theory HOL_Specific

 imports Base Main

 begin

 chapter {* Isabelle/HOL \label{ch:hol} *}

 section {* Higher-Order Logic *}

 text {* Isabelle/HOL is based on Higher-Order Logic, a polymorphic

   version of Church's Simple Theory of Types.  HOL can be best

   understood as a simply-typed version of classical set theory.  The

   logic was first implemented in Gordon's HOL system

   \cite{mgordon-hol}.  It extends Church's original logic

   \cite{church40} by explicit type variables (naive polymorphism) and

   a sound axiomatization scheme for new types based on subsets of

   existing types.

   Andrews's book \cite{andrews86} is a full description of the

   original Church-style higher-order logic, with proofs of correctness

   and completeness wrt.\ certain set-theoretic interpretations.  The

   particular extensions of Gordon-style HOL are explained semantically

   in two chapters of the 1993 HOL book \cite{pitts93}.

   Experience with HOL over decades has demonstrated that higher-order

   logic is widely applicable in many areas of mathematics and computer

   science.  In a sense, Higher-Order Logic is simpler than First-Order

   Logic, because there are fewer restrictions and special cases.  Note

   that HOL is \emph{weaker} than FOL with axioms for ZF set theory,

   which is traditionally considered the standard foundation of regular

   mathematics, but for most applications this does not matter.  If you

   prefer ML to Lisp, you will probably prefer HOL to ZF.

   \medskip The syntax of HOL follows @{text "\<lambda>"}-calculus and

   functional programming.  Function application is curried.  To apply

   the function @{text f} of type @{text "\<tau>\<^sub>1 \<Rightarrow> \<tau>\<^sub>2 \<Rightarrow> \<tau>\<^sub>3"} to the

   arguments @{text a} and @{text b} in HOL, you simply write @{text "f

   a b"} (as in ML or Haskell).  There is no ``apply'' operator; the

   existing application of the Pure @{text "\<lambda>"}-calculus is re-used.

   Note that in HOL @{text "f (a, b)"} means ``@{text "f"} applied to

   the pair @{text "(a, b)"} (which is notation for @{text "Pair a

   b"}).  The latter typically introduces extra formal efforts that can

   be avoided by currying functions by default.  Explicit tuples are as

   infrequent in HOL formalizations as in good ML or Haskell programs.

   \medskip Isabelle/HOL has a distinct feel, compared to other

   object-logics like Isabelle/ZF.  It identifies object-level types

   with meta-level types, taking advantage of the default

   type-inference mechanism of Isabelle/Pure.  HOL fully identifies

   object-level functions with meta-level functions, with native

   abstraction and application.

   These identifications allow Isabelle to support HOL particularly

   nicely, but they also mean that HOL requires some sophistication

   from the user.  In particular, an understanding of Hindley-Milner

   type-inference with type-classes, which are both used extensively in

   the standard libraries and applications.  Beginners can set

   @{attribute show_types} or even @{attribute show_sorts} to get more

   explicit information about the result of type-inference.  *}

 section {* Inductive and coinductive definitions \label{sec:hol-inductive} *}

 text {* An \emph{inductive definition} specifies the least predicate

   or set @{text R} closed under given rules: applying a rule to

   elements of @{text R} yields a result within @{text R}.  For

   example, a structural operational semantics is an inductive

   definition of an evaluation relation.

   Dually, a \emph{coinductive definition} specifies the greatest

   predicate or set @{text R} that is consistent with given rules:

   every element of @{text R} can be seen as arising by applying a rule

   to elements of @{text R}.  An important example is using

   bisimulation relations to formalise equivalence of processes and

   infinite data structures.

   Both inductive and coinductive definitions are based on the

   Knaster-Tarski fixed-point theorem for complete lattices.  The

   collection of introduction rules given by the user determines a

   functor on subsets of set-theoretic relations.  The required

   monotonicity of the recursion scheme is proven as a prerequisite to

   the fixed-point definition and the resulting consequences.  This

   works by pushing inclusion through logical connectives and any other

   operator that might be wrapped around recursive occurrences of the

   defined relation: there must be a monotonicity theorem of the form

   @{text "A \<le> B \<Longrightarrow> \<M> A \<le> \<M> B"}, for each premise @{text "\<M> R t"} in an

   introduction rule.  The default rule declarations of Isabelle/HOL

   already take care of most common situations.

   \begin{matharray}{rcl}

     @{command_def (HOL) "inductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

     @{command_def (HOL) "inductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

     @{command_def (HOL) "coinductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

     @{command_def (HOL) "coinductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

     @{attribute_def (HOL) mono} & : & @{text attribute} \\

   \end{matharray}

   @{rail "

     (@@{command (HOL) inductive} | @@{command (HOL) inductive_set} |

       @@{command (HOL) coinductive} | @@{command (HOL) coinductive_set})

     @{syntax target}? \\

     @{syntax \"fixes\"} (@'for' @{syntax \"fixes\"})? (@'where' clauses)? \\

     (@'monos' @{syntax thmrefs})?

     ;

     clauses: (@{syntax thmdecl}? @{syntax prop} + '|')

     ;

     @@{attribute (HOL) mono} (() | 'add' | 'del')

   "}

   \begin{description}

   \item @{command (HOL) "inductive"} and @{command (HOL)

   "coinductive"} define (co)inductive predicates from the introduction

   rules.

   The propositions given as @{text "clauses"} in the @{keyword

   "where"} part are either rules of the usual @{text "\<And>/\<Longrightarrow>"} format

   (with arbitrary nesting), or equalities using @{text "\<equiv>"}.  The

   latter specifies extra-logical abbreviations in the sense of

   @{command_ref abbreviation}.  Introducing abstract syntax

   simultaneously with the actual introduction rules is occasionally

   useful for complex specifications.

   The optional @{keyword "for"} part contains a list of parameters of

   the (co)inductive predicates that remain fixed throughout the

   definition, in contrast to arguments of the relation that may vary

   in each occurrence within the given @{text "clauses"}.

   The optional @{keyword "monos"} declaration contains additional

   \emph{monotonicity theorems}, which are required for each operator

   applied to a recursive set in the introduction rules.

   \item @{command (HOL) "inductive_set"} and @{command (HOL)

   "coinductive_set"} are wrappers for to the previous commands for

   native HOL predicates.  This allows to define (co)inductive sets,

   where multiple arguments are simulated via tuples.

   \item @{attribute (HOL) mono} declares monotonicity rules in the

   context.  These rule are involved in the automated monotonicity

   proof of the above inductive and coinductive definitions.

   \end{description}

 *}

 subsection {* Derived rules *}

 text {* A (co)inductive definition of @{text R} provides the following

   main theorems:

   \begin{description}

   \item @{text R.intros} is the list of introduction rules as proven

   theorems, for the recursive predicates (or sets).  The rules are

   also available individually, using the names given them in the

   theory file;

   \item @{text R.cases} is the case analysis (or elimination) rule;

   \item @{text R.induct} or @{text R.coinduct} is the (co)induction

   rule.

   \end{description}

   When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are

   defined simultaneously, the list of introduction rules is called

   @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are

   called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list

   of mutual induction rules is called @{text

   "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}.

 *}

 subsection {* Monotonicity theorems *}

 text {* The context maintains a default set of theorems that are used

   in monotonicity proofs.  New rules can be declared via the

   @{attribute (HOL) mono} attribute.  See the main Isabelle/HOL

   sources for some examples.  The general format of such monotonicity

   theorems is as follows:

   \begin{itemize}

   \item Theorems of the form @{text "A \<le> B \<Longrightarrow> \<M> A \<le> \<M> B"}, for proving

   monotonicity of inductive definitions whose introduction rules have

   premises involving terms such as @{text "\<M> R t"}.

   \item Monotonicity theorems for logical operators, which are of the

   general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}.  For example, in

   the case of the operator @{text "\<or>"}, the corresponding theorem is

   \[

   \infer{@{text "P\<^sub>1 \<or> P\<^sub>2 \<longrightarrow> Q\<^sub>1 \<or> Q\<^sub>2"}}{@{text "P\<^sub>1 \<longrightarrow> Q\<^sub>1"} & @{text "P\<^sub>2 \<longrightarrow> Q\<^sub>2"}}

   \]

   \item De Morgan style equations for reasoning about the ``polarity''

   of expressions, e.g.

   \[

   @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad

   @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}

   \]

   \item Equations for reducing complex operators to more primitive

   ones whose monotonicity can easily be proved, e.g.

   \[

   @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad

   @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}

   \]

   \end{itemize}

 *}

 subsubsection {* Examples *}

 text {* The finite powerset operator can be defined inductively like this: *}

 inductive_set Fin :: "'a set \<Rightarrow> 'a set set" for A :: "'a set"

 where

   empty: "{} \<in> Fin A"

 | insert: "a \<in> A \<Longrightarrow> B \<in> Fin A \<Longrightarrow> insert a B \<in> Fin A"

 text {* The accessible part of a relation is defined as follows: *}

 inductive acc :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"

   for r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"  (infix "\<prec>" 50)

 where acc: "(\<And>y. y \<prec> x \<Longrightarrow> acc r y) \<Longrightarrow> acc r x"

 text {* Common logical connectives can be easily characterized as

 non-recursive inductive definitions with parameters, but without

 arguments. *}

 inductive AND for A B :: bool

 where "A \<Longrightarrow> B \<Longrightarrow> AND A B"

 inductive OR for A B :: bool

 where "A \<Longrightarrow> OR A B"

   | "B \<Longrightarrow> OR A B"

 inductive EXISTS for B :: "'a \<Rightarrow> bool"

 where "B a \<Longrightarrow> EXISTS B"

 text {* Here the @{text "cases"} or @{text "induct"} rules produced by

   the @{command inductive} package coincide with the expected

   elimination rules for Natural Deduction.  Already in the original

   article by Gerhard Gentzen \cite{Gentzen:1935} there is a hint that

   each connective can be characterized by its introductions, and the

   elimination can be constructed systematically. *}

 section {* Recursive functions \label{sec:recursion} *}

 text {*

   \begin{matharray}{rcl}

     @{command_def (HOL) "primrec"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

     @{command_def (HOL) "fun"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

     @{command_def (HOL) "function"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\

     @{command_def (HOL) "termination"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\

   \end{matharray}

   @{rail "

     @@{command (HOL) primrec} @{syntax target}? @{syntax \"fixes\"} @'where' equations

     ;

     (@@{command (HOL) fun} | @@{command (HOL) function}) @{syntax target}? functionopts?

       @{syntax \"fixes\"} \\ @'where' equations

     ;

     equations: (@{syntax thmdecl}? @{syntax prop} + '|')

     ;

     functionopts: '(' (('sequential' | 'domintros') + ',') ')'

     ;

     @@{command (HOL) termination} @{syntax term}?

   "}

   \begin{description}

   \item @{command (HOL) "primrec"} defines primitive recursive

   functions over datatypes (see also @{command_ref (HOL) datatype} and

   @{command_ref (HOL) rep_datatype}).  The given @{text equations}

   specify reduction rules that are produced by instantiating the

   generic combinator for primitive recursion that is available for

   each datatype.

   Each equation needs to be of the form:

   @{text [display] "f x\<^sub>1 \<dots> x\<^sub>m (C y\<^sub>1 \<dots> y\<^sub>k) z\<^sub>1 \<dots> z\<^sub>n = rhs"}

   such that @{text C} is a datatype constructor, @{text rhs} contains

   only the free variables on the left-hand side (or from the context),

   and all recursive occurrences of @{text "f"} in @{text "rhs"} are of

   the form @{text "f \<dots> y\<^sub>i \<dots>"} for some @{text i}.  At most one

   reduction rule for each constructor can be given.  The order does

   not matter.  For missing constructors, the function is defined to

   return a default value, but this equation is made difficult to

   access for users.

   The reduction rules are declared as @{attribute simp} by default,

   which enables standard proof methods like @{method simp} and

   @{method auto} to normalize expressions of @{text "f"} applied to

   datatype constructions, by simulating symbolic computation via

   rewriting.

   \item @{command (HOL) "function"} defines functions by general

   wellfounded recursion. A detailed description with examples can be

   found in \cite{isabelle-function}. The function is specified by a

   set of (possibly conditional) recursive equations with arbitrary

   pattern matching. The command generates proof obligations for the

   completeness and the compatibility of patterns.

   The defined function is considered partial, and the resulting

   simplification rules (named @{text "f.psimps"}) and induction rule

   (named @{text "f.pinduct"}) are guarded by a generated domain

   predicate @{text "f_dom"}. The @{command (HOL) "termination"}

   command can then be used to establish that the function is total.

   \item @{command (HOL) "fun"} is a shorthand notation for ``@{command

   (HOL) "function"}~@{text "(sequential)"}, followed by automated

   proof attempts regarding pattern matching and termination.  See

   \cite{isabelle-function} for further details.

   \item @{command (HOL) "termination"}~@{text f} commences a

   termination proof for the previously defined function @{text f}.  If

   this is omitted, the command refers to the most recent function

   definition.  After the proof is closed, the recursive equations and

   the induction principle is established.

   \end{description}

   Recursive definitions introduced by the @{command (HOL) "function"}

   command accommodate reasoning by induction (cf.\ @{method induct}):

   rule @{text "f.induct"} refers to a specific induction rule, with

   parameters named according to the user-specified equations. Cases

   are numbered starting from 1.  For @{command (HOL) "primrec"}, the

   induction principle coincides with structural recursion on the

   datatype where the recursion is carried out.

   The equations provided by these packages may be referred later as

   theorem list @{text "f.simps"}, where @{text f} is the (collective)

   name of the functions defined.  Individual equations may be named

   explicitly as well.

   The @{command (HOL) "function"} command accepts the following

   options.

   \begin{description}

   \item @{text sequential} enables a preprocessor which disambiguates

   overlapping patterns by making them mutually disjoint.  Earlier

   equations take precedence over later ones.  This allows to give the

   specification in a format very similar to functional programming.

   Note that the resulting simplification and induction rules

   correspond to the transformed specification, not the one given

   originally. This usually means that each equation given by the user

   may result in several theorems.  Also note that this automatic

   transformation only works for ML-style datatype patterns.

   \item @{text domintros} enables the automated generation of

   introduction rules for the domain predicate. While mostly not

   needed, they can be helpful in some proofs about partial functions.

   \end{description}

 *}

 subsubsection {* Example: evaluation of expressions *}

 text {* Subsequently, we define mutual datatypes for arithmetic and

   boolean expressions, and use @{command primrec} for evaluation

   functions that follow the same recursive structure. *}

 datatype 'a aexp =

     IF "'a bexp"  "'a aexp"  "'a aexp"

   | Sum "'a aexp"  "'a aexp"

   | Diff "'a aexp"  "'a aexp"

   | Var 'a

   | Num nat

 and 'a bexp =

     Less "'a aexp"  "'a aexp"

   | And "'a bexp"  "'a bexp"

   | Neg "'a bexp"

 text {* \medskip Evaluation of arithmetic and boolean expressions *}

 primrec evala :: "('a \<Rightarrow> nat) \<Rightarrow> 'a aexp \<Rightarrow> nat"

   and evalb :: "('a \<Rightarrow> nat) \<Rightarrow> 'a bexp \<Rightarrow> bool"

 where

   "evala env (IF b a1 a2) = (if evalb env b then evala env a1 else evala env a2)"

 | "evala env (Sum a1 a2) = evala env a1 + evala env a2"

 | "evala env (Diff a1 a2) = evala env a1 - evala env a2"

 | "evala env (Var v) = env v"

 | "evala env (Num n) = n"

 | "evalb env (Less a1 a2) = (evala env a1 < evala env a2)"

 | "evalb env (And b1 b2) = (evalb env b1 \<and> evalb env b2)"

 | "evalb env (Neg b) = (\<not> evalb env b)"

 text {* Since the value of an expression depends on the value of its

   variables, the functions @{const evala} and @{const evalb} take an

   additional parameter, an \emph{environment} that maps variables to

   their values.

   \medskip Substitution on expressions can be defined similarly.  The

   mapping @{text f} of type @{typ "'a \<Rightarrow> 'a aexp"} given as a

   parameter is lifted canonically on the types @{typ "'a aexp"} and

   @{typ "'a bexp"}, respectively.

 *}

 primrec substa :: "('a \<Rightarrow> 'b aexp) \<Rightarrow> 'a aexp \<Rightarrow> 'b aexp"

   and substb :: "('a \<Rightarrow> 'b aexp) \<Rightarrow> 'a bexp \<Rightarrow> 'b bexp"

 where

   "substa f (IF b a1 a2) = IF (substb f b) (substa f a1) (substa f a2)"

 | "substa f (Sum a1 a2) = Sum (substa f a1) (substa f a2)"

 | "substa f (Diff a1 a2) = Diff (substa f a1) (substa f a2)"

 | "substa f (Var v) = f v"

 | "substa f (Num n) = Num n"

 | "substb f (Less a1 a2) = Less (substa f a1) (substa f a2)"

 | "substb f (And b1 b2) = And (substb f b1) (substb f b2)"

 | "substb f (Neg b) = Neg (substb f b)"

 text {* In textbooks about semantics one often finds substitution

   theorems, which express the relationship between substitution and

   evaluation.  For @{typ "'a aexp"} and @{typ "'a bexp"}, we can prove

   such a theorem by mutual induction, followed by simplification.

 *}

 lemma subst_one:

   "evala env (substa (Var (v := a')) a) = evala (env (v := evala env a')) a"

   "evalb env (substb (Var (v := a')) b) = evalb (env (v := evala env a')) b"

   by (induct a and b) simp_all

 lemma subst_all:

   "evala env (substa s a) = evala (\<lambda>x. evala env (s x)) a"

   "evalb env (substb s b) = evalb (\<lambda>x. evala env (s x)) b"

   by (induct a and b) simp_all

 subsubsection {* Example: a substitution function for terms *}

 text {* Functions on datatypes with nested recursion are also defined

   by mutual primitive recursion. *}

 datatype ('a, 'b) "term" = Var 'a | App 'b "('a, 'b) term list"

 text {* A substitution function on type @{typ "('a, 'b) term"} can be

   defined as follows, by working simultaneously on @{typ "('a, 'b)

   term list"}: *}

 primrec subst_term :: "('a \<Rightarrow> ('a, 'b) term) \<Rightarrow> ('a, 'b) term \<Rightarrow> ('a, 'b) term" and

   subst_term_list :: "('a \<Rightarrow> ('a, 'b) term) \<Rightarrow> ('a, 'b) term list \<Rightarrow> ('a, 'b) term list"

 where

   "subst_term f (Var a) = f a"

 | "subst_term f (App b ts) = App b (subst_term_list f ts)"

 | "subst_term_list f [] = []"

 | "subst_term_list f (t # ts) = subst_term f t # subst_term_list f ts"

 text {* The recursion scheme follows the structure of the unfolded

   definition of type @{typ "('a, 'b) term"}.  To prove properties of this

   substitution function, mutual induction is needed:

 *}

 lemma "subst_term (subst_term f1 \<circ> f2) t = subst_term f1 (subst_term f2 t)" and

   "subst_term_list (subst_term f1 \<circ> f2) ts = subst_term_list f1 (subst_term_list f2 ts)"

   by (induct t and ts) simp_all

 subsubsection {* Example: a map function for infinitely branching trees *}

 text {* Defining functions on infinitely branching datatypes by

   primitive recursion is just as easy.

 *}

 datatype 'a tree = Atom 'a | Branch "nat \<Rightarrow> 'a tree"

 primrec map_tree :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a tree \<Rightarrow> 'b tree"

 where

   "map_tree f (Atom a) = Atom (f a)"

 | "map_tree f (Branch ts) = Branch (\<lambda>x. map_tree f (ts x))"

 text {* Note that all occurrences of functions such as @{text ts}

   above must be applied to an argument.  In particular, @{term

   "map_tree f \<circ> ts"} is not allowed here. *}

 text {* Here is a simple composition lemma for @{term map_tree}: *}

 lemma "map_tree g (map_tree f t) = map_tree (g \<circ> f) t"

   by (induct t) simp_all

 subsection {* Proof methods related to recursive definitions *}

 text {*

   \begin{matharray}{rcl}

     @{method_def (HOL) pat_completeness} & : & @{text method} \\

     @{method_def (HOL) relation} & : & @{text method} \\

     @{method_def (HOL) lexicographic_order} & : & @{text method} \\

     @{method_def (HOL) size_change} & : & @{text method} \\

   \end{matharray}

   @{rail "

     @@{method (HOL) relation} @{syntax term}

     ;

     @@{method (HOL) lexicographic_order} (@{syntax clasimpmod} * )

     ;

     @@{method (HOL) size_change} ( orders (@{syntax clasimpmod} * ) )

     ;

     orders: ( 'max' | 'min' | 'ms' ) *

   "}

   \begin{description}

   \item @{method (HOL) pat_completeness} is a specialized method to

   solve goals regarding the completeness of pattern matching, as

   required by the @{command (HOL) "function"} package (cf.\

   \cite{isabelle-function}).

   \item @{method (HOL) relation}~@{text R} introduces a termination

   proof using the relation @{text R}.  The resulting proof state will

   contain goals expressing that @{text R} is wellfounded, and that the

   arguments of recursive calls decrease with respect to @{text R}.

   Usually, this method is used as the initial proof step of manual

   termination proofs.

   \item @{method (HOL) "lexicographic_order"} attempts a fully

   automated termination proof by searching for a lexicographic

   combination of size measures on the arguments of the function. The

   method accepts the same arguments as the @{method auto} method,

   which it uses internally to prove local descents.  The @{syntax

   clasimpmod} modifiers are accepted (as for @{method auto}).

   In case of failure, extensive information is printed, which can help

   to analyse the situation (cf.\ \cite{isabelle-function}).

   \item @{method (HOL) "size_change"} also works on termination goals,

   using a variation of the size-change principle, together with a

   graph decomposition technique (see \cite{krauss_phd} for details).

   Three kinds of orders are used internally: @{text max}, @{text min},

   and @{text ms} (multiset), which is only available when the theory

   @{text Multiset} is loaded. When no order kinds are given, they are

   tried in order. The search for a termination proof uses SAT solving

   internally.

   For local descent proofs, the @{syntax clasimpmod} modifiers are

   accepted (as for @{method auto}).

   \end{description}

 *}

 subsection {* Functions with explicit partiality *}

 text {*

   \begin{matharray}{rcl}

     @{command_def (HOL) "partial_function"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

     @{attribute_def (HOL) "partial_function_mono"} & : & @{text attribute} \\

   \end{matharray}

   @{rail "

     @@{command (HOL) partial_function} @{syntax target}?

       '(' @{syntax nameref} ')' @{syntax \"fixes\"} \\

       @'where' @{syntax thmdecl}? @{syntax prop}

   "}

   \begin{description}

   \item @{command (HOL) "partial_function"}~@{text "(mode)"} defines

   recursive functions based on fixpoints in complete partial

   orders. No termination proof is required from the user or

   constructed internally. Instead, the possibility of non-termination

   is modelled explicitly in the result type, which contains an

   explicit bottom element.

   Pattern matching and mutual recursion are currently not supported.

   Thus, the specification consists of a single function described by a

   single recursive equation.

   There are no fixed syntactic restrictions on the body of the

   function, but the induced functional must be provably monotonic

   wrt.\ the underlying order.  The monotonicitity proof is performed

   internally, and the definition is rejected when it fails. The proof

   can be influenced by declaring hints using the

   @{attribute (HOL) partial_function_mono} attribute.

   The mandatory @{text mode} argument specifies the mode of operation

   of the command, which directly corresponds to a complete partial

   order on the result type. By default, the following modes are

   defined:

   \begin{description}

   \item @{text option} defines functions that map into the @{type

   option} type. Here, the value @{term None} is used to model a

   non-terminating computation. Monotonicity requires that if @{term

   None} is returned by a recursive call, then the overall result

   must also be @{term None}. This is best achieved through the use of

   the monadic operator @{const "Option.bind"}.

   \item @{text tailrec} defines functions with an arbitrary result

   type and uses the slightly degenerated partial order where @{term

   "undefined"} is the bottom element.  Now, monotonicity requires that

   if @{term undefined} is returned by a recursive call, then the

   overall result must also be @{term undefined}. In practice, this is

   only satisfied when each recursive call is a tail call, whose result

   is directly returned. Thus, this mode of operation allows the

   definition of arbitrary tail-recursive functions.

   \end{description}

   Experienced users may define new modes by instantiating the locale

   @{const "partial_function_definitions"} appropriately.

   \item @{attribute (HOL) partial_function_mono} declares rules for

   use in the internal monononicity proofs of partial function

   definitions.

   \end{description}

 *}

 subsection {* Old-style recursive function definitions (TFL) *}

 text {*

   The old TFL commands @{command (HOL) "recdef"} and @{command (HOL)

   "recdef_tc"} for defining recursive are mostly obsolete; @{command

   (HOL) "function"} or @{command (HOL) "fun"} should be used instead.

   \begin{matharray}{rcl}

     @{command_def (HOL) "recdef"} & : & @{text "theory \<rightarrow> theory)"} \\

     @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & @{text "theory \<rightarrow> proof(prove)"} \\

   \end{matharray}

   @{rail "

     @@{command (HOL) recdef} ('(' @'permissive' ')')? \\

       @{syntax name} @{syntax term} (@{syntax prop} +) hints?

     ;

     recdeftc @{syntax thmdecl}? tc

     ;

     hints: '(' @'hints' ( recdefmod * ) ')'

     ;

     recdefmod: (('recdef_simp' | 'recdef_cong' | 'recdef_wf')

       (() | 'add' | 'del') ':' @{syntax thmrefs}) | @{syntax clasimpmod}

     ;

     tc: @{syntax nameref} ('(' @{syntax nat} ')')?

   "}

   \begin{description}

   \item @{command (HOL) "recdef"} defines general well-founded

   recursive functions (using the TFL package), see also

   \cite{isabelle-HOL}.  The ``@{text "(permissive)"}'' option tells

   TFL to recover from failed proof attempts, returning unfinished

   results.  The @{text recdef_simp}, @{text recdef_cong}, and @{text

   recdef_wf} hints refer to auxiliary rules to be used in the internal

   automated proof process of TFL.  Additional @{syntax clasimpmod}

   declarations may be given to tune the context of the Simplifier

   (cf.\ \secref{sec:simplifier}) and Classical reasoner (cf.\

   \secref{sec:classical}).

   \item @{command (HOL) "recdef_tc"}~@{text "c (i)"} recommences the

   proof for leftover termination condition number @{text i} (default

   1) as generated by a @{command (HOL) "recdef"} definition of

   constant @{text c}.

   Note that in most cases, @{command (HOL) "recdef"} is able to finish

   its internal proofs without manual intervention.

   \end{description}

   \medskip Hints for @{command (HOL) "recdef"} may be also declared

   globally, using the following attributes.

   \begin{matharray}{rcl}

     @{attribute_def (HOL) recdef_simp} & : & @{text attribute} \\

     @{attribute_def (HOL) recdef_cong} & : & @{text attribute} \\

     @{attribute_def (HOL) recdef_wf} & : & @{text attribute} \\

   \end{matharray}

   @{rail "

     (@@{attribute (HOL) recdef_simp} | @@{attribute (HOL) recdef_cong} |

       @@{attribute (HOL) recdef_wf}) (() | 'add' | 'del')

   "}

 *}

 section {* Datatypes \label{sec:hol-datatype} *}

 text {*

   \begin{matharray}{rcl}

     @{command_def (HOL) "datatype"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def (HOL) "rep_datatype"} & : & @{text "theory \<rightarrow> proof(prove)"} \\

   \end{matharray}

   @{rail "

     @@{command (HOL) datatype} (spec + @'and')

     ;

     @@{command (HOL) rep_datatype} ('(' (@{syntax name} +) ')')? (@{syntax term} +)

     ;

     spec: @{syntax parname}? @{syntax typespec} @{syntax mixfix}? '=' (cons + '|')

     ;

     cons: @{syntax name} (@{syntax type} * ) @{syntax mixfix}?

   "}

   \begin{description}

   \item @{command (HOL) "datatype"} defines inductive datatypes in

   HOL.

   \item @{command (HOL) "rep_datatype"} represents existing types as

   datatypes.

   For foundational reasons, some basic types such as @{typ nat}, @{typ

   "'a \<times> 'b"}, @{typ "'a + 'b"}, @{typ bool} and @{typ unit} are

   introduced by more primitive means using @{command_ref typedef}.  To

   recover the rich infrastructure of @{command datatype} (e.g.\ rules

   for @{method cases} and @{method induct} and the primitive recursion

   combinators), such types may be represented as actual datatypes

   later.  This is done by specifying the constructors of the desired

   type, and giving a proof of the induction rule, distinctness and

   injectivity of constructors.

   For example, see @{file "~~/src/HOL/Sum_Type.thy"} for the

   representation of the primitive sum type as fully-featured datatype.

   \end{description}

   The generated rules for @{method induct} and @{method cases} provide

   case names according to the given constructors, while parameters are

   named after the types (see also \secref{sec:cases-induct}).

   See \cite{isabelle-HOL} for more details on datatypes, but beware of

   the old-style theory syntax being used there!  Apart from proper

   proof methods for case-analysis and induction, there are also

   emulations of ML tactics @{method (HOL) case_tac} and @{method (HOL)

   induct_tac} available, see \secref{sec:hol-induct-tac}; these admit

   to refer directly to the internal structure of subgoals (including

   internally bound parameters).

 *}

 subsubsection {* Examples *}

 text {* We define a type of finite sequences, with slightly different

   names than the existing @{typ "'a list"} that is already in @{theory

   Main}: *}

 datatype 'a seq = Empty | Seq 'a "'a seq"

 text {* We can now prove some simple lemma by structural induction: *}

 lemma "Seq x xs \<noteq> xs"

 proof (induct xs arbitrary: x)

   case Empty

   txt {* This case can be proved using the simplifier: the freeness

     properties of the datatype are already declared as @{attribute

     simp} rules. *}

   show "Seq x Empty \<noteq> Empty"

     by simp

 next

   case (Seq y ys)

   txt {* The step case is proved similarly. *}

   show "Seq x (Seq y ys) \<noteq> Seq y ys"

     using `Seq y ys \<noteq> ys` by simp

 qed

 text {* Here is a more succinct version of the same proof: *}

 lemma "Seq x xs \<noteq> xs"

   by (induct xs arbitrary: x) simp_all

 section {* Records \label{sec:hol-record} *}

 text {*

   In principle, records merely generalize the concept of tuples, where

   components may be addressed by labels instead of just position.  The

   logical infrastructure of records in Isabelle/HOL is slightly more

   advanced, though, supporting truly extensible record schemes.  This

   admits operations that are polymorphic with respect to record

   extension, yielding ``object-oriented'' effects like (single)

   inheritance.  See also \cite{NaraschewskiW-TPHOLs98} for more

   details on object-oriented verification and record subtyping in HOL.

 *}

 subsection {* Basic concepts *}

 text {*

   Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records

   at the level of terms and types.  The notation is as follows:

   \begin{center}

   \begin{tabular}{l|l|l}

     & record terms & record types \\ \hline

     fixed & @{text "\<lparr>x = a, y = b\<rparr>"} & @{text "\<lparr>x :: A, y :: B\<rparr>"} \\

     schematic & @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} &

       @{text "\<lparr>x :: A, y :: B, \<dots> :: M\<rparr>"} \\

   \end{tabular}

   \end{center}

   \noindent The ASCII representation of @{text "\<lparr>x = a\<rparr>"} is @{text

   "(| x = a |)"}.

   A fixed record @{text "\<lparr>x = a, y = b\<rparr>"} has field @{text x} of value

   @{text a} and field @{text y} of value @{text b}.  The corresponding

   type is @{text "\<lparr>x :: A, y :: B\<rparr>"}, assuming that @{text "a :: A"}

   and @{text "b :: B"}.

   A record scheme like @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} contains fields

   @{text x} and @{text y} as before, but also possibly further fields

   as indicated by the ``@{text "\<dots>"}'' notation (which is actually part

   of the syntax).  The improper field ``@{text "\<dots>"}'' of a record

   scheme is called the \emph{more part}.  Logically it is just a free

   variable, which is occasionally referred to as ``row variable'' in

   the literature.  The more part of a record scheme may be

   instantiated by zero or more further components.  For example, the

   previous scheme may get instantiated to @{text "\<lparr>x = a, y = b, z =

   c, \<dots> = m'\<rparr>"}, where @{text m'} refers to a different more part.

   Fixed records are special instances of record schemes, where

   ``@{text "\<dots>"}'' is properly terminated by the @{text "() :: unit"}

   element.  In fact, @{text "\<lparr>x = a, y = b\<rparr>"} is just an abbreviation

   for @{text "\<lparr>x = a, y = b, \<dots> = ()\<rparr>"}.

   \medskip Two key observations make extensible records in a simply

   typed language like HOL work out:

   \begin{enumerate}

   \item the more part is internalized, as a free term or type

   variable,

   \item field names are externalized, they cannot be accessed within

   the logic as first-class values.

   \end{enumerate}

   \medskip In Isabelle/HOL record types have to be defined explicitly,

   fixing their field names and types, and their (optional) parent

   record.  Afterwards, records may be formed using above syntax, while

   obeying the canonical order of fields as given by their declaration.

   The record package provides several standard operations like

   selectors and updates.  The common setup for various generic proof

   tools enable succinct reasoning patterns.  See also the Isabelle/HOL

   tutorial \cite{isabelle-hol-book} for further instructions on using

   records in practice.

 *}

 subsection {* Record specifications *}

 text {*

   \begin{matharray}{rcl}

     @{command_def (HOL) "record"} & : & @{text "theory \<rightarrow> theory"} \\

   \end{matharray}

   @{rail "

     @@{command (HOL) record} @{syntax typespec_sorts} '=' \\

       (@{syntax type} '+')? (@{syntax constdecl} +)

   "}

   \begin{description}

   \item @{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1

   \<dots> c\<^sub>n :: \<sigma>\<^sub>n"} defines extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"},

   derived from the optional parent record @{text "\<tau>"} by adding new

   field components @{text "c\<^sub>i :: \<sigma>\<^sub>i"} etc.

   The type variables of @{text "\<tau>"} and @{text "\<sigma>\<^sub>i"} need to be

   covered by the (distinct) parameters @{text "\<alpha>\<^sub>1, \<dots>,

   \<alpha>\<^sub>m"}.  Type constructor @{text t} has to be new, while @{text

   \<tau>} needs to specify an instance of an existing record type.  At

   least one new field @{text "c\<^sub>i"} has to be specified.

   Basically, field names need to belong to a unique record.  This is

   not a real restriction in practice, since fields are qualified by

   the record name internally.

   The parent record specification @{text \<tau>} is optional; if omitted

   @{text t} becomes a root record.  The hierarchy of all records

   declared within a theory context forms a forest structure, i.e.\ a

   set of trees starting with a root record each.  There is no way to

   merge multiple parent records!

   For convenience, @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} is made a

   type abbreviation for the fixed record type @{text "\<lparr>c\<^sub>1 ::

   \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<rparr>"}, likewise is @{text

   "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m, \<zeta>) t_scheme"} made an abbreviation for

   @{text "\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n, \<dots> ::

   \<zeta>\<rparr>"}.

   \end{description}

 *}

 subsection {* Record operations *}

 text {*

   Any record definition of the form presented above produces certain

   standard operations.  Selectors and updates are provided for any

   field, including the improper one ``@{text more}''.  There are also

   cumulative record constructor functions.  To simplify the

   presentation below, we assume for now that @{text "(\<alpha>\<^sub>1, \<dots>,

   \<alpha>\<^sub>m) t"} is a root record with fields @{text "c\<^sub>1 ::

   \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n"}.

   \medskip \textbf{Selectors} and \textbf{updates} are available for

   any field (including ``@{text more}''):

   \begin{matharray}{lll}

     @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\

     @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\

   \end{matharray}

   There is special syntax for application of updates: @{text "r\<lparr>x :=

   a\<rparr>"} abbreviates term @{text "x_update a r"}.  Further notation for

   repeated updates is also available: @{text "r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<lparr>z :=

   c\<rparr>"} may be written @{text "r\<lparr>x := a, y := b, z := c\<rparr>"}.  Note that

   because of postfix notation the order of fields shown here is

   reverse than in the actual term.  Since repeated updates are just

   function applications, fields may be freely permuted in @{text "\<lparr>x

   := a, y := b, z := c\<rparr>"}, as far as logical equality is concerned.

   Thus commutativity of independent updates can be proven within the

   logic for any two fields, but not as a general theorem.

   \medskip The \textbf{make} operation provides a cumulative record

   constructor function:

   \begin{matharray}{lll}

     @{text "t.make"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\

   \end{matharray}

   \medskip We now reconsider the case of non-root records, which are

   derived of some parent.  In general, the latter may depend on

   another parent as well, resulting in a list of \emph{ancestor

   records}.  Appending the lists of fields of all ancestors results in

   a certain field prefix.  The record package automatically takes care

   of this by lifting operations over this context of ancestor fields.

   Assuming that @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} has ancestor

   fields @{text "b\<^sub>1 :: \<rho>\<^sub>1, \<dots>, b\<^sub>k :: \<rho>\<^sub>k"},

   the above record operations will get the following types:

   \medskip

   \begin{tabular}{lll}

     @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\

     @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow>

       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow>

       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\

     @{text "t.make"} & @{text "::"} & @{text "\<rho>\<^sub>1 \<Rightarrow> \<dots> \<rho>\<^sub>k \<Rightarrow> \<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow>

       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\

   \end{tabular}

   \medskip

   \noindent Some further operations address the extension aspect of a

   derived record scheme specifically: @{text "t.fields"} produces a

   record fragment consisting of exactly the new fields introduced here

   (the result may serve as a more part elsewhere); @{text "t.extend"}

   takes a fixed record and adds a given more part; @{text

   "t.truncate"} restricts a record scheme to a fixed record.

   \medskip

   \begin{tabular}{lll}

     @{text "t.fields"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\

     @{text "t.extend"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr> \<Rightarrow>

       \<zeta> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\

     @{text "t.truncate"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\

   \end{tabular}

   \medskip

   \noindent Note that @{text "t.make"} and @{text "t.fields"} coincide

   for root records.

 *}

 subsection {* Derived rules and proof tools *}

 text {*

   The record package proves several results internally, declaring

   these facts to appropriate proof tools.  This enables users to

   reason about record structures quite conveniently.  Assume that

   @{text t} is a record type as specified above.

   \begin{enumerate}

   \item Standard conversions for selectors or updates applied to

   record constructor terms are made part of the default Simplifier

   context; thus proofs by reduction of basic operations merely require

   the @{method simp} method without further arguments.  These rules

   are available as @{text "t.simps"}, too.

   \item Selectors applied to updated records are automatically reduced

   by an internal simplification procedure, which is also part of the

   standard Simplifier setup.

   \item Inject equations of a form analogous to @{prop "(x, y) = (x',

   y') \<equiv> x = x' \<and> y = y'"} are declared to the Simplifier and Classical

   Reasoner as @{attribute iff} rules.  These rules are available as

   @{text "t.iffs"}.

   \item The introduction rule for record equality analogous to @{text

   "x r = x r' \<Longrightarrow> y r = y r' \<dots> \<Longrightarrow> r = r'"} is declared to the Simplifier,

   and as the basic rule context as ``@{attribute intro}@{text "?"}''.

   The rule is called @{text "t.equality"}.

   \item Representations of arbitrary record expressions as canonical

   constructor terms are provided both in @{method cases} and @{method

   induct} format (cf.\ the generic proof methods of the same name,

   \secref{sec:cases-induct}).  Several variations are available, for

   fixed records, record schemes, more parts etc.

   The generic proof methods are sufficiently smart to pick the most

   sensible rule according to the type of the indicated record

   expression: users just need to apply something like ``@{text "(cases

   r)"}'' to a certain proof problem.

   \item The derived record operations @{text "t.make"}, @{text

   "t.fields"}, @{text "t.extend"}, @{text "t.truncate"} are \emph{not}

   treated automatically, but usually need to be expanded by hand,

   using the collective fact @{text "t.defs"}.

   \end{enumerate}

 *}

 subsubsection {* Examples *}

 text {* See @{file "~~/src/HOL/ex/Records.thy"}, for example. *}

 section {* Adhoc tuples *}

 text {*

   \begin{matharray}{rcl}

     @{attribute_def (HOL) split_format}@{text "\<^sup>*"} & : & @{text attribute} \\

   \end{matharray}

   @{rail "

     @@{attribute (HOL) split_format} ('(' 'complete' ')')?

   "}

   \begin{description}

   \item @{attribute (HOL) split_format}\ @{text "(complete)"} causes

   arguments in function applications to be represented canonically

   according to their tuple type structure.

   Note that this operation tends to invent funny names for new local

   parameters introduced.

   \end{description}

 *}

 section {* Typedef axiomatization \label{sec:hol-typedef} *}

 text {* A Gordon/HOL-style type definition is a certain axiom scheme

   that identifies a new type with a subset of an existing type.  More

   precisely, the new type is defined by exhibiting an existing type

   @{text \<tau>}, a set @{text "A :: \<tau> set"}, and a theorem that proves

   @{prop "\<exists>x. x \<in> A"}.  Thus @{text A} is a non-empty subset of @{text

   \<tau>}, and the new type denotes this subset.  New functions are

   postulated that establish an isomorphism between the new type and

   the subset.  In general, the type @{text \<tau>} may involve type

   variables @{text "\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n"} which means that the type definition

   produces a type constructor @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t"} depending on

   those type arguments.

   The axiomatization can be considered a ``definition'' in the sense

   of the particular set-theoretic interpretation of HOL

   \cite{pitts93}, where the universe of types is required to be

   downwards-closed wrt.\ arbitrary non-empty subsets.  Thus genuinely

   new types introduced by @{command "typedef"} stay within the range

   of HOL models by construction.  Note that @{command_ref

   type_synonym} from Isabelle/Pure merely introduces syntactic

   abbreviations, without any logical significance.

   \begin{matharray}{rcl}

     @{command_def (HOL) "typedef"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\

   \end{matharray}

   @{rail "

     @@{command (HOL) typedef} alt_name? abs_type '=' rep_set

     ;

     alt_name: '(' (@{syntax name} | @'open' | @'open' @{syntax name}) ')'

     ;

     abs_type: @{syntax typespec_sorts} @{syntax mixfix}?

     ;

     rep_set: @{syntax term} (@'morphisms' @{syntax name} @{syntax name})?

   "}

   \begin{description}

   \item @{command (HOL) "typedef"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = A"}

   axiomatizes a type definition in the background theory of the

   current context, depending on a non-emptiness result of the set

   @{text A} that needs to be proven here.  The set @{text A} may

   contain type variables @{text "\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n"} as specified on the LHS,

   but no term variables.

   Even though a local theory specification, the newly introduced type

   constructor cannot depend on parameters or assumptions of the

   context: this is structurally impossible in HOL.  In contrast, the

   non-emptiness proof may use local assumptions in unusual situations,

   which could result in different interpretations in target contexts:

   the meaning of the bijection between the representing set @{text A}

   and the new type @{text t} may then change in different application

   contexts.

   By default, @{command (HOL) "typedef"} defines both a type

   constructor @{text t} for the new type, and a term constant @{text

   t} for the representing set within the old type.  Use the ``@{text

   "(open)"}'' option to suppress a separate constant definition

   altogether.  The injection from type to set is called @{text Rep_t},

   its inverse @{text Abs_t}, unless explicit @{keyword (HOL)

   "morphisms"} specification provides alternative names.

   The core axiomatization uses the locale predicate @{const

   type_definition} as defined in Isabelle/HOL.  Various basic

   consequences of that are instantiated accordingly, re-using the

   locale facts with names derived from the new type constructor.  Thus

   the generic @{thm type_definition.Rep} is turned into the specific

   @{text "Rep_t"}, for example.

   Theorems @{thm type_definition.Rep}, @{thm

   type_definition.Rep_inverse}, and @{thm type_definition.Abs_inverse}

   provide the most basic characterization as a corresponding

   injection/surjection pair (in both directions).  The derived rules

   @{thm type_definition.Rep_inject} and @{thm

   type_definition.Abs_inject} provide a more convenient version of

   injectivity, suitable for automated proof tools (e.g.\ in

   declarations involving @{attribute simp} or @{attribute iff}).

   Furthermore, the rules @{thm type_definition.Rep_cases}~/ @{thm

   type_definition.Rep_induct}, and @{thm type_definition.Abs_cases}~/

   @{thm type_definition.Abs_induct} provide alternative views on

   surjectivity.  These rules are already declared as set or type rules

   for the generic @{method cases} and @{method induct} methods,

   respectively.

   An alternative name for the set definition (and other derived

   entities) may be specified in parentheses; the default is to use

   @{text t} directly.

   \end{description}

   \begin{warn}

   If you introduce a new type axiomatically, i.e.\ via @{command_ref

   typedecl} and @{command_ref axiomatization}, the minimum requirement

   is that it has a non-empty model, to avoid immediate collapse of the

   HOL logic.  Moreover, one needs to demonstrate that the

   interpretation of such free-form axiomatizations can coexist with

   that of the regular @{command_def typedef} scheme, and any extension

   that other people might have introduced elsewhere (e.g.\ in HOLCF

   \cite{MuellerNvOS99}).

   \end{warn}

 *}

 subsubsection {* Examples *}

 text {* Type definitions permit the introduction of abstract data

   types in a safe way, namely by providing models based on already

   existing types.  Given some abstract axiomatic description @{text P}

   of a type, this involves two steps:

   \begin{enumerate}

   \item Find an appropriate type @{text \<tau>} and subset @{text A} which

   has the desired properties @{text P}, and make a type definition

   based on this representation.

   \item Prove that @{text P} holds for @{text \<tau>} by lifting @{text P}

   from the representation.

   \end{enumerate}

   You can later forget about the representation and work solely in

   terms of the abstract properties @{text P}.

   \medskip The following trivial example pulls a three-element type

   into existence within the formal logical environment of HOL. *}

 typedef three = "{(True, True), (True, False), (False, True)}"

   by blast

 definition "One = Abs_three (True, True)"

 definition "Two = Abs_three (True, False)"

 definition "Three = Abs_three (False, True)"

 lemma three_distinct: "One \<noteq> Two"  "One \<noteq> Three"  "Two \<noteq> Three"

   by (simp_all add: One_def Two_def Three_def Abs_three_inject three_def)

 lemma three_cases:

   fixes x :: three obtains "x = One" | "x = Two" | "x = Three"

   by (cases x) (auto simp: One_def Two_def Three_def Abs_three_inject three_def)

 text {* Note that such trivial constructions are better done with

   derived specification mechanisms such as @{command datatype}: *}

 datatype three' = One' | Two' | Three'

 text {* This avoids re-doing basic definitions and proofs from the

   primitive @{command typedef} above. *}

 section {* Functorial structure of types *}

 text {*

   \begin{matharray}{rcl}

     @{command_def (HOL) "enriched_type"} & : & @{text "local_theory \<rightarrow> proof(prove)"}

   \end{matharray}

   @{rail "

     @@{command (HOL) enriched_type} (@{syntax name} ':')? @{syntax term}

     ;

   "}

   \begin{description}

   \item @{command (HOL) "enriched_type"}~@{text "prefix: m"} allows to

   prove and register properties about the functorial structure of type

   constructors.  These properties then can be used by other packages

   to deal with those type constructors in certain type constructions.

   Characteristic theorems are noted in the current local theory.  By

   default, they are prefixed with the base name of the type

   constructor, an explicit prefix can be given alternatively.

   The given term @{text "m"} is considered as \emph{mapper} for the

   corresponding type constructor and must conform to the following

   type pattern:

   \begin{matharray}{lll}

     @{text "m"} & @{text "::"} &

       @{text "\<sigma>\<^isub>1 \<Rightarrow> \<dots> \<sigma>\<^isub>k \<Rightarrow> (\<^vec>\<alpha>\<^isub>n) t \<Rightarrow> (\<^vec>\<beta>\<^isub>n) t"} \\

   \end{matharray}

   \noindent where @{text t} is the type constructor, @{text

   "\<^vec>\<alpha>\<^isub>n"} and @{text "\<^vec>\<beta>\<^isub>n"} are distinct

   type variables free in the local theory and @{text "\<sigma>\<^isub>1"},

   \ldots, @{text "\<sigma>\<^isub>k"} is a subsequence of @{text "\<alpha>\<^isub>1 \<Rightarrow>

   \<beta>\<^isub>1"}, @{text "\<beta>\<^isub>1 \<Rightarrow> \<alpha>\<^isub>1"}, \ldots,

   @{text "\<alpha>\<^isub>n \<Rightarrow> \<beta>\<^isub>n"}, @{text "\<beta>\<^isub>n \<Rightarrow>

   \<alpha>\<^isub>n"}.

   \end{description}

 *}

 section {* Quotient types *}

 text {*

   The quotient package defines a new quotient type given a raw type

   and a partial equivalence relation.

   It also includes automation for transporting definitions and theorems.

   It can automatically produce definitions and theorems on the quotient type,

   given the corresponding constants and facts on the raw type.

   \begin{matharray}{rcl}

     @{command_def (HOL) "quotient_type"} & : & @{text "local_theory \<rightarrow> proof(prove)"}\\

     @{command_def (HOL) "quotient_definition"} & : & @{text "local_theory \<rightarrow> proof(prove)"}\\

     @{command_def (HOL) "print_quotmaps"} & : & @{text "context \<rightarrow>"}\\

     @{command_def (HOL) "print_quotients"} & : & @{text "context \<rightarrow>"}\\

     @{command_def (HOL) "print_quotconsts"} & : & @{text "context \<rightarrow>"}\\

   \end{matharray}

   @{rail "

     @@{command (HOL) quotient_type} (spec + @'and');

     spec: @{syntax typespec} @{syntax mixfix}? '=' \\

      @{syntax type} '/' ('partial' ':')? @{syntax term}; 

   "}

   @{rail "

     @@{command (HOL) quotient_definition} constdecl? @{syntax thmdecl}? \\

     @{syntax term} 'is' @{syntax term};

     constdecl: @{syntax name} ('::' @{syntax type})? @{syntax mixfix}?

   "}

   \begin{description}

   \item @{command (HOL) "quotient_type"} defines quotient types.

   \item @{command (HOL) "quotient_definition"} defines a constant on the quotient type.

   \item @{command (HOL) "print_quotmaps"} prints quotient map functions.

   \item @{command (HOL) "print_quotients"} prints quotients.

   \item @{command (HOL) "print_quotconsts"} prints quotient constants.

   \end{description}

 *}

 section {* Coercive subtyping *}

 text {*

   \begin{matharray}{rcl}

     @{attribute_def (HOL) coercion} & : & @{text attribute} \\

     @{attribute_def (HOL) coercion_enabled} & : & @{text attribute} \\

     @{attribute_def (HOL) coercion_map} & : & @{text attribute} \\

   \end{matharray}

   @{rail "

     @@{attribute (HOL) coercion} (@{syntax term})?

     ;

   "}

   @{rail "

     @@{attribute (HOL) coercion_map} (@{syntax term})?

     ;

   "}

   Coercive subtyping allows the user to omit explicit type conversions,

   also called \emph{coercions}.  Type inference will add them as

   necessary when parsing a term. See

   \cite{traytel-berghofer-nipkow-2011} for details.

   \begin{description}

   \item @{attribute (HOL) "coercion"}~@{text "f"} registers a new

   coercion function @{text "f :: \<sigma>\<^isub>1 \<Rightarrow>

   \<sigma>\<^isub>2"} where @{text "\<sigma>\<^isub>1"} and @{text

   "\<sigma>\<^isub>2"} are nullary type constructors. Coercions are

   composed by the inference algorithm if needed. Note that the type

   inference algorithm is complete only if the registered coercions form

   a lattice.

   \item @{attribute (HOL) "coercion_map"}~@{text "map"} registers a new

   map function to lift coercions through type constructors. The function

   @{text "map"} must conform to the following type pattern

   \begin{matharray}{lll}

     @{text "map"} & @{text "::"} &

       @{text "f\<^isub>1 \<Rightarrow> \<dots> \<Rightarrow> f\<^isub>n \<Rightarrow> (\<alpha>\<^isub>1, \<dots>, \<alpha>\<^isub>n) t \<Rightarrow> (\<beta>\<^isub>1, \<dots>, \<beta>\<^isub>n) t"} \\

   \end{matharray}

   where @{text "t"} is a type constructor and @{text "f\<^isub>i"} is of

   type @{text "\<alpha>\<^isub>i \<Rightarrow> \<beta>\<^isub>i"} or

   @{text "\<beta>\<^isub>i \<Rightarrow> \<alpha>\<^isub>i"}.

   Registering a map function overwrites any existing map function for

   this particular type constructor.

   \item @{attribute (HOL) "coercion_enabled"} enables the coercion

   inference algorithm.

   \end{description}

 *}

 section {* Arithmetic proof support *}

 text {*

   \begin{matharray}{rcl}

     @{method_def (HOL) arith} & : & @{text method} \\

     @{attribute_def (HOL) arith} & : & @{text attribute} \\

     @{attribute_def (HOL) arith_split} & : & @{text attribute} \\

   \end{matharray}

   The @{method (HOL) arith} method decides linear arithmetic problems

   (on types @{text nat}, @{text int}, @{text real}).  Any current

   facts are inserted into the goal before running the procedure.

   The @{attribute (HOL) arith} attribute declares facts that are

   always supplied to the arithmetic provers implicitly.

   The @{attribute (HOL) arith_split} attribute declares case split

   rules to be expanded before @{method (HOL) arith} is invoked.

   Note that a simpler (but faster) arithmetic prover is

   already invoked by the Simplifier.

 *}

 section {* Intuitionistic proof search *}

 text {*

   \begin{matharray}{rcl}

     @{method_def (HOL) iprover} & : & @{text method} \\

   \end{matharray}

   @{rail "

     @@{method (HOL) iprover} ( @{syntax rulemod} * )

   "}

   The @{method (HOL) iprover} method performs intuitionistic proof

   search, depending on specifically declared rules from the context,

   or given as explicit arguments.  Chained facts are inserted into the

   goal before commencing proof search.

   Rules need to be classified as @{attribute (Pure) intro},

   @{attribute (Pure) elim}, or @{attribute (Pure) dest}; here the

   ``@{text "!"}'' indicator refers to ``safe'' rules, which may be

   applied aggressively (without considering back-tracking later).

   Rules declared with ``@{text "?"}'' are ignored in proof search (the

   single-step @{method (Pure) rule} method still observes these).  An

   explicit weight annotation may be given as well; otherwise the

   number of rule premises will be taken into account here.

 *}

 section {* Model Elimination and Resolution *}

 text {*

   \begin{matharray}{rcl}

     @{method_def (HOL) "meson"} & : & @{text method} \\

     @{method_def (HOL) "metis"} & : & @{text method} \\

   \end{matharray}

   @{rail "

     @@{method (HOL) meson} @{syntax thmrefs}?

     ;

     @@{method (HOL) metis} ( '(' ('partial_types' | 'full_types' | 'no_types'

                                   | @{syntax name}) ')' )? @{syntax thmrefs}?

   "}

   The @{method (HOL) meson} method implements Loveland's model elimination

   procedure \cite{loveland-78}. See @{file "~~/src/HOL/ex/Meson_Test.thy"} for

   examples.

   The @{method (HOL) metis} method combines ordered resolution and ordered

   paramodulation to find first-order (or mildly higher-order) proofs. The first

   optional argument specifies a type encoding; see the Sledgehammer manual

   \cite{isabelle-sledgehammer} for details. The @{file

   "~~/src/HOL/Metis_Examples"} directory contains several small theories

   developed to a large extent using Metis.

 *}

 section {* Coherent Logic *}

 text {*

   \begin{matharray}{rcl}

     @{method_def (HOL) "coherent"} & : & @{text method} \\

   \end{matharray}

   @{rail "

     @@{method (HOL) coherent} @{syntax thmrefs}?

   "}

   The @{method (HOL) coherent} method solves problems of

   \emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers

   applications in confluence theory, lattice theory and projective

   geometry.  See @{file "~~/src/HOL/ex/Coherent.thy"} for some

   examples.

 *}

 section {* Proving propositions *}

 text {*

   In addition to the standard proof methods, a number of diagnosis

   tools search for proofs and provide an Isar proof snippet on success.

   These tools are available via the following commands.

   \begin{matharray}{rcl}

     @{command_def (HOL) "solve_direct"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\

     @{command_def (HOL) "try"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\

     @{command_def (HOL) "try_methods"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\

     @{command_def (HOL) "sledgehammer"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\

     @{command_def (HOL) "sledgehammer_params"} & : & @{text "theory \<rightarrow> theory"}

   \end{matharray}

   @{rail "

     @@{command (HOL) try}

     ;

     @@{command (HOL) try_methods} ( ( ( 'simp' | 'intro' | 'elim' | 'dest' ) ':' @{syntax thmrefs} ) + ) ?

       @{syntax nat}?

     ;

     @@{command (HOL) sledgehammer} ( '[' args ']' )? facts? @{syntax nat}?

     ;

     @@{command (HOL) sledgehammer_params} ( ( '[' args ']' ) ? )

     ;

     args: ( @{syntax name} '=' value + ',' )

     ;

     facts: '(' ( ( ( ( 'add' | 'del' ) ':' ) ? @{syntax thmrefs} ) + ) ? ')'

     ;

   "} % FIXME check args "value"

   \begin{description}

   \item @{command (HOL) "solve_direct"} checks whether the current subgoals can

     be solved directly by an existing theorem. Duplicate lemmas can be detected

     in this way.

   \item @{command (HOL) "try_methods"} attempts to prove a subgoal using a combination

     of standard proof methods (@{text auto}, @{text simp}, @{text blast}, etc.).

     Additional facts supplied via @{text "simp:"}, @{text "intro:"},

     @{text "elim:"}, and @{text "dest:"} are passed to the appropriate proof

     methods.

   \item @{command (HOL) "try"} attempts to prove or disprove a subgoal

     using a combination of provers and disprovers (@{text "solve_direct"},

     @{text "quickcheck"}, @{text "try_methods"}, @{text "sledgehammer"},

     @{text "nitpick"}).

   \item @{command (HOL) "sledgehammer"} attempts to prove a subgoal using external

     automatic provers (resolution provers and SMT solvers). See the Sledgehammer

     manual \cite{isabelle-sledgehammer} for details.

   \item @{command (HOL) "sledgehammer_params"} changes

     @{command (HOL) "sledgehammer"} configuration options persistently.

   \end{description}

 *}

 section {* Checking and refuting propositions *}

 text {*

   Identifying incorrect propositions usually involves evaluation of

   particular assignments and systematic counterexample search.  This

   is supported by the following commands.

   \begin{matharray}{rcl}

     @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

     @{command_def (HOL) "quickcheck"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\

     @{command_def (HOL) "refute"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\

     @{command_def (HOL) "nitpick"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\

     @{command_def (HOL) "quickcheck_params"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def (HOL) "refute_params"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def (HOL) "nitpick_params"} & : & @{text "theory \<rightarrow> theory"}

   \end{matharray}

   @{rail "

     @@{command (HOL) value} ( '[' name ']' )? modes? @{syntax term}

     ;

     (@@{command (HOL) quickcheck} | @@{command (HOL) refute} | @@{command (HOL) nitpick})

       ( '[' args ']' )? @{syntax nat}?

     ;

     (@@{command (HOL) quickcheck_params} | @@{command (HOL) refute_params} |

       @@{command (HOL) nitpick_params}) ( '[' args ']' )?

     ;

     modes: '(' (@{syntax name} +) ')'

     ;

     args: ( @{syntax name} '=' value + ',' )

     ;

   "} % FIXME check "value"

   \begin{description}

   \item @{command (HOL) "value"}~@{text t} evaluates and prints a

     term; optionally @{text modes} can be specified, which are

     appended to the current print mode; see \secref{sec:print-modes}.

     Internally, the evaluation is performed by registered evaluators,

     which are invoked sequentially until a result is returned.

     Alternatively a specific evaluator can be selected using square

     brackets; typical evaluators use the current set of code equations

     to normalize and include @{text simp} for fully symbolic

     evaluation using the simplifier, @{text nbe} for

     \emph{normalization by evaluation} and \emph{code} for code

     generation in SML.

   \item @{command (HOL) "quickcheck"} tests the current goal for

     counterexamples using a series of assignments for its

     free variables; by default the first subgoal is tested, an other

     can be selected explicitly using an optional goal index.

     Assignments can be chosen exhausting the search space upto a given

     size, or using a fixed number of random assignments in the search space,

     or exploring the search space symbolically using narrowing.

     By default, quickcheck uses exhaustive testing.

     A number of configuration options are supported for

     @{command (HOL) "quickcheck"}, notably:

     \begin{description}

     \item[@{text tester}] specifies which testing approach to apply.

       There are three testers, @{text exhaustive},

       @{text random}, and @{text narrowing}.

       An unknown configuration option is treated as an argument to tester,

       making @{text "tester ="} optional.

       When multiple testers are given, these are applied in parallel. 

       If no tester is specified, quickcheck uses the testers that are

       set active, i.e., configurations

       @{text quickcheck_exhaustive_active}, @{text quickcheck_random_active},

       @{text quickcheck_narrowing_active} are set to true.

     \item[@{text size}] specifies the maximum size of the search space

     for assignment values.

     \item[@{text eval}] takes a term or a list of terms and evaluates

       these terms under the variable assignment found by quickcheck.

     \item[@{text iterations}] sets how many sets of assignments are

     generated for each particular size.

     \item[@{text no_assms}] specifies whether assumptions in

     structured proofs should be ignored.

     \item[@{text timeout}] sets the time limit in seconds.

     \item[@{text default_type}] sets the type(s) generally used to

     instantiate type variables.

     \item[@{text report}] if set quickcheck reports how many tests

     fulfilled the preconditions.

     \item[@{text quiet}] if not set quickcheck informs about the

     current size for assignment values.

     \item[@{text expect}] can be used to check if the user's

     expectation was met (@{text no_expectation}, @{text

     no_counterexample}, or @{text counterexample}).

     \end{description}

     These option can be given within square brackets.

   \item @{command (HOL) "quickcheck_params"} changes

     @{command (HOL) "quickcheck"} configuration options persistently.

   \item @{command (HOL) "refute"} tests the current goal for

     counterexamples using a reduction to SAT. The following configuration

     options are supported:

     \begin{description}

     \item[@{text minsize}] specifies the minimum size (cardinality) of the

       models to search for.

     \item[@{text maxsize}] specifies the maximum size (cardinality) of the

       models to search for. Nonpositive values mean $\infty$.

     \item[@{text maxvars}] specifies the maximum number of Boolean variables

     to use when transforming the term into a propositional formula.

     Nonpositive values mean $\infty$.

     \item[@{text satsolver}] specifies the SAT solver to use.

     \item[@{text no_assms}] specifies whether assumptions in

     structured proofs should be ignored.

     \item[@{text maxtime}] sets the time limit in seconds.

     \item[@{text expect}] can be used to check if the user's

     expectation was met (@{text genuine}, @{text potential},

     @{text none}, or @{text unknown}).

     \end{description}

     These option can be given within square brackets.

   \item @{command (HOL) "refute_params"} changes

     @{command (HOL) "refute"} configuration options persistently.

   \item @{command (HOL) "nitpick"} tests the current goal for counterexamples

     using a reduction to first-order relational logic. See the Nitpick manual

     \cite{isabelle-nitpick} for details.

   \item @{command (HOL) "nitpick_params"} changes

     @{command (HOL) "nitpick"} configuration options persistently.

   \end{description}

 *}

 section {* Unstructured case analysis and induction \label{sec:hol-induct-tac} *}

 text {*

   The following tools of Isabelle/HOL support cases analysis and

   induction in unstructured tactic scripts; see also

   \secref{sec:cases-induct} for proper Isar versions of similar ideas.

   \begin{matharray}{rcl}

     @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & @{text method} \\

     @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\

     @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\

     @{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

   \end{matharray}

   @{rail "

     @@{method (HOL) case_tac} @{syntax goal_spec}? @{syntax term} rule?

     ;

     @@{method (HOL) induct_tac} @{syntax goal_spec}? (@{syntax insts} * @'and') rule?

     ;

     @@{method (HOL) ind_cases} (@{syntax prop}+) (@'for' (@{syntax name}+))?

     ;

     @@{command (HOL) inductive_cases} (@{syntax thmdecl}? (@{syntax prop}+) + @'and')

     ;

     rule: 'rule' ':' @{syntax thmref}

   "}

   \begin{description}

   \item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit

   to reason about inductive types.  Rules are selected according to

   the declarations by the @{attribute cases} and @{attribute induct}

   attributes, cf.\ \secref{sec:cases-induct}.  The @{command (HOL)

   datatype} package already takes care of this.

   These unstructured tactics feature both goal addressing and dynamic

   instantiation.  Note that named rule cases are \emph{not} provided

   as would be by the proper @{method cases} and @{method induct} proof

   methods (see \secref{sec:cases-induct}).  Unlike the @{method

   induct} method, @{method induct_tac} does not handle structured rule

   statements, only the compact object-logic conclusion of the subgoal

   being addressed.

   \item @{method (HOL) ind_cases} and @{command (HOL)

   "inductive_cases"} provide an interface to the internal @{ML_text

   mk_cases} operation.  Rules are simplified in an unrestricted

   forward manner.

   While @{method (HOL) ind_cases} is a proof method to apply the

   result immediately as elimination rules, @{command (HOL)

   "inductive_cases"} provides case split theorems at the theory level

   for later use.  The @{keyword "for"} argument of the @{method (HOL)

   ind_cases} method allows to specify a list of variables that should

   be generalized before applying the resulting rule.

   \end{description}

 *}

 section {* Executable code *}

 text {* For validation purposes, it is often useful to \emph{execute}

   specifications.  In principle, execution could be simulated by

   Isabelle's inference kernel, i.e. by a combination of resolution and

   simplification.  Unfortunately, this approach is rather inefficient.

   A more efficient way of executing specifications is to translate

   them into a functional programming language such as ML.

   Isabelle provides two generic frameworks to support code generation

   from executable specifications.  Isabelle/HOL instantiates these

   mechanisms in a way that is amenable to end-user applications.

 *}

 subsection {* The new code generator (F. Haftmann) *}

 text {* This framework generates code from functional programs

   (including overloading using type classes) to SML \cite{SML}, OCaml

   \cite{OCaml}, Haskell \cite{haskell-revised-report} and Scala

   \cite{scala-overview-tech-report}.  Conceptually, code generation is

   split up in three steps: \emph{selection} of code theorems,

   \emph{translation} into an abstract executable view and

   \emph{serialization} to a specific \emph{target language}.

   Inductive specifications can be executed using the predicate

   compiler which operates within HOL.  See \cite{isabelle-codegen} for

   an introduction.

   \begin{matharray}{rcl}

     @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

     @{attribute_def (HOL) code} & : & @{text attribute} \\

     @{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

     @{attribute_def (HOL) code_inline} & : & @{text attribute} \\

     @{attribute_def (HOL) code_post} & : & @{text attribute} \\

     @{command_def (HOL) "print_codeproc"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

     @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

     @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

     @{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def (HOL) "code_reflect"} & : & @{text "theory \<rightarrow> theory"}

   \end{matharray}

   @{rail "

     @@{command (HOL) export_code} ( constexpr + ) \\

        ( ( @'in' target ( @'module_name' @{syntax string} ) ? \\

         ( @'file' ( @{syntax string} | '-' ) ) ? ( '(' args ')' ) ?) + ) ?

     ;

     const: @{syntax term}

     ;

     constexpr: ( const | 'name._' | '_' )

     ;

     typeconstructor: @{syntax nameref}

     ;

     class: @{syntax nameref}

     ;

     target: 'SML' | 'OCaml' | 'Haskell' | 'Scala'

     ;

     @@{attribute (HOL) code} ( 'del' | 'abstype' | 'abstract' )?

     ;

     @@{command (HOL) code_abort} ( const + )

     ;

     @@{command (HOL) code_datatype} ( const + )

     ;

     @@{attribute (HOL) code_inline} ( 'del' ) ?

     ;

     @@{attribute (HOL) code_post} ( 'del' ) ?

     ;

     @@{command (HOL) code_thms} ( constexpr + ) ?

     ;

     @@{command (HOL) code_deps} ( constexpr + ) ?

     ;

     @@{command (HOL) code_const} (const + @'and') \\

       ( ( '(' target ( syntax ? + @'and' ) ')' ) + )

     ;

     @@{command (HOL) code_type} (typeconstructor + @'and') \\

       ( ( '(' target ( syntax ? + @'and' ) ')' ) + )

     ;

     @@{command (HOL) code_class} (class + @'and') \\

       ( ( '(' target \\ ( @{syntax string} ? + @'and' ) ')' ) + )

     ;

     @@{command (HOL) code_instance} (( typeconstructor '::' class ) + @'and') \\

       ( ( '(' target ( '-' ? + @'and' ) ')' ) + )

     ;

     @@{command (HOL) code_reserved} target ( @{syntax string} + )

     ;

     @@{command (HOL) code_monad} const const target

     ;

     @@{command (HOL) code_include} target ( @{syntax string} ( @{syntax string} | '-') )

     ;

     @@{command (HOL) code_modulename} target ( ( @{syntax string} @{syntax string} ) + )

     ;

     @@{command (HOL) code_reflect} @{syntax string} \\

       ( @'datatypes' ( @{syntax string} '=' ( '_' | ( @{syntax string} + '|' ) + @'and' ) ) ) ? \\

       ( @'functions' ( @{syntax string} + ) ) ? ( @'file' @{syntax string} ) ?

     ;

     syntax: @{syntax string} | ( @'infix' | @'infixl' | @'infixr' ) @{syntax nat} @{syntax string}

   "}

   \begin{description}

   \item @{command (HOL) "export_code"} generates code for a given list

   of constants in the specified target language(s).  If no

   serialization instruction is given, only abstract code is generated

   internally.

   Constants may be specified by giving them literally, referring to

   all executable contants within a certain theory by giving @{text

   "name.*"}, or referring to \emph{all} executable constants currently

   available by giving @{text "*"}.

   By default, for each involved theory one corresponding name space

   module is generated.  Alternativly, a module name may be specified

   after the @{keyword "module_name"} keyword; then \emph{all} code is

   placed in this module.

   For \emph{SML}, \emph{OCaml} and \emph{Scala} the file specification

   refers to a single file; for \emph{Haskell}, it refers to a whole

   directory, where code is generated in multiple files reflecting the

   module hierarchy.  Omitting the file specification denotes standard

   output.

   Serializers take an optional list of arguments in parentheses.  For

   \emph{SML} and \emph{OCaml}, ``@{text no_signatures}`` omits

   explicit module signatures.

   For \emph{Haskell} a module name prefix may be given using the

   ``@{text "root:"}'' argument; ``@{text string_classes}'' adds a

   ``@{verbatim "deriving (Read, Show)"}'' clause to each appropriate

   datatype declaration.

   \item @{attribute (HOL) code} explicitly selects (or with option

   ``@{text "del"}'' deselects) a code equation for code generation.

   Usually packages introducing code equations provide a reasonable

   default setup for selection.  Variants @{text "code abstype"} and

   @{text "code abstract"} declare abstract datatype certificates or

   code equations on abstract datatype representations respectively.

   \item @{command (HOL) "code_abort"} declares constants which are not

   required to have a definition by means of code equations; if needed

   these are implemented by program abort instead.

   \item @{command (HOL) "code_datatype"} specifies a constructor set

   for a logical type.

   \item @{command (HOL) "print_codesetup"} gives an overview on

   selected code equations and code generator datatypes.

   \item @{attribute (HOL) code_inline} declares (or with option

   ``@{text "del"}'' removes) inlining theorems which are applied as

   rewrite rules to any code equation during preprocessing.

   \item @{attribute (HOL) code_post} declares (or with option ``@{text

   "del"}'' removes) theorems which are applied as rewrite rules to any

   result of an evaluation.

   \item @{command (HOL) "print_codeproc"} prints the setup of the code

   generator preprocessor.

   \item @{command (HOL) "code_thms"} prints a list of theorems

   representing the corresponding program containing all given

   constants after preprocessing.

   \item @{command (HOL) "code_deps"} visualizes dependencies of

   theorems representing the corresponding program containing all given

   constants after preprocessing.

   \item @{command (HOL) "code_const"} associates a list of constants

   with target-specific serializations; omitting a serialization

   deletes an existing serialization.

   \item @{command (HOL) "code_type"} associates a list of type

   constructors with target-specific serializations; omitting a

   serialization deletes an existing serialization.

   \item @{command (HOL) "code_class"} associates a list of classes

   with target-specific class names; omitting a serialization deletes

   an existing serialization.  This applies only to \emph{Haskell}.

   \item @{command (HOL) "code_instance"} declares a list of type

   constructor / class instance relations as ``already present'' for a

   given target.  Omitting a ``@{text "-"}'' deletes an existing

   ``already present'' declaration.  This applies only to

   \emph{Haskell}.

   \item @{command (HOL) "code_reserved"} declares a list of names as

   reserved for a given target, preventing it to be shadowed by any

   generated code.

   \item @{command (HOL) "code_monad"} provides an auxiliary mechanism

   to generate monadic code for Haskell.

   \item @{command (HOL) "code_include"} adds arbitrary named content

   (``include'') to generated code.  A ``@{text "-"}'' as last argument

   will remove an already added ``include''.

   \item @{command (HOL) "code_modulename"} declares aliasings from one

   module name onto another.

   \item @{command (HOL) "code_reflect"} without a ``@{text "file"}''

   argument compiles code into the system runtime environment and

   modifies the code generator setup that future invocations of system

   runtime code generation referring to one of the ``@{text

   "datatypes"}'' or ``@{text "functions"}'' entities use these precompiled

   entities.  With a ``@{text "file"}'' argument, the corresponding code

   is generated into that specified file without modifying the code

   generator setup.

   \end{description}

 *}

 subsection {* The old code generator (S. Berghofer) *}

 text {* This framework generates code from both functional and

   relational programs to SML, as explained below.

   \begin{matharray}{rcl}

     @{command_def "code_module"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def "code_library"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def "consts_code"} & : & @{text "theory \<rightarrow> theory"} \\

     @{command_def "types_code"} & : & @{text "theory \<rightarrow> theory"} \\

     @{attribute_def code} & : & @{text attribute} \\

   \end{matharray}

   @{rail "

   ( @@{command code_module} | @@{command code_library} ) modespec? @{syntax name}? \\

     ( @'file' name ) ? ( @'imports' ( @{syntax name} + ) ) ? \\

     @'contains' ( ( @{syntax name} '=' @{syntax term} ) + | @{syntax term} + )

   ;

   modespec: '(' ( @{syntax name} * ) ')'

   ;

   @@{command (HOL) consts_code} (codespec +)

   ;

   codespec: const template attachment ?

   ;

   @@{command (HOL) types_code} (tycodespec +)

   ;

   tycodespec: @{syntax name} template attachment ?

   ;

   const: @{syntax term}

   ;

   template: '(' @{syntax string} ')'

   ;

   attachment: 'attach' modespec? '{' @{syntax text} '}'

   ;

   @@{attribute code} name?

   "}

 *}

 subsubsection {* Invoking the code generator *}

 text {* The code generator is invoked via the @{command code_module}

   and @{command code_library} commands, which correspond to

   \emph{incremental} and \emph{modular} code generation, respectively.

   \begin{description}

   \item [Modular] For each theory, an ML structure is generated,

   containing the code generated from the constants defined in this

   theory.

   \item [Incremental] All the generated code is emitted into the same

   structure.  This structure may import code from previously generated

   structures, which can be specified via @{keyword "imports"}.

   Moreover, the generated structure may also be referred to in later

   invocations of the code generator.

   \end{description}

   After the @{command code_module} and @{command code_library}

   keywords, the user may specify an optional list of ``modes'' in

   parentheses. These can be used to instruct the code generator to

   emit additional code for special purposes, e.g.\ functions for

   converting elements of generated datatypes to Isabelle terms, or

   test data generators. The list of modes is followed by a module

   name.  The module name is optional for modular code generation, but

   must be specified for incremental code generation.

   The code can either be written to a file, in which case a file name

   has to be specified after the @{keyword "file"} keyword, or be loaded

   directly into Isabelle's ML environment. In the latter case, the

   @{command ML} theory command can be used to inspect the results

   interactively, for example.

   The terms from which to generate code can be specified after the

   @{keyword "contains"} keyword, either as a list of bindings, or just

   as a list of terms. In the latter case, the code generator just

   produces code for all constants and types occuring in the term, but

   does not bind the compiled terms to ML identifiers.

   Here is an example:

 *}

 code_module Test

 contains test = "foldl op + (0 :: int) [1, 2, 3, 4, 5]"

 text {* \noindent This binds the result of compiling the given term to

   the ML identifier @{ML Test.test}.  *}

 ML {* @{assert} (Test.test = 15) *}

 subsubsection {* Configuring the code generator *}

 text {* When generating code for a complex term, the code generator

   recursively calls itself for all subterms.  When it arrives at a

   constant, the default strategy of the code generator is to look up

   its definition and try to generate code for it.  Constants which

   have no definitions that are immediately executable, may be

   associated with a piece of ML code manually using the @{command_ref

   consts_code} command.  It takes a list whose elements consist of a

   constant (given in usual term syntax -- an explicit type constraint

   accounts for overloading), and a mixfix template describing the ML

   code. The latter is very much the same as the mixfix templates used

   when declaring new constants.  The most notable difference is that

   terms may be included in the ML template using antiquotation

   brackets @{verbatim "{"}@{verbatim "*"}~@{text "..."}~@{verbatim

   "*"}@{verbatim "}"}.

   A similar mechanism is available for types: @{command_ref

   types_code} associates type constructors with specific ML code.

   For example, the following declarations copied from @{file

   "~~/src/HOL/Product_Type.thy"} describe how the product type of

   Isabelle/HOL should be compiled to ML.  *}

 typedecl ('a, 'b) prod

 consts Pair :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) prod"

 types_code prod  ("(_ */ _)")

 consts_code Pair  ("(_,/ _)")

 text {* Sometimes, the code associated with a constant or type may

   need to refer to auxiliary functions, which have to be emitted when

   the constant is used. Code for such auxiliary functions can be

   declared using @{keyword "attach"}. For example, the @{const wfrec}

   function can be implemented as follows:

 *}

 consts_code wfrec  ("\<module>wfrec?")  (* FIXME !? *)

 attach {* fun wfrec f x = f (wfrec f) x *}

 text {* If the code containing a call to @{const wfrec} resides in an

   ML structure different from the one containing the function

   definition attached to @{const wfrec}, the name of the ML structure

   (followed by a ``@{text "."}'')  is inserted in place of ``@{text

   "\<module>"}'' in the above template.  The ``@{text "?"}''  means that

   the code generator should ignore the first argument of @{const

   wfrec}, i.e.\ the termination relation, which is usually not

   executable.

   \medskip Another possibility of configuring the code generator is to

   register theorems to be used for code generation. Theorems can be

   registered via the @{attribute code} attribute. It takes an optional

   name as an argument, which indicates the format of the

   theorem. Currently supported formats are equations (this is the

   default when no name is specified) and horn clauses (this is

   indicated by the name \texttt{ind}). The left-hand sides of

   equations may only contain constructors and distinct variables,

   whereas horn clauses must have the same format as introduction rules

   of inductive definitions.

   The following example specifies three equations from which to

   generate code for @{term "op <"} on natural numbers (see also

   @{"file" "~~/src/HOL/Nat.thy"}).  *}

 lemma [code]: "(Suc m < Suc n) = (m < n)"

   and [code]: "((n::nat) < 0) = False"

   and [code]: "(0 < Suc n) = True" by simp_all

 subsubsection {* Specific HOL code generators *}

 text {* The basic code generator framework offered by Isabelle/Pure

   has already been extended with additional code generators for

   specific HOL constructs. These include datatypes, recursive

   functions and inductive relations. The code generator for inductive

   relations can handle expressions of the form @{text "(t\<^sub>1, \<dots>, t\<^sub>n) \<in>

   r"}, where @{text "r"} is an inductively defined relation. If at

   least one of the @{text "t\<^sub>i"} is a dummy pattern ``@{text "_"}'',

   the above expression evaluates to a sequence of possible answers. If

   all of the @{text "t\<^sub>i"} are proper terms, the expression evaluates

   to a boolean value.

   The following example demonstrates this for beta-reduction on lambda

   terms (see also @{"file" "~~/src/HOL/Proofs/Lambda/Lambda.thy"}).

 *}

 datatype dB =

     Var nat

   | App dB dB  (infixl "\<degree>" 200)

   | Abs dB

 primrec lift :: "dB \<Rightarrow> nat \<Rightarrow> dB"

 where

     "lift (Var i) k = (if i < k then Var i else Var (i + 1))"

   | "lift (s \<degree> t) k = lift s k \<degree> lift t k"

   | "lift (Abs s) k = Abs (lift s (k + 1))"

 primrec subst :: "dB \<Rightarrow> dB \<Rightarrow> nat \<Rightarrow> dB"  ("_[_'/_]" [300, 0, 0] 300)

 where

     "(Var i)[s/k] =

       (if k < i then Var (i - 1) else if i = k then s else Var i)"

   | "(t \<degree> u)[s/k] = t[s/k] \<degree> u[s/k]"

   | "(Abs t)[s/k] = Abs (t[lift s 0 / k+1])"

 inductive beta :: "dB \<Rightarrow> dB \<Rightarrow> bool"  (infixl "\<rightarrow>\<^sub>\<beta>" 50)

 where

     beta: "Abs s \<degree> t \<rightarrow>\<^sub>\<beta> s[t/0]"

   | appL: "s \<rightarrow>\<^sub>\<beta> t \<Longrightarrow> s \<degree> u \<rightarrow>\<^sub>\<beta> t \<degree> u"

   | appR: "s \<rightarrow>\<^sub>\<beta> t \<Longrightarrow> u \<degree> s \<rightarrow>\<^sub>\<beta> u \<degree> t"

   | abs: "s \<rightarrow>\<^sub>\<beta> t \<Longrightarrow> Abs s \<rightarrow>\<^sub>\<beta> Abs t"

 code_module Test

 contains

   test1 = "Abs (Var 0) \<degree> Var 0 \<rightarrow>\<^sub>\<beta> Var 0"

   test2 = "Abs (Abs (Var 0 \<degree> Var 0) \<degree> (Abs (Var 0) \<degree> Var 0)) \<rightarrow>\<^sub>\<beta> _"

 text {*

   In the above example, @{ML Test.test1} evaluates to a boolean,

   whereas @{ML Test.test2} is a lazy sequence whose elements can be

   inspected separately.

 *}

 ML {* @{assert} Test.test1 *}

 ML {* val results = DSeq.list_of Test.test2 *}

 ML {* @{assert} (length results = 2) *}

 text {*

   \medskip The theory underlying the HOL code generator is described

   more detailed in \cite{Berghofer-Nipkow:2002}. More examples that

   illustrate the usage of the code generator can be found e.g.\ in

   @{"file" "~~/src/HOL/MicroJava/J/JListExample.thy"} and @{"file"

   "~~/src/HOL/MicroJava/JVM/JVMListExample.thy"}.

 *}

 section {* Definition by specification \label{sec:hol-specification} *}

 text {*

   \begin{matharray}{rcl}

     @{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\

     @{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\

   \end{matharray}

   @{rail "

   (@@{command (HOL) specification} | @@{command (HOL) ax_specification})

     '(' (decl +) ')' \\ (@{syntax thmdecl}? @{syntax prop} +)

   ;

   decl: ((@{syntax name} ':')? @{syntax term} '(' @'overloaded' ')'?)

   "}

   \begin{description}

   \item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a

   goal stating the existence of terms with the properties specified to

   hold for the constants given in @{text decls}.  After finishing the

   proof, the theory will be augmented with definitions for the given

   constants, as well as with theorems stating the properties for these

   constants.

   \item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up

   a goal stating the existence of terms with the properties specified

   to hold for the constants given in @{text decls}.  After finishing

   the proof, the theory will be augmented with axioms expressing the

   properties given in the first place.

   \item @{text decl} declares a constant to be defined by the

   specification given.  The definition for the constant @{text c} is

   bound to the name @{text c_def} unless a theorem name is given in

   the declaration.  Overloaded constants should be declared as such.

   \end{description}

   Whether to use @{command (HOL) "specification"} or @{command (HOL)

   "ax_specification"} is to some extent a matter of style.  @{command

   (HOL) "specification"} introduces no new axioms, and so by

   construction cannot introduce inconsistencies, whereas @{command

   (HOL) "ax_specification"} does introduce axioms, but only after the

   user has explicitly proven it to be safe.  A practical issue must be

   considered, though: After introducing two constants with the same

   properties using @{command (HOL) "specification"}, one can prove

   that the two constants are, in fact, equal.  If this might be a

   problem, one should use @{command (HOL) "ax_specification"}.

 *}

 end