(* Title: HOL/Imperative_HOL/ex/Imperative_Reverse.thy
Author: Lukas Bulwahn, TU Muenchen
*)
header {* An imperative in-place reversal on arrays *}
theory Imperative_Reverse
imports Subarray Imperative_HOL
begin
fun swap :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> unit Heap" where
"swap a i j = do {
x \<leftarrow> Array.nth a i;
y \<leftarrow> Array.nth a j;
Array.upd i y a;
Array.upd j x a;
return ()
}"
fun rev :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> unit Heap" where
"rev a i j = (if (i < j) then do {
swap a i j;
rev a (i + 1) (j - 1)
}
else return ())"
declare swap.simps [simp del] rev.simps [simp del]
lemma swap_pointwise: assumes "crel (swap a i j) h h' r"
shows "Array.get h' a ! k = (if k = i then Array.get h a ! j
else if k = j then Array.get h a ! i
else Array.get h a ! k)"
using assms unfolding swap.simps
by (elim crel_elims)
(auto simp: length_def)
lemma rev_pointwise: assumes "crel (rev a i j) h h' r"
shows "Array.get h' a ! k = (if k < i then Array.get h a ! k
else if j < k then Array.get h a ! k
else Array.get h a ! (j - (k - i)))" (is "?P a i j h h'")
using assms proof (induct a i j arbitrary: h h' rule: rev.induct)
case (1 a i j h h'')
thus ?case
proof (cases "i < j")
case True
with 1[unfolded rev.simps[of a i j]]
obtain h' where
swp: "crel (swap a i j) h h' ()"
and rev: "crel (rev a (i + 1) (j - 1)) h' h'' ()"
by (auto elim: crel_elims)
from rev 1 True
have eq: "?P a (i + 1) (j - 1) h' h''" by auto
have "k < i \<or> i = k \<or> (i < k \<and> k < j) \<or> j = k \<or> j < k" by arith
with True show ?thesis
by (elim disjE) (auto simp: eq swap_pointwise[OF swp])
next
case False
with 1[unfolded rev.simps[of a i j]]
show ?thesis
by (cases "k = j") (auto elim: crel_elims)
qed
qed
lemma rev_length:
assumes "crel (rev a i j) h h' r"
shows "Array.length h a = Array.length h' a"
using assms
proof (induct a i j arbitrary: h h' rule: rev.induct)
case (1 a i j h h'')
thus ?case
proof (cases "i < j")
case True
with 1[unfolded rev.simps[of a i j]]
obtain h' where
swp: "crel (swap a i j) h h' ()"
and rev: "crel (rev a (i + 1) (j - 1)) h' h'' ()"
by (auto elim: crel_elims)
from swp rev 1 True show ?thesis
unfolding swap.simps
by (elim crel_elims) fastsimp
next
case False
with 1[unfolded rev.simps[of a i j]]
show ?thesis
by (auto elim: crel_elims)
qed
qed
lemma rev2_rev': assumes "crel (rev a i j) h h' u"
assumes "j < Array.length h a"
shows "subarray i (j + 1) a h' = List.rev (subarray i (j + 1) a h)"
proof -
{
fix k
assume "k < Suc j - i"
with rev_pointwise[OF assms(1)] have "Array.get h' a ! (i + k) = Array.get h a ! (j - k)"
by auto
}
with assms(2) rev_length[OF assms(1)] show ?thesis
unfolding subarray_def Array.length_def
by (auto simp add: length_sublist' rev_nth min_def nth_sublist' intro!: nth_equalityI)
qed
lemma rev2_rev:
assumes "crel (rev a 0 (Array.length h a - 1)) h h' u"
shows "Array.get h' a = List.rev (Array.get h a)"
using rev2_rev'[OF assms] rev_length[OF assms] assms
by (cases "Array.length h a = 0", auto simp add: Array.length_def
subarray_def sublist'_all rev.simps[where j=0] elim!: crel_elims)
(drule sym[of "List.length (Array.get h a)"], simp)
export_code rev checking SML SML_imp OCaml? OCaml_imp? Haskell? Scala?
end