Theory State_Monad

theory State_Monad
imports Monad_Syntax
(*  Title:      HOL/Library/State_Monad.thy
    Author:     Florian Haftmann, TU Muenchen

section {* Combinator syntax for generic, open state monads (single-threaded monads) *}

theory State_Monad
imports Main Monad_Syntax

subsection {* Motivation *}

text {*
  The logic HOL has no notion of constructor classes, so it is not
  possible to model monads the Haskell way in full genericity in
  However, this theory provides substantial support for a very common
  class of monads: \emph{state monads} (or \emph{single-threaded
  monads}, since a state is transformed single-threadedly).

  To enter from the Haskell world,
  @{url ""} makes
  a good motivating start.  Here we just sketch briefly how those
  monads enter the game of Isabelle/HOL.

subsection {* State transformations and combinators *}

text {*
  We classify functions operating on states into two categories:


    \item[transformations] with type signature @{text "σ => σ'"},
      transforming a state.

    \item[``yielding'' transformations] with type signature @{text "σ
      => α × σ'"}, ``yielding'' a side result while transforming a

    \item[queries] with type signature @{text "σ => α"}, computing a
      result dependent on a state.


  By convention we write @{text "σ"} for types representing states and
  @{text "α"}, @{text "β"}, @{text "γ"}, @{text "…"} for types
  representing side results.  Type changes due to transformations are
  not excluded in our scenario.

  We aim to assert that values of any state type @{text "σ"} are used
  in a single-threaded way: after application of a transformation on a
  value of type @{text "σ"}, the former value should not be used
  again.  To achieve this, we use a set of monad combinators:

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

text {*
  Given two transformations @{term f} and @{term g}, they may be
  directly composed using the @{term "op o>"} combinator, forming a
  forward composition: @{prop "(f o> g) s = f (g s)"}.

  After any yielding transformation, we bind the side result
  immediately using a lambda abstraction.  This is the purpose of the
  @{term "op o->"} combinator: @{prop "(f o-> (λx. g)) s = (let (x, s')
  = f s in g s')"}.

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

  Naturally, a computation may yield a side result by pairing it to
  the state from the left; we introduce the suggestive abbreviation
  @{term return} for this purpose.

  The most crucial distinction to Haskell is that we do not need to
  introduce distinguished type constructors for different kinds of
  state.  This has two consequences:


    \item The monad model does not state anything about the kind of
       state; the model for the state is completely orthogonal and may
       be specified completely independently.

    \item There is no distinguished type constructor encapsulating
       away the state transformation, i.e.~transformations may be
       applied directly without using any lifting or providing and
       dropping units (``open monad'').

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


subsection {* Monad laws *}

text {*
  The common monadic laws hold and may also be used as normalization
  rules for monadic expressions:

lemmas monad_simp = Pair_scomp scomp_Pair id_fcomp fcomp_id
  scomp_scomp scomp_fcomp fcomp_scomp fcomp_assoc

text {*
  Evaluation of monadic expressions by force:

lemmas monad_collapse = monad_simp fcomp_apply scomp_apply split_beta

subsection {* Do-syntax *}

nonterminal sdo_binds and sdo_bind

  "_sdo_block" :: "sdo_binds => 'a" ("exec {//(2  _)//}" [12] 62)
  "_sdo_bind" :: "[pttrn, 'a] => sdo_bind" ("(_ <-/ _)" 13)
  "_sdo_let" :: "[pttrn, 'a] => sdo_bind" ("(2let _ =/ _)" [1000, 13] 13)
  "_sdo_then" :: "'a => sdo_bind" ("_" [14] 13)
  "_sdo_final" :: "'a => sdo_binds" ("_")
  "_sdo_cons" :: "[sdo_bind, sdo_binds] => sdo_binds" ("_;//_" [13, 12] 12)

syntax (xsymbols)
  "_sdo_bind"  :: "[pttrn, 'a] => sdo_bind" ("(_ \<leftarrow>/ _)" 13)

  "_sdo_block (_sdo_cons (_sdo_bind p t) (_sdo_final e))"
    == "CONST scomp t (λp. e)"
  "_sdo_block (_sdo_cons (_sdo_then t) (_sdo_final e))"
    => "CONST fcomp t e"
  "_sdo_final (_sdo_block (_sdo_cons (_sdo_then t) (_sdo_final e)))"
    <= "_sdo_final (CONST fcomp t e)"
  "_sdo_block (_sdo_cons (_sdo_then t) e)"
    <= "CONST fcomp t (_sdo_block e)"
  "_sdo_block (_sdo_cons (_sdo_let p t) bs)"
    == "let p = t in _sdo_block bs"
  "_sdo_block (_sdo_cons b (_sdo_cons c cs))"
    == "_sdo_block (_sdo_cons b (_sdo_final (_sdo_block (_sdo_cons c cs))))"
  "_sdo_cons (_sdo_let p t) (_sdo_final s)"
    == "_sdo_final (let p = t in s)"
  "_sdo_block (_sdo_final e)" => "e"

text {*
  For an example, see @{file "~~/src/HOL/Proofs/Extraction/Higman.thy"}.