author haftmann Sat Sep 06 14:02:36 2008 +0200 (2008-09-06) changeset 28145 af3923ed4786 parent 27487 c8a6ce181805 child 29799 7c7f759c438e permissions -rw-r--r--
dropped "run" marker in monad syntax
     1 (*  Title:      HOL/Library/State_Monad.thy

     2     ID:         $Id$

     3     Author:     Florian Haftmann, TU Muenchen

     4 *)

     5

     6 header {* Combinator syntax for generic, open state monads (single threaded monads) *}

     7

     8 theory State_Monad

     9 imports Plain "~~/src/HOL/List"

    10 begin

    11

    12 subsection {* Motivation *}

    13

    14 text {*

    15   The logic HOL has no notion of constructor classes, so

    16   it is not possible to model monads the Haskell way

    17   in full genericity in Isabelle/HOL.

    18

    19   However, this theory provides substantial support for

    20   a very common class of monads: \emph{state monads}

    21   (or \emph{single-threaded monads}, since a state

    22   is transformed single-threaded).

    23

    24   To enter from the Haskell world,

    25   \url{http://www.engr.mun.ca/~theo/Misc/haskell_and_monads.htm}

    26   makes a good motivating start.  Here we just sketch briefly

    27   how those monads enter the game of Isabelle/HOL.

    28 *}

    29

    30 subsection {* State transformations and combinators *}

    31

    32 text {*

    33   We classify functions operating on states into two categories:

    34

    35   \begin{description}

    36     \item[transformations]

    37       with type signature @{text "\<sigma> \<Rightarrow> \<sigma>'"},

    38       transforming a state.

    39     \item[yielding'' transformations]

    40       with type signature @{text "\<sigma> \<Rightarrow> \<alpha> \<times> \<sigma>'"},

    41       yielding'' a side result while transforming a state.

    42     \item[queries]

    43       with type signature @{text "\<sigma> \<Rightarrow> \<alpha>"},

    44       computing a result dependent on a state.

    45   \end{description}

    46

    47   By convention we write @{text "\<sigma>"} for types representing states

    48   and @{text "\<alpha>"}, @{text "\<beta>"}, @{text "\<gamma>"}, @{text "\<dots>"}

    49   for types representing side results.  Type changes due

    50   to transformations are not excluded in our scenario.

    51

    52   We aim to assert that values of any state type @{text "\<sigma>"}

    53   are used in a single-threaded way: after application

    54   of a transformation on a value of type @{text "\<sigma>"}, the

    55   former value should not be used again.  To achieve this,

    56   we use a set of monad combinators:

    57 *}

    58

    59 notation fcomp (infixl "o>" 60)

    60 notation (xsymbols) fcomp (infixl "o>" 60)

    61 notation scomp (infixl "o->" 60)

    62 notation (xsymbols) scomp (infixl "o\<rightarrow>" 60)

    63

    64 abbreviation (input)

    65   "return \<equiv> Pair"

    66

    67 text {*

    68   Given two transformations @{term f} and @{term g}, they

    69   may be directly composed using the @{term "op o>"} combinator,

    70   forming a forward composition: @{prop "(f o> g) s = f (g s)"}.

    71

    72   After any yielding transformation, we bind the side result

    73   immediately using a lambda abstraction.  This

    74   is the purpose of the @{term "op o\<rightarrow>"} combinator:

    75   @{prop "(f o\<rightarrow> (\<lambda>x. g)) s = (let (x, s') = f s in g s')"}.

    76

    77   For queries, the existing @{term "Let"} is appropriate.

    78

    79   Naturally, a computation may yield a side result by pairing

    80   it to the state from the left;  we introduce the

    81   suggestive abbreviation @{term return} for this purpose.

    82

    83   The most crucial distinction to Haskell is that we do

    84   not need to introduce distinguished type constructors

    85   for different kinds of state.  This has two consequences:

    86   \begin{itemize}

    87     \item The monad model does not state anything about

    88        the kind of state; the model for the state is

    89        completely orthogonal and may be

    90        specified completely independently.

    91     \item There is no distinguished type constructor

    92        encapsulating away the state transformation, i.e.~transformations

    93        may be applied directly without using any lifting

    94        or providing and dropping units (open monad'').

    95     \item The type of states may change due to a transformation.

    96   \end{itemize}

    97 *}

    98

    99

   100 subsection {* Monad laws *}

   101

   102 text {*

   103   The common monadic laws hold and may also be used

   104   as normalization rules for monadic expressions:

   105 *}

   106

   107 lemmas monad_simp = Pair_scomp scomp_Pair id_fcomp fcomp_id

   108   scomp_scomp scomp_fcomp fcomp_scomp fcomp_assoc

   109

   110 text {*

   111   Evaluation of monadic expressions by force:

   112 *}

   113

   114 lemmas monad_collapse = monad_simp fcomp_apply scomp_apply split_beta

   115

   116

   117 subsection {* Syntax *}

   118

   119 text {*

   120   We provide a convenient do-notation for monadic expressions

   121   well-known from Haskell.  @{const Let} is printed

   122   specially in do-expressions.

   123 *}

   124

   125 nonterminals do_expr

   126

   127 syntax

   128   "_do" :: "do_expr \<Rightarrow> 'a"

   129     ("do _ done"  12)

   130   "_scomp" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"

   131     ("_ <- _;// _" [1000, 13, 12] 12)

   132   "_fcomp" :: "'a \<Rightarrow> do_expr \<Rightarrow> do_expr"

   133     ("_;// _" [13, 12] 12)

   134   "_let" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"

   135     ("let _ = _;// _" [1000, 13, 12] 12)

   136   "_done" :: "'a \<Rightarrow> do_expr"

   137     ("_"  12)

   138

   139 syntax (xsymbols)

   140   "_scomp" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr"

   141     ("_ \<leftarrow> _;// _" [1000, 13, 12] 12)

   142

   143 translations

   144   "_do f" => "f"

   145   "_scomp x f g" => "f o\<rightarrow> (\<lambda>x. g)"

   146   "_fcomp f g" => "f o> g"

   147   "_let x t f" => "CONST Let t (\<lambda>x. f)"

   148   "_done f" => "f"

   149

   150 print_translation {*

   151 let

   152   fun dest_abs_eta (Abs (abs as (_, ty, _))) =

   153         let

   154           val (v, t) = Syntax.variant_abs abs;

   155         in (Free (v, ty), t) end

   156     | dest_abs_eta t =

   157         let

   158           val (v, t) = Syntax.variant_abs ("", dummyT, t $Bound 0);   159 in (Free (v, dummyT), t) end;   160 fun unfold_monad (Const (@{const_syntax scomp}, _)$ f $g) =   161 let   162 val (v, g') = dest_abs_eta g;   163 in Const ("_scomp", dummyT)$ v $f$ unfold_monad g' end

   164     | unfold_monad (Const (@{const_syntax fcomp}, _) $f$ g) =

   165         Const ("_fcomp", dummyT) $f$ unfold_monad g

   166     | unfold_monad (Const (@{const_syntax Let}, _) $f$ g) =

   167         let

   168           val (v, g') = dest_abs_eta g;

   169         in Const ("_let", dummyT) $v$ f $unfold_monad g' end   170 | unfold_monad (Const (@{const_syntax Pair}, _)$ f) =

   171         Const ("return", dummyT) $f   172 | unfold_monad f = f;   173 fun contains_scomp (Const (@{const_syntax scomp}, _)$ _ $_) = true   174 | contains_scomp (Const (@{const_syntax fcomp}, _)$ _ $t) =   175 contains_scomp t   176 | contains_scomp (Const (@{const_syntax Let}, _)$ _ $Abs (_, _, t)) =   177 contains_scomp t;   178 fun scomp_monad_tr' (f::g::ts) = list_comb   179 (Const ("_do", dummyT)$ unfold_monad (Const (@{const_syntax scomp}, dummyT) $f$ g), ts);

   180   fun fcomp_monad_tr' (f::g::ts) = if contains_scomp g then list_comb

   181       (Const ("_do", dummyT) $unfold_monad (Const (@{const_syntax fcomp}, dummyT)$ f $g), ts)   182 else raise Match;   183 fun Let_monad_tr' (f :: (g as Abs (_, _, g')) :: ts) = if contains_scomp g' then list_comb   184 (Const ("_do", dummyT)$ unfold_monad (Const (@{const_syntax Let}, dummyT) $f$ g), ts)

   185     else raise Match;

   186 in [

   187   (@{const_syntax scomp}, scomp_monad_tr'),

   188   (@{const_syntax fcomp}, fcomp_monad_tr'),

   189   (@{const_syntax Let}, Let_monad_tr')

   190 ] end;

   191 *}

   192

   193 text {*

   194   For an example, see HOL/ex/Random.thy.

   195 *}

   196

   197 end