used a better implementation of \index in Latex; added more to the theorem section
theory General
imports Base FirstSteps
begin
chapter {* Isabelle -- The Good, the Bad and the Ugly *}
text {*
Isabelle is build around a few central ideas. One is the LCF-approach to
theorem proving where there is a small trusted core and everything else is
build on top of this trusted core. The central data structures involved
in this core are certified terms and types as well as theorems.
*}
section {* Terms and Types *}
text {*
In Isabelle there are certified terms (respectively types) and uncertified
terms. The latter are called just terms. One way to construct them 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>"}
will show the term @{term "(a::nat) + b = c"}, but printed using the internal
representation corresponding to the datatype @{ML_type "term"} defined as follows:
*}
ML_val %linenosgray{*datatype term =
Const of string * typ
| Free of string * typ
| Var of indexname * typ
| Bound of int
| Abs of string * typ * term
| $ of term * term *}
text {*
This datatype implements lambda-terms typed in Church-style.
As can be seen in Line 5, terms use the usual de Bruijn index mechanism
for representing bound variables. For
example in
@{ML_response_fake [display, gray]
"@{term \"\<lambda>x y. x y\"}"
"Abs (\"x\", \"'a \<Rightarrow> 'b\", Abs (\"y\", \"'a\", Bound 1 $ Bound 0))"}
the indices refer to the number of Abstractions (@{ML Abs}) that we need to
skip until we hit the @{ML Abs} that binds the corresponding
variable. Constructing a term with dangling de Bruijn indices is possible,
but will be flagged as ill-formed when you try to typecheck or certify it
(see Section~\ref{sec:typechecking}). Note that the names of bound variables
are kept at abstractions for printing purposes, and so should be treated
only as ``comments''. Application in Isabelle is realised with the
term-constructor @{ML $}.
Isabelle makes a distinction between \emph{free} variables (term-constructor
@{ML Free} and written on the user level in blue colour) and
\emph{schematic} variables (term-constructor @{ML Var} and written with a
leading question mark). Consider the following two examples
@{ML_response_fake [display, gray]
"let
val v1 = Var ((\"x\", 3), @{typ bool})
val v2 = Var ((\"x1\", 3), @{typ bool})
val v3 = Free (\"x\", @{typ bool})
in
string_of_terms @{context} [v1, v2, v3]
|> tracing
end"
"?x3, ?x1.3, x"}
When constructing terms, you are usually concerned with free variables (as
mentioned earlier, you cannot construct schematic variables using the
antiquotation @{text "@{term \<dots>}"}). If you deal with theorems, you have to,
however, observe the distinction. The reason is that only schematic
varaibles can be instantiated with terms when a theorem is applied. A
similar distinction between free and schematic variables holds for types
(see below).
\begin{readmore}
Terms and types are described in detail in \isccite{sec:terms}. Their
definition and many useful operations are implemented in @{ML_file "Pure/term.ML"}.
For constructing terms involving HOL constants, many helper functions are defined
in @{ML_file "HOL/Tools/hologic.ML"}.
\end{readmore}
Constructing terms via antiquotations has the advantage that only typable
terms can be constructed. For example
@{ML_response_fake_both [display,gray]
"@{term \"x x\"}"
"Type unification failed: Occurs check!"}
raises a typing error, while it perfectly ok to construct the term
@{ML_response_fake [display,gray]
"let
val omega = Free (\"x\", @{typ nat}) $ Free (\"x\", @{typ nat})
in
tracing (string_of_term @{context} omega)
end"
"x x"}
with the raw ML-constructors.
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 printing depth to a higher
value:
@{ML [display,gray] "print_depth 50"}
\end{exercise}
The antiquotation @{text "@{prop \<dots>}"} constructs terms by inserting the
usually 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 a 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).
The purpose of the @{text "Trueprop"}-coercion is to embed formulae of
an object logic, for example HOL, into the meta-logic of Isabelle. It
is needed whenever a term is constructed that will be proved as a theorem.
As already seen above, types can be constructed using the antiquotation
@{text "@{typ \<dots>}"}. For example:
@{ML_response_fake [display,gray] "@{typ \"bool \<Rightarrow> nat\"}" "bool \<Rightarrow> nat"}
The corresponding datatype is
*}
ML_val{*datatype typ =
Type of string * typ list
| TFree of string * sort
| TVar of indexname * sort *}
text {*
Like with terms, there is the distinction between free type
variables (term-constructor @{ML "TFree"} and schematic
type variables (term-constructor @{ML "TVar"}). A type constant,
like @{typ "int"} or @{typ bool}, are types with an empty list
of argument types. However, it is a bit difficult to show an
example, because Isabelle always pretty-prints types (unlike terms).
Here is a contrived example:
@{ML_response [display, gray]
"if Type (\"bool\", []) = @{typ \"bool\"} then true else false"
"true"}
\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\label{sec:terms_types_manually} *}
text {*
While antiquotations are very convenient for constructing terms, they can
only construct fixed terms (remember they are ``linked'' at compile-time).
However, you often need to construct terms manually. For example, a
function that returns the implication @{text "\<And>(x::nat). P x \<Longrightarrow> Q x"} taking
@{term P} and @{term Q} as arguments can only be written as:
*}
ML{*fun make_imp P Q =
let
val x = Free ("x", @{typ nat})
in
Logic.all x (Logic.mk_implies (P $ x, Q $ x))
end *}
text {*
The reason is that you cannot pass the arguments @{term P} and @{term Q}
into an antiquotation.\footnote{At least not at the moment.} For example
the following does \emph{not} work.
*}
ML{*fun make_wrong_imp P Q = @{prop "\<And>(x::nat). P x \<Longrightarrow> Q x"} *}
text {*
To see this, apply @{text "@{term S}"} and @{text "@{term T}"}
to both functions. With @{ML make_imp} you obtain the intended term involving
the given arguments
@{ML_response [display,gray] "make_imp @{term S} @{term T}"
"Const \<dots> $
Abs (\"x\", Type (\"nat\",[]),
Const \<dots> $ (Free (\"S\",\<dots>) $ \<dots>) $ (Free (\"T\",\<dots>) $ \<dots>))"}
whereas with @{ML make_wrong_imp} you obtain a term involving the @{term "P"}
and @{text "Q"} from the antiquotation.
@{ML_response [display,gray] "make_wrong_imp @{term S} @{term T}"
"Const \<dots> $
Abs (\"x\", \<dots>,
Const \<dots> $ (Const \<dots> $ (Free (\"P\",\<dots>) $ \<dots>)) $
(Const \<dots> $ (Free (\"Q\",\<dots>) $ \<dots>)))"}
There are a number of handy functions that are frequently used for
constructing terms. One is the function @{ML_ind list_comb}, which takes a term
and a list of terms as arguments, and produces as output the term
list applied to the term. For example
@{ML_response_fake [display,gray]
"let
val trm = @{term \"P::nat\"}
val args = [@{term \"True\"}, @{term \"False\"}]
in
list_comb (trm, args)
end"
"Free (\"P\", \"nat\") $ Const (\"True\", \"bool\") $ Const (\"False\", \"bool\")"}
Another handy function is @{ML_ind lambda}, which abstracts a variable
in a term. For example
@{ML_response_fake [display,gray]
"let
val x_nat = @{term \"x::nat\"}
val trm = @{term \"(P::nat \<Rightarrow> bool) x\"}
in
lambda x_nat trm
end"
"Abs (\"x\", \"nat\", Free (\"P\", \"bool \<Rightarrow> bool\") $ Bound 0)"}
In this example, @{ML lambda} produces a de Bruijn index (i.e.~@{ML "Bound 0"}),
and an abstraction. It also records the type of the abstracted
variable and for printing purposes also its name. Note that because of the
typing annotation on @{text "P"}, the variable @{text "x"} in @{text "P x"}
is of the same type as the abstracted variable. If it is of different type,
as in
@{ML_response_fake [display,gray]
"let
val x_int = @{term \"x::int\"}
val trm = @{term \"(P::nat \<Rightarrow> bool) x\"}
in
lambda x_int trm
end"
"Abs (\"x\", \"int\", Free (\"P\", \"nat \<Rightarrow> bool\") $ Free (\"x\", \"nat\"))"}
then the variable @{text "Free (\"x\", \"int\")"} is \emph{not} abstracted.
This is a fundamental principle
of Church-style typing, where variables with the same name still differ, if they
have different type.
There is also the function @{ML_ind subst_free} with which terms can be
replaced by other terms. For example below, we will replace in @{term
"(f::nat \<Rightarrow> nat \<Rightarrow> nat) 0 x"} the subterm @{term "(f::nat \<Rightarrow> nat \<Rightarrow> nat) 0"} by
@{term y}, and @{term x} by @{term True}.
@{ML_response_fake [display,gray]
"let
val sub1 = (@{term \"(f::nat \<Rightarrow> nat \<Rightarrow> nat) 0\"}, @{term \"y::nat \<Rightarrow> nat\"})
val sub2 = (@{term \"x::nat\"}, @{term \"True\"})
val trm = @{term \"((f::nat \<Rightarrow> nat \<Rightarrow> nat) 0) x\"}
in
subst_free [sub1, sub2] trm
end"
"Free (\"y\", \"nat \<Rightarrow> nat\") $ Const (\"True\", \"bool\")"}
As can be seen, @{ML subst_free} does not take typability into account.
However it takes alpha-equivalence into account:
@{ML_response_fake [display, gray]
"let
val sub = (@{term \"(\<lambda>y::nat. y)\"}, @{term \"x::nat\"})
val trm = @{term \"(\<lambda>x::nat. x)\"}
in
subst_free [sub] trm
end"
"Free (\"x\", \"nat\")"}
Similarly the function @{ML_ind subst_bounds}, replaces lose bound
variables with terms. To see how this function works, let us implement a
function that strips off the outermost quantifiers in a term.
*}
ML{*fun strip_alls (Const ("All", _) $ Abs (n, T, t)) =
strip_alls t |>> cons (Free (n, T))
| strip_alls t = ([], t) *}
text {*
The function returns a pair consisting of the stripped off variables and
the body of the universal quantifications. For example
@{ML_response_fake [display, gray]
"strip_alls @{term \"\<forall>x y. x = (y::bool)\"}"
"([Free (\"x\", \"bool\"), Free (\"y\", \"bool\")],
Const (\"op =\", \<dots>) $ Bound 1 $ Bound 0)"}
After calling @{ML strip_alls}, you obtain a term with lose bound variables. With
the function @{ML subst_bounds}, you can replace these lose @{ML_ind
Bound}s with the stripped off variables.
@{ML_response_fake [display, gray, linenos]
"let
val (vrs, trm) = strip_alls @{term \"\<forall>x y. x = (y::bool)\"}
in
subst_bounds (rev vrs, trm)
|> string_of_term @{context}
|> tracing
end"
"x = y"}
Note that in Line 4 we had to reverse the list of variables that @{ML strip_alls}
returned. The reason is that the head of the list the function @{ML subst_bounds}
takes is the replacement for @{ML "Bound 0"}, the next element for @{ML "Bound 1"}
and so on.
There are many convenient functions that construct specific HOL-terms. For
example @{ML_ind mk_eq in HOLogic} constructs an equality out of two terms.
The types needed in this equality are calculated from the type of the
arguments. For example
@{ML_response_fake [gray,display]
"let
val eq = HOLogic.mk_eq (@{term \"True\"}, @{term \"False\"})
in
string_of_term @{context} eq
|> tracing
end"
"True = False"}
*}
text {*
\begin{readmore}
There are many functions in @{ML_file "Pure/term.ML"}, @{ML_file
"Pure/logic.ML"} and @{ML_file "HOL/Tools/hologic.ML"} that make such manual
constructions of terms and types easier.
\end{readmore}
When constructing terms manually, there are a few subtle issues with
constants. They usually crop up when pattern matching terms or types, or
when constructing them. While it is perfectly ok to write the function
@{text is_true} as follows
*}
ML{*fun is_true @{term True} = true
| is_true _ = false*}
text {*
this does not work for picking out @{text "\<forall>"}-quantified terms. Because
the function
*}
ML{*fun is_all (@{term All} $ _) = true
| is_all _ = false*}
text {*
will not correctly match the formula @{prop[source] "\<forall>x::nat. P x"}:
@{ML_response [display,gray] "is_all @{term \"\<forall>x::nat. P x\"}" "false"}
The problem is that the @{text "@term"}-antiquotation in the pattern
fixes the type of the constant @{term "All"} to be @{typ "('a \<Rightarrow> bool) \<Rightarrow> bool"} for
an arbitrary, but fixed type @{typ "'a"}. A properly working alternative
for this function is
*}
ML{*fun is_all (Const ("All", _) $ _) = true
| is_all _ = false*}
text {*
because now
@{ML_response [display,gray] "is_all @{term \"\<forall>x::nat. P x\"}" "true"}
matches correctly (the first wildcard in the pattern matches any type and the
second any term).
However there is still a problem: consider the similar function that
attempts to pick out @{text "Nil"}-terms:
*}
ML{*fun is_nil (Const ("Nil", _)) = true
| is_nil _ = false *}
text {*
Unfortunately, also this function does \emph{not} work as expected, since
@{ML_response [display,gray] "is_nil @{term \"Nil\"}" "false"}
The problem is that on the ML-level the name of a constant is more
subtle than you might expect. The function @{ML is_all} worked correctly,
because @{term "All"} is such a fundamental constant, which can be referenced
by @{ML "Const (\"All\", some_type)" for some_type}. However, if you look at
@{ML_response [display,gray] "@{term \"Nil\"}" "Const (\"List.list.Nil\", \<dots>)"}
the name of the constant @{text "Nil"} depends on the theory in which the
term constructor is defined (@{text "List"}) and also in which datatype
(@{text "list"}). Even worse, some constants have a name involving
type-classes. Consider for example the constants for @{term "zero"} and
\mbox{@{text "(op *)"}}:
@{ML_response [display,gray] "(@{term \"0::nat\"}, @{term \"(op *)\"})"
"(Const (\"HOL.zero_class.zero\", \<dots>),
Const (\"HOL.times_class.times\", \<dots>))"}
While you could use the complete name, for example
@{ML "Const (\"List.list.Nil\", some_type)" for some_type}, for referring to or
matching against @{text "Nil"}, this would make the code rather brittle.
The reason is that the theory and the name of the datatype can easily change.
To make the code more robust, it is better to use the antiquotation
@{text "@{const_name \<dots>}"}. With this antiquotation you can harness the
variable parts of the constant's name. Therefore a function for
matching against constants that have a polymorphic type should
be written as follows.
*}
ML{*fun is_nil_or_all (Const (@{const_name "Nil"}, _)) = true
| is_nil_or_all (Const (@{const_name "All"}, _) $ _) = true
| is_nil_or_all _ = false *}
text {*
The antiquotation for properly referencing type constants is @{text "@{type_name \<dots>}"}.
For example
@{ML_response [display,gray]
"@{type_name \"list\"}" "\"List.list\""}
\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 "Sign.extern_const"} or
@{ML_ind Long_Name.base_name}. For example:
@{ML_response [display,gray] "Sign.extern_const @{theory} \"List.list.Nil\"" "\"Nil\""}
The difference between both functions is that @{ML extern_const in Sign} returns
the smallest name that is still unique, whereas @{ML base_name in Long_Name} always
strips off all qualifiers.
\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}
Although types of terms can often be inferred, there are many
situations where you need to construct types manually, especially
when defining constants. For example the function returning a function
type is as follows:
*}
ML{*fun make_fun_type ty1 ty2 = Type ("fun", [ty1, ty2]) *}
text {* This can be equally written with the combinator @{ML_ind "-->"} as: *}
ML{*fun make_fun_type ty1 ty2 = ty1 --> ty2 *}
text {*
If you want to construct a function type with more than one argument
type, then you can use @{ML_ind "--->"}.
*}
ML{*fun make_fun_types tys ty = tys ---> ty *}
text {*
A handy function for manipulating terms is @{ML_ind map_types}: it takes a
function and applies it to every type in a term. You can, for example,
change every @{typ nat} in a term into an @{typ int} using the function:
*}
ML{*fun nat_to_int ty =
(case ty of
@{typ nat} => @{typ int}
| Type (s, tys) => Type (s, map nat_to_int tys)
| _ => ty)*}
text {*
Here is an example:
@{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\")"}
If you want to obtain the list of free type-variables of a term, you
can use the function @{ML_ind add_tfrees in Term}
(similarly @{ML_ind add_tvars in Term} for the schematic type-variables).
One would expect that such functions
take a term as input and return a list of types. But their type is actually
@{text[display] "Term.term -> (string * Term.sort) list -> (string * Term.sort) list"}
that is they take, besides a term, also a list of type-variables as input.
So in order to obtain the list of type-variables of a term you have to
call them as follows
@{ML_response [gray,display]
"Term.add_tfrees @{term \"(a, b)\"} []"
"[(\"'b\", [\"HOL.type\"]), (\"'a\", [\"HOL.type\"])]"}
The reason for this definition is that @{ML add_tfrees in Term} can
be easily folded over a list of terms. Similarly for all functions
named @{text "add_*"} in @{ML_file "Pure/term.ML"}.
\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 "n"} might be one)
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 produces the
number representing their sum.
\end{exercise}
\begin{exercise}\label{ex:debruijn}
Implement the function, which we below name deBruijn\footnote{According to
personal communication of de Bruijn to Dyckhoff.}, that depends on a natural
number n$>$0 and constructs terms of the form:
\begin{center}
\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
{\it rhs n} & $\dn$ & {\large$\bigwedge$}{\it i=1\ldots n. P\,i}\\
{\it lhs n} & $\dn$ & {\large$\bigwedge$}{\it i=1\ldots n. P\,i = P (i + 1 mod n)}
$\longrightarrow$ {\it rhs n}\\
{\it deBruijn n} & $\dn$ & {\it lhs n} $\longrightarrow$ {\it rhs n}\\
\end{tabular}
\end{center}
This function returns for n=3 the term
\begin{center}
\begin{tabular}{l}
(P 1 = P 2 $\longrightarrow$ P 1 $\wedge$ P 2 $\wedge$ P 3) $\wedge$\\
(P 2 = P 3 $\longrightarrow$ P 1 $\wedge$ P 2 $\wedge$ P 3) $\wedge$\\
(P 3 = P 1 $\longrightarrow$ P 1 $\wedge$ P 2 $\wedge$ P 3) $\longrightarrow$ P 1 $\wedge$ P 2 $\wedge$ P 3
\end{tabular}
\end{center}
Make sure you use the functions defined in @{ML_file "HOL/Tools/hologic.ML"}
for constructing the terms for the logical connectives.
\end{exercise}
*}
section {* Type-Checking\label{sec:typechecking} *}
text {*
Remember Isabelle follows the Church-style typing for terms, i.e., a term contains
enough typing information (constants, free variables and abstractions all have typing
information) so that it is always clear what the type of a term is.
Given a well-typed term, the function @{ML_ind type_of} returns the
type of a term. Consider for example:
@{ML_response [display,gray]
"type_of (@{term \"f::nat \<Rightarrow> bool\"} $ @{term \"x::nat\"})" "bool"}
To calculate the type, this function traverses the whole term and will
detect any typing inconsistency. For example changing the type of the variable
@{term "x"} from @{typ "nat"} to @{typ "int"} will result in the error message:
@{ML_response_fake [display,gray]
"type_of (@{term \"f::nat \<Rightarrow> bool\"} $ @{term \"x::int\"})"
"*** Exception- TYPE (\"type_of: type mismatch in application\" \<dots>"}
Since the complete traversal might sometimes be too costly and
not necessary, there is the function @{ML_ind fastype_of}, which
also returns the type of a term.
@{ML_response [display,gray]
"fastype_of (@{term \"f::nat \<Rightarrow> bool\"} $ @{term \"x::nat\"})" "bool"}
However, efficiency is gained on the expense of skipping some tests. You
can see this in the following example
@{ML_response [display,gray]
"fastype_of (@{term \"f::nat \<Rightarrow> bool\"} $ @{term \"x::int\"})" "bool"}
where no error is detected.
Sometimes it is a bit inconvenient to construct a term with
complete typing annotations, especially in cases where the typing
information is redundant. A short-cut is to use the ``place-holder''
type @{ML_ind dummyT} and then let type-inference figure out the
complete type. An example is as follows:
@{ML_response_fake [display,gray]
"let
val c = Const (@{const_name \"plus\"}, dummyT)
val o = @{term \"1::nat\"}
val v = Free (\"x\", dummyT)
in
Syntax.check_term @{context} (c $ o $ v)
end"
"Const (\"HOL.plus_class.plus\", \"nat \<Rightarrow> nat \<Rightarrow> nat\") $
Const (\"HOL.one_class.one\", \"nat\") $ Free (\"x\", \"nat\")"}
Instead of giving explicitly the type for the constant @{text "plus"} and the free
variable @{text "x"}, type-inference fills in the missing information.
\begin{readmore}
See @{ML_file "Pure/Syntax/syntax.ML"} where more functions about reading,
checking and pretty-printing of terms are defined. Functions related to
type-inference are implemented in @{ML_file "Pure/type.ML"} and
@{ML_file "Pure/type_infer.ML"}.
\end{readmore}
\footnote{\bf FIXME: say something about sorts.}
\footnote{\bf FIXME: give a ``readmore''.}
\begin{exercise}
Check that the function defined in Exercise~\ref{fun:revsum} returns a
result that type-checks. See what happens to the solutions of this
exercise given in Appendix \ref{ch:solutions} when they receive an
ill-typed term as input.
\end{exercise}
*}
section {* Certified Terms and Certified Types *}
text {*
You can freely construct and manipulate @{ML_type "term"}s and @{ML_type
typ}es, since they are just arbitrary unchecked trees. However, you
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_ind 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''.
Certification is always relative to a theory context. For example you can
write:
@{ML_response_fake [display,gray]
"cterm_of @{theory} @{term \"(a::nat) + b = c\"}"
"a + b = c"}
This can also be written 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"}
In Isabelle not just terms need to be certified, but also types. For example,
you obtain the certified type for the Isabelle type @{typ "nat \<Rightarrow> bool"} on
the ML-level as follows:
@{ML_response_fake [display,gray]
"ctyp_of @{theory} (@{typ nat} --> @{typ bool})"
"nat \<Rightarrow> bool"}
or with the antiquotation:
@{ML_response_fake [display,gray]
"@{ctyp \"nat \<Rightarrow> bool\"}"
"nat \<Rightarrow> bool"}
Since certified terms are, unlike terms, abstract objects, we cannot
pattern-match against them. However, we can construct them. For example
the function @{ML_ind capply in Thm} produces a certified application.
@{ML_response_fake [display,gray]
"Thm.capply @{cterm \"P::nat \<Rightarrow> bool\"} @{cterm \"3::nat\"}"
"P 3"}
Similarly @{ML_ind list_comb in Drule} applies a list of @{ML_type cterm}s.
@{ML_response_fake [display,gray]
"let
val chead = @{cterm \"P::unit \<Rightarrow> nat \<Rightarrow> bool\"}
val cargs = [@{cterm \"()\"}, @{cterm \"3::nat\"}]
in
Drule.list_comb (chead, cargs)
end"
"P () 3"}
\begin{readmore}
For functions related to @{ML_type cterm}s and @{ML_type ctyp}s see
the files @{ML_file "Pure/thm.ML"}, @{ML_file "Pure/more_thm.ML"} and
@{ML_file "Pure/drule.ML"}.
\end{readmore}
*}
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. This follows the tradition of the LCF approach
\cite{GordonMilnerWadsworth79}.
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:
*}
ML{*val my_thm =
let
val assm1 = @{cprop "\<And>(x::nat). P x \<Longrightarrow> Q x"}
val assm2 = @{cprop "(P::nat \<Rightarrow> bool) t"}
val Pt_implies_Qt =
assume assm1
|> forall_elim @{cterm "t::nat"}
val Qt = implies_elim Pt_implies_Qt (assume assm2)
in
Qt
|> implies_intr assm2
|> implies_intr assm1
end*}
text {*
If we print out the value of @{ML my_thm} then we see only the
final statement of the theorem.
@{ML_response_fake [display, gray]
"string_of_thm @{context} my_thm |> tracing"
"\<lbrakk>\<And>x. P x \<Longrightarrow> Q x; P t\<rbrakk> \<Longrightarrow> Q t"}
However, internally the code-snippet constructs the following
proof.
\[
\infer[(@{text "\<Longrightarrow>"}$-$intro)]{\vdash @{prop "(\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> P t \<Longrightarrow> Q t"}}
{\infer[(@{text "\<Longrightarrow>"}$-$intro)]{@{prop "\<And>x. P x \<Longrightarrow> Q x"} \vdash @{prop "P t \<Longrightarrow> Q t"}}
{\infer[(@{text "\<Longrightarrow>"}$-$elim)]{@{prop "\<And>x. P x \<Longrightarrow> Q x"}, @{prop "P t"} \vdash @{prop "Q t"}}
{\infer[(@{text "\<And>"}$-$elim)]{@{prop "\<And>x. P x \<Longrightarrow> Q x"} \vdash @{prop "P t \<Longrightarrow> Q t"}}
{\infer[(assume)]{@{prop "\<And>x. P x \<Longrightarrow> Q x"} \vdash @{prop "\<And>x. P x \<Longrightarrow> Q x"}}{}}
&
\infer[(assume)]{@{prop "P t"} \vdash @{prop "P t"}}{}
}
}
}
\]
While we obtained a theorem as result, this theorem is not
yet stored in Isabelle's theorem database. Consequently, it cannot be
referenced later on. One way to store it in the theorem database is
by using the function @{ML_ind note in LocalTheory}.
*}
local_setup %gray {*
LocalTheory.note Thm.theoremK
((@{binding "my_thm"}, []), [my_thm]) #> snd *}
text {*
The first argument @{ML_ind theoremK in Thm} is a kind indicator, which
classifies the theorem. For a theorem arising from a definition we should
state @{ML_ind definitionK in Thm}, instead. The second argument is the
name under which we stroe the theorem or theorems. The third contains is
a list of (theorem) attributes. Above it is empty, but if we want to store
the therem and at the same time add it to the simpset we have to declare.
*}
local_setup %gray {*
LocalTheory.note Thm.theoremK
((@{binding "my_thm_simp"}, [Attrib.internal (K Simplifier.simp_add)]), [my_thm]) #> snd *}
text {*
Now @{thm [source] my_thm} can be referenced with the \isacommand{thm}-command
on the user-level of Isabelle
\begin{isabelle}
\isacommand{thm}~@{text "my_thm"}\isanewline
@{text ">"}~@{prop "\<lbrakk>\<And>x. P x \<Longrightarrow> Q x; P t\<rbrakk> \<Longrightarrow> Q t"}
\end{isabelle}
or with the @{text "@{thm \<dots>}"}-antiquotation on the ML-level. Note that the
theorem does not have any meta-variables that would be present if we proved
this theorem on the user-level. As we shall see later on, we have to provide
this information explicitly.
There is a multitude of functions that manage or manipulate theorems. For example
we can test theorems for (alpha) equality. Suppose you proved the following three
facts.
*}
lemma
shows thm1: "\<forall>x. P x"
and thm2: "\<forall>y. P y"
and thm3: "\<forall>y. Q y" sorry
text {*
Testing for equality using the function @{ML_ind eq_thm in Thm} produces:
@{ML_response [display,gray]
"(Thm.eq_thm (@{thm thm1}, @{thm thm2}),
Thm.eq_thm (@{thm thm2}, @{thm thm3}))"
"(true, false)"}
Many functions destruct a theorem into a @{ML_type cterm}. For example
@{ML_ind lhs_of in Thm} and @{ML_ind rhs_of in Thm} return the left and
righ-hand side, respectively, of a meta-equality.
@{ML_response_fake [display,gray]
"let
val eq = @{thm True_def}
in
(Thm.lhs_of eq, Thm.rhs_of eq)
|> pairself (tracing o string_of_cterm @{context})
end"
"True
(\<lambda>x. x) = (\<lambda>x. x)"}
Other function produce immediately a term that can be pattern-matched.
Suppose the following theorem.
*}
lemma foo_test:
shows "A \<Longrightarrow> B \<Longrightarrow> C" sorry
text {*
@{ML_response_fake [display,gray]
"let
val thm = @{thm foo_test}
in
(Thm.prems_of thm, [Thm.concl_of thm])
|> pairself (tracing o string_of_terms @{context})
end"
"?A, ?B
?C"}
\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"}, @{ML_file "Pure/more_thm.ML"} and @{ML_file "Pure/drule.ML"}.
\end{readmore}
(FIXME: handy functions working on theorems, like @{ML_ind rulify in ObjectLogic} and so on)
(FIXME: @{ML_ind prove in Goal})
(FIXME: how to add case-names to goal states - maybe in the
next section)
(FIXME: example for how to add theorem styles)
*}
ML {*
fun strip_assums_all (params, Const("all",_) $ Abs(a, T, t)) =
strip_assums_all ((a, T)::params, t)
| strip_assums_all (params, B) = (params, B)
fun style_parm_premise i ctxt t =
let val prems = Logic.strip_imp_prems t in
if i <= length prems
then let val (params,t) = strip_assums_all([], nth prems (i - 1))
in subst_bounds(map Free params, t) end
else error ("Not enough premises for prem" ^ string_of_int i ^
" in propositon: " ^ string_of_term ctxt t)
end;
*}
ML {*
strip_assums_all ([], @{term "\<And>x y. A x y"})
*}
setup %gray {*
TermStyle.add_style "no_all_prem1" (style_parm_premise 1) #>
TermStyle.add_style "no_all_prem2" (style_parm_premise 2)
*}
lemma
shows "A \<Longrightarrow> B"
and "C \<Longrightarrow> D"
oops
section {* Setups (TBD) *}
text {*
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 {* Theorem Attributes\label{sec:attributes} *}
text {*
Theorem attributes are @{text "[symmetric]"}, @{text "[THEN \<dots>]"}, @{text
"[simp]"} and so on. Such attributes are \emph{neither} tags \emph{nor} flags
annotated to theorems, but functions that do further processing once a
theorem is proved. In particular, it is not possible to find out
what are all theorems that have a given attribute in common, unless of course
the function behind the attribute stores the theorems in a retrievable
data structure.
If you want to print out all currently known attributes a theorem can have,
you can use the Isabelle command
\begin{isabelle}
\isacommand{print\_attributes}\\
@{text "> COMP: direct composition with rules (no lifting)"}\\
@{text "> HOL.dest: declaration of Classical destruction rule"}\\
@{text "> HOL.elim: declaration of Classical elimination rule"}\\
@{text "> \<dots>"}
\end{isabelle}
The theorem attributes fall roughly into two categories: the first category manipulates
the proved theorem (for example @{text "[symmetric]"} and @{text "[THEN \<dots>]"}), and the second
stores the proved theorem somewhere as data (for example @{text "[simp]"}, which adds
the theorem to the current simpset).
To explain how to write your own attribute, let us start with an extremely simple
version of the attribute @{text "[symmetric]"}. The purpose of this attribute is
to produce the ``symmetric'' version of an equation. The main function behind
this attribute is
*}
ML{*val my_symmetric = Thm.rule_attribute (fn _ => fn thm => thm RS @{thm sym})*}
text {*
where the function @{ML_ind rule_attribute in Thm} expects a function taking a
context (which we ignore in the code above) and a theorem (@{text thm}), and
returns another theorem (namely @{text thm} resolved with the theorem
@{thm [source] sym}: @{thm sym[no_vars]}; the function @{ML_ind "RS"}
is explained in Section~\ref{sec:simpletacs}). The function
@{ML rule_attribute in Thm} then returns an attribute.
Before we can use the attribute, we need to set it up. This can be done
using the Isabelle command \isacommand{attribute\_setup} as follows:
*}
attribute_setup %gray my_sym = {* Scan.succeed my_symmetric *}
"applying the sym rule"
text {*
Inside the @{text "\<verbopen> \<dots> \<verbclose>"}, we have to specify a parser
for the theorem attribute. Since the attribute does not expect any further
arguments (unlike @{text "[THEN \<dots>]"}, for example), we use the parser @{ML
Scan.succeed}. Later on we will also consider attributes taking further
arguments. An example for the attribute @{text "[my_sym]"} is the proof
*}
lemma test[my_sym]: "2 = Suc (Suc 0)" by simp
text {*
which stores the theorem @{thm test} under the name @{thm [source] test}. You
can see this, if you query the lemma:
\begin{isabelle}
\isacommand{thm}~@{text "test"}\\
@{text "> "}~@{thm test}
\end{isabelle}
We can also use the attribute when referring to this theorem:
\begin{isabelle}
\isacommand{thm}~@{text "test[my_sym]"}\\
@{text "> "}~@{thm test[my_sym]}
\end{isabelle}
An alternative for setting up an attribute is the function @{ML_ind setup in Attrib}.
So instead of using \isacommand{attribute\_setup}, you can also set up the
attribute as follows:
*}
ML{*Attrib.setup @{binding "my_sym"} (Scan.succeed my_symmetric)
"applying the sym rule" *}
text {*
This gives a function from @{ML_type "Context.theory -> Context.theory"}, which
can be used for example with \isacommand{setup}.
As an example of a slightly more complicated theorem attribute, we implement
our own version of @{text "[THEN \<dots>]"}. This attribute will take a list of theorems
as argument and resolve the proved theorem with this list (one theorem
after another). The code for this attribute is
*}
ML{*fun MY_THEN thms =
Thm.rule_attribute (fn _ => fn thm => foldl ((op RS) o swap) thm thms)*}
text {*
where @{ML swap} swaps the components of a pair. The setup of this theorem
attribute uses the parser @{ML thms in Attrib}, which parses a list of
theorems.
*}
attribute_setup %gray MY_THEN = {* Attrib.thms >> MY_THEN *}
"resolving the list of theorems with the proved theorem"
text {*
You can, for example, use this theorem attribute to turn an equation into a
meta-equation:
\begin{isabelle}
\isacommand{thm}~@{text "test[MY_THEN eq_reflection]"}\\
@{text "> "}~@{thm test[MY_THEN eq_reflection]}
\end{isabelle}
If you need the symmetric version as a meta-equation, you can write
\begin{isabelle}
\isacommand{thm}~@{text "test[MY_THEN sym eq_reflection]"}\\
@{text "> "}~@{thm test[MY_THEN sym eq_reflection]}
\end{isabelle}
It is also possible to combine different theorem attributes, as in:
\begin{isabelle}
\isacommand{thm}~@{text "test[my_sym, MY_THEN eq_reflection]"}\\
@{text "> "}~@{thm test[my_sym, MY_THEN eq_reflection]}
\end{isabelle}
However, here also a weakness of the concept
of theorem attributes shows through: since theorem attributes can be
arbitrary functions, they do not commute in general. If you try
\begin{isabelle}
\isacommand{thm}~@{text "test[MY_THEN eq_reflection, my_sym]"}\\
@{text "> "}~@{text "exception THM 1 raised: RSN: no unifiers"}
\end{isabelle}
you get an exception indicating that the theorem @{thm [source] sym}
does not resolve with meta-equations.
The purpose of @{ML_ind rule_attribute in Thm} is to directly manipulate
theorems. Another usage of theorem attributes is to add and delete theorems
from stored data. For example the theorem attribute @{text "[simp]"} adds
or deletes a theorem from the current simpset. For these applications, you
can use @{ML_ind declaration_attribute in Thm}. To illustrate this function,
let us introduce a theorem list.
*}
ML{*structure MyThms = Named_Thms
(val name = "attr_thms"
val description = "Theorems for an Attribute") *}
text {*
We are going to modify this list by adding and deleting theorems.
For this we use the two functions @{ML MyThms.add_thm} and
@{ML MyThms.del_thm}. You can turn them into attributes
with the code
*}
ML{*val my_add = Thm.declaration_attribute MyThms.add_thm
val my_del = Thm.declaration_attribute MyThms.del_thm *}
text {*
and set up the attributes as follows
*}
attribute_setup %gray my_thms = {* Attrib.add_del my_add my_del *}
"maintaining a list of my_thms"
text {*
The parser @{ML_ind add_del in Attrib} is a predefined parser for
adding and deleting lemmas. Now if you prove the next lemma
and attach to it the attribute @{text "[my_thms]"}
*}
lemma trueI_2[my_thms]: "True" by simp
text {*
then you can see it is added to the initially empty list.
@{ML_response_fake [display,gray]
"MyThms.get @{context}"
"[\"True\"]"}
You can also add theorems using the command \isacommand{declare}.
*}
declare test[my_thms] trueI_2[my_thms add]
text {*
With this attribute, the @{text "add"} operation is the default and does
not need to be explicitly given. These three declarations will cause the
theorem list to be updated as:
@{ML_response_fake [display,gray]
"MyThms.get @{context}"
"[\"True\", \"Suc (Suc 0) = 2\"]"}
The theorem @{thm [source] trueI_2} only appears once, since the
function @{ML_ind add_thm in Thm} tests for duplicates, before extending
the list. Deletion from the list works as follows:
*}
declare test[my_thms del]
text {* After this, the theorem list is again:
@{ML_response_fake [display,gray]
"MyThms.get @{context}"
"[\"True\"]"}
We used in this example two functions declared as @{ML_ind
declaration_attribute in Thm}, but there can be any number of them. We just
have to change the parser for reading the arguments accordingly.
\footnote{\bf FIXME What are: @{text "theory_attributes"}, @{text "proof_attributes"}?}
\begin{readmore}
FIXME: @{ML_file "Pure/more_thm.ML"}; parsers for attributes is in
@{ML_file "Pure/Isar/attrib.ML"}...also explained in the chapter about
parsing.
\end{readmore}
*}
section {* Theories, Contexts and Local Theories (TBD) *}
text {*
There are theories, proof contexts and local theories (in this order, if you
want to order them).
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.
*}
section {* Storing Theorems\label{sec:storing} (TBD) *}
text {* @{ML_ind add_thms_dynamic in PureThy} *}
local_setup %gray {*
LocalTheory.note Thm.theoremK
((@{binding "allI_alt"}, []), [@{thm allI}]) #> snd *}
(*
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 {* Pretty-Printing\label{sec:pretty} *}
text {*
So far we printed out only plain strings without any formatting except for
occasional explicit line breaks using @{text [quotes] "\\n"}. This is
sufficient for ``quick-and-dirty'' printouts. For something more
sophisticated, Isabelle includes an infrastructure for properly formatting
text. Most of its functions do not operate on @{ML_type string}s, but on
instances of the pretty type:
@{ML_type_ind [display, gray] "Pretty.T"}
The function @{ML str in Pretty} transforms a (plain) string into such a pretty
type. For example
@{ML_response_fake [display,gray]
"Pretty.str \"test\"" "String (\"test\", 4)"}
where the result indicates that we transformed a string with length 4. Once
you have a pretty type, you can, for example, control where linebreaks may
occur in case the text wraps over a line, or with how much indentation a
text should be printed. However, if you want to actually output the
formatted text, you have to transform the pretty type back into a @{ML_type
string}. This can be done with the function @{ML_ind string_of in Pretty}. In what
follows we will use the following wrapper function for printing a pretty
type:
*}
ML{*fun pprint prt = tracing (Pretty.string_of prt)*}
text {*
The point of the pretty-printing infrastructure is to give hints about how to
layout text and let Isabelle do the actual layout. Let us first explain
how you can insert places where a line break can occur. For this assume the
following function that replicates a string n times:
*}
ML{*fun rep n str = implode (replicate n str) *}
text {*
and suppose we want to print out the string:
*}
ML{*val test_str = rep 8 "fooooooooooooooobaaaaaaaaaaaar "*}
text {*
We deliberately chose a large string so that it spans over more than one line.
If we print out the string using the usual ``quick-and-dirty'' method, then
we obtain the ugly output:
@{ML_response_fake [display,gray]
"tracing test_str"
"fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar foooooooooo
ooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaa
aaaaaaar fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar fo
oooooooooooooobaaaaaaaaaaaar"}
We obtain the same if we just use the function @{ML pprint}.
@{ML_response_fake [display,gray]
"pprint (Pretty.str test_str)"
"fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar foooooooooo
ooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaa
aaaaaaar fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar fo
oooooooooooooobaaaaaaaaaaaar"}
However by using pretty types you have the ability to indicate possible
linebreaks for example at each whitespace. You can achieve this with the
function @{ML_ind breaks in Pretty}, which expects a list of pretty types
and inserts a possible line break in between every two elements in this
list. To print this list of pretty types as a single string, we concatenate
them with the function @{ML_ind blk in Pretty} as follows:
@{ML_response_fake [display,gray]
"let
val ptrs = map Pretty.str (space_explode \" \" test_str)
in
pprint (Pretty.blk (0, Pretty.breaks ptrs))
end"
"fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar
fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar
fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar
fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar"}
Here the layout of @{ML test_str} is much more pleasing to the
eye. The @{ML "0"} in @{ML_ind blk in Pretty} stands for no indentation
of the printed string. You can increase the indentation and obtain
@{ML_response_fake [display,gray]
"let
val ptrs = map Pretty.str (space_explode \" \" test_str)
in
pprint (Pretty.blk (3, Pretty.breaks ptrs))
end"
"fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar
fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar
fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar
fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar"}
where starting from the second line the indent is 3. If you want
that every line starts with the same indent, you can use the
function @{ML_ind indent in Pretty} as follows:
@{ML_response_fake [display,gray]
"let
val ptrs = map Pretty.str (space_explode \" \" test_str)
in
pprint (Pretty.indent 10 (Pretty.blk (0, Pretty.breaks ptrs)))
end"
" fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar
fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar
fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar
fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar"}
If you want to print out a list of items separated by commas and
have the linebreaks handled properly, you can use the function
@{ML_ind commas in Pretty}. For example
@{ML_response_fake [display,gray]
"let
val ptrs = map (Pretty.str o string_of_int) (99998 upto 100020)
in
pprint (Pretty.blk (0, Pretty.commas ptrs))
end"
"99998, 99999, 100000, 100001, 100002, 100003, 100004, 100005, 100006,
100007, 100008, 100009, 100010, 100011, 100012, 100013, 100014, 100015,
100016, 100017, 100018, 100019, 100020"}
where @{ML upto} generates a list of integers. You can print out this
list as a ``set'', that means enclosed inside @{text [quotes] "{"} and
@{text [quotes] "}"}, and separated by commas using the function
@{ML_ind enum in Pretty}. For example
*}
text {*
@{ML_response_fake [display,gray]
"let
val ptrs = map (Pretty.str o string_of_int) (99998 upto 100020)
in
pprint (Pretty.enum \",\" \"{\" \"}\" ptrs)
end"
"{99998, 99999, 100000, 100001, 100002, 100003, 100004, 100005, 100006,
100007, 100008, 100009, 100010, 100011, 100012, 100013, 100014, 100015,
100016, 100017, 100018, 100019, 100020}"}
As can be seen, this function prints out the ``set'' so that starting
from the second, each new line has an indentation of 2.
If you print out something that goes beyond the capabilities of the
standard functions, you can do relatively easily the formatting
yourself. Assume you want to print out a list of items where like in ``English''
the last two items are separated by @{text [quotes] "and"}. For this you can
write the function
*}
ML %linenosgray{*fun and_list [] = []
| and_list [x] = [x]
| and_list xs =
let
val (front, last) = split_last xs
in
(Pretty.commas front) @
[Pretty.brk 1, Pretty.str "and", Pretty.brk 1, last]
end *}
text {*
where Line 7 prints the beginning of the list and Line
8 the last item. We have to use @{ML "Pretty.brk 1"} in order
to insert a space (of length 1) before the
@{text [quotes] "and"}. This space is also a place where a line break
can occur. We do the same after the @{text [quotes] "and"}. This gives you
for example
@{ML_response_fake [display,gray]
"let
val ptrs1 = map (Pretty.str o string_of_int) (1 upto 22)
val ptrs2 = map (Pretty.str o string_of_int) (10 upto 28)
in
pprint (Pretty.blk (0, and_list ptrs1));
pprint (Pretty.blk (0, and_list ptrs2))
end"
"1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21
and 22
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27 and
28"}
Next we like to pretty-print a term and its type. For this we use the
function @{text "tell_type"}.
*}
ML %linenosgray{*fun tell_type ctxt t =
let
fun pstr s = Pretty.breaks (map Pretty.str (space_explode " " s))
val ptrm = Pretty.quote (Syntax.pretty_term ctxt t)
val pty = Pretty.quote (Syntax.pretty_typ ctxt (fastype_of t))
in
pprint (Pretty.blk (0,
(pstr "The term " @ [ptrm] @ pstr " has type "
@ [pty, Pretty.str "."])))
end*}
text {*
In Line 3 we define a function that inserts possible linebreaks in places
where a space is. In Lines 4 and 5 we pretty-print the term and its type
using the functions @{ML_ind pretty_term in Syntax} and @{ML_ind
pretty_typ in Syntax}. We also use the function @{ML_ind quote in
Pretty} in order to enclose the term and type inside quotation marks. In
Line 9 we add a period right after the type without the possibility of a
line break, because we do not want that a line break occurs there.
Now you can write
@{ML_response_fake [display,gray]
"tell_type @{context} @{term \"min (Suc 0)\"}"
"The term \"min (Suc 0)\" has type \"nat \<Rightarrow> nat\"."}
To see the proper line breaking, you can try out the function on a bigger term
and type. For example:
@{ML_response_fake [display,gray]
"tell_type @{context} @{term \"op = (op = (op = (op = (op = op =))))\"}"
"The term \"op = (op = (op = (op = (op = op =))))\" has type
\"((((('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool) \<Rightarrow> bool) \<Rightarrow> bool) \<Rightarrow> bool) \<Rightarrow> bool\"."}
The function @{ML_ind big_list in Pretty} helps with printing long lists.
It is used for example in the command \isacommand{print\_theorems}. An
example is as follows.
@{ML_response_fake [display,gray]
"let
val pstrs = map (Pretty.str o string_of_int) (4 upto 10)
in
pprint (Pretty.big_list \"header\" pstrs)
end"
"header
4
5
6
7
8
9
10"}
Like @{ML blk in Pretty}, the function @{ML_ind chunks in Pretty} prints out
a list of items, but automatically inserts forced breaks between each item.
Compare
@{ML_response_fake [display,gray]
"let
val a_and_b = [Pretty.str \"a\", Pretty.str \"b\"]
in
pprint (Pretty.blk (0, a_and_b));
pprint (Pretty.chunks a_and_b)
end"
"ab
a
b"}
\footnote{\bf What happens with printing big formulae?}
\begin{readmore}
The general infrastructure for pretty-printing is implemented in the file
@{ML_file "Pure/General/pretty.ML"}. The file @{ML_file "Pure/Syntax/syntax.ML"}
contains pretty-printing functions for terms, types, theorems and so on.
@{ML_file "Pure/General/markup.ML"}
\end{readmore}
*}
(*
text {*
printing into the goal buffer as part of the proof state
*}
text {* writing into the goal buffer *}
ML {* fun my_hook interactive state =
(interactive ? Proof.goal_message (fn () => Pretty.str
"foo")) state
*}
setup %gray {* Context.theory_map (Specification.add_theorem_hook my_hook) *}
lemma "False"
oops
*)
(*
ML {*
fun setmp_show_all_types f =
setmp show_all_types
(! show_types orelse ! show_sorts orelse ! show_all_types) f;
val ctxt = @{context};
val t1 = @{term "undefined::nat"};
val t2 = @{term "Suc y"};
val pty = Pretty.block (Pretty.breaks
[(setmp show_question_marks false o setmp_show_all_types)
(Syntax.pretty_term ctxt) t1,
Pretty.str "=", Syntax.pretty_term ctxt t2]);
pty |> Pretty.string_of
*}
text {* the infrastructure of Syntax-pretty-term makes sure it is printed nicely *}
ML {* Pretty.mark Markup.hilite (Pretty.str "foo") |> Pretty.string_of |> tracing *}
ML {* (Pretty.str "bar") |> Pretty.string_of |> tracing *}
*)
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) *}
end