--- a/src/HOL/IsaMakefile Wed Feb 27 21:41:07 2008 +0100
+++ b/src/HOL/IsaMakefile Wed Feb 27 21:41:08 2008 +0100
@@ -233,7 +233,9 @@
Library/Code_Index.thy Library/Code_Char.thy Library/Code_Char_chr.thy \
Library/Code_Integer.thy Library/Code_Message.thy \
Library/Abstract_Rat.thy Library/Univ_Poly.thy\
- Library/Numeral_Type.thy Library/Boolean_Algebra.thy
+ Library/Numeral_Type.thy Library/Boolean_Algebra.thy Library/Countable.thy \
+ Library/RType.thy Library/Heap.thy Library/Heap_Monad.thy Library/Array.thy \
+ Library/Ref.thy Library/Imperative_HOL.thy
@cd Library; $(ISATOOL) usedir $(OUT)/HOL Library
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Array.thy Wed Feb 27 21:41:08 2008 +0100
@@ -0,0 +1,130 @@
+(* Title: HOL/Library/Array.thy
+ ID: $Id$
+ Author: John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen
+*)
+
+header {* Monadic arrays *}
+
+theory Array
+imports Heap_Monad
+begin
+
+subsection {* Primitives *}
+
+definition
+ new :: "nat \<Rightarrow> 'a\<Colon>heap \<Rightarrow> 'a array Heap" where
+ [code del]: "new n x = Heap_Monad.heap (Heap.array n x)"
+
+definition
+ of_list :: "'a\<Colon>heap list \<Rightarrow> 'a array Heap" where
+ [code del]: "of_list xs = Heap_Monad.heap (Heap.array_of_list xs)"
+
+definition
+ length :: "'a\<Colon>heap array \<Rightarrow> nat Heap" where
+ [code del]: "length arr = Heap_Monad.heap (\<lambda>h. (Heap.length arr h, h))"
+
+definition
+ nth :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a Heap"
+where
+ [code del]: "nth a i = (do len \<leftarrow> length a;
+ (if i < len
+ then Heap_Monad.heap (\<lambda>h. (get_array a h ! i, h))
+ else raise (''array lookup: index out of range''))
+ done)"
+
+-- {* FIXME adjustion for List theory *}
+no_syntax
+ nth :: "'a list \<Rightarrow> nat \<Rightarrow> 'a" (infixl "!" 100)
+
+abbreviation
+ nth_list :: "'a list \<Rightarrow> nat \<Rightarrow> 'a" (infixl "!" 100)
+where
+ "nth_list \<equiv> List.nth"
+
+definition
+ upd :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a\<Colon>heap array Heap"
+where
+ [code del]: "upd i x a = (do len \<leftarrow> length a;
+ (if i < len
+ then Heap_Monad.heap (\<lambda>h. ((), Heap.upd a i x h))
+ else raise (''array update: index out of range''));
+ return a
+ done)"
+
+lemma upd_return:
+ "upd i x a \<guillemotright> return a = upd i x a"
+ unfolding upd_def by (simp add: monad_simp)
+
+
+subsection {* Derivates *}
+
+definition
+ map_entry :: "nat \<Rightarrow> ('a\<Colon>heap \<Rightarrow> 'a) \<Rightarrow> 'a array \<Rightarrow> 'a array Heap"
+where
+ "map_entry i f a = (do
+ x \<leftarrow> nth a i;
+ upd i (f x) a
+ done)"
+
+definition
+ swap :: "nat \<Rightarrow> 'a \<Rightarrow> 'a\<Colon>heap array \<Rightarrow> 'a Heap"
+where
+ "swap i x a = (do
+ y \<leftarrow> nth a i;
+ upd i x a;
+ return x
+ done)"
+
+definition
+ make :: "nat \<Rightarrow> (nat \<Rightarrow> 'a\<Colon>heap) \<Rightarrow> 'a array Heap"
+where
+ "make n f = of_list (map f [0 ..< n])"
+
+definition
+ freeze :: "'a\<Colon>heap array \<Rightarrow> 'a list Heap"
+where
+ "freeze a = (do
+ n \<leftarrow> length a;
+ mapM (nth a) [0..<n]
+ done)"
+
+definition
+ map :: "('a\<Colon>heap \<Rightarrow> 'a) \<Rightarrow> 'a array \<Rightarrow> 'a array Heap"
+where
+ "map f a = (do
+ n \<leftarrow> length a;
+ foldM (\<lambda>n. map_entry n f) [0..<n] a
+ done)"
+
+hide (open) const new map -- {* avoid clashed with some popular names *}
+
+
+subsection {* Converting arrays to lists *}
+
+primrec list_of_aux :: "nat \<Rightarrow> ('a\<Colon>heap) array \<Rightarrow> 'a list \<Rightarrow> 'a list Heap" where
+ "list_of_aux 0 a xs = return xs"
+ | "list_of_aux (Suc n) a xs = (do
+ x \<leftarrow> Array.nth a n;
+ list_of_aux n a (x#xs)
+ done)"
+
+definition list_of :: "('a\<Colon>heap) array \<Rightarrow> 'a list Heap" where
+ "list_of a = (do n \<leftarrow> Array.length a;
+ list_of_aux n a []
+ done)"
+
+
+subsection {* Properties *}
+
+lemma array_make [code func]:
+ "Array.new n x = make n (\<lambda>_. x)"
+ by (induct n) (simp_all add: make_def new_def Heap_Monad.heap_def
+ monad_simp array_of_list_replicate [symmetric]
+ map_replicate_trivial replicate_append_same
+ of_list_def)
+
+lemma array_of_list_make [code func]:
+ "of_list xs = make (List.length xs) (\<lambda>n. xs ! n)"
+ unfolding make_def map_nth ..
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Heap.thy Wed Feb 27 21:41:08 2008 +0100
@@ -0,0 +1,450 @@
+(* Title: HOL/Library/Heap.thy
+ ID: $Id$
+ Author: John Matthews, Galois Connections; Alexander Krauss, TU Muenchen
+*)
+
+header {* A polymorphic heap based on cantor encodings *}
+
+theory Heap
+imports Main Countable RType
+begin
+
+subsection {* Representable types *}
+
+text {* The type class of representable types *}
+
+class heap = rtype + countable
+
+text {* Instances for common HOL types *}
+
+instance nat :: heap ..
+
+instance "*" :: (heap, heap) heap ..
+
+instance "+" :: (heap, heap) heap ..
+
+instance list :: (heap) heap ..
+
+instance option :: (heap) heap ..
+
+instance int :: heap ..
+
+instance set :: ("{heap, finite}") heap ..
+
+instance message_string :: countable
+ by (rule countable_classI [of "message_string_case to_nat"])
+ (auto split: message_string.splits)
+
+instance message_string :: heap ..
+
+text {* Reflected types themselves are heap-representable *}
+
+instantiation rtype :: countable
+begin
+
+lemma list_size_size_append:
+ "list_size size (xs @ ys) = list_size size xs + list_size size ys"
+ by (induct xs, auto)
+
+lemma rtype_size: "t = RType.RType c ts \<Longrightarrow> t' \<in> set ts \<Longrightarrow> size t' < size t"
+ by (frule split_list) (auto simp add: list_size_size_append)
+
+function to_nat_rtype :: "rtype \<Rightarrow> nat" where
+ "to_nat_rtype (RType.RType c ts) = to_nat (to_nat c, to_nat (map to_nat_rtype ts))"
+by pat_completeness auto
+
+termination by (relation "measure (\<lambda>x. size x)")
+ (simp, simp only: in_measure rtype_size)
+
+instance proof (rule countable_classI)
+ fix t t' :: rtype
+ have "(\<forall>t'. to_nat_rtype t = to_nat_rtype t' \<longrightarrow> t = t')
+ \<and> (\<forall>ts'. map to_nat_rtype ts = map to_nat_rtype ts' \<longrightarrow> ts = ts')"
+ proof (induct rule: rtype.induct)
+ case (RType c ts) show ?case
+ proof (rule allI, rule impI)
+ fix t'
+ assume hyp: "to_nat_rtype (rtype.RType c ts) = to_nat_rtype t'"
+ then obtain c' ts' where t': "t' = (rtype.RType c' ts')"
+ by (cases t') auto
+ with RType hyp have "c = c'" and "ts = ts'" by simp_all
+ with t' show "rtype.RType c ts = t'" by simp
+ qed
+ next
+ case Nil_rtype then show ?case by simp
+ next
+ case (Cons_rtype t ts) then show ?case by auto
+ qed
+ then have "to_nat_rtype t = to_nat_rtype t' \<Longrightarrow> t = t'" by auto
+ moreover assume "to_nat_rtype t = to_nat_rtype t'"
+ ultimately show "t = t'" by simp
+qed
+
+end
+
+instance rtype :: heap ..
+
+
+subsection {* A polymorphic heap with dynamic arrays and references *}
+
+types addr = nat -- "untyped heap references"
+
+datatype 'a array = Array addr
+datatype 'a ref = Ref addr -- "note the phantom type 'a "
+
+primrec addr_of_array :: "'a array \<Rightarrow> addr" where
+ "addr_of_array (Array x) = x"
+
+primrec addr_of_ref :: "'a ref \<Rightarrow> addr" where
+ "addr_of_ref (Ref x) = x"
+
+lemma addr_of_array_inj [simp]:
+ "addr_of_array a = addr_of_array a' \<longleftrightarrow> a = a'"
+ by (cases a, cases a') simp_all
+
+lemma addr_of_ref_inj [simp]:
+ "addr_of_ref r = addr_of_ref r' \<longleftrightarrow> r = r'"
+ by (cases r, cases r') simp_all
+
+instance array :: (type) countable
+ by (rule countable_classI [of addr_of_array]) simp
+
+instance ref :: (type) countable
+ by (rule countable_classI [of addr_of_ref]) simp
+
+setup {*
+ Sign.add_const_constraint (@{const_name Array}, SOME @{typ "nat \<Rightarrow> 'a\<Colon>heap array"})
+ #> Sign.add_const_constraint (@{const_name Ref}, SOME @{typ "nat \<Rightarrow> 'a\<Colon>heap ref"})
+ #> Sign.add_const_constraint (@{const_name addr_of_array}, SOME @{typ "'a\<Colon>heap array \<Rightarrow> nat"})
+ #> Sign.add_const_constraint (@{const_name addr_of_ref}, SOME @{typ "'a\<Colon>heap ref \<Rightarrow> nat"})
+*}
+
+types heap_rep = nat -- "representable values"
+
+record heap =
+ arrays :: "rtype \<Rightarrow> addr \<Rightarrow> heap_rep list"
+ refs :: "rtype \<Rightarrow> addr \<Rightarrow> heap_rep"
+ lim :: addr
+
+definition empty :: heap where
+ "empty = \<lparr>arrays = (\<lambda>_. arbitrary), refs = (\<lambda>_. arbitrary), lim = 0\<rparr>" -- "why arbitrary?"
+
+
+subsection {* Imperative references and arrays *}
+
+text {*
+ References and arrays are developed in parallel,
+ but keeping them seperate makes some later proofs simpler.
+*}
+
+subsubsection {* Primitive operations *}
+
+definition
+ new_ref :: "heap \<Rightarrow> ('a\<Colon>heap) ref \<times> heap" where
+ "new_ref h = (let l = lim h in (Ref l, h\<lparr>lim := l + 1\<rparr>))"
+
+definition
+ new_array :: "heap \<Rightarrow> ('a\<Colon>heap) array \<times> heap" where
+ "new_array h = (let l = lim h in (Array l, h\<lparr>lim := l + 1\<rparr>))"
+
+definition
+ ref_present :: "'a\<Colon>heap ref \<Rightarrow> heap \<Rightarrow> bool" where
+ "ref_present r h \<longleftrightarrow> addr_of_ref r < lim h"
+
+definition
+ array_present :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> bool" where
+ "array_present a h \<longleftrightarrow> addr_of_array a < lim h"
+
+definition
+ get_ref :: "'a\<Colon>heap ref \<Rightarrow> heap \<Rightarrow> 'a" where
+ "get_ref r h = from_nat (refs h (RTYPE('a)) (addr_of_ref r))"
+
+definition
+ get_array :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> 'a list" where
+ "get_array a h = map from_nat (arrays h (RTYPE('a)) (addr_of_array a))"
+
+definition
+ set_ref :: "'a\<Colon>heap ref \<Rightarrow> 'a \<Rightarrow> heap \<Rightarrow> heap" where
+ "set_ref r x =
+ refs_update (\<lambda>h. h( RTYPE('a) := ((h (RTYPE('a))) (addr_of_ref r:=to_nat x))))"
+
+definition
+ set_array :: "'a\<Colon>heap array \<Rightarrow> 'a list \<Rightarrow> heap \<Rightarrow> heap" where
+ "set_array a x =
+ arrays_update (\<lambda>h. h( RTYPE('a) := ((h (RTYPE('a))) (addr_of_array a:=map to_nat x))))"
+
+subsubsection {* Interface operations *}
+
+definition
+ ref :: "'a \<Rightarrow> heap \<Rightarrow> 'a\<Colon>heap ref \<times> heap" where
+ "ref x h = (let (r, h') = new_ref h;
+ h'' = set_ref r x h'
+ in (r, h''))"
+
+definition
+ array :: "nat \<Rightarrow> 'a \<Rightarrow> heap \<Rightarrow> 'a\<Colon>heap array \<times> heap" where
+ "array n x h = (let (r, h') = new_array h;
+ h'' = set_array r (replicate n x) h'
+ in (r, h''))"
+
+definition
+ array_of_list :: "'a list \<Rightarrow> heap \<Rightarrow> 'a\<Colon>heap array \<times> heap" where
+ "array_of_list xs h = (let (r, h') = new_array h;
+ h'' = set_array r xs h'
+ in (r, h''))"
+
+definition
+ upd :: "'a\<Colon>heap array \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> heap \<Rightarrow> heap" where
+ "upd a i x h = set_array a ((get_array a h)[i:=x]) h"
+
+definition
+ length :: "'a\<Colon>heap array \<Rightarrow> heap \<Rightarrow> nat" where
+ "length a h = size (get_array a h)"
+
+definition
+ array_ran :: "('a\<Colon>heap) option array \<Rightarrow> heap \<Rightarrow> 'a set" where
+ "array_ran a h = {e. Some e \<in> set (get_array a h)}"
+ -- {*FIXME*}
+
+
+subsubsection {* Reference equality *}
+
+text {*
+ The following relations are useful for comparing arrays and references.
+*}
+
+definition
+ noteq_refs :: "('a\<Colon>heap) ref \<Rightarrow> ('b\<Colon>heap) ref \<Rightarrow> bool" (infix "=!=" 70)
+where
+ "r =!= s \<longleftrightarrow> RTYPE('a) \<noteq> RTYPE('b) \<or> addr_of_ref r \<noteq> addr_of_ref s"
+
+definition
+ noteq_arrs :: "('a\<Colon>heap) array \<Rightarrow> ('b\<Colon>heap) array \<Rightarrow> bool" (infix "=!!=" 70)
+where
+ "r =!!= s \<longleftrightarrow> RTYPE('a) \<noteq> RTYPE('b) \<or> addr_of_array r \<noteq> addr_of_array s"
+
+lemma noteq_refs_sym: "r =!= s \<Longrightarrow> s =!= r"
+ and noteq_arrs_sym: "a =!!= b \<Longrightarrow> b =!!= a"
+ and unequal_refs [simp]: "r \<noteq> r' \<longleftrightarrow> r =!= r'" -- "same types!"
+ and unequal_arrs [simp]: "a \<noteq> a' \<longleftrightarrow> a =!!= a'"
+unfolding noteq_refs_def noteq_arrs_def by auto
+
+lemma present_new_ref: "ref_present r h \<Longrightarrow> r =!= fst (ref v h)"
+ by (simp add: ref_present_def new_ref_def ref_def Let_def noteq_refs_def)
+
+lemma present_new_arr: "array_present a h \<Longrightarrow> a =!!= fst (array v x h)"
+ by (simp add: array_present_def noteq_arrs_def new_array_def array_def Let_def)
+
+
+subsubsection {* Properties of heap containers *}
+
+text {* Properties of imperative arrays *}
+
+text {* FIXME: Does there exist a "canonical" array axiomatisation in
+the literature? *}
+
+lemma array_get_set_eq [simp]: "get_array r (set_array r x h) = x"
+ by (simp add: get_array_def set_array_def)
+
+lemma array_get_set_neq [simp]: "r =!!= s \<Longrightarrow> get_array r (set_array s x h) = get_array r h"
+ by (simp add: noteq_arrs_def get_array_def set_array_def)
+
+lemma set_array_same [simp]:
+ "set_array r x (set_array r y h) = set_array r x h"
+ by (simp add: set_array_def)
+
+lemma array_set_set_swap:
+ "r =!!= r' \<Longrightarrow> set_array r x (set_array r' x' h) = set_array r' x' (set_array r x h)"
+ by (simp add: Let_def expand_fun_eq noteq_arrs_def set_array_def)
+
+lemma array_ref_set_set_swap:
+ "set_array r x (set_ref r' x' h) = set_ref r' x' (set_array r x h)"
+ by (simp add: Let_def expand_fun_eq set_array_def set_ref_def)
+
+lemma get_array_upd_eq [simp]:
+ "get_array a (upd a i v h) = (get_array a h) [i := v]"
+ by (simp add: upd_def)
+
+lemma nth_upd_array_neq_array [simp]:
+ "a =!!= b \<Longrightarrow> get_array a (upd b j v h) ! i = get_array a h ! i"
+ by (simp add: upd_def noteq_arrs_def)
+
+lemma get_arry_array_upd_elem_neqIndex [simp]:
+ "i \<noteq> j \<Longrightarrow> get_array a (upd a j v h) ! i = get_array a h ! i"
+ by simp
+
+lemma length_upd_eq [simp]:
+ "length a (upd a i v h) = length a h"
+ by (simp add: length_def upd_def)
+
+lemma length_upd_neq [simp]:
+ "length a (upd b i v h) = length a h"
+ by (simp add: upd_def length_def set_array_def get_array_def)
+
+lemma upd_swap_neqArray:
+ "a =!!= a' \<Longrightarrow>
+ upd a i v (upd a' i' v' h)
+ = upd a' i' v' (upd a i v h)"
+apply (unfold upd_def)
+apply simp
+apply (subst array_set_set_swap, assumption)
+apply (subst array_get_set_neq)
+apply (erule noteq_arrs_sym)
+apply (simp)
+done
+
+lemma upd_swap_neqIndex:
+ "\<lbrakk> i \<noteq> i' \<rbrakk> \<Longrightarrow> upd a i v (upd a i' v' h) = upd a i' v' (upd a i v h)"
+by (auto simp add: upd_def array_set_set_swap list_update_swap)
+
+lemma get_array_init_array_list:
+ "get_array (fst (array_of_list ls h)) (snd (array_of_list ls' h)) = ls'"
+ by (simp add: Let_def split_def array_of_list_def)
+
+lemma set_array:
+ "set_array (fst (array_of_list ls h))
+ new_ls (snd (array_of_list ls h))
+ = snd (array_of_list new_ls h)"
+ by (simp add: Let_def split_def array_of_list_def)
+
+lemma array_present_upd [simp]:
+ "array_present a (upd b i v h) = array_present a h"
+ by (simp add: upd_def array_present_def set_array_def get_array_def)
+
+lemma array_of_list_replicate:
+ "array_of_list (replicate n x) = array n x"
+ by (simp add: expand_fun_eq array_of_list_def array_def)
+
+
+text {* Properties of imperative references *}
+
+lemma next_ref_fresh [simp]:
+ assumes "(r, h') = new_ref h"
+ shows "\<not> ref_present r h"
+ using assms by (cases h) (auto simp add: new_ref_def ref_present_def Let_def)
+
+lemma next_ref_present [simp]:
+ assumes "(r, h') = new_ref h"
+ shows "ref_present r h'"
+ using assms by (cases h) (auto simp add: new_ref_def ref_present_def Let_def)
+
+lemma ref_get_set_eq [simp]: "get_ref r (set_ref r x h) = x"
+ by (simp add: get_ref_def set_ref_def)
+
+lemma ref_get_set_neq [simp]: "r =!= s \<Longrightarrow> get_ref r (set_ref s x h) = get_ref r h"
+ by (simp add: noteq_refs_def get_ref_def set_ref_def)
+
+(* FIXME: We need some infrastructure to infer that locally generated
+ new refs (by new_ref(_no_init), new_array(')) are distinct
+ from all existing refs.
+*)
+
+lemma ref_set_get: "set_ref r (get_ref r h) h = h"
+apply (simp add: set_ref_def get_ref_def)
+oops
+
+lemma set_ref_same[simp]:
+ "set_ref r x (set_ref r y h) = set_ref r x h"
+ by (simp add: set_ref_def)
+
+lemma ref_set_set_swap:
+ "r =!= r' \<Longrightarrow> set_ref r x (set_ref r' x' h) = set_ref r' x' (set_ref r x h)"
+ by (simp add: Let_def expand_fun_eq noteq_refs_def set_ref_def)
+
+lemma ref_new_set: "fst (ref v (set_ref r v' h)) = fst (ref v h)"
+ by (simp add: ref_def new_ref_def set_ref_def Let_def)
+
+lemma ref_get_new [simp]:
+ "get_ref (fst (ref v h)) (snd (ref v' h)) = v'"
+ by (simp add: ref_def Let_def split_def)
+
+lemma ref_set_new [simp]:
+ "set_ref (fst (ref v h)) new_v (snd (ref v h)) = snd (ref new_v h)"
+ by (simp add: ref_def Let_def split_def)
+
+lemma ref_get_new_neq: "r =!= (fst (ref v h)) \<Longrightarrow>
+ get_ref r (snd (ref v h)) = get_ref r h"
+ by (simp add: get_ref_def set_ref_def ref_def Let_def new_ref_def noteq_refs_def)
+
+lemma lim_set_ref [simp]:
+ "lim (set_ref r v h) = lim h"
+ by (simp add: set_ref_def)
+
+lemma ref_present_new_ref [simp]:
+ "ref_present r h \<Longrightarrow> ref_present r (snd (ref v h))"
+ by (simp add: new_ref_def ref_present_def ref_def Let_def)
+
+lemma ref_present_set_ref [simp]:
+ "ref_present r (set_ref r' v h) = ref_present r h"
+ by (simp add: set_ref_def ref_present_def)
+
+lemma array_ranI: "\<lbrakk> Some b = get_array a h ! i; i < Heap.length a h \<rbrakk> \<Longrightarrow> b \<in> array_ran a h"
+unfolding array_ran_def Heap.length_def by simp
+
+lemma array_ran_upd_array_Some:
+ assumes "cl \<in> array_ran a (Heap.upd a i (Some b) h)"
+ shows "cl \<in> array_ran a h \<or> cl = b"
+proof -
+ have "set (get_array a h[i := Some b]) \<subseteq> insert (Some b) (set (get_array a h))" by (rule set_update_subset_insert)
+ with assms show ?thesis
+ unfolding array_ran_def Heap.upd_def by fastsimp
+qed
+
+lemma array_ran_upd_array_None:
+ assumes "cl \<in> array_ran a (Heap.upd a i None h)"
+ shows "cl \<in> array_ran a h"
+proof -
+ have "set (get_array a h[i := None]) \<subseteq> insert None (set (get_array a h))" by (rule set_update_subset_insert)
+ with assms show ?thesis
+ unfolding array_ran_def Heap.upd_def by auto
+qed
+
+
+text {* Non-interaction between imperative array and imperative references *}
+
+lemma get_array_set_ref [simp]: "get_array a (set_ref r v h) = get_array a h"
+ by (simp add: get_array_def set_ref_def)
+
+lemma nth_set_ref [simp]: "get_array a (set_ref r v h) ! i = get_array a h ! i"
+ by simp
+
+lemma get_ref_upd [simp]: "get_ref r (upd a i v h) = get_ref r h"
+ by (simp add: get_ref_def set_array_def upd_def)
+
+lemma new_ref_upd: "fst (ref v (upd a i v' h)) = fst (ref v h)"
+ by (simp add: set_array_def get_array_def Let_def ref_new_set upd_def ref_def new_ref_def)
+
+(*not actually true ???
+lemma upd_set_ref_swap: "upd a i v (set_ref r v' h) = set_ref r v' (upd a i v h)"
+apply (case_tac a)
+apply (simp add: Let_def upd_def)
+apply auto
+done*)
+
+lemma length_new_ref[simp]:
+ "length a (snd (ref v h)) = length a h"
+ by (simp add: get_array_def set_ref_def length_def new_ref_def ref_def Let_def)
+
+lemma get_array_new_ref [simp]:
+ "get_array a (snd (ref v h)) = get_array a h"
+ by (simp add: new_ref_def ref_def set_ref_def get_array_def Let_def)
+
+lemma get_array_new_ref [simp]:
+ "get_array a (snd (ref v h)) ! i = get_array a h ! i"
+ by (simp add: get_array_def new_ref_def ref_def set_ref_def Let_def)
+
+lemma ref_present_upd [simp]:
+ "ref_present r (upd a i v h) = ref_present r h"
+ by (simp add: upd_def ref_present_def set_array_def get_array_def)
+
+lemma array_present_set_ref [simp]:
+ "array_present a (set_ref r v h) = array_present a h"
+ by (simp add: array_present_def set_ref_def)
+
+lemma array_present_new_ref [simp]:
+ "array_present a h \<Longrightarrow> array_present a (snd (ref v h))"
+ by (simp add: array_present_def new_ref_def ref_def Let_def)
+
+hide (open) const empty array array_of_list upd length ref
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Heap_Monad.thy Wed Feb 27 21:41:08 2008 +0100
@@ -0,0 +1,277 @@
+(* Title: HOL/Library/Heap_Monad.thy
+ ID: $Id$
+ Author: John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen
+*)
+
+header {* A monad with a polymorphic heap *}
+
+theory Heap_Monad
+imports Heap
+begin
+
+subsection {* The monad *}
+
+subsubsection {* Monad combinators *}
+
+datatype exception = Exn
+
+text {* Monadic heap actions either produce values
+ and transform the heap, or fail *}
+datatype 'a Heap = Heap "heap \<Rightarrow> ('a + exception) \<times> heap"
+
+primrec
+ execute :: "'a Heap \<Rightarrow> heap \<Rightarrow> ('a + exception) \<times> heap" where
+ "execute (Heap f) = f"
+lemmas [code del] = execute.simps
+
+lemma Heap_execute [simp]:
+ "Heap (execute f) = f" by (cases f) simp_all
+
+lemma Heap_eqI:
+ "(\<And>h. execute f h = execute g h) \<Longrightarrow> f = g"
+ by (cases f, cases g) (auto simp: expand_fun_eq)
+
+lemma Heap_eqI':
+ "(\<And>h. (\<lambda>x. execute (f x) h) = (\<lambda>y. execute (g y) h)) \<Longrightarrow> f = g"
+ by (auto simp: expand_fun_eq intro: Heap_eqI)
+
+lemma Heap_strip: "(\<And>f. PROP P f) \<equiv> (\<And>g. PROP P (Heap g))"
+proof
+ fix g :: "heap \<Rightarrow> ('a + exception) \<times> heap"
+ assume "\<And>f. PROP P f"
+ then show "PROP P (Heap g)" .
+next
+ fix f :: "'a Heap"
+ assume assm: "\<And>g. PROP P (Heap g)"
+ then have "PROP P (Heap (execute f))" .
+ then show "PROP P f" by simp
+qed
+
+definition
+ heap :: "(heap \<Rightarrow> 'a \<times> heap) \<Rightarrow> 'a Heap" where
+ [code del]: "heap f = Heap (\<lambda>h. apfst Inl (f h))"
+
+lemma execute_heap [simp]:
+ "execute (heap f) h = apfst Inl (f h)"
+ by (simp add: heap_def)
+
+definition
+ run :: "'a Heap \<Rightarrow> 'a Heap" where
+ run_drop [code del]: "run f = f"
+
+definition
+ bindM :: "'a Heap \<Rightarrow> ('a \<Rightarrow> 'b Heap) \<Rightarrow> 'b Heap" (infixl ">>=" 54) where
+ [code del]: "f >>= g = Heap (\<lambda>h. case execute f h of
+ (Inl x, h') \<Rightarrow> execute (g x) h'
+ | r \<Rightarrow> r)"
+
+notation
+ bindM (infixl "\<guillemotright>=" 54)
+
+abbreviation
+ chainM :: "'a Heap \<Rightarrow> 'b Heap \<Rightarrow> 'b Heap" (infixl ">>" 54) where
+ "f >> g \<equiv> f >>= (\<lambda>_. g)"
+
+notation
+ chainM (infixl "\<guillemotright>" 54)
+
+definition
+ return :: "'a \<Rightarrow> 'a Heap" where
+ [code del]: "return x = heap (Pair x)"
+
+lemma execute_return [simp]:
+ "execute (return x) h = apfst Inl (x, h)"
+ by (simp add: return_def)
+
+definition
+ raise :: "string \<Rightarrow> 'a Heap" where -- {* the string is just decoration *}
+ [code del]: "raise s = Heap (Pair (Inr Exn))"
+
+notation (latex output)
+ "raise" ("\<^raw:{\textsf{raise}}>")
+
+lemma execute_raise [simp]:
+ "execute (raise s) h = (Inr Exn, h)"
+ by (simp add: raise_def)
+
+
+subsubsection {* do-syntax *}
+
+text {*
+ We provide a convenient do-notation for monadic expressions
+ well-known from Haskell. @{const Let} is printed
+ specially in do-expressions.
+*}
+
+nonterminals do_expr
+
+syntax
+ "_do" :: "do_expr \<Rightarrow> 'a"
+ ("(do (_)//done)" [12] 100)
+ "_bindM" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
+ ("_ <- _;//_" [1000, 13, 12] 12)
+ "_chainM" :: "'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
+ ("_;//_" [13, 12] 12)
+ "_let" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
+ ("let _ = _;//_" [1000, 13, 12] 12)
+ "_nil" :: "'a \<Rightarrow> do_expr"
+ ("_" [12] 12)
+
+syntax (xsymbols)
+ "_bindM" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
+ ("_ \<leftarrow> _;//_" [1000, 13, 12] 12)
+syntax (latex output)
+ "_do" :: "do_expr \<Rightarrow> 'a"
+ ("(\<^raw:{\textsf{do}}> (_))" [12] 100)
+ "_let" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
+ ("\<^raw:\textsf{let}> _ = _;//_" [1000, 13, 12] 12)
+notation (latex output)
+ "return" ("\<^raw:{\textsf{return}}>")
+
+translations
+ "_do f" => "CONST run f"
+ "_bindM x f g" => "f \<guillemotright>= (\<lambda>x. g)"
+ "_chainM f g" => "f \<guillemotright> g"
+ "_let x t f" => "CONST Let t (\<lambda>x. f)"
+ "_nil f" => "f"
+
+print_translation {*
+let
+ fun dest_abs_eta (Abs (abs as (_, ty, _))) =
+ let
+ val (v, t) = Syntax.variant_abs abs;
+ in ((v, ty), t) end
+ | dest_abs_eta t =
+ let
+ val (v, t) = Syntax.variant_abs ("", dummyT, t $ Bound 0);
+ in ((v, dummyT), t) end
+ fun unfold_monad (Const (@{const_syntax bindM}, _) $ f $ g) =
+ let
+ val ((v, ty), g') = dest_abs_eta g;
+ val v_used = fold_aterms
+ (fn Free (w, _) => (fn s => s orelse v = w) | _ => I) g' false;
+ in if v_used then
+ Const ("_bindM", dummyT) $ Free (v, ty) $ f $ unfold_monad g'
+ else
+ Const ("_chainM", dummyT) $ f $ unfold_monad g'
+ end
+ | unfold_monad (Const (@{const_syntax chainM}, _) $ f $ g) =
+ Const ("_chainM", dummyT) $ f $ unfold_monad g
+ | unfold_monad (Const (@{const_syntax Let}, _) $ f $ g) =
+ let
+ val ((v, ty), g') = dest_abs_eta g;
+ in Const ("_let", dummyT) $ Free (v, ty) $ f $ unfold_monad g' end
+ | unfold_monad (Const (@{const_syntax Pair}, _) $ f) =
+ Const ("return", dummyT) $ f
+ | unfold_monad f = f;
+ fun tr' (f::ts) =
+ list_comb (Const ("_do", dummyT) $ unfold_monad f, ts)
+in [(@{const_syntax "run"}, tr')] end;
+*}
+
+subsubsection {* Plain evaluation *}
+
+definition
+ evaluate :: "'a Heap \<Rightarrow> 'a"
+where
+ [code del]: "evaluate f = (case execute f Heap.empty
+ of (Inl x, _) \<Rightarrow> x)"
+
+
+subsection {* Monad properties *}
+
+subsubsection {* Superfluous runs *}
+
+text {* @{term run} is just a doodle *}
+
+lemma run_simp [simp]:
+ "\<And>f. run (run f) = run f"
+ "\<And>f g. run f \<guillemotright>= g = f \<guillemotright>= g"
+ "\<And>f g. run f \<guillemotright> g = f \<guillemotright> g"
+ "\<And>f g. f \<guillemotright>= (\<lambda>x. run g) = f \<guillemotright>= (\<lambda>x. g)"
+ "\<And>f g. f \<guillemotright> run g = f \<guillemotright> g"
+ "\<And>f. f = run g \<longleftrightarrow> f = g"
+ "\<And>f. run f = g \<longleftrightarrow> f = g"
+ unfolding run_drop by rule+
+
+subsubsection {* Monad laws *}
+
+lemma return_bind: "return x \<guillemotright>= f = f x"
+ by (simp add: bindM_def return_def)
+
+lemma bind_return: "f \<guillemotright>= return = f"
+proof (rule Heap_eqI)
+ fix h
+ show "execute (f \<guillemotright>= return) h = execute f h"
+ by (auto simp add: bindM_def return_def split: sum.splits prod.splits)
+qed
+
+lemma bind_bind: "(f \<guillemotright>= g) \<guillemotright>= h = f \<guillemotright>= (\<lambda>x. g x \<guillemotright>= h)"
+ by (rule Heap_eqI) (auto simp add: bindM_def split: split: sum.splits prod.splits)
+
+lemma bind_bind': "f \<guillemotright>= (\<lambda>x. g x \<guillemotright>= h x) = f \<guillemotright>= (\<lambda>x. g x \<guillemotright>= (\<lambda>y. return (x, y))) \<guillemotright>= (\<lambda>(x, y). h x y)"
+ by (rule Heap_eqI) (auto simp add: bindM_def split: split: sum.splits prod.splits)
+
+lemma raise_bind: "raise e \<guillemotright>= f = raise e"
+ by (simp add: raise_def bindM_def)
+
+
+lemmas monad_simp = return_bind bind_return bind_bind raise_bind
+
+
+subsection {* Generic combinators *}
+
+definition
+ liftM :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b Heap"
+where
+ "liftM f = return o f"
+
+definition
+ compM :: "('a \<Rightarrow> 'b Heap) \<Rightarrow> ('b \<Rightarrow> 'c Heap) \<Rightarrow> 'a \<Rightarrow> 'c Heap" (infixl ">>==" 54)
+where
+ "(f >>== g) = (\<lambda>x. f x \<guillemotright>= g)"
+
+notation
+ compM (infixl "\<guillemotright>==" 54)
+
+lemma liftM_collapse: "liftM f x = return (f x)"
+ by (simp add: liftM_def)
+
+lemma liftM_compM: "liftM f \<guillemotright>== g = g o f"
+ by (auto intro: Heap_eqI' simp add: expand_fun_eq liftM_def compM_def bindM_def)
+
+lemma compM_return: "f \<guillemotright>== return = f"
+ by (simp add: compM_def monad_simp)
+
+lemma compM_compM: "(f \<guillemotright>== g) \<guillemotright>== h = f \<guillemotright>== (g \<guillemotright>== h)"
+ by (simp add: compM_def monad_simp)
+
+lemma liftM_bind:
+ "(\<lambda>x. liftM f x \<guillemotright>= liftM g) = liftM (\<lambda>x. g (f x))"
+ by (rule Heap_eqI') (simp add: monad_simp liftM_def bindM_def)
+
+lemma liftM_comp:
+ "liftM f o g = liftM (f o g)"
+ by (rule Heap_eqI') (simp add: liftM_def)
+
+lemmas monad_simp' = monad_simp liftM_compM compM_return
+ compM_compM liftM_bind liftM_comp
+
+primrec
+ mapM :: "('a \<Rightarrow> 'b Heap) \<Rightarrow> 'a list \<Rightarrow> 'b list Heap"
+where
+ "mapM f [] = return []"
+ | "mapM f (x#xs) = do y \<leftarrow> f x;
+ ys \<leftarrow> mapM f xs;
+ return (y # ys)
+ done"
+
+primrec
+ foldM :: "('a \<Rightarrow> 'b \<Rightarrow> 'b Heap) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b Heap"
+where
+ "foldM f [] s = return s"
+ | "foldM f (x#xs) s = f x s \<guillemotright>= foldM f xs"
+
+hide (open) const heap execute
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Imperative_HOL.thy Wed Feb 27 21:41:08 2008 +0100
@@ -0,0 +1,12 @@
+(* Title: HOL/Library/Imperative_HOL.thy
+ ID: $Id$
+ Author: John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen
+*)
+
+header {* Entry point into monadic imperative HOL *}
+
+theory Imperative_HOL
+imports Array Ref
+begin
+
+end
--- a/src/HOL/Library/Library.thy Wed Feb 27 21:41:07 2008 +0100
+++ b/src/HOL/Library/Library.thy Wed Feb 27 21:41:08 2008 +0100
@@ -8,18 +8,22 @@
Binomial
Boolean_Algebra
Char_ord
+ Code_Char_chr
Code_Index
+ Code_Integer
Code_Message
Coinductive_List
Commutative_Ring
Continuity
+ Countable
Dense_Linear_Order
Efficient_Nat
- (*Eval*)
+ Eval
Eval_Witness
Executable_Set
FuncSet
GCD
+ Imperative_HOL
Infinite_Set
ListSpace
Multiset
@@ -30,8 +34,6 @@
OptionalSugar
Parity
Permutation
- Code_Integer
- Code_Char_chr
Primes
Quicksort
Quotient
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Ref.thy Wed Feb 27 21:41:08 2008 +0100
@@ -0,0 +1,56 @@
+(* Title: HOL/Library/Ref.thy
+ ID: $Id$
+ Author: John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen
+*)
+
+header {* Monadic references *}
+
+theory Ref
+imports Heap_Monad
+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 {* Primitives *}
+
+definition
+ new :: "'a\<Colon>heap \<Rightarrow> 'a ref Heap" where
+ [code del]: "new v = Heap_Monad.heap (Heap.ref v)"
+
+definition
+ lookup :: "'a\<Colon>heap ref \<Rightarrow> 'a Heap" ("!_" 61) where
+ [code del]: "lookup r = Heap_Monad.heap (\<lambda>h. (get_ref r h, h))"
+
+definition
+ update :: "'a ref \<Rightarrow> ('a\<Colon>heap) \<Rightarrow> unit Heap" ("_ := _" 62) where
+ [code del]: "update r e = Heap_Monad.heap (\<lambda>h. ((), set_ref r e 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)"
+
+hide (open) const new lookup update change
+
+subsection {* Properties *}
+
+lemma lookup_chain:
+ "(!r \<guillemotright> f) = f"
+ by (cases f)
+ (auto simp add: Let_def bindM_def lookup_def expand_fun_eq)
+
+lemma update_change [code func]:
+ "r := e = Ref.change (\<lambda>_. e) r \<guillemotright> return ()"
+ by (auto simp add: monad_simp change_def lookup_chain)
+
+end
\ No newline at end of file