(*  Title:      HOL/Imperative_HOL/Heap.thy

     Author:     John Matthews, Galois Connections; Alexander Krauss, TU Muenchen

 *)

 header {* A polymorphic heap based on cantor encodings *}

 theory Heap

 imports Main Countable

 begin

 subsection {* Representable types *}

 text {* The type class of representable types *}

 class heap = typerep + countable

 instance nat :: heap ..

 instance prod :: (heap, heap) heap ..

 instance sum :: (heap, heap) heap ..

 instance list :: (heap) heap ..

 instance option :: (heap) heap ..

 instance int :: heap ..

 instance String.literal :: countable

   by (rule countable_classI [of "String.literal_case to_nat"])

    (auto split: String.literal.splits)

 instance String.literal :: heap ..

 instance typerep :: heap ..

 subsection {* A polymorphic heap with dynamic arrays and references *}

 subsubsection {* Type definitions *}

 types addr = nat -- "untyped heap references"

 types heap_rep = nat -- "representable values"

 record heap =

   arrays :: "typerep \<Rightarrow> addr \<Rightarrow> heap_rep list"

   refs :: "typerep \<Rightarrow> addr \<Rightarrow> heap_rep"

   lim  :: addr

 definition empty :: heap where

   "empty = \<lparr>arrays = (\<lambda>_ _. []), refs = (\<lambda>_ _. 0), lim = 0\<rparr>" -- "why undefined?"

 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

 text {* Syntactic convenience *}

 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"})

 *}

 subsection {* Imperative references and arrays *}

 text {*

   References and arrays are developed in parallel,

   but keeping them separate makes some later proofs simpler.

 *}

 subsubsection {* Primitive operations *}

 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 (TYPEREP('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 (TYPEREP('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(TYPEREP('a) := ((h (TYPEREP('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(TYPEREP('a) := ((h(TYPEREP('a))) (addr_of_array a:=map to_nat x))))"

 definition ref :: "'a \<Rightarrow> heap \<Rightarrow> 'a\<Colon>heap ref \<times> heap" where

   "ref x h = (let

      l = lim h;

      r = Ref l;

      h'' = set_ref r x (h\<lparr>lim := l + 1\<rparr>)

    in (r, h''))"

 definition array :: "'a list \<Rightarrow> heap \<Rightarrow> 'a\<Colon>heap array \<times> heap" where

   "array xs h = (let

      l = lim h;

      r = Array l;

      h'' = set_array r xs (h\<lparr>lim := l + 1\<rparr>)

    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)"

 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> TYPEREP('a) \<noteq> TYPEREP('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> TYPEREP('a) \<noteq> TYPEREP('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 noteq_refs_irrefl: "r =!= r \<Longrightarrow> False"

   unfolding noteq_refs_def by auto

 lemma present_new_ref: "ref_present r h \<Longrightarrow> r =!= fst (ref v h)"

   by (simp add: ref_present_def ref_def Let_def noteq_refs_def)

 lemma present_new_arr: "array_present a h \<Longrightarrow> a =!!= fst (array xs h)"

   by (simp add: array_present_def noteq_arrs_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 o_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 ls h)) (snd (array ls' h)) = ls'"

   by (simp add: Let_def split_def array_def)

 lemma set_array:

   "set_array (fst (array ls h))

      new_ls (snd (array ls h))

        = snd (array new_ls h)"

   by (simp add: Let_def split_def array_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') = ref x h"

   shows "\<not> ref_present r h"

   using assms by (cases h) (auto simp add: ref_def ref_present_def Let_def)

 lemma next_ref_present [simp]:

   assumes "(r, h') = ref x h"

   shows "ref_present r h'"

   using assms by (cases h) (auto simp add: ref_def set_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 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 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: 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 noteq_refsI: "\<lbrakk> ref_present r h; \<not>ref_present r' h \<rbrakk>  \<Longrightarrow> r =!= r'"

   unfolding noteq_refs_def ref_present_def

   by auto

 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)

 text {*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

 oops

 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  ref_def Let_def)

 lemma get_array_new_ref [simp]: 

   "get_array a (snd (ref v h)) = get_array a h"

   by (simp add: ref_def set_ref_def get_array_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 ref_def Let_def)

 hide_const (open) empty upd length array ref

 end