (elim crel_ifE crel_returnE) auto

lemma quicksort_sorts:

  assumes "crel (quicksort a l r) h h' rs"

  assumes l_r_length: "l < Array.length h a" "r < Array.length h a" 

  shows "sorted (subarray l (r + 1) a h')"

  using assms

proof (induct a l r arbitrary: h h' rs rule: quicksort.induct)

  case (1 a l r h h' rs)

  note cr = `crel (quicksort a l r) h h' rs`

  thus ?case

  proof (cases "r > l")

    case False

    hence "l \<ge> r + 1 \<or> l = r" by arith 

    with length_remains[OF cr] 1(5) show ?thesis

      by (auto simp add: subarray_Nil subarray_single)

  next

    case True

    { 

      fix h1 h2 p

      assume part: "crel (partition a l r) h h1 p"

      assume qs1: "crel (quicksort a l (p - 1)) h1 h2 ()"

      assume qs2: "crel (quicksort a (p + 1) r) h2 h' ()"

      from partition_returns_index_in_bounds [OF part True]

      have pivot: "l\<le> p \<and> p \<le> r" .

     note length_remains = length_remains[OF qs2] length_remains[OF qs1] partition_length_remains[OF part]

      from quicksort_outer_remains [OF qs2] quicksort_outer_remains [OF qs1] pivot quicksort_is_skip[OF qs1]

      have pivot_unchanged: "get_array a h1 ! p = get_array a h' ! p" by (cases p, auto)

        (*-- First of all, by induction hypothesis both sublists are sorted. *)

      from 1(1)[OF True pivot qs1] length_remains pivot 1(5) 

      have IH1: "sorted (subarray l p a h2)"  by (cases p, auto simp add: subarray_Nil)

      from quicksort_outer_remains [OF qs2] length_remains

      have left_subarray_remains: "subarray l p a h2 = subarray l p a h'"

        by (simp add: subarray_eq_samelength_iff)

      with IH1 have IH1': "sorted (subarray l p a h')" by simp

      from 1(2)[OF True pivot qs2] pivot 1(5) length_remains

      have IH2: "sorted (subarray (p + 1) (r + 1) a h')"

        by (cases "Suc p \<le> r", auto simp add: subarray_Nil)

           (* -- Secondly, both sublists remain partitioned. *)

      from partition_partitions[OF part True]

      have part_conds1: "\<forall>j. j \<in> set (subarray l p a h1) \<longrightarrow> j \<le> get_array a h1 ! p "

        and part_conds2: "\<forall>j. j \<in> set (subarray (p + 1) (r + 1) a h1) \<longrightarrow> get_array a h1 ! p \<le> j"

        by (auto simp add: all_in_set_subarray_conv)

      from quicksort_outer_remains [OF qs1] quicksort_permutes [OF qs1] True

        length_remains 1(5) pivot multiset_of_sublist [of l p "get_array a h1" "get_array a h2"]

      have multiset_partconds1: "multiset_of (subarray l p a h2) = multiset_of (subarray l p a h1)"

        unfolding Array.length_def subarray_def by (cases p, auto)

      with left_subarray_remains part_conds1 pivot_unchanged

      have part_conds2': "\<forall>j. j \<in> set (subarray l p a h') \<longrightarrow> j \<le> get_array a h' ! p"

        by (simp, subst set_of_multiset_of[symmetric], simp)

          (* -- These steps are the analogous for the right sublist \<dots> *)

      from quicksort_outer_remains [OF qs1] length_remains

      have right_subarray_remains: "subarray (p + 1) (r + 1) a h1 = subarray (p + 1) (r + 1) a h2"

        by (auto simp add: subarray_eq_samelength_iff)

      from quicksort_outer_remains [OF qs2] quicksort_permutes [OF qs2] True

        length_remains 1(5) pivot multiset_of_sublist [of "p + 1" "r + 1" "get_array a h2" "get_array a h'"]

      have multiset_partconds2: "multiset_of (subarray (p + 1) (r + 1) a h') = multiset_of (subarray (p + 1) (r + 1) a h2)"

        unfolding Array.length_def subarray_def by auto

      with right_subarray_remains part_conds2 pivot_unchanged

      have part_conds1': "\<forall>j. j \<in> set (subarray (p + 1) (r + 1) a h') \<longrightarrow> get_array a h' ! p \<le> j"

        by (simp, subst set_of_multiset_of[symmetric], simp)

          (* -- Thirdly and finally, we show that the array is sorted

          following from the facts above. *)

      from True pivot 1(5) length_remains have "subarray l (r + 1) a h' = subarray l p a h' @ [get_array a h' ! p] @ subarray (p + 1) (r + 1) a h'"

        by (simp add: subarray_nth_array_Cons, cases "l < p") (auto simp add: subarray_append subarray_Nil)

      with IH1' IH2 part_conds1' part_conds2' pivot have ?thesis

        unfolding subarray_def

        apply (auto simp add: sorted_append sorted_Cons all_in_set_sublist'_conv)

        by (auto simp add: set_sublist' dest: le_trans [of _ "get_array a h' ! p"])

    }

    with True cr show ?thesis

      unfolding quicksort.simps [of a l r]

      by (elim crel_ifE crel_returnE crel_bindE crel_assertE) auto

  qed

qed

lemma quicksort_is_sort:

  assumes crel: "crel (quicksort a 0 (Array.length h a - 1)) h h' rs"

  shows "get_array a h' = sort (get_array a h)"

proof (cases "get_array a h = []")

  case True

  with quicksort_is_skip[OF crel] show ?thesis

  unfolding Array.length_def by simp

next

  case False

  from quicksort_sorts [OF crel] False have "sorted (sublist' 0 (List.length (get_array a h)) (get_array a h'))"

    unfolding Array.length_def subarray_def by auto

  with length_remains[OF crel] have "sorted (get_array a h')"

    unfolding Array.length_def by simp

  with quicksort_permutes [OF crel] properties_for_sort show ?thesis by fastsimp

qed

subsection {* No Errors in quicksort *}

text {* We have proved that quicksort sorts (if no exceptions occur).

We will now show that exceptions do not occur. *}

lemma success_part1I: 

  assumes "l < Array.length h a" "r < Array.length h a"

  shows "success (part1 a l r p) h"

  using assms

proof (induct a l r p arbitrary: h rule: part1.induct)

  case (1 a l r p)

  thus ?case unfolding part1.simps [of a l r]

  apply (auto intro!: success_intros del: success_ifI simp add: not_le)

  apply (auto intro!: crel_intros crel_swapI)

  done

qed

lemma success_bindI' [success_intros]: (*FIXME move*)

  assumes "success f h"

  assumes "\<And>h' r. crel f h h' r \<Longrightarrow> success (g r) h'"

  shows "success (f \<guillemotright>= g) h"

using assms(1) proof (rule success_crelE)

  fix h' r

  assume "crel f h h' r"

  moreover with assms(2) have "success (g r) h'" .

  ultimately show "success (f \<guillemotright>= g) h" by (rule success_bind_crelI)

qed

lemma success_partitionI:

  assumes "l < r" "l < Array.length h a" "r < Array.length h a"

  shows "success (partition a l r) h"

using assms unfolding partition.simps swap_def

apply (auto intro!: success_bindI' success_ifI success_returnI success_nthI success_updI success_part1I elim!: crel_bindE crel_updE crel_nthE crel_returnE simp add:)

apply (frule part_length_remains)

apply (frule part_returns_index_in_bounds)

apply auto

apply (frule part_length_remains)

apply (frule part_returns_index_in_bounds)

apply auto

apply (frule part_length_remains)

apply auto

done

lemma success_quicksortI:

  assumes "l < Array.length h a" "r < Array.length h a"

  shows "success (quicksort a l r) h"

using assms

proof (induct a l r arbitrary: h rule: quicksort.induct)

  case (1 a l ri h)

  thus ?case

    unfolding quicksort.simps [of a l ri]

    apply (auto intro!: success_ifI success_bindI' success_returnI success_nthI success_updI success_assertI success_partitionI)

    apply (frule partition_returns_index_in_bounds)

    apply auto

    apply (frule partition_returns_index_in_bounds)

    apply auto

    apply (auto elim!: crel_assertE dest!: partition_length_remains length_remains)

    apply (subgoal_tac "Suc r \<le> ri \<or> r = ri") 

    apply (erule disjE)

    apply auto

    unfolding quicksort.simps [of a "Suc ri" ri]

    apply (auto intro!: success_ifI success_returnI)

    done

qed

subsection {* Example *}

definition "qsort a = do {

    k \<leftarrow> Array.len a;

    quicksort a 0 (k - 1);

    return a

  }"

code_reserved SML upto

ML {* @{code qsort} (Array.fromList [42, 2, 3, 5, 0, 1705, 8, 3, 15]) () *}

export_code qsort in SML_imp module_name QSort file -

export_code qsort in OCaml module_name QSort file -

export_code qsort in OCaml_imp module_name QSort file -

export_code qsort in Haskell module_name QSort file -

end