haftmann@30674
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(* Author: Lukas Bulwahn, TU Muenchen *)
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haftmann@30674
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haftmann@30674
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theory Imperative_Quicksort
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haftmann@29396
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imports "~~/src/HOL/Imperative_HOL/Imperative_HOL" Subarray Multiset Efficient_Nat
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bulwahn@27656
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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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haftmann@28145
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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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haftmann@28145
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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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haftmann@28145
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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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bulwahn@27656
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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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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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haftmann@28145
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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 1(1)[OF rec_condition True rec1]
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show ?thesis by fastsimp
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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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haftmann@28145
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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 swp rec_condition have
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haftmann@28013
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"\<forall>i. i < l \<or> r < i \<longrightarrow> get_array a h ! i = get_array a h1 ! i"
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haftmann@28145
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unfolding swap_def
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by (elim crelE crel_nth crel_upd crel_return) auto
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with 1(2) [OF rec_condition False rec2] 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_partitions:
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assumes "crel (part1 a l r p) h h' rs"
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shows "(\<forall>i. l \<le> i \<and> i < rs \<longrightarrow> get_array (a::nat array) h' ! i \<le> p)
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\<and> (\<forall>i. rs < i \<and> i \<le> r \<longrightarrow> get_array a h' ! i \<ge> p)"
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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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bulwahn@27656
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show ?case
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bulwahn@27656
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proof (cases "r \<le> l")
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case True (* Terminating case *)
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with cr have "rs = r"
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haftmann@28145
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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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with True
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show ?thesis by auto
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next
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case False (* recursive case *)
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note lr = 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 lr cr
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have rec1: "crel (part1 a (l + 1) r p) h h' rs"
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haftmann@28145
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unfolding part1.simps[of a l r p]
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bulwahn@27656
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by (elim crelE crel_nth crel_if crel_return) auto
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from True part_outer_remains[OF rec1] have a_l: "get_array a h' ! l \<le> p"
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by fastsimp
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have "\<forall>i. (l \<le> i = (l = i \<or> Suc l \<le> i))" by arith
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with 1(1)[OF False True rec1] a_l show ?thesis
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by auto
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next
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case False
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bulwahn@27656
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with lr cr
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obtain h1 where swp: "crel (swap a l r) h h1 ()"
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bulwahn@27656
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and rec2: "crel (part1 a l (r - 1) p) h1 h' rs"
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haftmann@28145
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unfolding part1.simps[of a l r p]
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bulwahn@27656
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by (elim crelE crel_nth crel_if crel_return) auto
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bulwahn@27656
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from swp False have "get_array a h1 ! r \<ge> p"
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haftmann@28145
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unfolding swap_def
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bulwahn@27656
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by (auto simp add: Heap.length_def elim!: crelE crel_nth crel_upd crel_return)
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with part_outer_remains [OF rec2] lr have a_r: "get_array a h' ! r \<ge> p"
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bulwahn@27656
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by fastsimp
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have "\<forall>i. (i \<le> r = (i = r \<or> i \<le> r - 1))" by arith
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with 1(2)[OF lr False rec2] a_r show ?thesis
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by auto
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qed
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bulwahn@27656
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qed
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qed
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fun partition :: "nat array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat Heap"
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where
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"partition a left right = (do
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pivot \<leftarrow> nth a right;
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middle \<leftarrow> part1 a left (right - 1) pivot;
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v \<leftarrow> nth a middle;
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m \<leftarrow> return (if (v \<le> pivot) then (middle + 1) else middle);
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swap a m right;
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return m
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done)"
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declare partition.simps[simp del]
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lemma partition_permutes:
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assumes "crel (partition a l r) h h' rs"
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shows "multiset_of (get_array a h')
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bulwahn@27656
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= multiset_of (get_array a h)"
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bulwahn@27656
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proof -
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bulwahn@27656
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from assms part_permutes swap_permutes show ?thesis
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haftmann@28145
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unfolding partition.simps
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bulwahn@27656
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by (elim crelE crel_return crel_nth crel_if crel_upd) auto
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qed
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bulwahn@27656
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bulwahn@27656
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lemma partition_length_remains:
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bulwahn@27656
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assumes "crel (partition a l r) h h' rs"
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bulwahn@27656
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shows "Heap.length a h = Heap.length a h'"
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proof -
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bulwahn@27656
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from assms part_length_remains show ?thesis
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haftmann@28145
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unfolding partition.simps swap_def
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bulwahn@27656
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by (elim crelE crel_return crel_nth crel_if crel_upd) auto
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qed
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bulwahn@27656
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lemma partition_outer_remains:
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bulwahn@27656
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assumes "crel (partition a l r) h h' rs"
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bulwahn@27656
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assumes "l < r"
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bulwahn@27656
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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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bulwahn@27656
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proof -
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bulwahn@27656
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from assms part_outer_remains part_returns_index_in_bounds show ?thesis
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haftmann@28145
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unfolding partition.simps swap_def
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bulwahn@27656
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by (elim crelE crel_return crel_nth crel_if crel_upd) fastsimp
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bulwahn@27656
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qed
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bulwahn@27656
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lemma partition_returns_index_in_bounds:
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bulwahn@27656
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assumes crel: "crel (partition a l r) h h' rs"
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bulwahn@27656
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assumes "l < r"
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bulwahn@27656
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shows "l \<le> rs \<and> rs \<le> r"
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proof -
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bulwahn@27656
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from crel obtain middle h'' p where part: "crel (part1 a l (r - 1) p) h h'' middle"
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bulwahn@27656
|
265 |
and rs_equals: "rs = (if get_array a h'' ! middle \<le> get_array a h ! r then middle + 1
|
bulwahn@27656
|
266 |
else middle)"
|
haftmann@28145
|
267 |
unfolding partition.simps
|
bulwahn@27656
|
268 |
by (elim crelE crel_return crel_nth crel_if crel_upd) simp
|
bulwahn@27656
|
269 |
from `l < r` have "l \<le> r - 1" by arith
|
bulwahn@27656
|
270 |
from part_returns_index_in_bounds[OF part this] rs_equals `l < r` show ?thesis by auto
|
bulwahn@27656
|
271 |
qed
|
bulwahn@27656
|
272 |
|
bulwahn@27656
|
273 |
lemma partition_partitions:
|
bulwahn@27656
|
274 |
assumes crel: "crel (partition a l r) h h' rs"
|
bulwahn@27656
|
275 |
assumes "l < r"
|
bulwahn@27656
|
276 |
shows "(\<forall>i. l \<le> i \<and> i < rs \<longrightarrow> get_array (a::nat array) h' ! i \<le> get_array a h' ! rs) \<and>
|
bulwahn@27656
|
277 |
(\<forall>i. rs < i \<and> i \<le> r \<longrightarrow> get_array a h' ! rs \<le> get_array a h' ! i)"
|
bulwahn@27656
|
278 |
proof -
|
bulwahn@27656
|
279 |
let ?pivot = "get_array a h ! r"
|
bulwahn@27656
|
280 |
from crel obtain middle h1 where part: "crel (part1 a l (r - 1) ?pivot) h h1 middle"
|
bulwahn@27656
|
281 |
and swap: "crel (swap a rs r) h1 h' ()"
|
bulwahn@27656
|
282 |
and rs_equals: "rs = (if get_array a h1 ! middle \<le> ?pivot then middle + 1
|
bulwahn@27656
|
283 |
else middle)"
|
haftmann@28145
|
284 |
unfolding partition.simps
|
bulwahn@27656
|
285 |
by (elim crelE crel_return crel_nth crel_if crel_upd) simp
|
bulwahn@27656
|
286 |
from swap have h'_def: "h' = Heap.upd a r (get_array a h1 ! rs)
|
bulwahn@27656
|
287 |
(Heap.upd a rs (get_array a h1 ! r) h1)"
|
haftmann@28145
|
288 |
unfolding swap_def
|
bulwahn@27656
|
289 |
by (elim crelE crel_return crel_nth crel_upd) simp
|
bulwahn@27656
|
290 |
from swap have in_bounds: "r < Heap.length a h1 \<and> rs < Heap.length a h1"
|
haftmann@28145
|
291 |
unfolding swap_def
|
bulwahn@27656
|
292 |
by (elim crelE crel_return crel_nth crel_upd) simp
|
bulwahn@27656
|
293 |
from swap have swap_length_remains: "Heap.length a h1 = Heap.length a h'"
|
haftmann@28145
|
294 |
unfolding swap_def by (elim crelE crel_return crel_nth crel_upd) auto
|
bulwahn@27656
|
295 |
from `l < r` have "l \<le> r - 1" by simp
|
bulwahn@27656
|
296 |
note middle_in_bounds = part_returns_index_in_bounds[OF part this]
|
bulwahn@27656
|
297 |
from part_outer_remains[OF part] `l < r`
|
bulwahn@27656
|
298 |
have "get_array a h ! r = get_array a h1 ! r"
|
bulwahn@27656
|
299 |
by fastsimp
|
bulwahn@27656
|
300 |
with swap
|
bulwahn@27656
|
301 |
have right_remains: "get_array a h ! r = get_array a h' ! rs"
|
haftmann@28145
|
302 |
unfolding swap_def
|
bulwahn@27656
|
303 |
by (auto simp add: Heap.length_def elim!: crelE crel_return crel_nth crel_upd) (cases "r = rs", auto)
|
bulwahn@27656
|
304 |
from part_partitions [OF part]
|
bulwahn@27656
|
305 |
show ?thesis
|
bulwahn@27656
|
306 |
proof (cases "get_array a h1 ! middle \<le> ?pivot")
|
bulwahn@27656
|
307 |
case True
|
bulwahn@27656
|
308 |
with rs_equals have rs_equals: "rs = middle + 1" by simp
|
bulwahn@27656
|
309 |
{
|
bulwahn@27656
|
310 |
fix i
|
bulwahn@27656
|
311 |
assume i_is_left: "l \<le> i \<and> i < rs"
|
bulwahn@27656
|
312 |
with swap_length_remains in_bounds middle_in_bounds rs_equals `l < r`
|
bulwahn@27656
|
313 |
have i_props: "i < Heap.length a h'" "i \<noteq> r" "i \<noteq> rs" by auto
|
bulwahn@27656
|
314 |
from i_is_left rs_equals have "l \<le> i \<and> i < middle \<or> i = middle" by arith
|
bulwahn@27656
|
315 |
with part_partitions[OF part] right_remains True
|
bulwahn@27656
|
316 |
have "get_array a h1 ! i \<le> get_array a h' ! rs" by fastsimp
|
bulwahn@27656
|
317 |
with i_props h'_def in_bounds have "get_array a h' ! i \<le> get_array a h' ! rs"
|
bulwahn@27656
|
318 |
unfolding Heap.upd_def Heap.length_def by simp
|
bulwahn@27656
|
319 |
}
|
bulwahn@27656
|
320 |
moreover
|
bulwahn@27656
|
321 |
{
|
bulwahn@27656
|
322 |
fix i
|
bulwahn@27656
|
323 |
assume "rs < i \<and> i \<le> r"
|
bulwahn@27656
|
324 |
|
bulwahn@27656
|
325 |
hence "(rs < i \<and> i \<le> r - 1) \<or> (rs < i \<and> i = r)" by arith
|
bulwahn@27656
|
326 |
hence "get_array a h' ! rs \<le> get_array a h' ! i"
|
bulwahn@27656
|
327 |
proof
|
bulwahn@27656
|
328 |
assume i_is: "rs < i \<and> i \<le> r - 1"
|
bulwahn@27656
|
329 |
with swap_length_remains in_bounds middle_in_bounds rs_equals
|
bulwahn@27656
|
330 |
have i_props: "i < Heap.length a h'" "i \<noteq> r" "i \<noteq> rs" by auto
|
bulwahn@27656
|
331 |
from part_partitions[OF part] rs_equals right_remains i_is
|
bulwahn@27656
|
332 |
have "get_array a h' ! rs \<le> get_array a h1 ! i"
|
bulwahn@27656
|
333 |
by fastsimp
|
bulwahn@27656
|
334 |
with i_props h'_def show ?thesis by fastsimp
|
bulwahn@27656
|
335 |
next
|
bulwahn@27656
|
336 |
assume i_is: "rs < i \<and> i = r"
|
bulwahn@27656
|
337 |
with rs_equals have "Suc middle \<noteq> r" by arith
|
bulwahn@27656
|
338 |
with middle_in_bounds `l < r` have "Suc middle \<le> r - 1" by arith
|
bulwahn@27656
|
339 |
with part_partitions[OF part] right_remains
|
bulwahn@27656
|
340 |
have "get_array a h' ! rs \<le> get_array a h1 ! (Suc middle)"
|
bulwahn@27656
|
341 |
by fastsimp
|
bulwahn@27656
|
342 |
with i_is True rs_equals right_remains h'_def
|
bulwahn@27656
|
343 |
show ?thesis using in_bounds
|
bulwahn@27656
|
344 |
unfolding Heap.upd_def Heap.length_def
|
bulwahn@27656
|
345 |
by auto
|
bulwahn@27656
|
346 |
qed
|
bulwahn@27656
|
347 |
}
|
bulwahn@27656
|
348 |
ultimately show ?thesis by auto
|
bulwahn@27656
|
349 |
next
|
bulwahn@27656
|
350 |
case False
|
bulwahn@27656
|
351 |
with rs_equals have rs_equals: "middle = rs" by simp
|
bulwahn@27656
|
352 |
{
|
bulwahn@27656
|
353 |
fix i
|
bulwahn@27656
|
354 |
assume i_is_left: "l \<le> i \<and> i < rs"
|
bulwahn@27656
|
355 |
with swap_length_remains in_bounds middle_in_bounds rs_equals
|
bulwahn@27656
|
356 |
have i_props: "i < Heap.length a h'" "i \<noteq> r" "i \<noteq> rs" by auto
|
bulwahn@27656
|
357 |
from part_partitions[OF part] rs_equals right_remains i_is_left
|
bulwahn@27656
|
358 |
have "get_array a h1 ! i \<le> get_array a h' ! rs" by fastsimp
|
bulwahn@27656
|
359 |
with i_props h'_def have "get_array a h' ! i \<le> get_array a h' ! rs"
|
bulwahn@27656
|
360 |
unfolding Heap.upd_def by simp
|
bulwahn@27656
|
361 |
}
|
bulwahn@27656
|
362 |
moreover
|
bulwahn@27656
|
363 |
{
|
bulwahn@27656
|
364 |
fix i
|
bulwahn@27656
|
365 |
assume "rs < i \<and> i \<le> r"
|
bulwahn@27656
|
366 |
hence "(rs < i \<and> i \<le> r - 1) \<or> i = r" by arith
|
bulwahn@27656
|
367 |
hence "get_array a h' ! rs \<le> get_array a h' ! i"
|
bulwahn@27656
|
368 |
proof
|
bulwahn@27656
|
369 |
assume i_is: "rs < i \<and> i \<le> r - 1"
|
bulwahn@27656
|
370 |
with swap_length_remains in_bounds middle_in_bounds rs_equals
|
bulwahn@27656
|
371 |
have i_props: "i < Heap.length a h'" "i \<noteq> r" "i \<noteq> rs" by auto
|
bulwahn@27656
|
372 |
from part_partitions[OF part] rs_equals right_remains i_is
|
bulwahn@27656
|
373 |
have "get_array a h' ! rs \<le> get_array a h1 ! i"
|
bulwahn@27656
|
374 |
by fastsimp
|
bulwahn@27656
|
375 |
with i_props h'_def show ?thesis by fastsimp
|
bulwahn@27656
|
376 |
next
|
bulwahn@27656
|
377 |
assume i_is: "i = r"
|
bulwahn@27656
|
378 |
from i_is False rs_equals right_remains h'_def
|
bulwahn@27656
|
379 |
show ?thesis using in_bounds
|
bulwahn@27656
|
380 |
unfolding Heap.upd_def Heap.length_def
|
bulwahn@27656
|
381 |
by auto
|
bulwahn@27656
|
382 |
qed
|
bulwahn@27656
|
383 |
}
|
bulwahn@27656
|
384 |
ultimately
|
bulwahn@27656
|
385 |
show ?thesis by auto
|
bulwahn@27656
|
386 |
qed
|
bulwahn@27656
|
387 |
qed
|
bulwahn@27656
|
388 |
|
bulwahn@27656
|
389 |
|
bulwahn@27656
|
390 |
function quicksort :: "nat array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> unit Heap"
|
bulwahn@27656
|
391 |
where
|
bulwahn@27656
|
392 |
"quicksort arr left right =
|
bulwahn@27656
|
393 |
(if (right > left) then
|
bulwahn@27656
|
394 |
do
|
bulwahn@27656
|
395 |
pivotNewIndex \<leftarrow> partition arr left right;
|
bulwahn@27656
|
396 |
pivotNewIndex \<leftarrow> assert (\<lambda>x. left \<le> x \<and> x \<le> right) pivotNewIndex;
|
bulwahn@27656
|
397 |
quicksort arr left (pivotNewIndex - 1);
|
bulwahn@27656
|
398 |
quicksort arr (pivotNewIndex + 1) right
|
bulwahn@27656
|
399 |
done
|
bulwahn@27656
|
400 |
else return ())"
|
bulwahn@27656
|
401 |
by pat_completeness auto
|
bulwahn@27656
|
402 |
|
bulwahn@27656
|
403 |
(* For termination, we must show that the pivotNewIndex is between left and right *)
|
bulwahn@27656
|
404 |
termination
|
bulwahn@27656
|
405 |
by (relation "measure (\<lambda>(a, l, r). (r - l))") auto
|
bulwahn@27656
|
406 |
|
bulwahn@27656
|
407 |
declare quicksort.simps[simp del]
|
bulwahn@27656
|
408 |
|
bulwahn@27656
|
409 |
|
bulwahn@27656
|
410 |
lemma quicksort_permutes:
|
bulwahn@27656
|
411 |
assumes "crel (quicksort a l r) h h' rs"
|
bulwahn@27656
|
412 |
shows "multiset_of (get_array a h')
|
bulwahn@27656
|
413 |
= multiset_of (get_array a h)"
|
bulwahn@27656
|
414 |
using assms
|
bulwahn@27656
|
415 |
proof (induct a l r arbitrary: h h' rs rule: quicksort.induct)
|
bulwahn@27656
|
416 |
case (1 a l r h h' rs)
|
bulwahn@27656
|
417 |
with partition_permutes show ?case
|
haftmann@28145
|
418 |
unfolding quicksort.simps [of a l r]
|
bulwahn@27656
|
419 |
by (elim crel_if crelE crel_assert crel_return) auto
|
bulwahn@27656
|
420 |
qed
|
bulwahn@27656
|
421 |
|
bulwahn@27656
|
422 |
lemma length_remains:
|
bulwahn@27656
|
423 |
assumes "crel (quicksort a l r) h h' rs"
|
bulwahn@27656
|
424 |
shows "Heap.length a h = Heap.length a h'"
|
bulwahn@27656
|
425 |
using assms
|
bulwahn@27656
|
426 |
proof (induct a l r arbitrary: h h' rs rule: quicksort.induct)
|
bulwahn@27656
|
427 |
case (1 a l r h h' rs)
|
bulwahn@27656
|
428 |
with partition_length_remains show ?case
|
haftmann@28145
|
429 |
unfolding quicksort.simps [of a l r]
|
bulwahn@27656
|
430 |
by (elim crel_if crelE crel_assert crel_return) auto
|
bulwahn@27656
|
431 |
qed
|
bulwahn@27656
|
432 |
|
bulwahn@27656
|
433 |
lemma quicksort_outer_remains:
|
bulwahn@27656
|
434 |
assumes "crel (quicksort a l r) h h' rs"
|
bulwahn@27656
|
435 |
shows "\<forall>i. i < l \<or> r < i \<longrightarrow> get_array (a::nat array) h ! i = get_array a h' ! i"
|
bulwahn@27656
|
436 |
using assms
|
bulwahn@27656
|
437 |
proof (induct a l r arbitrary: h h' rs rule: quicksort.induct)
|
bulwahn@27656
|
438 |
case (1 a l r h h' rs)
|
bulwahn@27656
|
439 |
note cr = `crel (quicksort a l r) h h' rs`
|
bulwahn@27656
|
440 |
thus ?case
|
bulwahn@27656
|
441 |
proof (cases "r > l")
|
bulwahn@27656
|
442 |
case False
|
bulwahn@27656
|
443 |
with cr have "h' = h"
|
bulwahn@27656
|
444 |
unfolding quicksort.simps [of a l r]
|
bulwahn@27656
|
445 |
by (elim crel_if crel_return) auto
|
bulwahn@27656
|
446 |
thus ?thesis by simp
|
bulwahn@27656
|
447 |
next
|
bulwahn@27656
|
448 |
case True
|
bulwahn@27656
|
449 |
{
|
bulwahn@27656
|
450 |
fix h1 h2 p ret1 ret2 i
|
bulwahn@27656
|
451 |
assume part: "crel (partition a l r) h h1 p"
|
bulwahn@27656
|
452 |
assume qs1: "crel (quicksort a l (p - 1)) h1 h2 ret1"
|
bulwahn@27656
|
453 |
assume qs2: "crel (quicksort a (p + 1) r) h2 h' ret2"
|
bulwahn@27656
|
454 |
assume pivot: "l \<le> p \<and> p \<le> r"
|
bulwahn@27656
|
455 |
assume i_outer: "i < l \<or> r < i"
|
bulwahn@27656
|
456 |
from partition_outer_remains [OF part True] i_outer
|
bulwahn@27656
|
457 |
have "get_array a h !i = get_array a h1 ! i" by fastsimp
|
bulwahn@27656
|
458 |
moreover
|
bulwahn@27656
|
459 |
with 1(1) [OF True pivot qs1] pivot i_outer
|
bulwahn@27656
|
460 |
have "get_array a h1 ! i = get_array a h2 ! i" by auto
|
bulwahn@27656
|
461 |
moreover
|
bulwahn@27656
|
462 |
with qs2 1(2) [of p h2 h' ret2] True pivot i_outer
|
bulwahn@27656
|
463 |
have "get_array a h2 ! i = get_array a h' ! i" by auto
|
bulwahn@27656
|
464 |
ultimately have "get_array a h ! i= get_array a h' ! i" by simp
|
bulwahn@27656
|
465 |
}
|
bulwahn@27656
|
466 |
with cr show ?thesis
|
haftmann@28145
|
467 |
unfolding quicksort.simps [of a l r]
|
bulwahn@27656
|
468 |
by (elim crel_if crelE crel_assert crel_return) auto
|
bulwahn@27656
|
469 |
qed
|
bulwahn@27656
|
470 |
qed
|
bulwahn@27656
|
471 |
|
bulwahn@27656
|
472 |
lemma quicksort_is_skip:
|
bulwahn@27656
|
473 |
assumes "crel (quicksort a l r) h h' rs"
|
bulwahn@27656
|
474 |
shows "r \<le> l \<longrightarrow> h = h'"
|
bulwahn@27656
|
475 |
using assms
|
haftmann@28145
|
476 |
unfolding quicksort.simps [of a l r]
|
bulwahn@27656
|
477 |
by (elim crel_if crel_return) auto
|
bulwahn@27656
|
478 |
|
bulwahn@27656
|
479 |
lemma quicksort_sorts:
|
bulwahn@27656
|
480 |
assumes "crel (quicksort a l r) h h' rs"
|
bulwahn@27656
|
481 |
assumes l_r_length: "l < Heap.length a h" "r < Heap.length a h"
|
bulwahn@27656
|
482 |
shows "sorted (subarray l (r + 1) a h')"
|
bulwahn@27656
|
483 |
using assms
|
bulwahn@27656
|
484 |
proof (induct a l r arbitrary: h h' rs rule: quicksort.induct)
|
bulwahn@27656
|
485 |
case (1 a l r h h' rs)
|
bulwahn@27656
|
486 |
note cr = `crel (quicksort a l r) h h' rs`
|
bulwahn@27656
|
487 |
thus ?case
|
bulwahn@27656
|
488 |
proof (cases "r > l")
|
bulwahn@27656
|
489 |
case False
|
bulwahn@27656
|
490 |
hence "l \<ge> r + 1 \<or> l = r" by arith
|
bulwahn@27656
|
491 |
with length_remains[OF cr] 1(5) show ?thesis
|
bulwahn@27656
|
492 |
by (auto simp add: subarray_Nil subarray_single)
|
bulwahn@27656
|
493 |
next
|
bulwahn@27656
|
494 |
case True
|
bulwahn@27656
|
495 |
{
|
bulwahn@27656
|
496 |
fix h1 h2 p
|
bulwahn@27656
|
497 |
assume part: "crel (partition a l r) h h1 p"
|
bulwahn@27656
|
498 |
assume qs1: "crel (quicksort a l (p - 1)) h1 h2 ()"
|
bulwahn@27656
|
499 |
assume qs2: "crel (quicksort a (p + 1) r) h2 h' ()"
|
bulwahn@27656
|
500 |
from partition_returns_index_in_bounds [OF part True]
|
bulwahn@27656
|
501 |
have pivot: "l\<le> p \<and> p \<le> r" .
|
bulwahn@27656
|
502 |
note length_remains = length_remains[OF qs2] length_remains[OF qs1] partition_length_remains[OF part]
|
bulwahn@27656
|
503 |
from quicksort_outer_remains [OF qs2] quicksort_outer_remains [OF qs1] pivot quicksort_is_skip[OF qs1]
|
bulwahn@27656
|
504 |
have pivot_unchanged: "get_array a h1 ! p = get_array a h' ! p" by (cases p, auto)
|
haftmann@28013
|
505 |
(*-- First of all, by induction hypothesis both sublists are sorted. *)
|
bulwahn@27656
|
506 |
from 1(1)[OF True pivot qs1] length_remains pivot 1(5)
|
bulwahn@27656
|
507 |
have IH1: "sorted (subarray l p a h2)" by (cases p, auto simp add: subarray_Nil)
|
bulwahn@27656
|
508 |
from quicksort_outer_remains [OF qs2] length_remains
|
bulwahn@27656
|
509 |
have left_subarray_remains: "subarray l p a h2 = subarray l p a h'"
|
bulwahn@27656
|
510 |
by (simp add: subarray_eq_samelength_iff)
|
bulwahn@27656
|
511 |
with IH1 have IH1': "sorted (subarray l p a h')" by simp
|
bulwahn@27656
|
512 |
from 1(2)[OF True pivot qs2] pivot 1(5) length_remains
|
bulwahn@27656
|
513 |
have IH2: "sorted (subarray (p + 1) (r + 1) a h')"
|
haftmann@28013
|
514 |
by (cases "Suc p \<le> r", auto simp add: subarray_Nil)
|
haftmann@28013
|
515 |
(* -- Secondly, both sublists remain partitioned. *)
|
bulwahn@27656
|
516 |
from partition_partitions[OF part True]
|
bulwahn@27656
|
517 |
have part_conds1: "\<forall>j. j \<in> set (subarray l p a h1) \<longrightarrow> j \<le> get_array a h1 ! p "
|
haftmann@28013
|
518 |
and part_conds2: "\<forall>j. j \<in> set (subarray (p + 1) (r + 1) a h1) \<longrightarrow> get_array a h1 ! p \<le> j"
|
haftmann@28013
|
519 |
by (auto simp add: all_in_set_subarray_conv)
|
bulwahn@27656
|
520 |
from quicksort_outer_remains [OF qs1] quicksort_permutes [OF qs1] True
|
haftmann@28013
|
521 |
length_remains 1(5) pivot multiset_of_sublist [of l p "get_array a h1" "get_array a h2"]
|
bulwahn@27656
|
522 |
have multiset_partconds1: "multiset_of (subarray l p a h2) = multiset_of (subarray l p a h1)"
|
bulwahn@27656
|
523 |
unfolding Heap.length_def subarray_def by (cases p, auto)
|
bulwahn@27656
|
524 |
with left_subarray_remains part_conds1 pivot_unchanged
|
bulwahn@27656
|
525 |
have part_conds2': "\<forall>j. j \<in> set (subarray l p a h') \<longrightarrow> j \<le> get_array a h' ! p"
|
haftmann@28013
|
526 |
by (simp, subst set_of_multiset_of[symmetric], simp)
|
haftmann@28013
|
527 |
(* -- These steps are the analogous for the right sublist \<dots> *)
|
bulwahn@27656
|
528 |
from quicksort_outer_remains [OF qs1] length_remains
|
bulwahn@27656
|
529 |
have right_subarray_remains: "subarray (p + 1) (r + 1) a h1 = subarray (p + 1) (r + 1) a h2"
|
bulwahn@27656
|
530 |
by (auto simp add: subarray_eq_samelength_iff)
|
bulwahn@27656
|
531 |
from quicksort_outer_remains [OF qs2] quicksort_permutes [OF qs2] True
|
haftmann@28013
|
532 |
length_remains 1(5) pivot multiset_of_sublist [of "p + 1" "r + 1" "get_array a h2" "get_array a h'"]
|
bulwahn@27656
|
533 |
have multiset_partconds2: "multiset_of (subarray (p + 1) (r + 1) a h') = multiset_of (subarray (p + 1) (r + 1) a h2)"
|
haftmann@28013
|
534 |
unfolding Heap.length_def subarray_def by auto
|
bulwahn@27656
|
535 |
with right_subarray_remains part_conds2 pivot_unchanged
|
bulwahn@27656
|
536 |
have part_conds1': "\<forall>j. j \<in> set (subarray (p + 1) (r + 1) a h') \<longrightarrow> get_array a h' ! p \<le> j"
|
haftmann@28013
|
537 |
by (simp, subst set_of_multiset_of[symmetric], simp)
|
haftmann@28013
|
538 |
(* -- Thirdly and finally, we show that the array is sorted
|
haftmann@28013
|
539 |
following from the facts above. *)
|
bulwahn@27656
|
540 |
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'"
|
bulwahn@27656
|
541 |
by (simp add: subarray_nth_array_Cons, cases "l < p") (auto simp add: subarray_append subarray_Nil)
|
bulwahn@27656
|
542 |
with IH1' IH2 part_conds1' part_conds2' pivot have ?thesis
|
bulwahn@27656
|
543 |
unfolding subarray_def
|
bulwahn@27656
|
544 |
apply (auto simp add: sorted_append sorted_Cons all_in_set_sublist'_conv)
|
bulwahn@27656
|
545 |
by (auto simp add: set_sublist' dest: le_trans [of _ "get_array a h' ! p"])
|
bulwahn@27656
|
546 |
}
|
bulwahn@27656
|
547 |
with True cr show ?thesis
|
haftmann@28145
|
548 |
unfolding quicksort.simps [of a l r]
|
bulwahn@27656
|
549 |
by (elim crel_if crel_return crelE crel_assert) auto
|
bulwahn@27656
|
550 |
qed
|
bulwahn@27656
|
551 |
qed
|
bulwahn@27656
|
552 |
|
bulwahn@27656
|
553 |
|
bulwahn@27656
|
554 |
lemma quicksort_is_sort:
|
bulwahn@27656
|
555 |
assumes crel: "crel (quicksort a 0 (Heap.length a h - 1)) h h' rs"
|
bulwahn@27656
|
556 |
shows "get_array a h' = sort (get_array a h)"
|
bulwahn@27656
|
557 |
proof (cases "get_array a h = []")
|
bulwahn@27656
|
558 |
case True
|
bulwahn@27656
|
559 |
with quicksort_is_skip[OF crel] show ?thesis
|
bulwahn@27656
|
560 |
unfolding Heap.length_def by simp
|
bulwahn@27656
|
561 |
next
|
bulwahn@27656
|
562 |
case False
|
bulwahn@27656
|
563 |
from quicksort_sorts [OF crel] False have "sorted (sublist' 0 (List.length (get_array a h)) (get_array a h'))"
|
bulwahn@27656
|
564 |
unfolding Heap.length_def subarray_def by auto
|
bulwahn@27656
|
565 |
with length_remains[OF crel] have "sorted (get_array a h')"
|
bulwahn@27656
|
566 |
unfolding Heap.length_def by simp
|
bulwahn@27656
|
567 |
with quicksort_permutes [OF crel] properties_for_sort show ?thesis by fastsimp
|
bulwahn@27656
|
568 |
qed
|
bulwahn@27656
|
569 |
|
bulwahn@27656
|
570 |
subsection {* No Errors in quicksort *}
|
bulwahn@27656
|
571 |
text {* We have proved that quicksort sorts (if no exceptions occur).
|
bulwahn@27656
|
572 |
We will now show that exceptions do not occur. *}
|
bulwahn@27656
|
573 |
|
bulwahn@27656
|
574 |
lemma noError_part1:
|
bulwahn@27656
|
575 |
assumes "l < Heap.length a h" "r < Heap.length a h"
|
bulwahn@27656
|
576 |
shows "noError (part1 a l r p) h"
|
bulwahn@27656
|
577 |
using assms
|
bulwahn@27656
|
578 |
proof (induct a l r p arbitrary: h rule: part1.induct)
|
bulwahn@27656
|
579 |
case (1 a l r p)
|
bulwahn@27656
|
580 |
thus ?case
|
haftmann@28145
|
581 |
unfolding part1.simps [of a l r] swap_def
|
bulwahn@27656
|
582 |
by (auto intro!: noError_if noErrorI noError_return noError_nth noError_upd elim!: crelE crel_upd crel_nth crel_return)
|
bulwahn@27656
|
583 |
qed
|
bulwahn@27656
|
584 |
|
bulwahn@27656
|
585 |
lemma noError_partition:
|
bulwahn@27656
|
586 |
assumes "l < r" "l < Heap.length a h" "r < Heap.length a h"
|
bulwahn@27656
|
587 |
shows "noError (partition a l r) h"
|
bulwahn@27656
|
588 |
using assms
|
haftmann@28145
|
589 |
unfolding partition.simps swap_def
|
bulwahn@27656
|
590 |
apply (auto intro!: noError_if noErrorI noError_return noError_nth noError_upd noError_part1 elim!: crelE crel_upd crel_nth crel_return)
|
bulwahn@27656
|
591 |
apply (frule part_length_remains)
|
bulwahn@27656
|
592 |
apply (frule part_returns_index_in_bounds)
|
bulwahn@27656
|
593 |
apply auto
|
bulwahn@27656
|
594 |
apply (frule part_length_remains)
|
bulwahn@27656
|
595 |
apply (frule part_returns_index_in_bounds)
|
bulwahn@27656
|
596 |
apply auto
|
bulwahn@27656
|
597 |
apply (frule part_length_remains)
|
bulwahn@27656
|
598 |
apply auto
|
bulwahn@27656
|
599 |
done
|
bulwahn@27656
|
600 |
|
bulwahn@27656
|
601 |
lemma noError_quicksort:
|
bulwahn@27656
|
602 |
assumes "l < Heap.length a h" "r < Heap.length a h"
|
bulwahn@27656
|
603 |
shows "noError (quicksort a l r) h"
|
bulwahn@27656
|
604 |
using assms
|
bulwahn@27656
|
605 |
proof (induct a l r arbitrary: h rule: quicksort.induct)
|
bulwahn@27656
|
606 |
case (1 a l ri h)
|
bulwahn@27656
|
607 |
thus ?case
|
haftmann@28145
|
608 |
unfolding quicksort.simps [of a l ri]
|
bulwahn@27656
|
609 |
apply (auto intro!: noError_if noErrorI noError_return noError_nth noError_upd noError_assert noError_partition)
|
bulwahn@27656
|
610 |
apply (frule partition_returns_index_in_bounds)
|
bulwahn@27656
|
611 |
apply auto
|
bulwahn@27656
|
612 |
apply (frule partition_returns_index_in_bounds)
|
bulwahn@27656
|
613 |
apply auto
|
bulwahn@27656
|
614 |
apply (auto elim!: crel_assert dest!: partition_length_remains length_remains)
|
bulwahn@27656
|
615 |
apply (subgoal_tac "Suc r \<le> ri \<or> r = ri")
|
bulwahn@27656
|
616 |
apply (erule disjE)
|
bulwahn@27656
|
617 |
apply auto
|
haftmann@28145
|
618 |
unfolding quicksort.simps [of a "Suc ri" ri]
|
bulwahn@27656
|
619 |
apply (auto intro!: noError_if noError_return)
|
bulwahn@27656
|
620 |
done
|
bulwahn@27656
|
621 |
qed
|
bulwahn@27656
|
622 |
|
haftmann@27674
|
623 |
|
haftmann@27674
|
624 |
subsection {* Example *}
|
haftmann@27674
|
625 |
|
haftmann@27674
|
626 |
definition "qsort a = do
|
haftmann@27674
|
627 |
k \<leftarrow> length a;
|
haftmann@27674
|
628 |
quicksort a 0 (k - 1);
|
haftmann@27674
|
629 |
return a
|
haftmann@27674
|
630 |
done"
|
haftmann@27674
|
631 |
|
haftmann@27674
|
632 |
ML {* @{code qsort} (Array.fromList [42, 2, 3, 5, 0, 1705, 8, 3, 15]) () *}
|
haftmann@27674
|
633 |
|
haftmann@31872
|
634 |
(*export_code qsort in SML_imp module_name QSort*)
|
haftmann@29730
|
635 |
export_code qsort in OCaml module_name QSort file -
|
haftmann@31872
|
636 |
(*export_code qsort in OCaml_imp module_name QSort file -*)
|
haftmann@29730
|
637 |
export_code qsort in Haskell module_name QSort file -
|
haftmann@27674
|
638 |
|
bulwahn@27656
|
639 |
end |