theory Advanced

imports Base FirstSteps

begin

(*<*)

setup{*

open_file_with_prelude 

  "Advanced_Code.thy"

*}

(*>*)

chapter {* Advanced Isabelle *}

text {*

  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 Isabelle programming. 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\label{sec:theories} (TBD) *}

text {*

  definitions, syntax declarations, axioms, proofs etc. If a definition is

  stated, 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}

  \begin{center}

  \begin{tikzpicture}

  \node[top color=white, bottom color=gray!30, draw=black!100, drop shadow] {A};

  \end{tikzpicture}

  \end{center}

  \footnote{\bf FIXME: list append in merge operations can cause 

  exponential blowups.}

*}

section {* Setups (TBD) *}

text {*

  @{ML Sign.declare_const}

  In the previous section we used \isacommand{setup} in order to make

  a theorem attribute known to Isabelle. 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 theory attribute has been stored.

  This is a fundamental principle in Isabelle. A similar situation occurs 

  for example with declaring constants. The function that declares a 

  constant on the ML-level is @{ML_ind  add_consts_i in Sign}. 

  If you write\footnote{Recall that ML-code  needs to be 

  enclosed in \isacommand{ML}~@{text "\<verbopen> \<dots> \<verbclose>"}.} 

*}  

ML{*Sign.add_consts_i [(@{binding "BAR"}, @{typ "nat"}, NoSyn)] @{theory} *}

text {*

  for declaring the constant @{text "BAR"} with type @{typ nat} and 

  run the code, then you indeed 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 [quotes] "BAR"}\\

  @{text "> \"BAR\" :: \"'a\""}

  \end{isabelle}

  you do 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 {* Sign.add_consts_i [(@{binding "BAR"}, @{typ "nat"}, NoSyn)] *}

text {* 

  Now 

  \begin{isabelle}

  \isacommand{term}~@{text [quotes] "BAR"}\\

  @{text "> \"BAR\" :: \"nat\""}

  \end{isabelle}

  returns a (black) constant with the type @{typ nat}.

  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 {* Contexts (TBD) *}

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"}

*}

(*

setup {*

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

*}

lemma "bar = (1::nat)"

  oops

setup {*   

  Sign.add_consts_i [("foo", @{typ "nat"},NoSyn)] 

 #> PureThy.add_defs false [((@{binding "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 {* 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 = I, typ = I, term = I, fact = I}*}

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 {* Managing Name Spaces (TBD) *}

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

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

text {* 

  @{ML_ind "Binding.str_of"} returns the string with markup;

  @{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