--- a/src/HOL/Imperative_HOL/Heap.thy Mon Jul 05 15:36:37 2010 +0200
+++ b/src/HOL/Imperative_HOL/Heap.thy Mon Jul 05 16:46:23 2010 +0200
@@ -37,7 +37,10 @@
subsection {* A polymorphic heap with dynamic arrays and references *}
-subsubsection {* Type definitions *}
+text {*
+ References and arrays are developed in parallel,
+ but keeping them separate makes some later proofs simpler.
+*}
types addr = nat -- "untyped heap references"
types heap_rep = nat -- "representable values"
@@ -82,283 +85,4 @@
#> 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