--- a/ProgTutorial/General.thy Thu Nov 19 14:11:50 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,2571 +0,0 @@
-theory General
-imports Base FirstSteps
-begin
-
-(*<*)
-setup{*
-open_file_with_prelude
- "General_Code.thy"
- ["theory General", "imports Base FirstSteps", "begin"]
-*}
-(*>*)
-
-
-chapter {* Isabelle Essentials *}
-
-text {*
- Isabelle is build around a few central ideas. One central idea 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
- \cite{GordonMilnerWadsworth79}. The fundamental data
- structures involved in this core are certified terms and certified types,
- as well as theorems.
-*}
-
-
-section {* Terms and Types *}
-
-text {*
- In Isabelle, there are certified terms and uncertified terms (respectively types).
- Uncertified terms are often just called 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>"}
-
- constructs the term @{term "(a::nat) + b = c"}. The resulting term is printed using
- the internal representation corresponding to the datatype @{ML_type_ind "term"},
- which is 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 Church-style lambda-terms, where types are
- explicitly recorded in variables, constants and abstractions. 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
- variables 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 \<Rightarrow> 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. The coercion
- 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"} and printed with
- a leading question mark). A type constant,
- like @{typ "int"} or @{typ bool}, are types with an empty list
- of argument types. However, it needs a bit of effort to show an
- example, because Isabelle always pretty prints types (unlike terms).
- Using just the antiquotation @{text "@{typ \"bool\"}"} we only see
-
- @{ML_response [display, gray]
- "@{typ \"bool\"}"
- "bool"}
-
- that is the pretty printed version of @{text "bool"}. However, in PolyML
- (version 5.3) it is easy to install your own pretty printer. With the
- function below we mimic the behaviour of the usual pretty printer for
- datatypes (it uses pretty-printing functions which will be explained in more
- detail in Section~\ref{sec:pretty}).
-*}
-
-ML{*local
- fun pp_pair (x, y) = Pretty.list "(" ")" [x, y]
- fun pp_list xs = Pretty.list "[" "]" xs
- fun pp_str s = Pretty.str s
- fun pp_qstr s = Pretty.quote (pp_str s)
- fun pp_int i = pp_str (string_of_int i)
- fun pp_sort S = pp_list (map pp_qstr S)
- fun pp_constr a args = Pretty.block [pp_str a, Pretty.brk 1, args]
-in
-fun raw_pp_typ (TVar ((a, i), S)) =
- pp_constr "TVar" (pp_pair (pp_pair (pp_qstr a, pp_int i), pp_sort S))
- | raw_pp_typ (TFree (a, S)) =
- pp_constr "TFree" (pp_pair (pp_qstr a, pp_sort S))
- | raw_pp_typ (Type (a, tys)) =
- pp_constr "Type" (pp_pair (pp_qstr a, pp_list (map raw_pp_typ tys)))
-end*}
-
-text {*
- We can install this pretty printer with the function
- @{ML_ind addPrettyPrinter in PolyML} as follows.
-*}
-
-ML{*PolyML.addPrettyPrinter
- (fn _ => fn _ => ml_pretty o Pretty.to_ML o raw_pp_typ)*}
-
-text {*
- Now the type bool is printed out in full detail.
-
- @{ML_response [display,gray]
- "@{typ \"bool\"}"
- "Type (\"bool\", [])"}
-
- When printing out a list-type
-
- @{ML_response [display,gray]
- "@{typ \"'a list\"}"
- "Type (\"List.list\", [TFree (\"'a\", [\"HOL.type\"])])"}
-
- we can see the full name of the type is actually @{text "List.list"}, indicating
- that it is defined in the theory @{text "List"}. However, one has to be
- careful with names of types, because even if
- @{text "fun"}, @{text "bool"} and @{text "nat"} are defined in the
- theories @{text "HOL"} and @{text "Nat"}, respectively, they are
- still represented by their simple name.
-
- @{ML_response [display,gray]
- "@{typ \"bool \<Rightarrow> nat\"}"
- "Type (\"fun\", [Type (\"bool\", []), Type (\"nat\", [])])"}
-
- We can restore the usual behaviour of Isabelle's pretty printer
- with the code
-*}
-
-ML{*PolyML.addPrettyPrinter
- (fn _ => fn _ => ml_pretty o Pretty.to_ML o Proof_Display.pp_typ Pure.thy)*}
-
-text {*
- After that the types for booleans, lists and so on are printed out again
- the standard Isabelle way.
-
- @{ML_response_fake [display, gray]
- "@{typ \"bool\"};
-@{typ \"'a list\"}"
- "\"bool\"
-\"'a List.list\""}
-
- \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 in Term}, which
- takes as argument a term and a list of terms, 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 in Term}, 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, where 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 in Term} 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 in Term}, 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 t =
-let
- fun aux (x, T, t) = strip_alls t |>> cons (Free (x, T))
-in
- case t of
- Const ("All", _) $ Abs body => aux body
- | _ => ([], t)
-end*}
-
-text {*
- The function returns a pair consisting of the stripped off variables and
- the body of the universal quantification. 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 in Term}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.
-
- Notice also that this function might introduce name clashes, since we
- substitute just a variable with the name recorded in an abstraction. This
- name is by no means unique. If clashes need to be avoided, then we should
- use the function @{ML_ind dest_abs in Term}, which returns the body where
- the lose de Bruijn index is replaced by a unique free variable. For example
-
-
- @{ML_response_fake [display,gray]
- "let
- val body = Bound 0 $ Free (\"x\", @{typ nat})
-in
- Term.dest_abs (\"x\", @{typ \"nat \<Rightarrow> bool\"}, body)
-end"
- "(\"xa\", Free (\"xa\", \"nat \<Rightarrow> bool\") $ Free (\"x\", \"nat\"))"}
-
- There are also many convenient functions that construct specific HOL-terms
- in the structure @{ML_struct HOLogic}. 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 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 "-->" in Term} 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 "--->" in Term}.
-*}
-
-ML{*fun make_fun_types tys ty = tys ---> ty *}
-
-text {*
- A handy function for manipulating terms is @{ML_ind map_types in Term}: 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 that 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, 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.\footnote{Thanks to Roy
- Dyckhoff for suggesting this exercise and working out the details.}
- \end{exercise}
-*}
-
-section {* Unification and Matching *}
-
-text {*
- As seen earlier, Isabelle's terms and types may contain schematic term variables
- (term-constructor @{ML Var}) and schematic type variables (term-constructor
- @{ML TVar}). These variables stand for unknown entities, which can be made
- more concrete by instantiations. Such instantiations might be a result of
- unification or matching. While in case of types, unification and matching is
- relatively straightforward, in case of terms the algorithms are
- substantially more complicated, because terms need higher-order versions of
- the unification and matching algorithms. Below we shall use the
- antiquotations @{text "@{typ_pat \<dots>}"} and @{text "@{term_pat \<dots>}"} from
- Section~\ref{sec:antiquote} in order to construct examples involving
- schematic variables.
-
- Let us begin with describing the unification and matching function for
- types. Both return type environments (ML-type @{ML_type "Type.tyenv"})
- which map schematic type variables to types and sorts. Below we use the
- function @{ML_ind typ_unify in Sign} from the structure @{ML_struct Sign}
- for unifying the types @{text "?'a * ?'b"} and @{text "?'b list *
- nat"}. This will produce the mapping, or type environment, @{text "[?'a :=
- ?'b list, ?'b := nat]"}.
-*}
-
-ML %linenosgray{*val (tyenv_unif, _) = let
- val ty1 = @{typ_pat "?'a * ?'b"}
- val ty2 = @{typ_pat "?'b list * nat"}
-in
- Sign.typ_unify @{theory} (ty1, ty2) (Vartab.empty, 0)
-end*}
-
-text {*
- The environment @{ML_ind "Vartab.empty"} in line 5 stands for the empty type
- environment, which is needed for starting the unification without any
- (pre)instantiations. The @{text 0} is an integer index that will be explained
- below. In case of failure @{ML typ_unify in Sign} will throw the exception
- @{text TUNIFY}. We can print out the resulting type environment bound to
- @{ML tyenv_unif} with the built-in function @{ML_ind dest in Vartab} from the
- structure @{ML_struct Vartab}.
-
- @{ML_response_fake [display,gray]
- "Vartab.dest tyenv_unif"
- "[((\"'a\", 0), ([\"HOL.type\"], \"?'b List.list\")),
- ((\"'b\", 0), ([\"HOL.type\"], \"nat\"))]"}
-*}
-
-text_raw {*
- \begin{figure}[t]
- \begin{minipage}{\textwidth}
- \begin{isabelle}*}
-ML{*fun pretty_helper aux env =
- env |> Vartab.dest
- |> map ((fn (s1, s2) => s1 ^ " := " ^ s2) o aux)
- |> commas
- |> enclose "[" "]"
- |> tracing
-
-fun pretty_tyenv ctxt tyenv =
-let
- fun get_typs (v, (s, T)) = (TVar (v, s), T)
- val print = pairself (Syntax.string_of_typ ctxt)
-in
- pretty_helper (print o get_typs) tyenv
-end*}
-text_raw {*
- \end{isabelle}
- \end{minipage}
- \caption{A pretty printing function for type environments, which are
- produced by unification and matching.\label{fig:prettyenv}}
- \end{figure}
-*}
-
-text {*
- The first components in this list stand for the schematic type variables and
- the second are the associated sorts and types. In this example the sort is
- the default sort @{text "HOL.type"}. Instead of @{ML "Vartab.dest"}, we will
- use in what follows our own pretty-printing function from
- Figure~\ref{fig:prettyenv} for @{ML_type "Type.tyenv"}s. For the type
- environment in the example this function prints out the more legible:
-
-
- @{ML_response_fake [display, gray]
- "pretty_tyenv @{context} tyenv_unif"
- "[?'a := ?'b list, ?'b := nat]"}
-
- The way the unification function @{ML typ_unify in Sign} is implemented
- using an initial type environment and initial index makes it easy to
- unify more than two terms. For example
-*}
-
-ML %linenosgray{*val (tyenvs, _) = let
- val tys1 = (@{typ_pat "?'a"}, @{typ_pat "?'b list"})
- val tys2 = (@{typ_pat "?'b"}, @{typ_pat "nat"})
-in
- fold (Sign.typ_unify @{theory}) [tys1, tys2] (Vartab.empty, 0)
-end*}
-
-text {*
- The index @{text 0} in Line 5 is the maximal index of the schematic type
- variables occurring in @{text tys1} and @{text tys2}. This index will be
- increased whenever a new schematic type variable is introduced during
- unification. This is for example the case when two schematic type variables
- have different, incomparable sorts. Then a new schematic type variable is
- introduced with the combined sorts. To show this let us assume two sorts,
- say @{text "s1"} and @{text "s2"}, which we attach to the schematic type
- variables @{text "?'a"} and @{text "?'b"}. Since we do not make any
- assumption about the sorts, they are incomparable.
-*}
-
-ML{*val (tyenv, index) = let
- val ty1 = @{typ_pat "?'a::s1"}
- val ty2 = @{typ_pat "?'b::s2"}
-in
- Sign.typ_unify @{theory} (ty1, ty2) (Vartab.empty, 0)
-end*}
-
-text {*
- To print out the result type environment we switch on the printing
- of sort information by setting @{ML_ind show_sorts in Syntax} to
- true. This allows us to inspect the typing environment.
-
- @{ML_response_fake [display,gray]
- "pretty_tyenv @{context} tyenv"
- "[?'a::s1 := ?'a1::{s1, s2}, ?'b::s2 := ?'a1::{s1, s2}]"}
-
- As can be seen, the type variables @{text "?'a"} and @{text "?'b"} are instantiated
- with a new type variable @{text "?'a1"} with sort @{text "{s1, s2}"}. Since a new
- type variable has been introduced the @{ML index}, originally being @{text 0},
- has been increased to @{text 1}.
-
- @{ML_response [display,gray]
- "index"
- "1"}
-
- Let us now return to the unification problem @{text "?'a * ?'b"} and
- @{text "?'b list * nat"} from the beginning of this section, and the
- calculated type environment @{ML tyenv_unif}:
-
- @{ML_response_fake [display, gray]
- "pretty_tyenv @{context} tyenv_unif"
- "[?'a := ?'b list, ?'b := nat]"}
-
- Observe that the type environment which the function @{ML typ_unify in
- Sign} returns is in \emph{not} an instantiation in fully solved form: while @{text
- "?'b"} is instantiated to @{typ nat}, this is not propagated to the
- instantiation for @{text "?'a"}. In unification theory, this is often
- called an instantiation in \emph{triangular form}. These triangular
- instantiations, or triangular type environments, are used because of
- performance reasons. To apply such a type environment to a type, say @{text "?'a *
- ?'b"}, you should use the function @{ML_ind norm_type in Envir}:
-
- @{ML_response_fake [display, gray]
- "Envir.norm_type tyenv_unif @{typ_pat \"?'a * ?'b\"}"
- "nat list * nat"}
-
- Matching of types can be done with the function @{ML_ind typ_match in Sign}
- also from the structure @{ML_struct Sign}. This function returns a @{ML_type
- Type.tyenv} as well, but might raise the exception @{text TYPE_MATCH} in case
- of failure. For example
-*}
-
-ML{*val tyenv_match = let
- val pat = @{typ_pat "?'a * ?'b"}
- and ty = @{typ_pat "bool list * nat"}
-in
- Sign.typ_match @{theory} (pat, ty) Vartab.empty
-end*}
-
-text {*
- Printing out the calculated matcher gives
-
- @{ML_response_fake [display,gray]
- "pretty_tyenv @{context} tyenv_match"
- "[?'a := bool list, ?'b := nat]"}
-
- Unlike unification, which uses the function @{ML norm_type in Envir},
- applying the matcher to a type needs to be done with the function
- @{ML_ind subst_type in Envir}. For example
-
- @{ML_response_fake [display, gray]
- "Envir.subst_type tyenv_match @{typ_pat \"?'a * ?'b\"}"
- "bool list * nat"}
-
- Be careful to observe the difference: use always
- @{ML subst_type in Envir} for matchers and @{ML norm_type in Envir}
- for unifiers. To show the difference, let us calculate the
- following matcher:
-*}
-
-ML{*val tyenv_match' = let
- val pat = @{typ_pat "?'a * ?'b"}
- and ty = @{typ_pat "?'b list * nat"}
-in
- Sign.typ_match @{theory} (pat, ty) Vartab.empty
-end*}
-
-text {*
- Now @{ML tyenv_unif} is equal to @{ML tyenv_match'}. If we apply
- @{ML norm_type in Envir} to the type @{text "?'a * ?'b"} we obtain
-
- @{ML_response_fake [display, gray]
- "Envir.norm_type tyenv_match' @{typ_pat \"?'a * ?'b\"}"
- "nat list * nat"}
-
- which does not solve the matching problem, and if
- we apply @{ML subst_type in Envir} to the same type we obtain
-
- @{ML_response_fake [display, gray]
- "Envir.subst_type tyenv_unif @{typ_pat \"?'a * ?'b\"}"
- "?'b list * nat"}
-
- which does not solve the unification problem.
-
- \begin{readmore}
- Unification and matching for types is implemented
- in @{ML_file "Pure/type.ML"}. The ``interface'' functions for them
- are in @{ML_file "Pure/sign.ML"}. Matching and unification produce type environments
- as results. These are implemented in @{ML_file "Pure/envir.ML"}.
- This file also includes the substitution and normalisation functions,
- which apply a type environment to a type. Type environments are lookup
- tables which are implemented in @{ML_file "Pure/term_ord.ML"}.
- \end{readmore}
-*}
-
-text {*
- Unification and matching of terms is substantially more complicated than the
- type-case. The reason is that terms have abstractions and, in this context,
- unification or matching modulo plain equality is often not meaningful.
- Nevertheless, Isabelle implements the function @{ML_ind
- first_order_match in Pattern} for terms. This matching function returns a
- type environment and a term environment. To pretty print the latter we use
- the function @{text "pretty_env"}:
-*}
-
-ML{*fun pretty_env ctxt env =
-let
- fun get_trms (v, (T, t)) = (Var (v, T), t)
- val print = pairself (string_of_term ctxt)
-in
- pretty_helper (print o get_trms) env
-end*}
-
-text {*
- As can be seen from the @{text "get_trms"}-function, a term environment associates
- a schematic term variable with a type and a term. An example of a first-order
- matching problem is the term @{term "P (\<lambda>a b. Q b a)"} and the pattern
- @{text "?X ?Y"}.
-*}
-
-ML{*val (_, fo_env) = let
- val fo_pat = @{term_pat "(?X::(nat\<Rightarrow>nat\<Rightarrow>nat)\<Rightarrow>bool) ?Y"}
- val trm_a = @{term "P::(nat\<Rightarrow>nat\<Rightarrow>nat)\<Rightarrow>bool"}
- val trm_b = @{term "\<lambda>a b. (Q::nat\<Rightarrow>nat\<Rightarrow>nat) b a"}
- val init = (Vartab.empty, Vartab.empty)
-in
- Pattern.first_order_match @{theory} (fo_pat, trm_a $ trm_b) init
-end *}
-
-text {*
- In this example we annotated explicitly types because then
- the type environment is empty and can be ignored. The
- resulting term environment is
-
- @{ML_response_fake [display, gray]
- "pretty_env @{context} fo_env"
- "[?X := P, ?Y := \<lambda>a b. Q b a]"}
-
- The matcher can be applied to a term using the function @{ML_ind subst_term
- in Envir} (remember the same convention for types applies to terms: @{ML
- subst_term in Envir} is for matchers and @{ML norm_term in Envir} for
- unifiers). The function @{ML subst_term in Envir} expects a type environment,
- which is set to empty in the example below, and a term environment.
-
- @{ML_response_fake [display, gray]
- "let
- val trm = @{term_pat \"(?X::(nat\<Rightarrow>nat\<Rightarrow>nat)\<Rightarrow>bool) ?Y\"}
-in
- Envir.subst_term (Vartab.empty, fo_env) trm
- |> string_of_term @{context}
- |> tracing
-end"
- "P (\<lambda>a b. Q b a)"}
-
- First-order matching is useful for matching against applications and
- variables. It can deal also with abstractions and a limited form of
- alpha-equivalence, but this kind of matching should be used with care, since
- it is not clear whether the result is meaningful. A meaningful example is
- matching @{text "\<lambda>x. P x"} against the pattern @{text "\<lambda>y. ?X y"}. In this
- case, first-order matching produces @{text "[?X := P]"}.
-
- @{ML_response_fake [display, gray]
- "let
- val fo_pat = @{term_pat \"\<lambda>y. (?X::nat\<Rightarrow>bool) y\"}
- val trm = @{term \"\<lambda>x. (P::nat\<Rightarrow>bool) x\"}
- val init = (Vartab.empty, Vartab.empty)
-in
- Pattern.first_order_match @{theory} (fo_pat, trm) init
- |> snd
- |> pretty_env @{context}
-end"
- "[?X := P]"}
-*}
-
-text {*
- Unification of abstractions is more thoroughly studied in the context
- of higher-order pattern unification and higher-order pattern matching. A
- \emph{\index*{pattern}} is an abstraction term whose ``head symbol'' (that is the
- first symbol under an abstraction) is either a constant, a schematic or a free
- variable. If it is a schematic variable then it can be only applied with
- distinct bound variables. This excludes terms where a schematic variable is an
- argument of another one and where a schematic variable is applied
- twice with the same bound variable. The function @{ML_ind pattern in Pattern}
- in the structure @{ML_struct Pattern} tests whether a term satisfies these
- restrictions.
-
- @{ML_response [display, gray]
- "let
- val trm_list =
- [@{term_pat \"?X\"}, @{term_pat \"a\"},
- @{term_pat \"\<lambda>a b. ?X a b\"}, @{term_pat \"\<lambda>a b. (op +) a b\"},
- @{term_pat \"\<lambda>a. (op +) a ?Y\"}, @{term_pat \"?X ?Y\"},
- @{term_pat \"\<lambda>a b. ?X a b ?Y\"}, @{term_pat \"\<lambda>a. ?X a a\"}]
-in
- map Pattern.pattern trm_list
-end"
- "[true, true, true, true, true, false, false, false]"}
-
- The point of the restriction to patterns is that unification and matching
- are decidable and produce most general unifiers, respectively matchers.
- In this way, matching and unification can be implemented as functions that
- produce a type and term environment (unification actually returns a
- record of type @{ML_type Envir.env} containing a maxind, a type environment
- and a term environment). The corresponding functions are @{ML_ind match in Pattern},
- and @{ML_ind unify in Pattern} both implemented in the structure
- @{ML_struct Pattern}. An example for higher-order pattern unification is
-
- @{ML_response_fake [display, gray]
- "let
- val trm1 = @{term_pat \"\<lambda>x y. g (?X y x) (f (?Y x))\"}
- val trm2 = @{term_pat \"\<lambda>u v. g u (f u)\"}
- val init = Envir.empty 0
- val env = Pattern.unify @{theory} (trm1, trm2) init
-in
- pretty_env @{context} (Envir.term_env env)
-end"
- "[?X := \<lambda>y x. x, ?Y := \<lambda>x. x]"}
-
- The function @{ML_ind "Envir.empty"} generates a record with a specified
- max-index for the schematic variables (in the example the index is @{text
- 0}) and empty type and term environments. The function @{ML_ind
- "Envir.term_env"} pulls out the term environment from the result record. The
- function for type environment is @{ML_ind "Envir.type_env"}. An assumption of
- this function is that the terms to be unified have already the same type. In
- case of failure, the exceptions that are raised are either @{text Pattern},
- @{text MATCH} or @{text Unif}.
-
- As mentioned before, unrestricted higher-order unification, respectively
- higher-order matching, is in general undecidable and might also not posses a
- single most general solution. Therefore Isabelle implements the unification
- function @{ML_ind unifiers in Unify} so that it returns a lazy list of
- potentially infinite unifiers. An example is as follows
-*}
-
-ML{*val uni_seq =
-let
- val trm1 = @{term_pat "?X ?Y"}
- val trm2 = @{term "f a"}
- val init = Envir.empty 0
-in
- Unify.unifiers (@{theory}, init, [(trm1, trm2)])
-end *}
-
-text {*
- The unifiers can be extracted from the lazy sequence using the
- function @{ML_ind "Seq.pull"}. In the example we obtain three
- unifiers @{text "un1\<dots>un3"}.
-*}
-
-ML{*val SOME ((un1, _), next1) = Seq.pull uni_seq;
-val SOME ((un2, _), next2) = Seq.pull next1;
-val SOME ((un3, _), next3) = Seq.pull next2;
-val NONE = Seq.pull next3 *}
-
-text {*
- \footnote{\bf FIXME: what is the list of term pairs in the unifier: flex-flex pairs?}
-
- We can print them out as follows.
-
- @{ML_response_fake [display, gray]
- "pretty_env @{context} (Envir.term_env un1);
-pretty_env @{context} (Envir.term_env un2);
-pretty_env @{context} (Envir.term_env un3)"
- "[?X := \<lambda>a. a, ?Y := f a]
-[?X := f, ?Y := a]
-[?X := \<lambda>b. f a]"}
-
-
- In case of failure the function @{ML_ind unifiers in Unify} does not raise
- an exception, rather returns the empty sequence. For example
-
- @{ML_response [display, gray]
-"let
- val trm1 = @{term \"a\"}
- val trm2 = @{term \"b\"}
- val init = Envir.empty 0
-in
- Unify.unifiers (@{theory}, init, [(trm1, trm2)])
- |> Seq.pull
-end"
-"NONE"}
-
- In order to find a
- reasonable solution for a unification problem, Isabelle also tries first to
- solve the problem by higher-order pattern unification.
-
- For higher-order
- matching the function is called @{ML_ind matchers in Unify} implemented
- in the structure @{ML_struct Unify}. Also this
- function returns sequences with possibly more than one matcher.
- Like @{ML unifiers in Unify}, this function does not raise an exception
- in case of failure, but returns an empty sequence. It also first tries
- out whether the matching problem can be solved by first-order matching.
-
- \begin{readmore}
- Unification and matching of higher-order patterns is implemented in
- @{ML_file "Pure/pattern.ML"}. This file also contains a first-order matcher
- for terms. Full higher-order unification is implemented
- in @{ML_file "Pure/unify.ML"}. It uses lazy sequences which are implemented
- in @{ML_file "Pure/General/seq.ML"}.
- \end{readmore}
-*}
-
-
-
-
-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 in Term} 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 in Term}, 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 in Term} 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 in Thm}, 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\"}
- val plus = Const (@{const_name plus}, [natT, natT] ---> natT)
-in
- cterm_of @{theory} (plus $ 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 the function @{ML_ind list_comb in Drule} from the structure @{ML_struct 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.
-
-
- 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]
- "tracing (string_of_thm @{context} my_thm)"
- "\<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 on the user level. One way to store it in the theorem database is
- by using the function @{ML_ind note in Local_Theory}.\footnote{\bf FIXME: make sure a pointer
- to the section about local-setup is given earlier.}
-*}
-
-local_setup %gray {*
- Local_Theory.note ((@{binding "my_thm"}, []), [my_thm]) #> snd *}
-
-text {*
- The fourth argument of @{ML note in Local_Theory} is the list of theorems we
- want to store under a name. We can store more than one under a single name.
- The first argument @{ML_ind theoremK in Thm} is
- a kind indicator, which classifies the theorem. There are several such kind
- indicators: for a theorem arising from a definition you should use @{ML_ind
- definitionK in Thm}, and for
- ``normal'' theorems the kinds @{ML_ind theoremK in Thm} or @{ML_ind lemmaK
- in Thm}. The second argument of @{ML note in Local_Theory} is the name under
- which we store the theorem or theorems. The third argument can contain a
- list of theorem attributes, which we will explain in detail in
- Section~\ref{sec:attributes}. Below we just use one such attribute for
- adding the theorem to the simpset:
-*}
-
-local_setup %gray {*
- Local_Theory.note ((@{binding "my_thm_simp"},
- [Attrib.internal (K Simplifier.simp_add)]), [my_thm]) #> snd *}
-
-text {*
- Note that we have to use another name under which the theorem is stored,
- since Isabelle does not allow us to call @{ML_ind note in Local_Theory} twice
- with the same name. The attribute needs to be wrapped inside the function @{ML_ind
- internal in Attrib} from the structure @{ML_struct Attrib}. If we use the function
- @{ML get_thm_names_from_ss} from
- the previous chapter, we can check whether the theorem has actually been
- added.
-
-
- @{ML_response [display,gray]
- "let
- fun pred s = match_string \"my_thm_simp\" s
-in
- exists pred (get_thm_names_from_ss @{simpset})
-end"
- "true"}
-
- The main point of storing the theorems @{thm [source] my_thm} and @{thm
- [source] my_thm_simp} is that they can now also 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. Otherwise the
- user has no access to these theorems.
-
- Recall that Isabelle does not let you call @{ML note in Local_Theory} twice
- with the same theorem name. In effect, once a theorem is stored under a name,
- this association is fixed. While this is a ``safety-net'' to make sure a
- theorem name refers to a particular theorem or collection of theorems, it is
- also a bit too restrictive in cases where a theorem name should refer to a
- dynamically expanding list of theorems (like a simpset). Therefore Isabelle
- also implements a mechanism where a theorem name can refer to a custom theorem
- list. For this you can use the function @{ML_ind add_thms_dynamic in PureThy}.
- To see how it works let us assume we defined our own theorem list @{text MyThmList}.
-*}
-
-ML{*structure MyThmList = Generic_Data
- (type T = thm list
- val empty = []
- val extend = I
- val merge = merge Thm.eq_thm_prop)
-
-fun update thm = Context.theory_map (MyThmList.map (Thm.add_thm thm))*}
-
-text {*
- The function @{ML update} allows us to update the theorem list, for example
- by adding the theorem @{thm [source] TrueI}.
-*}
-
-setup %gray {* update @{thm TrueI} *}
-
-text {*
- We can now install the theorem list so that it is visible to the user and
- can be refered to by a theorem name. For this need to call
- @{ML_ind add_thms_dynamic in PureThy}
-*}
-
-setup %gray {*
- PureThy.add_thms_dynamic (@{binding "mythmlist"}, MyThmList.get)
-*}
-
-text {*
- with a name and a function that accesses the theorem list. Now if the
- user issues the command
-
- \begin{isabelle}
- \isacommand{thm}~@{text "mythmlist"}\\
- @{text "> True"}
- \end{isabelle}
-
- the current content of the theorem list is displayed. If more theorems are stored in
- the list, say
-*}
-
-setup %gray {* update @{thm FalseE} *}
-
-text {*
- then the same command produces
-
- \begin{isabelle}
- \isacommand{thm}~@{text "mythmlist"}\\
- @{text "> False \<Longrightarrow> ?P"}\\
- @{text "> True"}
- \end{isabelle}
-
- There is a multitude of functions in the structures @{ML_struct Thm} and @{ML_struct Drule}
- for managing or manipulating theorems. For example
- we can test theorems for alpha equality. Suppose you proved the following three
- theorems.
-*}
-
-lemma
- shows thm1: "\<forall>x. P x"
- and thm2: "\<forall>y. P y"
- and thm3: "\<forall>y. Q y" sorry
-
-text {*
- Testing them for alpha 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 theorems into @{ML_type cterm}s. For example
- the functions @{ML_ind lhs_of in Thm} and @{ML_ind rhs_of in Thm} return
- the left and right-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 (string_of_cterm @{context})
-end"
- "(True, (\<lambda>x. x) = (\<lambda>x. x))"}
-
- Other function produce terms that can be pattern-matched.
- Suppose the following two theorems.
-*}
-
-lemma
- shows foo_test1: "A \<Longrightarrow> B \<Longrightarrow> C"
- and foo_test2: "A \<longrightarrow> B \<longrightarrow> C" sorry
-
-text {*
- We can destruct them into premises and conclusions as follows.
-
- @{ML_response_fake [display,gray]
-"let
- val ctxt = @{context}
- fun prems_and_concl thm =
- [\"Premises: \" ^ (string_of_terms ctxt (Thm.prems_of thm))] @
- [\"Conclusion: \" ^ (string_of_term ctxt (Thm.concl_of thm))]
- |> cat_lines
- |> tracing
-in
- prems_and_concl @{thm foo_test1};
- prems_and_concl @{thm foo_test2}
-end"
-"Premises: ?A, ?B
-Conclusion: ?C
-Premises:
-Conclusion: ?A \<longrightarrow> ?B \<longrightarrow> ?C"}
-
- Note that in the second case, there is no premise.
-
- \begin{readmore}
- 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}
-
- Although we will explain the simplifier in more detail as tactic in
- Section~\ref{sec:simplifier}, the simplifier
- can be used to work directly over theorems, for example to unfold definitions. To show
- this, we build the theorem @{term "True \<equiv> True"} (Line 1) and then
- unfold the constant @{term "True"} according to its definition (Line 2).
-
- @{ML_response_fake [display,gray,linenos]
- "Thm.reflexive @{cterm \"True\"}
- |> Simplifier.rewrite_rule [@{thm True_def}]
- |> string_of_thm @{context}
- |> tracing"
- "(\<lambda>x. x) = (\<lambda>x. x) \<equiv> (\<lambda>x. x) = (\<lambda>x. x)"}
-
- Often it is necessary to transform theorems to and from the object
- logic, that is replacing all @{text "\<longrightarrow>"} and @{text "\<forall>"} by @{text "\<Longrightarrow>"}
- and @{text "\<And>"}, or the other way around. A reason for such a transformation
- might be stating a definition. The reason is that definitions can only be
- stated using object logic connectives, while theorems using the connectives
- from the meta logic are more convenient for reasoning. Therefore there are
- some build in functions which help with these transformations. The function
- @{ML_ind rulify in ObjectLogic}
- replaces all object connectives by equivalents in the meta logic. For example
-
- @{ML_response_fake [display, gray]
- "ObjectLogic.rulify @{thm foo_test2}"
- "\<lbrakk>?A; ?B\<rbrakk> \<Longrightarrow> ?C"}
-
- The transformation in the other direction can be achieved with function
- @{ML_ind atomize in ObjectLogic} and the following code.
-
- @{ML_response_fake [display,gray]
- "let
- val thm = @{thm foo_test1}
- val meta_eq = ObjectLogic.atomize (cprop_of thm)
-in
- MetaSimplifier.rewrite_rule [meta_eq] thm
-end"
- "?A \<longrightarrow> ?B \<longrightarrow> ?C"}
-
- In this code the function @{ML atomize in ObjectLogic} produces
- a meta-equation between the given theorem and the theorem transformed
- into the object logic. The result is the theorem with object logic
- connectives. However, in order to completely transform a theorem
- involving meta variables, such as @{thm [source] list.induct}, which
- is of the form
-
- @{thm [display] list.induct}
-
- we have to first abstract over the meta variables @{text "?P"} and
- @{text "?list"}. For this we can use the function
- @{ML_ind forall_intr_vars in Drule}. This allows us to implement the
- following function for atomizing a theorem.
-*}
-
-ML{*fun atomize_thm thm =
-let
- val thm' = forall_intr_vars thm
- val thm'' = ObjectLogic.atomize (cprop_of thm')
-in
- MetaSimplifier.rewrite_rule [thm''] thm'
-end*}
-
-text {*
- This function produces for the theorem @{thm [source] list.induct}
-
- @{ML_response_fake [display, gray]
- "atomize_thm @{thm list.induct}"
- "\<forall>P list. P [] \<longrightarrow> (\<forall>a list. P list \<longrightarrow> P (a # list)) \<longrightarrow> P list"}
-
- \footnote{\bf FIXME: say someting about @{ML_ind standard in Drule}.}
-
- Theorems can also be produced from terms by giving an explicit proof.
- One way to achieve this is by using the function @{ML_ind prove in Goal}
- in the structure @{ML_struct Goal}. For example below we use this function
- to prove the term @{term "P \<Longrightarrow> P"}.
-
- @{ML_response_fake [display,gray]
- "let
- val trm = @{term \"P \<Longrightarrow> P::bool\"}
- val tac = K (atac 1)
-in
- Goal.prove @{context} [\"P\"] [] trm tac
-end"
- "?P \<Longrightarrow> ?P"}
-
- This function takes first a context and second a list of strings. This list
- specifies which variables should be turned into schematic variables once the term
- is proved. The fourth argument is the term to be proved. The fifth is a
- corresponding proof given in form of a tactic (we explain tactics in
- Chapter~\ref{chp:tactical}). In the code above, the tactic proves the theorem
- by assumption. As before this code will produce a theorem, but the theorem
- is not yet stored in the theorem database.
-
- While the LCF-approach of going through interfaces ensures soundness in Isabelle, there
- is the function @{ML_ind make_thm in Skip_Proof} in the structure
- @{ML_struct Skip_Proof} that allows us to turn any proposition into a theorem.
- Potentially making the system unsound. This is sometimes useful for developing
- purposes, or when explicit proof construction should be omitted due to performace
- reasons. An example of this function is as follows:
-
- @{ML_response_fake [display, gray]
- "Skip_Proof.make_thm @{theory} @{prop \"True = False\"}"
- "True = False"}
-
- The function @{ML make_thm in Skip_Proof} however only works if
- the ``quick-and-dirty'' mode is switched on.
-
- Theorems also contain auxiliary data, such as the name of the theorem, its
- kind, the names for cases and so on. This data is stored in a string-string
- list and can be retrieved with the function @{ML_ind get_tags in
- Thm}. Assume you prove the following lemma.
-*}
-
-lemma foo_data:
- shows "P \<Longrightarrow> P \<Longrightarrow> P" by assumption
-
-text {*
- The auxiliary data of this lemma can be retrieved using the function
- @{ML_ind get_tags in Thm}. So far the the auxiliary data of this lemma is
-
- @{ML_response_fake [display,gray]
- "Thm.get_tags @{thm foo_data}"
- "[(\"name\", \"General.foo_data\"), (\"kind\", \"lemma\")]"}
-
- consisting of a name and a kind. When we store lemmas in the theorem database,
- we might want to explicitly extend this data by attaching case names to the
- two premises of the lemma. This can be done with the function @{ML_ind name in Rule_Cases}
- from the structure @{ML_struct Rule_Cases}.
-*}
-
-local_setup %gray {*
- Local_Theory.note ((@{binding "foo_data'"}, []),
- [(Rule_Cases.name ["foo_case_one", "foo_case_two"]
- @{thm foo_data})]) #> snd *}
-
-text {*
- The data of the theorem @{thm [source] foo_data'} is then as follows:
-
- @{ML_response_fake [display,gray]
- "Thm.get_tags @{thm foo_data'}"
- "[(\"name\", \"General.foo_data'\"), (\"kind\", \"lemma\"),
- (\"case_names\", \"foo_case_one;foo_case_two\")]"}
-
- You can observe the case names of this lemma on the user level when using
- the proof methods @{text cases} and @{text induct}. In the proof below
-*}
-
-lemma
- shows "Q \<Longrightarrow> Q \<Longrightarrow> Q"
-proof (cases rule: foo_data')
-
-(*<*)oops(*>*)
-text_raw{*
-\begin{tabular}{@ {}l}
-\isacommand{print\_cases}\\
-@{text "> cases:"}\\
-@{text "> foo_case_one:"}\\
-@{text "> let \"?case\" = \"?P\""}\\
-@{text "> foo_case_two:"}\\
-@{text "> let \"?case\" = \"?P\""}
-\end{tabular}*}
-
-text {*
- we can proceed by analysing the cases @{text "foo_case_one"} and
- @{text "foo_case_two"}. While if the theorem has no names, then
- the cases have standard names @{text 1}, @{text 2} and so
- on. This can be seen in the proof below.
-*}
-
-lemma
- shows "Q \<Longrightarrow> Q \<Longrightarrow> Q"
-proof (cases rule: foo_data)
-
-(*<*)oops(*>*)
-text_raw{*
-\begin{tabular}{@ {}l}
-\isacommand{print\_cases}\\
-@{text "> cases:"}\\
-@{text "> 1:"}\\
-@{text "> let \"?case\" = \"?P\""}\\
-@{text "> 2:"}\\
-@{text "> let \"?case\" = \"?P\""}
-\end{tabular}*}
-
-
-text {*
- One great feature of Isabelle is its document preparation system, where
- proved theorems can be quoted in documents referencing directly their
- formalisation. This helps tremendously to minimise cut-and-paste errors. However,
- sometimes the verbatim quoting is not what is wanted or what can be shown to
- readers. For such situations Isabelle allows the installation of \emph{\index*{theorem
- styles}}. These are, roughly speaking, functions from terms to terms. The input
- term stands for the theorem to be presented; the output can be constructed to
- ones wishes. Let us, for example, assume we want to quote theorems without
- leading @{text \<forall>}-quantifiers. For this we can implement the following function
- that strips off @{text "\<forall>"}s.
-*}
-
-ML %linenosgray{*fun strip_allq (Const (@{const_name "All"}, _) $ Abs body) =
- Term.dest_abs body |> snd |> strip_allq
- | strip_allq (Const (@{const_name "Trueprop"}, _) $ t) =
- strip_allq t
- | strip_allq t = t*}
-
-text {*
- We use in Line 2 the function @{ML_ind dest_abs in Term} for deconstructing abstractions,
- since this function deals correctly with potential name clashes. This function produces
- a pair consisting of the variable and the body of the abstraction. We are only interested
- in the body, which we feed into the recursive call. In Line 3 and 4, we also
- have to explicitly strip of the outermost @{term Trueprop}-coercion. Now we can
- install this function as the theorem style named @{text "my_strip_allq"}.
-*}
-
-setup %gray {*
- Term_Style.setup "my_strip_allq" (Scan.succeed (K strip_allq))
-*}
-
-text {*
- We can test this theorem style with the following theorem
-*}
-
-theorem style_test:
- shows "\<forall>x y z. (x, x) = (y, z)" sorry
-
-text {*
- Now printing out in a document the theorem @{thm [source] style_test} normally
- using @{text "@{thm \<dots>}"} produces
-
- \begin{isabelle}
- @{text "@{thm style_test}"}\\
- @{text ">"}~@{thm style_test}
- \end{isabelle}
-
- as expected. But with the theorem style @{text "@{thm (my_strip_allq) \<dots>}"}
- we obtain
-
- \begin{isabelle}
- @{text "@{thm (my_strip_allq) style_test}"}\\
- @{text ">"}~@{thm (my_strip_allq) style_test}\\
- \end{isabelle}
-
- without the leading quantifiers. We can improve this theorem style by explicitly
- giving a list of strings that should be used for the replacement of the
- variables. For this we implement the function which takes a list of strings
- and uses them as name in the outermost abstractions.
-*}
-
-ML{*fun rename_allq [] t = t
- | rename_allq (x::xs) (Const (@{const_name "All"}, U) $ Abs (_, T, t)) =
- Const (@{const_name "All"}, U) $ Abs (x, T, rename_allq xs t)
- | rename_allq xs (Const (@{const_name "Trueprop"}, U) $ t) =
- rename_allq xs t
- | rename_allq _ t = t*}
-
-text {*
- We can now install a the modified theorem style as follows
-*}
-
-setup %gray {* let
- val parser = Scan.repeat Args.name
- fun action xs = K (rename_allq xs #> strip_allq)
-in
- Term_Style.setup "my_strip_allq2" (parser >> action)
-end *}
-
-text {*
- The parser reads a list of names. In the function @{text action} we first
- call @{ML rename_allq} with the parsed list, then we call @{ML strip_allq}
- on the resulting term. We can now suggest, for example, two variables for
- stripping off the first two @{text \<forall>}-quantifiers.
-
-
- \begin{isabelle}
- @{text "@{thm (my_strip_allq2 x' x'') style_test}"}\\
- @{text ">"}~@{thm (my_strip_allq2 x' x'') style_test}
- \end{isabelle}
-
- Such theorem styles allow one to print out theorems in documents formatted to
- ones heart content. Next we explain theorem attributes, which is another
- mechanism for dealing with theorems.
-
- \begin{readmore}
- Theorem styles are implemented in @{ML_file "Pure/Thy/term_style.ML"}.
- \end{readmore}
-*}
-
-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 of
- theorems. 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
- theorems (for example @{text "[symmetric]"} and @{text "[THEN \<dots>]"}), and the second
- stores theorems somewhere as data (for example @{text "[simp]"}, which adds
- theorems 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 in Drule}
- 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 instance), we use the parser @{ML
- Scan.succeed}. 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 "theory -> theory"}, which
- can be used for example with \isacommand{setup} or with
- @{ML "Context.>> o Context.map_theory"}.\footnote{\bf FIXME: explain what happens here.}
-
- 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 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 => fold (curry ((op RS) o swap)) thms thm)*}
-
-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 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\label{sec:theories} (TBD) *}
-
-
-text {*
- Theories are the most fundamental building blocks in Isabelle. They
- contain 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 theort can be querried 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}
-
-
- 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"}
-
- \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 {*
- @{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 {* 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 hanging
- 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 FIXME: 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 {* 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 {* Summary *}
-
-text {*
- \begin{conventions}
- \begin{itemize}
- \item Start with a proper context and then pass it around
- through all your functions.
- \end{itemize}
- \end{conventions}
-*}
-
-end