Theory Heap

(*  Title:      HOL/Imperative_HOL/Heap.thy
    Author:     John Matthews, Galois Connections; Alexander Krauss, TU Muenchen
*)

section ‹A polymorphic heap based on cantor encodings›

theory Heap
imports Main "HOL-Library.Countable"
begin

subsection ‹Representable types›

text ‹The type class of representable types›

class heap = typerep + countable

instance unit :: heap ..

instance bool :: heap ..

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 :: heap ..

instance char :: heap ..

instance typerep :: heap ..


subsection ‹A polymorphic heap with dynamic arrays and references›

text ‹
  References and arrays are developed in parallel,
  but keeping them separate makes some later proofs simpler.
›

type_synonym addr = nat ― ‹untyped heap references›
type_synonym heap_rep = nat ― ‹representable values›

record heap =
  arrays :: "typerep ⇒ addr ⇒ heap_rep list"
  refs :: "typerep ⇒ addr ⇒ heap_rep"
  lim  :: addr

definition empty :: heap where
  "empty = ⦇arrays = (λ_ _. []), refs = (λ_ _. 0), lim = 0⦈"

datatype 'a array = Array addr ― ‹note the phantom type 'a›
datatype 'a ref = Ref addr ― ‹note the phantom type 'a›

primrec addr_of_array :: "'a array ⇒ addr" where
  "addr_of_array (Array x) = x"

primrec addr_of_ref :: "'a ref ⇒ addr" where
  "addr_of_ref (Ref x) = x"

lemma addr_of_array_inj [simp]:
  "addr_of_array a = addr_of_array a' ⟷ a = a'"
  by (cases a, cases a') simp_all

lemma addr_of_ref_inj [simp]:
  "addr_of_ref r = addr_of_ref r' ⟷ 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

instance array :: (type) heap ..

instance ref :: (type) heap ..


text ‹Syntactic convenience›

setup ‹
  Sign.add_const_constraint (\<^const_name>‹Array›, SOME \<^typ>‹nat ⇒ 'a::heap array›)
  #> Sign.add_const_constraint (\<^const_name>‹Ref›, SOME \<^typ>‹nat ⇒ 'a::heap ref›)
  #> Sign.add_const_constraint (\<^const_name>‹addr_of_array›, SOME \<^typ>‹'a::heap array ⇒ nat›)
  #> Sign.add_const_constraint (\<^const_name>‹addr_of_ref›, SOME \<^typ>‹'a::heap ref ⇒ nat›)
›

hide_const (open) empty

end