author | haftmann |
Sat, 18 Nov 2006 00:20:27 +0100 | |
changeset 21418 | 4bc2882f80af |
parent 21404 | eb85850d3eb7 |
child 21601 | 6588b947d631 |
permissions | -rw-r--r-- |
21192 | 1 |
(* Title: HOL/Library/State_Monad.thy |
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ID: $Id$ |
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Author: Florian Haftmann, TU Muenchen |
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*) |
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header {* Combinators syntax for generic, open state monads (single threaded monads) *} |
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theory State_Monad |
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imports Main |
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begin |
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section {* Generic, open state monads *} |
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subsection {* Motivation *} |
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text {* |
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The logic HOL has no notion of constructor classes, so |
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it is not possible to model monads the Haskell way |
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in full genericity in Isabelle/HOL. |
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However, this theory provides substantial support for |
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a very common class of monads: \emph{state monads} |
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(or \emph{single-threaded monads}, since a state |
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is transformed single-threaded). |
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To enter from the Haskell world, |
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\url{http://www.engr.mun.ca/~theo/Misc/haskell_and_monads.htm} |
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makes a good motivating start. Here we just sketch briefly |
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how those monads enter the game of Isabelle/HOL. |
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*} |
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subsection {* State transformations and combinators *} |
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(*<*) |
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typedecl \<alpha> |
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typedecl \<beta> |
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typedecl \<gamma> |
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typedecl \<sigma> |
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typedecl \<sigma>' |
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(*>*) |
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text {* |
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We classify functions operating on states into two categories: |
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\begin{description} |
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\item[transformations] |
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with type signature @{typ "\<sigma> \<Rightarrow> \<sigma>'"}, |
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transforming a state. |
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\item[``yielding'' transformations] |
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with type signature @{typ "\<sigma> \<Rightarrow> \<alpha> \<times> \<sigma>'"}, |
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``yielding'' a side result while transforming a state. |
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\item[queries] |
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with type signature @{typ "\<sigma> \<Rightarrow> \<alpha>"}, |
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computing a result dependent on a state. |
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\end{description} |
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By convention we write @{typ "\<sigma>"} for types representing states |
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and @{typ "\<alpha>"}, @{typ "\<beta>"}, @{typ "\<gamma>"}, @{text "\<dots>"} |
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for types representing side results. Type changes due |
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to transformations are not excluded in our scenario. |
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We aim to assert that values of any state type @{typ "\<sigma>"} |
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are used in a single-threaded way: after application |
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of a transformation on a value of type @{typ "\<sigma>"}, the |
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former value should not be used again. To achieve this, |
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we use a set of monad combinators: |
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*} |
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definition |
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mbind :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" |
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(infixl ">>=" 60) where |
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"f >>= g = split g \<circ> f" |
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more robust syntax for definition/abbreviation/notation;
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parents:
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definition |
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fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" |
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(infixl ">>" 60) where |
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"f >> g = g \<circ> f" |
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more robust syntax for definition/abbreviation/notation;
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parents:
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more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
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definition |
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more robust syntax for definition/abbreviation/notation;
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parents:
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run :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where |
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"run f = f" |
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print_ast_translation {*[ |
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(Sign.const_syntax_name (the_context ()) "State_Monad.run", fn (f::ts) => Syntax.mk_appl f ts) |
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]*} |
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syntax (xsymbols) |
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mbind :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" |
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(infixl "\<guillemotright>=" 60) |
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fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" |
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(infixl "\<guillemotright>" 60) |
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abbreviation (input) |
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"return \<equiv> Pair" |
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text {* |
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Given two transformations @{term f} and @{term g}, they |
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may be directly composed using the @{term "op \<guillemotright>"} combinator, |
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forming a forward composition: @{prop "(f \<guillemotright> g) s = f (g s)"}. |
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After any yielding transformation, we bind the side result |
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immediately using a lambda abstraction. This |
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is the purpose of the @{term "op \<guillemotright>="} combinator: |
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@{prop "(f \<guillemotright>= (\<lambda>x. g)) s = (let (x, s') = f s in g s')"}. |
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For queries, the existing @{term "Let"} is appropriate. |
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Naturally, a computation may yield a side result by pairing |
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it to the state from the left; we introduce the |
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suggestive abbreviation @{term return} for this purpose. |
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The @{const run} ist just a marker. |
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The most crucial distinction to Haskell is that we do |
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not need to introduce distinguished type constructors |
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for different kinds of state. This has two consequences: |
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\begin{itemize} |
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\item The monad model does not state anything about |
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the kind of state; the model for the state is |
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completely orthogonal and has to (or may) be |
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specified completely independent. |
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\item There is no distinguished type constructor |
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encapsulating away the state transformation, i.e.~transformations |
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may be applied directly without using any lifting |
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or providing and dropping units (``open monad''). |
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\item The type of states may change due to a transformation. |
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\end{itemize} |
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*} |
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subsection {* Obsolete runs *} |
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text {* |
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@{term run} is just a doodle and should not occur nested: |
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*} |
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lemma run_simp [simp]: |
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"\<And>f. run (run f) = run f" |
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"\<And>f g. run f \<guillemotright>= g = f \<guillemotright>= g" |
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"\<And>f g. run f \<guillemotright> g = f \<guillemotright> g" |
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"\<And>f g. f \<guillemotright>= (\<lambda>x. run g) = f \<guillemotright>= (\<lambda>x. g)" |
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"\<And>f g. f \<guillemotright> run g = f \<guillemotright> g" |
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"\<And>f. f = run f \<longleftrightarrow> True" |
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"\<And>f. run f = f \<longleftrightarrow> True" |
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unfolding run_def by rule+ |
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subsection {* Monad laws *} |
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text {* |
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The common monadic laws hold and may also be used |
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as normalization rules for monadic expressions: |
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*} |
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lemma |
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return_mbind [simp]: "return x \<guillemotright>= f = f x" |
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unfolding mbind_def by (simp add: expand_fun_eq) |
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lemma |
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mbind_return [simp]: "x \<guillemotright>= return = x" |
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unfolding mbind_def by (simp add: expand_fun_eq split_Pair) |
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lemma |
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id_fcomp [simp]: "id \<guillemotright> f = f" |
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unfolding fcomp_def by simp |
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lemma |
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fcomp_id [simp]: "f \<guillemotright> id = f" |
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unfolding fcomp_def by simp |
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lemma |
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mbind_mbind [simp]: "(f \<guillemotright>= g) \<guillemotright>= h = f \<guillemotright>= (\<lambda>x. g x \<guillemotright>= h)" |
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unfolding mbind_def by (simp add: split_def expand_fun_eq) |
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lemma |
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mbind_fcomp [simp]: "(f \<guillemotright>= g) \<guillemotright> h = f \<guillemotright>= (\<lambda>x. g x \<guillemotright> h)" |
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unfolding mbind_def fcomp_def by (simp add: split_def expand_fun_eq) |
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lemma |
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fcomp_mbind [simp]: "(f \<guillemotright> g) \<guillemotright>= h = f \<guillemotright> (g \<guillemotright>= h)" |
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unfolding mbind_def fcomp_def by (simp add: split_def expand_fun_eq) |
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lemma |
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fcomp_fcomp [simp]: "(f \<guillemotright> g) \<guillemotright> h = f \<guillemotright> (g \<guillemotright> h)" |
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unfolding fcomp_def o_assoc .. |
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lemmas monad_simp = run_simp return_mbind mbind_return id_fcomp fcomp_id |
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mbind_mbind mbind_fcomp fcomp_mbind fcomp_fcomp |
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text {* |
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Evaluation of monadic expressions by force: |
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*} |
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lemmas monad_collapse = monad_simp o_apply o_assoc split_Pair split_comp |
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mbind_def fcomp_def run_def |
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subsection {* 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] 12) |
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"_mbind" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr" |
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("_ <- _;// _" [1000, 13, 12] 12) |
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"_fcomp" :: "'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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"_mbind" :: "pttrn \<Rightarrow> 'a \<Rightarrow> do_expr \<Rightarrow> do_expr" |
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("_ \<leftarrow> _;// _" [1000, 13, 12] 12) |
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translations |
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"_do f" => "State_Monad.run f" |
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"_mbind x f g" => "f \<guillemotright>= (\<lambda>x. g)" |
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"_fcomp f g" => "f \<guillemotright> g" |
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"_let x t f" => "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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val syntax_name = Sign.const_syntax_name (the_context ()); |
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val name_mbind = syntax_name "State_Monad.mbind"; |
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val name_fcomp = syntax_name "State_Monad.fcomp"; |
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fun unfold_monad (t as Const (name, _) $ f $ g) = |
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if name = name_mbind then let |
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val ([(v, ty)], g') = Term.strip_abs_eta 1 g; |
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in Const ("_mbind", dummyT) $ Free (v, ty) $ f $ unfold_monad g' end |
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else if name = name_fcomp then |
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Const ("_fcomp", dummyT) $ f $ unfold_monad g |
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else t |
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| unfold_monad (Const ("Let", _) $ f $ g) = |
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let |
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val ([(v, ty)], g') = Term.strip_abs_eta 1 g; |
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in Const ("_let", dummyT) $ Free (v, ty) $ f $ unfold_monad g' end |
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| unfold_monad (Const ("Pair", _) $ f) = |
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Const ("return", dummyT) $ f |
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| unfold_monad f = f; |
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fun tr' (f::ts) = |
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list_comb (Const ("_do", dummyT) $ unfold_monad f, ts) |
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in [ |
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(syntax_name "State_Monad.run", tr') |
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] end; |
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*} |
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subsection {* Combinators *} |
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definition |
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lift :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'b \<times> 'c" |
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"lift f x = return (f x)" |
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fun |
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list :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b" where |
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"list f [] = id" |
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"list f (x#xs) = (do f x; list f xs done)" |
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lemmas [code] = list.simps |
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fun list_yield :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'c list \<times> 'b" where |
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"list_yield f [] = return []" |
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"list_yield f (x#xs) = (do y \<leftarrow> f x; ys \<leftarrow> list_yield f xs; return (y#ys) done)" |
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lemmas [code] = list_yield.simps |
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text {* combinator properties *} |
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lemma list_append [simp]: |
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"list f (xs @ ys) = list f xs \<guillemotright> list f ys" |
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by (induct xs) (simp_all del: id_apply) (*FIXME*) |
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lemma list_cong [fundef_cong, recdef_cong]: |
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"\<lbrakk> \<And>x. x \<in> set xs \<Longrightarrow> f x = g x; xs = ys \<rbrakk> |
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\<Longrightarrow> list f xs = list g ys" |
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proof (induct f xs arbitrary: g ys rule: list.induct) |
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case 1 then show ?case by simp |
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next |
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case (2 f x xs g) |
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from 2 have "\<And>y. y \<in> set (x # xs) \<Longrightarrow> f y = g y" by auto |
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then have "\<And>y. y \<in> set xs \<Longrightarrow> f y = g y" by auto |
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with 2 have "list f xs = list g xs" by auto |
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with 2 have "list f (x # xs) = list g (x # xs)" by auto |
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with 2 show "list f (x # xs) = list g ys" by auto |
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qed |
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lemma list_yield_cong [fundef_cong, recdef_cong]: |
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"\<lbrakk> \<And>x. x \<in> set xs \<Longrightarrow> f x = g x; xs = ys \<rbrakk> |
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\<Longrightarrow> list_yield f xs = list_yield g ys" |
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proof (induct f xs arbitrary: g ys rule: list_yield.induct) |
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case 1 then show ?case by simp |
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next |
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case (2 f x xs g) |
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from 2 have "\<And>y. y \<in> set (x # xs) \<Longrightarrow> f y = g y" by auto |
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then have "\<And>y. y \<in> set xs \<Longrightarrow> f y = g y" by auto |
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with 2 have "list_yield f xs = list_yield g xs" by auto |
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with 2 have "list_yield f (x # xs) = list_yield g (x # xs)" by auto |
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with 2 show "list_yield f (x # xs) = list_yield g ys" by auto |
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qed |
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text {* |
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still waiting for extensions@{text "\<dots>"} |
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*} |
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text {* |
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For an example, see HOL/ex/CodeRandom.thy (more examples coming soon). |
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*} |
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end |