src/HOL/Imperative_HOL/Heap_Monad.thy
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(*  Title:      HOL/Library/Heap_Monad.thy
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    Author:     John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen
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*)
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header {* A monad with a polymorphic heap *}
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theory Heap_Monad
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imports Heap
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begin
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subsection {* The monad *}
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subsubsection {* Monad combinators *}
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datatype exception = Exn
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text {* Monadic heap actions either produce values
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  and transform the heap, or fail *}
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datatype 'a Heap = Heap "heap \<Rightarrow> ('a + exception) \<times> heap"
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primrec
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  execute :: "'a Heap \<Rightarrow> heap \<Rightarrow> ('a + exception) \<times> heap" where
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  "execute (Heap f) = f"
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lemmas [code del] = execute.simps
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lemma Heap_execute [simp]:
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  "Heap (execute f) = f" by (cases f) simp_all
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lemma Heap_eqI:
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  "(\<And>h. execute f h = execute g h) \<Longrightarrow> f = g"
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    by (cases f, cases g) (auto simp: expand_fun_eq)
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lemma Heap_eqI':
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  "(\<And>h. (\<lambda>x. execute (f x) h) = (\<lambda>y. execute (g y) h)) \<Longrightarrow> f = g"
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    by (auto simp: expand_fun_eq intro: Heap_eqI)
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lemma Heap_strip: "(\<And>f. PROP P f) \<equiv> (\<And>g. PROP P (Heap g))"
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proof
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  fix g :: "heap \<Rightarrow> ('a + exception) \<times> heap" 
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  assume "\<And>f. PROP P f"
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  then show "PROP P (Heap g)" .
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next
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  fix f :: "'a Heap" 
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  assume assm: "\<And>g. PROP P (Heap g)"
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  then have "PROP P (Heap (execute f))" .
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  then show "PROP P f" by simp
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qed
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definition
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  heap :: "(heap \<Rightarrow> 'a \<times> heap) \<Rightarrow> 'a Heap" where
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  [code del]: "heap f = Heap (\<lambda>h. apfst Inl (f h))"
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lemma execute_heap [simp]:
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  "execute (heap f) h = apfst Inl (f h)"
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  by (simp add: heap_def)
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definition
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  bindM :: "'a Heap \<Rightarrow> ('a \<Rightarrow> 'b Heap) \<Rightarrow> 'b Heap" (infixl ">>=" 54) where
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  [code del]: "f >>= g = Heap (\<lambda>h. case execute f h of
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                  (Inl x, h') \<Rightarrow> execute (g x) h'
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                | r \<Rightarrow> r)"
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notation
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  bindM (infixl "\<guillemotright>=" 54)
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abbreviation
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  chainM :: "'a Heap \<Rightarrow> 'b Heap \<Rightarrow> 'b Heap"  (infixl ">>" 54) where
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  "f >> g \<equiv> f >>= (\<lambda>_. g)"
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notation
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  chainM (infixl "\<guillemotright>" 54)
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definition
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  return :: "'a \<Rightarrow> 'a Heap" where
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  [code del]: "return x = heap (Pair x)"
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lemma execute_return [simp]:
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  "execute (return x) h = apfst Inl (x, h)"
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  by (simp add: return_def)
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definition
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  raise :: "string \<Rightarrow> 'a Heap" where -- {* the string is just decoration *}
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  [code del]: "raise s = Heap (Pair (Inr Exn))"
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notation (latex output)
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  "raise" ("\<^raw:{\textsf{raise}}>")
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lemma execute_raise [simp]:
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  "execute (raise s) h = (Inr Exn, h)"
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  by (simp add: raise_def)
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subsubsection {* do-syntax *}
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text {*
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  We provide a convenient do-notation for monadic expressions
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  well-known from Haskell.  @{const Let} is printed
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  specially in do-expressions.
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*}
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nonterminals do_expr
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syntax
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  "_do" :: "do_expr \<Rightarrow> 'a"
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    ("(do (_)//done)" [12] 100)
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  "_bindM" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
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    ("_ <- _;//_" [1000, 13, 12] 12)
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  "_chainM" :: "'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
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    ("_;//_" [13, 12] 12)
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  "_let" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
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    ("let _ = _;//_" [1000, 13, 12] 12)
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  "_nil" :: "'a \<Rightarrow> do_expr"
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    ("_" [12] 12)
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syntax (xsymbols)
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  "_bindM" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
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    ("_ \<leftarrow> _;//_" [1000, 13, 12] 12)
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syntax (latex output)
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  "_do" :: "do_expr \<Rightarrow> 'a"
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    ("(\<^raw:{\textsf{do}}> (_))" [12] 100)
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  "_let" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"
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    ("\<^raw:\textsf{let}> _ = _;//_" [1000, 13, 12] 12)
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notation (latex output)
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  "return" ("\<^raw:{\textsf{return}}>")
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translations
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  "_do f" => "f"
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  "_bindM x f g" => "f \<guillemotright>= (\<lambda>x. g)"
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  "_chainM f g" => "f \<guillemotright> g"
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  "_let x t f" => "CONST Let t (\<lambda>x. f)"
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  "_nil f" => "f"
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print_translation {*
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let
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  fun dest_abs_eta (Abs (abs as (_, ty, _))) =
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        let
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          val (v, t) = Syntax.variant_abs abs;
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        in (Free (v, ty), t) end
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    | dest_abs_eta t =
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        let
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          val (v, t) = Syntax.variant_abs ("", dummyT, t $ Bound 0);
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        in (Free (v, dummyT), t) end;
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  fun unfold_monad (Const (@{const_syntax bindM}, _) $ f $ g) =
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        let
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          val (v, g') = dest_abs_eta g;
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          val vs = fold_aterms (fn Free (v, _) => insert (op =) v | _ => I) v [];
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          val v_used = fold_aterms
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            (fn Free (w, _) => (fn s => s orelse member (op =) vs w) | _ => I) g' false;
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        in if v_used then
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          Const ("_bindM", dummyT) $ v $ f $ unfold_monad g'
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        else
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          Const ("_chainM", dummyT) $ f $ unfold_monad g'
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        end
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    | unfold_monad (Const (@{const_syntax chainM}, _) $ f $ g) =
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        Const ("_chainM", dummyT) $ f $ unfold_monad g
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    | unfold_monad (Const (@{const_syntax Let}, _) $ f $ g) =
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        let
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          val (v, g') = dest_abs_eta g;
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        in Const ("_let", dummyT) $ v $ f $ unfold_monad g' end
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    | unfold_monad (Const (@{const_syntax Pair}, _) $ f) =
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        Const (@{const_syntax return}, dummyT) $ f
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    | unfold_monad f = f;
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  fun contains_bindM (Const (@{const_syntax bindM}, _) $ _ $ _) = true
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    | contains_bindM (Const (@{const_syntax Let}, _) $ _ $ Abs (_, _, t)) =
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        contains_bindM t;
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  fun bindM_monad_tr' (f::g::ts) = list_comb
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    (Const ("_do", dummyT) $ unfold_monad (Const (@{const_syntax bindM}, dummyT) $ f $ g), ts);
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  fun Let_monad_tr' (f :: (g as Abs (_, _, g')) :: ts) = if contains_bindM g' then list_comb
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      (Const ("_do", dummyT) $ unfold_monad (Const (@{const_syntax Let}, dummyT) $ f $ g), ts)
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    else raise Match;
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in [
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  (@{const_syntax bindM}, bindM_monad_tr'),
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  (@{const_syntax Let}, Let_monad_tr')
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] end;
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*}
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subsection {* Monad properties *}
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subsubsection {* Monad laws *}
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lemma return_bind: "return x \<guillemotright>= f = f x"
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  by (simp add: bindM_def return_def)
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lemma bind_return: "f \<guillemotright>= return = f"
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proof (rule Heap_eqI)
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  fix h
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  show "execute (f \<guillemotright>= return) h = execute f h"
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    by (auto simp add: bindM_def return_def split: sum.splits prod.splits)
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qed
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lemma bind_bind: "(f \<guillemotright>= g) \<guillemotright>= h = f \<guillemotright>= (\<lambda>x. g x \<guillemotright>= h)"
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  by (rule Heap_eqI) (auto simp add: bindM_def split: split: sum.splits prod.splits)
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lemma bind_bind': "f \<guillemotright>= (\<lambda>x. g x \<guillemotright>= h x) = f \<guillemotright>= (\<lambda>x. g x \<guillemotright>= (\<lambda>y. return (x, y))) \<guillemotright>= (\<lambda>(x, y). h x y)"
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  by (rule Heap_eqI) (auto simp add: bindM_def split: split: sum.splits prod.splits)
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lemma raise_bind: "raise e \<guillemotright>= f = raise e"
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  by (simp add: raise_def bindM_def)
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lemmas monad_simp = return_bind bind_return bind_bind raise_bind
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subsection {* Generic combinators *}
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definition
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  liftM :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b Heap"
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where
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  "liftM f = return o f"
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definition
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  compM :: "('a \<Rightarrow> 'b Heap) \<Rightarrow> ('b \<Rightarrow> 'c Heap) \<Rightarrow> 'a \<Rightarrow> 'c Heap" (infixl ">>==" 54)
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where
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  "(f >>== g) = (\<lambda>x. f x \<guillemotright>= g)"
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notation
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  compM (infixl "\<guillemotright>==" 54)
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lemma liftM_collapse: "liftM f x = return (f x)"
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  by (simp add: liftM_def)
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lemma liftM_compM: "liftM f \<guillemotright>== g = g o f"
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  by (auto intro: Heap_eqI' simp add: expand_fun_eq liftM_def compM_def bindM_def)
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lemma compM_return: "f \<guillemotright>== return = f"
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  by (simp add: compM_def monad_simp)
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lemma compM_compM: "(f \<guillemotright>== g) \<guillemotright>== h = f \<guillemotright>== (g \<guillemotright>== h)"
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  by (simp add: compM_def monad_simp)
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lemma liftM_bind:
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  "(\<lambda>x. liftM f x \<guillemotright>= liftM g) = liftM (\<lambda>x. g (f x))"
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  by (rule Heap_eqI') (simp add: monad_simp liftM_def bindM_def)
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lemma liftM_comp:
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  "liftM f o g = liftM (f o g)"
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  by (rule Heap_eqI') (simp add: liftM_def)
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lemmas monad_simp' = monad_simp liftM_compM compM_return
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  compM_compM liftM_bind liftM_comp
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primrec 
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  mapM :: "('a \<Rightarrow> 'b Heap) \<Rightarrow> 'a list \<Rightarrow> 'b list Heap"
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where
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  "mapM f [] = return []"
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  | "mapM f (x#xs) = do y \<leftarrow> f x;
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                        ys \<leftarrow> mapM f xs;
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                        return (y # ys)
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                     done"
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primrec
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  foldM :: "('a \<Rightarrow> 'b \<Rightarrow> 'b Heap) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b Heap"
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where
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  "foldM f [] s = return s"
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  | "foldM f (x#xs) s = f x s \<guillemotright>= foldM f xs"
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definition
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  assert :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a Heap"
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where
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  "assert P x = (if P x then return x else raise (''assert''))"
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lemma assert_cong [fundef_cong]:
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  assumes "P = P'"
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  assumes "\<And>x. P' x \<Longrightarrow> f x = f' x"
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  shows "(assert P x >>= f) = (assert P' x >>= f')"
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  using assms by (auto simp add: assert_def return_bind raise_bind)
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hide (open) const heap execute
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subsection {* Code generator setup *}
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subsubsection {* Logical intermediate layer *}
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definition
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  Fail :: "String.literal \<Rightarrow> exception"
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where
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  [code del]: "Fail s = Exn"
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definition
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  raise_exc :: "exception \<Rightarrow> 'a Heap"
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where
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  [code del]: "raise_exc e = raise []"
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lemma raise_raise_exc [code, code inline]:
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  "raise s = raise_exc (Fail (STR s))"
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  unfolding Fail_def raise_exc_def raise_def ..
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hide (open) const Fail raise_exc
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subsubsection {* SML and OCaml *}
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code_type Heap (SML "unit/ ->/ _")
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code_const Heap (SML "raise/ (Fail/ \"bare Heap\")")
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code_const "op \<guillemotright>=" (SML "!(fn/ f'_/ =>/ fn/ ()/ =>/ f'_/ (_/ ())/ ())")
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code_const return (SML "!(fn/ ()/ =>/ _)")
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code_const "Heap_Monad.Fail" (SML "Fail")
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code_const "Heap_Monad.raise_exc" (SML "!(fn/ ()/ =>/ raise/ _)")
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code_type Heap (OCaml "_")
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code_const Heap (OCaml "failwith/ \"bare Heap\"")
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code_const "op \<guillemotright>=" (OCaml "!(fun/ f'_/ ()/ ->/ f'_/ (_/ ())/ ())")
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code_const return (OCaml "!(fun/ ()/ ->/ _)")
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code_const "Heap_Monad.Fail" (OCaml "Failure")
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code_const "Heap_Monad.raise_exc" (OCaml "!(fun/ ()/ ->/ raise/ _)")
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setup {*
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let
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open Code_Thingol;
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fun imp_program naming =
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  let
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    fun is_const c = case lookup_const naming c
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     of SOME c' => (fn c'' => c' = c'')
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      | NONE => K false;
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    val is_bindM = is_const @{const_name bindM};
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    val is_return = is_const @{const_name return};
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    val dummy_name = "X";
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    val dummy_type = ITyVar dummy_name;
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    val dummy_case_term = IVar "";
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    (*assumption: dummy values are not relevant for serialization*)
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    val unitt = case lookup_const naming @{const_name Unity}
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     of SOME unit' => IConst (unit', (([], []), []))
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      | NONE => error ("Must include " ^ @{const_name Unity} ^ " in generated constants.");
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    fun dest_abs ((v, ty) `|=> t, _) = ((v, ty), t)
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      | dest_abs (t, ty) =
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          let
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            val vs = fold_varnames cons t [];
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            val v = Name.variant vs "x";
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            val ty' = (hd o fst o unfold_fun) ty;
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          in ((v, ty'), t `$ IVar v) end;
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    fun force (t as IConst (c, _) `$ t') = if is_return c
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          then t' else t `$ unitt
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      | force t = t `$ unitt;
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    fun tr_bind' [(t1, _), (t2, ty2)] =
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      let
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        val ((v, ty), t) = dest_abs (t2, ty2);
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      in ICase (((force t1, ty), [(IVar v, tr_bind'' t)]), dummy_case_term) end
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    and tr_bind'' t = case unfold_app t
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         of (IConst (c, (_, ty1 :: ty2 :: _)), [x1, x2]) => if is_bindM c
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              then tr_bind' [(x1, ty1), (x2, ty2)]
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              else force t
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          | _ => force t;
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    fun imp_monad_bind'' ts = (dummy_name, dummy_type) `|=> ICase (((IVar dummy_name, dummy_type),
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      [(unitt, tr_bind' ts)]), dummy_case_term)
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    and imp_monad_bind' (const as (c, (_, tys))) ts = if is_bindM c then case (ts, tys)
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       of ([t1, t2], ty1 :: ty2 :: _) => imp_monad_bind'' [(t1, ty1), (t2, ty2)]
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        | ([t1, t2, t3], ty1 :: ty2 :: _) => imp_monad_bind'' [(t1, ty1), (t2, ty2)] `$ t3
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        | (ts, _) => imp_monad_bind (eta_expand 2 (const, ts))
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      else IConst const `$$ map imp_monad_bind ts
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    and imp_monad_bind (IConst const) = imp_monad_bind' const []
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      | imp_monad_bind (t as IVar _) = t
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      | imp_monad_bind (t as _ `$ _) = (case unfold_app t
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         of (IConst const, ts) => imp_monad_bind' const ts
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          | (t, ts) => imp_monad_bind t `$$ map imp_monad_bind ts)
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      | imp_monad_bind (v_ty `|=> t) = v_ty `|=> imp_monad_bind t
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      | imp_monad_bind (ICase (((t, ty), pats), t0)) = ICase
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          (((imp_monad_bind t, ty),
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            (map o pairself) imp_monad_bind pats),
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              imp_monad_bind t0);
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  in (Graph.map_nodes o map_terms_stmt) imp_monad_bind end;
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in
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Code_Target.extend_target ("SML_imp", ("SML", imp_program))
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#> Code_Target.extend_target ("OCaml_imp", ("OCaml", imp_program))
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end
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*}
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code_reserved OCaml Failure raise
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subsubsection {* Haskell *}
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text {* Adaption layer *}
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code_include Haskell "Heap"
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{*import qualified Control.Monad;
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import qualified Control.Monad.ST;
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import qualified Data.STRef;
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import qualified Data.Array.ST;
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type RealWorld = Control.Monad.ST.RealWorld;
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type ST s a = Control.Monad.ST.ST s a;
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type STRef s a = Data.STRef.STRef s a;
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type STArray s a = Data.Array.ST.STArray s Int a;
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newSTRef = Data.STRef.newSTRef;
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readSTRef = Data.STRef.readSTRef;
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writeSTRef = Data.STRef.writeSTRef;
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newArray :: (Int, Int) -> a -> ST s (STArray s a);
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newArray = Data.Array.ST.newArray;
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newListArray :: (Int, Int) -> [a] -> ST s (STArray s a);
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newListArray = Data.Array.ST.newListArray;
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lengthArray :: STArray s a -> ST s Int;
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lengthArray a = Control.Monad.liftM snd (Data.Array.ST.getBounds a);
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readArray :: STArray s a -> Int -> ST s a;
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readArray = Data.Array.ST.readArray;
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writeArray :: STArray s a -> Int -> a -> ST s ();
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writeArray = Data.Array.ST.writeArray;*}
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code_reserved Haskell Heap
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text {* Monad *}
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code_type Heap (Haskell "Heap.ST/ Heap.RealWorld/ _")
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code_const Heap (Haskell "error/ \"bare Heap\"")
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code_monad "op \<guillemotright>=" Haskell
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code_const return (Haskell "return")
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code_const "Heap_Monad.Fail" (Haskell "_")
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code_const "Heap_Monad.raise_exc" (Haskell "error")
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end