theory FirstSteps

imports Base

begin

chapter {* First Steps *}

text {*

  Isabelle programming is done in ML.  Just like lemmas and proofs, ML-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 starting with

  \begin{center}

  \begin{tabular}{@ {}l}

  \isacommand{theory} FirstSteps\\

  \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\smallskip

\isa{\isacommand{ML}

\isacharverbatimopen\isanewline

\hspace{5mm}@{ML "3 + 4"}\isanewline

\isacharverbatimclose\isanewline

@{text "> 7"}\smallskip}

  The @{text [quotes] "> 7"} indicates the response Isabelle prints out when the 

  \isacommand{ML}-command is evaluated. Like ``normal'' Isabelle proof 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, and even those can be

  undone when the proof script is retracted. For better readability, we will in what 

  follows drop the \isacommand{ML} \isa{\isacharverbatimopen \ldots \isacharverbatimclose} 

  whenever we show code and its response.

  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} FirstSteps\\

  \isacommand{imports} Main\\

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

  \isacommand{begin}\\

  \ldots

  \end{tabular}

  \end{center}

*}

section {* Debugging and Printing *}

text {*

  During development you might find it necessary to inspect some data

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

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

  @{ML_response_fake [display] "warning \"any string\"" "\"any string\""}

  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 @{ML makestring}:

  @{ML_response_fake [display] "warning (makestring 1)" "\"1\""} 

  However @{ML makestring} only works if the type of what is converted is monomorphic 

  and is not a function.

  The function @{ML "warning"} 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 tracing} is more appropriate. This function writes all output into

  a separate tracing buffer. For example

  @{ML_response_fake [display] "tracing \"foo\"" "\"foo\""}

  It is also possible to redirect the ``channel'' where the string @{text "foo"} is 

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

  amounts of trace output. This redirection 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 "redirect_tracing"} with @{ML "(TextIO.openOut \"foo.bar\")"} 

  will cause that all tracing information is printed into the file @{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. By

  this we mean definitions, theorems, terms and so on. This kind of reference is

  realised with antiquotations.  For example, one can print out the name of the current

  theory by typing

  @{ML_response [display] "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 with

  the function @{ML "Context.theory_name"}. 

  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, as its name suggest, \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 name.

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

  @{ML_response_fake [display] "@{thm allI}" "(\<And>x. ?P x) \<Longrightarrow> \<forall>x. ?P x"}

  avoid explicit bindings for theorems such as

*}

ML{*val allI = thm "allI" *}

text {*

  These bindings were difficult to maintain and also could accidentally

  be overwritten by the user. This usually broke definitional

  packages. Antiquotations solve this problem, since they are ``linked''

  statically at compile-time. However, that also sometimes limits their

  usefulness. 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 

  \mbox{@{text "@{term \<dots>}"}}. For example

  @{ML_response [display] "@{term \"(a::nat) + b = c\"}" 

                          "Const (\"op =\", \<dots>)  $ (Const (\"HOL.plus_class.plus\", \<dots>) $ \<dots> $ \<dots>) $ \<dots>"}

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

  representation of this term. This internal representation corresponds to the 

  datatype @{ML_type "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. 

  \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 function to set the limit to a value high 

  enough:

  \end{exercise}

  @{ML [display] "print_depth 50"}

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

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

  Consider for example the pairs

  @{ML_response [display] "(@{term \"P x\"}, @{prop \"P x\"})" "(Free (\"P\", \<dots>) $ Free (\"x\", \<dots>),

 Const (\"Trueprop\", \<dots>) $ (Free (\"P\", \<dots>) $ Free (\"x\", \<dots>)))"}

  where an coercion is inserted in the second component and 

  @{ML_response [display] "(@{term \"P x \<Longrightarrow> Q x\"}, @{prop \"P x \<Longrightarrow> Q x\"})" 

  "(Const (\"==>\", \<dots>) $ \<dots> $ \<dots>, Const (\"==>\", \<dots>) $ \<dots> $ \<dots>)"}

  where it is not (since it is already constructed using the meta-implication). 

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

  @{ML_response_fake [display] "@{typ \"bool \<Rightarrow> nat\"}" "bool \<Rightarrow> nat"}

  \begin{readmore}

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

  definition and many useful operations can also 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 (remember they are ``linked'' at compile-time). 

  However, one often needs to construct terms

  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 type @{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 (P $ x, 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 \emph{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. With @{ML make_imp} we obtain the correct term involving 

  @{term "S"}, @{text "T"} and  @{text "@{typ nat}"} 

  @{ML_response [display] "make_imp @{term S} @{term T} @{typ nat}" 

    "Const \<dots> $ 

    Abs (\"x\", Type (\"nat\",[]),

      Const \<dots> $ (Free (\"S\",\<dots>) $ \<dots>) $ 

                  (Free (\"T\",\<dots>) $ \<dots>))"}

  With @{ML make_wrong_imp} we obtain an incorrect term involving the @{term "P"} 

  and @{text "Q"} from the antiquotation.

  @{ML_response [display] "make_wrong_imp @{term S} @{term T} @{typ nat}" 

    "Const \<dots> $ 

    Abs (\"x\", \<dots>,

      Const \<dots> $ (Const \<dots> $ (Free (\"P\",\<dots>) $ \<dots>)) $ 

                  (Const \<dots> $ (Free (\"Q\",\<dots>) $ \<dots>)))"}

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

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

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

  \begin{center}

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

  \end{center}

  The extra prefixes @{text zero_class} and @{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 Isabelle provides the

  antiquotation @{text "@{const_name \<dots>}"} which does the expansion

  automatically, for example:

  @{ML_response_fake [display] "@{const_name \"Nil\"}" "List.list.Nil"}

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

  Similarly, types can be constructed manually, for example as follows:

*} 

ML{*fun make_fun_type tau1 tau2 = Type ("fun",[tau1,tau2]) *}

text {*

  which can be equally written as 

*}

ML{*fun make_fun_type tau1 tau2 = tau1 --> tau2 *}

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 

  and types 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 @{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 converts 

  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 use

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

  For example we can write

  @{ML_response_fake [display] "cterm_of @{theory} @{term \"a + b = c\"}" "a + b = c"}

  or use the antiquotation

  @{ML_response_fake [display] "@{cterm \"(a::nat) + b = c\"}" "a + b = c"}

  Attempting

  @{ML_response_fake_both [display] "@{cterm \"1 + True\"}" "Type unification failed \<dots>"}

  yields an error. A slightly more elaborate example is

@{ML_response_fake [display] 

"let

  val natT = @{typ \"nat\"}

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

in

  cterm_of @{theory} 

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

  $ zero $ zero)

end" "0 + 0"}

  \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 are abstract objects of type @{ML_type thm} 

  that can only be built by going through interfaces. In effect all proofs 

  are 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_response_fake [display]

"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" "\<lbrakk>\<And>x. P x \<Longrightarrow> Q x; P t\<rbrakk> \<Longrightarrow> Q t"}

  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 the functions @{text "assume"}, @{text "forall_elim"} etc 

  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: