theory FirstSteps

begin

chapter {* First Steps *}

text {* 

  Isabelle programming is done in Standard ML.

  Just like lemmas and proofs, code in Isabelle is part of a 

  theory. If you want to follow the code written in this chapter, we 

  assume you are working inside the theory defined by

  \begin{center}

  \begin{tabular}{@ {}l}

  \isacommand{theory} CookBook\\

  \isacommand{imports} Main\\

  \isacommand{begin}\\

  \ldots

  \end{tabular}

  \end{center}

*}

section {* Including ML-Code *}

text {*

  The easiest and quickest way to include code in a theory is

  by using the \isacommand{ML} command. For example

*}

ML {*

  3 + 4

*}

text {*

  Expressions inside \isacommand{ML} commands are immediately evaluated,

  like ``normal'' Isabelle proof scripts, 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. However on such \isacommand{ML}

  commands the undo operation behaves slightly counter-intuitive, because 

  if you define

*}

ML {*

  val foo = true

*}

text {*

  then Isabelle's undo operation has no effect on the definition of 

  @{ML "foo"}. 

  Once a portion of code is relatively stable, one usually wants to 

  export it to a separate ML-file. Such files can then be included in a 

  theory by using \isacommand{uses} in the header of the theory, like

  \begin{center}

  \begin{tabular}{@ {}l}

  \isacommand{theory} CookBook\\

  \isacommand{imports} Main\\

  \isacommand{uses} @{ML_text "\"file_to_be_included.ML\""} @{text "\<dots>"}\\

  \isacommand{begin}\\

  \ldots

  \end{tabular}

  \end{center}

*}

section {* Debugging and Printing *}

text {*

  During developments you might find it necessary to quickly inspect some data

  in your code. This can be done in a ``quick-and-dirty'' fashion using 

  the function @{ML "warning"}. For example 

*}

ML {* warning "any string" *}

text {*

  will print out @{ML "\"any string\""} inside the response buffer of Isabelle.

  If you develop under PolyML, then there is a convenient, though again 

  ``quick-and-dirty'', method for converting values into strings, 

  for example: 

*}

ML {* warning (makestring 1) *}

text {*

  However this only works if the type of what is converted is monomorphic and not 

  a function.

*}

text {* 

  The funtion warning should only be used for testing purposes, because

  any output this funtion generates will be overwritten, as soon as an error 

  is raised. 

  Therefore for printing anything more serious and elaborate, the function @{ML tracing}

  should be used. This function writes all output into a separate buffer.

*}

ML {* tracing "foo" *}

text {* 

  It is also possible to redirect the channel where the @{ML_text "foo"} is 

  printed to a separate file, e.g. to prevent Proof General from choking on massive 

  amounts of trace output. This rediretion can be achieved using 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 @{ML_text "redirect_tracing"} with @{ML_text "(TextIO.openOut \"foo.bar\")"} 

  will cause that all tracing information is printed into the file @{ML_text "foo.bar"}.

*}

section {* Antiquotations *}

text {*

  The main advantage of embedding all code 

  in a theory is that the code can contain references to entities defined 

  on the logical level of Isabelle. This is done using antiquotations.

  For example, one can print out the name of 

  the current theory by typing

*}

ML {* Context.theory_name @{theory} *}

text {* 

  where @{text "@{theory}"} is an antiquotation that is substituted with the

  current theory (remember that we assumed we are inside the theory CookBook). 

  The name of this theory can be extracted using the function @{ML "Context.theory_name"}. 

  So the code above returns the string @{ML "\"CookBook\""}.

  Note, however, that antiquotations are statically scoped, that is the value is

  determined at ``compile-time'', not ``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 @{ML "\"CookBook\""}, no matter where the

  function is called. Operationally speaking,  @{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 name.

  In a similar way you can use antiquotations to refer to theorems or simpsets:

*}

ML {* @{thm allI} *}

ML {* @{simpset} *}

text {*

  In the course of this introduction, we will learn more about 

  these antiquotations: they greatly simplify Isabelle programming since one

  can directly access all kinds of logical elements from ML.

*}

section {* Terms and Types *}

text {*

  One way to construct terms of Isabelle on the ML-level is by using the antiquotation 

  @{text "@{term \<dots>}"}:

*}

ML {* @{term "(a::nat) + b = c"} *}

text {*

  This will show the term @{term "(a::nat) + b = c"}, but printed out using the internal

  representation of this term. This internal representation corresponds to the 

  datatype @{ML_text "term"}.

  The internal representation of terms uses the usual de-Bruijn index mechanism where bound 

  variables are represented by the constructor @{ML Bound}. The index in @{ML Bound} refers to

  the number of Abstractions (@{ML Abs}) we have to skip until we hit the @{ML Abs} that

  binds the corresponding variable. However, in Isabelle the names of bound variables are 

  kept at abstractions for printing purposes, and so should be treated only as comments. 

  \begin{readmore}

  Terms are described in detail in \isccite{sec:terms}. Their

  definition and many useful operations can be found in @{ML_file "Pure/term.ML"}.

  \end{readmore}

  Sometimes the internal representation of terms can be surprisingly different

  from what you see at the user level, because the layers of

  parsing/type checking/pretty printing can be quite elaborate. 

*}

text {*

  \begin{exercise}

  Look at the internal term representation of the following terms, and

  find out why they are represented like this.

  \begin{itemize}

  \item @{term "case x of 0 \<Rightarrow> 0 | Suc y \<Rightarrow> y"}  

  \item @{term "\<lambda>(x,y). P y x"}  

  \item @{term "{ [x::int] | x. x \<le> -2 }"}  

  \end{itemize}

  Hint: The third term is already quite big, and the pretty printer

  may omit parts of it by default. If you want to see all of it, you

  can use the following ML funtion to set the limit to a value high 

  enough:

  \end{exercise}

*}

ML {* print_depth 50 *}

text {*

  The antiquotation @{text "@{prop \<dots>}"} constructs terms of propositional type, 

  inserting the invisible @{text "Trueprop"} coercions whenever necessary. 

  Consider for example

*}

ML {* @{term "P x"} ; @{prop "P x"} *}

text {* which inserts the coercion in the latter case and *}

ML {* @{term "P x \<Longrightarrow> Q x"} ; @{prop "P x \<Longrightarrow> Q x"} *}

text {* 

  which does not. 

  Types can be constructed using the antiquotation @{text "@{typ \<dots>}"}. For example

  *}

ML {* @{typ "bool \<Rightarrow> nat"} *}

text {*

  (FIXME: Unlike the term antiquotation, @{text "@{typ \<dots>}"} does not print the

  internal representation. Is there a reason for this, that needs to be explained 

  here?)

  \begin{readmore}

  Types are described in detail in \isccite{sec:types}. Their

  definition and many useful operations can be found in @{ML_file "Pure/type.ML"}.

  \end{readmore}

  *}

section {* Constructing Terms and Types Manually *} 

text {*

  While antiquotations are very convenient for constructing terms and types, 

  they can only construct fixed terms. Unfortunately, one often needs to construct them 

  dynamically. For example, a function that returns the implication 

  @{text "\<And>(x::\<tau>). P x \<Longrightarrow> Q x"} taking @{term P}, @{term Q} and the typ @{term "\<tau>"}

  as arguments can only be written as

*}

ML {*

  fun make_imp P Q tau =

  let

    val x = Free ("x",tau)

  in Logic.all x (Logic.mk_implies (HOLogic.mk_Trueprop (P $ x),

                                    HOLogic.mk_Trueprop (Q $ x)))

  end

*}

text {*

  The reason is that one cannot pass the arguments @{term P}, @{term Q} and 

  @{term "tau"} into an antiquotation. For example the following does not work.

*}

ML {*

  fun make_wrong_imp P Q tau = @{prop "\<And>x. P x \<Longrightarrow> Q x"} 

*}

text {*

  To see this apply @{text "@{term S}"}, @{text "@{term T}"} and @{text "@{typ nat}"} 

  to both functions. 

  One tricky point in constructing terms by hand is to obtain the 

  fully qualified name for constants. For example the names for @{text "zero"} or 

  @{text "+"} are more complex than one first expects, namely 

  \begin{center}

  @{ML_text "HOL.zero_class.zero"} and @{ML_text "HOL.plus_class.plus"}. 

  \end{center}

  The extra prefixes @{ML_text zero_class} and @{ML_text plus_class} are present because 

  these constants are defined within type classes; the prefix @{text "HOL"} indicates in 

  which theory they are defined. Guessing such internal names can sometimes be quite hard. 

  Therefore Isabellle provides the antiquotation @{text "@{const_name \<dots>}"} which does the 

  expansion automatically, for example:

*}

text {*

  (FIXME: Is it useful to explain @{text "@{const_syntax}"}?)

  (FIXME: how to construct types manually)

  \begin{readmore}

  There are many functions in @{ML_file "Pure/logic.ML"} and

  @{ML_file "HOL/hologic.ML"} that make such manual constructions of terms 

  easier.\end{readmore}

  Have a look at these files and try to solve the following two exercises:

*}

text {*

  \begin{exercise}\label{fun:revsum}

  Write a function @{ML_text "rev_sum : term -> term"} that takes a

  term of the form @{text "t\<^isub>1 + t\<^isub>2 + \<dots> + t\<^isub>n"} (whereby @{text "i"} might be zero)

  and returns the reversed sum @{text "t\<^isub>n + \<dots> + t\<^isub>2 + t\<^isub>1"}. Assume

  the @{text "t\<^isub>i"} can be arbitrary expressions and also note that @{text "+"} 

  associates to the left. Try your function on some examples. 

  \end{exercise}

  \begin{exercise}\label{fun:makesum}

  Write a function which takes two terms representing natural numbers

  in unary (like @{term "Suc (Suc (Suc 0))"}), and produce the unary

  number representing their sum.

  \end{exercise}

*}

section {* Type Checking *}

text {* 

  We can freely construct and manipulate terms, since they are just

  arbitrary unchecked trees. However, we eventually want to see if a

  term is well-formed, or type checks, relative to a theory.

  Type checking is done via the function @{ML cterm_of}, which turns 

  a @{ML_type term} into a  @{ML_type cterm}, a \emph{certified} term. 

  Unlike @{ML_type term}s, which are just trees, @{ML_type

  "cterm"}s are abstract objects that are guaranteed to be

  type-correct, and that can only be constructed via the official

  interfaces.

  Type checking is always relative to a theory context. For now we can use

  the @{ML "@{theory}"} antiquotation to get hold of the current theory.

  For example we can write:

*}

ML {* cterm_of @{theory} @{term "(a::nat) + b = c"} *}

text {* and *}

ML {*

  let

    val natT = @{typ "nat"}

    val zero = @{term "0::nat"}

  in

    cterm_of @{theory} 

        (Const (@{const_name plus}, natT --> natT --> natT) 

         $ zero $ zero)

  end

*}

text {*

  \begin{exercise}

  Check that the function defined in Exercise~\ref{fun:revsum} returns a

  result that type checks.

  \end{exercise}

*}

section {* Theorems *}

text {*

  Just like @{ML_type cterm}s, theorems (of type @{ML_type thm}) are

  abstract objects that can only be built by going through the kernel

  interfaces, which means that all your proofs will be checked. 

  To see theorems in ``action'', let us give a proof for the following statement

*}

  lemma 

   assumes assm\<^isub>1: "\<And>(x::nat). P x \<Longrightarrow> Q x" 

   and     assm\<^isub>2: "P t"

   shows "Q t" (*<*)oops(*>*) 

text {*

  on the ML level:\footnote{Note that @{text "|>"} is reverse

  application. This combinator, and several variants are defined in

  @{ML_file "Pure/General/basics.ML"}.}

*}

ML {*

let

  val thy = @{theory}

  val assm1 = cterm_of thy @{prop "\<And>(x::nat). P x \<Longrightarrow> Q x"}

  val assm2 = cterm_of thy @{prop "((P::nat\<Rightarrow>bool) t)"}

  val Pt_implies_Qt = 

        assume assm1

        |> forall_elim (cterm_of thy @{term "t::nat"});

  val Qt = implies_elim Pt_implies_Qt (assume assm2);

in

  Qt 

  |> implies_intr assm2

  |> implies_intr assm1

end

*}

text {*

  This code-snippet constructs the following proof:

  \[

  \infer[(@{text "\<Longrightarrow>"}$-$intro)]{\vdash @{prop "(\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> P t \<Longrightarrow> Q t"}}

    {\infer[(@{text "\<Longrightarrow>"}$-$intro)]{@{prop "\<And>x. P x \<Longrightarrow> Q x"} \vdash @{prop "P t \<Longrightarrow> Q t"}}

       {\infer[(@{text "\<Longrightarrow>"}$-$elim)]{@{prop "\<And>x. P x \<Longrightarrow> Q x"}, @{prop "P t"} \vdash @{prop "Q t"}}

          {\infer[(@{text "\<And>"}$-$elim)]{@{prop "\<And>x. P x \<Longrightarrow> Q x"} \vdash @{prop "P t \<Longrightarrow> Q t"}}

                 {\infer[(assume)]{@{prop "\<And>x. P x \<Longrightarrow> Q x"} \vdash @{prop "\<And>x. P x \<Longrightarrow> Q x"}}{}}

                 &

           \infer[(assume)]{@{prop "P t"} \vdash @{prop "P t"}}{}

          }

       }

    }

  \]

  \begin{readmore}

  For how the functions @{text "assume"}, @{text "forall_elim"} etc work

  see \isccite{sec:thms}. The basic functions for theorems are defined in

  @{ML_file "Pure/thm.ML"}. 

  \end{readmore}

*}

section {* Tactical Reasoning *}

text {*

  The goal-oriented tactical style reasoning of the ML level is similar 

  to the @{text apply}-style at the user level, i.e.~the reasoning is centred 

  around a \emph{goal}, which is modified in a sequence of proof steps 

  until it is solved.

  A goal (or goal state) is a special @{ML_type thm}, which by

  convention is an implication of the form:

  @{text[display] "A\<^isub>1 \<Longrightarrow> \<dots> \<Longrightarrow> A\<^isub>n \<Longrightarrow> #(C)"}

  where @{term C} is the goal to be proved and the @{term "A\<^isub>i"} are the open subgoals. 

  Since the goal @{term C} can potentially be an implication, there is a

  @{text "#"} wrapped around it, which prevents that premises are 

  misinterpreted as open subgoals. The protection @{text "# :: prop \<Rightarrow>

  prop"} is just the identity function and used as a syntactic marker. 

  (FIXME: maybe show how this is printed on the screen) 

  \begin{readmore}