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,