diff -r a0b280dd4bc7 -r 520127b708e6 ProgTutorial/FirstSteps.thy --- a/ProgTutorial/FirstSteps.thy Tue Jul 20 13:34:44 2010 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1445 +0,0 @@ -theory FirstSteps -imports Base -begin - -(*<*) -setup{* -open_file_with_prelude - "FirstSteps_Code.thy" - ["theory FirstSteps", "imports Main", "begin"] -*} -(*>*) - -chapter {* First Steps\label{chp:firststeps} *} - -text {* - \begin{flushright} - {\em - ``We will most likely never realize the full importance of painting the Tower,\\ - that it is the essential element in the conservation of metal works and the\\ - more meticulous the paint job, the longer the tower shall endure.''} \\[1ex] - Gustave Eiffel, In his book {\em The 300-Meter Tower}.\footnote{The Eiffel Tower has been - re-painted 18 times since its initial construction, an average of once every - seven years. It takes more than one year for a team of 25 painters to paint the tower - from top to bottom.} - \end{flushright} - - \medskip - Isabelle programming is done in ML. Just like lemmas and proofs, ML-code for - Isabelle must be part of a theory. If you want to follow the code given in - this chapter, we assume you are working inside the theory starting with - - \begin{quote} - \begin{tabular}{@ {}l} - \isacommand{theory} FirstSteps\\ - \isacommand{imports} Main\\ - \isacommand{begin}\\ - \ldots - \end{tabular} - \end{quote} - - We also generally assume you are working with the logic HOL. The examples - that will be given might need to be adapted if you work in a different logic. -*} - -section {* Including ML-Code\label{sec:include} *} - -text {* - The easiest and quickest way to include code in a theory is by using the - \isacommand{ML}-command. For example: - - \begin{isabelle} - \begin{graybox} - \isacommand{ML}~@{text "\"}\isanewline - \hspace{5mm}@{ML "3 + 4"}\isanewline - @{text "\"}\isanewline - @{text "> 7"}\smallskip - \end{graybox} - \end{isabelle} - - Like normal Isabelle scripts, \isacommand{ML}-commands can be evaluated by - using the advance and undo buttons of your Isabelle environment. The code - inside the \isacommand{ML}-command can also contain value and function - bindings, for example -*} - -ML %gray {* - val r = Unsynchronized.ref 0 - fun f n = n + 1 -*} - -text {* - and even those can be undone when the proof script is retracted. As - mentioned in the Introduction, we will drop the \isacommand{ML}~@{text - "\ \ \"} scaffolding whenever we show code. The lines - prefixed with @{text [quotes] ">"} are not part of the code, rather they - indicate what the response is when the code is evaluated. There are also - the commands \isacommand{ML\_val} and \isacommand{ML\_prf} for including - ML-code. The first evaluates the given code, but any effect on the theory, - in which the code is embedded, is suppressed. The second needs to be used if - ML-code is defined inside a proof. For example - - \begin{quote} - \begin{isabelle} - \isacommand{lemma}~@{text "test:"}\isanewline - \isacommand{shows}~@{text [quotes] "True"}\isanewline - \isacommand{ML\_prf}~@{text "\"}~@{ML "writeln \"Trivial!\""}~@{text "\"}\isanewline - \isacommand{oops} - \end{isabelle} - \end{quote} - - However, both commands will only play minor roles in this tutorial (we will - always arrange that the ML-code is defined outside proofs). - - - - - Once a portion of code is relatively stable, you usually want to export it - to a separate ML-file. Such files can then be included somewhere inside a - theory by using the command \isacommand{use}. For example - - \begin{quote} - \begin{tabular}{@ {}l} - \isacommand{theory} FirstSteps\\ - \isacommand{imports} Main\\ - \isacommand{uses}~@{text "(\"file_to_be_included.ML\")"} @{text "\"}\\ - \isacommand{begin}\\ - \ldots\\ - \isacommand{use}~@{text "\"file_to_be_included.ML\""}\\ - \ldots - \end{tabular} - \end{quote} - - The \isacommand{uses}-command in the header of the theory is needed in order - to indicate the dependency of the theory on the ML-file. Alternatively, the - file can be included by just writing in the header - - \begin{quote} - \begin{tabular}{@ {}l} - \isacommand{theory} FirstSteps\\ - \isacommand{imports} Main\\ - \isacommand{uses} @{text "\"file_to_be_included.ML\""} @{text "\"}\\ - \isacommand{begin}\\ - \ldots - \end{tabular} - \end{quote} - - Note that no parentheses are given in this case. Note also that the included - ML-file should not contain any \isacommand{use} itself. Otherwise Isabelle - is unable to record all file dependencies, which is a nuisance if you have - to track down errors. -*} - -section {* Printing and Debugging\label{sec:printing} *} - -text {* - During development you might find it necessary to inspect data in your - code. This can be done in a ``quick-and-dirty'' fashion using the function - @{ML_ind writeln in Output}. For example - - @{ML_response_eq [display,gray] "writeln \"any string\"" "\"any string\"" with "(op =)"} - - will print out @{text [quotes] "any string"} inside the response buffer of - Isabelle. This function expects a string as argument. If you develop under - PolyML, then there is a convenient, though again ``quick-and-dirty'', method - for converting values into strings, namely the antiquotation - @{text "@{make_string}"}: - - @{ML_response_fake [display,gray] "writeln (@{make_string} 1)" "\"1\""} - - However, @{text "@{makes_tring}"} only works if the type of what - is converted is monomorphic and not a function. - - The function @{ML "writeln"} should only be used for testing purposes, - because any output this function generates will be overwritten as soon as an - error is raised. For printing anything more serious and elaborate, the - function @{ML_ind tracing in Output} is more appropriate. This function writes all - output into a separate tracing buffer. For example: - - @{ML_response_eq [display,gray] "tracing \"foo\"" "\"foo\"" with "(op =)"} - - It is also possible to redirect the ``channel'' where the string @{text - "foo"} is printed to a separate file, e.g., in order to prevent ProofGeneral from - choking on massive amounts of trace output. This redirection can be achieved - with the code: -*} - -ML{*val strip_specials = -let - fun strip ("\^A" :: _ :: cs) = strip cs - | strip (c :: cs) = c :: strip cs - | strip [] = []; -in - implode o strip o explode -end - -fun redirect_tracing stream = - Output.tracing_fn := (fn s => - (TextIO.output (stream, (strip_specials s)); - TextIO.output (stream, "\n"); - TextIO.flushOut stream)) *} - -text {* - Calling now - - @{ML [display,gray] "redirect_tracing (TextIO.openOut \"foo.bar\")"} - - will cause that all tracing information is printed into the file @{text "foo.bar"}. - - You can print out error messages with the function @{ML_ind error in Library}; for - example: - - @{ML_response_fake [display,gray] - "if 0=1 then true else (error \"foo\")" -"Exception- ERROR \"foo\" raised -At command \"ML\"."} - - This function raises the exception @{text ERROR}, which will then - be displayed by the infrastructure. Note that this exception is meant - for ``user-level'' error messages seen by the ``end-user''. - - For messages where you want to indicate a genuine program error, then - use the exception @{text Fail}. Here the infrastructure indicates that this - is a low-level exception, and also prints the source position of the ML - raise statement. - - - \footnote{\bf FIXME Mention how to work with @{ML_ind debug in Toplevel} and - @{ML_ind profiling in Toplevel}.} - -*} - -(* FIXME*) -(* -ML {* reset Toplevel.debug *} - -ML {* fun dodgy_fun () = (raise TYPE ("",[],[]); 1) *} - -ML {* fun innocent () = dodgy_fun () *} -ML {* exception_trace (fn () => cterm_of @{theory} (Bound 0)) *} -ML {* exception_trace (fn () => innocent ()) *} - -ML {* Toplevel.program (fn () => cterm_of @{theory} (Bound 0)) *} - -ML {* Toplevel.program (fn () => innocent ()) *} -*) - -text {* - %Kernel exceptions TYPE, TERM, THM also have their place in situations - %where kernel-like operations are involved. The printing is similar as for - %Fail, although there is some special treatment when Toplevel.debug is - %enabled. - - Most often you want to inspect data of Isabelle's basic data structures, - namely @{ML_type term}, @{ML_type typ}, @{ML_type cterm}, @{ML_type ctyp} - and @{ML_type thm}. Isabelle contains elaborate pretty-printing functions, - which we will explain in more detail in Section \ref{sec:pretty}. For now - we just use the functions @{ML_ind writeln in Pretty} from the structure - @{ML_struct Pretty} and @{ML_ind pretty_term in Syntax} from the structure - @{ML_struct Syntax}. For more convenience, we bind them to the toplevel. -*} - -ML{*val string_of_term = Syntax.string_of_term*} -ML{*val pretty_term = Syntax.pretty_term*} -ML{*val pwriteln = Pretty.writeln*} - -text {* - They can now be used as follows - - @{ML_response_fake [display,gray] - "pwriteln (pretty_term @{context} @{term \"1::nat\"})" - "\"1\""} - - If there is more than one term to be printed, you can use the - function @{ML_ind enum in Pretty} to separate them. -*} - -ML{*fun string_of_terms ctxt ts = - commas (map (string_of_term ctxt) ts)*} -ML{*fun pretty_terms ctxt ts = - Pretty.enum "," "" "" (map (pretty_term ctxt) ts)*} - -text {* - You can also print out terms together with their typing information. - For this you need to set the reference @{ML_ind show_types in Syntax} - to @{ML true}. -*} - -ML{*show_types := true*} - -text {* - Now @{ML pretty_term} prints out - - @{ML_response_fake [display, gray] - "pwriteln (pretty_term @{context} @{term \"(1::nat, x)\"})" - "(1::nat, x::'a)"} - - where @{text 1} and @{text x} are displayed with their inferred type. - Even more type information can be printed by setting - the reference @{ML_ind show_all_types in Syntax} to @{ML true}. - In this case we obtain -*} -(*<*)ML %linenos {*show_all_types := true*} -(*>*) -text {* - @{ML_response_fake [display, gray] - "pwriteln (pretty_term @{context} @{term \"(1::nat, x)\"})" - "(Pair::nat \ 'a \ nat \ 'a) (1::nat) (x::'a)"} - - where @{term Pair} is the term-constructor for products. - Other references that influence printing of terms are - @{ML_ind show_brackets in Syntax} and @{ML_ind show_sorts in Syntax}. -*} -(*<*)ML %linenos {*show_types := false; show_all_types := false*} -(*>*) -text {* - A @{ML_type cterm} can be printed with the following function. -*} - -ML{*fun string_of_cterm ctxt ct = - string_of_term ctxt (term_of ct)*} -ML{*fun pretty_cterm ctxt ct = - pretty_term ctxt (term_of ct)*} - -text {* - Here the function @{ML_ind term_of in Thm} extracts the @{ML_type - term} from a @{ML_type cterm}. More than one @{ML_type cterm}s can be - printed again with @{ML enum in Pretty}. -*} - -ML{*fun string_of_cterms ctxt cts = - commas (map (string_of_cterm ctxt) cts)*} -ML{*fun pretty_cterms ctxt cts = - Pretty.enum "," "" "" (map (pretty_cterm ctxt) cts)*} - -text {* - The easiest way to get the string of a theorem is to transform it - into a @{ML_type term} using the function @{ML_ind prop_of in Thm}. -*} - -ML{*fun pretty_thm ctxt thm = - pretty_term ctxt (prop_of thm)*} - -text {* - Theorems include schematic variables, such as @{text "?P"}, - @{text "?Q"} and so on. They are needed in Isabelle in order to able to - instantiate theorems when they are applied. For example the theorem - @{thm [source] conjI} shown below can be used for any (typable) - instantiation of @{text "?P"} and @{text "?Q"}. - - @{ML_response_fake [display, gray] - "pwriteln (pretty_thm @{context} @{thm conjI})" - "\?P; ?Q\ \ ?P \ ?Q"} - - However, in order to improve the readability when printing theorems, we - convert these schematic variables into free variables using the function - @{ML_ind import in Variable}. This is similar to statements like @{text - "conjI[no_vars]"} on Isabelle's user-level. -*} - -ML{*fun no_vars ctxt thm = -let - val ((_, [thm']), _) = Variable.import true [thm] ctxt -in - thm' -end - -fun pretty_thm_no_vars ctxt thm = - pretty_term ctxt (prop_of (no_vars ctxt thm))*} - - -text {* - With this function, theorem @{thm [source] conjI} is now printed as follows: - - @{ML_response_fake [display, gray] - "pwriteln (pretty_thm_no_vars @{context} @{thm conjI})" - "\P; Q\ \ P \ Q"} - - Again the function @{ML commas} helps with printing more than one theorem. -*} - -ML{*fun pretty_thms ctxt thms = - Pretty.enum "," "" "" (map (pretty_thm ctxt) thms) - -fun pretty_thms_no_vars ctxt thms = - Pretty.enum "," "" "" (map (pretty_thm_no_vars ctxt) thms)*} - -text {* - The printing functions for types are -*} - -ML{*fun pretty_typ ctxt ty = Syntax.pretty_typ ctxt ty -fun pretty_typs ctxt tys = Pretty.commas (map (pretty_typ ctxt) tys)*} - -text {* - respectively ctypes -*} - -ML{*fun pretty_ctyp ctxt cty = pretty_typ ctxt (typ_of cty) -fun pretty_ctyps ctxt ctys = Pretty.commas (map (pretty_ctyp ctxt) ctys)*} - -text {* - \begin{readmore} - The simple conversion functions from Isabelle's main datatypes to - @{ML_type string}s are implemented in @{ML_file "Pure/Syntax/syntax.ML"}. - The references that change the printing information are declared in - @{ML_file "Pure/Syntax/printer.ML"}. - \end{readmore} - - Note that for printing out several ``parcels'' of information that belong - together, like a warning message consisting of a term and its type, you - should try to print these parcels together in a single string. Therefore do - \emph{not} print out information as - -@{ML_response_fake [display,gray] -"writeln \"First half,\"; -writeln \"and second half.\"" -"First half, -and second half."} - - but as a single string with appropriate formatting. For example - -@{ML_response_fake [display,gray] -"writeln (\"First half,\" ^ \"\\n\" ^ \"and second half.\")" -"First half, -and second half."} - - To ease this kind of string manipulations, there are a number - of library functions in Isabelle. For example, the function - @{ML_ind cat_lines in Library} concatenates a list of strings - and inserts newlines in between each element. - - @{ML_response_fake [display, gray] - "writeln (cat_lines [\"foo\", \"bar\"])" - "foo -bar"} - - Section \ref{sec:pretty} will explain the infrastructure that Isabelle - provides for more elaborate pretty printing. - - \begin{readmore} - Most of the basic string functions of Isabelle are defined in - @{ML_file "Pure/library.ML"}. - \end{readmore} -*} - - -section {* Combinators\label{sec:combinators} *} - -text {* - For beginners perhaps the most puzzling parts in the existing code of - Isabelle are the combinators. At first they seem to greatly obstruct the - comprehension of code, but after getting familiar with them and handled with - care, they actually ease the understanding and also the programming. - - The simplest combinator is @{ML_ind I in Library}, which is just the - identity function defined as -*} - -ML{*fun I x = x*} - -text {* - Another simple combinator is @{ML_ind K in Library}, defined as -*} - -ML{*fun K x = fn _ => x*} - -text {* - @{ML K} ``wraps'' a function around @{text "x"} that ignores its argument. As a - result, @{ML K} defines a constant function always returning @{text x}. - - The next combinator is reverse application, @{ML_ind "|>" in Basics}, defined as: -*} - -ML{*fun x |> f = f x*} - -text {* While just syntactic sugar for the usual function application, - the purpose of this combinator is to implement functions in a - ``waterfall fashion''. Consider for example the function *} - -ML %linenosgray{*fun inc_by_five x = - x |> (fn x => x + 1) - |> (fn x => (x, x)) - |> fst - |> (fn x => x + 4)*} - -text {* - which increments its argument @{text x} by 5. It does this by first - incrementing the argument by 1 (Line 2); then storing the result in a pair - (Line 3); taking the first component of the pair (Line 4) and finally - incrementing the first component by 4 (Line 5). This kind of cascading - manipulations of values is quite common when dealing with theories. The - reverse application allows you to read what happens in a top-down - manner. This kind of coding should be familiar, if you have been exposed to - Haskell's {\it do}-notation. Writing the function @{ML inc_by_five} using - the reverse application is much clearer than writing -*} - -ML{*fun inc_by_five x = fst ((fn x => (x, x)) (x + 1)) + 4*} - -text {* or *} - -ML{*fun inc_by_five x = - ((fn x => x + 4) o fst o (fn x => (x, x)) o (fn x => x + 1)) x*} - -text {* and typographically more economical than *} - -ML{*fun inc_by_five x = -let val y1 = x + 1 - val y2 = (y1, y1) - val y3 = fst y2 - val y4 = y3 + 4 -in y4 end*} - -text {* - Another reason why the let-bindings in the code above are better to be - avoided: it is more than easy to get the intermediate values wrong, not to - mention the nightmares the maintenance of this code causes! - - In Isabelle a ``real world'' example for a function written in - the waterfall fashion might be the following code: -*} - -ML %linenosgray{*fun apply_fresh_args f ctxt = - f |> fastype_of - |> binder_types - |> map (pair "z") - |> Variable.variant_frees ctxt [f] - |> map Free - |> curry list_comb f *} - -text {* - This function takes a term and a context as argument. If the term is of function - type, then @{ML "apply_fresh_args"} returns the term with distinct variables - applied to it. For example below three variables are applied to the term - @{term [show_types] "P::nat \ int \ unit \ bool"}: - - @{ML_response_fake [display,gray] -"let - val trm = @{term \"P::nat \ int \ unit \ bool\"} - val ctxt = @{context} -in - apply_fresh_args trm ctxt - |> string_of_term ctxt - |> tracing -end" - "P z za zb"} - - You can read off this behaviour from how @{ML apply_fresh_args} is coded: in - Line 2, the function @{ML_ind fastype_of in Term} calculates the type of the - term; @{ML_ind binder_types in Term} in the next line produces the list of - argument types (in the case above the list @{text "[nat, int, unit]"}); Line - 4 pairs up each type with the string @{text "z"}; the function @{ML_ind - variant_frees in Variable} generates for each @{text "z"} a unique name - avoiding the given @{text f}; the list of name-type pairs is turned into a - list of variable terms in Line 6, which in the last line is applied by the - function @{ML_ind list_comb in Term} to the original term. In this last step we have - to use the function @{ML_ind curry in Library}, because @{ML list_comb} - expects the function and the variables list as a pair. - - Functions like @{ML apply_fresh_args} are often needed when constructing - terms involving fresh variables. For this the infrastructure helps - tremendously to avoid any name clashes. Consider for example: - - @{ML_response_fake [display,gray] -"let - val trm = @{term \"za::'a \ 'b \ 'c\"} - val ctxt = @{context} -in - apply_fresh_args trm ctxt - |> string_of_term ctxt - |> tracing -end" - "za z zb"} - - where the @{text "za"} is correctly avoided. - - The combinator @{ML_ind "#>" in Basics} is the reverse function composition. - It can be used to define the following function -*} - -ML{*val inc_by_six = - (fn x => x + 1) #> - (fn x => x + 2) #> - (fn x => x + 3)*} - -text {* - which is the function composed of first the increment-by-one function and - then increment-by-two, followed by increment-by-three. Again, the reverse - function composition allows you to read the code top-down. This combinator - is often used for setup functions inside the - \isacommand{setup}-command. These functions have to be of type @{ML_type - "theory -> theory"}. More than one such setup function can be composed with - @{ML "#>"}. For example -*} - -setup %gray {* let - val (ival1, setup_ival1) = Attrib.config_int "ival1" (K 1) - val (ival2, setup_ival2) = Attrib.config_int "ival2" (K 2) -in - setup_ival1 #> - setup_ival2 -end *} - -text {* - after this the configuration values @{text ival1} and @{text ival2} are known - in the current theory and can be manipulated by the user (for more information - about configuration values see Section~\ref{sec:storing}, for more about setup - functions see Section~\ref{sec:theories}). - - The remaining combinators we describe in this section add convenience for the - ``waterfall method'' of writing functions. The combinator @{ML_ind tap in - Basics} allows you to get hold of an intermediate result (to do some - side-calculations for instance). The function - *} - -ML %linenosgray{*fun inc_by_three x = - x |> (fn x => x + 1) - |> tap (fn x => tracing (PolyML.makestring x)) - |> (fn x => x + 2)*} - -text {* - increments the argument first by @{text "1"} and then by @{text "2"}. In the - middle (Line 3), however, it uses @{ML tap} for printing the ``plus-one'' - intermediate result. The function @{ML tap} can only be used for - side-calculations, because any value that is computed cannot be merged back - into the ``main waterfall''. To do this, you can use the next combinator. - - The combinator @{ML_ind "`" in Basics} (a backtick) is similar to @{ML tap}, - but applies a function to the value and returns the result together with the - value (as a pair). It is defined as -*} - -ML{*fun `f = fn x => (f x, x)*} - -text {* - An example for this combinator is the function -*} - -ML{*fun inc_as_pair x = - x |> `(fn x => x + 1) - |> (fn (x, y) => (x, y + 1))*} - -text {* - which takes @{text x} as argument, and then increments @{text x}, but also keeps - @{text x}. The intermediate result is therefore the pair @{ML "(x + 1, x)" - for x}. After that, the function increments the right-hand component of the - pair. So finally the result will be @{ML "(x + 1, x + 1)" for x}. - - The combinators @{ML_ind "|>>" in Basics} and @{ML_ind "||>" in Basics} are - defined for functions manipulating pairs. The first applies the function to - the first component of the pair, defined as -*} - -ML{*fun (x, y) |>> f = (f x, y)*} - -text {* - and the second combinator to the second component, defined as -*} - -ML{*fun (x, y) ||> f = (x, f y)*} - -text {* - These two functions can, for example, be used to avoid explicit @{text "lets"} for - intermediate values in functions that return pairs. As an example, suppose you - want to separate a list of integers into two lists according to a - threshold. If the threshold is @{ML "5"}, the list @{ML "[1,6,2,5,3,4]"} - should be separated as @{ML "([1,2,3,4], [6,5])"}. Such a function can be - implemented as -*} - -ML{*fun separate i [] = ([], []) - | separate i (x::xs) = - let - val (los, grs) = separate i xs - in - if i <= x then (los, x::grs) else (x::los, grs) - end*} - -text {* - where the return value of the recursive call is bound explicitly to - the pair @{ML "(los, grs)" for los grs}. However, this function - can be implemented more concisely as -*} - -ML{*fun separate i [] = ([], []) - | separate i (x::xs) = - if i <= x - then separate i xs ||> cons x - else separate i xs |>> cons x*} - -text {* - avoiding the explicit @{text "let"}. While in this example the gain in - conciseness is only small, in more complicated situations the benefit of - avoiding @{text "lets"} can be substantial. - - With the combinator @{ML_ind "|->" in Basics} you can re-combine the - elements from a pair. This combinator is defined as -*} - -ML{*fun (x, y) |-> f = f x y*} - -text {* - and can be used to write the following roundabout version - of the @{text double} function: -*} - -ML{*fun double x = - x |> (fn x => (x, x)) - |-> (fn x => fn y => x + y)*} - -text {* - The combinator @{ML_ind ||>> in Basics} plays a central rôle whenever your - task is to update a theory and the update also produces a side-result (for - example a theorem). Functions for such tasks return a pair whose second - component is the theory and the fist component is the side-result. Using - @{ML ||>>}, you can do conveniently the update and also - accumulate the side-results. Consider the following simple function. -*} - -ML %linenosgray{*fun acc_incs x = - x |> (fn x => ("", x)) - ||>> (fn x => (x, x + 1)) - ||>> (fn x => (x, x + 1)) - ||>> (fn x => (x, x + 1))*} - -text {* - The purpose of Line 2 is to just pair up the argument with a dummy value (since - @{ML ||>>} operates on pairs). Each of the next three lines just increment - the value by one, but also nest the intermediate results to the left. For example - - @{ML_response [display,gray] - "acc_incs 1" - "((((\"\", 1), 2), 3), 4)"} - - You can continue this chain with: - - @{ML_response [display,gray] - "acc_incs 1 ||>> (fn x => (x, x + 2))" - "(((((\"\", 1), 2), 3), 4), 6)"} - - \footnote{\bf FIXME: maybe give a ``real world'' example for this combinator.} -*} - -text {* - Recall that @{ML "|>"} is the reverse function application. Recall also that - the related reverse function composition is @{ML "#>"}. In fact all the - combinators @{ML "|->"}, @{ML "|>>"} , @{ML "||>"} and @{ML "||>>"} - described above have related combinators for function composition, namely - @{ML_ind "#->" in Basics}, @{ML_ind "#>>" in Basics}, @{ML_ind "##>" in - Basics} and @{ML_ind "##>>" in Basics}. Using @{ML "#->"}, for example, the - function @{text double} can also be written as: -*} - -ML{*val double = - (fn x => (x, x)) - #-> (fn x => fn y => x + y)*} - - -text {* - When using combinators for writing functions in waterfall fashion, it is - sometimes necessary to do some ``plumbing'' in order to fit functions - together. We have already seen such plumbing in the function @{ML - apply_fresh_args}, where @{ML curry} is needed for making the function @{ML - list_comb}, which works over pairs, to fit with the combinator @{ML "|>"}. Such - plumbing is also needed in situations where a function operates over lists, - but one calculates only with a single element. An example is the function - @{ML_ind check_terms in Syntax}, whose purpose is to simultaneously type-check - a list of terms. Consider the code: - - @{ML_response_fake [display, gray] - "let - val ctxt = @{context} -in - map (Syntax.parse_term ctxt) [\"m + n\", \"m * n\", \"m - (n::nat)\"] - |> Syntax.check_terms ctxt - |> string_of_terms ctxt - |> tracing -end" - "m + n, m * n, m - n"} -*} - -text {* - In this example we obtain three terms (using the function - @{ML_ind parse_term in Syntax}) whose variables @{text m} and @{text n} - are of type @{typ "nat"}. If you have only a single term, then @{ML - check_terms in Syntax} needs plumbing. This can be done with the function - @{ML_ind singleton in Library}.\footnote{There is already a function @{ML check_term in - Syntax} in the file @{ML_file "Pure/Syntax/syntax.ML"} that is implemented - in terms of @{ML singleton} and @{ML check_terms in Syntax}.} For example - - @{ML_response_fake [display, gray, linenos] - "let - val ctxt = @{context} -in - Syntax.parse_term ctxt \"m - (n::nat)\" - |> singleton (Syntax.check_terms ctxt) - |> string_of_term ctxt - |> tracing -end" - "m - n"} - - where in Line 5, the function operating over lists fits with the - single term generated in Line 4. - - \begin{readmore} - The most frequently used combinators are defined in the files @{ML_file - "Pure/library.ML"} - and @{ML_file "Pure/General/basics.ML"}. Also \isccite{sec:ML-linear-trans} - contains further information about combinators. - \end{readmore} - - \footnote{\bf FIXME: find a good exercise for combinators} - \begin{exercise} - Find out what the combinator @{ML "K I"} does. - \end{exercise} - - - \footnote{\bf FIXME: say something about calling conventions} -*} - - -section {* ML-Antiquotations\label{sec:antiquote} *} - -text {* - Recall from Section \ref{sec:include} that code in Isabelle is always - embedded in a theory. The main advantage of this is that the code can - contain references to entities defined on the logical level of Isabelle. By - this we mean references to definitions, theorems, terms and so on. These - reference are realised in Isabelle with ML-antiquotations, often just called - antiquotations.\footnote{Note that there are two kinds of antiquotations in - Isabelle, which have very different purposes and infrastructures. The first - kind, described in this section, are \emph{\index*{ML-antiquotation}}. They - are used to refer to entities (like terms, types etc) from Isabelle's logic - layer inside ML-code. The other kind of antiquotations are - \emph{document}\index{document antiquotation} antiquotations. They are used - only in the text parts of Isabelle and their purpose is to print logical - entities inside \LaTeX-documents. Document antiquotations are part of the - user level and therefore we are not interested in them in this Tutorial, - except in Appendix \ref{rec:docantiquotations} where we show how to - implement your own document antiquotations.} Syntactically antiquotations - are indicated by the @{ML_text @}-sign followed by text wrapped in @{text - "{\}"}. For example, one can print out the name of the current theory with - the code - - @{ML_response [display,gray] "Context.theory_name @{theory}" "\"FirstSteps\""} - - where @{text "@{theory}"} is an antiquotation that is substituted with the - current theory (remember that we assumed we are inside the theory - @{text FirstSteps}). The name of this theory can be extracted using - the function @{ML_ind theory_name in Context}. - - Note, however, that antiquotations are statically linked, that is their value is - determined at ``compile-time'', not at ``run-time''. For example the function -*} - -ML{*fun not_current_thyname () = Context.theory_name @{theory} *} - -text {* - does \emph{not} return the name of the current theory, if it is run in a - different theory. Instead, the code above defines the constant function - that always returns the string @{text [quotes] "FirstSteps"}, no matter where the - function is called. Operationally speaking, the antiquotation @{text "@{theory}"} is - \emph{not} replaced with code that will look up the current theory in - some data structure and return it. Instead, it is literally - replaced with the value representing the theory. - - Another important antiquotation is @{text "@{context}"}. (What the - difference between a theory and a context is will be described in Chapter - \ref{chp:advanced}.) A context is for example needed in order to use the - function @{ML print_abbrevs in ProofContext} that list of all currently - defined abbreviations. - - @{ML_response_fake [display, gray] - "ProofContext.print_abbrevs @{context}" -"Code_Evaluation.valtermify \ \x. (x, \u. Code_Evaluation.termify x) -INTER \ INFI -Inter \ Inf -\"} - - You can also use antiquotations to refer to proved theorems: - @{text "@{thm \}"} for a single theorem - - @{ML_response_fake [display,gray] "@{thm allI}" "(\x. ?P x) \ \x. ?P x"} - - and @{text "@{thms \}"} for more than one - -@{ML_response_fake [display,gray] - "@{thms conj_ac}" -"(?P \ ?Q) = (?Q \ ?P) -(?P \ ?Q \ ?R) = (?Q \ ?P \ ?R) -((?P \ ?Q) \ ?R) = (?P \ ?Q \ ?R)"} - - The thm-antiquotations can also be used for manipulating theorems. For - example, if you need the version of te theorem @{thm [source] refl} that - has a meta-equality instead of an equality, you can write - -@{ML_response_fake [display,gray] - "@{thm refl[THEN eq_reflection]}" - "?x \ ?x"} - - The point of these antiquotations is that referring to theorems in this way - makes your code independent from what theorems the user might have stored - under this name (this becomes especially important when you deal with - theorem lists; see Section \ref{sec:storing}). - - It is also possible to prove lemmas with the antiquotation @{text "@{lemma \ by \}"} - whose first argument is a statement (possibly many of them separated by @{text "and"}) - and the second is a proof. For example -*} - -ML{*val foo_thm = @{lemma "True" and "False \ P" by simp_all} *} - -ML {* -pretty_thms_no_vars -*} - -text {* - The result can be printed out as follows. - - @{ML_response_fake [gray,display] -"foo_thm |> pretty_thms_no_vars @{context} - |> pwriteln" - "True, False \ P"} - - You can also refer to the current simpset via an antiquotation. To illustrate - this we implement the function that extracts the theorem names stored in a - simpset. -*} - -ML{*fun get_thm_names_from_ss simpset = -let - val {simps,...} = MetaSimplifier.dest_ss simpset -in - map #1 simps -end*} - -text {* - The function @{ML_ind dest_ss in MetaSimplifier} returns a record containing all - information stored in the simpset, but here we are only interested in the names of the - simp-rules. Now you can feed in the current simpset into this function. - The current simpset can be referred to using the antiquotation @{text "@{simpset}"}. - - @{ML_response_fake [display,gray] - "get_thm_names_from_ss @{simpset}" - "[\"Nat.of_nat_eq_id\", \"Int.of_int_eq_id\", \"Nat.One_nat_def\", \]"} - - Again, this way of referencing simpsets makes you independent from additions - of lemmas to the simpset by the user, which can potentially cause loops in your - code. - - It is also possible to define your own antiquotations. But you should - exercise care when introducing new ones, as they can also make your code - also difficult to read. In the next chapter we describe how to construct - terms with the (build in) antiquotation @{text "@{term \}"}. A restriction - of this antiquotation is that it does not allow you to use schematic - variables in terms. If you want to have an antiquotation that does not have - this restriction, you can implement your own using the function @{ML_ind - inline in ML_Antiquote} from the structure @{ML_struct ML_Antiquote}. The code - for the antiquotation @{text "term_pat"} is as follows. -*} - -ML %linenosgray{*let - val parser = Args.context -- Scan.lift Args.name_source - - fun term_pat (ctxt, str) = - str |> ProofContext.read_term_pattern ctxt - |> ML_Syntax.print_term - |> ML_Syntax.atomic -in - ML_Antiquote.inline "term_pat" (parser >> term_pat) -end*} - -text {* - The parser in Line 2 provides us with a context and a string; this string is - transformed into a term using the function @{ML_ind read_term_pattern in - ProofContext} (Line 5); the next two lines transform the term into a string - so that the ML-system can understand it. (All these functions will be explained - in more detail in later sections.) An example for this antiquotation is: - - @{ML_response_fake [display,gray] - "@{term_pat \"Suc (?x::nat)\"}" - "Const (\"Suc\", \"nat \ nat\") $ Var ((\"x\", 0), \"nat\")"} - - which shows the internal representation of the term @{text "Suc ?x"}. Similarly - we can write an antiquotation for type patterns. -*} - -ML{*let - val parser = Args.context -- Scan.lift Args.name_source - - fun typ_pat (ctxt, str) = - str |> Syntax.parse_typ ctxt - |> ML_Syntax.print_typ - |> ML_Syntax.atomic -in - ML_Antiquote.inline "typ_pat" (parser >> typ_pat) -end*} - -text {* - \begin{readmore} - The file @{ML_file "Pure/ML/ml_antiquote.ML"} contains the the definitions - for most antiquotations. Most of the basic operations on ML-syntax are implemented - in @{ML_file "Pure/ML/ml_syntax.ML"}. - \end{readmore} -*} - -section {* Storing Data in Isabelle\label{sec:storing} *} - -text {* - Isabelle provides mechanisms for storing (and retrieving) arbitrary - data. Before we delve into the details, let us digress a bit. Conventional - wisdom has it that the type-system of ML ensures that an - @{ML_type "'a list"}, say, can only hold elements of the same type, namely - @{ML_type "'a"}. Despite this wisdom, however, it is possible to implement a - universal type in ML, although by some arguably accidental features of ML. - This universal type can be used to store data of different type into a single list. - In fact, it allows one to inject and to project data of \emph{arbitrary} type. This is - in contrast to datatypes, which only allow injection and projection of data for - some \emph{fixed} collection of types. In light of the conventional wisdom cited - above it is important to keep in mind that the universal type does not - destroy type-safety of ML: storing and accessing the data can only be done - in a type-safe manner. - - \begin{readmore} - In Isabelle the universal type is implemented as the type @{ML_type - Universal.universal} in the file - @{ML_file "Pure/ML-Systems/universal.ML"}. - \end{readmore} - - We will show the usage of the universal type by storing an integer and - a boolean into a single list. Let us first define injection and projection - functions for booleans and integers into and from the type @{ML_type Universal.universal}. -*} - -ML{*local - val fn_int = Universal.tag () : int Universal.tag - val fn_bool = Universal.tag () : bool Universal.tag -in - val inject_int = Universal.tagInject fn_int; - val inject_bool = Universal.tagInject fn_bool; - val project_int = Universal.tagProject fn_int; - val project_bool = Universal.tagProject fn_bool -end*} - -text {* - Using the injection functions, we can inject the integer @{ML_text "13"} - and the boolean value @{ML_text "true"} into @{ML_type Universal.universal}, and - then store them in a @{ML_type "Universal.universal list"} as follows: -*} - -ML{*val foo_list = -let - val thirteen = inject_int 13 - val truth_val = inject_bool true -in - [thirteen, truth_val] -end*} - -text {* - The data can be retrieved with the projection functions defined above. - - @{ML_response_fake [display, gray] -"project_int (nth foo_list 0); -project_bool (nth foo_list 1)" -"13 -true"} - - Notice that we access the integer as an integer and the boolean as - a boolean. If we attempt to access the integer as a boolean, then we get - a runtime error. - - @{ML_response_fake [display, gray] -"project_bool (nth foo_list 0)" -"*** Exception- Match raised"} - - This runtime error is the reason why ML is still type-sound despite - containing a universal type. - - Now, Isabelle heavily uses this mechanism for storing all sorts of - data: theorem lists, simpsets, facts etc. Roughly speaking, there are two - places where data can be stored in Isabelle: in \emph{theories} and in \emph{proof - contexts}. Data such as simpsets are ``global'' and therefore need to be stored - in a theory (simpsets need to be maintained across proofs and even across - theories). On the other hand, data such as facts change inside a proof and - are only relevant to the proof at hand. Therefore such data needs to be - maintained inside a proof context, which represents ``local'' data. - - For theories and proof contexts there are, respectively, the functors - @{ML_funct_ind Theory_Data} and @{ML_funct_ind Proof_Data} that help - with the data storage. Below we show how to implement a table in which you - can store theorems and look them up according to a string key. The - intention in this example is to be able to look up introduction rules for logical - connectives. Such a table might be useful in an automatic proof procedure - and therefore it makes sense to store this data inside a theory. - Consequently we use the functor @{ML_funct Theory_Data}. - The code for the table is: -*} - -ML %linenosgray{*structure Data = Theory_Data - (type T = thm Symtab.table - val empty = Symtab.empty - val extend = I - val merge = Symtab.merge (K true))*} - -text {* - In order to store data in a theory, we have to specify the type of the data - (Line 2). In this case we specify the type @{ML_type "thm Symtab.table"}, - which stands for a table in which @{ML_type string}s can be looked up - producing an associated @{ML_type thm}. We also have to specify four - functions to use this functor: namely how to initialise the data storage - (Line 3), how to extend it (Line 4) and how two - tables should be merged (Line 5). These functions correspond roughly to the - operations performed on theories and we just give some sensible - defaults.\footnote{\bf FIXME: Say more about the - assumptions of these operations.} The result structure @{ML_text Data} - contains functions for accessing the table (@{ML Data.get}) and for updating - it (@{ML Data.map}). There is also the functions @{ML Data.put}, which however is - not relevant here. Below we define two - auxiliary functions, which help us with accessing the table. -*} - -ML{*val lookup = Symtab.lookup o Data.get -fun update k v = Data.map (Symtab.update (k, v))*} - -text {* - Since we want to store introduction rules associated with their - logical connective, we can fill the table as follows. -*} - -setup %gray {* - update "op &" @{thm conjI} #> - update "op -->" @{thm impI} #> - update "All" @{thm allI} -*} - -text {* - The use of the command \isacommand{setup} makes sure the table in the - \emph{current} theory is updated (this is explained further in - section~\ref{sec:theories}). The lookup can now be performed as follows. - - @{ML_response_fake [display, gray] -"lookup @{theory} \"op &\"" -"SOME \"\?P; ?Q\ \ ?P \ ?Q\""} - - An important point to note is that these tables (and data in general) - need to be treated in a purely functional fashion. Although - we can update the table as follows -*} - -setup %gray {* update "op &" @{thm TrueI} *} - -text {* - and accordingly, @{ML lookup} now produces the introduction rule for @{term "True"} - -@{ML_response_fake [display, gray] -"lookup @{theory} \"op &\"" -"SOME \"True\""} - - there are no references involved. This is one of the most fundamental - coding conventions for programming in Isabelle. References - interfere with the multithreaded execution model of Isabelle and also - defeat its undo-mechanism. To see the latter, consider the - following data container where we maintain a reference to a list of - integers. -*} - -ML{*structure WrongRefData = Theory_Data - (type T = (int list) Unsynchronized.ref - val empty = Unsynchronized.ref [] - val extend = I - val merge = fst)*} - -text {* - We initialise the reference with the empty list. Consequently a first - lookup produces @{ML "ref []" in Unsynchronized}. - - @{ML_response_fake [display,gray] - "WrongRefData.get @{theory}" - "ref []"} - - For updating the reference we use the following function -*} - -ML{*fun ref_update n = WrongRefData.map - (fn r => let val _ = r := n::(!r) in r end)*} - -text {* - which takes an integer and adds it to the content of the reference. - As before, we update the reference with the command - \isacommand{setup}. -*} - -setup %gray {* ref_update 1 *} - -text {* - A lookup in the current theory gives then the expected list - @{ML "ref [1]" in Unsynchronized}. - - @{ML_response_fake [display,gray] - "WrongRefData.get @{theory}" - "ref [1]"} - - So far everything is as expected. But, the trouble starts if we attempt to - backtrack to the ``point'' before the \isacommand{setup}-command. There, we - would expect that the list is empty again. But since it is stored in a - reference, Isabelle has no control over it. So it is not empty, but still - @{ML "ref [1]" in Unsynchronized}. Adding to the trouble, if we execute the - \isacommand{setup}-command again, we do not obtain @{ML "ref [1]" in - Unsynchronized}, but - - @{ML_response_fake [display,gray] - "WrongRefData.get @{theory}" - "ref [1, 1]"} - - Now imagine how often you go backwards and forwards in your proof scripts. - By using references in Isabelle code, you are bound to cause all - hell to break loose. Therefore observe the coding convention: - Do not use references for storing data! - - \begin{readmore} - The functors for data storage are defined in @{ML_file "Pure/context.ML"}. - Isabelle contains implementations of several container data structures, - including association lists in @{ML_file "Pure/General/alist.ML"}, - directed graphs in @{ML_file "Pure/General/graph.ML"}, and - tables and symtables in @{ML_file "Pure/General/table.ML"}. - \end{readmore} - - Storing data in a proof context is done in a similar fashion. As mentioned - before, the corresponding functor is @{ML_funct_ind Proof_Data}. With the - following code we can store a list of terms in a proof context. -*} - -ML{*structure Data = Proof_Data - (type T = term list - fun init _ = [])*} - -text {* - The init-function we have to specify must produce a list for when a context - is initialised (possibly taking the theory into account from which the - context is derived). We choose here to just return the empty list. Next - we define two auxiliary functions for updating the list with a given - term and printing the list. -*} - -ML{*fun update trm = Data.map (fn trms => trm::trms) - -fun print ctxt = - case (Data.get ctxt) of - [] => tracing "Empty!" - | trms => tracing (string_of_terms ctxt trms)*} - -text {* - Next we start with the context generated by the antiquotation - @{text "@{context}"} and update it in various ways. - - @{ML_response_fake [display,gray] -"let - val ctxt0 = @{context} - val ctxt1 = ctxt0 |> update @{term \"False\"} - |> update @{term \"True \ True\"} - val ctxt2 = ctxt0 |> update @{term \"1::nat\"} - val ctxt3 = ctxt2 |> update @{term \"2::nat\"} -in - print ctxt0; - print ctxt1; - print ctxt2; - print ctxt3 -end" -"Empty! -True \ True, False -1 -2, 1"} - - Many functions in Isabelle manage and update data in a similar - fashion. Consequently, such calculations with contexts occur frequently in - Isabelle code, although the ``context flow'' is usually only linear. - Note also that the calculation above has no effect on the underlying - theory. Once we throw away the contexts, we have no access to their - associated data. This is different for theories, where the command - \isacommand{setup} registers the data with the current and future - theories, and therefore one can access the data potentially - indefinitely. - - For convenience there is an abstract layer, namely the type @{ML_type Context.generic}, - for treating theories and proof contexts more uniformly. This type is defined as follows -*} - -ML_val{*datatype generic = - Theory of theory -| Proof of proof*} - -text {* - \footnote{\bf FIXME: say more about generic contexts.} - - There are two special instances of the data storage mechanism described - above. The first instance implements named theorem lists using the functor - @{ML_funct_ind Named_Thms}. This is because storing theorems in a list - is such a common task. To obtain a named theorem list, you just declare -*} - -ML{*structure FooRules = Named_Thms - (val name = "foo" - val description = "Theorems for foo") *} - -text {* - and set up the @{ML_struct FooRules} with the command -*} - -setup %gray {* FooRules.setup *} - -text {* - This code declares a data container where the theorems are stored, - an attribute @{text foo} (with the @{text add} and @{text del} options - for adding and deleting theorems) and an internal ML-interface for retrieving and - modifying the theorems. - Furthermore, the theorems are made available on the user-level under the name - @{text foo}. For example you can declare three lemmas to be a member of the - theorem list @{text foo} by: -*} - -lemma rule1[foo]: "A" sorry -lemma rule2[foo]: "B" sorry -lemma rule3[foo]: "C" sorry - -text {* and undeclare the first one by: *} - -declare rule1[foo del] - -text {* You can query the remaining ones with: - - \begin{isabelle} - \isacommand{thm}~@{text "foo"}\\ - @{text "> ?C"}\\ - @{text "> ?B"} - \end{isabelle} - - On the ML-level, we can add theorems to the list with @{ML FooRules.add_thm}: -*} - -setup %gray {* Context.theory_map (FooRules.add_thm @{thm TrueI}) *} - -text {* - The rules in the list can be retrieved using the function - @{ML FooRules.get}: - - @{ML_response_fake [display,gray] - "FooRules.get @{context}" - "[\"True\", \"?C\",\"?B\"]"} - - Note that this function takes a proof context as argument. This might be - confusing, since the theorem list is stored as theory data. It becomes clear by knowing - that the proof context contains the information about the current theory and so the function - can access the theorem list in the theory via the context. - - \begin{readmore} - For more information about named theorem lists see - @{ML_file "Pure/Tools/named_thms.ML"}. - \end{readmore} - - The second special instance of the data storage mechanism are configuration - values. They are used to enable users to configure tools without having to - resort to the ML-level (and also to avoid references). Assume you want the - user to control three values, say @{text bval} containing a boolean, @{text - ival} containing an integer and @{text sval} containing a string. These - values can be declared by -*} - -ML{*val (bval, setup_bval) = Attrib.config_bool "bval" (K false) -val (ival, setup_ival) = Attrib.config_int "ival" (K 0) -val (sval, setup_sval) = Attrib.config_string "sval" (K "some string") *} - -text {* - where each value needs to be given a default. To enable these values on the - user-level, they need to be set up with -*} - -setup %gray {* - setup_bval #> - setup_ival #> - setup_sval -*} - -text {* - The user can now manipulate the values from the user-level of Isabelle - with the command -*} - -declare [[bval = true, ival = 3]] - -text {* - On the ML-level these values can be retrieved using the - function @{ML_ind get in Config} from a proof context - - @{ML_response [display,gray] - "Config.get @{context} bval" - "true"} - - or directly from a theory using the function @{ML_ind get_global in Config} - - @{ML_response [display,gray] - "Config.get_global @{theory} bval" - "true"} - - It is also possible to manipulate the configuration values - from the ML-level with the functions @{ML_ind put in Config} - and @{ML_ind put_global in Config}. For example - - @{ML_response [display,gray] - "let - val ctxt = @{context} - val ctxt' = Config.put sval \"foo\" ctxt - val ctxt'' = Config.put sval \"bar\" ctxt' -in - (Config.get ctxt sval, - Config.get ctxt' sval, - Config.get ctxt'' sval) -end" - "(\"some string\", \"foo\", \"bar\")"} - - \begin{readmore} - For more information about configuration values see - the files @{ML_file "Pure/Isar/attrib.ML"} and - @{ML_file "Pure/config.ML"}. - \end{readmore} -*} - -section {* Summary *} - -text {* - This chapter describes the combinators that are used in Isabelle, as well - as a simple printing infrastructure for @{ML_type term}, @{ML_type cterm} - and @{ML_type thm}. The section on ML-antiquotations shows how to refer - statically to entities from the logic level of Isabelle. Isabelle also - contains mechanisms for storing arbitrary data in theory and proof - contexts. - - \begin{conventions} - \begin{itemize} - \item Print messages that belong together in a single string. - \item Do not use references in Isabelle code. - \end{itemize} - \end{conventions} - -*} - - -(**************************************************) -(* bak *) -(**************************************************) - -(* -section {* Do Not Try This At Home! *} - -ML {* val my_thms = ref ([]: thm list) *} - -attribute_setup my_thm_bad = - {* Scan.succeed (Thm.declaration_attribute (fn th => fn ctxt => - (my_thms := th :: ! my_thms; ctxt))) *} "bad attribute" - -declare sym [my_thm_bad] -declare refl [my_thm_bad] - -ML "!my_thms" -*) -end