(*:maxLineLen=78:*)
theory HOL_Specific
imports
Main
"HOL-Library.Old_Datatype"
"HOL-Library.Old_Recdef"
"HOL-Library.Adhoc_Overloading"
"HOL-Library.Dlist"
"HOL-Library.FSet"
Base
begin
chapter \<open>Higher-Order Logic\<close>
text \<open>
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>\<open>weaker\<close> 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 \<open>\<lambda>\<close>-calculus and functional programming.
Function application is curried. To apply the function \<open>f\<close> of type \<open>\<tau>\<^sub>1 \<Rightarrow>
\<tau>\<^sub>2 \<Rightarrow> \<tau>\<^sub>3\<close> to the arguments \<open>a\<close> and \<open>b\<close> in HOL, you simply write \<open>f a b\<close> (as
in ML or Haskell). There is no ``apply'' operator; the existing application
of the Pure \<open>\<lambda>\<close>-calculus is re-used. Note that in HOL \<open>f (a, b)\<close> means ``\<open>f\<close>
applied to the pair \<open>(a, b)\<close> (which is notation for \<open>Pair a b\<close>). 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.
\<close>
chapter \<open>Derived specification elements\<close>
section \<open>Inductive and coinductive definitions \label{sec:hol-inductive}\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def (HOL) "inductive"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def (HOL) "inductive_set"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def (HOL) "coinductive"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def (HOL) "coinductive_set"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def "print_inductives"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
@{attribute_def (HOL) mono} & : & \<open>attribute\<close> \\
\end{matharray}
An \<^emph>\<open>inductive definition\<close> specifies the least predicate or set \<open>R\<close> closed
under given rules: applying a rule to elements of \<open>R\<close> yields a result within
\<open>R\<close>. For example, a structural operational semantics is an inductive
definition of an evaluation relation.
Dually, a \<^emph>\<open>coinductive definition\<close> specifies the greatest predicate or set
\<open>R\<close> that is consistent with given rules: every element of \<open>R\<close> can be seen as
arising by applying a rule to elements of \<open>R\<close>. 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 \<open>A \<le> B \<Longrightarrow> \<M> A \<le> \<M>
B\<close>, for each premise \<open>\<M> R t\<close> in an introduction rule. The default rule
declarations of Isabelle/HOL already take care of most common situations.
\<^rail>\<open>
(@@{command (HOL) inductive} | @@{command (HOL) inductive_set} |
@@{command (HOL) coinductive} | @@{command (HOL) coinductive_set})
@{syntax vars} @{syntax for_fixes} \<newline>
(@'where' @{syntax multi_specs})? (@'monos' @{syntax thms})?
;
@@{command print_inductives} ('!'?)
;
@@{attribute (HOL) mono} (() | 'add' | 'del')
\<close>
\<^descr> @{command (HOL) "inductive"} and @{command (HOL) "coinductive"} define
(co)inductive predicates from the introduction rules.
The propositions given as \<open>clauses\<close> in the @{keyword "where"} part are
either rules of the usual \<open>\<And>/\<Longrightarrow>\<close> format (with arbitrary nesting), or
equalities using \<open>\<equiv>\<close>. 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 \<open>clauses\<close>.
The optional @{keyword "monos"} declaration contains additional
\<^emph>\<open>monotonicity theorems\<close>, which are required for each operator applied to a
recursive set in the introduction rules.
\<^descr> @{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.
\<^descr> @{command "print_inductives"} prints (co)inductive definitions and
monotonicity rules; the ``\<open>!\<close>'' option indicates extra verbosity.
\<^descr> @{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.
\<close>
subsection \<open>Derived rules\<close>
text \<open>
A (co)inductive definition of \<open>R\<close> provides the following main theorems:
\<^descr> \<open>R.intros\<close> 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;
\<^descr> \<open>R.cases\<close> is the case analysis (or elimination) rule;
\<^descr> \<open>R.induct\<close> or \<open>R.coinduct\<close> is the (co)induction rule;
\<^descr> \<open>R.simps\<close> is the equation unrolling the fixpoint of the predicate one
step.
When several predicates \<open>R\<^sub>1, \<dots>, R\<^sub>n\<close> are defined simultaneously, the list
of introduction rules is called \<open>R\<^sub>1_\<dots>_R\<^sub>n.intros\<close>, the case analysis rules
are called \<open>R\<^sub>1.cases, \<dots>, R\<^sub>n.cases\<close>, and the list of mutual induction rules
is called \<open>R\<^sub>1_\<dots>_R\<^sub>n.inducts\<close>.
\<close>
subsection \<open>Monotonicity theorems\<close>
text \<open>
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:
\<^item> Theorems of the form \<open>A \<le> B \<Longrightarrow> \<M> A \<le> \<M> B\<close>, for proving monotonicity of
inductive definitions whose introduction rules have premises involving terms
such as \<open>\<M> R t\<close>.
\<^item> Monotonicity theorems for logical operators, which are of the general form
\<open>(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>\<close>. For example, in the case of the operator \<open>\<or>\<close>,
the corresponding theorem is
\[
\infer{\<open>P\<^sub>1 \<or> P\<^sub>2 \<longrightarrow> Q\<^sub>1 \<or> Q\<^sub>2\<close>}{\<open>P\<^sub>1 \<longrightarrow> Q\<^sub>1\<close> & \<open>P\<^sub>2 \<longrightarrow> Q\<^sub>2\<close>}
\]
\<^item> De Morgan style equations for reasoning about the ``polarity'' of
expressions, e.g.
\[
\<^prop>\<open>\<not> \<not> P \<longleftrightarrow> P\<close> \qquad\qquad
\<^prop>\<open>\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q\<close>
\]
\<^item> Equations for reducing complex operators to more primitive ones whose
monotonicity can easily be proved, e.g.
\[
\<^prop>\<open>(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q\<close> \qquad\qquad
\<^prop>\<open>Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x\<close>
\]
\<close>
subsubsection \<open>Examples\<close>
text \<open>The finite powerset operator can be defined inductively like this:\<close>
(*<*)experiment begin(*>*)
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 \<open>The accessible part of a relation is defined as follows:\<close>
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"
(*<*)end(*>*)
text \<open>
Common logical connectives can be easily characterized as non-recursive
inductive definitions with parameters, but without arguments.
\<close>
(*<*)experiment begin(*>*)
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"
(*<*)end(*>*)
text \<open>
Here the \<open>cases\<close> or \<open>induct\<close> 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.
\<close>
section \<open>Recursive functions \label{sec:recursion}\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def (HOL) "primrec"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def (HOL) "fun"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def (HOL) "function"} & : & \<open>local_theory \<rightarrow> proof(prove)\<close> \\
@{command_def (HOL) "termination"} & : & \<open>local_theory \<rightarrow> proof(prove)\<close> \\
@{command_def (HOL) "fun_cases"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
\end{matharray}
\<^rail>\<open>
@@{command (HOL) primrec} @{syntax specification}
;
(@@{command (HOL) fun} | @@{command (HOL) function}) opts? @{syntax specification}
;
opts: '(' (('sequential' | 'domintros') + ',') ')'
;
@@{command (HOL) termination} @{syntax term}?
;
@@{command (HOL) fun_cases} (@{syntax thmdecl}? @{syntax prop} + @'and')
\<close>
\<^descr> @{command (HOL) "primrec"} defines primitive recursive functions over
datatypes (see also @{command_ref (HOL) datatype}). The given \<open>equations\<close>
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 \<open>C\<close> is a datatype constructor, \<open>rhs\<close> contains only the free
variables on the left-hand side (or from the context), and all recursive
occurrences of \<open>f\<close> in \<open>rhs\<close> are of the form \<open>f \<dots> y\<^sub>i \<dots>\<close> for some \<open>i\<close>. 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 \<open>f\<close> applied to datatype constructions, by
simulating symbolic computation via rewriting.
\<^descr> @{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 \<open>f.psimps\<close>) and induction rule (named \<open>f.pinduct\<close>) are guarded
by a generated domain predicate \<open>f_dom\<close>. The @{command (HOL) "termination"}
command can then be used to establish that the function is total.
\<^descr> @{command (HOL) "fun"} is a shorthand notation for ``@{command (HOL)
"function"}~\<open>(sequential)\<close>'', followed by automated proof attempts regarding
pattern matching and termination. See @{cite "isabelle-function"} for
further details.
\<^descr> @{command (HOL) "termination"}~\<open>f\<close> commences a termination proof for the
previously defined function \<open>f\<close>. 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.
\<^descr> @{command (HOL) "fun_cases"} generates specialized elimination rules for
function equations. It expects one or more function equations and produces
rules that eliminate the given equalities, following the cases given in the
function definition.
Recursive definitions introduced by the @{command (HOL) "function"} command
accommodate reasoning by induction (cf.\ @{method induct}): rule \<open>f.induct\<close>
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 \<open>f.simps\<close>, where \<open>f\<close> 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.
\<^descr> \<open>sequential\<close> 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.
\<^descr> \<open>domintros\<close> 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.
\<close>
subsubsection \<open>Example: evaluation of expressions\<close>
text \<open>
Subsequently, we define mutual datatypes for arithmetic and boolean
expressions, and use @{command primrec} for evaluation functions that follow
the same recursive structure.
\<close>
(*<*)experiment begin(*>*)
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 \<open>\<^medskip> Evaluation of arithmetic and boolean expressions\<close>
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 \<open>
Since the value of an expression depends on the value of its variables, the
functions \<^const>\<open>evala\<close> and \<^const>\<open>evalb\<close> take an additional parameter, an
\<^emph>\<open>environment\<close> that maps variables to their values.
\<^medskip>
Substitution on expressions can be defined similarly. The mapping \<open>f\<close> of
type \<^typ>\<open>'a \<Rightarrow> 'a aexp\<close> given as a parameter is lifted canonically on the
types \<^typ>\<open>'a aexp\<close> and \<^typ>\<open>'a bexp\<close>, respectively.
\<close>
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 \<open>
In textbooks about semantics one often finds substitution theorems, which
express the relationship between substitution and evaluation. For \<^typ>\<open>'a
aexp\<close> and \<^typ>\<open>'a bexp\<close>, we can prove such a theorem by mutual
induction, followed by simplification.
\<close>
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
(*<*)end(*>*)
subsubsection \<open>Example: a substitution function for terms\<close>
text \<open>Functions on datatypes with nested recursion are also defined
by mutual primitive recursion.\<close>
(*<*)experiment begin(*>*)
datatype ('a, 'b) "term" = Var 'a | App 'b "('a, 'b) term list"
text \<open>
A substitution function on type \<^typ>\<open>('a, 'b) term\<close> can be defined as
follows, by working simultaneously on \<^typ>\<open>('a, 'b) term list\<close>:
\<close>
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 \<open>
The recursion scheme follows the structure of the unfolded definition of
type \<^typ>\<open>('a, 'b) term\<close>. To prove properties of this substitution
function, mutual induction is needed:
\<close>
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 rule: subst_term.induct subst_term_list.induct) simp_all
(*<*)end(*>*)
subsubsection \<open>Example: a map function for infinitely branching trees\<close>
text \<open>Defining functions on infinitely branching datatypes by primitive
recursion is just as easy.\<close>
(*<*)experiment begin(*>*)
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 \<open>
Note that all occurrences of functions such as \<open>ts\<close> above must be applied to
an argument. In particular, \<^term>\<open>map_tree f \<circ> ts\<close> is not allowed here.
\<^medskip>
Here is a simple composition lemma for \<^term>\<open>map_tree\<close>:
\<close>
lemma "map_tree g (map_tree f t) = map_tree (g \<circ> f) t"
by (induct t) simp_all
(*<*)end(*>*)
subsection \<open>Proof methods related to recursive definitions\<close>
text \<open>
\begin{matharray}{rcl}
@{method_def (HOL) pat_completeness} & : & \<open>method\<close> \\
@{method_def (HOL) relation} & : & \<open>method\<close> \\
@{method_def (HOL) lexicographic_order} & : & \<open>method\<close> \\
@{method_def (HOL) size_change} & : & \<open>method\<close> \\
@{method_def (HOL) induction_schema} & : & \<open>method\<close> \\
\end{matharray}
\<^rail>\<open>
@@{method (HOL) relation} @{syntax term}
;
@@{method (HOL) lexicographic_order} (@{syntax clasimpmod} * )
;
@@{method (HOL) size_change} ( orders (@{syntax clasimpmod} * ) )
;
@@{method (HOL) induction_schema}
;
orders: ( 'max' | 'min' | 'ms' ) *
\<close>
\<^descr> @{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"}).
\<^descr> @{method (HOL) relation}~\<open>R\<close> introduces a termination proof using the
relation \<open>R\<close>. The resulting proof state will contain goals expressing that
\<open>R\<close> is wellfounded, and that the arguments of recursive calls decrease with
respect to \<open>R\<close>. Usually, this method is used as the initial proof step of
manual termination proofs.
\<^descr> @{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"}).
\<^descr> @{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: \<open>max\<close>, \<open>min\<close>, and \<open>ms\<close> (multiset), which is only available
when the theory \<open>Multiset\<close> 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}).
\<^descr> @{method (HOL) induction_schema} derives user-specified induction rules
from well-founded induction and completeness of patterns. This factors out
some operations that are done internally by the function package and makes
them available separately. See \<^file>\<open>~~/src/HOL/ex/Induction_Schema.thy\<close> for
examples.
\<close>
subsection \<open>Functions with explicit partiality\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def (HOL) "partial_function"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{attribute_def (HOL) "partial_function_mono"} & : & \<open>attribute\<close> \\
\end{matharray}
\<^rail>\<open>
@@{command (HOL) partial_function} '(' @{syntax name} ')'
@{syntax specification}
\<close>
\<^descr> @{command (HOL) "partial_function"}~\<open>(mode)\<close> 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 monotonicity 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 \<open>mode\<close> 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:
\<^descr> \<open>option\<close> defines functions that map into the \<^type>\<open>option\<close> type. Here,
the value \<^term>\<open>None\<close> is used to model a non-terminating computation.
Monotonicity requires that if \<^term>\<open>None\<close> is returned by a recursive
call, then the overall result must also be \<^term>\<open>None\<close>. This is best
achieved through the use of the monadic operator \<^const>\<open>Option.bind\<close>.
\<^descr> \<open>tailrec\<close> defines functions with an arbitrary result type and uses the
slightly degenerated partial order where \<^term>\<open>undefined\<close> is the bottom
element. Now, monotonicity requires that if \<^term>\<open>undefined\<close> is returned
by a recursive call, then the overall result must also be \<^term>\<open>undefined\<close>. 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.
Experienced users may define new modes by instantiating the locale \<^const>\<open>partial_function_definitions\<close> appropriately.
\<^descr> @{attribute (HOL) partial_function_mono} declares rules for use in the
internal monotonicity proofs of partial function definitions.
\<close>
subsection \<open>Old-style recursive function definitions (TFL)\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def (HOL) "recdef"} & : & \<open>theory \<rightarrow> theory)\<close> \\
\end{matharray}
The old TFL command @{command (HOL) "recdef"} for defining recursive is
mostly obsolete; @{command (HOL) "function"} or @{command (HOL) "fun"}
should be used instead.
\<^rail>\<open>
@@{command (HOL) recdef} ('(' @'permissive' ')')? \<newline>
@{syntax name} @{syntax term} (@{syntax prop} +) hints?
;
hints: '(' @'hints' ( recdefmod * ) ')'
;
recdefmod: (('recdef_simp' | 'recdef_cong' | 'recdef_wf')
(() | 'add' | 'del') ':' @{syntax thms}) | @{syntax clasimpmod}
\<close>
\<^descr> @{command (HOL) "recdef"} defines general well-founded recursive functions
(using the TFL package). The ``\<open>(permissive)\<close>'' option tells TFL to recover
from failed proof attempts, returning unfinished results. The \<open>recdef_simp\<close>,
\<open>recdef_cong\<close>, and \<open>recdef_wf\<close> 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}).
\<^medskip>
Hints for @{command (HOL) "recdef"} may be also declared globally, using the
following attributes.
\begin{matharray}{rcl}
@{attribute_def (HOL) recdef_simp} & : & \<open>attribute\<close> \\
@{attribute_def (HOL) recdef_cong} & : & \<open>attribute\<close> \\
@{attribute_def (HOL) recdef_wf} & : & \<open>attribute\<close> \\
\end{matharray}
\<^rail>\<open>
(@@{attribute (HOL) recdef_simp} | @@{attribute (HOL) recdef_cong} |
@@{attribute (HOL) recdef_wf}) (() | 'add' | 'del')
\<close>
\<close>
section \<open>Adhoc overloading of constants\<close>
text \<open>
\begin{tabular}{rcll}
@{command_def "adhoc_overloading"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{command_def "no_adhoc_overloading"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{attribute_def "show_variants"} & : & \<open>attribute\<close> & default \<open>false\<close> \\
\end{tabular}
\<^medskip>
Adhoc overloading allows to overload a constant depending on its type.
Typically this involves the introduction of an uninterpreted constant (used
for input and output) and the addition of some variants (used internally).
For examples see \<^file>\<open>~~/src/HOL/ex/Adhoc_Overloading_Examples.thy\<close> and
\<^file>\<open>~~/src/HOL/Library/Monad_Syntax.thy\<close>.
\<^rail>\<open>
(@@{command adhoc_overloading} | @@{command no_adhoc_overloading})
(@{syntax name} (@{syntax term} + ) + @'and')
\<close>
\<^descr> @{command "adhoc_overloading"}~\<open>c v\<^sub>1 ... v\<^sub>n\<close> associates variants with an
existing constant.
\<^descr> @{command "no_adhoc_overloading"} is similar to @{command
"adhoc_overloading"}, but removes the specified variants from the present
context.
\<^descr> @{attribute "show_variants"} controls printing of variants of overloaded
constants. If enabled, the internally used variants are printed instead of
their respective overloaded constants. This is occasionally useful to check
whether the system agrees with a user's expectations about derived variants.
\<close>
section \<open>Definition by specification \label{sec:hol-specification}\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def (HOL) "specification"} & : & \<open>theory \<rightarrow> proof(prove)\<close> \\
\end{matharray}
\<^rail>\<open>
@@{command (HOL) specification} '(' (decl +) ')' \<newline>
(@{syntax thmdecl}? @{syntax prop} +)
;
decl: (@{syntax name} ':')? @{syntax term} ('(' @'overloaded' ')')?
\<close>
\<^descr> @{command (HOL) "specification"}~\<open>decls \<phi>\<close> sets up a goal stating the
existence of terms with the properties specified to hold for the constants
given in \<open>decls\<close>. 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.
\<open>decl\<close> declares a constant to be defined by the specification given. The
definition for the constant \<open>c\<close> is bound to the name \<open>c_def\<close> unless a
theorem name is given in the declaration. Overloaded constants should be
declared as such.
\<close>
section \<open>Old-style datatypes \label{sec:hol-datatype}\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def (HOL) "old_rep_datatype"} & : & \<open>theory \<rightarrow> proof(prove)\<close> \\
\end{matharray}
\<^rail>\<open>
@@{command (HOL) old_rep_datatype} ('(' (@{syntax name} +) ')')? (@{syntax term} +)
;
spec: @{syntax typespec_sorts} @{syntax mixfix}? '=' (cons + '|')
;
cons: @{syntax name} (@{syntax type} * ) @{syntax mixfix}?
\<close>
\<^descr> @{command (HOL) "old_rep_datatype"} represents existing types as
old-style datatypes.
These commands are mostly obsolete; @{command (HOL) "datatype"} should be
used instead.
See @{cite "isabelle-datatypes"} for more details on datatypes. 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).
\<close>
subsubsection \<open>Examples\<close>
text \<open>
We define a type of finite sequences, with slightly different names than the
existing \<^typ>\<open>'a list\<close> that is already in \<^theory>\<open>Main\<close>:
\<close>
(*<*)experiment begin(*>*)
datatype 'a seq = Empty | Seq 'a "'a seq"
text \<open>We can now prove some simple lemma by structural induction:\<close>
lemma "Seq x xs \<noteq> xs"
proof (induct xs arbitrary: x)
case Empty
txt \<open>This case can be proved using the simplifier: the freeness
properties of the datatype are already declared as @{attribute
simp} rules.\<close>
show "Seq x Empty \<noteq> Empty"
by simp
next
case (Seq y ys)
txt \<open>The step case is proved similarly.\<close>
show "Seq x (Seq y ys) \<noteq> Seq y ys"
using \<open>Seq y ys \<noteq> ys\<close> by simp
qed
text \<open>Here is a more succinct version of the same proof:\<close>
lemma "Seq x xs \<noteq> xs"
by (induct xs arbitrary: x) simp_all
(*<*)end(*>*)
section \<open>Records \label{sec:hol-record}\<close>
text \<open>
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.
\<close>
subsection \<open>Basic concepts\<close>
text \<open>
Isabelle/HOL supports both \<^emph>\<open>fixed\<close> and \<^emph>\<open>schematic\<close> 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 & \<open>\<lparr>x = a, y = b\<rparr>\<close> & \<open>\<lparr>x :: A, y :: B\<rparr>\<close> \\
schematic & \<open>\<lparr>x = a, y = b, \<dots> = m\<rparr>\<close> &
\<open>\<lparr>x :: A, y :: B, \<dots> :: M\<rparr>\<close> \\
\end{tabular}
\end{center}
The ASCII representation of \<open>\<lparr>x = a\<rparr>\<close> is \<open>(| x = a |)\<close>.
A fixed record \<open>\<lparr>x = a, y = b\<rparr>\<close> has field \<open>x\<close> of value \<open>a\<close> and field \<open>y\<close> of
value \<open>b\<close>. The corresponding type is \<open>\<lparr>x :: A, y :: B\<rparr>\<close>, assuming that \<open>a ::
A\<close> and \<open>b :: B\<close>.
A record scheme like \<open>\<lparr>x = a, y = b, \<dots> = m\<rparr>\<close> contains fields \<open>x\<close> and \<open>y\<close> as
before, but also possibly further fields as indicated by the ``\<open>\<dots>\<close>''
notation (which is actually part of the syntax). The improper field ``\<open>\<dots>\<close>''
of a record scheme is called the \<^emph>\<open>more part\<close>. 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 \<open>\<lparr>x = a, y = b, z = c, \<dots> = m'\<rparr>\<close>, where \<open>m'\<close> refers to a
different more part. Fixed records are special instances of record schemes,
where ``\<open>\<dots>\<close>'' is properly terminated by the \<open>() :: unit\<close> element. In fact,
\<open>\<lparr>x = a, y = b\<rparr>\<close> is just an abbreviation for \<open>\<lparr>x = a, y = b, \<dots> = ()\<rparr>\<close>.
\<^medskip>
Two key observations make extensible records in a simply typed language like
HOL work out:
\<^enum> the more part is internalized, as a free term or type variable,
\<^enum> field names are externalized, they cannot be accessed within the logic as
first-class values.
\<^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.
\<close>
subsection \<open>Record specifications\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def (HOL) "record"} & : & \<open>theory \<rightarrow> theory\<close> \\
@{command_def (HOL) "print_record"} & : & \<open>context \<rightarrow>\<close> \\
\end{matharray}
\<^rail>\<open>
@@{command (HOL) record} @{syntax "overloaded"}? @{syntax typespec_sorts} '=' \<newline>
(@{syntax type} '+')? (constdecl +)
;
constdecl: @{syntax name} '::' @{syntax type} @{syntax mixfix}?
;
@@{command (HOL) print_record} modes? @{syntax typespec_sorts}
;
modes: '(' (@{syntax name} +) ')'
\<close>
\<^descr> @{command (HOL) "record"}~\<open>(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1 \<dots> c\<^sub>n :: \<sigma>\<^sub>n\<close>
defines extensible record type \<open>(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t\<close>, derived from the optional
parent record \<open>\<tau>\<close> by adding new field components \<open>c\<^sub>i :: \<sigma>\<^sub>i\<close> etc.
The type variables of \<open>\<tau>\<close> and \<open>\<sigma>\<^sub>i\<close> need to be covered by the (distinct)
parameters \<open>\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m\<close>. Type constructor \<open>t\<close> has to be new, while \<open>\<tau>\<close>
needs to specify an instance of an existing record type. At least one new
field \<open>c\<^sub>i\<close> 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 \<open>\<tau>\<close> is optional; if omitted \<open>t\<close> 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, \<open>(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t\<close> is made a type abbreviation for the fixed
record type \<open>\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<rparr>\<close>, likewise is \<open>(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m, \<zeta>)
t_scheme\<close> made an abbreviation for \<open>\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n, \<dots> :: \<zeta>\<rparr>\<close>.
\<^descr> @{command (HOL) "print_record"}~\<open>(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t\<close> prints the definition of
record \<open>(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t\<close>. Optionally \<open>modes\<close> can be specified, which are
appended to the current print mode; see \secref{sec:print-modes}.
\<close>
subsection \<open>Record operations\<close>
text \<open>
Any record definition of the form presented above produces certain standard
operations. Selectors and updates are provided for any field, including the
improper one ``\<open>more\<close>''. There are also cumulative record constructor
functions. To simplify the presentation below, we assume for now that \<open>(\<alpha>\<^sub>1,
\<dots>, \<alpha>\<^sub>m) t\<close> is a root record with fields \<open>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<close>.
\<^medskip>
\<^bold>\<open>Selectors\<close> and \<^bold>\<open>updates\<close> are available for any
field (including ``\<open>more\<close>''):
\begin{matharray}{lll}
\<open>c\<^sub>i\<close> & \<open>::\<close> & \<open>\<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i\<close> \\
\<open>c\<^sub>i_update\<close> & \<open>::\<close> & \<open>\<sigma>\<^sub>i \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>\<close> \\
\end{matharray}
There is special syntax for application of updates: \<open>r\<lparr>x := a\<rparr>\<close> abbreviates
term \<open>x_update a r\<close>. Further notation for repeated updates is also
available: \<open>r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<lparr>z := c\<rparr>\<close> may be written \<open>r\<lparr>x := a, y := b, z
:= c\<rparr>\<close>. 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 \<open>\<lparr>x := a, y := b, z := c\<rparr>\<close>,
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 \<^bold>\<open>make\<close> operation provides a cumulative record constructor function:
\begin{matharray}{lll}
\<open>t.make\<close> & \<open>::\<close> & \<open>\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>\<close> \\
\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>\<open>ancestor records\<close>. 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 \<open>(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t\<close> has ancestor fields \<open>b\<^sub>1 ::
\<rho>\<^sub>1, \<dots>, b\<^sub>k :: \<rho>\<^sub>k\<close>, the above record operations will get the following
types:
\<^medskip>
\begin{tabular}{lll}
\<open>c\<^sub>i\<close> & \<open>::\<close> & \<open>\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i\<close> \\
\<open>c\<^sub>i_update\<close> & \<open>::\<close> & \<open>\<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>\<close> \\
\<open>t.make\<close> & \<open>::\<close> & \<open>\<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>\<close> \\
\end{tabular}
\<^medskip>
Some further operations address the extension aspect of a derived record
scheme specifically: \<open>t.fields\<close> produces a record fragment consisting of
exactly the new fields introduced here (the result may serve as a more part
elsewhere); \<open>t.extend\<close> takes a fixed record and adds a given more part;
\<open>t.truncate\<close> restricts a record scheme to a fixed record.
\<^medskip>
\begin{tabular}{lll}
\<open>t.fields\<close> & \<open>::\<close> & \<open>\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>\<close> \\
\<open>t.extend\<close> & \<open>::\<close> & \<open>\<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>\<close> \\
\<open>t.truncate\<close> & \<open>::\<close> & \<open>\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>\<close> \\
\end{tabular}
\<^medskip>
Note that \<open>t.make\<close> and \<open>t.fields\<close> coincide for root records.
\<close>
subsection \<open>Derived rules and proof tools\<close>
text \<open>
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 \<open>t\<close> is a record type as specified
above.
\<^enum> 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 \<open>t.simps\<close>,
too.
\<^enum> Selectors applied to updated records are automatically reduced by an
internal simplification procedure, which is also part of the standard
Simplifier setup.
\<^enum> Inject equations of a form analogous to \<^prop>\<open>(x, y) = (x', y') \<equiv> x = x'
\<and> y = y'\<close> are declared to the Simplifier and Classical Reasoner as
@{attribute iff} rules. These rules are available as \<open>t.iffs\<close>.
\<^enum> The introduction rule for record equality analogous to \<open>x r = x r' \<Longrightarrow> y r =
y r' \<dots> \<Longrightarrow> r = r'\<close> is declared to the Simplifier, and as the basic rule
context as ``@{attribute intro}\<open>?\<close>''. The rule is called \<open>t.equality\<close>.
\<^enum> 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 ``\<open>(cases r)\<close>'' to a certain proof problem.
\<^enum> The derived record operations \<open>t.make\<close>, \<open>t.fields\<close>, \<open>t.extend\<close>,
\<open>t.truncate\<close> are \<^emph>\<open>not\<close> treated automatically, but usually need to be
expanded by hand, using the collective fact \<open>t.defs\<close>.
\<close>
subsubsection \<open>Examples\<close>
text \<open>See \<^file>\<open>~~/src/HOL/ex/Records.thy\<close>, for example.\<close>
section \<open>Semantic subtype definitions \label{sec:hol-typedef}\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def (HOL) "typedef"} & : & \<open>local_theory \<rightarrow> proof(prove)\<close> \\
\end{matharray}
A type definition identifies a new type with a non-empty subset of an
existing type. More precisely, the new type is defined by exhibiting an
existing type \<open>\<tau>\<close>, a set \<open>A :: \<tau> set\<close>, and proving \<^prop>\<open>\<exists>x. x \<in> A\<close>. Thus
\<open>A\<close> is a non-empty subset of \<open>\<tau>\<close>, 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 \<open>\<tau>\<close> may involve type variables \<open>\<alpha>\<^sub>1, \<dots>,
\<alpha>\<^sub>n\<close> which means that the type definition produces a type constructor \<open>(\<alpha>\<^sub>1,
\<dots>, \<alpha>\<^sub>n) t\<close> depending on those type arguments.
\<^rail>\<open>
@@{command (HOL) typedef} @{syntax "overloaded"}? abs_type '=' rep_set
;
@{syntax_def "overloaded"}: ('(' @'overloaded' ')')
;
abs_type: @{syntax typespec_sorts} @{syntax mixfix}?
;
rep_set: @{syntax term} (@'morphisms' @{syntax name} @{syntax name})?
\<close>
To understand the concept of type definition better, we need to recount its
somewhat complex history. The HOL logic goes back to the ``Simple Theory of
Types'' (STT) of A. Church @{cite "church40"}, which is further explained in
the book by P. Andrews @{cite "andrews86"}. The overview article by W.
Farmer @{cite "Farmer:2008"} points out the ``seven virtues'' of this
relatively simple family of logics. STT has only ground types, without
polymorphism and without type definitions.
\<^medskip>
M. Gordon @{cite "Gordon:1985:HOL"} augmented Church's STT by adding
schematic polymorphism (type variables and type constructors) and a facility
to introduce new types as semantic subtypes from existing types. This
genuine extension of the logic was explained semantically by A. Pitts in the
book of the original Cambridge HOL88 system @{cite "pitts93"}. Type
definitions work in this setting, because the general model-theory of STT is
restricted to models that ensure that the universe of type interpretations
is closed by forming subsets (via predicates taken from the logic).
\<^medskip>
Isabelle/HOL goes beyond Gordon-style HOL by admitting overloaded constant
definitions @{cite "Wenzel:1997:TPHOL" and "Haftmann-Wenzel:2006:classes"},
which are actually a concept of Isabelle/Pure and do not depend on
particular set-theoretic semantics of HOL. Over many years, there was no
formal checking of semantic type definitions in Isabelle/HOL versus
syntactic constant definitions in Isabelle/Pure. So the @{command typedef}
command was described as ``axiomatic'' in the sense of
\secref{sec:axiomatizations}, only with some local checks of the given type
and its representing set.
Recent clarification of overloading in the HOL logic proper @{cite
"Kuncar-Popescu:2015"} demonstrates how the dissimilar concepts of constant
definitions versus type definitions may be understood uniformly. This
requires an interpretation of Isabelle/HOL that substantially reforms the
set-theoretic model of A. Pitts @{cite "pitts93"}, by taking a schematic
view on polymorphism and interpreting only ground types in the set-theoretic
sense of HOL88. Moreover, type-constructors may be explicitly overloaded,
e.g.\ by making the subset depend on type-class parameters (cf.\
\secref{sec:class}). This is semantically like a dependent type: the meaning
relies on the operations provided by different type-class instances.
\<^descr> @{command (HOL) "typedef"}~\<open>(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = A\<close> defines a new type \<open>(\<alpha>\<^sub>1,
\<dots>, \<alpha>\<^sub>n) t\<close> from the set \<open>A\<close> over an existing type. The set \<open>A\<close> may contain
type variables \<open>\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n\<close> as specified on the LHS, but no term variables.
Non-emptiness of \<open>A\<close> needs to be proven on the spot, in order to turn the
internal conditional characterization into usable theorems.
The ``\<open>(overloaded)\<close>'' option allows the @{command "typedef"} specification
to depend on constants that are not (yet) specified and thus left open as
parameters, e.g.\ type-class parameters.
Within a local theory specification, the newly introduced type constructor
cannot depend on parameters or assumptions of the context: this is
syntactically impossible in HOL. The non-emptiness proof may formally depend
on local assumptions, but this has little practical relevance.
For @{command (HOL) "typedef"}~\<open>t = A\<close> the newly introduced type \<open>t\<close> is
accompanied by a pair of morphisms to relate it to the representing set over
the old type. By default, the injection from type to set is called \<open>Rep_t\<close>
and its inverse \<open>Abs_t\<close>: An explicit @{keyword (HOL) "morphisms"}
specification allows to provide alternative names.
The logical characterization of @{command typedef} uses the predicate of
locale \<^const>\<open>type_definition\<close> that is 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
theorem @{thm type_definition.Rep} is turned into the specific \<open>Rep_t\<close>, 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.
\<close>
subsubsection \<open>Examples\<close>
text \<open>
The following trivial example pulls a three-element type into existence
within the formal logical environment of Isabelle/HOL.\<close>
(*<*)experiment begin(*>*)
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)
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)
(*<*)end(*>*)
text \<open>Note that such trivial constructions are better done with
derived specification mechanisms such as @{command datatype}:\<close>
(*<*)experiment begin(*>*)
datatype three = One | Two | Three
(*<*)end(*>*)
text \<open>This avoids re-doing basic definitions and proofs from the
primitive @{command typedef} above.\<close>
section \<open>Functorial structure of types\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def (HOL) "functor"} & : & \<open>local_theory \<rightarrow> proof(prove)\<close>
\end{matharray}
\<^rail>\<open>
@@{command (HOL) functor} (@{syntax name} ':')? @{syntax term}
\<close>
\<^descr> @{command (HOL) "functor"}~\<open>prefix: m\<close> 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 \<open>m\<close> is considered as \<^emph>\<open>mapper\<close> for the corresponding type
constructor and must conform to the following type pattern:
\begin{matharray}{lll}
\<open>m\<close> & \<open>::\<close> &
\<open>\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>k \<Rightarrow> (\<^vec>\<alpha>\<^sub>n) t \<Rightarrow> (\<^vec>\<beta>\<^sub>n) t\<close> \\
\end{matharray}
where \<open>t\<close> is the type constructor, \<open>\<^vec>\<alpha>\<^sub>n\<close> and \<open>\<^vec>\<beta>\<^sub>n\<close> are
distinct type variables free in the local theory and \<open>\<sigma>\<^sub>1\<close>, \ldots, \<open>\<sigma>\<^sub>k\<close> is
a subsequence of \<open>\<alpha>\<^sub>1 \<Rightarrow> \<beta>\<^sub>1\<close>, \<open>\<beta>\<^sub>1 \<Rightarrow> \<alpha>\<^sub>1\<close>, \ldots, \<open>\<alpha>\<^sub>n \<Rightarrow> \<beta>\<^sub>n\<close>, \<open>\<beta>\<^sub>n \<Rightarrow> \<alpha>\<^sub>n\<close>.
\<close>
section \<open>Quotient types with lifting and transfer\<close>
text \<open>
The quotient package defines a new quotient type given a raw type and a
partial equivalence relation (\secref{sec:quotient-type}). The package also
historically includes automation for transporting definitions and theorems
(\secref{sec:old-quotient}), but most of this automation was superseded by
the Lifting (\secref{sec:lifting}) and Transfer (\secref{sec:transfer})
packages.
\<close>
subsection \<open>Quotient type definition \label{sec:quotient-type}\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def (HOL) "quotient_type"} & : & \<open>local_theory \<rightarrow> proof(prove)\<close>\\
\end{matharray}
\<^rail>\<open>
@@{command (HOL) quotient_type} @{syntax "overloaded"}? \<newline>
@{syntax typespec} @{syntax mixfix}? '=' quot_type \<newline>
quot_morphisms? quot_parametric?
;
quot_type: @{syntax type} '/' ('partial' ':')? @{syntax term}
;
quot_morphisms: @'morphisms' @{syntax name} @{syntax name}
;
quot_parametric: @'parametric' @{syntax thm}
\<close>
\<^descr> @{command (HOL) "quotient_type"} defines a new quotient type \<open>\<tau>\<close>. The
injection from a quotient type to a raw type is called \<open>rep_\<tau>\<close>, its inverse
\<open>abs_\<tau>\<close> unless explicit @{keyword (HOL) "morphisms"} specification provides
alternative names. @{command (HOL) "quotient_type"} requires the user to
prove that the relation is an equivalence relation (predicate \<open>equivp\<close>),
unless the user specifies explicitly \<open>partial\<close> in which case the obligation
is \<open>part_equivp\<close>. A quotient defined with \<open>partial\<close> is weaker in the sense
that less things can be proved automatically.
The command internally proves a Quotient theorem and sets up the Lifting
package by the command @{command (HOL) setup_lifting}. Thus the Lifting and
Transfer packages can be used also with quotient types defined by @{command
(HOL) "quotient_type"} without any extra set-up. The parametricity theorem
for the equivalence relation R can be provided as an extra argument of the
command and is passed to the corresponding internal call of @{command (HOL)
setup_lifting}. This theorem allows the Lifting package to generate a
stronger transfer rule for equality.
\<close>
subsection \<open>Lifting package \label{sec:lifting}\<close>
text \<open>
The Lifting package allows users to lift terms of the raw type to the
abstract type, which is a necessary step in building a library for an
abstract type. Lifting defines a new constant by combining coercion
functions (\<^term>\<open>Abs\<close> and \<^term>\<open>Rep\<close>) with the raw term. It also proves an
appropriate transfer rule for the Transfer (\secref{sec:transfer}) package
and, if possible, an equation for the code generator.
The Lifting package provides two main commands: @{command (HOL)
"setup_lifting"} for initializing the package to work with a new type, and
@{command (HOL) "lift_definition"} for lifting constants. The Lifting
package works with all four kinds of type abstraction: type copies,
subtypes, total quotients and partial quotients.
Theoretical background can be found in @{cite
"Huffman-Kuncar:2013:lifting_transfer"}.
\begin{matharray}{rcl}
@{command_def (HOL) "setup_lifting"} & : & \<open>local_theory \<rightarrow> local_theory\<close>\\
@{command_def (HOL) "lift_definition"} & : & \<open>local_theory \<rightarrow> proof(prove)\<close>\\
@{command_def (HOL) "lifting_forget"} & : & \<open>local_theory \<rightarrow> local_theory\<close>\\
@{command_def (HOL) "lifting_update"} & : & \<open>local_theory \<rightarrow> local_theory\<close>\\
@{command_def (HOL) "print_quot_maps"} & : & \<open>context \<rightarrow>\<close>\\
@{command_def (HOL) "print_quotients"} & : & \<open>context \<rightarrow>\<close>\\
@{attribute_def (HOL) "quot_map"} & : & \<open>attribute\<close> \\
@{attribute_def (HOL) "relator_eq_onp"} & : & \<open>attribute\<close> \\
@{attribute_def (HOL) "relator_mono"} & : & \<open>attribute\<close> \\
@{attribute_def (HOL) "relator_distr"} & : & \<open>attribute\<close> \\
@{attribute_def (HOL) "quot_del"} & : & \<open>attribute\<close> \\
@{attribute_def (HOL) "lifting_restore"} & : & \<open>attribute\<close> \\
\end{matharray}
\<^rail>\<open>
@@{command (HOL) setup_lifting} @{syntax thm} @{syntax thm}? \<newline>
(@'parametric' @{syntax thm})?
;
@@{command (HOL) lift_definition} ('(' 'code_dt' ')')? \<newline>
@{syntax name} '::' @{syntax type} @{syntax mixfix}? 'is' @{syntax term} \<newline>
(@'parametric' (@{syntax thm}+))?
;
@@{command (HOL) lifting_forget} @{syntax name}
;
@@{command (HOL) lifting_update} @{syntax name}
;
@@{attribute (HOL) lifting_restore}
@{syntax thm} (@{syntax thm} @{syntax thm})?
\<close>
\<^descr> @{command (HOL) "setup_lifting"} Sets up the Lifting package to work with
a user-defined type. The command supports two modes.
\<^enum> The first one is a low-level mode when the user must provide as a first
argument of @{command (HOL) "setup_lifting"} a quotient theorem \<^term>\<open>Quotient R Abs Rep T\<close>. The package configures a transfer rule for
equality, a domain transfer rules and sets up the @{command_def (HOL)
"lift_definition"} command to work with the abstract type. An optional
theorem \<^term>\<open>reflp R\<close>, which certifies that the equivalence relation R
is total, can be provided as a second argument. This allows the package to
generate stronger transfer rules. And finally, the parametricity theorem
for \<^term>\<open>R\<close> can be provided as a third argument. This allows the package
to generate a stronger transfer rule for equality.
Users generally will not prove the \<open>Quotient\<close> theorem manually for new
types, as special commands exist to automate the process.
\<^enum> When a new subtype is defined by @{command (HOL) typedef}, @{command
(HOL) "lift_definition"} can be used in its second mode, where only the
\<^term>\<open>type_definition\<close> theorem \<^term>\<open>type_definition Rep Abs A\<close> is
used as an argument of the command. The command internally proves the
corresponding \<^term>\<open>Quotient\<close> theorem and registers it with @{command
(HOL) setup_lifting} using its first mode.
For quotients, the command @{command (HOL) quotient_type} can be used. The
command defines a new quotient type and similarly to the previous case, the
corresponding Quotient theorem is proved and registered by @{command (HOL)
setup_lifting}.
\<^medskip>
The command @{command (HOL) "setup_lifting"} also sets up the code generator
for the new type. Later on, when a new constant is defined by @{command
(HOL) "lift_definition"}, the Lifting package proves and registers a code
equation (if there is one) for the new constant.
\<^descr> @{command (HOL) "lift_definition"} \<open>f :: \<tau>\<close> @{keyword (HOL) "is"} \<open>t\<close>
Defines a new function \<open>f\<close> with an abstract type \<open>\<tau>\<close> in terms of a
corresponding operation \<open>t\<close> on a representation type. More formally, if \<open>t
:: \<sigma>\<close>, then the command builds a term \<open>F\<close> as a corresponding combination of
abstraction and representation functions such that \<open>F :: \<sigma> \<Rightarrow> \<tau>\<close> and defines
\<open>f \<equiv> F t\<close>. The term \<open>t\<close> does not have to be necessarily a constant but it
can be any term.
The command opens a proof and the user must discharge a respectfulness proof
obligation. For a type copy, i.e.\ a typedef with \<open>UNIV\<close>, the obligation is
discharged automatically. The proof goal is presented in a user-friendly,
readable form. A respectfulness theorem in the standard format \<open>f.rsp\<close> and a
transfer rule \<open>f.transfer\<close> for the Transfer package are generated by the
package.
The user can specify a parametricity theorems for \<open>t\<close> after the keyword
@{keyword "parametric"}, which allows the command to generate parametric
transfer rules for \<open>f\<close>.
For each constant defined through trivial quotients (type copies or
subtypes) \<open>f.rep_eq\<close> is generated. The equation is a code certificate that
defines \<open>f\<close> using the representation function.
For each constant \<open>f.abs_eq\<close> is generated. The equation is unconditional for
total quotients. The equation defines \<open>f\<close> using the abstraction function.
\<^medskip>
Integration with [@{attribute code} abstract]: For subtypes (e.g.\
corresponding to a datatype invariant, such as \<^typ>\<open>'a dlist\<close>), @{command
(HOL) "lift_definition"} uses a code certificate theorem \<open>f.rep_eq\<close> as a
code equation. Because of the limitation of the code generator, \<open>f.rep_eq\<close>
cannot be used as a code equation if the subtype occurs inside the result
type rather than at the top level (e.g.\ function returning \<^typ>\<open>'a dlist
option\<close> vs. \<^typ>\<open>'a dlist\<close>).
In this case, an extension of @{command (HOL) "lift_definition"} can be
invoked by specifying the flag \<open>code_dt\<close>. This extension enables code
execution through series of internal type and lifting definitions if the
return type \<open>\<tau>\<close> meets the following inductive conditions:
\<^descr> \<open>\<tau>\<close> is a type variable
\<^descr> \<open>\<tau> = \<tau>\<^sub>1 \<dots> \<tau>\<^sub>n \<kappa>\<close>, where \<open>\<kappa>\<close> is an abstract type constructor and \<open>\<tau>\<^sub>1 \<dots>
\<tau>\<^sub>n\<close> do not contain abstract types (i.e.\ \<^typ>\<open>int dlist\<close> is allowed
whereas \<^typ>\<open>int dlist dlist\<close> not)
\<^descr> \<open>\<tau> = \<tau>\<^sub>1 \<dots> \<tau>\<^sub>n \<kappa>\<close>, \<open>\<kappa>\<close> is a type constructor that was defined as a
(co)datatype whose constructor argument types do not contain either
non-free datatypes or the function type.
Integration with [@{attribute code} equation]: For total quotients,
@{command (HOL) "lift_definition"} uses \<open>f.abs_eq\<close> as a code equation.
\<^descr> @{command (HOL) lifting_forget} and @{command (HOL) lifting_update} These
two commands serve for storing and deleting the set-up of the Lifting
package and corresponding transfer rules defined by this package. This is
useful for hiding of type construction details of an abstract type when the
construction is finished but it still allows additions to this construction
when this is later necessary.
Whenever the Lifting package is set up with a new abstract type \<open>\<tau>\<close> by
@{command_def (HOL) "lift_definition"}, the package defines a new bundle
that is called \<open>\<tau>.lifting\<close>. This bundle already includes set-up for the
Lifting package. The new transfer rules introduced by @{command (HOL)
"lift_definition"} can be stored in the bundle by the command @{command
(HOL) "lifting_update"} \<open>\<tau>.lifting\<close>.
The command @{command (HOL) "lifting_forget"} \<open>\<tau>.lifting\<close> deletes set-up of
the Lifting package for \<open>\<tau>\<close> and deletes all the transfer rules that were
introduced by @{command (HOL) "lift_definition"} using \<open>\<tau>\<close> as an abstract
type.
The stored set-up in a bundle can be reintroduced by the Isar commands for
including a bundle (@{command "include"}, @{keyword "includes"} and
@{command "including"}).
\<^descr> @{command (HOL) "print_quot_maps"} prints stored quotient map theorems.
\<^descr> @{command (HOL) "print_quotients"} prints stored quotient theorems.
\<^descr> @{attribute (HOL) quot_map} registers a quotient map theorem, a theorem
showing how to ``lift'' quotients over type constructors. E.g.\ \<^term>\<open>Quotient R Abs Rep T \<Longrightarrow> Quotient (rel_set R) (image Abs) (image Rep)
(rel_set T)\<close>. For examples see \<^file>\<open>~~/src/HOL/Lifting_Set.thy\<close> or
\<^file>\<open>~~/src/HOL/Lifting.thy\<close>. This property is proved automatically if the
involved type is BNF without dead variables.
\<^descr> @{attribute (HOL) relator_eq_onp} registers a theorem that shows that a
relator applied to an equality restricted by a predicate \<^term>\<open>P\<close> (i.e.\
\<^term>\<open>eq_onp P\<close>) is equal to a predicator applied to the \<^term>\<open>P\<close>. The
combinator \<^const>\<open>eq_onp\<close> is used for internal encoding of proper subtypes.
Such theorems allows the package to hide \<open>eq_onp\<close> from a user in a
user-readable form of a respectfulness theorem. For examples see
\<^file>\<open>~~/src/HOL/Lifting_Set.thy\<close> or \<^file>\<open>~~/src/HOL/Lifting.thy\<close>. This property
is proved automatically if the involved type is BNF without dead variables.
\<^descr> @{attribute (HOL) "relator_mono"} registers a property describing a
monotonicity of a relator. E.g.\ \<^prop>\<open>A \<le> B \<Longrightarrow> rel_set A \<le> rel_set B\<close>.
This property is needed for proving a stronger transfer rule in
@{command_def (HOL) "lift_definition"} when a parametricity theorem for the
raw term is specified and also for the reflexivity prover. For examples see
\<^file>\<open>~~/src/HOL/Lifting_Set.thy\<close> or \<^file>\<open>~~/src/HOL/Lifting.thy\<close>. This property
is proved automatically if the involved type is BNF without dead variables.
\<^descr> @{attribute (HOL) "relator_distr"} registers a property describing a
distributivity of the relation composition and a relator. E.g.\ \<open>rel_set R
\<circ>\<circ> rel_set S = rel_set (R \<circ>\<circ> S)\<close>. This property is needed for proving a
stronger transfer rule in @{command_def (HOL) "lift_definition"} when a
parametricity theorem for the raw term is specified. When this equality does
not hold unconditionally (e.g.\ for the function type), the user can
specified each direction separately and also register multiple theorems with
different set of assumptions. This attribute can be used only after the
monotonicity property was already registered by @{attribute (HOL)
"relator_mono"}. For examples see \<^file>\<open>~~/src/HOL/Lifting_Set.thy\<close> or
\<^file>\<open>~~/src/HOL/Lifting.thy\<close>. This property is proved automatically if the
involved type is BNF without dead variables.
\<^descr> @{attribute (HOL) quot_del} deletes a corresponding Quotient theorem from
the Lifting infrastructure and thus de-register the corresponding quotient.
This effectively causes that @{command (HOL) lift_definition} will not do
any lifting for the corresponding type. This attribute is rather used for
low-level manipulation with set-up of the Lifting package because @{command
(HOL) lifting_forget} is preferred for normal usage.
\<^descr> @{attribute (HOL) lifting_restore} \<open>Quotient_thm pcr_def pcr_cr_eq_thm\<close>
registers the Quotient theorem \<open>Quotient_thm\<close> in the Lifting infrastructure
and thus sets up lifting for an abstract type \<open>\<tau>\<close> (that is defined by
\<open>Quotient_thm\<close>). Optional theorems \<open>pcr_def\<close> and \<open>pcr_cr_eq_thm\<close> can be
specified to register the parametrized correspondence relation for \<open>\<tau>\<close>.
E.g.\ for \<^typ>\<open>'a dlist\<close>, \<open>pcr_def\<close> is \<open>pcr_dlist A \<equiv> list_all2 A \<circ>\<circ>
cr_dlist\<close> and \<open>pcr_cr_eq_thm\<close> is \<open>pcr_dlist (=) = (=)\<close>. This attribute
is rather used for low-level manipulation with set-up of the Lifting package
because using of the bundle \<open>\<tau>.lifting\<close> together with the commands @{command
(HOL) lifting_forget} and @{command (HOL) lifting_update} is preferred for
normal usage.
\<^descr> Integration with the BNF package @{cite "isabelle-datatypes"}: As already
mentioned, the theorems that are registered by the following attributes are
proved and registered automatically if the involved type is BNF without dead
variables: @{attribute (HOL) quot_map}, @{attribute (HOL) relator_eq_onp},
@{attribute (HOL) "relator_mono"}, @{attribute (HOL) "relator_distr"}. Also
the definition of a relator and predicator is provided automatically.
Moreover, if the BNF represents a datatype, simplification rules for a
predicator are again proved automatically.
\<close>
subsection \<open>Transfer package \label{sec:transfer}\<close>
text \<open>
\begin{matharray}{rcl}
@{method_def (HOL) "transfer"} & : & \<open>method\<close> \\
@{method_def (HOL) "transfer'"} & : & \<open>method\<close> \\
@{method_def (HOL) "transfer_prover"} & : & \<open>method\<close> \\
@{attribute_def (HOL) "Transfer.transferred"} & : & \<open>attribute\<close> \\
@{attribute_def (HOL) "untransferred"} & : & \<open>attribute\<close> \\
@{method_def (HOL) "transfer_start"} & : & \<open>method\<close> \\
@{method_def (HOL) "transfer_prover_start"} & : & \<open>method\<close> \\
@{method_def (HOL) "transfer_step"} & : & \<open>method\<close> \\
@{method_def (HOL) "transfer_end"} & : & \<open>method\<close> \\
@{method_def (HOL) "transfer_prover_end"} & : & \<open>method\<close> \\
@{attribute_def (HOL) "transfer_rule"} & : & \<open>attribute\<close> \\
@{attribute_def (HOL) "transfer_domain_rule"} & : & \<open>attribute\<close> \\
@{attribute_def (HOL) "relator_eq"} & : & \<open>attribute\<close> \\
@{attribute_def (HOL) "relator_domain"} & : & \<open>attribute\<close> \\
\end{matharray}
\<^descr> @{method (HOL) "transfer"} method replaces the current subgoal with a
logically equivalent one that uses different types and constants. The
replacement of types and constants is guided by the database of transfer
rules. Goals are generalized over all free variables by default; this is
necessary for variables whose types change, but can be overridden for
specific variables with e.g. \<open>transfer fixing: x y z\<close>.
\<^descr> @{method (HOL) "transfer'"} is a variant of @{method (HOL) transfer} that
allows replacing a subgoal with one that is logically stronger (rather than
equivalent). For example, a subgoal involving equality on a quotient type
could be replaced with a subgoal involving equality (instead of the
corresponding equivalence relation) on the underlying raw type.
\<^descr> @{method (HOL) "transfer_prover"} method assists with proving a transfer
rule for a new constant, provided the constant is defined in terms of other
constants that already have transfer rules. It should be applied after
unfolding the constant definitions.
\<^descr> @{method (HOL) "transfer_start"}, @{method (HOL) "transfer_step"},
@{method (HOL) "transfer_end"}, @{method (HOL) "transfer_prover_start"} and
@{method (HOL) "transfer_prover_end"} methods are meant to be used for
debugging of @{method (HOL) "transfer"} and @{method (HOL)
"transfer_prover"}, which we can decompose as follows: @{method (HOL)
"transfer"} = (@{method (HOL) "transfer_start"}, @{method (HOL)
"transfer_step"}+, @{method (HOL) "transfer_end"}) and @{method (HOL)
"transfer_prover"} = (@{method (HOL) "transfer_prover_start"}, @{method
(HOL) "transfer_step"}+, @{method (HOL) "transfer_prover_end"}). For usage
examples see \<^file>\<open>~~/src/HOL/ex/Transfer_Debug.thy\<close>.
\<^descr> @{attribute (HOL) "untransferred"} proves the same equivalent theorem as
@{method (HOL) "transfer"} internally does.
\<^descr> @{attribute (HOL) Transfer.transferred} works in the opposite direction
than @{method (HOL) "transfer'"}. E.g.\ given the transfer relation \<open>ZN x n
\<equiv> (x = int n)\<close>, corresponding transfer rules and the theorem \<open>\<forall>x::int \<in>
{0..}. x < x + 1\<close>, the attribute would prove \<open>\<forall>n::nat. n < n + 1\<close>. The
attribute is still in experimental phase of development.
\<^descr> @{attribute (HOL) "transfer_rule"} attribute maintains a collection of
transfer rules, which relate constants at two different types. Typical
transfer rules may relate different type instances of the same polymorphic
constant, or they may relate an operation on a raw type to a corresponding
operation on an abstract type (quotient or subtype). For example:
\<open>((A ===> B) ===> list_all2 A ===> list_all2 B) map map\<close> \\
\<open>(cr_int ===> cr_int ===> cr_int) (\<lambda>(x,y) (u,v). (x+u, y+v)) plus\<close>
Lemmas involving predicates on relations can also be registered using the
same attribute. For example:
\<open>bi_unique A \<Longrightarrow> (list_all2 A ===> (=)) distinct distinct\<close> \\
\<open>\<lbrakk>bi_unique A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (rel_prod A B)\<close>
Preservation of predicates on relations (\<open>bi_unique, bi_total, right_unique,
right_total, left_unique, left_total\<close>) with the respect to a relator is
proved automatically if the involved type is BNF @{cite
"isabelle-datatypes"} without dead variables.
\<^descr> @{attribute (HOL) "transfer_domain_rule"} attribute maintains a collection
of rules, which specify a domain of a transfer relation by a predicate.
E.g.\ given the transfer relation \<open>ZN x n \<equiv> (x = int n)\<close>, one can register
the following transfer domain rule: \<open>Domainp ZN = (\<lambda>x. x \<ge> 0)\<close>. The rules
allow the package to produce more readable transferred goals, e.g.\ when
quantifiers are transferred.
\<^descr> @{attribute (HOL) relator_eq} attribute collects identity laws for
relators of various type constructors, e.g. \<^term>\<open>rel_set (=) = (=)\<close>.
The @{method (HOL) transfer} method uses these lemmas to infer
transfer rules for non-polymorphic constants on the fly. For examples see
\<^file>\<open>~~/src/HOL/Lifting_Set.thy\<close> or \<^file>\<open>~~/src/HOL/Lifting.thy\<close>. This property
is proved automatically if the involved type is BNF without dead variables.
\<^descr> @{attribute_def (HOL) "relator_domain"} attribute collects rules
describing domains of relators by predicators. E.g.\ \<^term>\<open>Domainp
(rel_set T) = (\<lambda>A. Ball A (Domainp T))\<close>. This allows the package to lift
transfer domain rules through type constructors. For examples see
\<^file>\<open>~~/src/HOL/Lifting_Set.thy\<close> or \<^file>\<open>~~/src/HOL/Lifting.thy\<close>. This property
is proved automatically if the involved type is BNF without dead variables.
Theoretical background can be found in @{cite
"Huffman-Kuncar:2013:lifting_transfer"}.
\<close>
subsection \<open>Old-style definitions for quotient types \label{sec:old-quotient}\<close>
text \<open>
\begin{matharray}{rcl}
@{command_def (HOL) "quotient_definition"} & : & \<open>local_theory \<rightarrow> proof(prove)\<close>\\
@{command_def (HOL) "print_quotmapsQ3"} & : & \<open>context \<rightarrow>\<close>\\
@{command_def (HOL) "print_quotientsQ3"} & : & \<open>context \<rightarrow>\<close>\\
@{command_def (HOL) "print_quotconsts"} & : & \<open>context \<rightarrow>\<close>\\
@{method_def (HOL) "lifting"} & : & \<open>method\<close> \\
@{method_def (HOL) "lifting_setup"} & : & \<open>method\<close> \\
@{method_def (HOL) "descending"} & : & \<open>method\<close> \\
@{method_def (HOL) "descending_setup"} & : & \<open>method\<close> \\
@{method_def (HOL) "partiality_descending"} & : & \<open>method\<close> \\
@{method_def (HOL) "partiality_descending_setup"} & : & \<open>method\<close> \\
@{method_def (HOL) "regularize"} & : & \<open>method\<close> \\
@{method_def (HOL) "injection"} & : & \<open>method\<close> \\
@{method_def (HOL) "cleaning"} & : & \<open>method\<close> \\
@{attribute_def (HOL) "quot_thm"} & : & \<open>attribute\<close> \\
@{attribute_def (HOL) "quot_lifted"} & : & \<open>attribute\<close> \\
@{attribute_def (HOL) "quot_respect"} & : & \<open>attribute\<close> \\
@{attribute_def (HOL) "quot_preserve"} & : & \<open>attribute\<close> \\
\end{matharray}
\<^rail>\<open>
@@{command (HOL) quotient_definition} constdecl? @{syntax thmdecl}? \<newline>
@{syntax term} 'is' @{syntax term}
;
constdecl: @{syntax name} ('::' @{syntax type})? @{syntax mixfix}?
;
@@{method (HOL) lifting} @{syntax thms}?
;
@@{method (HOL) lifting_setup} @{syntax thms}?
\<close>
\<^descr> @{command (HOL) "quotient_definition"} defines a constant on the quotient
type.
\<^descr> @{command (HOL) "print_quotmapsQ3"} prints quotient map functions.
\<^descr> @{command (HOL) "print_quotientsQ3"} prints quotients.
\<^descr> @{command (HOL) "print_quotconsts"} prints quotient constants.
\<^descr> @{method (HOL) "lifting"} and @{method (HOL) "lifting_setup"} methods
match the current goal with the given raw theorem to be lifted producing
three new subgoals: regularization, injection and cleaning subgoals.
@{method (HOL) "lifting"} tries to apply the heuristics for automatically
solving these three subgoals and leaves only the subgoals unsolved by the
heuristics to the user as opposed to @{method (HOL) "lifting_setup"} which
leaves the three subgoals unsolved.
\<^descr> @{method (HOL) "descending"} and @{method (HOL) "descending_setup"} try to
guess a raw statement that would lift to the current subgoal. Such statement
is assumed as a new subgoal and @{method (HOL) "descending"} continues in
the same way as @{method (HOL) "lifting"} does. @{method (HOL) "descending"}
tries to solve the arising regularization, injection and cleaning subgoals
with the analogous method @{method (HOL) "descending_setup"} which leaves
the four unsolved subgoals.
\<^descr> @{method (HOL) "partiality_descending"} finds the regularized theorem that
would lift to the current subgoal, lifts it and leaves as a subgoal. This
method can be used with partial equivalence quotients where the non
regularized statements would not be true. @{method (HOL)
"partiality_descending_setup"} leaves the injection and cleaning subgoals
unchanged.
\<^descr> @{method (HOL) "regularize"} applies the regularization heuristics to the
current subgoal.
\<^descr> @{method (HOL) "injection"} applies the injection heuristics to the
current goal using the stored quotient respectfulness theorems.
\<^descr> @{method (HOL) "cleaning"} applies the injection cleaning heuristics to
the current subgoal using the stored quotient preservation theorems.
\<^descr> @{attribute (HOL) quot_lifted} attribute tries to automatically transport
the theorem to the quotient type. The attribute uses all the defined
quotients types and quotient constants often producing undesired results or
theorems that cannot be lifted.
\<^descr> @{attribute (HOL) quot_respect} and @{attribute (HOL) quot_preserve}
attributes declare a theorem as a respectfulness and preservation theorem
respectively. These are stored in the local theory store and used by the
@{method (HOL) "injection"} and @{method (HOL) "cleaning"} methods
respectively.
\<^descr> @{attribute (HOL) quot_thm} declares that a certain theorem is a quotient
extension theorem. Quotient extension theorems allow for quotienting inside
container types. Given a polymorphic type that serves as a container, a map
function defined for this container using @{command (HOL) "functor"} and a
relation map defined for for the container type, the quotient extension
theorem should be \<^term>\<open>Quotient3 R Abs Rep \<Longrightarrow> Quotient3 (rel_map R) (map
Abs) (map Rep)\<close>. Quotient extension theorems are stored in a database and
are used all the steps of lifting theorems.
\<close>
chapter \<open>Proof tools\<close>
section \<open>Proving propositions\<close>
text \<open>
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"}\<open>\<^sup>*\<close> & : & \<open>proof \<rightarrow>\<close> \\
@{command_def (HOL) "try"}\<open>\<^sup>*\<close> & : & \<open>proof \<rightarrow>\<close> \\
@{command_def (HOL) "try0"}\<open>\<^sup>*\<close> & : & \<open>proof \<rightarrow>\<close> \\
@{command_def (HOL) "sledgehammer"}\<open>\<^sup>*\<close> & : & \<open>proof \<rightarrow>\<close> \\
@{command_def (HOL) "sledgehammer_params"} & : & \<open>theory \<rightarrow> theory\<close>
\end{matharray}
\<^rail>\<open>
@@{command (HOL) try}
;
@@{command (HOL) try0} ( ( ( 'simp' | 'intro' | 'elim' | 'dest' ) ':' @{syntax thms} ) + ) ?
@{syntax nat}?
;
@@{command (HOL) sledgehammer} ( '[' args ']' )? facts? @{syntax nat}?
;
@@{command (HOL) sledgehammer_params} ( ( '[' args ']' ) ? )
;
args: ( @{syntax name} '=' value + ',' )
;
facts: '(' ( ( ( ( 'add' | 'del' ) ':' ) ? @{syntax thms} ) + ) ? ')'
\<close> % FIXME check args "value"
\<^descr> @{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.
\<^descr> @{command (HOL) "try0"} attempts to prove a subgoal using a combination of
standard proof methods (@{method auto}, @{method simp}, @{method blast},
etc.). Additional facts supplied via \<open>simp:\<close>, \<open>intro:\<close>, \<open>elim:\<close>, and \<open>dest:\<close>
are passed to the appropriate proof methods.
\<^descr> @{command (HOL) "try"} attempts to prove or disprove a subgoal using a
combination of provers and disprovers (@{command (HOL) "solve_direct"},
@{command (HOL) "quickcheck"}, @{command (HOL) "try0"}, @{command (HOL)
"sledgehammer"}, @{command (HOL) "nitpick"}).
\<^descr> @{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.
\<^descr> @{command (HOL) "sledgehammer_params"} changes @{command (HOL)
"sledgehammer"} configuration options persistently.
\<close>
section \<open>Checking and refuting propositions\<close>
text \<open>
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"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
@{command_def (HOL) "values"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
@{command_def (HOL) "quickcheck"}\<open>\<^sup>*\<close> & : & \<open>proof \<rightarrow>\<close> \\
@{command_def (HOL) "nitpick"}\<open>\<^sup>*\<close> & : & \<open>proof \<rightarrow>\<close> \\
@{command_def (HOL) "quickcheck_params"} & : & \<open>theory \<rightarrow> theory\<close> \\
@{command_def (HOL) "nitpick_params"} & : & \<open>theory \<rightarrow> theory\<close> \\
@{command_def (HOL) "quickcheck_generator"} & : & \<open>theory \<rightarrow> theory\<close> \\
@{command_def (HOL) "find_unused_assms"} & : & \<open>context \<rightarrow>\<close>
\end{matharray}
\<^rail>\<open>
@@{command (HOL) value} ( '[' @{syntax name} ']' )? modes? @{syntax term}
;
@@{command (HOL) values} modes? @{syntax nat}? @{syntax term}
;
(@@{command (HOL) quickcheck} | @@{command (HOL) nitpick})
( '[' args ']' )? @{syntax nat}?
;
(@@{command (HOL) quickcheck_params} |
@@{command (HOL) nitpick_params}) ( '[' args ']' )?
;
@@{command (HOL) quickcheck_generator} @{syntax name} \<newline>
'operations:' ( @{syntax term} +)
;
@@{command (HOL) find_unused_assms} @{syntax name}?
;
modes: '(' (@{syntax name} +) ')'
;
args: ( @{syntax name} '=' value + ',' )
\<close> % FIXME check "value"
\<^descr> @{command (HOL) "value"}~\<open>t\<close> evaluates and prints a term; optionally
\<open>modes\<close> can be specified, which are appended to the current print mode; see
\secref{sec:print-modes}. Evaluation is tried first using ML, falling back
to normalization by evaluation if this fails. Alternatively a specific
evaluator can be selected using square brackets; typical evaluators use the
current set of code equations to normalize and include \<open>simp\<close> for fully
symbolic evaluation using the simplifier, \<open>nbe\<close> for \<^emph>\<open>normalization by
evaluation\<close> and \<^emph>\<open>code\<close> for code generation in SML.
\<^descr> @{command (HOL) "values"}~\<open>t\<close> enumerates a set comprehension by evaluation
and prints its values up to the given number of solutions; optionally
\<open>modes\<close> can be specified, which are appended to the current print mode; see
\secref{sec:print-modes}.
\<^descr> @{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 up to 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:
\<^descr>[\<open>tester\<close>] specifies which testing approach to apply. There are three
testers, \<open>exhaustive\<close>, \<open>random\<close>, and \<open>narrowing\<close>. An unknown configuration
option is treated as an argument to tester, making \<open>tester =\<close> 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 @{attribute quickcheck_exhaustive_active},
@{attribute quickcheck_random_active}, @{attribute
quickcheck_narrowing_active} are set to true.
\<^descr>[\<open>size\<close>] specifies the maximum size of the search space for assignment
values.
\<^descr>[\<open>genuine_only\<close>] sets quickcheck only to return genuine counterexample,
but not potentially spurious counterexamples due to underspecified
functions.
\<^descr>[\<open>abort_potential\<close>] sets quickcheck to abort once it found a potentially
spurious counterexample and to not continue to search for a further
genuine counterexample. For this option to be effective, the
\<open>genuine_only\<close> option must be set to false.
\<^descr>[\<open>eval\<close>] takes a term or a list of terms and evaluates these terms under
the variable assignment found by quickcheck. This option is currently only
supported by the default (exhaustive) tester.
\<^descr>[\<open>iterations\<close>] sets how many sets of assignments are generated for each
particular size.
\<^descr>[\<open>no_assms\<close>] specifies whether assumptions in structured proofs should be
ignored.
\<^descr>[\<open>locale\<close>] specifies how to process conjectures in a locale context,
i.e.\ they can be interpreted or expanded. The option is a
whitespace-separated list of the two words \<open>interpret\<close> and \<open>expand\<close>. The
list determines the order they are employed. The default setting is to
first use interpretations and then test the expanded conjecture. The
option is only provided as attribute declaration, but not as parameter to
the command.
\<^descr>[\<open>timeout\<close>] sets the time limit in seconds.
\<^descr>[\<open>default_type\<close>] sets the type(s) generally used to instantiate type
variables.
\<^descr>[\<open>report\<close>] if set quickcheck reports how many tests fulfilled the
preconditions.
\<^descr>[\<open>use_subtype\<close>] if set quickcheck automatically lifts conjectures to
registered subtypes if possible, and tests the lifted conjecture.
\<^descr>[\<open>quiet\<close>] if set quickcheck does not output anything while testing.
\<^descr>[\<open>verbose\<close>] if set quickcheck informs about the current size and
cardinality while testing.
\<^descr>[\<open>expect\<close>] can be used to check if the user's expectation was met
(\<open>no_expectation\<close>, \<open>no_counterexample\<close>, or \<open>counterexample\<close>).
These option can be given within square brackets.
Using the following type classes, the testers generate values and convert
them back into Isabelle terms for displaying counterexamples.
\<^descr>[\<open>exhaustive\<close>] The parameters of the type classes \<^class>\<open>exhaustive\<close> and
\<^class>\<open>full_exhaustive\<close> implement the testing. They take a testing
function as a parameter, which takes a value of type \<^typ>\<open>'a\<close> and
optionally produces a counterexample, and a size parameter for the test
values. In \<^class>\<open>full_exhaustive\<close>, the testing function parameter
additionally expects a lazy term reconstruction in the type \<^typ>\<open>Code_Evaluation.term\<close> of the tested value.
The canonical implementation for \<open>exhaustive\<close> testers calls the given
testing function on all values up to the given size and stops as soon as a
counterexample is found.
\<^descr>[\<open>random\<close>] The operation \<^const>\<open>Quickcheck_Random.random\<close> of the type
class \<^class>\<open>random\<close> generates a pseudo-random value of the given size
and a lazy term reconstruction of the value in the type \<^typ>\<open>Code_Evaluation.term\<close>. A pseudo-randomness generator is defined in theory
\<^theory>\<open>HOL.Random\<close>.
\<^descr>[\<open>narrowing\<close>] implements Haskell's Lazy Smallcheck @{cite
"runciman-naylor-lindblad"} using the type classes \<^class>\<open>narrowing\<close> and
\<^class>\<open>partial_term_of\<close>. Variables in the current goal are initially
represented as symbolic variables. If the execution of the goal tries to
evaluate one of them, the test engine replaces it with refinements
provided by \<^const>\<open>narrowing\<close>. Narrowing views every value as a
sum-of-products which is expressed using the operations \<^const>\<open>Quickcheck_Narrowing.cons\<close> (embedding a value), \<^const>\<open>Quickcheck_Narrowing.apply\<close> (product) and \<^const>\<open>Quickcheck_Narrowing.sum\<close> (sum). The refinement should enable further
evaluation of the goal.
For example, \<^const>\<open>narrowing\<close> for the list type \<^typ>\<open>'a :: narrowing list\<close>
can be recursively defined as
\<^term>\<open>Quickcheck_Narrowing.sum (Quickcheck_Narrowing.cons [])
(Quickcheck_Narrowing.apply
(Quickcheck_Narrowing.apply
(Quickcheck_Narrowing.cons (#))
narrowing)
narrowing)\<close>.
If a symbolic variable of type \<^typ>\<open>_ list\<close> is evaluated, it is
replaced by (i)~the empty list \<^term>\<open>[]\<close> and (ii)~by a non-empty list
whose head and tail can then be recursively refined if needed.
To reconstruct counterexamples, the operation \<^const>\<open>partial_term_of\<close>
transforms \<open>narrowing\<close>'s deep representation of terms to the type \<^typ>\<open>Code_Evaluation.term\<close>. The deep representation models symbolic variables
as \<^const>\<open>Quickcheck_Narrowing.Narrowing_variable\<close>, which are normally
converted to \<^const>\<open>Code_Evaluation.Free\<close>, and refined values as \<^term>\<open>Quickcheck_Narrowing.Narrowing_constructor i args\<close>, where \<^term>\<open>i ::
integer\<close> denotes the index in the sum of refinements. In the above
example for lists, \<^term>\<open>0\<close> corresponds to \<^term>\<open>[]\<close> and \<^term>\<open>1\<close>
to \<^term>\<open>(#)\<close>.
The command @{command (HOL) "code_datatype"} sets up \<^const>\<open>partial_term_of\<close> such that the \<^term>\<open>i\<close>-th refinement is interpreted as
the \<^term>\<open>i\<close>-th constructor, but it does not ensures consistency with
\<^const>\<open>narrowing\<close>.
\<^descr> @{command (HOL) "quickcheck_params"} changes @{command (HOL) "quickcheck"}
configuration options persistently.
\<^descr> @{command (HOL) "quickcheck_generator"} creates random and exhaustive
value generators for a given type and operations. It generates values by
using the operations as if they were constructors of that type.
\<^descr> @{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.
\<^descr> @{command (HOL) "nitpick_params"} changes @{command (HOL) "nitpick"}
configuration options persistently.
\<^descr> @{command (HOL) "find_unused_assms"} finds potentially superfluous
assumptions in theorems using quickcheck. It takes the theory name to be
checked for superfluous assumptions as optional argument. If not provided,
it checks the current theory. Options to the internal quickcheck invocations
can be changed with common configuration declarations.
\<close>
section \<open>Coercive subtyping\<close>
text \<open>
\begin{matharray}{rcl}
@{attribute_def (HOL) coercion} & : & \<open>attribute\<close> \\
@{attribute_def (HOL) coercion_delete} & : & \<open>attribute\<close> \\
@{attribute_def (HOL) coercion_enabled} & : & \<open>attribute\<close> \\
@{attribute_def (HOL) coercion_map} & : & \<open>attribute\<close> \\
@{attribute_def (HOL) coercion_args} & : & \<open>attribute\<close> \\
\end{matharray}
Coercive subtyping allows the user to omit explicit type conversions, also
called \<^emph>\<open>coercions\<close>. Type inference will add them as necessary when parsing
a term. See @{cite "traytel-berghofer-nipkow-2011"} for details.
\<^rail>\<open>
@@{attribute (HOL) coercion} (@{syntax term})
;
@@{attribute (HOL) coercion_delete} (@{syntax term})
;
@@{attribute (HOL) coercion_map} (@{syntax term})
;
@@{attribute (HOL) coercion_args} (@{syntax const}) (('+' | '0' | '-')+)
\<close>
\<^descr> @{attribute (HOL) "coercion"}~\<open>f\<close> registers a new coercion function \<open>f ::
\<sigma>\<^sub>1 \<Rightarrow> \<sigma>\<^sub>2\<close> where \<open>\<sigma>\<^sub>1\<close> and \<open>\<sigma>\<^sub>2\<close> are type constructors without arguments.
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.
\<^descr> @{attribute (HOL) "coercion_delete"}~\<open>f\<close> deletes a preceding declaration
(using @{attribute (HOL) "coercion"}) of the function \<open>f :: \<sigma>\<^sub>1 \<Rightarrow> \<sigma>\<^sub>2\<close> as a
coercion.
\<^descr> @{attribute (HOL) "coercion_map"}~\<open>map\<close> registers a new map function to
lift coercions through type constructors. The function \<open>map\<close> must conform to
the following type pattern
\begin{matharray}{lll}
\<open>map\<close> & \<open>::\<close> &
\<open>f\<^sub>1 \<Rightarrow> \<dots> \<Rightarrow> f\<^sub>n \<Rightarrow> (\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t \<Rightarrow> (\<beta>\<^sub>1, \<dots>, \<beta>\<^sub>n) t\<close> \\
\end{matharray}
where \<open>t\<close> is a type constructor and \<open>f\<^sub>i\<close> is of type \<open>\<alpha>\<^sub>i \<Rightarrow> \<beta>\<^sub>i\<close> or \<open>\<beta>\<^sub>i \<Rightarrow>
\<alpha>\<^sub>i\<close>. Registering a map function overwrites any existing map function for
this particular type constructor.
\<^descr> @{attribute (HOL) "coercion_args"} can be used to disallow coercions to be
inserted in certain positions in a term. For example, given the constant \<open>c
:: \<sigma>\<^sub>1 \<Rightarrow> \<sigma>\<^sub>2 \<Rightarrow> \<sigma>\<^sub>3 \<Rightarrow> \<sigma>\<^sub>4\<close> and the list of policies \<open>- + 0\<close> as arguments,
coercions will not be inserted in the first argument of \<open>c\<close> (policy \<open>-\<close>);
they may be inserted in the second argument (policy \<open>+\<close>) even if the
constant \<open>c\<close> itself is in a position where coercions are disallowed; the
third argument inherits the allowance of coercsion insertion from the
position of the constant \<open>c\<close> (policy \<open>0\<close>). The standard usage of policies is
the definition of syntatic constructs (usually extralogical, i.e., processed
and stripped during type inference), that should not be destroyed by the
insertion of coercions (see, for example, the setup for the case syntax in
\<^theory>\<open>HOL.Ctr_Sugar\<close>).
\<^descr> @{attribute (HOL) "coercion_enabled"} enables the coercion inference
algorithm.
\<close>
section \<open>Arithmetic proof support\<close>
text \<open>
\begin{matharray}{rcl}
@{method_def (HOL) arith} & : & \<open>method\<close> \\
@{attribute_def (HOL) arith} & : & \<open>attribute\<close> \\
@{attribute_def (HOL) arith_split} & : & \<open>attribute\<close> \\
\end{matharray}
\<^descr> @{method (HOL) arith} decides linear arithmetic problems (on types \<open>nat\<close>,
\<open>int\<close>, \<open>real\<close>). Any current facts are inserted into the goal before running
the procedure.
\<^descr> @{attribute (HOL) arith} declares facts that are supplied to the
arithmetic provers implicitly.
\<^descr> @{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.
\<close>
section \<open>Intuitionistic proof search\<close>
text \<open>
\begin{matharray}{rcl}
@{method_def (HOL) iprover} & : & \<open>method\<close> \\
\end{matharray}
\<^rail>\<open>
@@{method (HOL) iprover} (@{syntax rulemod} *)
\<close>
\<^descr> @{method (HOL) iprover} 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 ``\<open>!\<close>'' indicator refers to
``safe'' rules, which may be applied aggressively (without considering
back-tracking later). Rules declared with ``\<open>?\<close>'' 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.
\<close>
section \<open>Model Elimination and Resolution\<close>
text \<open>
\begin{matharray}{rcl}
@{method_def (HOL) "meson"} & : & \<open>method\<close> \\
@{method_def (HOL) "metis"} & : & \<open>method\<close> \\
\end{matharray}
\<^rail>\<open>
@@{method (HOL) meson} @{syntax thms}?
;
@@{method (HOL) metis}
('(' ('partial_types' | 'full_types' | 'no_types' | @{syntax name}) ')')?
@{syntax thms}?
\<close>
\<^descr> @{method (HOL) meson} implements Loveland's model elimination procedure
@{cite "loveland-78"}. See \<^file>\<open>~~/src/HOL/ex/Meson_Test.thy\<close> for examples.
\<^descr> @{method (HOL) metis} 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 directory
\<^dir>\<open>~~/src/HOL/Metis_Examples\<close> contains several small theories developed to a
large extent using @{method (HOL) metis}.
\<close>
section \<open>Algebraic reasoning via Gr\"obner bases\<close>
text \<open>
\begin{matharray}{rcl}
@{method_def (HOL) "algebra"} & : & \<open>method\<close> \\
@{attribute_def (HOL) algebra} & : & \<open>attribute\<close> \\
\end{matharray}
\<^rail>\<open>
@@{method (HOL) algebra}
('add' ':' @{syntax thms})?
('del' ':' @{syntax thms})?
;
@@{attribute (HOL) algebra} (() | 'add' | 'del')
\<close>
\<^descr> @{method (HOL) algebra} performs algebraic reasoning via Gr\"obner bases,
see also @{cite "Chaieb-Wenzel:2007"} and @{cite \<open>\S3.2\<close> "Chaieb-thesis"}.
The method handles deals with two main classes of problems:
\<^enum> Universal problems over multivariate polynomials in a
(semi)-ring/field/idom; the capabilities of the method are augmented
according to properties of these structures. For this problem class the
method is only complete for algebraically closed fields, since the
underlying method is based on Hilbert's Nullstellensatz, where the
equivalence only holds for algebraically closed fields.
The problems can contain equations \<open>p = 0\<close> or inequations \<open>q \<noteq> 0\<close> anywhere
within a universal problem statement.
\<^enum> All-exists problems of the following restricted (but useful) form:
@{text [display] "\<forall>x\<^sub>1 \<dots> x\<^sub>n.
e\<^sub>1(x\<^sub>1, \<dots>, x\<^sub>n) = 0 \<and> \<dots> \<and> e\<^sub>m(x\<^sub>1, \<dots>, x\<^sub>n) = 0 \<longrightarrow>
(\<exists>y\<^sub>1 \<dots> y\<^sub>k.
p\<^sub>1\<^sub>1(x\<^sub>1, \<dots> ,x\<^sub>n) * y\<^sub>1 + \<dots> + p\<^sub>1\<^sub>k(x\<^sub>1, \<dots>, x\<^sub>n) * y\<^sub>k = 0 \<and>
\<dots> \<and>
p\<^sub>t\<^sub>1(x\<^sub>1, \<dots>, x\<^sub>n) * y\<^sub>1 + \<dots> + p\<^sub>t\<^sub>k(x\<^sub>1, \<dots>, x\<^sub>n) * y\<^sub>k = 0)"}
Here \<open>e\<^sub>1, \<dots>, e\<^sub>n\<close> and the \<open>p\<^sub>i\<^sub>j\<close> are multivariate polynomials only in
the variables mentioned as arguments.
The proof method is preceded by a simplification step, which may be modified
by using the form \<open>(algebra add: ths\<^sub>1 del: ths\<^sub>2)\<close>. This acts like
declarations for the Simplifier (\secref{sec:simplifier}) on a private
simpset for this tool.
\<^descr> @{attribute algebra} (as attribute) manages the default collection of
pre-simplification rules of the above proof method.
\<close>
subsubsection \<open>Example\<close>
text \<open>
The subsequent example is from geometry: collinearity is invariant by
rotation.
\<close>
(*<*)experiment begin(*>*)
type_synonym point = "int \<times> int"
fun collinear :: "point \<Rightarrow> point \<Rightarrow> point \<Rightarrow> bool" where
"collinear (Ax, Ay) (Bx, By) (Cx, Cy) \<longleftrightarrow>
(Ax - Bx) * (By - Cy) = (Ay - By) * (Bx - Cx)"
lemma collinear_inv_rotation:
assumes "collinear (Ax, Ay) (Bx, By) (Cx, Cy)" and "c\<^sup>2 + s\<^sup>2 = 1"
shows "collinear (Ax * c - Ay * s, Ay * c + Ax * s)
(Bx * c - By * s, By * c + Bx * s) (Cx * c - Cy * s, Cy * c + Cx * s)"
using assms by (algebra add: collinear.simps)
(*<*)end(*>*)
text \<open>
See also \<^file>\<open>~~/src/HOL/ex/Groebner_Examples.thy\<close>.
\<close>
section \<open>Coherent Logic\<close>
text \<open>
\begin{matharray}{rcl}
@{method_def (HOL) "coherent"} & : & \<open>method\<close> \\
\end{matharray}
\<^rail>\<open>
@@{method (HOL) coherent} @{syntax thms}?
\<close>
\<^descr> @{method (HOL) coherent} solves problems of \<^emph>\<open>Coherent Logic\<close> @{cite
"Bezem-Coquand:2005"}, which covers applications in confluence theory,
lattice theory and projective geometry. See \<^file>\<open>~~/src/HOL/ex/Coherent.thy\<close>
for some examples.
\<close>
section \<open>Unstructured case analysis and induction \label{sec:hol-induct-tac}\<close>
text \<open>
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}\<open>\<^sup>*\<close> & : & \<open>method\<close> \\
@{method_def (HOL) induct_tac}\<open>\<^sup>*\<close> & : & \<open>method\<close> \\
@{method_def (HOL) ind_cases}\<open>\<^sup>*\<close> & : & \<open>method\<close> \\
@{command_def (HOL) "inductive_cases"}\<open>\<^sup>*\<close> & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
\end{matharray}
\<^rail>\<open>
@@{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}+) @{syntax for_fixes}
;
@@{command (HOL) inductive_cases} (@{syntax thmdecl}? (@{syntax prop}+) + @'and')
;
rule: 'rule' ':' @{syntax thm}
\<close>
\<^descr> @{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>\<open>not\<close> 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.
\<^descr> @{method (HOL) ind_cases} and @{command (HOL) "inductive_cases"} provide
an interface to the internal \<^ML_text>\<open>mk_cases\<close> 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.
\<close>
section \<open>Adhoc tuples\<close>
text \<open>
\begin{matharray}{rcl}
@{attribute_def (HOL) split_format}\<open>\<^sup>*\<close> & : & \<open>attribute\<close> \\
\end{matharray}
\<^rail>\<open>
@@{attribute (HOL) split_format} ('(' 'complete' ')')?
\<close>
\<^descr> @{attribute (HOL) split_format}\ \<open>(complete)\<close> 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.
\<close>
chapter \<open>Executable code\<close>
text \<open>
For validation purposes, it is often useful to \<^emph>\<open>execute\<close> 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 a generic framework to support code generation from
executable specifications. Isabelle/HOL instantiates these mechanisms in a
way that is amenable to end-user applications. Code can be generated for
functional programs (including overloading using type classes) targeting 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>\<open>selection\<close> of code theorems,
\<^emph>\<open>translation\<close> into an abstract executable view and \<^emph>\<open>serialization\<close> to a
specific \<^emph>\<open>target language\<close>. 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"}\<open>\<^sup>*\<close> & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
@{attribute_def (HOL) code} & : & \<open>attribute\<close> \\
@{command_def (HOL) "code_datatype"} & : & \<open>theory \<rightarrow> theory\<close> \\
@{command_def (HOL) "print_codesetup"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
@{attribute_def (HOL) code_unfold} & : & \<open>attribute\<close> \\
@{attribute_def (HOL) code_post} & : & \<open>attribute\<close> \\
@{attribute_def (HOL) code_abbrev} & : & \<open>attribute\<close> \\
@{command_def (HOL) "print_codeproc"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
@{command_def (HOL) "code_thms"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
@{command_def (HOL) "code_deps"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
@{command_def (HOL) "code_reserved"} & : & \<open>theory \<rightarrow> theory\<close> \\
@{command_def (HOL) "code_printing"} & : & \<open>theory \<rightarrow> theory\<close> \\
@{command_def (HOL) "code_identifier"} & : & \<open>theory \<rightarrow> theory\<close> \\
@{command_def (HOL) "code_monad"} & : & \<open>theory \<rightarrow> theory\<close> \\
@{command_def (HOL) "code_reflect"} & : & \<open>theory \<rightarrow> theory\<close> \\
@{command_def (HOL) "code_pred"} & : & \<open>theory \<rightarrow> proof(prove)\<close>
\end{matharray}
\<^rail>\<open>
@@{command (HOL) export_code} @'open'? \<newline> (const_expr+) (export_target*)
;
export_target:
@'in' target (@'module_name' @{syntax string})? \<newline>
(@'file' @{syntax string})? ('(' args ')')?
;
target: 'SML' | 'OCaml' | 'Haskell' | 'Scala' | 'Eval'
;
const_expr: (const | 'name._' | '_')
;
const: @{syntax term}
;
type_constructor: @{syntax name}
;
class: @{syntax name}
;
@@{attribute (HOL) code} ('equation' | 'nbe' | 'abstype' | 'abstract'
| 'del' | 'drop:' (const+) | 'abort:' (const+))?
;
@@{command (HOL) code_datatype} (const+)
;
@@{attribute (HOL) code_unfold} 'del'?
;
@@{attribute (HOL) code_post} 'del'?
;
@@{attribute (HOL) code_abbrev} 'del'?
;
@@{command (HOL) code_thms} (const_expr+)
;
@@{command (HOL) code_deps} (const_expr+)
;
@@{command (HOL) code_reserved} target (@{syntax string}+)
;
symbol_const: @'constant' const
;
symbol_type_constructor: @'type_constructor' type_constructor
;
symbol_class: @'type_class' class
;
symbol_class_relation: @'class_relation' class ('<' | '\<subseteq>') class
;
symbol_class_instance: @'class_instance' type_constructor @'::' class
;
symbol_module: @'code_module' name
;
syntax: @{syntax string} | (@'infix' | @'infixl' | @'infixr')
@{syntax nat} @{syntax string}
;
printing_const: symbol_const ('\<rightharpoonup>' | '=>') \<newline>
('(' target ')' syntax ? + @'and')
;
printing_type_constructor: symbol_type_constructor ('\<rightharpoonup>' | '=>') \<newline>
('(' target ')' syntax ? + @'and')
;
printing_class: symbol_class ('\<rightharpoonup>' | '=>') \<newline>
('(' target ')' @{syntax string} ? + @'and')
;
printing_class_relation: symbol_class_relation ('\<rightharpoonup>' | '=>') \<newline>
('(' target ')' @{syntax string} ? + @'and')
;
printing_class_instance: symbol_class_instance ('\<rightharpoonup>'| '=>') \<newline>
('(' target ')' '-' ? + @'and')
;
printing_module: symbol_module ('\<rightharpoonup>' | '=>') \<newline>
('(' target ')' \<newline>
(@{syntax string}
(@'for' ((symbol_const | symbol_type_constructor | symbol_class | symbol_class_relation | symbol_class_instance)+))?)? + @'and')
;
@@{command (HOL) code_printing} ((printing_const | printing_type_constructor
| printing_class | printing_class_relation | printing_class_instance
| printing_module) + '|')
;
@@{command (HOL) code_identifier} ((symbol_const | symbol_type_constructor
| symbol_class | symbol_class_relation | symbol_class_instance
| symbol_module ) ('\<rightharpoonup>' | '=>') \<newline>
('(' target ')' @{syntax string} ? + @'and') + '|')
;
@@{command (HOL) code_monad} const const target
;
@@{command (HOL) code_reflect} @{syntax string} \<newline>
(@'datatypes' (@{syntax string} '=' ('_' | (@{syntax string} + '|') + @'and')))? \<newline>
(@'functions' (@{syntax string} +))? (@'file' @{syntax string})?
;
@@{command (HOL) code_pred} \<newline> ('(' @'modes' ':' modedecl ')')? \<newline> const
;
modedecl: (modes | ((const ':' modes) \<newline>
(@'and' ((const ':' modes @'and')+))?))
;
modes: mode @'as' const
\<close>
\<^descr> @{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 constants within a certain theory by giving \<open>name._\<close>, or
referring to \<^emph>\<open>all\<close> executable constants currently available by giving \<open>_\<close>.
By default, exported identifiers are minimized per module. This can be
suppressed by prepending @{keyword "open"} before the list of constants.
By default, for each involved theory one corresponding name space module is
generated. Alternatively, a module name may be specified after the @{keyword
"module_name"} keyword; then \<^emph>\<open>all\<close> code is placed in this module.
By default, generated code is treated as theory export which can be
explicitly retrieved using @{tool_ref export}. For diagnostic purposes
generated code can also be written to the file system using @{keyword "file"};
for \<^emph>\<open>SML\<close>, \<^emph>\<open>OCaml\<close> and \<^emph>\<open>Scala\<close> the file specification refers to a single
file; for \<^emph>\<open>Haskell\<close>, it refers to a whole directory, where code is
generated in multiple files reflecting the module hierarchy.
Passing an empty file specification denotes standard output.
Serializers take an optional list of arguments in parentheses.
\<^item> For \<^emph>\<open>Haskell\<close> a module name prefix may be given using the ``\<open>root:\<close>''
argument; ``\<open>string_classes\<close>'' adds a ``\<^verbatim>\<open>deriving (Read, Show)\<close>'' clause
to each appropriate datatype declaration.
\<^item> For \<^emph>\<open>Scala\<close>, ``\<open>case_insensitive\<close>'' avoids name clashes on
case-insensitive file systems.
\<^descr> @{attribute (HOL) code} declares code equations for code generation.
Variant \<open>code equation\<close> declares a conventional equation as code equation.
Variants \<open>code abstype\<close> and \<open>code abstract\<close> declare abstract datatype
certificates or code equations on abstract datatype representations
respectively.
Vanilla \<open>code\<close> falls back to \<open>code equation\<close> or \<open>code abstract\<close>
depending on the syntactic shape of the underlying equation.
Variant \<open>code del\<close> deselects a code equation for code generation.
Variant \<open>code nbe\<close> accepts also non-left-linear equations for
\<^emph>\<open>normalization by evaluation\<close> only.
Variants \<open>code drop:\<close> and \<open>code abort:\<close> take a list of constant as arguments
and drop all code equations declared for them. In the case of \<open>abort\<close>,
these constants then do not require to a specification by means of
code equations; if needed these are implemented by program abort (exception)
instead.
Packages declaring code equations usually provide a reasonable default
setup.
\<^descr> @{command (HOL) "code_datatype"} specifies a constructor set for a logical
type.
\<^descr> @{command (HOL) "print_codesetup"} gives an overview on selected code
equations and code generator datatypes.
\<^descr> @{attribute (HOL) code_unfold} declares (or with option ``\<open>del\<close>'' removes)
theorems which during preprocessing are applied as rewrite rules to any code
equation or evaluation input.
\<^descr> @{attribute (HOL) code_post} declares (or with option ``\<open>del\<close>'' removes)
theorems which are applied as rewrite rules to any result of an evaluation.
\<^descr> @{attribute (HOL) code_abbrev} declares (or with option ``\<open>del\<close>'' removes)
equations which are applied as rewrite rules to any result of an evaluation
and symmetrically during preprocessing to any code equation or evaluation
input.
\<^descr> @{command (HOL) "print_codeproc"} prints the setup of the code generator
preprocessor.
\<^descr> @{command (HOL) "code_thms"} prints a list of theorems representing the
corresponding program containing all given constants after preprocessing.
\<^descr> @{command (HOL) "code_deps"} visualizes dependencies of theorems
representing the corresponding program containing all given constants after
preprocessing.
\<^descr> @{command (HOL) "code_reserved"} declares a list of names as reserved for
a given target, preventing it to be shadowed by any generated code.
\<^descr> @{command (HOL) "code_printing"} associates a series of symbols
(constants, type constructors, classes, class relations, instances, module
names) with target-specific serializations; omitting a serialization deletes
an existing serialization.
\<^descr> @{command (HOL) "code_monad"} provides an auxiliary mechanism to generate
monadic code for Haskell.
\<^descr> @{command (HOL) "code_identifier"} associates a a series of symbols
(constants, type constructors, classes, class relations, instances, module
names) with target-specific hints how these symbols shall be named. These
hints gain precedence over names for symbols with no hints at all.
Conflicting hints are subject to name disambiguation. \<^emph>\<open>Warning:\<close> It is at
the discretion of the user to ensure that name prefixes of identifiers in
compound statements like type classes or datatypes are still the same.
\<^descr> @{command (HOL) "code_reflect"} without a ``\<open>file\<close>'' 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 ``\<open>datatypes\<close>'' or ``\<open>functions\<close>'' entities use these precompiled
entities. With a ``\<open>file\<close>'' argument, the corresponding code is generated
into that specified file without modifying the code generator setup.
\<^descr> @{command (HOL) "code_pred"} creates code equations for a predicate given
a set of introduction rules. Optional mode annotations determine which
arguments are supposed to be input or output. If alternative introduction
rules are declared, one must prove a corresponding elimination rule.
\<close>
end