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

section ‹A monad with a polymorphic heap and primitive reasoning infrastructure›

imports
Heap
begin

text ‹Monadic heap actions either produce values
and transform the heap, or fail›
datatype 'a Heap = Heap "heap ⇒ ('a × heap) option"

declare [[code drop: "Code_Evaluation.term_of :: 'a::typerep Heap ⇒ Code_Evaluation.term"]]

primrec execute :: "'a Heap ⇒ heap ⇒ ('a × heap) option" where
[code del]: "execute (Heap f) = f"

lemma Heap_cases [case_names succeed fail]:
fixes f and h
assumes succeed: "⋀x h'. execute f h = Some (x, h') ⟹ P"
assumes fail: "execute f h = None ⟹ P"
shows P
using assms by (cases "execute f h") auto

lemma Heap_execute [simp]:
"Heap (execute f) = f" by (cases f) simp_all

lemma Heap_eqI:
"(⋀h. execute f h = execute g h) ⟹ f = g"
by (cases f, cases g) (auto simp: fun_eq_iff)

named_theorems execute_simps "simplification rules for execute"

lemma execute_Let [execute_simps]:
"execute (let x = t in f x) = (let x = t in execute (f x))"

subsubsection ‹Specialised lifters›

definition tap :: "(heap ⇒ 'a) ⇒ 'a Heap" where
[code del]: "tap f = Heap (λh. Some (f h, h))"

lemma execute_tap [execute_simps]:
"execute (tap f) h = Some (f h, h)"

definition heap :: "(heap ⇒ 'a × heap) ⇒ 'a Heap" where
[code del]: "heap f = Heap (Some ∘ f)"

lemma execute_heap [execute_simps]:
"execute (heap f) = Some ∘ f"

definition guard :: "(heap ⇒ bool) ⇒ (heap ⇒ 'a × heap) ⇒ 'a Heap" where
[code del]: "guard P f = Heap (λh. if P h then Some (f h) else None)"

lemma execute_guard [execute_simps]:
"¬ P h ⟹ execute (guard P f) h = None"
"P h ⟹ execute (guard P f) h = Some (f h)"

subsubsection ‹Predicate classifying successful computations›

definition success :: "'a Heap ⇒ heap ⇒ bool" where
"success f h ⟷ execute f h ≠ None"

lemma successI:
"execute f h ≠ None ⟹ success f h"

lemma successE:
assumes "success f h"
obtains r h' where "execute f h = Some (r, h')"
using assms by (auto simp: success_def)

named_theorems success_intros "introduction rules for success"

lemma success_tapI [success_intros]:
"success (tap f) h"
by (rule successI) (simp add: execute_simps)

lemma success_heapI [success_intros]:
"success (heap f) h"
by (rule successI) (simp add: execute_simps)

lemma success_guardI [success_intros]:
"P h ⟹ success (guard P f) h"
by (rule successI) (simp add: execute_guard)

lemma success_LetI [success_intros]:
"x = t ⟹ success (f x) h ⟹ success (let x = t in f x) h"

lemma success_ifI:
"(c ⟹ success t h) ⟹ (¬ c ⟹ success e h) ⟹
success (if c then t else e) h"

subsubsection ‹Predicate for a simple relational calculus›

text ‹
The ‹effect› predicate states that when a computation ‹c›
runs with the heap ‹h› will result in return value ‹r›
and a heap ‹h'›, i.e.~no exception occurs.
›

definition effect :: "'a Heap ⇒ heap ⇒ heap ⇒ 'a ⇒ bool" where
effect_def: "effect c h h' r ⟷ execute c h = Some (r, h')"

lemma effectI:
"execute c h = Some (r, h') ⟹ effect c h h' r"

lemma effectE:
assumes "effect c h h' r"
obtains "r = fst (the (execute c h))"
and "h' = snd (the (execute c h))"
and "success c h"
proof (rule that)
from assms have *: "execute c h = Some (r, h')" by (simp add: effect_def)
then show "success c h" by (simp add: success_def)
from * have "fst (the (execute c h)) = r" and "snd (the (execute c h)) = h'"
by simp_all
then show "r = fst (the (execute c h))"
and "h' = snd (the (execute c h))" by simp_all
qed

lemma effect_success:
"effect c h h' r ⟹ success c h"

lemma success_effectE:
assumes "success c h"
obtains r h' where "effect c h h' r"
using assms by (auto simp add: effect_def success_def)

lemma effect_deterministic:
assumes "effect f h h' a"
and "effect f h h'' b"
shows "a = b" and "h' = h''"
using assms unfolding effect_def by auto

named_theorems effect_intros "introduction rules for effect"
and effect_elims "elimination rules for effect"

lemma effect_LetI [effect_intros]:
assumes "x = t" "effect (f x) h h' r"
shows "effect (let x = t in f x) h h' r"
using assms by simp

lemma effect_LetE [effect_elims]:
assumes "effect (let x = t in f x) h h' r"
obtains "effect (f t) h h' r"
using assms by simp

lemma effect_ifI:
assumes "c ⟹ effect t h h' r"
and "¬ c ⟹ effect e h h' r"
shows "effect (if c then t else e) h h' r"
by (cases c) (simp_all add: assms)

lemma effect_ifE:
assumes "effect (if c then t else e) h h' r"
obtains "c" "effect t h h' r"
| "¬ c" "effect e h h' r"
using assms by (cases c) simp_all

lemma effect_tapI [effect_intros]:
assumes "h' = h" "r = f h"
shows "effect (tap f) h h' r"
by (rule effectI) (simp add: assms execute_simps)

lemma effect_tapE [effect_elims]:
assumes "effect (tap f) h h' r"
obtains "h' = h" and "r = f h"
using assms by (rule effectE) (auto simp add: execute_simps)

lemma effect_heapI [effect_intros]:
assumes "h' = snd (f h)" "r = fst (f h)"
shows "effect (heap f) h h' r"
by (rule effectI) (simp add: assms execute_simps)

lemma effect_heapE [effect_elims]:
assumes "effect (heap f) h h' r"
obtains "h' = snd (f h)" and "r = fst (f h)"
using assms by (rule effectE) (simp add: execute_simps)

lemma effect_guardI [effect_intros]:
assumes "P h" "h' = snd (f h)" "r = fst (f h)"
shows "effect (guard P f) h h' r"
by (rule effectI) (simp add: assms execute_simps)

lemma effect_guardE [effect_elims]:
assumes "effect (guard P f) h h' r"
obtains "h' = snd (f h)" "r = fst (f h)" "P h"
using assms by (rule effectE)
(auto simp add: execute_simps elim!: successE, cases "P h", auto simp add: execute_simps)

definition return :: "'a ⇒ 'a Heap" where
[code del]: "return x = heap (Pair x)"

lemma execute_return [execute_simps]:
"execute (return x) = Some ∘ Pair x"

lemma success_returnI [success_intros]:
"success (return x) h"
by (rule successI) (simp add: execute_simps)

lemma effect_returnI [effect_intros]:
"h = h' ⟹ effect (return x) h h' x"
by (rule effectI) (simp add: execute_simps)

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

definition raise :: "String.literal ⇒ 'a Heap" ― ‹the literal is just decoration›
where "raise s = Heap (λ_. None)"

code_datatype raise ― ‹avoid \<^const>‹Heap› formally›

lemma execute_raise [execute_simps]:
"execute (raise s) = (λ_. None)"

lemma effect_raiseE [effect_elims]:
assumes "effect (raise x) h h' r"
obtains "False"
using assms by (rule effectE) (simp add: success_def execute_simps)

definition bind :: "'a Heap ⇒ ('a ⇒ 'b Heap) ⇒ 'b Heap" where
[code del]: "bind f g = Heap (λh. case execute f h of
Some (x, h') ⇒ execute (g x) h'
| None ⇒ None)"

lemma execute_bind [execute_simps]:
"execute f h = Some (x, h') ⟹ execute (f ⤜ g) h = execute (g x) h'"
"execute f h = None ⟹ execute (f ⤜ g) h = None"

lemma execute_bind_case:
"execute (f ⤜ g) h = (case (execute f h) of
Some (x, h') ⇒ execute (g x) h' | None ⇒ None)"

lemma execute_bind_success:
"success f h ⟹ execute (f ⤜ g) h = execute (g (fst (the (execute f h)))) (snd (the (execute f h)))"
by (cases f h rule: Heap_cases) (auto elim: successE simp add: bind_def)

lemma success_bind_executeI:
"execute f h = Some (x, h') ⟹ success (g x) h' ⟹ success (f ⤜ g) h"
by (auto intro!: successI elim: successE simp add: bind_def)

lemma success_bind_effectI [success_intros]:
"effect f h h' x ⟹ success (g x) h' ⟹ success (f ⤜ g) h"
by (auto simp add: effect_def success_def bind_def)

lemma effect_bindI [effect_intros]:
assumes "effect f h h' r" "effect (g r) h' h'' r'"
shows "effect (f ⤜ g) h h'' r'"
using assms
apply (auto intro!: effectI elim!: effectE successE)
apply (subst execute_bind, simp_all)
done

lemma effect_bindE [effect_elims]:
assumes "effect (f ⤜ g) h h'' r'"
obtains h' r where "effect f h h' r" "effect (g r) h' h'' r'"
using assms by (auto simp add: effect_def bind_def split: option.split_asm)

lemma execute_bind_eq_SomeI:
assumes "execute f h = Some (x, h')"
and "execute (g x) h' = Some (y, h'')"
shows "execute (f ⤜ g) h = Some (y, h'')"
using assms by (simp add: bind_def)

lemma return_bind [simp]: "return x ⤜ f = f x"
by (rule Heap_eqI) (simp add: execute_simps)

lemma bind_return [simp]: "f ⤜ return = f"
by (rule Heap_eqI) (simp add: bind_def execute_simps split: option.splits)

lemma bind_bind [simp]: "(f ⤜ g) ⤜ k = (f :: 'a Heap) ⤜ (λx. g x ⤜ k)"
by (rule Heap_eqI) (simp add: bind_def execute_simps split: option.splits)

lemma raise_bind [simp]: "raise e ⤜ f = raise e"
by (rule Heap_eqI) (simp add: execute_simps)

subsection ‹Generic combinators›

subsubsection ‹Assertions›

definition assert :: "('a ⇒ bool) ⇒ 'a ⇒ 'a Heap" where
"assert P x = (if P x then return x else raise STR ''assert'')"

lemma execute_assert [execute_simps]:
"P x ⟹ execute (assert P x) h = Some (x, h)"
"¬ P x ⟹ execute (assert P x) h = None"

lemma success_assertI [success_intros]:
"P x ⟹ success (assert P x) h"
by (rule successI) (simp add: execute_assert)

lemma effect_assertI [effect_intros]:
"P x ⟹ h' = h ⟹ r = x ⟹ effect (assert P x) h h' r"
by (rule effectI) (simp add: execute_assert)

lemma effect_assertE [effect_elims]:
assumes "effect (assert P x) h h' r"
obtains "P x" "r = x" "h' = h"
using assms by (rule effectE) (cases "P x", simp_all add: execute_assert success_def)

lemma assert_cong [fundef_cong]:
assumes "P = P'"
assumes "⋀x. P' x ⟹ f x = f' x"
shows "(assert P x ⤜ f) = (assert P' x ⤜ f')"
by (rule Heap_eqI) (insert assms, simp add: assert_def)

subsubsection ‹Plain lifting›

definition lift :: "('a ⇒ 'b) ⇒ 'a ⇒ 'b Heap" where
"lift f = return o f"

lemma lift_collapse [simp]:
"lift f x = return (f x)"

lemma bind_lift:
"(f ⤜ lift g) = (f ⤜ (λx. return (g x)))"

subsubsection ‹Iteration -- warning: this is rarely useful!›

primrec fold_map :: "('a ⇒ 'b Heap) ⇒ 'a list ⇒ 'b list Heap" where
"fold_map f [] = return []"
| "fold_map f (x # xs) = do {
y ← f x;
ys ← fold_map f xs;
return (y # ys)
}"

lemma fold_map_append:
"fold_map f (xs @ ys) = fold_map f xs ⤜ (λxs. fold_map f ys ⤜ (λys. return (xs @ ys)))"
by (induct xs) simp_all

lemma execute_fold_map_unchanged_heap [execute_simps]:
assumes "⋀x. x ∈ set xs ⟹ ∃y. execute (f x) h = Some (y, h)"
shows "execute (fold_map f xs) h =
Some (List.map (λx. fst (the (execute (f x) h))) xs, h)"
using assms proof (induct xs)
case Nil show ?case by (simp add: execute_simps)
next
case (Cons x xs)
from Cons.prems obtain y
where y: "execute (f x) h = Some (y, h)" by auto
moreover from Cons.prems Cons.hyps have "execute (fold_map f xs) h =
Some (map (λx. fst (the (execute (f x) h))) xs, h)" by auto
ultimately show ?case by (simp, simp only: execute_bind(1), simp add: execute_simps)
qed

subsection ‹Partial function definition setup›

definition Heap_ord :: "'a Heap ⇒ 'a Heap ⇒ bool" where
"Heap_ord = img_ord execute (fun_ord option_ord)"

definition Heap_lub :: "'a Heap set ⇒ 'a Heap" where
"Heap_lub = img_lub execute Heap (fun_lub (flat_lub None))"

lemma Heap_lub_empty: "Heap_lub {} = Heap Map.empty"
by(simp add: Heap_lub_def img_lub_def fun_lub_def flat_lub_def)

lemma heap_interpretation: "partial_function_definitions Heap_ord Heap_lub"
proof -
have "partial_function_definitions (fun_ord option_ord) (fun_lub (flat_lub None))"
by (rule partial_function_lift) (rule flat_interpretation)
then have "partial_function_definitions (img_ord execute (fun_ord option_ord))
(img_lub execute Heap (fun_lub (flat_lub None)))"
by (rule partial_function_image) (auto intro: Heap_eqI)
then show "partial_function_definitions Heap_ord Heap_lub"
by (simp only: Heap_ord_def Heap_lub_def)
qed

interpretation heap: partial_function_definitions Heap_ord Heap_lub
rewrites "Heap_lub {} ≡ Heap Map.empty"

(λf:: 'a => ('b * 'c) option. ∀h h' r. f h = Some (r, h') ⟶ P x h h' r)"
fix A :: "('a ⇒ ('b * 'c) option) set"
assume ch: "Complete_Partial_Order.chain option.le_fun A"
and IH: "∀f∈A. ∀h h' r. f h = Some (r, h') ⟶ P x h h' r"
from ch have ch': "⋀x. Complete_Partial_Order.chain option_ord {y. ∃f∈A. y = f x}" by (rule chain_fun)
show "∀h h' r. option.lub_fun A h = Some (r, h') ⟶ P x h h' r"
proof (intro allI impI)
fix h h' r assume "option.lub_fun A h = Some (r, h')"
from flat_lub_in_chain[OF ch' this[unfolded fun_lub_def]]
have "Some (r, h') ∈ {y. ∃f∈A. y = f h}" by simp
then have "∃f∈A. f h = Some (r, h')" by auto
with IH show "P x h h' r" by auto
qed
qed

"heap.admissible (λf. ∀x h h' r. effect (f x) h h' r ⟶ P x h h' r)"
fix x
show "ccpo.admissible Heap_lub Heap_ord (λa. ∀h h' r. effect a h h' r ⟶ P x h h' r)"
unfolding Heap_ord_def Heap_lub_def
show "option.admissible ((λa. ∀h h' r. effect a h h' r ⟶ P x h h' r) ∘ Heap)"
unfolding comp_def effect_def execute.simps
qed

lemma fixp_induct_heap:
fixes F :: "'c ⇒ 'c" and
U :: "'c ⇒ 'b ⇒ 'a Heap" and
C :: "('b ⇒ 'a Heap) ⇒ 'c" and
P :: "'b ⇒ heap ⇒ heap ⇒ 'a ⇒ bool"
assumes mono: "⋀x. monotone (fun_ord Heap_ord) Heap_ord (λf. U (F (C f)) x)"
assumes eq: "f ≡ C (ccpo.fixp (fun_lub Heap_lub) (fun_ord Heap_ord) (λf. U (F (C f))))"
assumes inverse2: "⋀f. U (C f) = f"
assumes step: "⋀f x h h' r. (⋀x h h' r. effect (U f x) h h' r ⟹ P x h h' r)
⟹ effect (U (F f) x) h h' r ⟹ P x h h' r"
assumes defined: "effect (U f x) h h' r"
shows "P x h h' r"
using step defined heap.fixp_induct_uc[of U F C, OF mono eq inverse2 admissible_heap, of P]
unfolding effect_def execute.simps
by blast

declaration ‹Partial_Function.init "heap" \<^term>‹heap.fixp_fun›
\<^term>‹heap.mono_body› @{thm heap.fixp_rule_uc} @{thm heap.fixp_induct_uc}
(SOME @{thm fixp_induct_heap})›

abbreviation "mono_Heap ≡ monotone (fun_ord Heap_ord) Heap_ord"

lemma Heap_ordI:
assumes "⋀h. execute x h = None ∨ execute x h = execute y h"
shows "Heap_ord x y"
using assms unfolding Heap_ord_def img_ord_def fun_ord_def flat_ord_def
by blast

lemma Heap_ordE:
assumes "Heap_ord x y"
obtains "execute x h = None" | "execute x h = execute y h"
using assms unfolding Heap_ord_def img_ord_def fun_ord_def flat_ord_def
by atomize_elim blast

lemma bind_mono [partial_function_mono]:
assumes mf: "mono_Heap B" and mg: "⋀y. mono_Heap (λf. C y f)"
shows "mono_Heap (λf. B f ⤜ (λy. C y f))"
proof (rule monotoneI)
fix f g :: "'a ⇒ 'b Heap" assume fg: "fun_ord Heap_ord f g"
from mf
have 1: "Heap_ord (B f) (B g)" by (rule monotoneD) (rule fg)
from mg
have 2: "⋀y'. Heap_ord (C y' f) (C y' g)" by (rule monotoneD) (rule fg)

have "Heap_ord (B f ⤜ (λy. C y f)) (B g ⤜ (λy. C y f))"
(is "Heap_ord ?L ?R")
proof (rule Heap_ordI)
fix h
from 1 show "execute ?L h = None ∨ execute ?L h = execute ?R h"
by (rule Heap_ordE[where h = h]) (auto simp: execute_bind_case)
qed
also
have "Heap_ord (B g ⤜ (λy'. C y' f)) (B g ⤜ (λy'. C y' g))"
(is "Heap_ord ?L ?R")
proof (rule Heap_ordI)
fix h
show "execute ?L h = None ∨ execute ?L h = execute ?R h"
proof (cases "execute (B g) h")
case None
then have "execute ?L h = None" by (auto simp: execute_bind_case)
thus ?thesis ..
next
case Some
then obtain r h' where "execute (B g) h = Some (r, h')"
by (metis surjective_pairing)
then have "execute ?L h = execute (C r f) h'"
"execute ?R h = execute (C r g) h'"
by (auto simp: execute_bind_case)
with 2[of r] show ?thesis by (auto elim: Heap_ordE)
qed
qed
finally (heap.leq_trans)
show "Heap_ord (B f ⤜ (λy. C y f)) (B g ⤜ (λy'. C y' g))" .
qed

subsection ‹Code generator setup›

subsubsection ‹SML and OCaml›

code_printing type_constructor Heap ⇀ (SML) "(unit/ ->/ _)"
code_printing constant bind ⇀ (SML) "!(fn/ f'_/ =>/ fn/ ()/ =>/ f'_/ (_/ ())/ ())"
code_printing constant return ⇀ (SML) "!(fn/ ()/ =>/ _)"
code_printing constant Heap_Monad.raise ⇀ (SML) "!(raise/ Fail/ _)"

code_printing type_constructor Heap ⇀ (OCaml) "(unit/ ->/ _)"
code_printing constant bind ⇀ (OCaml) "!(fun/ f'_/ ()/ ->/ f'_/ (_/ ())/ ())"
code_printing constant return ⇀ (OCaml) "!(fun/ ()/ ->/ _)"
code_printing constant Heap_Monad.raise ⇀ (OCaml) "failwith"

‹
module Heap(ST, RealWorld, STRef, newSTRef, readSTRef, writeSTRef,
STArray, newArray, newListArray, newFunArray, lengthArray, readArray, writeArray) where

import qualified Data.Array.ST

type STArray s a = Data.Array.ST.STArray s Integer a

newArray :: Integer -> a -> ST s (STArray s a)
newArray k = Data.Array.ST.newArray (0, k - 1)

newListArray :: [a] -> ST s (STArray s a)
newListArray xs = Data.Array.ST.newListArray (0, (fromInteger . toInteger . length) xs - 1) xs

newFunArray :: Integer -> (Integer -> a) -> ST s (STArray s a)
newFunArray k f = Data.Array.ST.newListArray (0, k - 1) (map f [0..k-1])

lengthArray :: STArray s a -> ST s Integer
lengthArray a = liftM (\(_, l) -> l + 1) (Data.Array.ST.getBounds a)

readArray :: STArray s a -> Integer -> ST s a

writeArray :: STArray s a -> Integer -> a -> ST s ()
writeArray = Data.Array.ST.writeArray›

code_printing type_constructor Heap ⇀ (Haskell) "Heap.ST/ Heap.RealWorld/ _"
code_printing constant return ⇀ (Haskell) "return"

subsubsection ‹Scala›

code_printing code_module "Heap" ⇀ (Scala)
‹object Heap {
def bind[A, B](f: Unit => A, g: A => Unit => B): Unit => B = (_: Unit) => g(f(()))(())
}

class Ref[A](x: A) {
var value = x
}

object Ref {
def apply[A](x: A): Ref[A] =
new Ref[A](x)
def lookup[A](r: Ref[A]): A =
r.value
def update[A](r: Ref[A], x: A): Unit =
{ r.value = x }
}

object Array {
class T[A](n: Int)
{
val array: Array[AnyRef] = new Array[AnyRef](n)
def apply(i: Int): A = array(i).asInstanceOf[A]
def update(i: Int, x: A): Unit = array(i) = x.asInstanceOf[AnyRef]
def length: Int = array.length
def toList: List[A] = array.toList.asInstanceOf[List[A]]
override def toString: String = array.mkString("Array.T(", ",", ")")
}
def init[A](n: Int)(f: Int => A): T[A] = {
val a = new T[A](n)
for (i <- 0 until n) a(i) = f(i)
a
}
def make[A](n: BigInt)(f: BigInt => A): T[A] = init(n.toInt)((i: Int) => f(BigInt(i)))
def alloc[A](n: BigInt)(x: A): T[A] = init(n.toInt)(_ => x)
def len[A](a: T[A]): BigInt = BigInt(a.length)
def nth[A](a: T[A], n: BigInt): A = a(n.toInt)
def upd[A](a: T[A], n: BigInt, x: A): Unit = a.update(n.toInt, x)
def freeze[A](a: T[A]): List[A] = a.toList
}

›

code_reserved Scala Heap Ref Array

code_printing type_constructor Heap ⇀ (Scala) "(Unit/ =>/ _)"
code_printing constant bind ⇀ (Scala) "Heap.bind"
code_printing constant return ⇀ (Scala) "('_: Unit)/ =>/ _"
code_printing constant Heap_Monad.raise ⇀ (Scala) "!sys.error((_))"

subsubsection ‹Target variants with less units›

setup ‹

let

open Code_Thingol;

val imp_program =
let
val is_bind = curry (=) \<^const_name>‹bind›;
val is_return = curry (=) \<^const_name>‹return›;
val dummy_name = "";
val dummy_case_term = IVar NONE;
(*assumption: dummy values are not relevant for serialization*)
val unitT = \<^type_name>‹unit› `%% [];
val unitt =
IConst { sym = Code_Symbol.Constant \<^const_name>‹Unity›, typargs = [], dicts = [], dom = [],
annotation = NONE, range = unitT };
fun dest_abs ((v, ty) `|=> (t, _), _) = ((v, ty), t)
| dest_abs (t, ty) =
let
val vs = fold_varnames cons t [];
val v = singleton (Name.variant_list vs) "x";
val ty' = (hd o fst o unfold_fun) ty;
in ((SOME v, ty'), t `\$ IVar (SOME v)) end;
fun force (t as IConst { sym = Code_Symbol.Constant c, ... } `\$ t') = if is_return c
then t' else t `\$ unitt
| force t = t `\$ unitt;
fun tr_bind'' [(t1, _), (t2, ty2)] =
let
val ((v, ty), t) = dest_abs (t2, ty2);
in ICase { term = force t1, typ = ty, clauses = [(IVar v, tr_bind' t)], primitive = dummy_case_term } end
and tr_bind' t = case unfold_app t
of (IConst { sym = Code_Symbol.Constant c, dom = ty1 :: ty2 :: _, ... }, [x1, x2]) => if is_bind c
then tr_bind'' [(x1, ty1), (x2, ty2)]
else force t
| _ => force t;
fun imp_monad_bind'' ts = (SOME dummy_name, unitT) `|=>
(ICase { term = IVar (SOME dummy_name), typ = unitT, clauses = [(unitt, tr_bind'' ts)], primitive = dummy_case_term }, unitT)
fun imp_monad_bind' (const as { sym = Code_Symbol.Constant c, dom = dom, ... }) ts = if is_bind c then case (ts, dom)
of ([t1, t2], ty1 :: ty2 :: _) => imp_monad_bind'' [(t1, ty1), (t2, ty2)]
| ([t1, t2, t3], ty1 :: ty2 :: _) => imp_monad_bind'' [(t1, ty1), (t2, ty2)] `\$ t3
| (ts, _) => imp_monad_bind (saturated_application 2 (const, ts))
else IConst const `\$\$ map imp_monad_bind ts
| imp_monad_bind (t as IVar _) = t
| imp_monad_bind (t as _ `\$ _) = (case unfold_app t
of (IConst const, ts) => imp_monad_bind' const ts
| imp_monad_bind (ICase { term = t, typ = ty, clauses = clauses, primitive = t0 }) =
ICase { term = imp_monad_bind t, typ = ty,
clauses = (map o apply2) imp_monad_bind clauses, primitive = imp_monad_bind t0 };

in (Code_Symbol.Graph.map o K o map_terms_stmt) imp_monad_bind end;

in