author | wenzelm |
Mon, 16 Mar 2009 18:24:30 +0100 | |
changeset 30549 | d2d7874648bd |
parent 29793 | 86cac1fab613 |
permissions | -rw-r--r-- |
27679 | 1 |
theory ImperativeQuicksort |
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imports "~~/src/HOL/Imperative_HOL/Imperative_HOL" Subarray Multiset Efficient_Nat |
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begin |
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text {* We prove QuickSort correct in the Relational Calculus. *} |
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definition swap :: "nat array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> unit Heap" |
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where |
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"swap arr i j = ( |
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do |
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x \<leftarrow> nth arr i; |
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y \<leftarrow> nth arr j; |
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upd i y arr; |
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upd j x arr; |
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return () |
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done)" |
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lemma swap_permutes: |
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assumes "crel (swap a i j) h h' rs" |
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shows "multiset_of (get_array a h') |
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= multiset_of (get_array a h)" |
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using assms |
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unfolding swap_def |
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by (auto simp add: Heap.length_def multiset_of_swap dest: sym [of _ "h'"] elim!: crelE crel_nth crel_return crel_upd) |
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function part1 :: "nat array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat Heap" |
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where |
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"part1 a left right p = ( |
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if (right \<le> left) then return right |
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else (do |
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v \<leftarrow> nth a left; |
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(if (v \<le> p) then (part1 a (left + 1) right p) |
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else (do swap a left right; |
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part1 a left (right - 1) p done)) |
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done))" |
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by pat_completeness auto |
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termination |
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by (relation "measure (\<lambda>(_,l,r,_). r - l )") auto |
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declare part1.simps[simp del] |
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lemma part_permutes: |
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assumes "crel (part1 a l r p) h h' rs" |
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shows "multiset_of (get_array a h') |
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= multiset_of (get_array a h)" |
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using assms |
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proof (induct a l r p arbitrary: h h' rs rule:part1.induct) |
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case (1 a l r p h h' rs) |
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thus ?case |
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unfolding part1.simps [of a l r p] |
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by (elim crelE crel_if crel_return crel_nth) (auto simp add: swap_permutes) |
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qed |
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lemma part_returns_index_in_bounds: |
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assumes "crel (part1 a l r p) h h' rs" |
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assumes "l \<le> r" |
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shows "l \<le> rs \<and> rs \<le> r" |
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using assms |
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proof (induct a l r p arbitrary: h h' rs rule:part1.induct) |
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case (1 a l r p h h' rs) |
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note cr = `crel (part1 a l r p) h h' rs` |
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show ?case |
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proof (cases "r \<le> l") |
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case True (* Terminating case *) |
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with cr `l \<le> r` show ?thesis |
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unfolding part1.simps[of a l r p] |
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by (elim crelE crel_if crel_return crel_nth) auto |
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next |
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case False (* recursive case *) |
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note rec_condition = this |
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let ?v = "get_array a h ! l" |
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show ?thesis |
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proof (cases "?v \<le> p") |
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case True |
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with cr False |
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have rec1: "crel (part1 a (l + 1) r p) h h' rs" |
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unfolding part1.simps[of a l r p] |
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by (elim crelE crel_nth crel_if crel_return) auto |
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from rec_condition have "l + 1 \<le> r" by arith |
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from 1(1)[OF rec_condition True rec1 `l + 1 \<le> r`] |
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show ?thesis by simp |
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next |
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case False |
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with rec_condition cr |
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obtain h1 where swp: "crel (swap a l r) h h1 ()" |
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and rec2: "crel (part1 a l (r - 1) p) h1 h' rs" |
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unfolding part1.simps[of a l r p] |
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by (elim crelE crel_nth crel_if crel_return) auto |
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from rec_condition have "l \<le> r - 1" by arith |
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from 1(2) [OF rec_condition False rec2 `l \<le> r - 1`] show ?thesis by fastsimp |
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qed |
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qed |
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qed |
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lemma part_length_remains: |
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assumes "crel (part1 a l r p) h h' rs" |
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shows "Heap.length a h = Heap.length a h'" |
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using assms |
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proof (induct a l r p arbitrary: h h' rs rule:part1.induct) |
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case (1 a l r p h h' rs) |
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note cr = `crel (part1 a l r p) h h' rs` |
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show ?case |
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proof (cases "r \<le> l") |
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case True (* Terminating case *) |
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with cr show ?thesis |
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unfolding part1.simps[of a l r p] |
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by (elim crelE crel_if crel_return crel_nth) auto |
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next |
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case False (* recursive case *) |
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with cr 1 show ?thesis |
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unfolding part1.simps [of a l r p] swap_def |
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by (auto elim!: crelE crel_if crel_nth crel_return crel_upd) fastsimp |
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qed |
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qed |
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lemma part_outer_remains: |
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assumes "crel (part1 a l r p) h h' rs" |
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shows "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array (a::nat array) h ! i = get_array a h' ! i" |
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using assms |
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proof (induct a l r p arbitrary: h h' rs rule:part1.induct) |
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case (1 a l r p h h' rs) |
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note cr = `crel (part1 a l r p) h h' rs` |
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125 |
|
d4f6e64ee7cc
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bulwahn
parents:
diff
changeset
|
126 |
show ?case |
d4f6e64ee7cc
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bulwahn
parents:
diff
changeset
|
127 |
proof (cases "r \<le> l") |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
128 |
case True (* Terminating case *) |
d4f6e64ee7cc
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bulwahn
parents:
diff
changeset
|
129 |
with cr show ?thesis |
28145 | 130 |
unfolding part1.simps[of a l r p] |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
131 |
by (elim crelE crel_if crel_return crel_nth) auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
132 |
next |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
133 |
case False (* recursive case *) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
134 |
note rec_condition = this |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
135 |
let ?v = "get_array a h ! l" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
136 |
show ?thesis |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
137 |
proof (cases "?v \<le> p") |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
138 |
case True |
d4f6e64ee7cc
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bulwahn
parents:
diff
changeset
|
139 |
with cr False |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
140 |
have rec1: "crel (part1 a (l + 1) r p) h h' rs" |
28145 | 141 |
unfolding part1.simps[of a l r p] |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
142 |
by (elim crelE crel_nth crel_if crel_return) auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
143 |
from 1(1)[OF rec_condition True rec1] |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
144 |
show ?thesis by fastsimp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
145 |
next |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
146 |
case False |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
147 |
with rec_condition cr |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
148 |
obtain h1 where swp: "crel (swap a l r) h h1 ()" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
149 |
and rec2: "crel (part1 a l (r - 1) p) h1 h' rs" |
28145 | 150 |
unfolding part1.simps[of a l r p] |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
151 |
by (elim crelE crel_nth crel_if crel_return) auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
152 |
from swp rec_condition have |
28013 | 153 |
"\<forall>i. i < l \<or> r < i \<longrightarrow> get_array a h ! i = get_array a h1 ! i" |
28145 | 154 |
unfolding swap_def |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
155 |
by (elim crelE crel_nth crel_upd crel_return) auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
156 |
with 1(2) [OF rec_condition False rec2] show ?thesis by fastsimp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
157 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
158 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
159 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
160 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
161 |
|
d4f6e64ee7cc
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bulwahn
parents:
diff
changeset
|
162 |
lemma part_partitions: |
d4f6e64ee7cc
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bulwahn
parents:
diff
changeset
|
163 |
assumes "crel (part1 a l r p) h h' rs" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
164 |
shows "(\<forall>i. l \<le> i \<and> i < rs \<longrightarrow> get_array (a::nat array) h' ! i \<le> p) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
165 |
\<and> (\<forall>i. rs < i \<and> i \<le> r \<longrightarrow> get_array a h' ! i \<ge> p)" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
166 |
using assms |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
167 |
proof (induct a l r p arbitrary: h h' rs rule:part1.induct) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
168 |
case (1 a l r p h h' rs) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
169 |
note cr = `crel (part1 a l r p) h h' rs` |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
170 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
171 |
show ?case |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
172 |
proof (cases "r \<le> l") |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
173 |
case True (* Terminating case *) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
174 |
with cr have "rs = r" |
28145 | 175 |
unfolding part1.simps[of a l r p] |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
176 |
by (elim crelE crel_if crel_return crel_nth) auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
177 |
with True |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
178 |
show ?thesis by auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
179 |
next |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
180 |
case False (* recursive case *) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
181 |
note lr = this |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
182 |
let ?v = "get_array a h ! l" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
183 |
show ?thesis |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
184 |
proof (cases "?v \<le> p") |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
185 |
case True |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
186 |
with lr cr |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
187 |
have rec1: "crel (part1 a (l + 1) r p) h h' rs" |
28145 | 188 |
unfolding part1.simps[of a l r p] |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
189 |
by (elim crelE crel_nth crel_if crel_return) auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
190 |
from True part_outer_remains[OF rec1] have a_l: "get_array a h' ! l \<le> p" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
191 |
by fastsimp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
192 |
have "\<forall>i. (l \<le> i = (l = i \<or> Suc l \<le> i))" by arith |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
193 |
with 1(1)[OF False True rec1] a_l show ?thesis |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
194 |
by auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
195 |
next |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
196 |
case False |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
197 |
with lr cr |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
198 |
obtain h1 where swp: "crel (swap a l r) h h1 ()" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
199 |
and rec2: "crel (part1 a l (r - 1) p) h1 h' rs" |
28145 | 200 |
unfolding part1.simps[of a l r p] |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
201 |
by (elim crelE crel_nth crel_if crel_return) auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
202 |
from swp False have "get_array a h1 ! r \<ge> p" |
28145 | 203 |
unfolding swap_def |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
204 |
by (auto simp add: Heap.length_def elim!: crelE crel_nth crel_upd crel_return) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
205 |
with part_outer_remains [OF rec2] lr have a_r: "get_array a h' ! r \<ge> p" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
206 |
by fastsimp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
207 |
have "\<forall>i. (i \<le> r = (i = r \<or> i \<le> r - 1))" by arith |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
208 |
with 1(2)[OF lr False rec2] a_r show ?thesis |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
209 |
by auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
210 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
211 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
212 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
213 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
214 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
215 |
fun partition :: "nat array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat Heap" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
216 |
where |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
217 |
"partition a left right = (do |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
218 |
pivot \<leftarrow> nth a right; |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
219 |
middle \<leftarrow> part1 a left (right - 1) pivot; |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
220 |
v \<leftarrow> nth a middle; |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
221 |
m \<leftarrow> return (if (v \<le> pivot) then (middle + 1) else middle); |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
222 |
swap a m right; |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
223 |
return m |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
224 |
done)" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
225 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
226 |
declare partition.simps[simp del] |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
227 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
228 |
lemma partition_permutes: |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
229 |
assumes "crel (partition a l r) h h' rs" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
230 |
shows "multiset_of (get_array a h') |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
231 |
= multiset_of (get_array a h)" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
232 |
proof - |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
233 |
from assms part_permutes swap_permutes show ?thesis |
28145 | 234 |
unfolding partition.simps |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
235 |
by (elim crelE crel_return crel_nth crel_if crel_upd) auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
236 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
237 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
238 |
lemma partition_length_remains: |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
239 |
assumes "crel (partition a l r) h h' rs" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
240 |
shows "Heap.length a h = Heap.length a h'" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
241 |
proof - |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
242 |
from assms part_length_remains show ?thesis |
28145 | 243 |
unfolding partition.simps swap_def |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
244 |
by (elim crelE crel_return crel_nth crel_if crel_upd) auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
245 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
246 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
247 |
lemma partition_outer_remains: |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
248 |
assumes "crel (partition a l r) h h' rs" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
249 |
assumes "l < r" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
250 |
shows "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array (a::nat array) h ! i = get_array a h' ! i" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
251 |
proof - |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
252 |
from assms part_outer_remains part_returns_index_in_bounds show ?thesis |
28145 | 253 |
unfolding partition.simps swap_def |
27656
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bulwahn
parents:
diff
changeset
|
254 |
by (elim crelE crel_return crel_nth crel_if crel_upd) fastsimp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
255 |
qed |
d4f6e64ee7cc
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bulwahn
parents:
diff
changeset
|
256 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
257 |
lemma partition_returns_index_in_bounds: |
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parents:
diff
changeset
|
258 |
assumes crel: "crel (partition a l r) h h' rs" |
d4f6e64ee7cc
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bulwahn
parents:
diff
changeset
|
259 |
assumes "l < r" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
260 |
shows "l \<le> rs \<and> rs \<le> r" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
261 |
proof - |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
262 |
from crel obtain middle h'' p where part: "crel (part1 a l (r - 1) p) h h'' middle" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
263 |
and rs_equals: "rs = (if get_array a h'' ! middle \<le> get_array a h ! r then middle + 1 |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
264 |
else middle)" |
28145 | 265 |
unfolding partition.simps |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
266 |
by (elim crelE crel_return crel_nth crel_if crel_upd) simp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
267 |
from `l < r` have "l \<le> r - 1" by arith |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
268 |
from part_returns_index_in_bounds[OF part this] rs_equals `l < r` show ?thesis by auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
269 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
270 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
271 |
lemma partition_partitions: |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
272 |
assumes crel: "crel (partition a l r) h h' rs" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
273 |
assumes "l < r" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
274 |
shows "(\<forall>i. l \<le> i \<and> i < rs \<longrightarrow> get_array (a::nat array) h' ! i \<le> get_array a h' ! rs) \<and> |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
275 |
(\<forall>i. rs < i \<and> i \<le> r \<longrightarrow> get_array a h' ! rs \<le> get_array a h' ! i)" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
276 |
proof - |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
277 |
let ?pivot = "get_array a h ! r" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
278 |
from crel obtain middle h1 where part: "crel (part1 a l (r - 1) ?pivot) h h1 middle" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
279 |
and swap: "crel (swap a rs r) h1 h' ()" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
280 |
and rs_equals: "rs = (if get_array a h1 ! middle \<le> ?pivot then middle + 1 |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
281 |
else middle)" |
28145 | 282 |
unfolding partition.simps |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
283 |
by (elim crelE crel_return crel_nth crel_if crel_upd) simp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
284 |
from swap have h'_def: "h' = Heap.upd a r (get_array a h1 ! rs) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
285 |
(Heap.upd a rs (get_array a h1 ! r) h1)" |
28145 | 286 |
unfolding swap_def |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
287 |
by (elim crelE crel_return crel_nth crel_upd) simp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
288 |
from swap have in_bounds: "r < Heap.length a h1 \<and> rs < Heap.length a h1" |
28145 | 289 |
unfolding swap_def |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
290 |
by (elim crelE crel_return crel_nth crel_upd) simp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
291 |
from swap have swap_length_remains: "Heap.length a h1 = Heap.length a h'" |
28145 | 292 |
unfolding swap_def by (elim crelE crel_return crel_nth crel_upd) auto |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
293 |
from `l < r` have "l \<le> r - 1" by simp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
294 |
note middle_in_bounds = part_returns_index_in_bounds[OF part this] |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
295 |
from part_outer_remains[OF part] `l < r` |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
296 |
have "get_array a h ! r = get_array a h1 ! r" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
297 |
by fastsimp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
298 |
with swap |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
299 |
have right_remains: "get_array a h ! r = get_array a h' ! rs" |
28145 | 300 |
unfolding swap_def |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
301 |
by (auto simp add: Heap.length_def elim!: crelE crel_return crel_nth crel_upd) (cases "r = rs", auto) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
302 |
from part_partitions [OF part] |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
303 |
show ?thesis |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
304 |
proof (cases "get_array a h1 ! middle \<le> ?pivot") |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
305 |
case True |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
306 |
with rs_equals have rs_equals: "rs = middle + 1" by simp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
307 |
{ |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
308 |
fix i |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
309 |
assume i_is_left: "l \<le> i \<and> i < rs" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
310 |
with swap_length_remains in_bounds middle_in_bounds rs_equals `l < r` |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
311 |
have i_props: "i < Heap.length a h'" "i \<noteq> r" "i \<noteq> rs" by auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
312 |
from i_is_left rs_equals have "l \<le> i \<and> i < middle \<or> i = middle" by arith |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
313 |
with part_partitions[OF part] right_remains True |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
314 |
have "get_array a h1 ! i \<le> get_array a h' ! rs" by fastsimp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
315 |
with i_props h'_def in_bounds have "get_array a h' ! i \<le> get_array a h' ! rs" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
316 |
unfolding Heap.upd_def Heap.length_def by simp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
317 |
} |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
318 |
moreover |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
319 |
{ |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
320 |
fix i |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
321 |
assume "rs < i \<and> i \<le> r" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
322 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
323 |
hence "(rs < i \<and> i \<le> r - 1) \<or> (rs < i \<and> i = r)" by arith |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
324 |
hence "get_array a h' ! rs \<le> get_array a h' ! i" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
325 |
proof |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
326 |
assume i_is: "rs < i \<and> i \<le> r - 1" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
327 |
with swap_length_remains in_bounds middle_in_bounds rs_equals |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
328 |
have i_props: "i < Heap.length a h'" "i \<noteq> r" "i \<noteq> rs" by auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
329 |
from part_partitions[OF part] rs_equals right_remains i_is |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
330 |
have "get_array a h' ! rs \<le> get_array a h1 ! i" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
331 |
by fastsimp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
332 |
with i_props h'_def show ?thesis by fastsimp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
333 |
next |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
334 |
assume i_is: "rs < i \<and> i = r" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
335 |
with rs_equals have "Suc middle \<noteq> r" by arith |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
336 |
with middle_in_bounds `l < r` have "Suc middle \<le> r - 1" by arith |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
337 |
with part_partitions[OF part] right_remains |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
338 |
have "get_array a h' ! rs \<le> get_array a h1 ! (Suc middle)" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
339 |
by fastsimp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
340 |
with i_is True rs_equals right_remains h'_def |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
341 |
show ?thesis using in_bounds |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
342 |
unfolding Heap.upd_def Heap.length_def |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
343 |
by auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
344 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
345 |
} |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
346 |
ultimately show ?thesis by auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
347 |
next |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
348 |
case False |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
349 |
with rs_equals have rs_equals: "middle = rs" by simp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
350 |
{ |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
351 |
fix i |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
352 |
assume i_is_left: "l \<le> i \<and> i < rs" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
353 |
with swap_length_remains in_bounds middle_in_bounds rs_equals |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
354 |
have i_props: "i < Heap.length a h'" "i \<noteq> r" "i \<noteq> rs" by auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
355 |
from part_partitions[OF part] rs_equals right_remains i_is_left |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
356 |
have "get_array a h1 ! i \<le> get_array a h' ! rs" by fastsimp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
357 |
with i_props h'_def have "get_array a h' ! i \<le> get_array a h' ! rs" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
358 |
unfolding Heap.upd_def by simp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
359 |
} |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
360 |
moreover |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
361 |
{ |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
362 |
fix i |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
363 |
assume "rs < i \<and> i \<le> r" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
364 |
hence "(rs < i \<and> i \<le> r - 1) \<or> i = r" by arith |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
365 |
hence "get_array a h' ! rs \<le> get_array a h' ! i" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
366 |
proof |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
367 |
assume i_is: "rs < i \<and> i \<le> r - 1" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
368 |
with swap_length_remains in_bounds middle_in_bounds rs_equals |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
369 |
have i_props: "i < Heap.length a h'" "i \<noteq> r" "i \<noteq> rs" by auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
370 |
from part_partitions[OF part] rs_equals right_remains i_is |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
371 |
have "get_array a h' ! rs \<le> get_array a h1 ! i" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
372 |
by fastsimp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
373 |
with i_props h'_def show ?thesis by fastsimp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
374 |
next |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
375 |
assume i_is: "i = r" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
376 |
from i_is False rs_equals right_remains h'_def |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
377 |
show ?thesis using in_bounds |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
378 |
unfolding Heap.upd_def Heap.length_def |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
379 |
by auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
380 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
381 |
} |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
382 |
ultimately |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
383 |
show ?thesis by auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
384 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
385 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
386 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
387 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
388 |
function quicksort :: "nat array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> unit Heap" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
389 |
where |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
390 |
"quicksort arr left right = |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
391 |
(if (right > left) then |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
392 |
do |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
393 |
pivotNewIndex \<leftarrow> partition arr left right; |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
394 |
pivotNewIndex \<leftarrow> assert (\<lambda>x. left \<le> x \<and> x \<le> right) pivotNewIndex; |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
395 |
quicksort arr left (pivotNewIndex - 1); |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
396 |
quicksort arr (pivotNewIndex + 1) right |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
397 |
done |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
398 |
else return ())" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
399 |
by pat_completeness auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
400 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
401 |
(* For termination, we must show that the pivotNewIndex is between left and right *) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
402 |
termination |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
403 |
by (relation "measure (\<lambda>(a, l, r). (r - l))") auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
404 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
405 |
declare quicksort.simps[simp del] |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
406 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
407 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
408 |
lemma quicksort_permutes: |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
409 |
assumes "crel (quicksort a l r) h h' rs" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
410 |
shows "multiset_of (get_array a h') |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
411 |
= multiset_of (get_array a h)" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
412 |
using assms |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
413 |
proof (induct a l r arbitrary: h h' rs rule: quicksort.induct) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
414 |
case (1 a l r h h' rs) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
415 |
with partition_permutes show ?case |
28145 | 416 |
unfolding quicksort.simps [of a l r] |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
417 |
by (elim crel_if crelE crel_assert crel_return) auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
418 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
419 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
420 |
lemma length_remains: |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
421 |
assumes "crel (quicksort a l r) h h' rs" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
422 |
shows "Heap.length a h = Heap.length a h'" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
423 |
using assms |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
424 |
proof (induct a l r arbitrary: h h' rs rule: quicksort.induct) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
425 |
case (1 a l r h h' rs) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
426 |
with partition_length_remains show ?case |
28145 | 427 |
unfolding quicksort.simps [of a l r] |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
428 |
by (elim crel_if crelE crel_assert crel_return) auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
429 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
430 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
431 |
lemma quicksort_outer_remains: |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
432 |
assumes "crel (quicksort a l r) h h' rs" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
433 |
shows "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array (a::nat array) h ! i = get_array a h' ! i" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
434 |
using assms |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
435 |
proof (induct a l r arbitrary: h h' rs rule: quicksort.induct) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
436 |
case (1 a l r h h' rs) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
437 |
note cr = `crel (quicksort a l r) h h' rs` |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
438 |
thus ?case |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
439 |
proof (cases "r > l") |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
440 |
case False |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
441 |
with cr have "h' = h" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
442 |
unfolding quicksort.simps [of a l r] |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
443 |
by (elim crel_if crel_return) auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
444 |
thus ?thesis by simp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
445 |
next |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
446 |
case True |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
447 |
{ |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
448 |
fix h1 h2 p ret1 ret2 i |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
449 |
assume part: "crel (partition a l r) h h1 p" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
450 |
assume qs1: "crel (quicksort a l (p - 1)) h1 h2 ret1" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
451 |
assume qs2: "crel (quicksort a (p + 1) r) h2 h' ret2" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
452 |
assume pivot: "l \<le> p \<and> p \<le> r" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
453 |
assume i_outer: "i < l \<or> r < i" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
454 |
from partition_outer_remains [OF part True] i_outer |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
455 |
have "get_array a h !i = get_array a h1 ! i" by fastsimp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
456 |
moreover |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
457 |
with 1(1) [OF True pivot qs1] pivot i_outer |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
458 |
have "get_array a h1 ! i = get_array a h2 ! i" by auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
459 |
moreover |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
460 |
with qs2 1(2) [of p h2 h' ret2] True pivot i_outer |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
461 |
have "get_array a h2 ! i = get_array a h' ! i" by auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
462 |
ultimately have "get_array a h ! i= get_array a h' ! i" by simp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
463 |
} |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
464 |
with cr show ?thesis |
28145 | 465 |
unfolding quicksort.simps [of a l r] |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
466 |
by (elim crel_if crelE crel_assert crel_return) auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
467 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
468 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
469 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
470 |
lemma quicksort_is_skip: |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
471 |
assumes "crel (quicksort a l r) h h' rs" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
472 |
shows "r \<le> l \<longrightarrow> h = h'" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
473 |
using assms |
28145 | 474 |
unfolding quicksort.simps [of a l r] |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
475 |
by (elim crel_if crel_return) auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
476 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
477 |
lemma quicksort_sorts: |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
478 |
assumes "crel (quicksort a l r) h h' rs" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
479 |
assumes l_r_length: "l < Heap.length a h" "r < Heap.length a h" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
480 |
shows "sorted (subarray l (r + 1) a h')" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
481 |
using assms |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
482 |
proof (induct a l r arbitrary: h h' rs rule: quicksort.induct) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
483 |
case (1 a l r h h' rs) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
484 |
note cr = `crel (quicksort a l r) h h' rs` |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
485 |
thus ?case |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
486 |
proof (cases "r > l") |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
487 |
case False |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
488 |
hence "l \<ge> r + 1 \<or> l = r" by arith |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
489 |
with length_remains[OF cr] 1(5) show ?thesis |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
490 |
by (auto simp add: subarray_Nil subarray_single) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
491 |
next |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
492 |
case True |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
493 |
{ |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
494 |
fix h1 h2 p |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
495 |
assume part: "crel (partition a l r) h h1 p" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
496 |
assume qs1: "crel (quicksort a l (p - 1)) h1 h2 ()" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
497 |
assume qs2: "crel (quicksort a (p + 1) r) h2 h' ()" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
498 |
from partition_returns_index_in_bounds [OF part True] |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
499 |
have pivot: "l\<le> p \<and> p \<le> r" . |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
500 |
note length_remains = length_remains[OF qs2] length_remains[OF qs1] partition_length_remains[OF part] |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
501 |
from quicksort_outer_remains [OF qs2] quicksort_outer_remains [OF qs1] pivot quicksort_is_skip[OF qs1] |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
502 |
have pivot_unchanged: "get_array a h1 ! p = get_array a h' ! p" by (cases p, auto) |
28013 | 503 |
(*-- First of all, by induction hypothesis both sublists are sorted. *) |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
504 |
from 1(1)[OF True pivot qs1] length_remains pivot 1(5) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
505 |
have IH1: "sorted (subarray l p a h2)" by (cases p, auto simp add: subarray_Nil) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
506 |
from quicksort_outer_remains [OF qs2] length_remains |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
507 |
have left_subarray_remains: "subarray l p a h2 = subarray l p a h'" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
508 |
by (simp add: subarray_eq_samelength_iff) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
509 |
with IH1 have IH1': "sorted (subarray l p a h')" by simp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
510 |
from 1(2)[OF True pivot qs2] pivot 1(5) length_remains |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
511 |
have IH2: "sorted (subarray (p + 1) (r + 1) a h')" |
28013 | 512 |
by (cases "Suc p \<le> r", auto simp add: subarray_Nil) |
513 |
(* -- Secondly, both sublists remain partitioned. *) |
|
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
514 |
from partition_partitions[OF part True] |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
515 |
have part_conds1: "\<forall>j. j \<in> set (subarray l p a h1) \<longrightarrow> j \<le> get_array a h1 ! p " |
28013 | 516 |
and part_conds2: "\<forall>j. j \<in> set (subarray (p + 1) (r + 1) a h1) \<longrightarrow> get_array a h1 ! p \<le> j" |
517 |
by (auto simp add: all_in_set_subarray_conv) |
|
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
518 |
from quicksort_outer_remains [OF qs1] quicksort_permutes [OF qs1] True |
28013 | 519 |
length_remains 1(5) pivot multiset_of_sublist [of l p "get_array a h1" "get_array a h2"] |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
520 |
have multiset_partconds1: "multiset_of (subarray l p a h2) = multiset_of (subarray l p a h1)" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
521 |
unfolding Heap.length_def subarray_def by (cases p, auto) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
522 |
with left_subarray_remains part_conds1 pivot_unchanged |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
523 |
have part_conds2': "\<forall>j. j \<in> set (subarray l p a h') \<longrightarrow> j \<le> get_array a h' ! p" |
28013 | 524 |
by (simp, subst set_of_multiset_of[symmetric], simp) |
525 |
(* -- These steps are the analogous for the right sublist \<dots> *) |
|
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
526 |
from quicksort_outer_remains [OF qs1] length_remains |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
527 |
have right_subarray_remains: "subarray (p + 1) (r + 1) a h1 = subarray (p + 1) (r + 1) a h2" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
528 |
by (auto simp add: subarray_eq_samelength_iff) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
529 |
from quicksort_outer_remains [OF qs2] quicksort_permutes [OF qs2] True |
28013 | 530 |
length_remains 1(5) pivot multiset_of_sublist [of "p + 1" "r + 1" "get_array a h2" "get_array a h'"] |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
531 |
have multiset_partconds2: "multiset_of (subarray (p + 1) (r + 1) a h') = multiset_of (subarray (p + 1) (r + 1) a h2)" |
28013 | 532 |
unfolding Heap.length_def subarray_def by auto |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
533 |
with right_subarray_remains part_conds2 pivot_unchanged |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
534 |
have part_conds1': "\<forall>j. j \<in> set (subarray (p + 1) (r + 1) a h') \<longrightarrow> get_array a h' ! p \<le> j" |
28013 | 535 |
by (simp, subst set_of_multiset_of[symmetric], simp) |
536 |
(* -- Thirdly and finally, we show that the array is sorted |
|
537 |
following from the facts above. *) |
|
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
538 |
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'" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
539 |
by (simp add: subarray_nth_array_Cons, cases "l < p") (auto simp add: subarray_append subarray_Nil) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
540 |
with IH1' IH2 part_conds1' part_conds2' pivot have ?thesis |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
541 |
unfolding subarray_def |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
542 |
apply (auto simp add: sorted_append sorted_Cons all_in_set_sublist'_conv) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
543 |
by (auto simp add: set_sublist' dest: le_trans [of _ "get_array a h' ! p"]) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
544 |
} |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
545 |
with True cr show ?thesis |
28145 | 546 |
unfolding quicksort.simps [of a l r] |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
547 |
by (elim crel_if crel_return crelE crel_assert) auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
548 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
549 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
550 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
551 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
552 |
lemma quicksort_is_sort: |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
553 |
assumes crel: "crel (quicksort a 0 (Heap.length a h - 1)) h h' rs" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
554 |
shows "get_array a h' = sort (get_array a h)" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
555 |
proof (cases "get_array a h = []") |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
556 |
case True |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
557 |
with quicksort_is_skip[OF crel] show ?thesis |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
558 |
unfolding Heap.length_def by simp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
559 |
next |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
560 |
case False |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
561 |
from quicksort_sorts [OF crel] False have "sorted (sublist' 0 (List.length (get_array a h)) (get_array a h'))" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
562 |
unfolding Heap.length_def subarray_def by auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
563 |
with length_remains[OF crel] have "sorted (get_array a h')" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
564 |
unfolding Heap.length_def by simp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
565 |
with quicksort_permutes [OF crel] properties_for_sort show ?thesis by fastsimp |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
566 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
567 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
568 |
subsection {* No Errors in quicksort *} |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
569 |
text {* We have proved that quicksort sorts (if no exceptions occur). |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
570 |
We will now show that exceptions do not occur. *} |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
571 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
572 |
lemma noError_part1: |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
573 |
assumes "l < Heap.length a h" "r < Heap.length a h" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
574 |
shows "noError (part1 a l r p) h" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
575 |
using assms |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
576 |
proof (induct a l r p arbitrary: h rule: part1.induct) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
577 |
case (1 a l r p) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
578 |
thus ?case |
28145 | 579 |
unfolding part1.simps [of a l r] swap_def |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
580 |
by (auto intro!: noError_if noErrorI noError_return noError_nth noError_upd elim!: crelE crel_upd crel_nth crel_return) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
581 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
582 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
583 |
lemma noError_partition: |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
584 |
assumes "l < r" "l < Heap.length a h" "r < Heap.length a h" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
585 |
shows "noError (partition a l r) h" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
586 |
using assms |
28145 | 587 |
unfolding partition.simps swap_def |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
588 |
apply (auto intro!: noError_if noErrorI noError_return noError_nth noError_upd noError_part1 elim!: crelE crel_upd crel_nth crel_return) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
589 |
apply (frule part_length_remains) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
590 |
apply (frule part_returns_index_in_bounds) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
591 |
apply auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
592 |
apply (frule part_length_remains) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
593 |
apply (frule part_returns_index_in_bounds) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
594 |
apply auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
595 |
apply (frule part_length_remains) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
596 |
apply auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
597 |
done |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
598 |
|
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
599 |
lemma noError_quicksort: |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
600 |
assumes "l < Heap.length a h" "r < Heap.length a h" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
601 |
shows "noError (quicksort a l r) h" |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
602 |
using assms |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
603 |
proof (induct a l r arbitrary: h rule: quicksort.induct) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
604 |
case (1 a l ri h) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
605 |
thus ?case |
28145 | 606 |
unfolding quicksort.simps [of a l ri] |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
607 |
apply (auto intro!: noError_if noErrorI noError_return noError_nth noError_upd noError_assert noError_partition) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
608 |
apply (frule partition_returns_index_in_bounds) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
609 |
apply auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
610 |
apply (frule partition_returns_index_in_bounds) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
611 |
apply auto |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
612 |
apply (auto elim!: crel_assert dest!: partition_length_remains length_remains) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
613 |
apply (subgoal_tac "Suc r \<le> ri \<or> r = ri") |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
614 |
apply (erule disjE) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
615 |
apply auto |
28145 | 616 |
unfolding quicksort.simps [of a "Suc ri" ri] |
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
617 |
apply (auto intro!: noError_if noError_return) |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
618 |
done |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
619 |
qed |
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
620 |
|
27674 | 621 |
|
622 |
subsection {* Example *} |
|
623 |
||
624 |
definition "qsort a = do |
|
625 |
k \<leftarrow> length a; |
|
626 |
quicksort a 0 (k - 1); |
|
627 |
return a |
|
628 |
done" |
|
629 |
||
630 |
ML {* @{code qsort} (Array.fromList [42, 2, 3, 5, 0, 1705, 8, 3, 15]) () *} |
|
631 |
||
29793 | 632 |
export_code qsort in SML_imp module_name QSort |
633 |
export_code qsort in OCaml module_name QSort file - |
|
634 |
export_code qsort in OCaml_imp module_name QSort file - |
|
635 |
export_code qsort in Haskell module_name QSort file - |
|
27674 | 636 |
|
27656
d4f6e64ee7cc
added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents:
diff
changeset
|
637 |
end |