# Theory HOL-Imperative_HOL.Ref

```(*  Title:      HOL/Imperative_HOL/Ref.thy
Author:     John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen
*)

theory Ref
imports Array
begin

text ‹
Imperative reference operations; modeled after their ML counterparts.
See 🌐‹https://caml.inria.fr/pub/docs/manual-caml-light/node14.15.html›
and 🌐‹https://www.smlnj.org/doc/Conversion/top-level-comparison.html›.
›

subsection ‹Primitives›

definition present :: "heap ⇒ 'a::heap ref ⇒ bool" where
"present h r ⟷ addr_of_ref r < lim h"

definition get :: "heap ⇒ 'a::heap ref ⇒ 'a" where
"get h = from_nat ∘ refs h TYPEREP('a) ∘ addr_of_ref"

definition set :: "'a::heap ref ⇒ 'a ⇒ heap ⇒ heap" where
"set r x = refs_update
(λh. h(TYPEREP('a) := ((h (TYPEREP('a))) (addr_of_ref r := to_nat x))))"

definition alloc :: "'a ⇒ heap ⇒ 'a::heap ref × heap" where
"alloc x h = (let
l = lim h;
r = Ref l
in (r, set r x (h⦇lim := l + 1⦈)))"

definition noteq :: "'a::heap ref ⇒ 'b::heap ref ⇒ bool" (infix "=!=" 70) where
"r =!= s ⟷ TYPEREP('a) ≠ TYPEREP('b) ∨ addr_of_ref r ≠ addr_of_ref s"

definition ref :: "'a::heap ⇒ 'a ref Heap" where
[code del]: "ref v = Heap_Monad.heap (alloc v)"

definition lookup :: "'a::heap ref ⇒ 'a Heap" ("!_" 61) where
[code del]: "lookup r = Heap_Monad.tap (λh. get h r)"

definition update :: "'a ref ⇒ 'a::heap ⇒ unit Heap" ("_ := _" 62) where
[code del]: "update r v = Heap_Monad.heap (λh. ((), set r v h))"

definition change :: "('a::heap ⇒ 'a) ⇒ 'a ref ⇒ 'a Heap" where
"change f r = do {
x ← ! r;
let y = f x;
r := y;
return y
}"

subsection ‹Properties›

text ‹Primitives›

lemma noteq_sym: "r =!= s ⟹ s =!= r"
and unequal [simp]: "r ≠ r' ⟷ r =!= r'" ― ‹same types!›
by (auto simp add: noteq_def)

lemma noteq_irrefl: "r =!= r ⟹ False"
by (auto simp add: noteq_def)

lemma present_alloc_neq: "present h r ⟹ r =!= fst (alloc v h)"
by (simp add: present_def alloc_def noteq_def Let_def)

lemma next_fresh [simp]:
assumes "(r, h') = alloc x h"
shows "¬ present h r"
using assms by (cases h) (auto simp add: alloc_def present_def Let_def)

lemma next_present [simp]:
assumes "(r, h') = alloc x h"
shows "present h' r"
using assms by (cases h) (auto simp add: alloc_def set_def present_def Let_def)

lemma get_set_eq [simp]:
"get (set r x h) r = x"
by (simp add: get_def set_def)

lemma get_set_neq [simp]:
"r =!= s ⟹ get (set s x h) r = get h r"
by (simp add: noteq_def get_def set_def)

lemma set_same [simp]:
"set r x (set r y h) = set r x h"
by (simp add: set_def)

lemma not_present_alloc [simp]:
"¬ present h (fst (alloc v h))"
by (simp add: present_def alloc_def Let_def)

lemma set_set_swap:
"r =!= r' ⟹ set r x (set r' x' h) = set r' x' (set r x h)"
by (simp add: noteq_def set_def fun_eq_iff)

lemma alloc_set:
"fst (alloc x (set r x' h)) = fst (alloc x h)"
by (simp add: alloc_def set_def Let_def)

lemma get_alloc [simp]:
"get (snd (alloc x h)) (fst (alloc x' h)) = x"
by (simp add: alloc_def Let_def)

lemma set_alloc [simp]:
"set (fst (alloc v h)) v' (snd (alloc v h)) = snd (alloc v' h)"
by (simp add: alloc_def Let_def)

lemma get_alloc_neq: "r =!= fst (alloc v h) ⟹
get (snd (alloc v h)) r  = get h r"
by (simp add: get_def set_def alloc_def Let_def noteq_def)

lemma lim_set [simp]:
"lim (set r v h) = lim h"
by (simp add: set_def)

lemma present_alloc [simp]:
"present h r ⟹ present (snd (alloc v h)) r"
by (simp add: present_def alloc_def Let_def)

lemma present_set [simp]:
"present (set r v h) = present h"
by (simp add: present_def fun_eq_iff)

lemma noteq_I:
"present h r ⟹ ¬ present h r' ⟹ r =!= r'"
by (auto simp add: noteq_def present_def)

lemma execute_ref [execute_simps]:
"execute (ref v) h = Some (alloc v h)"
by (simp add: ref_def execute_simps)

lemma success_refI [success_intros]:
"success (ref v) h"
by (auto intro: success_intros simp add: ref_def)

lemma effect_refI [effect_intros]:
assumes "(r, h') = alloc v h"
shows "effect (ref v) h h' r"
by (rule effectI) (insert assms, simp add: execute_simps)

lemma effect_refE [effect_elims]:
assumes "effect (ref v) h h' r"
obtains "get h' r = v" and "present h' r" and "¬ present h r"
using assms by (rule effectE) (simp add: execute_simps)

lemma execute_lookup [execute_simps]:
"Heap_Monad.execute (lookup r) h = Some (get h r, h)"
by (simp add: lookup_def execute_simps)

lemma success_lookupI [success_intros]:
"success (lookup r) h"
by (auto intro: success_intros  simp add: lookup_def)

lemma effect_lookupI [effect_intros]:
assumes "h' = h" "x = get h r"
shows "effect (!r) h h' x"
by (rule effectI) (insert assms, simp add: execute_simps)

lemma effect_lookupE [effect_elims]:
assumes "effect (!r) h h' x"
obtains "h' = h" "x = get h r"
using assms by (rule effectE) (simp add: execute_simps)

lemma execute_update [execute_simps]:
"Heap_Monad.execute (update r v) h = Some ((), set r v h)"
by (simp add: update_def execute_simps)

lemma success_updateI [success_intros]:
"success (update r v) h"
by (auto intro: success_intros  simp add: update_def)

lemma effect_updateI [effect_intros]:
assumes "h' = set r v h"
shows "effect (r := v) h h' x"
by (rule effectI) (insert assms, simp add: execute_simps)

lemma effect_updateE [effect_elims]:
assumes "effect (r' := v) h h' r"
obtains "h' = set r' v h"
using assms by (rule effectE) (simp add: execute_simps)

lemma execute_change [execute_simps]:
"Heap_Monad.execute (change f r) h = Some (f (get h r), set r (f (get h r)) h)"
by (simp add: change_def bind_def Let_def execute_simps)

lemma success_changeI [success_intros]:
"success (change f r) h"
by (auto intro!: success_intros effect_intros simp add: change_def)

lemma effect_changeI [effect_intros]:
assumes "h' = set r (f (get h r)) h" "x = f (get h r)"
shows "effect (change f r) h h' x"
by (rule effectI) (insert assms, simp add: execute_simps)

lemma effect_changeE [effect_elims]:
assumes "effect (change f r') h h' r"
obtains "h' = set r' (f (get h r')) h" "r = f (get h r')"
using assms by (rule effectE) (simp add: execute_simps)

lemma lookup_chain:
"(!r ⪢ f) = f"
by (rule Heap_eqI) (auto simp add: lookup_def execute_simps intro: execute_bind)

lemma update_change [code]:
"r := e = change (λ_. e) r ⪢ return ()"
by (rule Heap_eqI) (simp add: change_def lookup_chain)

text ‹Non-interaction between imperative arrays and imperative references›

lemma array_get_set [simp]:
"Array.get (set r v h) = Array.get h"
by (simp add: Array.get_def set_def fun_eq_iff)

lemma get_update [simp]:
"get (Array.update a i v h) r = get h r"
by (simp add: get_def Array.update_def Array.set_def)

lemma alloc_update:
"fst (alloc v (Array.update a i v' h)) = fst (alloc v h)"
by (simp add: Array.update_def Array.get_def Array.set_def alloc_def Let_def)

lemma update_set_swap:
"Array.update a i v (set r v' h) = set r v' (Array.update a i v h)"
by (simp add: Array.update_def Array.get_def Array.set_def set_def)

lemma length_alloc [simp]:
"Array.length (snd (alloc v h)) a = Array.length h a"
by (simp add: Array.length_def Array.get_def alloc_def set_def Let_def)

lemma array_get_alloc [simp]:
"Array.get (snd (alloc v h)) = Array.get h"
by (simp add: Array.get_def alloc_def set_def Let_def fun_eq_iff)

lemma present_update [simp]:
"present (Array.update a i v h) = present h"
by (simp add: Array.update_def Array.set_def fun_eq_iff present_def)

lemma array_present_set [simp]:
"Array.present (set r v h) = Array.present h"
by (simp add: Array.present_def set_def fun_eq_iff)

lemma array_present_alloc [simp]:
"Array.present h a ⟹ Array.present (snd (alloc v h)) a"
by (simp add: Array.present_def alloc_def Let_def)

lemma set_array_set_swap:
"Array.set a xs (set r x' h) = set r x' (Array.set a xs h)"
by (simp add: Array.set_def set_def)

hide_const (open) present get set alloc noteq lookup update change

subsection ‹Code generator setup›

text ‹Intermediate operation avoids invariance problem in ‹Scala› (similar to value restriction)›

definition ref' where
[code del]: "ref' = ref"

lemma [code]:
"ref x = ref' x"
by (simp add: ref'_def)

text ‹SML / Eval›

code_printing type_constructor ref ⇀ (SML) "_/ ref"
code_printing type_constructor ref ⇀ (Eval) "_/ Unsynchronized.ref"
code_printing constant Ref ⇀ (SML) "raise/ (Fail/ \"bare Ref\")"
code_printing constant ref' ⇀ (SML) "(fn/ ()/ =>/ ref/ _)"
code_printing constant ref' ⇀ (Eval) "(fn/ ()/ =>/ Unsynchronized.ref/ _)"
code_printing constant Ref.lookup ⇀ (SML) "(fn/ ()/ =>/ !/ _)"
code_printing constant Ref.update ⇀ (SML) "(fn/ ()/ =>/ _/ :=/ _)"
code_printing constant "HOL.equal :: 'a ref ⇒ 'a ref ⇒ bool" ⇀ (SML) infixl 6 "="

code_reserved Eval Unsynchronized

text ‹OCaml›

code_printing type_constructor ref ⇀ (OCaml) "_/ ref"
code_printing constant Ref ⇀ (OCaml) "failwith/ \"bare Ref\""
code_printing constant ref' ⇀ (OCaml) "(fun/ ()/ ->/ ref/ _)"
code_printing constant Ref.lookup ⇀ (OCaml) "(fun/ ()/ ->/ !/ _)"
code_printing constant Ref.update ⇀ (OCaml) "(fun/ ()/ ->/ _/ :=/ _)"
code_printing constant "HOL.equal :: 'a ref ⇒ 'a ref ⇒ bool" ⇀ (OCaml) infixl 4 "="

code_reserved OCaml ref

code_printing type_constructor ref ⇀ (Haskell) "Heap.STRef/ Heap.RealWorld/ _"
code_printing constant Ref ⇀ (Haskell) "error/ \"bare Ref\""
code_printing constant ref' ⇀ (Haskell) "Heap.newSTRef"
code_printing constant Ref.update ⇀ (Haskell) "Heap.writeSTRef"
code_printing constant "HOL.equal :: 'a ref ⇒ 'a ref ⇒ bool" ⇀ (Haskell) infix 4 "=="
code_printing class_instance ref :: HOL.equal ⇀ (Haskell) -

text ‹Scala›

code_printing type_constructor ref ⇀ (Scala) "!Ref[_]"
code_printing constant Ref ⇀ (Scala) "!sys.error(\"bare Ref\")"
code_printing constant ref' ⇀ (Scala) "('_: Unit)/ =>/ Ref((_))"
code_printing constant Ref.lookup ⇀ (Scala) "('_: Unit)/ =>/ Ref.lookup((_))"
code_printing constant Ref.update ⇀ (Scala) "('_: Unit)/ =>/ Ref.update((_), (_))"
code_printing constant "HOL.equal :: 'a ref ⇒ 'a ref ⇒ bool" ⇀ (Scala) infixl 5 "=="

end
```