(* Title: HOL/Library/Ref.thy
Author: John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen
*)
header {* Monadic references *}
theory Ref
imports Array
begin
text {*
Imperative reference operations; modeled after their ML counterparts.
See http://caml.inria.fr/pub/docs/manual-caml-light/node14.15.html
and http://www.smlnj.org/doc/Conversion/top-level-comparison.html
*}
subsection {* Primitive layer *}
definition present :: "heap \<Rightarrow> 'a\<Colon>heap ref \<Rightarrow> bool" where
"present h r \<longleftrightarrow> addr_of_ref r < lim h"
definition get :: "heap \<Rightarrow> 'a\<Colon>heap ref \<Rightarrow> 'a" where
"get h = from_nat \<circ> refs h TYPEREP('a) \<circ> addr_of_ref"
definition set :: "'a\<Colon>heap ref \<Rightarrow> 'a \<Rightarrow> heap \<Rightarrow> heap" where
"set r x = refs_update
(\<lambda>h. h(TYPEREP('a) := ((h (TYPEREP('a))) (addr_of_ref r := to_nat x))))"
definition alloc :: "'a \<Rightarrow> heap \<Rightarrow> 'a\<Colon>heap ref \<times> heap" where
"alloc x h = (let
l = lim h;
r = Ref l
in (r, set r x (h\<lparr>lim := l + 1\<rparr>)))"
definition noteq :: "'a\<Colon>heap ref \<Rightarrow> 'b\<Colon>heap ref \<Rightarrow> bool" (infix "=!=" 70) where
"r =!= s \<longleftrightarrow> TYPEREP('a) \<noteq> TYPEREP('b) \<or> addr_of_ref r \<noteq> addr_of_ref s"
lemma noteq_sym: "r =!= s \<Longrightarrow> s =!= r"
and unequal [simp]: "r \<noteq> r' \<longleftrightarrow> r =!= r'" -- "same types!"
by (auto simp add: noteq_def)
lemma noteq_irrefl: "r =!= r \<Longrightarrow> False"
by (auto simp add: noteq_def)
lemma present_alloc_neq: "present h r \<Longrightarrow> r =!= fst (alloc v h)"
by (simp add: present_def alloc_def noteq_def Let_def)
lemma next_fresh [simp]:
assumes "(r, h') = alloc x h"
shows "\<not> present h r"
using assms by (cases h) (auto simp add: alloc_def present_def Let_def)
lemma next_present [simp]:
assumes "(r, h') = alloc x h"
shows "present h' r"
using assms by (cases h) (auto simp add: alloc_def set_def present_def Let_def)
lemma get_set_eq [simp]:
"get (set r x h) r = x"
by (simp add: get_def set_def)
lemma get_set_neq [simp]:
"r =!= s \<Longrightarrow> get (set s x h) r = get h r"
by (simp add: noteq_def get_def set_def)
lemma set_same [simp]:
"set r x (set r y h) = set r x h"
by (simp add: set_def)
lemma set_set_swap:
"r =!= r' \<Longrightarrow> set r x (set r' x' h) = set r' x' (set r x h)"
by (simp add: noteq_def set_def expand_fun_eq)
lemma alloc_set:
"fst (alloc x (set r x' h)) = fst (alloc x h)"
by (simp add: alloc_def set_def Let_def)
lemma get_alloc [simp]:
"get (snd (alloc x h)) (fst (alloc x' h)) = x"
by (simp add: alloc_def Let_def)
lemma set_alloc [simp]:
"set (fst (alloc v h)) v' (snd (alloc v h)) = snd (alloc v' h)"
by (simp add: alloc_def Let_def)
lemma get_alloc_neq: "r =!= fst (alloc v h) \<Longrightarrow>
get (snd (alloc v h)) r = get h r"
by (simp add: get_def set_def alloc_def Let_def noteq_def)
lemma lim_set [simp]:
"lim (set r v h) = lim h"
by (simp add: set_def)
lemma present_alloc [simp]:
"present h r \<Longrightarrow> present (snd (alloc v h)) r"
by (simp add: present_def alloc_def Let_def)
lemma present_set [simp]:
"present (set r v h) = present h"
by (simp add: present_def expand_fun_eq)
lemma noteq_I:
"present h r \<Longrightarrow> \<not> present h r' \<Longrightarrow> r =!= r'"
by (auto simp add: noteq_def present_def)
subsection {* Primitives *}
definition ref :: "'a\<Colon>heap \<Rightarrow> 'a ref Heap" where
[code del]: "ref v = Heap_Monad.heap (alloc v)"
definition lookup :: "'a\<Colon>heap ref \<Rightarrow> 'a Heap" ("!_" 61) where
[code del]: "lookup r = Heap_Monad.heap (\<lambda>h. (get h r, h))"
definition update :: "'a ref \<Rightarrow> 'a\<Colon>heap \<Rightarrow> unit Heap" ("_ := _" 62) where
[code del]: "update r v = Heap_Monad.heap (\<lambda>h. ((), set r v h))"
subsection {* Derivates *}
definition change :: "('a\<Colon>heap \<Rightarrow> 'a) \<Rightarrow> 'a ref \<Rightarrow> 'a Heap" where
"change f r = (do
x \<leftarrow> ! r;
let y = f x;
r := y;
return y
done)"
subsection {* Properties *}
lemma lookup_chain:
"(!r \<guillemotright> f) = f"
by (rule Heap_eqI) (simp add: lookup_def)
lemma update_change [code]:
"r := e = change (\<lambda>_. e) r \<guillemotright> return ()"
by (rule Heap_eqI) (simp add: change_def lookup_chain)
text {* Non-interaction between imperative array and imperative references *}
lemma get_array_set [simp]:
"get_array a (set r v h) = get_array a h"
by (simp add: get_array_def set_def)
lemma nth_set [simp]:
"get_array a (set r v h) ! i = get_array a h ! i"
by simp
lemma get_change [simp]:
"get (Array.change a i v h) r = get h r"
by (simp add: get_def Array.change_def set_array_def)
lemma alloc_change:
"fst (alloc v (Array.change a i v' h)) = fst (alloc v h)"
by (simp add: Array.change_def get_array_def set_array_def alloc_def Let_def)
lemma change_set_swap:
"Array.change a i v (set r v' h) = set r v' (Array.change a i v h)"
by (simp add: Array.change_def get_array_def set_array_def set_def)
lemma length_alloc [simp]:
"Array.length a (snd (alloc v h)) = Array.length a h"
by (simp add: Array.length_def get_array_def alloc_def set_def Let_def)
lemma get_array_alloc [simp]:
"get_array a (snd (alloc v h)) = get_array a h"
by (simp add: get_array_def alloc_def set_def Let_def)
lemma present_change [simp]:
"present (Array.change a i v h) = present h"
by (simp add: Array.change_def set_array_def expand_fun_eq present_def)
lemma array_present_set [simp]:
"array_present a (set r v h) = array_present a h"
by (simp add: array_present_def set_def)
lemma array_present_alloc [simp]:
"array_present a h \<Longrightarrow> array_present a (snd (alloc v h))"
by (simp add: array_present_def alloc_def Let_def)
lemma set_array_set_swap:
"set_array a xs (set r x' h) = set r x' (set_array a xs h)"
by (simp add: set_array_def set_def)
hide_const (open) present get set alloc lookup update change
subsection {* Code generator setup *}
subsubsection {* SML *}
code_type ref (SML "_/ Unsynchronized.ref")
code_const Ref (SML "raise/ (Fail/ \"bare Ref\")")
code_const ref (SML "(fn/ ()/ =>/ Unsynchronized.ref/ _)")
code_const Ref.lookup (SML "(fn/ ()/ =>/ !/ _)")
code_const Ref.update (SML "(fn/ ()/ =>/ _/ :=/ _)")
code_reserved SML ref
subsubsection {* OCaml *}
code_type ref (OCaml "_/ ref")
code_const Ref (OCaml "failwith/ \"bare Ref\")")
code_const ref (OCaml "(fn/ ()/ =>/ ref/ _)")
code_const Ref.lookup (OCaml "(fn/ ()/ =>/ !/ _)")
code_const Ref.update (OCaml "(fn/ ()/ =>/ _/ :=/ _)")
code_reserved OCaml ref
subsubsection {* Haskell *}
code_type ref (Haskell "Heap.STRef/ Heap.RealWorld/ _")
code_const Ref (Haskell "error/ \"bare Ref\"")
code_const ref (Haskell "Heap.newSTRef")
code_const Ref.lookup (Haskell "Heap.readSTRef")
code_const Ref.update (Haskell "Heap.writeSTRef")
end