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 given 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:

\begin{isabelle}
\begin{graybox}
\isacommand{ML}~@{text "\<verbopen>"}\isanewline
\hspace{5mm}@{ML "3 + 4"}\isanewline
@{text "\<verbclose>"}\isanewline
@{text "> 7"}\smallskip
\end{graybox}
\end{isabelle}

  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. As mentioned earlier, we will  
  drop the \isacommand{ML}~@{text "\<verbopen> \<dots> \<verbclose>"} 
  scaffolding whenever we show code.

  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 the \isacommand{uses}-command 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,gray] "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 using the function @{ML makestring}:

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

  However @{ML makestring} only works if the type of what is converted is monomorphic 
  and 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,gray] "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"}.

  You can print out error messages with the function @{ML error}, as in:

  @{ML_response_fake [display,gray] "if 0=1 then 1 else (error \"foo\")" "\"foo\""}

*}




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,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 "Context.theory_name"}. 

  Note, however, that antiquotations are statically linked, that is their 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 @{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 proved theorems:

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

  or simpsets:

  @{ML_response_fake [display,gray] 
"let
  val ({rules,...}, _) = MetaSimplifier.rep_ss @{simpset}
in
  map #name (Net.entries rules)
end" "[\"Nat.of_nat_eq_id\", \"Int.of_int_eq_id\", \"Nat.One_nat_def\", \<dots>]"}

  The code about simpsets extracts the theorem names stored in the
  current simpset.  We get hold of the current simpset with the antiquotation 
  @{text "@{simpset}"}.  The function @{ML rep_ss in MetaSimplifier} returns a record
  containing all information about the simpset. The rules of a simpset are
  stored in a \emph{discrimination net} (a datastructure for fast
  indexing). From this net we can extract the entries using the function @{ML
  Net.entries}.


  \begin{readmore}
  The infrastructure for simpsets is implemented in @{ML_file "Pure/meta_simplifier.ML"}
  and @{ML_file "Pure/simplifier.ML"}. Discrimination nets are implemented
  in @{ML_file "Pure/net.ML"}.
  \end{readmore}

  While antiquotations have many applications, they were originally introduced in order 
  to avoid explicit bindings for theorems such as:
*}

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

text {*
  These bindings are difficult to maintain and also can be accidentally
  overwritten by the user. This often breakes Isabelle
  packages. Antiquotations solve this problem, since they are ``linked''
  statically at compile-time.  However, this static linkage also limits their
  usefulness in cases where data needs to be build up dynamically. 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 th ML-level.

*}

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,gray] 
"@{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 are implemented 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:

  @{ML [display,gray] "print_depth 50"}
  \end{exercise}

  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,gray] "(@{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,gray] "(@{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 by a meta-implication). 

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

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

  \begin{readmore}
  Types are described in detail in \isccite{sec:types}. Their
  definition and many useful operations are implemented 
  in @{ML_file "Pure/type.ML"}.
  \end{readmore}
*}


section {* Constructing Terms and Types Manually *} 

text {*
  While antiquotations are very convenient for constructing terms, they can
  only construct fixed terms (remember they are ``linked'' at
  compile-time). See Recipe~\ref{rec:external} on Page~\pageref{rec:external}
  for a function that pattern-matches over terms and where the patterns are
  constructed using antiquotations.  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 intended term involving 
  @{term "S"}, @{text "T"} and  @{text "@{typ nat}"} 

  @{ML_response [display,gray] "make_imp @{term S} @{term T} @{typ nat}" 
    "Const \<dots> $ 
    Abs (\"x\", Type (\"nat\",[]),
      Const \<dots> $ (Free (\"S\",\<dots>) $ \<dots>) $ 
                  (Free (\"T\",\<dots>) $ \<dots>))"}

  whereas with @{ML make_wrong_imp} we obtain a term involving the @{term "P"} 
  and @{text "Q"} from the antiquotation.

  @{ML_response [display,gray] "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>)))"}

  (FIXME: expand the following point)

  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,gray] "@{const_name \"Nil\"}" "List.list.Nil"}

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

  Similarly, one can construct types manually. For example the function returning
  a function type is as follows:

*} 

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

text {* This 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/term.ML"}, @{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 notation (like @{term "Suc (Suc (Suc 0))"}), and produce the
  number representing their sum.
  \end{exercise}

  (FIXME: maybe should go)

*}

ML{*fun nat_to_int t =
  (case t of
     @{typ nat} => @{typ int}
   | Type (s, ts) => Type (s, map nat_to_int ts)
   | _ => t)*}

text {*

@{ML_response_fake [display,gray] 
"map_types nat_to_int @{term \"a = (1::nat)\"}" 
"Const (\"op =\", \"int \<Rightarrow> int \<Rightarrow> bool\")
           $ Free (\"a\", \"int\") $ Const (\"HOL.one_class.one\", \"int\")"}

*}

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 they can only be constructed via ``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,gray] "cterm_of @{theory} @{term \"a + b = c\"}" "a + b = c"}

  This can also be wirtten with an antiquotation

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

  Attempting to obtain the certified term for

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

  yields an error (since the term is not typable). A slightly more elaborate
  example that type-checks is


@{ML_response_fake [display,gray] 
"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}

  (FIXME: @{text "ctyp_of"}, @{ML fastype_of}, @{text dummyT})

*}

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. As a consequence, every proof 
  in Isabelle is correct by construction (FIXME reference LCF-philosophy)

  To see theorems in ``action'', let us give a proof on the ML-level 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 {*
  The corresponding ML-code is as follows:\footnote{Note that @{text "|>"} is reverse
  application. See Section~\ref{sec:combinators}.}

@{ML_response_fake [display,gray]
"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 {* Storing Theorems *}

section {* Theorem Attributes *}

section {* Operations on Constants (Names) *}

text {*

@{ML_response [display] "Sign.base_name \"List.list.Nil\"" "\"Nil\""}

  authentic syntax?

*}

ML {* @{const_name lfp} *}

text {*
  constants in case-patterns?

  In the meantime, lfp has been moved to the Inductive theory, so it is  
  no longer called Lfp.lfp. If a @{text "@{const_name}"} antiquotation had been  
  used, we would have gotten an error for this. Another advantage of the  
  antiquotation is that we can then just write @{text "@{const_name lfp}"} rather  
  than @{text "@{const_name Lfp.lfp}"} or whatever, and it expands to the correct  
  name.
  
*}

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 the code, but after getting familiar with them, they
  actually ease the understanding and also the programming.

  \begin{readmore}
  The most frequently used combinator 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}

  The simplest combinator is @{ML I}, which is just the identity function.
*}

ML{*fun I x = x*}

text {* Another simple combinator is @{ML K}, defined as *}

ML{*fun K x = fn _ => x*}

text {*
  @{ML K} ``wraps'' a function around the argument @{text "x"}. However, this 
  function ignores its argument. As a result @{ML K} defines a constant function 
  returning @{text x}.

  The next combinator is reverse application, @{ML "|>"}, 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 %linenumbers{*fun inc_by_five x =
  x |> (fn x => x + 1)
    |> (fn x => (x, x))
    |> fst
    |> (fn x => x + 4)*}

text {*
  which increments the 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 (for example by adding a definition, followed by
  lemmas and so on). It should also be familiar to anyone who has used Haskell's
  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!


  (FIXME: give a real world example involving theories)

  Similarly, the combinator @{ML "#>"} 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 one to read the code top-down.

  The remaining combinators described in this section add convenience for the
  ``waterfall method'' of writing functions. The combinator @{ML tap} allows
  one to get hold of an intermediate result (to do some side-calculations for
  instance). The function

 *}

ML %linenumbers{*fun inc_by_three x =
     x |> (fn x => x + 1)
       |> tap (fn x => tracing (makestring x))
       |> (fn x => x + 2)*}

text {* increments the argument first by one and then by two. In the middle (Line 3),
  however, it uses @{ML tap} for printing the ``plus-one'' intermediate 
  result inside the tracing buffer. 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, the next combinator
  can be used.

  The combinator @{ML "`"} is similar to @{ML tap}, but applies a function to the value
  and returns the result together with the value (as a pair). For example
  the function 
*}

ML{*fun inc_as_pair x =
     x |> `(fn x => x + 1)
       |> (fn (x, y) => (x, y + 1))*}

text {*
  takes @{text x} as argument, and then first 
  increments @{text x}, but also keeps @{text x}. The intermediate result is 
  therefore the pair @{ML "(x + 1, x)" for x}. The function then increments the 
  right-hand component of the pair. So finally the result will be 
  @{ML "(x + 1, x + 1)" for x}.

  The combinators @{ML "|>>"} and @{ML "||>"} 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 {*
  With the combinator @{ML "|->"} 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 version of the @{text double} function *}

ML{*fun double x =
      x |>  (fn x => (x, x))
        |-> (fn x => fn y => x + y)*}

text {*
  Recall that @{ML "|>"} is the reverse function applications. Recall also that the related 
  reverse function composition is @{ML "#>"}. In fact all the combinators @{ML "|->"},
  @{ML "|>>"} and @{ML "||>"} described above have related combinators for function
  composition, namely @{ML "#->"}, @{ML "#>>"} and @{ML "##>"}. 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 {*
  
  (FIXME: find a good exercise for combinators)
*}


(*<*)
setup {*
 Sign.add_consts_i [("bar", @{typ "nat"},NoSyn)] 
*}

lemma "bar = (1::nat)"
  oops

setup {*   
  Sign.add_consts_i [("foo", @{typ "nat"},NoSyn)] 
 #> PureThy.add_defs false [((Binding.name "foo_def", 
       Logic.mk_equals (Const ("FirstSteps.foo", @{typ "nat"}), @{term "1::nat"})), [])] 
 #> snd
*}

lemma "foo = (1::nat)"
  apply(simp add: foo_def)
  done

thm foo_def
(*>*)

section {* Misc *}

ML {*DatatypePackage.get_datatype @{theory} "List.list"*}

end