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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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|
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shows "(assert P x >>= f) = (assert P' x >>= f')"
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haftmann@28742
|
267 |
using assms by (auto simp add: assert_def return_bind raise_bind)
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haftmann@28742
|
268 |
|
haftmann@26170
|
269 |
hide (open) const heap execute
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haftmann@26170
|
270 |
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haftmann@26182
|
271 |
|
haftmann@26182
|
272 |
subsection {* Code generator setup *}
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haftmann@26182
|
273 |
|
haftmann@26182
|
274 |
subsubsection {* Logical intermediate layer *}
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haftmann@26182
|
275 |
|
haftmann@26182
|
276 |
definition
|
haftmann@31205
|
277 |
Fail :: "String.literal \<Rightarrow> exception"
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haftmann@26182
|
278 |
where
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haftmann@28562
|
279 |
[code del]: "Fail s = Exn"
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haftmann@26182
|
280 |
|
haftmann@26182
|
281 |
definition
|
haftmann@26182
|
282 |
raise_exc :: "exception \<Rightarrow> 'a Heap"
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haftmann@26182
|
283 |
where
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haftmann@28562
|
284 |
[code del]: "raise_exc e = raise []"
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haftmann@26182
|
285 |
|
haftmann@32061
|
286 |
lemma raise_raise_exc [code, code_unfold]:
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haftmann@26182
|
287 |
"raise s = raise_exc (Fail (STR s))"
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haftmann@26182
|
288 |
unfolding Fail_def raise_exc_def raise_def ..
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haftmann@26182
|
289 |
|
haftmann@26182
|
290 |
hide (open) const Fail raise_exc
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haftmann@26182
|
291 |
|
haftmann@26182
|
292 |
|
haftmann@27707
|
293 |
subsubsection {* SML and OCaml *}
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haftmann@26182
|
294 |
|
haftmann@26752
|
295 |
code_type Heap (SML "unit/ ->/ _")
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haftmann@26182
|
296 |
code_const Heap (SML "raise/ (Fail/ \"bare Heap\")")
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haftmann@27826
|
297 |
code_const "op \<guillemotright>=" (SML "!(fn/ f'_/ =>/ fn/ ()/ =>/ f'_/ (_/ ())/ ())")
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haftmann@27707
|
298 |
code_const return (SML "!(fn/ ()/ =>/ _)")
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haftmann@26182
|
299 |
code_const "Heap_Monad.Fail" (SML "Fail")
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haftmann@27707
|
300 |
code_const "Heap_Monad.raise_exc" (SML "!(fn/ ()/ =>/ raise/ _)")
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haftmann@26182
|
301 |
|
haftmann@26182
|
302 |
code_type Heap (OCaml "_")
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haftmann@26182
|
303 |
code_const Heap (OCaml "failwith/ \"bare Heap\"")
|
haftmann@27826
|
304 |
code_const "op \<guillemotright>=" (OCaml "!(fun/ f'_/ ()/ ->/ f'_/ (_/ ())/ ())")
|
haftmann@27707
|
305 |
code_const return (OCaml "!(fun/ ()/ ->/ _)")
|
haftmann@26182
|
306 |
code_const "Heap_Monad.Fail" (OCaml "Failure")
|
haftmann@27707
|
307 |
code_const "Heap_Monad.raise_exc" (OCaml "!(fun/ ()/ ->/ raise/ _)")
|
haftmann@27707
|
308 |
|
haftmann@31870
|
309 |
setup {*
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haftmann@27707
|
310 |
|
haftmann@31870
|
311 |
let
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haftmann@27707
|
312 |
|
haftmann@31870
|
313 |
open Code_Thingol;
|
haftmann@27707
|
314 |
|
haftmann@31870
|
315 |
fun imp_program naming =
|
haftmann@31870
|
316 |
|
haftmann@31870
|
317 |
let
|
haftmann@31870
|
318 |
fun is_const c = case lookup_const naming c
|
haftmann@31870
|
319 |
of SOME c' => (fn c'' => c' = c'')
|
haftmann@31870
|
320 |
| NONE => K false;
|
haftmann@31870
|
321 |
val is_bindM = is_const @{const_name bindM};
|
haftmann@31870
|
322 |
val is_return = is_const @{const_name return};
|
haftmann@31893
|
323 |
val dummy_name = "";
|
haftmann@31870
|
324 |
val dummy_type = ITyVar dummy_name;
|
haftmann@31893
|
325 |
val dummy_case_term = IVar NONE;
|
haftmann@31870
|
326 |
(*assumption: dummy values are not relevant for serialization*)
|
haftmann@31870
|
327 |
val unitt = case lookup_const naming @{const_name Unity}
|
haftmann@31870
|
328 |
of SOME unit' => IConst (unit', (([], []), []))
|
haftmann@31870
|
329 |
| NONE => error ("Must include " ^ @{const_name Unity} ^ " in generated constants.");
|
haftmann@31870
|
330 |
fun dest_abs ((v, ty) `|=> t, _) = ((v, ty), t)
|
haftmann@31870
|
331 |
| dest_abs (t, ty) =
|
haftmann@31870
|
332 |
let
|
haftmann@31870
|
333 |
val vs = fold_varnames cons t [];
|
haftmann@31870
|
334 |
val v = Name.variant vs "x";
|
haftmann@31870
|
335 |
val ty' = (hd o fst o unfold_fun) ty;
|
haftmann@31893
|
336 |
in ((SOME v, ty'), t `$ IVar (SOME v)) end;
|
haftmann@31870
|
337 |
fun force (t as IConst (c, _) `$ t') = if is_return c
|
haftmann@31870
|
338 |
then t' else t `$ unitt
|
haftmann@31870
|
339 |
| force t = t `$ unitt;
|
haftmann@31870
|
340 |
fun tr_bind' [(t1, _), (t2, ty2)] =
|
haftmann@31870
|
341 |
let
|
haftmann@31870
|
342 |
val ((v, ty), t) = dest_abs (t2, ty2);
|
haftmann@31870
|
343 |
in ICase (((force t1, ty), [(IVar v, tr_bind'' t)]), dummy_case_term) end
|
haftmann@31870
|
344 |
and tr_bind'' t = case unfold_app t
|
haftmann@31870
|
345 |
of (IConst (c, (_, ty1 :: ty2 :: _)), [x1, x2]) => if is_bindM c
|
haftmann@31870
|
346 |
then tr_bind' [(x1, ty1), (x2, ty2)]
|
haftmann@31870
|
347 |
else force t
|
haftmann@31870
|
348 |
| _ => force t;
|
haftmann@31893
|
349 |
fun imp_monad_bind'' ts = (SOME dummy_name, dummy_type) `|=> ICase (((IVar (SOME dummy_name), dummy_type),
|
haftmann@31870
|
350 |
[(unitt, tr_bind' ts)]), dummy_case_term)
|
haftmann@31870
|
351 |
and imp_monad_bind' (const as (c, (_, tys))) ts = if is_bindM c then case (ts, tys)
|
haftmann@31870
|
352 |
of ([t1, t2], ty1 :: ty2 :: _) => imp_monad_bind'' [(t1, ty1), (t2, ty2)]
|
haftmann@31870
|
353 |
| ([t1, t2, t3], ty1 :: ty2 :: _) => imp_monad_bind'' [(t1, ty1), (t2, ty2)] `$ t3
|
haftmann@31870
|
354 |
| (ts, _) => imp_monad_bind (eta_expand 2 (const, ts))
|
haftmann@31870
|
355 |
else IConst const `$$ map imp_monad_bind ts
|
haftmann@31870
|
356 |
and imp_monad_bind (IConst const) = imp_monad_bind' const []
|
haftmann@31870
|
357 |
| imp_monad_bind (t as IVar _) = t
|
haftmann@31870
|
358 |
| imp_monad_bind (t as _ `$ _) = (case unfold_app t
|
haftmann@31870
|
359 |
of (IConst const, ts) => imp_monad_bind' const ts
|
haftmann@31870
|
360 |
| (t, ts) => imp_monad_bind t `$$ map imp_monad_bind ts)
|
haftmann@31870
|
361 |
| imp_monad_bind (v_ty `|=> t) = v_ty `|=> imp_monad_bind t
|
haftmann@31870
|
362 |
| imp_monad_bind (ICase (((t, ty), pats), t0)) = ICase
|
haftmann@31870
|
363 |
(((imp_monad_bind t, ty),
|
haftmann@31870
|
364 |
(map o pairself) imp_monad_bind pats),
|
haftmann@31870
|
365 |
imp_monad_bind t0);
|
haftmann@31870
|
366 |
|
haftmann@31870
|
367 |
in (Graph.map_nodes o map_terms_stmt) imp_monad_bind end;
|
haftmann@27707
|
368 |
|
haftmann@27707
|
369 |
in
|
haftmann@27707
|
370 |
|
haftmann@31870
|
371 |
Code_Target.extend_target ("SML_imp", ("SML", imp_program))
|
haftmann@31870
|
372 |
#> Code_Target.extend_target ("OCaml_imp", ("OCaml", imp_program))
|
haftmann@27707
|
373 |
|
haftmann@27707
|
374 |
end
|
haftmann@31870
|
375 |
|
haftmann@27707
|
376 |
*}
|
haftmann@27707
|
377 |
|
haftmann@26182
|
378 |
code_reserved OCaml Failure raise
|
haftmann@26182
|
379 |
|
haftmann@26182
|
380 |
|
haftmann@26182
|
381 |
subsubsection {* Haskell *}
|
haftmann@26182
|
382 |
|
haftmann@26182
|
383 |
text {* Adaption layer *}
|
haftmann@26182
|
384 |
|
haftmann@29730
|
385 |
code_include Haskell "Heap"
|
haftmann@26182
|
386 |
{*import qualified Control.Monad;
|
haftmann@26182
|
387 |
import qualified Control.Monad.ST;
|
haftmann@26182
|
388 |
import qualified Data.STRef;
|
haftmann@26182
|
389 |
import qualified Data.Array.ST;
|
haftmann@26182
|
390 |
|
haftmann@27695
|
391 |
type RealWorld = Control.Monad.ST.RealWorld;
|
haftmann@26182
|
392 |
type ST s a = Control.Monad.ST.ST s a;
|
haftmann@26182
|
393 |
type STRef s a = Data.STRef.STRef s a;
|
haftmann@27673
|
394 |
type STArray s a = Data.Array.ST.STArray s Int a;
|
haftmann@26182
|
395 |
|
haftmann@26182
|
396 |
newSTRef = Data.STRef.newSTRef;
|
haftmann@26182
|
397 |
readSTRef = Data.STRef.readSTRef;
|
haftmann@26182
|
398 |
writeSTRef = Data.STRef.writeSTRef;
|
haftmann@26182
|
399 |
|
haftmann@27673
|
400 |
newArray :: (Int, Int) -> a -> ST s (STArray s a);
|
haftmann@26182
|
401 |
newArray = Data.Array.ST.newArray;
|
haftmann@26182
|
402 |
|
haftmann@27673
|
403 |
newListArray :: (Int, Int) -> [a] -> ST s (STArray s a);
|
haftmann@26182
|
404 |
newListArray = Data.Array.ST.newListArray;
|
haftmann@26182
|
405 |
|
haftmann@27673
|
406 |
lengthArray :: STArray s a -> ST s Int;
|
haftmann@27673
|
407 |
lengthArray a = Control.Monad.liftM snd (Data.Array.ST.getBounds a);
|
haftmann@26182
|
408 |
|
haftmann@27673
|
409 |
readArray :: STArray s a -> Int -> ST s a;
|
haftmann@26182
|
410 |
readArray = Data.Array.ST.readArray;
|
haftmann@26182
|
411 |
|
haftmann@27673
|
412 |
writeArray :: STArray s a -> Int -> a -> ST s ();
|
haftmann@26182
|
413 |
writeArray = Data.Array.ST.writeArray;*}
|
haftmann@26182
|
414 |
|
haftmann@29730
|
415 |
code_reserved Haskell Heap
|
haftmann@26182
|
416 |
|
haftmann@26182
|
417 |
text {* Monad *}
|
haftmann@26182
|
418 |
|
haftmann@29730
|
419 |
code_type Heap (Haskell "Heap.ST/ Heap.RealWorld/ _")
|
haftmann@27695
|
420 |
code_const Heap (Haskell "error/ \"bare Heap\"")
|
haftmann@28145
|
421 |
code_monad "op \<guillemotright>=" Haskell
|
haftmann@26182
|
422 |
code_const return (Haskell "return")
|
haftmann@26182
|
423 |
code_const "Heap_Monad.Fail" (Haskell "_")
|
haftmann@26182
|
424 |
code_const "Heap_Monad.raise_exc" (Haskell "error")
|
haftmann@26182
|
425 |
|
haftmann@26170
|
426 |
end
|