theory FirstSteps

imports Main

uses "antiquote_setup.ML"

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 {* Inluding 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 {*

  The expression inside \isacommand{ML} commands is 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 

*}

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 

  \begin{center}

  \begin{tabular}{@ {}l}

  \isacommand{theory} CookBook\\

  \isacommand{imports} Main\\

  \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.

  PolyML provides a convenient, though again ``quick-and-dirty'', method for 

  converting arbitrary values into strings, for example: 

*}

ML {* warning (makestring 1) *}

text {*

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

  a function.

*}

text {* 

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

*}

ML {* tracing "foo" *}

*}

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));

*}

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 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 types and theorems:

*}

ML {* @{typ "(int * nat) list"} *}

ML {* @{thm allI} *}

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.

*}

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}

*}

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"} *}

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

text {* which does not. *}

text {*

  dynamically. For example, a function that returns the implication 

  can only be written as

*}

ML {*

  fun make_imp P Q =

  let

    val x = @{term "x::nat"}

  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} and @{term Q} into 

*}

ML {*

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

*}

text {*

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

*}

ML {* @{const_name HOL.zero}; @{const_name  plus} *}

text {*

  \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}

  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 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"} *}

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 just 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 {*

  \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. 

  \begin{readmore}

  For more on goals see \isccite{sec:tactical-goals}. 

  \end{readmore}

  Tactics are functions that map a goal state to a (lazy)

  sequence of successor states, hence the type of a tactic is

  @{ML_type[display] "thm -> thm Seq.seq"}

  \begin{readmore}

  See @{ML_file "Pure/General/seq.ML"} for the implementation of lazy

  sequences. However in day-to-day Isabelle programming, one rarely 

  constructs sequences explicitly, but uses the predefined tactic 

  combinators (tacticals) instead (see @{ML_file "Pure/tctical.ML"}). 

  (FIXME: Pointer to the old reference manual)

  \end{readmore}

  While tatics can operate on the subgoals (the @{text "A\<^isub>i"} above), they 

  are expected to leave the conclusion @{term C} intact, with the 

  exception of possibly instantiating schematic variables. 

  To see how tactics work, let us transcribe a simple @{text apply}-style 

  proof from the tutorial \cite{isa-tutorial} into ML:

*}

lemma disj_swap: "P \<or> Q \<Longrightarrow> Q \<or> P"

apply (erule disjE)

 apply (rule disjI2)

 apply assumption

apply (rule disjI1)

apply assumption

done

text {*

  To start the proof, the function  @{ML "Goal.prove"}~@{text "ctxt params assms goal tac"}

  sets up a goal state for proving @{text goal} under the assumptions @{text assms} with

  the variables @{text params} that will be generalised once the

  goal is proved; @{text "tac"} is a function that returns a tactic having some funny 

  input parameters (FIXME by possibly having an explanation in the implementation-manual). 

*}