theory Advanced
imports Base First_Steps
begin

(*<*)
setup{*
open_file_with_prelude 
  "Advanced_Code.thy"
  ["theory Advanced", "imports Base First_Steps", "begin"]
*}
(*>*)


chapter {* Advanced Isabelle\label{chp:advanced} *}

text {*
   \begin{flushright}
  {\em All things are difficult before they are easy.} \\[1ex]
  proverb
  \end{flushright}

  \medskip
  While terms, types and theorems are the most basic data structures in
  Isabelle, there are a number of layers built on top of them. Most of these
  layers are concerned with storing and manipulating data. Handling them
  properly is an essential skill for programming on the ML-level of Isabelle. 
  The most basic layer are theories. They contain global data and
  can be seen as the ``long-term memory'' of Isabelle. There is usually only
  one theory active at each moment. Proof contexts and local theories, on the
  other hand, store local data for a task at hand. They act like the
  ``short-term memory'' and there can be many of them that are active in
  parallel.
*}

section {* Theories and Setups\label{sec:theories} *}

text {*
  Theories, as said above, are a basic layer of abstraction in
  Isabelle. They contain definitions, syntax declarations, axioms,
  theorems and much more.  For example, if a definition is made, it
  must be stored in a theory in order to be usable later on. Similar
  with proofs: once a proof is finished, the proved theorem needs to
  be stored in the theorem database of the theory in order to be
  usable. All relevant data of a theory can be queried as follows.

  \begin{isabelle}
  \isacommand{print\_theory}\\
  @{text "> names: Pure Code_Generator HOL \<dots>"}\\
  @{text "> classes: Inf < type \<dots>"}\\
  @{text "> default sort: type"}\\
  @{text "> syntactic types: #prop \<dots>"}\\
  @{text "> logical types: 'a \<times> 'b \<dots>"}\\
  @{text "> type arities: * :: (random, random) random \<dots>"}\\
  @{text "> logical constants: == :: 'a \<Rightarrow> 'a \<Rightarrow> prop \<dots>"}\\
  @{text "> abbreviations: \<dots>"}\\
  @{text "> axioms: \<dots>"}\\
  @{text "> oracles: \<dots>"}\\
  @{text "> definitions: \<dots>"}\\
  @{text "> theorems: \<dots>"}
  \end{isabelle}

  Functions acting on theories often end with the suffix @{text "_global"},
  for example the function @{ML read_term_global in Syntax} in the structure
  @{ML_struct Syntax}. The reason is to set them syntactically apart from
  functions acting on contexts or local theories (which will be discussed in
  the next sections). There is a tendency in the Isabelle development to prefer
  ``non-global'' operations, because they have some advantages, as we will also
  discuss later. However, some basic management of theories will very likely
  never go away.

  In the context of ML-programming, the most important command with theories
  is \isacommand{setup}. In the previous chapters we used it, for
  example, to make a theorem attribute known to Isabelle or to register a theorem
  under a name. What happens behind the scenes is that \isacommand{setup}
  expects a function of type @{ML_type "theory -> theory"}: the input theory
  is the current theory and the output the theory where the attribute has been
  registered or the theorem has been stored.  This is a fundamental principle
  in Isabelle. A similar situation arises with declaring constants. The
  function that declares a constant on the ML-level is @{ML_ind declare_const
  in Sign} in the structure @{ML_struct Sign}. To see how \isacommand{setup}
  works, consider the following code: 
*}  

ML{*let
  val thy = @{theory}
  val bar_const = ((@{binding "BAR"}, @{typ "nat"}), NoSyn)
in 
  Sign.declare_const @{context} bar_const thy  
end*}

text {*
  If you simply run this code\footnote{Recall that ML-code needs to be enclosed in
  \isacommand{ML}~@{text "\<verbopen> \<dots> \<verbclose>"}.} with the
  intention of declaring a constant @{text "BAR"} with type @{typ nat}, then 
  indeed you obtain a theory as result. But if you query the
  constant on the Isabelle level using the command \isacommand{term}

  \begin{isabelle}
  \isacommand{term}~@{text BAR}\\
  @{text "> \"BAR\" :: \"'a\""}
  \end{isabelle}

  you can see that you do \emph{not} obtain a constant of type @{typ nat}, but a free 
  variable (printed in blue) of polymorphic type. The problem is that the 
  ML-expression above did not ``register'' the declaration with the current theory. 
  This is what the command \isacommand{setup} is for. The constant is properly 
  declared with
*}

setup %gray {* fn thy => 
let
  val bar_const = ((@{binding "BAR"}, @{typ "nat"}), NoSyn)
  val (_, thy') = Sign.declare_const @{context} bar_const thy
in 
  thy'
end *}

text {* 
  where the declaration is actually applied to the current theory and
  
  \begin{isabelle}
  \isacommand{term}~@{text [quotes] "BAR"}\\
  @{text "> \"BAR\" :: \"nat\""}
  \end{isabelle}

  returns now a (black) constant with the type @{typ nat}, as expected.

  In a sense, \isacommand{setup} can be seen as a transaction that takes the
  current theory, applies an operation, and produces a new current theory. This
  means that we have to be careful to apply operations always to the most current
  theory, not to a \emph{stale} one. Consider again the function inside the
  \isacommand{setup}-command:

  \begin{isabelle}
  \begin{graybox}
  \isacommand{setup}~@{text "\<verbopen>"} @{ML
"fn thy => 
let
  val bar_const = ((@{binding \"BAR\"}, @{typ \"nat\"}), NoSyn)
  val (_, thy') = Sign.declare_const @{context} bar_const thy
in
  thy
end"}~@{text "\<verbclose>"}\isanewline
  @{text "> ERROR \"Stale theory encountered\""}
  \end{graybox}
  \end{isabelle}

  This time we erroneously return the original theory @{text thy}, instead of
  the modified one @{text thy'}. Such buggy code will always result into 
  a runtime error message about stale theories.

  However, sometimes it does make sense to work with two theories at the same
  time, especially in the context of parsing and typing. In the code below we
  use in Line 3 the function @{ML_ind copy in Theory} from the structure
  @{ML_struct Theory} for obtaining a new theory that contains the same
  data, but is unrelated to the existing theory.
*}

setup %graylinenos {* fn thy => 
let
  val tmp_thy = Theory.copy thy
  val foo_const = ((@{binding "FOO"}, @{typ "nat => nat"}), NoSyn)
  val (_, tmp_thy') = Sign.declare_const @{context} foo_const tmp_thy
  val trm1 = Syntax.read_term_global tmp_thy' "FOO baz"
  val trm2 = Syntax.read_term_global thy "FOO baz"
  val _ = writeln (@{make_string} trm1 ^ "\n" ^ @{make_string} trm2)
in
  thy
end *}

text {*
  That means we can make changes to the theory @{text tmp_thy} without
  affecting the current theory @{text thy}. In this case we declare in @{text
  "tmp_thy"} the constant @{text FOO} (Lines 4 and 5). The point of this code
  is that we next, in Lines 6 and 7, parse a string to become a term (both
  times the string is @{text [quotes] "FOO baz"}). But since we parse the string
  once in the context of the theory @{text tmp_thy'} in which @{text FOO} is
  declared to be a constant of type @{typ "nat \<Rightarrow>nat"} and once in the context 
  of @{text thy} where it is not, we obtain two different terms, namely 

  \begin{isabelle}
  \begin{graybox}
  @{text "> Const (\"Advanced.FOO\", \"nat \<Rightarrow> nat\") $ Free (\"baz\", \"nat\")"}\isanewline
  @{text "> Free (\"FOO\", \"'a \<Rightarrow> 'b\") $ Free (\"baz\", \"'a\")"}
  \end{graybox}
  \end{isabelle}

  There are two reasons for parsing a term in a temporary theory. One is to
  obtain fully qualified names for constants and the other is appropriate type 
  inference. This is relevant in situations where definitions are made later, 
  but parsing and type inference has to take already proceed as if the definitions 
  were already made.
*}

section {* Contexts (TBD) *}

text {*
  Contexts are arguably more important than theories, even though they only 
  contain ``short-term memory data''. The reason is that a vast number of
  functions in Isabelle depend one way or the other on contexts. Even such
  mundane operation like printing out a term make essential use of contexts.
  For this consider the following contrived proof which only purpose is to 
  fix two variables 
*}

lemma "True"
proof -
  txt_raw {*\mbox{}\\[-6mm]*} 
  ML_prf {* Variable.dest_fixes @{context} *} 
  txt_raw {*\mbox{}\\[-6mm]*}
  fix x y  
  txt_raw {*\mbox{}\\[-6mm]*}
  ML_prf {* Variable.dest_fixes @{context} *} 
  txt_raw {*\mbox{}\\[-6mm]*}
oops


(*
ML{*Proof_Context.debug := true*}
ML{*Proof_Context.verbose := true*}
*)

(*
lemma "True"
proof -
  { -- "\<And>x. _"
    fix x
    have "B x" sorry
    thm this
  }

  thm this

  { -- "A \<Longrightarrow> _"
    assume A
    have B sorry
    thm this
  }
   
  thm this

  { -- "\<And>x. x = _ \<Longrightarrow> _"
    def x \<equiv> a
    have "B x" sorry
  }

  thm this

oops
*)

section {* Local Theories (TBD) *}

text {*
  In contrast to an ordinary theory, which simply consists of a type
  signature, as well as tables for constants, axioms and theorems, a local
  theory contains additional context information, such as locally fixed
  variables and local assumptions that may be used by the package. The type
  @{ML_type local_theory} is identical to the type of \emph{proof contexts}
  @{ML_type "Proof.context"}, although not every proof context constitutes a
  valid local theory.

  @{ML "Context.>> o Context.map_theory"}
  @{ML_ind "Local_Theory.declaration"}

   A similar command is \isacommand{local\_setup}, which expects a function
  of type @{ML_type "local_theory -> local_theory"}. Later on we will also
  use the commands \isacommand{method\_setup} for installing methods in the
  current theory and \isacommand{simproc\_setup} for adding new simprocs to
  the current simpset.
*}


section {* Morphisms (TBD) *}

text {*
  Morphisms are arbitrary transformations over terms, types, theorems and bindings.
  They can be constructed using the function @{ML_ind morphism in Morphism},
  which expects a record with functions of type

  \begin{isabelle}
  \begin{tabular}{rl}
  @{text "binding:"} & @{text "binding -> binding"}\\
  @{text "typ:"}     & @{text "typ -> typ"}\\
  @{text "term:"}    & @{text "term -> term"}\\
  @{text "fact:"}    & @{text "thm list -> thm list"}
  \end{tabular}
  \end{isabelle}

  The simplest morphism is the  @{ML_ind identity in Morphism}-morphism defined as
*}

ML{*val identity = Morphism.morphism {binding = [], typ = [], term = [], fact = []}*}
  
text {*
  Morphisms can be composed with the function @{ML_ind "$>" in Morphism}
*}

ML{*fun trm_phi (Free (x, T)) = Var ((x, 0), T) 
  | trm_phi (Abs (x, T, t)) = Abs (x, T, trm_phi t)
  | trm_phi (t $ s) = (trm_phi t) $ (trm_phi s)
  | trm_phi t = t*}

ML{*val phi = Morphism.term_morphism trm_phi*}

ML{*Morphism.term phi @{term "P x y"}*}

text {*
  @{ML_ind term_morphism in Morphism}

  @{ML_ind term in Morphism},
  @{ML_ind thm in Morphism}

  \begin{readmore}
  Morphisms are implemented in the file @{ML_file "Pure/morphism.ML"}.
  \end{readmore}
*}

section {* Misc (TBD) *}

ML {*Datatype.get_info @{theory} "List.list"*}

text {* 
FIXME: association lists:
@{ML_file "Pure/General/alist.ML"}

FIXME: calling the ML-compiler

*}

section {* What Is In an Isabelle Name? (TBD) *}

text {*
  On the ML-level of Isabelle, you often have to work with qualified names.
  These are strings with some additional information, such as positional
  information and qualifiers. Such qualified names can be generated with the
  antiquotation @{text "@{binding \<dots>}"}. For example

  @{ML_response [display,gray]
  "@{binding \"name\"}"
  "name"}

  An example where a qualified name is needed is the function 
  @{ML_ind define in Local_Theory}.  This function is used below to define 
  the constant @{term "TrueConj"} as the conjunction @{term "True \<and> True"}.
*}

local_setup %gray {* 
  Local_Theory.define ((@{binding "TrueConj"}, NoSyn), 
      (Attrib.empty_binding, @{term "True \<and> True"})) #> snd *}

text {* 
  Now querying the definition you obtain:

  \begin{isabelle}
  \isacommand{thm}~@{text "TrueConj_def"}\\
  @{text "> "}~@{thm TrueConj_def}
  \end{isabelle}

  \begin{readmore}
  The basic operations on bindings are implemented in 
  @{ML_file "Pure/General/binding.ML"}.
  \end{readmore}

  \footnote{\bf FIXME give a better example why bindings are important}
  \footnote{\bf FIXME give a pointer to \isacommand{local\_setup}; if not, then explain
  why @{ML snd} is needed.}
  \footnote{\bf FIXME: There should probably a separate section on binding, long-names
  and sign.}

*}


ML {* Sign.intern_type @{theory} "list" *}
ML {* Sign.intern_const @{theory} "prod_fun" *}

text {*
  \footnote{\bf FIXME: Explain the following better; maybe put in a separate
  section and link with the comment in the antiquotation section.}

  Occasionally you have to calculate what the ``base'' name of a given
  constant is. For this you can use the function @{ML_ind  Long_Name.base_name}. For example:

  @{ML_response [display,gray] "Long_Name.base_name \"List.list.Nil\"" "\"Nil\""}

  \begin{readmore}
  Functions about naming are implemented in @{ML_file "Pure/General/name_space.ML"};
  functions about signatures in @{ML_file "Pure/sign.ML"}.
  \end{readmore}
*}

text {* 
  @{ML_ind "Binding.name_of"} returns the string without markup

  @{ML_ind "Binding.conceal"} 
*}

section {* Concurrency (TBD) *}

text {*
  @{ML_ind prove_future in Goal}
  @{ML_ind future_result in Goal}
  @{ML_ind fork_pri in Future}
*}

section {* Parse and Print Translations (TBD) *}

section {* Summary *}

text {*
  TBD
*}

end