src/HOL/Imperative_HOL/ex/Imperative_Quicksort.thy
 author haftmann Tue, 13 Jul 2010 12:05:20 +0200 changeset 37797 96551d6b1414 parent 37792 ba0bc31b90d7 parent 37796 08bd610b2583 child 37798 0b0570445a2a permissions -rw-r--r--
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```
(*  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> nth arr i;
y \<leftarrow> nth arr j;
upd i y arr;
upd j x arr;
return ()
}"

lemma crel_swapI [crel_intros]:
assumes "i < Array.length a h" "j < Array.length a h"
"x = get_array a h ! i" "y = get_array a h ! j"
"h' = Array.change a j x (Array.change 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> 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 a h = Array.length 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)
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> nth a right;
middle \<leftarrow> part1 a left (right - 1) pivot;
v \<leftarrow> 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 a h = Array.length a h'"
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.change a r (get_array a h1 ! rs)
(Array.change 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 a h1 \<and> rs < Array.length a h1"
unfolding swap_def
by (elim crel_bindE crel_returnE crel_nthE crel_updE) simp
from swap have swap_length_remains: "Array.length a h1 = Array.length a h'"
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 a h'" "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.change_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 a h'" "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.change_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 a h'" "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.change_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 a h'" "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.change_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 a h = Array.length 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_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 a h" "r < Array.length a h"
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'"
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"
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"
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 a h - 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 a h" "r < Array.length a h"
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 a h" "r < Array.length a h"
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 a h" "r < Array.length a h"
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> 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
```