# HG changeset patch # User Christian Urban # Date 1250847734 -7200 # Node ID efb5fff99c96c0c144e1b2f9290917a562802b6a # Parent d69214e47ef9d4d13b11143e63c4f7f5fea43401 split up the first-steps section into two chapters diff -r d69214e47ef9 -r efb5fff99c96 ProgTutorial/FirstSteps.thy --- a/ProgTutorial/FirstSteps.thy Thu Aug 20 23:30:51 2009 +0200 +++ b/ProgTutorial/FirstSteps.thy Fri Aug 21 11:42:14 2009 +0200 @@ -831,1638 +831,4 @@ own document antiquotations. *} -section {* Terms and Types *} - -text {* - One way to construct Isabelle terms, is by using the antiquotation - \mbox{@{text "@{term \}"}}. For example - - @{ML_response [display,gray] -"@{term \"(a::nat) + b = c\"}" -"Const (\"op =\", \) $ - (Const (\"HOL.plus_class.plus\", \) $ \ $ \) $ \"} - - will show the term @{term "(a::nat) + b = c"}, but printed using the internal - representation corresponding to the datatype @{ML_type "term"} defined as follows: -*} - -ML_val %linenosgray{*datatype term = - Const of string * typ -| Free of string * typ -| Var of indexname * typ -| Bound of int -| Abs of string * typ * term -| $ of term * term *} - -text {* - This datatype implements lambda-terms typed in Church-style. - As can be seen in Line 5, terms use the usual de Bruijn index mechanism - for representing bound variables. For - example in - - @{ML_response_fake [display, gray] - "@{term \"\x y. x y\"}" - "Abs (\"x\", \"'a \ '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 \}"}). If you deal with theorems, you have to, - however, observe the distinction. The reason is that only schematic - varaibles can be instantiated with terms when a theorem is applied. A - similar distinction between free and schematic variables holds for types - (see below). - - \begin{readmore} - Terms and types are described in detail in \isccite{sec:terms}. Their - definition and many useful operations are implemented in @{ML_file "Pure/term.ML"}. - For constructing terms involving HOL constants, many helper functions are defined - in @{ML_file "HOL/Tools/hologic.ML"}. - \end{readmore} - - Constructing terms via antiquotations has the advantage that only typable - terms can be constructed. For example - - @{ML_response_fake_both [display,gray] - "@{term \"x x\"}" - "Type unification failed: Occurs check!"} - - raises a typing error, while it perfectly ok to construct the term - - @{ML_response_fake [display,gray] -"let - val omega = Free (\"x\", @{typ nat}) $ Free (\"x\", @{typ nat}) -in - tracing (string_of_term @{context} omega) -end" - "x x"} - - with the raw ML-constructors. - - Sometimes the internal representation of terms can be surprisingly different - from what you see at the user-level, because the layers of - parsing/type-checking/pretty printing can be quite elaborate. - - \begin{exercise} - Look at the internal term representation of the following terms, and - find out why they are represented like this: - - \begin{itemize} - \item @{term "case x of 0 \ 0 | Suc y \ y"} - \item @{term "\(x,y). P y x"} - \item @{term "{ [x::int] | x. x \ -2 }"} - \end{itemize} - - Hint: The third term is already quite big, and the pretty printer - may omit parts of it by default. If you want to see all of it, you - can use the following ML-function to set the printing depth to a higher - value: - - @{ML [display,gray] "print_depth 50"} - \end{exercise} - - The antiquotation @{text "@{prop \}"} constructs terms by inserting the - usually invisible @{text "Trueprop"}-coercions whenever necessary. - Consider for example the pairs - -@{ML_response [display,gray] "(@{term \"P x\"}, @{prop \"P x\"})" -"(Free (\"P\", \) $ Free (\"x\", \), - Const (\"Trueprop\", \) $ (Free (\"P\", \) $ Free (\"x\", \)))"} - - where a coercion is inserted in the second component and - - @{ML_response [display,gray] "(@{term \"P x \ Q x\"}, @{prop \"P x \ Q x\"})" - "(Const (\"==>\", \) $ \ $ \, - Const (\"==>\", \) $ \ $ \)"} - - where it is not (since it is already constructed by a meta-implication). - The purpose of the @{text "Trueprop"}-coercion is to embed formulae of - an object logic, for example HOL, into the meta-logic of Isabelle. It - is needed whenever a term is constructed that will be proved as a theorem. - - As already seen above, types can be constructed using the antiquotation - @{text "@{typ \}"}. For example: - - @{ML_response_fake [display,gray] "@{typ \"bool \ nat\"}" "bool \ nat"} - - The corresponding datatype is -*} - -ML_val{*datatype typ = - Type of string * typ list -| TFree of string * sort -| TVar of indexname * sort *} - -text {* - Like with terms, there is the distinction between free type - variables (term-constructor @{ML "TFree"} and schematic - type variables (term-constructor @{ML "TVar"}). A type constant, - like @{typ "int"} or @{typ bool}, are types with an empty list - of argument types. However, it is a bit difficult to show an - example, because Isabelle always pretty-prints types (unlike terms). - Here is a contrived example: - - @{ML_response [display, gray] - "if Type (\"bool\", []) = @{typ \"bool\"} then true else false" - "true"} - - \begin{readmore} - Types are described in detail in \isccite{sec:types}. Their - definition and many useful operations are implemented - in @{ML_file "Pure/type.ML"}. - \end{readmore} -*} - -section {* Constructing Terms and Types Manually\label{sec:terms_types_manually} *} - -text {* - While antiquotations are very convenient for constructing terms, they can - only construct fixed terms (remember they are ``linked'' at compile-time). - However, you often need to construct terms dynamically. For example, a - function that returns the implication @{text "\(x::nat). P x \ 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 "\(x::nat). P x \ Q x"} *} - -text {* - To see this, apply @{text "@{term S}"} and @{text "@{term T}"} - to both functions. With @{ML make_imp} you obtain the intended term involving - the given arguments - - @{ML_response [display,gray] "make_imp @{term S} @{term T}" -"Const \ $ - Abs (\"x\", Type (\"nat\",[]), - Const \ $ (Free (\"S\",\) $ \) $ (Free (\"T\",\) $ \))"} - - 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 \ $ - Abs (\"x\", \, - Const \ $ (Const \ $ (Free (\"P\",\) $ \)) $ - (Const \ $ (Free (\"Q\",\) $ \)))"} - - There are a number of handy functions that are frequently used for - constructing terms. One is the function @{ML_ind list_comb}, which takes a term - and a list of terms as arguments, and produces as output the term - list applied to the term. For example - -@{ML_response_fake [display,gray] -"let - val trm = @{term \"P::nat\"} - val args = [@{term \"True\"}, @{term \"False\"}] -in - list_comb (trm, args) -end" -"Free (\"P\", \"nat\") $ Const (\"True\", \"bool\") $ Const (\"False\", \"bool\")"} - - Another handy function is @{ML_ind lambda}, which abstracts a variable - in a term. For example - - @{ML_response_fake [display,gray] -"let - val x_nat = @{term \"x::nat\"} - val trm = @{term \"(P::nat \ bool) x\"} -in - lambda x_nat trm -end" - "Abs (\"x\", \"nat\", Free (\"P\", \"bool \ bool\") $ Bound 0)"} - - In this example, @{ML lambda} produces a de Bruijn index (i.e.~@{ML "Bound 0"}), - and an abstraction. It also records the type of the abstracted - variable and for printing purposes also its name. Note that because of the - typing annotation on @{text "P"}, the variable @{text "x"} in @{text "P x"} - is of the same type as the abstracted variable. If it is of different type, - as in - - @{ML_response_fake [display,gray] -"let - val x_int = @{term \"x::int\"} - val trm = @{term \"(P::nat \ bool) x\"} -in - lambda x_int trm -end" - "Abs (\"x\", \"int\", Free (\"P\", \"nat \ bool\") $ Free (\"x\", \"nat\"))"} - - then the variable @{text "Free (\"x\", \"int\")"} is \emph{not} abstracted. - This is a fundamental principle - of Church-style typing, where variables with the same name still differ, if they - have different type. - - There is also the function @{ML_ind subst_free} with which terms can be - replaced by other terms. For example below, we will replace in @{term - "(f::nat \ nat \ nat) 0 x"} the subterm @{term "(f::nat \ nat \ nat) 0"} by - @{term y}, and @{term x} by @{term True}. - - @{ML_response_fake [display,gray] -"let - val sub1 = (@{term \"(f::nat \ nat \ nat) 0\"}, @{term \"y::nat \ nat\"}) - val sub2 = (@{term \"x::nat\"}, @{term \"True\"}) - val trm = @{term \"((f::nat \ nat \ nat) 0) x\"} -in - subst_free [sub1, sub2] trm -end" - "Free (\"y\", \"nat \ 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 \"(\y::nat. y)\"}, @{term \"x::nat\"}) - val trm = @{term \"(\x::nat. x)\"} -in - subst_free [sub] trm -end" - "Free (\"x\", \"nat\")"} - - Similarly the function @{ML_ind subst_bounds}, replaces lose bound - variables with terms. To see how this function works, let us implement a - function that strips off the outermost quantifiers in a term. -*} - -ML{*fun strip_alls (Const ("All", _) $ Abs (n, T, t)) = - strip_alls t |>> cons (Free (n, T)) - | strip_alls t = ([], t) *} - -text {* - The function returns a pair consisting of the stripped off variables and - the body of the universal quantifications. For example - - @{ML_response_fake [display, gray] - "strip_alls @{term \"\x y. x = (y::bool)\"}" -"([Free (\"x\", \"bool\"), Free (\"y\", \"bool\")], - Const (\"op =\", \) $ Bound 1 $ Bound 0)"} - - After calling @{ML strip_alls}, you obtain a term with lose bound variables. With - the function @{ML subst_bounds}, you can replace these lose @{ML_ind - Bound}s with the stripped off variables. - - @{ML_response_fake [display, gray, linenos] - "let - val (vrs, trm) = strip_alls @{term \"\x y. x = (y::bool)\"} -in - subst_bounds (rev vrs, trm) - |> string_of_term @{context} - |> tracing -end" - "x = y"} - - Note that in Line 4 we had to reverse the list of variables that @{ML strip_alls} - returned. The reason is that the head of the list the function @{ML subst_bounds} - takes is the replacement for @{ML "Bound 0"}, the next element for @{ML "Bound 1"} - and so on. - - There are many convenient functions that construct specific HOL-terms. For - example @{ML_ind mk_eq in HOLogic} constructs an equality out of two terms. - The types needed in this equality are calculated from the type of the - arguments. For example - -@{ML_response_fake [gray,display] -"let - val eq = HOLogic.mk_eq (@{term \"True\"}, @{term \"False\"}) -in - string_of_term @{context} eq - |> tracing -end" - "True = False"} -*} - -text {* - \begin{readmore} - There are many functions in @{ML_file "Pure/term.ML"}, @{ML_file - "Pure/logic.ML"} and @{ML_file "HOL/Tools/hologic.ML"} that make such manual - constructions of terms and types easier. - \end{readmore} - - When constructing terms manually, there are a few subtle issues with - constants. They usually crop up when pattern matching terms or types, or - when constructing them. While it is perfectly ok to write the function - @{text is_true} as follows -*} - -ML{*fun is_true @{term True} = true - | is_true _ = false*} - -text {* - this does not work for picking out @{text "\"}-quantified terms. Because - the function -*} - -ML{*fun is_all (@{term All} $ _) = true - | is_all _ = false*} - -text {* - will not correctly match the formula @{prop[source] "\x::nat. P x"}: - - @{ML_response [display,gray] "is_all @{term \"\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 \ bool) \ 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 \"\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\", \)"} - - 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\", \), - Const (\"HOL.times_class.times\", \))"} - - 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 \}"}. 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 \}"}. - For example - - @{ML_response [display,gray] - "@{type_name \"list\"}" "\"List.list\""} - - (FIXME: Explain the following better.) - - Occasionally you have to calculate what the ``base'' name of a given - constant is. For this you can use the function @{ML_ind "Sign.extern_const"} or - @{ML_ind Long_Name.base_name}. For example: - - @{ML_response [display,gray] "Sign.extern_const @{theory} \"List.list.Nil\"" "\"Nil\""} - - The difference between both functions is that @{ML extern_const in Sign} returns - the smallest name that is still unique, whereas @{ML base_name in Long_Name} always - strips off all qualifiers. - - \begin{readmore} - Functions about naming are implemented in @{ML_file "Pure/General/name_space.ML"}; - functions about signatures in @{ML_file "Pure/sign.ML"}. - \end{readmore} - - Although types of terms can often be inferred, there are many - situations where you need to construct types manually, especially - when defining constants. For example the function returning a function - type is as follows: - -*} - -ML{*fun make_fun_type ty1 ty2 = Type ("fun", [ty1, ty2]) *} - -text {* This can be equally written with the combinator @{ML_ind "-->"} as: *} - -ML{*fun make_fun_type ty1 ty2 = ty1 --> ty2 *} - -text {* - If you want to construct a function type with more than one argument - type, then you can use @{ML_ind "--->"}. -*} - -ML{*fun make_fun_types tys ty = tys ---> ty *} - -text {* - A handy function for manipulating terms is @{ML_ind map_types}: it takes a - function and applies it to every type in a term. You can, for example, - change every @{typ nat} in a term into an @{typ int} using the function: -*} - -ML{*fun nat_to_int ty = - (case ty of - @{typ nat} => @{typ int} - | Type (s, tys) => Type (s, map nat_to_int tys) - | _ => ty)*} - -text {* - Here is an example: - -@{ML_response_fake [display,gray] -"map_types nat_to_int @{term \"a = (1::nat)\"}" -"Const (\"op =\", \"int \ int \ 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 + \ + t\<^isub>n"} (whereby @{text "n"} might be one) - and returns the reversed sum @{text "t\<^isub>n + \ + t\<^isub>2 + t\<^isub>1"}. Assume - the @{text "t\<^isub>i"} can be arbitrary expressions and also note that @{text "+"} - associates to the left. Try your function on some examples. - \end{exercise} - - \begin{exercise}\label{fun:makesum} - Write a function which takes two terms representing natural numbers - in unary notation (like @{term "Suc (Suc (Suc 0))"}), and produces the - number representing their sum. - \end{exercise} - - \begin{exercise}\footnote{Personal communication of - de Bruijn to Dyckhoff.}\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} - - For n=3 this function returns the term - - \begin{center} - \begin{tabular}{l} - (P 1 = P 2 $\longrightarrow$ P 1 $\wedge$ P 2 $\wedge$ P 3) $\wedge$\\ - (P 2 = P 3 $\longrightarrow$ P 1 $\wedge$ P 2 $\wedge$ P 3) $\wedge$\\ - (P 3 = P 1 $\longrightarrow$ P 1 $\wedge$ P 2 $\wedge$ P 3) $\longrightarrow$ P 1 $\wedge$ P 2 $\wedge$ P 3 - \end{tabular} - \end{center} - - Make sure you use the functions defined in @{ML_file "HOL/Tools/hologic.ML"} - for constructing the terms for the logical connectives. - \end{exercise} -*} - - -section {* Type-Checking\label{sec:typechecking} *} - -text {* - - You can freely construct and manipulate @{ML_type "term"}s and @{ML_type - typ}es, since they are just arbitrary unchecked trees. However, you - eventually want to see if a term is well-formed, or type-checks, relative to - a theory. Type-checking is done via the function @{ML_ind cterm_of}, which - converts a @{ML_type term} into a @{ML_type cterm}, a \emph{certified} - term. Unlike @{ML_type term}s, which are just trees, @{ML_type "cterm"}s are - abstract objects that are guaranteed to be type-correct, and they can only - be constructed via ``official interfaces''. - - - Type-checking is always relative to a theory context. For now we use - the @{ML "@{theory}"} antiquotation to get hold of the current theory. - 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 \"} - - yields an error (since the term is not typable). A slightly more elaborate - example that type-checks is: - -@{ML_response_fake [display,gray] -"let - val natT = @{typ \"nat\"} - val zero = @{term \"0::nat\"} -in - cterm_of @{theory} - (Const (@{const_name plus}, natT --> natT --> natT) $ zero $ zero) -end" "0 + 0"} - - In Isabelle not just terms need to be certified, but also types. For example, - you obtain the certified type for the Isabelle type @{typ "nat \ bool"} on - the ML-level as follows: - - @{ML_response_fake [display,gray] - "ctyp_of @{theory} (@{typ nat} --> @{typ bool})" - "nat \ bool"} - - or with the antiquotation: - - @{ML_response_fake [display,gray] - "@{ctyp \"nat \ bool\"}" - "nat \ bool"} - - \begin{readmore} - For functions related to @{ML_type cterm}s and @{ML_type ctyp}s see - the file @{ML_file "Pure/thm.ML"}. - \end{readmore} - - Remember Isabelle follows the Church-style typing for terms, i.e., a term contains - enough typing information (constants, free variables and abstractions all have typing - information) so that it is always clear what the type of a term is. - Given a well-typed term, the function @{ML_ind type_of} returns the - type of a term. Consider for example: - - @{ML_response [display,gray] - "type_of (@{term \"f::nat \ 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 \ bool\"} $ @{term \"x::int\"})" - "*** Exception- TYPE (\"type_of: type mismatch in application\" \"} - - Since the complete traversal might sometimes be too costly and - not necessary, there is the function @{ML_ind fastype_of}, which - also returns the type of a term. - - @{ML_response [display,gray] - "fastype_of (@{term \"f::nat \ 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 \ bool\"} $ @{term \"x::int\"})" "bool"} - - where no error is detected. - - Sometimes it is a bit inconvenient to construct a term with - complete typing annotations, especially in cases where the typing - information is redundant. A short-cut is to use the ``place-holder'' - type @{ML_ind dummyT} and then let type-inference figure out the - complete type. An example is as follows: - - @{ML_response_fake [display,gray] -"let - val c = Const (@{const_name \"plus\"}, dummyT) - val o = @{term \"1::nat\"} - val v = Free (\"x\", dummyT) -in - Syntax.check_term @{context} (c $ o $ v) -end" -"Const (\"HOL.plus_class.plus\", \"nat \ nat \ 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} - - (FIXME: say something about sorts) - - \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 \ref{ch:solutions} when they receive an ill-typed term - as input. - \end{exercise} -*} - - -section {* Theorems *} - -text {* - Just like @{ML_type cterm}s, theorems are abstract objects of type @{ML_type thm} - that can only be built by going through interfaces. As a consequence, every proof - in Isabelle is correct by construction. This follows the tradition of the LCF approach - \cite{GordonMilnerWadsworth79}. - - - To see theorems in ``action'', let us give a proof on the ML-level for the following - statement: -*} - - lemma - assumes assm\<^isub>1: "\(x::nat). P x \ Q x" - and assm\<^isub>2: "P t" - shows "Q t" (*<*)oops(*>*) - -text {* - The corresponding ML-code is as follows: - -@{ML_response_fake [display,gray] -"let - val assm1 = @{cprop \"\(x::nat). P x \ Q x\"} - val assm2 = @{cprop \"(P::nat\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" "\\x. P x \ Q x; P t\ \ Q t"} - - This code-snippet constructs the following proof: - - \[ - \infer[(@{text "\"}$-$intro)]{\vdash @{prop "(\x. P x \ Q x) \ P t \ Q t"}} - {\infer[(@{text "\"}$-$intro)]{@{prop "\x. P x \ Q x"} \vdash @{prop "P t \ Q t"}} - {\infer[(@{text "\"}$-$elim)]{@{prop "\x. P x \ Q x"}, @{prop "P t"} \vdash @{prop "Q t"}} - {\infer[(@{text "\"}$-$elim)]{@{prop "\x. P x \ Q x"} \vdash @{prop "P t \ Q t"}} - {\infer[(assume)]{@{prop "\x. P x \ Q x"} \vdash @{prop "\x. P x \ Q x"}}{}} - & - \infer[(assume)]{@{prop "P t"} \vdash @{prop "P t"}}{} - } - } - } - \] - - However, while we obtained a theorem as result, this theorem is not - yet stored in Isabelle's theorem database. So it cannot be referenced later - on. How to store theorems will be explained in Section~\ref{sec:storing}. - - \begin{readmore} - For the functions @{text "assume"}, @{text "forall_elim"} etc - see \isccite{sec:thms}. The basic functions for theorems are defined in - @{ML_file "Pure/thm.ML"}. - \end{readmore} - - (FIXME: handy functions working on theorems, like @{ML_ind rulify in ObjectLogic} and so on) - - (FIXME: how to add case-names to goal states - maybe in the - next section) - - (FIXME: example for how to add theorem styles) -*} - -ML {* -fun strip_assums_all (params, Const("all",_) $ Abs(a, T, t)) = - strip_assums_all ((a, T)::params, t) - | strip_assums_all (params, B) = (params, B) - -fun style_parm_premise i ctxt t = - let val prems = Logic.strip_imp_prems t in - if i <= length prems - then let val (params,t) = strip_assums_all([], nth prems (i - 1)) - in subst_bounds(map Free params, t) end - else error ("Not enough premises for prem" ^ string_of_int i ^ - " in propositon: " ^ string_of_term ctxt t) - end; -*} - -ML {* -strip_assums_all ([], @{term "\x y. A x y"}) -*} - -setup %gray {* - TermStyle.add_style "no_all_prem1" (style_parm_premise 1) #> - TermStyle.add_style "no_all_prem2" (style_parm_premise 2) -*} - - -section {* Setups (TBD) *} - -text {* - In the previous section we used \isacommand{setup} in order to make - a theorem attribute known to Isabelle. What happens behind the scenes - is that \isacommand{setup} expects a function of type - @{ML_type "theory -> theory"}: the input theory is the current theory and the - output the theory where the theory attribute has been stored. - - This is a fundamental principle in Isabelle. A similar situation occurs - for example with declaring constants. The function that declares a - constant on the ML-level is @{ML_ind add_consts_i in Sign}. - If you write\footnote{Recall that ML-code needs to be - enclosed in \isacommand{ML}~@{text "\ \ \"}.} -*} - -ML{*Sign.add_consts_i [(@{binding "BAR"}, @{typ "nat"}, NoSyn)] @{theory} *} - -text {* - for declaring the constant @{text "BAR"} with type @{typ nat} and - run the code, then you indeed obtain a theory as result. But if you - query the constant on the Isabelle level using the command \isacommand{term} - - \begin{isabelle} - \isacommand{term}~@{text [quotes] "BAR"}\\ - @{text "> \"BAR\" :: \"'a\""} - \end{isabelle} - - you do not obtain a constant of type @{typ nat}, but a free variable (printed in - blue) of polymorphic type. The problem is that the ML-expression above did - not register the declaration with the current theory. This is what the command - \isacommand{setup} is for. The constant is properly declared with -*} - -setup %gray {* Sign.add_consts_i [(@{binding "BAR"}, @{typ "nat"}, NoSyn)] *} - -text {* - Now - - \begin{isabelle} - \isacommand{term}~@{text [quotes] "BAR"}\\ - @{text "> \"BAR\" :: \"nat\""} - \end{isabelle} - - returns a (black) constant with the type @{typ nat}. - - A similar command is \isacommand{local\_setup}, which expects a function - of type @{ML_type "local_theory -> local_theory"}. Later on we will also - use the commands \isacommand{method\_setup} for installing methods in the - current theory and \isacommand{simproc\_setup} for adding new simprocs to - the current simpset. -*} - -section {* Theorem Attributes\label{sec:attributes} *} - -text {* - Theorem attributes are @{text "[symmetric]"}, @{text "[THEN \]"}, @{text - "[simp]"} and so on. Such attributes are \emph{neither} tags \emph{nor} flags - annotated to theorems, but functions that do further processing once a - theorem is proved. In particular, it is not possible to find out - what are all theorems that have a given attribute in common, unless of course - the function behind the attribute stores the theorems in a retrievable - data structure. - - If you want to print out all currently known attributes a theorem can have, - you can use the Isabelle command - - \begin{isabelle} - \isacommand{print\_attributes}\\ - @{text "> COMP: direct composition with rules (no lifting)"}\\ - @{text "> HOL.dest: declaration of Classical destruction rule"}\\ - @{text "> HOL.elim: declaration of Classical elimination rule"}\\ - @{text "> \"} - \end{isabelle} - - The theorem attributes fall roughly into two categories: the first category manipulates - the proved theorem (for example @{text "[symmetric]"} and @{text "[THEN \]"}), and the second - stores the proved theorem somewhere as data (for example @{text "[simp]"}, which adds - the theorem to the current simpset). - - To explain how to write your own attribute, let us start with an extremely simple - version of the attribute @{text "[symmetric]"}. The purpose of this attribute is - to produce the ``symmetric'' version of an equation. The main function behind - this attribute is -*} - -ML{*val my_symmetric = Thm.rule_attribute (fn _ => fn thm => thm RS @{thm sym})*} - -text {* - where the function @{ML_ind rule_attribute in Thm} expects a function taking a - context (which we ignore in the code above) and a theorem (@{text thm}), and - returns another theorem (namely @{text thm} resolved with the theorem - @{thm [source] sym}: @{thm sym[no_vars]}; the function @{ML_ind "RS"} - is explained in Section~\ref{sec:simpletacs}). The function - @{ML rule_attribute in Thm} then returns an attribute. - - Before we can use the attribute, we need to set it up. This can be done - using the Isabelle command \isacommand{attribute\_setup} as follows: -*} - -attribute_setup %gray my_sym = {* Scan.succeed my_symmetric *} - "applying the sym rule" - -text {* - Inside the @{text "\ \ \"}, we have to specify a parser - for the theorem attribute. Since the attribute does not expect any further - arguments (unlike @{text "[THEN \]"}, for example), we use the parser @{ML - Scan.succeed}. Later on we will also consider attributes taking further - arguments. An example for the attribute @{text "[my_sym]"} is the proof -*} - -lemma test[my_sym]: "2 = Suc (Suc 0)" by simp - -text {* - which stores the theorem @{thm test} under the name @{thm [source] test}. You - can see this, if you query the lemma: - - \begin{isabelle} - \isacommand{thm}~@{text "test"}\\ - @{text "> "}~@{thm test} - \end{isabelle} - - We can also use the attribute when referring to this theorem: - - \begin{isabelle} - \isacommand{thm}~@{text "test[my_sym]"}\\ - @{text "> "}~@{thm test[my_sym]} - \end{isabelle} - - An alternative for setting up an attribute is the function @{ML_ind setup in Attrib}. - So instead of using \isacommand{attribute\_setup}, you can also set up the - attribute as follows: -*} - -ML{*Attrib.setup @{binding "my_sym"} (Scan.succeed my_symmetric) - "applying the sym rule" *} - -text {* - This gives a function from @{ML_type "Context.theory -> Context.theory"}, which - can be used for example with \isacommand{setup}. - - As an example of a slightly more complicated theorem attribute, we implement - our own version of @{text "[THEN \]"}. This attribute will take a list of theorems - as argument and resolve the proved theorem with this list (one theorem - after another). The code for this attribute is -*} - -ML{*fun MY_THEN thms = - Thm.rule_attribute (fn _ => fn thm => foldl ((op RS) o swap) thm thms)*} - -text {* - where @{ML swap} swaps the components of a pair. The setup of this theorem - attribute uses the parser @{ML thms in Attrib}, which parses a list of - theorems. -*} - -attribute_setup %gray MY_THEN = {* Attrib.thms >> MY_THEN *} - "resolving the list of theorems with the proved theorem" - -text {* - You can, for example, use this theorem attribute to turn an equation into a - meta-equation: - - \begin{isabelle} - \isacommand{thm}~@{text "test[MY_THEN eq_reflection]"}\\ - @{text "> "}~@{thm test[MY_THEN eq_reflection]} - \end{isabelle} - - If you need the symmetric version as a meta-equation, you can write - - \begin{isabelle} - \isacommand{thm}~@{text "test[MY_THEN sym eq_reflection]"}\\ - @{text "> "}~@{thm test[MY_THEN sym eq_reflection]} - \end{isabelle} - - It is also possible to combine different theorem attributes, as in: - - \begin{isabelle} - \isacommand{thm}~@{text "test[my_sym, MY_THEN eq_reflection]"}\\ - @{text "> "}~@{thm test[my_sym, MY_THEN eq_reflection]} - \end{isabelle} - - However, here also a weakness of the concept - of theorem attributes shows through: since theorem attributes can be - arbitrary functions, they do not in general commute. 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 reference containing a list - of theorems. -*} - -ML{*val my_thms = ref ([] : thm list)*} - -text {* - The purpose of this reference is to store a list of theorems. - We are going to modify it by adding and deleting theorems. - However, a word of warning: such references must not - be used in any code that is meant to be more than just for testing purposes! - Here it is only used to illustrate matters. We will show later how to store - data properly without using references. - - We need to provide two functions that add and delete theorems from this list. - For this we use the two functions: -*} - -ML{*fun my_thm_add thm ctxt = - (my_thms := Thm.add_thm thm (!my_thms); ctxt) - -fun my_thm_del thm ctxt = - (my_thms := Thm.del_thm thm (!my_thms); ctxt)*} - -text {* - These functions take a theorem and a context and, for what we are explaining - here it is sufficient that they just return the context unchanged. They change - however the reference @{ML my_thms}, whereby the function - @{ML_ind add_thm in Thm} adds a theorem if it is not already included in - the list, and @{ML_ind del_thm in Thm} deletes one (both functions use the - predicate @{ML_ind eq_thm_prop in Thm}, which compares theorems according to - their proved propositions modulo alpha-equivalence). - - You can turn functions @{ML my_thm_add} and @{ML my_thm_del} into - attributes with the code -*} - -ML{*val my_add = Thm.declaration_attribute my_thm_add -val my_del = Thm.declaration_attribute my_thm_del *} - -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 - rough test only!" - -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] - "!my_thms" "[\"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] - "!my_thms" - "[\"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] - "!my_thms" - "[\"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. - - However, as said at the beginning of this example, using references for storing theorems is - \emph{not} the received way of doing such things. The received way is to - start a ``data slot'', below called @{text MyThmsData}, generated by the functor - @{text GenericDataFun}: -*} - -ML{*structure MyThmsData = GenericDataFun - (type T = thm list - val empty = [] - val extend = I - fun merge _ = Thm.merge_thms) *} - -text {* - The type @{text "T"} of this data slot is @{ML_type "thm list"}.\footnote{FIXME: give a pointer - to where data slots are explained properly.} - To use this data slot, you only have to change @{ML my_thm_add} and - @{ML my_thm_del} to: -*} - -ML{*val my_thm_add = MyThmsData.map o Thm.add_thm -val my_thm_del = MyThmsData.map o Thm.del_thm*} - -text {* - where @{ML MyThmsData.map} updates the data appropriately. The - corresponding theorem attributes are -*} - -ML{*val my_add = Thm.declaration_attribute my_thm_add -val my_del = Thm.declaration_attribute my_thm_del *} - -text {* - and the setup is as follows -*} - -attribute_setup %gray my_thms2 = {* Attrib.add_del my_add my_del *} - "properly maintaining a list of my_thms" - -text {* - Initially, the data slot is empty - - @{ML_response_fake [display,gray] - "MyThmsData.get (Context.Proof @{context})" - "[]"} - - but if you prove -*} - -lemma three[my_thms2]: "3 = Suc (Suc (Suc 0))" by simp - -text {* - then the lemma is recorded. - - @{ML_response_fake [display,gray] - "MyThmsData.get (Context.Proof @{context})" - "[\"3 = Suc (Suc (Suc 0))\"]"} - - With theorem attribute @{text my_thms2} you can also nicely see why it - is important to - store data in a ``data slot'' and \emph{not} in a reference. Backtrack - to the point just before the lemma @{thm [source] three} was proved and - check the the content of @{ML_struct MyThmsData}: it should be empty. - The addition has been properly retracted. Now consider the proof: -*} - -lemma four[my_thms]: "4 = Suc (Suc (Suc (Suc 0)))" by simp - -text {* - Checking the content of @{ML my_thms} gives - - @{ML_response_fake [display,gray] - "!my_thms" - "[\"4 = Suc (Suc (Suc (Suc 0)))\", \"True\"]"} - - as expected, but if you backtrack before the lemma @{thm [source] four}, the - content of @{ML my_thms} is unchanged. The backtracking mechanism - of Isabelle is completely oblivious about what to do with references, but - properly treats ``data slots''! - - Since storing theorems in a list is such a common task, there is the special - functor @{ML_functor Named_Thms}, which does most of the work for you. To obtain - a named theorem list, you just declare -*} - -ML{*structure FooRules = Named_Thms - (val name = "foo" - val description = "Rules for foo") *} - -text {* - and set up the @{ML_struct FooRules} with the command -*} - -setup %gray {* FooRules.setup *} - -text {* - This code declares a data slot where the theorems are stored, - an attribute @{text foo} (with the @{text add} and @{text del} options - for adding and deleting theorems) and an internal ML interface to retrieve and - modify the theorems. - - Furthermore, the facts are made available on the user-level under the dynamic - fact name @{text foo}. For example you can declare three lemmas to be of the kind - @{text foo} by: -*} - -lemma rule1[foo]: "A" sorry -lemma rule2[foo]: "B" sorry -lemma rule3[foo]: "C" sorry - -text {* and undeclare the first one by: *} - -declare rule1[foo del] - -text {* and query the remaining ones with: - - \begin{isabelle} - \isacommand{thm}~@{text "foo"}\\ - @{text "> ?C"}\\ - @{text "> ?B"} - \end{isabelle} - - On the ML-level the rules marked with @{text "foo"} can be retrieved - using the function @{ML FooRules.get}: - - @{ML_response_fake [display,gray] "FooRules.get @{context}" "[\"?C\",\"?B\"]"} - - \begin{readmore} - For more information see @{ML_file "Pure/Tools/named_thms.ML"}. - \end{readmore} - - (FIXME What are: @{text "theory_attributes"}, @{text "proof_attributes"}?) - - - \begin{readmore} - FIXME: @{ML_file "Pure/more_thm.ML"}; parsers for attributes is in - @{ML_file "Pure/Isar/attrib.ML"}...also explained in the chapter about - parsing. - \end{readmore} -*} - - - -section {* Theories, Contexts and Local Theories (TBD) *} - -text {* - There are theories, proof contexts and local theories (in this order, if you - want to order them). - - In contrast to an ordinary theory, which simply consists of a type - signature, as well as tables for constants, axioms and theorems, a local - theory contains additional context information, such as locally fixed - variables and local assumptions that may be used by the package. The type - @{ML_type local_theory} is identical to the type of \emph{proof contexts} - @{ML_type "Proof.context"}, although not every proof context constitutes a - valid local theory. -*} - -(* -ML{*signature UNIVERSAL_TYPE = -sig - type t - - val embed: unit -> ('a -> t) * (t -> 'a option) -end*} - -ML{*structure U:> UNIVERSAL_TYPE = - struct - type t = exn - - fun 'a embed () = - let - exception E of 'a - fun project (e: t): 'a option = - case e of - E a => SOME a - | _ => NONE - in - (E, project) - end - end*} - -text {* - The idea is that type t is the universal type and that each call to embed - returns a new pair of functions (inject, project), where inject embeds a - value into the universal type and project extracts the value from the - universal type. A pair (inject, project) returned by embed works together in - that project u will return SOME v if and only if u was created by inject - v. If u was created by a different function inject', then project returns - NONE. - - in library.ML -*} - -ML_val{*structure Object = struct type T = exn end; *} - -ML{*functor Test (U: UNIVERSAL_TYPE): sig end = - struct - val (intIn: int -> U.t, intOut) = U.embed () - val r: U.t ref = ref (intIn 13) - val s1 = - case intOut (!r) of - NONE => "NONE" - | SOME i => Int.toString i - val (realIn: real -> U.t, realOut) = U.embed () - val () = r := realIn 13.0 - val s2 = - case intOut (!r) of - NONE => "NONE" - | SOME i => Int.toString i - val s3 = - case realOut (!r) of - NONE => "NONE" - | SOME x => Real.toString x - val () = tracing (concat [s1, " ", s2, " ", s3, "\n"]) - end*} - -ML_val{*structure t = Test(U) *} - -ML_val{*structure Datatab = TableFun(type key = int val ord = int_ord);*} - -ML {* LocalTheory.restore *} -ML {* LocalTheory.set_group *} -*) - -section {* Storing Theorems\label{sec:storing} (TBD) *} - -text {* @{ML_ind add_thms_dynamic in PureThy} *} - -local_setup %gray {* - LocalTheory.note Thm.theoremK - ((@{binding "allI_alt"}, []), [@{thm allI}]) #> snd *} - - -(* FIXME: some code below *) - -(*<*) -(* -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. - This infrastructure is loosely based on a paper by Oppen~\cite{Oppen80}. Most of - its functions do not operate on @{ML_type string}s, but on instances of the - 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 - -@{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 a possible - line break for example at each space. You can achieve this with the function - @{ML_ind breaks in Pretty}, which expects a list of pretty types and inserts a - possible line break in between every two elements in this list. To print - this list of pretty types as a single string, we concatenate them - with the function @{ML_ind blk in Pretty} as follows: - - -@{ML_response_fake [display,gray] -"let - val ptrs = map Pretty.str (space_explode \" \" test_str) -in - pprint (Pretty.blk (0, Pretty.breaks ptrs)) -end" -"fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar -fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar -fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar -fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar"} - - Here the layout of @{ML test_str} is much more pleasing to the - eye. The @{ML "0"} in @{ML_ind blk in Pretty} stands for no indentation - of the printed string. You can increase the indentation and obtain - -@{ML_response_fake [display,gray] -"let - val ptrs = map Pretty.str (space_explode \" \" test_str) -in - pprint (Pretty.blk (3, Pretty.breaks ptrs)) -end" -"fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar - fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar - fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar - fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar"} - - where starting from the second line the indent is 3. If you want - that every line starts with the same indent, you can use the - function @{ML_ind indent in Pretty} as follows: - -@{ML_response_fake [display,gray] -"let - val ptrs = map Pretty.str (space_explode \" \" test_str) -in - pprint (Pretty.indent 10 (Pretty.blk (0, Pretty.breaks ptrs))) -end" -" fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar - fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar - fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar - fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar"} - - If you want to print out a list of items separated by commas and - have the linebreaks handled properly, you can use the function - @{ML_ind commas in Pretty}. For example - -@{ML_response_fake [display,gray] -"let - val ptrs = map (Pretty.str o string_of_int) (99998 upto 100020) -in - pprint (Pretty.blk (0, Pretty.commas ptrs)) -end" -"99998, 99999, 100000, 100001, 100002, 100003, 100004, 100005, 100006, -100007, 100008, 100009, 100010, 100011, 100012, 100013, 100014, 100015, -100016, 100017, 100018, 100019, 100020"} - - where @{ML upto} generates a list of integers. You can print out this - list as a ``set'', that means enclosed inside @{text [quotes] "{"} and - @{text [quotes] "}"}, and separated by commas using the function - @{ML_ind enum in Pretty}. For example -*} - -text {* - -@{ML_response_fake [display,gray] -"let - val ptrs = map (Pretty.str o string_of_int) (99998 upto 100020) -in - pprint (Pretty.enum \",\" \"{\" \"}\" ptrs) -end" -"{99998, 99999, 100000, 100001, 100002, 100003, 100004, 100005, 100006, - 100007, 100008, 100009, 100010, 100011, 100012, 100013, 100014, 100015, - 100016, 100017, 100018, 100019, 100020}"} - - As can be seen, this function prints out the ``set'' so that starting - from the second, each new line as 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 ptrs = map (Pretty.str o string_of_int) (1 upto 22) -in - pprint (Pretty.blk (0, and_list ptrs)) -end" -"1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 -and 22"} - - 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 \ 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 \ 'a \ bool) \ bool) \ bool) \ bool) \ bool) \ bool\"."} - - - FIXME: TBD below -*} - -ML {* pprint (Pretty.big_list "header" (map (Pretty.str o string_of_int) (4 upto 10))) *} - -text {* -chunks inserts forced breaks (unlike blk where you have to do this yourself) -*} - -ML {* (Pretty.chunks [Pretty.str "a", Pretty.str "b"], - Pretty.blk (0, [Pretty.str "a", Pretty.str "b"])) *} - -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 |> priority -*} - -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 *} - - -ML {* Pretty.mark Markup.subgoal (Pretty.str "foo") |> Pretty.string_of |> tracing *} -ML {* (Pretty.str "bar") |> Pretty.string_of |> tracing *} - -text {* - Still to be done: - - What happens with 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 -*} - - -ML {* Pretty.mark Markup.hilite (Pretty.str "foo") |> Pretty.string_of |> tracing *} -ML {* (Pretty.str "bar") |> Pretty.string_of |> tracing *} - -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 - - -section {* Misc (TBD) *} - -ML {*Datatype.get_info @{theory} "List.list"*} - -section {* Name Space Issues (TBD) *} - - end diff -r d69214e47ef9 -r efb5fff99c96 ProgTutorial/General.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/ProgTutorial/General.thy Fri Aug 21 11:42:14 2009 +0200 @@ -0,0 +1,1648 @@ +theory General +imports Base FirstSteps +begin + +chapter {* General Infrastructure *} + +section {* Terms and Types *} + +text {* + One way to construct Isabelle terms, is by using the antiquotation + \mbox{@{text "@{term \}"}}. For example + + @{ML_response [display,gray] +"@{term \"(a::nat) + b = c\"}" +"Const (\"op =\", \) $ + (Const (\"HOL.plus_class.plus\", \) $ \ $ \) $ \"} + + will show the term @{term "(a::nat) + b = c"}, but printed using the internal + representation corresponding to the datatype @{ML_type "term"} defined as follows: +*} + +ML_val %linenosgray{*datatype term = + Const of string * typ +| Free of string * typ +| Var of indexname * typ +| Bound of int +| Abs of string * typ * term +| $ of term * term *} + +text {* + This datatype implements lambda-terms typed in Church-style. + As can be seen in Line 5, terms use the usual de Bruijn index mechanism + for representing bound variables. For + example in + + @{ML_response_fake [display, gray] + "@{term \"\x y. x y\"}" + "Abs (\"x\", \"'a \ '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 \}"}). If you deal with theorems, you have to, + however, observe the distinction. The reason is that only schematic + varaibles can be instantiated with terms when a theorem is applied. A + similar distinction between free and schematic variables holds for types + (see below). + + \begin{readmore} + Terms and types are described in detail in \isccite{sec:terms}. Their + definition and many useful operations are implemented in @{ML_file "Pure/term.ML"}. + For constructing terms involving HOL constants, many helper functions are defined + in @{ML_file "HOL/Tools/hologic.ML"}. + \end{readmore} + + Constructing terms via antiquotations has the advantage that only typable + terms can be constructed. For example + + @{ML_response_fake_both [display,gray] + "@{term \"x x\"}" + "Type unification failed: Occurs check!"} + + raises a typing error, while it perfectly ok to construct the term + + @{ML_response_fake [display,gray] +"let + val omega = Free (\"x\", @{typ nat}) $ Free (\"x\", @{typ nat}) +in + tracing (string_of_term @{context} omega) +end" + "x x"} + + with the raw ML-constructors. + + Sometimes the internal representation of terms can be surprisingly different + from what you see at the user-level, because the layers of + parsing/type-checking/pretty printing can be quite elaborate. + + \begin{exercise} + Look at the internal term representation of the following terms, and + find out why they are represented like this: + + \begin{itemize} + \item @{term "case x of 0 \ 0 | Suc y \ y"} + \item @{term "\(x,y). P y x"} + \item @{term "{ [x::int] | x. x \ -2 }"} + \end{itemize} + + Hint: The third term is already quite big, and the pretty printer + may omit parts of it by default. If you want to see all of it, you + can use the following ML-function to set the printing depth to a higher + value: + + @{ML [display,gray] "print_depth 50"} + \end{exercise} + + The antiquotation @{text "@{prop \}"} constructs terms by inserting the + usually invisible @{text "Trueprop"}-coercions whenever necessary. + Consider for example the pairs + +@{ML_response [display,gray] "(@{term \"P x\"}, @{prop \"P x\"})" +"(Free (\"P\", \) $ Free (\"x\", \), + Const (\"Trueprop\", \) $ (Free (\"P\", \) $ Free (\"x\", \)))"} + + where a coercion is inserted in the second component and + + @{ML_response [display,gray] "(@{term \"P x \ Q x\"}, @{prop \"P x \ Q x\"})" + "(Const (\"==>\", \) $ \ $ \, + Const (\"==>\", \) $ \ $ \)"} + + where it is not (since it is already constructed by a meta-implication). + The purpose of the @{text "Trueprop"}-coercion is to embed formulae of + an object logic, for example HOL, into the meta-logic of Isabelle. It + is needed whenever a term is constructed that will be proved as a theorem. + + As already seen above, types can be constructed using the antiquotation + @{text "@{typ \}"}. For example: + + @{ML_response_fake [display,gray] "@{typ \"bool \ nat\"}" "bool \ nat"} + + The corresponding datatype is +*} + +ML_val{*datatype typ = + Type of string * typ list +| TFree of string * sort +| TVar of indexname * sort *} + +text {* + Like with terms, there is the distinction between free type + variables (term-constructor @{ML "TFree"} and schematic + type variables (term-constructor @{ML "TVar"}). A type constant, + like @{typ "int"} or @{typ bool}, are types with an empty list + of argument types. However, it is a bit difficult to show an + example, because Isabelle always pretty-prints types (unlike terms). + Here is a contrived example: + + @{ML_response [display, gray] + "if Type (\"bool\", []) = @{typ \"bool\"} then true else false" + "true"} + + \begin{readmore} + Types are described in detail in \isccite{sec:types}. Their + definition and many useful operations are implemented + in @{ML_file "Pure/type.ML"}. + \end{readmore} +*} + +section {* Constructing Terms and Types Manually\label{sec:terms_types_manually} *} + +text {* + While antiquotations are very convenient for constructing terms, they can + only construct fixed terms (remember they are ``linked'' at compile-time). + However, you often need to construct terms dynamically. For example, a + function that returns the implication @{text "\(x::nat). P x \ 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 "\(x::nat). P x \ Q x"} *} + +text {* + To see this, apply @{text "@{term S}"} and @{text "@{term T}"} + to both functions. With @{ML make_imp} you obtain the intended term involving + the given arguments + + @{ML_response [display,gray] "make_imp @{term S} @{term T}" +"Const \ $ + Abs (\"x\", Type (\"nat\",[]), + Const \ $ (Free (\"S\",\) $ \) $ (Free (\"T\",\) $ \))"} + + 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 \ $ + Abs (\"x\", \, + Const \ $ (Const \ $ (Free (\"P\",\) $ \)) $ + (Const \ $ (Free (\"Q\",\) $ \)))"} + + There are a number of handy functions that are frequently used for + constructing terms. One is the function @{ML_ind list_comb}, which takes a term + and a list of terms as arguments, and produces as output the term + list applied to the term. For example + +@{ML_response_fake [display,gray] +"let + val trm = @{term \"P::nat\"} + val args = [@{term \"True\"}, @{term \"False\"}] +in + list_comb (trm, args) +end" +"Free (\"P\", \"nat\") $ Const (\"True\", \"bool\") $ Const (\"False\", \"bool\")"} + + Another handy function is @{ML_ind lambda}, which abstracts a variable + in a term. For example + + @{ML_response_fake [display,gray] +"let + val x_nat = @{term \"x::nat\"} + val trm = @{term \"(P::nat \ bool) x\"} +in + lambda x_nat trm +end" + "Abs (\"x\", \"nat\", Free (\"P\", \"bool \ bool\") $ Bound 0)"} + + In this example, @{ML lambda} produces a de Bruijn index (i.e.~@{ML "Bound 0"}), + and an abstraction. It also records the type of the abstracted + variable and for printing purposes also its name. Note that because of the + typing annotation on @{text "P"}, the variable @{text "x"} in @{text "P x"} + is of the same type as the abstracted variable. If it is of different type, + as in + + @{ML_response_fake [display,gray] +"let + val x_int = @{term \"x::int\"} + val trm = @{term \"(P::nat \ bool) x\"} +in + lambda x_int trm +end" + "Abs (\"x\", \"int\", Free (\"P\", \"nat \ bool\") $ Free (\"x\", \"nat\"))"} + + then the variable @{text "Free (\"x\", \"int\")"} is \emph{not} abstracted. + This is a fundamental principle + of Church-style typing, where variables with the same name still differ, if they + have different type. + + There is also the function @{ML_ind subst_free} with which terms can be + replaced by other terms. For example below, we will replace in @{term + "(f::nat \ nat \ nat) 0 x"} the subterm @{term "(f::nat \ nat \ nat) 0"} by + @{term y}, and @{term x} by @{term True}. + + @{ML_response_fake [display,gray] +"let + val sub1 = (@{term \"(f::nat \ nat \ nat) 0\"}, @{term \"y::nat \ nat\"}) + val sub2 = (@{term \"x::nat\"}, @{term \"True\"}) + val trm = @{term \"((f::nat \ nat \ nat) 0) x\"} +in + subst_free [sub1, sub2] trm +end" + "Free (\"y\", \"nat \ 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 \"(\y::nat. y)\"}, @{term \"x::nat\"}) + val trm = @{term \"(\x::nat. x)\"} +in + subst_free [sub] trm +end" + "Free (\"x\", \"nat\")"} + + Similarly the function @{ML_ind subst_bounds}, replaces lose bound + variables with terms. To see how this function works, let us implement a + function that strips off the outermost quantifiers in a term. +*} + +ML{*fun strip_alls (Const ("All", _) $ Abs (n, T, t)) = + strip_alls t |>> cons (Free (n, T)) + | strip_alls t = ([], t) *} + +text {* + The function returns a pair consisting of the stripped off variables and + the body of the universal quantifications. For example + + @{ML_response_fake [display, gray] + "strip_alls @{term \"\x y. x = (y::bool)\"}" +"([Free (\"x\", \"bool\"), Free (\"y\", \"bool\")], + Const (\"op =\", \) $ Bound 1 $ Bound 0)"} + + After calling @{ML strip_alls}, you obtain a term with lose bound variables. With + the function @{ML subst_bounds}, you can replace these lose @{ML_ind + Bound}s with the stripped off variables. + + @{ML_response_fake [display, gray, linenos] + "let + val (vrs, trm) = strip_alls @{term \"\x y. x = (y::bool)\"} +in + subst_bounds (rev vrs, trm) + |> string_of_term @{context} + |> tracing +end" + "x = y"} + + Note that in Line 4 we had to reverse the list of variables that @{ML strip_alls} + returned. The reason is that the head of the list the function @{ML subst_bounds} + takes is the replacement for @{ML "Bound 0"}, the next element for @{ML "Bound 1"} + and so on. + + There are many convenient functions that construct specific HOL-terms. For + example @{ML_ind mk_eq in HOLogic} constructs an equality out of two terms. + The types needed in this equality are calculated from the type of the + arguments. For example + +@{ML_response_fake [gray,display] +"let + val eq = HOLogic.mk_eq (@{term \"True\"}, @{term \"False\"}) +in + string_of_term @{context} eq + |> tracing +end" + "True = False"} +*} + +text {* + \begin{readmore} + There are many functions in @{ML_file "Pure/term.ML"}, @{ML_file + "Pure/logic.ML"} and @{ML_file "HOL/Tools/hologic.ML"} that make such manual + constructions of terms and types easier. + \end{readmore} + + When constructing terms manually, there are a few subtle issues with + constants. They usually crop up when pattern matching terms or types, or + when constructing them. While it is perfectly ok to write the function + @{text is_true} as follows +*} + +ML{*fun is_true @{term True} = true + | is_true _ = false*} + +text {* + this does not work for picking out @{text "\"}-quantified terms. Because + the function +*} + +ML{*fun is_all (@{term All} $ _) = true + | is_all _ = false*} + +text {* + will not correctly match the formula @{prop[source] "\x::nat. P x"}: + + @{ML_response [display,gray] "is_all @{term \"\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 \ bool) \ 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 \"\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\", \)"} + + 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\", \), + Const (\"HOL.times_class.times\", \))"} + + 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 \}"}. 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 \}"}. + For example + + @{ML_response [display,gray] + "@{type_name \"list\"}" "\"List.list\""} + + (FIXME: Explain the following better.) + + Occasionally you have to calculate what the ``base'' name of a given + constant is. For this you can use the function @{ML_ind "Sign.extern_const"} or + @{ML_ind Long_Name.base_name}. For example: + + @{ML_response [display,gray] "Sign.extern_const @{theory} \"List.list.Nil\"" "\"Nil\""} + + The difference between both functions is that @{ML extern_const in Sign} returns + the smallest name that is still unique, whereas @{ML base_name in Long_Name} always + strips off all qualifiers. + + \begin{readmore} + Functions about naming are implemented in @{ML_file "Pure/General/name_space.ML"}; + functions about signatures in @{ML_file "Pure/sign.ML"}. + \end{readmore} + + Although types of terms can often be inferred, there are many + situations where you need to construct types manually, especially + when defining constants. For example the function returning a function + type is as follows: + +*} + +ML{*fun make_fun_type ty1 ty2 = Type ("fun", [ty1, ty2]) *} + +text {* This can be equally written with the combinator @{ML_ind "-->"} as: *} + +ML{*fun make_fun_type ty1 ty2 = ty1 --> ty2 *} + +text {* + If you want to construct a function type with more than one argument + type, then you can use @{ML_ind "--->"}. +*} + +ML{*fun make_fun_types tys ty = tys ---> ty *} + +text {* + A handy function for manipulating terms is @{ML_ind map_types}: it takes a + function and applies it to every type in a term. You can, for example, + change every @{typ nat} in a term into an @{typ int} using the function: +*} + +ML{*fun nat_to_int ty = + (case ty of + @{typ nat} => @{typ int} + | Type (s, tys) => Type (s, map nat_to_int tys) + | _ => ty)*} + +text {* + Here is an example: + +@{ML_response_fake [display,gray] +"map_types nat_to_int @{term \"a = (1::nat)\"}" +"Const (\"op =\", \"int \ int \ 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 + \ + t\<^isub>n"} (whereby @{text "n"} might be one) + and returns the reversed sum @{text "t\<^isub>n + \ + t\<^isub>2 + t\<^isub>1"}. Assume + the @{text "t\<^isub>i"} can be arbitrary expressions and also note that @{text "+"} + associates to the left. Try your function on some examples. + \end{exercise} + + \begin{exercise}\label{fun:makesum} + Write a function which takes two terms representing natural numbers + in unary notation (like @{term "Suc (Suc (Suc 0))"}), and produces the + number representing their sum. + \end{exercise} + + \begin{exercise}\footnote{Personal communication of + de Bruijn to Dyckhoff.}\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} + + For n=3 this function returns the term + + \begin{center} + \begin{tabular}{l} + (P 1 = P 2 $\longrightarrow$ P 1 $\wedge$ P 2 $\wedge$ P 3) $\wedge$\\ + (P 2 = P 3 $\longrightarrow$ P 1 $\wedge$ P 2 $\wedge$ P 3) $\wedge$\\ + (P 3 = P 1 $\longrightarrow$ P 1 $\wedge$ P 2 $\wedge$ P 3) $\longrightarrow$ P 1 $\wedge$ P 2 $\wedge$ P 3 + \end{tabular} + \end{center} + + Make sure you use the functions defined in @{ML_file "HOL/Tools/hologic.ML"} + for constructing the terms for the logical connectives. + \end{exercise} +*} + + +section {* Type-Checking\label{sec:typechecking} *} + +text {* + + You can freely construct and manipulate @{ML_type "term"}s and @{ML_type + typ}es, since they are just arbitrary unchecked trees. However, you + eventually want to see if a term is well-formed, or type-checks, relative to + a theory. Type-checking is done via the function @{ML_ind cterm_of}, which + converts a @{ML_type term} into a @{ML_type cterm}, a \emph{certified} + term. Unlike @{ML_type term}s, which are just trees, @{ML_type "cterm"}s are + abstract objects that are guaranteed to be type-correct, and they can only + be constructed via ``official interfaces''. + + + Type-checking is always relative to a theory context. For now we use + the @{ML "@{theory}"} antiquotation to get hold of the current theory. + 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 \"} + + yields an error (since the term is not typable). A slightly more elaborate + example that type-checks is: + +@{ML_response_fake [display,gray] +"let + val natT = @{typ \"nat\"} + val zero = @{term \"0::nat\"} +in + cterm_of @{theory} + (Const (@{const_name plus}, natT --> natT --> natT) $ zero $ zero) +end" "0 + 0"} + + In Isabelle not just terms need to be certified, but also types. For example, + you obtain the certified type for the Isabelle type @{typ "nat \ bool"} on + the ML-level as follows: + + @{ML_response_fake [display,gray] + "ctyp_of @{theory} (@{typ nat} --> @{typ bool})" + "nat \ bool"} + + or with the antiquotation: + + @{ML_response_fake [display,gray] + "@{ctyp \"nat \ bool\"}" + "nat \ bool"} + + \begin{readmore} + For functions related to @{ML_type cterm}s and @{ML_type ctyp}s see + the file @{ML_file "Pure/thm.ML"}. + \end{readmore} + + Remember Isabelle follows the Church-style typing for terms, i.e., a term contains + enough typing information (constants, free variables and abstractions all have typing + information) so that it is always clear what the type of a term is. + Given a well-typed term, the function @{ML_ind type_of} returns the + type of a term. Consider for example: + + @{ML_response [display,gray] + "type_of (@{term \"f::nat \ 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 \ bool\"} $ @{term \"x::int\"})" + "*** Exception- TYPE (\"type_of: type mismatch in application\" \"} + + Since the complete traversal might sometimes be too costly and + not necessary, there is the function @{ML_ind fastype_of}, which + also returns the type of a term. + + @{ML_response [display,gray] + "fastype_of (@{term \"f::nat \ 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 \ bool\"} $ @{term \"x::int\"})" "bool"} + + where no error is detected. + + Sometimes it is a bit inconvenient to construct a term with + complete typing annotations, especially in cases where the typing + information is redundant. A short-cut is to use the ``place-holder'' + type @{ML_ind dummyT} and then let type-inference figure out the + complete type. An example is as follows: + + @{ML_response_fake [display,gray] +"let + val c = Const (@{const_name \"plus\"}, dummyT) + val o = @{term \"1::nat\"} + val v = Free (\"x\", dummyT) +in + Syntax.check_term @{context} (c $ o $ v) +end" +"Const (\"HOL.plus_class.plus\", \"nat \ nat \ 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} + + (FIXME: say something about sorts) + + \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 \ref{ch:solutions} when they receive an ill-typed term + as input. + \end{exercise} +*} + + +section {* Theorems *} + +text {* + Just like @{ML_type cterm}s, theorems are abstract objects of type @{ML_type thm} + that can only be built by going through interfaces. As a consequence, every proof + in Isabelle is correct by construction. This follows the tradition of the LCF approach + \cite{GordonMilnerWadsworth79}. + + + To see theorems in ``action'', let us give a proof on the ML-level for the following + statement: +*} + + lemma + assumes assm\<^isub>1: "\(x::nat). P x \ Q x" + and assm\<^isub>2: "P t" + shows "Q t" (*<*)oops(*>*) + +text {* + The corresponding ML-code is as follows: + +@{ML_response_fake [display,gray] +"let + val assm1 = @{cprop \"\(x::nat). P x \ Q x\"} + val assm2 = @{cprop \"(P::nat\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" "\\x. P x \ Q x; P t\ \ Q t"} + + This code-snippet constructs the following proof: + + \[ + \infer[(@{text "\"}$-$intro)]{\vdash @{prop "(\x. P x \ Q x) \ P t \ Q t"}} + {\infer[(@{text "\"}$-$intro)]{@{prop "\x. P x \ Q x"} \vdash @{prop "P t \ Q t"}} + {\infer[(@{text "\"}$-$elim)]{@{prop "\x. P x \ Q x"}, @{prop "P t"} \vdash @{prop "Q t"}} + {\infer[(@{text "\"}$-$elim)]{@{prop "\x. P x \ Q x"} \vdash @{prop "P t \ Q t"}} + {\infer[(assume)]{@{prop "\x. P x \ Q x"} \vdash @{prop "\x. P x \ Q x"}}{}} + & + \infer[(assume)]{@{prop "P t"} \vdash @{prop "P t"}}{} + } + } + } + \] + + However, while we obtained a theorem as result, this theorem is not + yet stored in Isabelle's theorem database. So it cannot be referenced later + on. How to store theorems will be explained in Section~\ref{sec:storing}. + + \begin{readmore} + For the functions @{text "assume"}, @{text "forall_elim"} etc + see \isccite{sec:thms}. The basic functions for theorems are defined in + @{ML_file "Pure/thm.ML"}. + \end{readmore} + + (FIXME: handy functions working on theorems, like @{ML_ind rulify in ObjectLogic} and so on) + + (FIXME: @{ML_ind prove in Goal}) + + (FIXME: how to add case-names to goal states - maybe in the + next section) + + (FIXME: example for how to add theorem styles) +*} + +ML {* +fun strip_assums_all (params, Const("all",_) $ Abs(a, T, t)) = + strip_assums_all ((a, T)::params, t) + | strip_assums_all (params, B) = (params, B) + +fun style_parm_premise i ctxt t = + let val prems = Logic.strip_imp_prems t in + if i <= length prems + then let val (params,t) = strip_assums_all([], nth prems (i - 1)) + in subst_bounds(map Free params, t) end + else error ("Not enough premises for prem" ^ string_of_int i ^ + " in propositon: " ^ string_of_term ctxt t) + end; +*} + +ML {* +strip_assums_all ([], @{term "\x y. A x y"}) +*} + +setup %gray {* + TermStyle.add_style "no_all_prem1" (style_parm_premise 1) #> + TermStyle.add_style "no_all_prem2" (style_parm_premise 2) +*} + +lemma + shows "A \ B" + and "C \ D" +oops + + +section {* Setups (TBD) *} + +text {* + In the previous section we used \isacommand{setup} in order to make + a theorem attribute known to Isabelle. What happens behind the scenes + is that \isacommand{setup} expects a function of type + @{ML_type "theory -> theory"}: the input theory is the current theory and the + output the theory where the theory attribute has been stored. + + This is a fundamental principle in Isabelle. A similar situation occurs + for example with declaring constants. The function that declares a + constant on the ML-level is @{ML_ind add_consts_i in Sign}. + If you write\footnote{Recall that ML-code needs to be + enclosed in \isacommand{ML}~@{text "\ \ \"}.} +*} + +ML{*Sign.add_consts_i [(@{binding "BAR"}, @{typ "nat"}, NoSyn)] @{theory} *} + +text {* + for declaring the constant @{text "BAR"} with type @{typ nat} and + run the code, then you indeed obtain a theory as result. But if you + query the constant on the Isabelle level using the command \isacommand{term} + + \begin{isabelle} + \isacommand{term}~@{text [quotes] "BAR"}\\ + @{text "> \"BAR\" :: \"'a\""} + \end{isabelle} + + you do not obtain a constant of type @{typ nat}, but a free variable (printed in + blue) of polymorphic type. The problem is that the ML-expression above did + not register the declaration with the current theory. This is what the command + \isacommand{setup} is for. The constant is properly declared with +*} + +setup %gray {* Sign.add_consts_i [(@{binding "BAR"}, @{typ "nat"}, NoSyn)] *} + +text {* + Now + + \begin{isabelle} + \isacommand{term}~@{text [quotes] "BAR"}\\ + @{text "> \"BAR\" :: \"nat\""} + \end{isabelle} + + returns a (black) constant with the type @{typ nat}. + + A similar command is \isacommand{local\_setup}, which expects a function + of type @{ML_type "local_theory -> local_theory"}. Later on we will also + use the commands \isacommand{method\_setup} for installing methods in the + current theory and \isacommand{simproc\_setup} for adding new simprocs to + the current simpset. +*} + +section {* Theorem Attributes\label{sec:attributes} *} + +text {* + Theorem attributes are @{text "[symmetric]"}, @{text "[THEN \]"}, @{text + "[simp]"} and so on. Such attributes are \emph{neither} tags \emph{nor} flags + annotated to theorems, but functions that do further processing once a + theorem is proved. In particular, it is not possible to find out + what are all theorems that have a given attribute in common, unless of course + the function behind the attribute stores the theorems in a retrievable + data structure. + + If you want to print out all currently known attributes a theorem can have, + you can use the Isabelle command + + \begin{isabelle} + \isacommand{print\_attributes}\\ + @{text "> COMP: direct composition with rules (no lifting)"}\\ + @{text "> HOL.dest: declaration of Classical destruction rule"}\\ + @{text "> HOL.elim: declaration of Classical elimination rule"}\\ + @{text "> \"} + \end{isabelle} + + The theorem attributes fall roughly into two categories: the first category manipulates + the proved theorem (for example @{text "[symmetric]"} and @{text "[THEN \]"}), and the second + stores the proved theorem somewhere as data (for example @{text "[simp]"}, which adds + the theorem to the current simpset). + + To explain how to write your own attribute, let us start with an extremely simple + version of the attribute @{text "[symmetric]"}. The purpose of this attribute is + to produce the ``symmetric'' version of an equation. The main function behind + this attribute is +*} + +ML{*val my_symmetric = Thm.rule_attribute (fn _ => fn thm => thm RS @{thm sym})*} + +text {* + where the function @{ML_ind rule_attribute in Thm} expects a function taking a + context (which we ignore in the code above) and a theorem (@{text thm}), and + returns another theorem (namely @{text thm} resolved with the theorem + @{thm [source] sym}: @{thm sym[no_vars]}; the function @{ML_ind "RS"} + is explained in Section~\ref{sec:simpletacs}). The function + @{ML rule_attribute in Thm} then returns an attribute. + + Before we can use the attribute, we need to set it up. This can be done + using the Isabelle command \isacommand{attribute\_setup} as follows: +*} + +attribute_setup %gray my_sym = {* Scan.succeed my_symmetric *} + "applying the sym rule" + +text {* + Inside the @{text "\ \ \"}, we have to specify a parser + for the theorem attribute. Since the attribute does not expect any further + arguments (unlike @{text "[THEN \]"}, for example), we use the parser @{ML + Scan.succeed}. Later on we will also consider attributes taking further + arguments. An example for the attribute @{text "[my_sym]"} is the proof +*} + +lemma test[my_sym]: "2 = Suc (Suc 0)" by simp + +text {* + which stores the theorem @{thm test} under the name @{thm [source] test}. You + can see this, if you query the lemma: + + \begin{isabelle} + \isacommand{thm}~@{text "test"}\\ + @{text "> "}~@{thm test} + \end{isabelle} + + We can also use the attribute when referring to this theorem: + + \begin{isabelle} + \isacommand{thm}~@{text "test[my_sym]"}\\ + @{text "> "}~@{thm test[my_sym]} + \end{isabelle} + + An alternative for setting up an attribute is the function @{ML_ind setup in Attrib}. + So instead of using \isacommand{attribute\_setup}, you can also set up the + attribute as follows: +*} + +ML{*Attrib.setup @{binding "my_sym"} (Scan.succeed my_symmetric) + "applying the sym rule" *} + +text {* + This gives a function from @{ML_type "Context.theory -> Context.theory"}, which + can be used for example with \isacommand{setup}. + + As an example of a slightly more complicated theorem attribute, we implement + our own version of @{text "[THEN \]"}. This attribute will take a list of theorems + as argument and resolve the proved theorem with this list (one theorem + after another). The code for this attribute is +*} + +ML{*fun MY_THEN thms = + Thm.rule_attribute (fn _ => fn thm => foldl ((op RS) o swap) thm thms)*} + +text {* + where @{ML swap} swaps the components of a pair. The setup of this theorem + attribute uses the parser @{ML thms in Attrib}, which parses a list of + theorems. +*} + +attribute_setup %gray MY_THEN = {* Attrib.thms >> MY_THEN *} + "resolving the list of theorems with the proved theorem" + +text {* + You can, for example, use this theorem attribute to turn an equation into a + meta-equation: + + \begin{isabelle} + \isacommand{thm}~@{text "test[MY_THEN eq_reflection]"}\\ + @{text "> "}~@{thm test[MY_THEN eq_reflection]} + \end{isabelle} + + If you need the symmetric version as a meta-equation, you can write + + \begin{isabelle} + \isacommand{thm}~@{text "test[MY_THEN sym eq_reflection]"}\\ + @{text "> "}~@{thm test[MY_THEN sym eq_reflection]} + \end{isabelle} + + It is also possible to combine different theorem attributes, as in: + + \begin{isabelle} + \isacommand{thm}~@{text "test[my_sym, MY_THEN eq_reflection]"}\\ + @{text "> "}~@{thm test[my_sym, MY_THEN eq_reflection]} + \end{isabelle} + + However, here also a weakness of the concept + of theorem attributes shows through: since theorem attributes can be + arbitrary functions, they do not in general commute. 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 reference containing a list + of theorems. +*} + +ML{*val my_thms = ref ([] : thm list)*} + +text {* + The purpose of this reference is to store a list of theorems. + We are going to modify it by adding and deleting theorems. + However, a word of warning: such references must not + be used in any code that is meant to be more than just for testing purposes! + Here it is only used to illustrate matters. We will show later how to store + data properly without using references. + + We need to provide two functions that add and delete theorems from this list. + For this we use the two functions: +*} + +ML{*fun my_thm_add thm ctxt = + (my_thms := Thm.add_thm thm (!my_thms); ctxt) + +fun my_thm_del thm ctxt = + (my_thms := Thm.del_thm thm (!my_thms); ctxt)*} + +text {* + These functions take a theorem and a context and, for what we are explaining + here it is sufficient that they just return the context unchanged. They change + however the reference @{ML my_thms}, whereby the function + @{ML_ind add_thm in Thm} adds a theorem if it is not already included in + the list, and @{ML_ind del_thm in Thm} deletes one (both functions use the + predicate @{ML_ind eq_thm_prop in Thm}, which compares theorems according to + their proved propositions modulo alpha-equivalence). + + You can turn functions @{ML my_thm_add} and @{ML my_thm_del} into + attributes with the code +*} + +ML{*val my_add = Thm.declaration_attribute my_thm_add +val my_del = Thm.declaration_attribute my_thm_del *} + +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 - rough test only!" + +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] + "!my_thms" "[\"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] + "!my_thms" + "[\"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] + "!my_thms" + "[\"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. + + However, as said at the beginning of this example, using references for storing theorems is + \emph{not} the received way of doing such things. The received way is to + start a ``data slot'', below called @{text MyThmsData}, generated by the functor + @{text GenericDataFun}: +*} + +ML{*structure MyThmsData = GenericDataFun + (type T = thm list + val empty = [] + val extend = I + fun merge _ = Thm.merge_thms) *} + +text {* + The type @{text "T"} of this data slot is @{ML_type "thm list"}.\footnote{FIXME: give a pointer + to where data slots are explained properly.} + To use this data slot, you only have to change @{ML my_thm_add} and + @{ML my_thm_del} to: +*} + +ML{*val my_thm_add = MyThmsData.map o Thm.add_thm +val my_thm_del = MyThmsData.map o Thm.del_thm*} + +text {* + where @{ML MyThmsData.map} updates the data appropriately. The + corresponding theorem attributes are +*} + +ML{*val my_add = Thm.declaration_attribute my_thm_add +val my_del = Thm.declaration_attribute my_thm_del *} + +text {* + and the setup is as follows +*} + +attribute_setup %gray my_thms2 = {* Attrib.add_del my_add my_del *} + "properly maintaining a list of my_thms" + +text {* + Initially, the data slot is empty + + @{ML_response_fake [display,gray] + "MyThmsData.get (Context.Proof @{context})" + "[]"} + + but if you prove +*} + +lemma three[my_thms2]: "3 = Suc (Suc (Suc 0))" by simp + +text {* + then the lemma is recorded. + + @{ML_response_fake [display,gray] + "MyThmsData.get (Context.Proof @{context})" + "[\"3 = Suc (Suc (Suc 0))\"]"} + + With theorem attribute @{text my_thms2} you can also nicely see why it + is important to + store data in a ``data slot'' and \emph{not} in a reference. Backtrack + to the point just before the lemma @{thm [source] three} was proved and + check the the content of @{ML_struct MyThmsData}: it should be empty. + The addition has been properly retracted. Now consider the proof: +*} + +lemma four[my_thms]: "4 = Suc (Suc (Suc (Suc 0)))" by simp + +text {* + Checking the content of @{ML my_thms} gives + + @{ML_response_fake [display,gray] + "!my_thms" + "[\"4 = Suc (Suc (Suc (Suc 0)))\", \"True\"]"} + + as expected, but if you backtrack before the lemma @{thm [source] four}, the + content of @{ML my_thms} is unchanged. The backtracking mechanism + of Isabelle is completely oblivious about what to do with references, but + properly treats ``data slots''! + + Since storing theorems in a list is such a common task, there is the special + functor @{ML_functor Named_Thms}, which does most of the work for you. To obtain + a named theorem list, you just declare +*} + +ML{*structure FooRules = Named_Thms + (val name = "foo" + val description = "Rules for foo") *} + +text {* + and set up the @{ML_struct FooRules} with the command +*} + +setup %gray {* FooRules.setup *} + +text {* + This code declares a data slot where the theorems are stored, + an attribute @{text foo} (with the @{text add} and @{text del} options + for adding and deleting theorems) and an internal ML interface to retrieve and + modify the theorems. + + Furthermore, the facts are made available on the user-level under the dynamic + fact name @{text foo}. For example you can declare three lemmas to be of the kind + @{text foo} by: +*} + +lemma rule1[foo]: "A" sorry +lemma rule2[foo]: "B" sorry +lemma rule3[foo]: "C" sorry + +text {* and undeclare the first one by: *} + +declare rule1[foo del] + +text {* and query the remaining ones with: + + \begin{isabelle} + \isacommand{thm}~@{text "foo"}\\ + @{text "> ?C"}\\ + @{text "> ?B"} + \end{isabelle} + + On the ML-level the rules marked with @{text "foo"} can be retrieved + using the function @{ML FooRules.get}: + + @{ML_response_fake [display,gray] "FooRules.get @{context}" "[\"?C\",\"?B\"]"} + + \begin{readmore} + For more information see @{ML_file "Pure/Tools/named_thms.ML"}. + \end{readmore} + + (FIXME What are: @{text "theory_attributes"}, @{text "proof_attributes"}?) + + + \begin{readmore} + FIXME: @{ML_file "Pure/more_thm.ML"}; parsers for attributes is in + @{ML_file "Pure/Isar/attrib.ML"}...also explained in the chapter about + parsing. + \end{readmore} +*} + + + +section {* Theories, Contexts and Local Theories (TBD) *} + +text {* + There are theories, proof contexts and local theories (in this order, if you + want to order them). + + In contrast to an ordinary theory, which simply consists of a type + signature, as well as tables for constants, axioms and theorems, a local + theory contains additional context information, such as locally fixed + variables and local assumptions that may be used by the package. The type + @{ML_type local_theory} is identical to the type of \emph{proof contexts} + @{ML_type "Proof.context"}, although not every proof context constitutes a + valid local theory. +*} + +(* +ML{*signature UNIVERSAL_TYPE = +sig + type t + + val embed: unit -> ('a -> t) * (t -> 'a option) +end*} + +ML{*structure U:> UNIVERSAL_TYPE = + struct + type t = exn + + fun 'a embed () = + let + exception E of 'a + fun project (e: t): 'a option = + case e of + E a => SOME a + | _ => NONE + in + (E, project) + end + end*} + +text {* + The idea is that type t is the universal type and that each call to embed + returns a new pair of functions (inject, project), where inject embeds a + value into the universal type and project extracts the value from the + universal type. A pair (inject, project) returned by embed works together in + that project u will return SOME v if and only if u was created by inject + v. If u was created by a different function inject', then project returns + NONE. + + in library.ML +*} + +ML_val{*structure Object = struct type T = exn end; *} + +ML{*functor Test (U: UNIVERSAL_TYPE): sig end = + struct + val (intIn: int -> U.t, intOut) = U.embed () + val r: U.t ref = ref (intIn 13) + val s1 = + case intOut (!r) of + NONE => "NONE" + | SOME i => Int.toString i + val (realIn: real -> U.t, realOut) = U.embed () + val () = r := realIn 13.0 + val s2 = + case intOut (!r) of + NONE => "NONE" + | SOME i => Int.toString i + val s3 = + case realOut (!r) of + NONE => "NONE" + | SOME x => Real.toString x + val () = tracing (concat [s1, " ", s2, " ", s3, "\n"]) + end*} + +ML_val{*structure t = Test(U) *} + +ML_val{*structure Datatab = TableFun(type key = int val ord = int_ord);*} + +ML {* LocalTheory.restore *} +ML {* LocalTheory.set_group *} +*) + +section {* Storing Theorems\label{sec:storing} (TBD) *} + +text {* @{ML_ind add_thms_dynamic in PureThy} *} + +local_setup %gray {* + LocalTheory.note Thm.theoremK + ((@{binding "allI_alt"}, []), [@{thm allI}]) #> snd *} + + +(* FIXME: some code below *) + +(*<*) +(* +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. + This infrastructure is loosely based on a paper by Oppen~\cite{Oppen80}. Most of + its functions do not operate on @{ML_type string}s, but on instances of the + 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 + +@{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 a possible + line break for example at each space. You can achieve this with the function + @{ML_ind breaks in Pretty}, which expects a list of pretty types and inserts a + possible line break in between every two elements in this list. To print + this list of pretty types as a single string, we concatenate them + with the function @{ML_ind blk in Pretty} as follows: + + +@{ML_response_fake [display,gray] +"let + val ptrs = map Pretty.str (space_explode \" \" test_str) +in + pprint (Pretty.blk (0, Pretty.breaks ptrs)) +end" +"fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar +fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar +fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar +fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar"} + + Here the layout of @{ML test_str} is much more pleasing to the + eye. The @{ML "0"} in @{ML_ind blk in Pretty} stands for no indentation + of the printed string. You can increase the indentation and obtain + +@{ML_response_fake [display,gray] +"let + val ptrs = map Pretty.str (space_explode \" \" test_str) +in + pprint (Pretty.blk (3, Pretty.breaks ptrs)) +end" +"fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar + fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar + fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar + fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar"} + + where starting from the second line the indent is 3. If you want + that every line starts with the same indent, you can use the + function @{ML_ind indent in Pretty} as follows: + +@{ML_response_fake [display,gray] +"let + val ptrs = map Pretty.str (space_explode \" \" test_str) +in + pprint (Pretty.indent 10 (Pretty.blk (0, Pretty.breaks ptrs))) +end" +" fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar + fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar + fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar + fooooooooooooooobaaaaaaaaaaaar fooooooooooooooobaaaaaaaaaaaar"} + + If you want to print out a list of items separated by commas and + have the linebreaks handled properly, you can use the function + @{ML_ind commas in Pretty}. For example + +@{ML_response_fake [display,gray] +"let + val ptrs = map (Pretty.str o string_of_int) (99998 upto 100020) +in + pprint (Pretty.blk (0, Pretty.commas ptrs)) +end" +"99998, 99999, 100000, 100001, 100002, 100003, 100004, 100005, 100006, +100007, 100008, 100009, 100010, 100011, 100012, 100013, 100014, 100015, +100016, 100017, 100018, 100019, 100020"} + + where @{ML upto} generates a list of integers. You can print out this + list as a ``set'', that means enclosed inside @{text [quotes] "{"} and + @{text [quotes] "}"}, and separated by commas using the function + @{ML_ind enum in Pretty}. For example +*} + +text {* + +@{ML_response_fake [display,gray] +"let + val ptrs = map (Pretty.str o string_of_int) (99998 upto 100020) +in + pprint (Pretty.enum \",\" \"{\" \"}\" ptrs) +end" +"{99998, 99999, 100000, 100001, 100002, 100003, 100004, 100005, 100006, + 100007, 100008, 100009, 100010, 100011, 100012, 100013, 100014, 100015, + 100016, 100017, 100018, 100019, 100020}"} + + As can be seen, this function prints out the ``set'' so that starting + from the second, each new line as 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 ptrs = map (Pretty.str o string_of_int) (1 upto 22) +in + pprint (Pretty.blk (0, and_list ptrs)) +end" +"1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 +and 22"} + + 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 \ 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 \ 'a \ bool) \ bool) \ bool) \ bool) \ bool) \ bool\"."} + + + FIXME: TBD below +*} + +ML {* pprint (Pretty.big_list "header" (map (Pretty.str o string_of_int) (4 upto 10))) *} + +text {* +chunks inserts forced breaks (unlike blk where you have to do this yourself) +*} + +ML {* (Pretty.chunks [Pretty.str "a", Pretty.str "b"], + Pretty.blk (0, [Pretty.str "a", Pretty.str "b"])) *} + +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 |> priority +*} + +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 *} + + +ML {* Pretty.mark Markup.subgoal (Pretty.str "foo") |> Pretty.string_of |> tracing *} +ML {* (Pretty.str "bar") |> Pretty.string_of |> tracing *} + +text {* + Still to be done: + + What happens with 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 +*} + + +ML {* Pretty.mark Markup.hilite (Pretty.str "foo") |> Pretty.string_of |> tracing *} +ML {* (Pretty.str "bar") |> Pretty.string_of |> tracing *} + +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 + + +section {* Misc (TBD) *} + +ML {*Datatype.get_info @{theory} "List.list"*} + +section {* Name Space Issues (TBD) *} + + +end diff -r d69214e47ef9 -r efb5fff99c96 ProgTutorial/ROOT.ML --- a/ProgTutorial/ROOT.ML Thu Aug 20 23:30:51 2009 +0200 +++ b/ProgTutorial/ROOT.ML Fri Aug 21 11:42:14 2009 +0200 @@ -5,6 +5,7 @@ use_thy "Intro"; use_thy "FirstSteps"; +use_thy "General"; use_thy "Parsing"; use_thy "Tactical"; diff -r d69214e47ef9 -r efb5fff99c96 ProgTutorial/Tactical.thy --- a/ProgTutorial/Tactical.thy Thu Aug 20 23:30:51 2009 +0200 +++ b/ProgTutorial/Tactical.thy Fri Aug 21 11:42:14 2009 +0200 @@ -49,41 +49,43 @@ THEN atac 1) end" "?P \ ?Q \ ?Q \ ?P"} - To start the proof, the function @{ML "Goal.prove"}~@{text "ctxt xs As C - tac"} sets up a goal state for proving the goal @{text C} - (that is @{prop "P \ Q \ Q \ P"} in the proof at hand) under the - assumptions @{text As} (happens to be empty) with the variables - @{text xs} that will be generalised once the goal is proved (in our case - @{text P} and @{text Q}). The @{text "tac"} is the tactic that proves the goal; - it can make use of the local assumptions (there are none in this example). - The tactics @{ML_ind etac}, @{ML_ind rtac} and @{ML_ind atac} - in the code above correspond - roughly to @{text erule}, @{text rule} and @{text assumption}, respectively. - The operator @{ML_ind THEN} strings the tactics together. + To start the proof, the function @{ML_ind "Goal.prove"}~@{text "ctxt xs As C + tac"} sets up a goal state for proving the goal @{text C} (that is @{prop "P + \ Q \ Q \ P"} in the proof at hand) under the assumptions @{text As} + (happens to be empty) with the variables @{text xs} that will be generalised + once the goal is proved (in our case @{text P} and @{text + Q}).\footnote{FIXME: explain prove earlier} The @{text "tac"} is the tactic + that proves the goal; it can make use of the local assumptions (there are + none in this example). The tactics @{ML_ind etac}, @{ML_ind rtac} and + @{ML_ind atac} in the code above correspond roughly to @{text erule}, @{text + rule} and @{text assumption}, respectively. The operator @{ML_ind THEN} + strings the tactics together. \begin{readmore} - To learn more about the function @{ML_ind prove in Goal} see \isccite{sec:results} - and the file @{ML_file "Pure/goal.ML"}. See @{ML_file + To learn more about the function @{ML_ind prove in Goal} see + \isccite{sec:results} and the file @{ML_file "Pure/goal.ML"}. See @{ML_file "Pure/tactic.ML"} and @{ML_file "Pure/tactical.ML"} for the code of basic tactics and tactic combinators; see also Chapters 3 and 4 in the old - Isabelle Reference Manual, and Chapter 3 in the Isabelle/Isar Implementation Manual. + Isabelle Reference Manual, and Chapter 3 in the Isabelle/Isar Implementation + Manual. \end{readmore} - Note that in the code above we use antiquotations for referencing the theorems. Many theorems - also have ML-bindings with the same name. Therefore, we could also just have - written @{ML "etac disjE 1"}, or in case where there is no ML-binding obtain - the theorem dynamically using the function @{ML thm}; for example - \mbox{@{ML "etac (thm \"disjE\") 1"}}. Both ways however are considered bad style! - The reason - is that the binding for @{ML disjE} can be re-assigned by the user and thus - one does not have complete control over which theorem is actually - applied. This problem is nicely prevented by using antiquotations, because - then the theorems are fixed statically at compile-time. + Note that in the code above we use antiquotations for referencing the + theorems. Many theorems also have ML-bindings with the same name. Therefore, + we could also just have written @{ML "etac disjE 1"}, or in case where there + is no ML-binding obtain the theorem dynamically using the function @{ML + thm}; for example \mbox{@{ML "etac (thm \"disjE\") 1"}}. Both ways however + are considered bad style! The reason is that the binding for @{ML disjE} can + be re-assigned by the user and thus one does not have complete control over + which theorem is actually applied. This problem is nicely prevented by using + antiquotations, because then the theorems are fixed statically at + compile-time. - During the development of automatic proof procedures, you will often find it - necessary to test a tactic on examples. This can be conveniently - done with the command \isacommand{apply}@{text "(tactic \ \ \)"}. + During the development of automatic proof procedures, you will often find it + necessary to test a tactic on examples. This can be conveniently done with + the command \isacommand{apply}@{text "(tactic \ \ \)"}. Consider the following sequence of tactics + *} ML{*val foo_tac = @@ -302,7 +304,6 @@ \end{figure} *} - text {* which prints out the given theorem (using the string-function defined in Section~\ref{sec:printing}) and then behaves like @{ML all_tac}. With this @@ -315,9 +316,7 @@ where @{term C} is the goal to be proved and the @{term "A\<^isub>i"} are the subgoals. So after setting up the lemma, the goal state is always of the form @{text "C \ #C"}; when the proof is finished we are left with @{text - "#C"}.\footnote{This only applies to single statements. If the lemma - contains more than one statement, then one has more such implications.} - Since the goal @{term C} can potentially be an implication, there is a + "#C"}. Since the goal @{term C} can potentially be an implication, there is a ``protector'' wrapped around it (the wrapper is the outermost constant @{text "Const (\"prop\", bool \ bool)"}; in the figure we make it visible as an @{text #}). This wrapper prevents that premises of @{text C} are @@ -1267,6 +1266,7 @@ simplification rules, congruence rules and simprocs.\label{fig:printss}} \end{figure} *} + text {* To see how they work, consider the function in Figure~\ref{fig:printss}, which prints out some parts of a simpset. If you use it to print out the components of the diff -r d69214e47ef9 -r efb5fff99c96 progtutorial.pdf Binary file progtutorial.pdf has changed