theory FirstSteps

imports Main

uses "antiquote_setup.ML"

begin

(*<*)

ML {*

local structure O = ThyOutput

in

  fun check_exists f = 

    if File.exists (Path.explode ("~~/src/" ^ f)) then ()

    else error ("Source file " ^ quote f ^ " does not exist.")

  val _ = O.add_commands

   [("ML_file", O.args (Scan.lift Args.name) (O.output (fn _ => fn name =>

         (check_exists name; Pretty.str name))))];

end

*}

(*>*)

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}

  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 

  can also contain value- and function bindings. However on such 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"}. 

  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 

*}

ML {*

  val _ = warning "any string"

*}

text {*

*}

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). 

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

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

*}

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 antoquotations: they greatly simplify Isabelle programming since one

  can directly access all kinds of logical elements from ML.

*}

section {* Terms *}

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 represenation corresponds to the 

  datatype defined in @{ML_file "Pure/term.ML"}.

  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

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

  \begin{readmore}

  Terms are described in detail in \ichcite{ch:logic}. Their

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

  \end{readmore}

  Sometimes the internal representation can be surprisingly different

  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 @{ML "print_depth 50"} to set the limit to a value high enough.

  \end{exercise}

*}

ML {*

*}

text {*

  often more complex than one first expects. Here, the extra prefixes

  @{text zero_class} and @{text plus_class} are present because the

  constants are defined within a type class. Guessing such internal

  names can be extremely hard, which is why the system provides

  another antiquotation: @{ML "@{const_name plus}"} gives just this

*}

text {*

*}

text {*

  \begin{exercise}

  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, and see if 

  \end{exercise}

*}

ML {* 

  fun rev_sum t =

  let

   fun dest_sum (Const (@{const_name plus}, _) $ u $ u') = 

                                                      u' :: dest_sum u

     | dest_sum u = [u]

   in

     foldl1 (HOLogic.mk_binop @{const_name plus}) (dest_sum t)

   end;

*}

text {*

  \begin{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}

*}

ML {*

  fun make_sum t1 t2 =

      HOLogic.mk_nat (HOLogic.dest_nat t1 + HOLogic.dest_nat t2)

*}

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 wellformed, 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"} *}

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. The

  basic rules of the Isabelle/Pure logical framework are defined in

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

  Using these rules, which are just ML functions, you can do simple

  natural deduction proofs on the ML level. For example, the 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 {*

  can be proved in ML like 

  this\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 {*

*}

text {*

  The goal-oriented tactical style is similar to the @{text apply}

  style at the user level. Reasoning is centered 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

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

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

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

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

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

*}

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

apply (erule disjE)

 apply (rule disjI2)

 apply assumption

apply (rule disjI1)

apply assumption

done

ML {*

let

  val ctxt = @{context}

  val goal = @{prop "P \<or> Q \<Longrightarrow> Q \<or> P"}

in

  Goal.prove ctxt ["P", "Q"] [] goal (fn _ => 

    eresolve_tac [disjE] 1

    THEN resolve_tac [disjI2] 1

    THEN assume_tac 1

    THEN resolve_tac [disjI1] 1

    THEN assume_tac 1)

end

*}

ML {*

*}

end