(* Title: HOL/Hoare/SepLogHeap.thy
Author: Tobias Nipkow
Copyright 2002 TUM
*)
section \<open>Heap abstractions for Separation Logic\<close>
text \<open>(at the moment only Path and List)\<close>
theory SepLogHeap
imports Main
begin
type_synonym heap = "(nat \ nat option)"
text\<open>\<open>Some\<close> means allocated, \<open>None\<close> means
free. Address \<open>0\<close> serves as the null reference.\<close>
subsection "Paths in the heap"
primrec Path :: "heap \ nat \ nat list \ nat \ bool"
where
"Path h x [] y = (x = y)"
| "Path h x (a#as) y = (x\0 \ a=x \ (\b. h x = Some b \ Path h b as y))"
lemma [iff]: "Path h 0 xs y = (xs = [] \ y = 0)"
by (cases xs) simp_all
lemma [simp]: "x\0 \ Path h x as z =
(as = [] \<and> z = x \<or> (\<exists>y bs. as = x#bs \<and> h x = Some y & Path h y bs z))"
by (cases as) auto
lemma [simp]: "\x. Path f x (as@bs) z = (\y. Path f x as y \ Path f y bs z)"
by (induct as) auto
lemma Path_upd[simp]:
"\x. u \ set as \ Path (f(u := v)) x as y = Path f x as y"
by (induct as) simp_all
subsection "Lists on the heap"
definition List :: "heap \ nat \ nat list \ bool"
where "List h x as = Path h x as 0"
lemma [simp]: "List h x [] = (x = 0)"
by (simp add: List_def)
lemma [simp]:
"List h x (a#as) = (x\0 \ a=x \ (\y. h x = Some y \ List h y as))"
by (simp add: List_def)
lemma [simp]: "List h 0 as = (as = [])"
by (cases as) simp_all
lemma List_non_null: "a\0 \
List h a as = (\<exists>b bs. as = a#bs \<and> h a = Some b \<and> List h b bs)"
by (cases as) simp_all
theorem notin_List_update[simp]:
"\x. a \ set as \ List (h(a := y)) x as = List h x as"
by (induct as) simp_all
lemma List_unique: "\x bs. List h x as \ List h x bs \ as = bs"
by (induct as) (auto simp add:List_non_null)
lemma List_unique1: "List h p as \ \!as. List h p as"
by (blast intro: List_unique)
lemma List_app: "\x. List h x (as@bs) = (\y. Path h x as y \ List h y bs)"
by (induct as) auto
lemma List_hd_not_in_tl[simp]: "List h b as \ h a = Some b \ a \ set as"
apply (clarsimp simp add:in_set_conv_decomp)
apply(frule List_app[THEN iffD1])
apply(fastforce dest: List_unique)
done
lemma List_distinct[simp]: "\x. List h x as \ distinct as"
by (induct as) (auto dest:List_hd_not_in_tl)
lemma list_in_heap: "\p. List h p ps \ set ps \ dom h"
by (induct ps) auto
lemma list_ortho_sum1[simp]:
"\p. \ List h1 p ps; dom h1 \ dom h2 = {}\ \ List (h1++h2) p ps"
by (induct ps) (auto simp add:map_add_def split:option.split)
lemma list_ortho_sum2[simp]:
"\p. \ List h2 p ps; dom h1 \ dom h2 = {}\ \ List (h1++h2) p ps"
by (induct ps) (auto simp add:map_add_def split:option.split)
end
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