theory HOL_Specific
imports Base "~~/src/HOL/Library/Old_Datatype" "~~/src/HOL/Library/Old_Recdef"
  "~~/src/Tools/Adhoc_Overloading"
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{weaker} than FOL with axioms for ZF set theory,
  which is traditionally considered the standard foundation of regular
  mathematics, but for most applications this does not matter.  If you
  prefer ML to Lisp, you will probably prefer HOL to ZF.

  \medskip The syntax of HOL follows @{text "\<lambda>"}-calculus and
  functional programming.  Function application is curried.  To apply
  the function @{text f} of type @{text "\<tau>\<^sub>1 \<Rightarrow> \<tau>\<^sub>2 \<Rightarrow> \<tau>\<^sub>3"} to the
  arguments @{text a} and @{text b} in HOL, you simply write @{text "f
  a b"} (as in ML or Haskell).  There is no ``apply'' operator; the
  existing application of the Pure @{text "\<lambda>"}-calculus is re-used.
  Note that in HOL @{text "f (a, b)"} means ``@{text "f"} applied to
  the pair @{text "(a, b)"} (which is notation for @{text "Pair a
  b"}).  The latter typically introduces extra formal efforts that can
  be avoided by currying functions by default.  Explicit tuples are as
  infrequent in HOL formalizations as in good ML or Haskell programs.

  \medskip Isabelle/HOL has a distinct feel, compared to other
  object-logics like Isabelle/ZF.  It identifies object-level types
  with meta-level types, taking advantage of the default
  type-inference mechanism of Isabelle/Pure.  HOL fully identifies
  object-level functions with meta-level functions, with native
  abstraction and application.

  These identifications allow Isabelle to support HOL particularly
  nicely, but they also mean that HOL requires some sophistication
  from the user.  In particular, an understanding of Hindley-Milner
  type-inference with type-classes, which are both used extensively in
  the standard libraries and applications.  Beginners can set
  @{attribute show_types} or even @{attribute show_sorts} to get more
  explicit information about the result of type-inference.\<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"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
    @{command_def (HOL) "inductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
    @{command_def (HOL) "coinductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
    @{command_def (HOL) "coinductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
    @{command_def "print_inductives"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
    @{attribute_def (HOL) mono} & : & @{text attribute} \\
  \end{matharray}

  An \emph{inductive definition} specifies the least predicate or set
  @{text R} closed under given rules: applying a rule to elements of
  @{text R} yields a result within @{text R}.  For example, a
  structural operational semantics is an inductive definition of an
  evaluation relation.

  Dually, a \emph{coinductive definition} specifies the greatest
  predicate or set @{text R} that is consistent with given rules:
  every element of @{text R} can be seen as arising by applying a rule
  to elements of @{text R}.  An important example is using
  bisimulation relations to formalise equivalence of processes and
  infinite data structures.

  Both inductive and coinductive definitions are based on the
  Knaster-Tarski fixed-point theorem for complete lattices.  The
  collection of introduction rules given by the user determines a
  functor on subsets of set-theoretic relations.  The required
  monotonicity of the recursion scheme is proven as a prerequisite to
  the fixed-point definition and the resulting consequences.  This
  works by pushing inclusion through logical connectives and any other
  operator that might be wrapped around recursive occurrences of the
  defined relation: there must be a monotonicity theorem of the form
  @{text "A \<le> B \<Longrightarrow> \<M> A \<le> \<M> B"}, for each premise @{text "\<M> R t"} in an
  introduction rule.  The default rule declarations of Isabelle/HOL
  already take care of most common situations.

  @{rail \<open>
    (@@{command (HOL) inductive} | @@{command (HOL) inductive_set} |
      @@{command (HOL) coinductive} | @@{command (HOL) coinductive_set})
    @{syntax target}? \<newline>
    @{syntax "fixes"} (@'for' @{syntax "fixes"})? (@'where' clauses)? \<newline>
    (@'monos' @{syntax thmrefs})?
    ;
    clauses: (@{syntax thmdecl}? @{syntax prop} + '|')
    ;
    @@{attribute (HOL) mono} (() | 'add' | 'del')
  \<close>}

  \begin{description}

  \item @{command (HOL) "inductive"} and @{command (HOL)
  "coinductive"} define (co)inductive predicates from the introduction
  rules.

  The propositions given as @{text "clauses"} in the @{keyword
  "where"} part are either rules of the usual @{text "\<And>/\<Longrightarrow>"} format
  (with arbitrary nesting), or equalities using @{text "\<equiv>"}.  The
  latter specifies extra-logical abbreviations in the sense of
  @{command_ref abbreviation}.  Introducing abstract syntax
  simultaneously with the actual introduction rules is occasionally
  useful for complex specifications.

  The optional @{keyword "for"} part contains a list of parameters of
  the (co)inductive predicates that remain fixed throughout the
  definition, in contrast to arguments of the relation that may vary
  in each occurrence within the given @{text "clauses"}.

  The optional @{keyword "monos"} declaration contains additional
  \emph{monotonicity theorems}, which are required for each operator
  applied to a recursive set in the introduction rules.

  \item @{command (HOL) "inductive_set"} and @{command (HOL)
  "coinductive_set"} are wrappers for to the previous commands for
  native HOL predicates.  This allows to define (co)inductive sets,
  where multiple arguments are simulated via tuples.

  \item @{command "print_inductives"} prints (co)inductive definitions and
  monotonicity rules.

  \item @{attribute (HOL) mono} declares monotonicity rules in the
  context.  These rule are involved in the automated monotonicity
  proof of the above inductive and coinductive definitions.

  \end{description}
\<close>

subsection \<open>Derived rules\<close>

text \<open>A (co)inductive definition of @{text R} provides the following
  main theorems:

  \begin{description}

  \item @{text R.intros} is the list of introduction rules as proven
  theorems, for the recursive predicates (or sets).  The rules are
  also available individually, using the names given them in the
  theory file;

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

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

  \item @{text R.simps} is the equation unrolling the fixpoint of the
  predicate one step.

  \end{description}

  When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are
  defined simultaneously, the list of introduction rules is called
  @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are
  called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list
  of mutual induction rules is called @{text
  "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}.
\<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:

  \begin{itemize}

  \item Theorems of the form @{text "A \<le> B \<Longrightarrow> \<M> A \<le> \<M> B"}, for proving
  monotonicity of inductive definitions whose introduction rules have
  premises involving terms such as @{text "\<M> R t"}.

  \item Monotonicity theorems for logical operators, which are of the
  general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}.  For example, in
  the case of the operator @{text "\<or>"}, the corresponding theorem is
  \[
  \infer{@{text "P\<^sub>1 \<or> P\<^sub>2 \<longrightarrow> Q\<^sub>1 \<or> Q\<^sub>2"}}{@{text "P\<^sub>1 \<longrightarrow> Q\<^sub>1"} & @{text "P\<^sub>2 \<longrightarrow> Q\<^sub>2"}}
  \]

  \item De Morgan style equations for reasoning about the ``polarity''
  of expressions, e.g.
  \[
  @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad
  @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}
  \]

  \item Equations for reducing complex operators to more primitive
  ones whose monotonicity can easily be proved, e.g.
  \[
  @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad
  @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}
  \]

  \end{itemize}
\<close>

subsubsection \<open>Examples\<close>

text \<open>The finite powerset operator can be defined inductively like this:\<close>

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"

text \<open>Common logical connectives can be easily characterized as
non-recursive inductive definitions with parameters, but without
arguments.\<close>

inductive AND for A B :: bool
where "A \<Longrightarrow> B \<Longrightarrow> AND A B"

inductive OR for A B :: bool
where "A \<Longrightarrow> OR A B"
  | "B \<Longrightarrow> OR A B"

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

text \<open>Here the @{text "cases"} or @{text "induct"} rules produced by
  the @{command inductive} package coincide with the expected
  elimination rules for Natural Deduction.  Already in the original
  article by Gerhard Gentzen @{cite "Gentzen:1935"} there is a hint that
  each connective can be characterized by its introductions, and the
  elimination can be constructed systematically.\<close>

section \<open>Recursive functions \label{sec:recursion}\<close>

text \<open>
  \begin{matharray}{rcl}
    @{command_def (HOL) "primrec"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
    @{command_def (HOL) "fun"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
    @{command_def (HOL) "function"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
    @{command_def (HOL) "termination"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
    @{command_def (HOL) "fun_cases"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
  \end{matharray}

  @{rail \<open>
    @@{command (HOL) primrec} @{syntax target}? @{syntax "fixes"} @'where' equations
    ;
    (@@{command (HOL) fun} | @@{command (HOL) function}) @{syntax target}? functionopts?
      @{syntax "fixes"} \<newline> @'where' equations
    ;

    equations: (@{syntax thmdecl}? @{syntax prop} + '|')
    ;
    functionopts: '(' (('sequential' | 'domintros') + ',') ')'
    ;
    @@{command (HOL) termination} @{syntax term}?
    ;
    @@{command (HOL) fun_cases} (@{syntax thmdecl}? @{syntax prop} + @'and')
  \<close>}

  \begin{description}

  \item @{command (HOL) "primrec"} defines primitive recursive
  functions over datatypes (see also @{command_ref (HOL) datatype}).
  The given @{text equations} specify reduction rules that are produced
  by instantiating the generic combinator for primitive recursion that
  is available for each datatype.

  Each equation needs to be of the form:

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

  such that @{text C} is a datatype constructor, @{text rhs} contains
  only the free variables on the left-hand side (or from the context),
  and all recursive occurrences of @{text "f"} in @{text "rhs"} are of
  the form @{text "f \<dots> y\<^sub>i \<dots>"} for some @{text i}.  At most one
  reduction rule for each constructor can be given.  The order does
  not matter.  For missing constructors, the function is defined to
  return a default value, but this equation is made difficult to
  access for users.

  The reduction rules are declared as @{attribute simp} by default,
  which enables standard proof methods like @{method simp} and
  @{method auto} to normalize expressions of @{text "f"} applied to
  datatype constructions, by simulating symbolic computation via
  rewriting.

  \item @{command (HOL) "function"} defines functions by general
  wellfounded recursion. A detailed description with examples can be
  found in @{cite "isabelle-function"}. The function is specified by a
  set of (possibly conditional) recursive equations with arbitrary
  pattern matching. The command generates proof obligations for the
  completeness and the compatibility of patterns.

  The defined function is considered partial, and the resulting
  simplification rules (named @{text "f.psimps"}) and induction rule
  (named @{text "f.pinduct"}) are guarded by a generated domain
  predicate @{text "f_dom"}. The @{command (HOL) "termination"}
  command can then be used to establish that the function is total.

  \item @{command (HOL) "fun"} is a shorthand notation for ``@{command
  (HOL) "function"}~@{text "(sequential)"}, followed by automated
  proof attempts regarding pattern matching and termination.  See
  @{cite "isabelle-function"} for further details.

  \item @{command (HOL) "termination"}~@{text f} commences a
  termination proof for the previously defined function @{text f}.  If
  this is omitted, the command refers to the most recent function
  definition.  After the proof is closed, the recursive equations and
  the induction principle is established.

  \item @{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.
  \end{description}

  Recursive definitions introduced by the @{command (HOL) "function"}
  command accommodate reasoning by induction (cf.\ @{method induct}):
  rule @{text "f.induct"} refers to a specific induction rule, with
  parameters named according to the user-specified equations. Cases
  are numbered starting from 1.  For @{command (HOL) "primrec"}, the
  induction principle coincides with structural recursion on the
  datatype where the recursion is carried out.

  The equations provided by these packages may be referred later as
  theorem list @{text "f.simps"}, where @{text f} is the (collective)
  name of the functions defined.  Individual equations may be named
  explicitly as well.

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

  \begin{description}

  \item @{text sequential} enables a preprocessor which disambiguates
  overlapping patterns by making them mutually disjoint.  Earlier
  equations take precedence over later ones.  This allows to give the
  specification in a format very similar to functional programming.
  Note that the resulting simplification and induction rules
  correspond to the transformed specification, not the one given
  originally. This usually means that each equation given by the user
  may result in several theorems.  Also note that this automatic
  transformation only works for ML-style datatype patterns.

  \item @{text domintros} enables the automated generation of
  introduction rules for the domain predicate. While mostly not
  needed, they can be helpful in some proofs about partial functions.

  \end{description}
\<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>

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 evala} and @{const evalb} take an
  additional parameter, an \emph{environment} that maps variables to
  their values.

  \medskip Substitution on expressions can be defined similarly.  The
  mapping @{text f} of type @{typ "'a \<Rightarrow> 'a aexp"} given as a
  parameter is lifted canonically on the types @{typ "'a aexp"} and
  @{typ "'a bexp"}, respectively.
\<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 "'a aexp"} and @{typ "'a bexp"}, 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

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>

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

text \<open>A substitution function on type @{typ "('a, 'b) term"} can be
  defined as follows, by working simultaneously on @{typ "('a, 'b)
  term list"}:\<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 "('a, 'b) term"}.  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

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>

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 @{text ts}
  above must be applied to an argument.  In particular, @{term
  "map_tree f \<circ> ts"} is not allowed here.\<close>

text \<open>Here is a simple composition lemma for @{term map_tree}:\<close>

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

subsection \<open>Proof methods related to recursive definitions\<close>

text \<open>
  \begin{matharray}{rcl}
    @{method_def (HOL) pat_completeness} & : & @{text method} \\
    @{method_def (HOL) relation} & : & @{text method} \\
    @{method_def (HOL) lexicographic_order} & : & @{text method} \\
    @{method_def (HOL) size_change} & : & @{text method} \\
    @{method_def (HOL) induction_schema} & : & @{text method} \\
  \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>}

  \begin{description}

  \item @{method (HOL) pat_completeness} is a specialized method to
  solve goals regarding the completeness of pattern matching, as
  required by the @{command (HOL) "function"} package (cf.\
  @{cite "isabelle-function"}).

  \item @{method (HOL) relation}~@{text R} introduces a termination
  proof using the relation @{text R}.  The resulting proof state will
  contain goals expressing that @{text R} is wellfounded, and that the
  arguments of recursive calls decrease with respect to @{text R}.
  Usually, this method is used as the initial proof step of manual
  termination proofs.

  \item @{method (HOL) "lexicographic_order"} attempts a fully
  automated termination proof by searching for a lexicographic
  combination of size measures on the arguments of the function. The
  method accepts the same arguments as the @{method auto} method,
  which it uses internally to prove local descents.  The @{syntax
  clasimpmod} modifiers are accepted (as for @{method auto}).

  In case of failure, extensive information is printed, which can help
  to analyse the situation (cf.\ @{cite "isabelle-function"}).

  \item @{method (HOL) "size_change"} also works on termination goals,
  using a variation of the size-change principle, together with a
  graph decomposition technique (see @{cite krauss_phd} for details).
  Three kinds of orders are used internally: @{text max}, @{text min},
  and @{text ms} (multiset), which is only available when the theory
  @{text Multiset} is loaded. When no order kinds are given, they are
  tried in order. The search for a termination proof uses SAT solving
  internally.

  For local descent proofs, the @{syntax clasimpmod} modifiers are
  accepted (as for @{method auto}).

  \item @{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 "~~/src/HOL/ex/Induction_Schema.thy"} for examples.

  \end{description}
\<close>

subsection \<open>Functions with explicit partiality\<close>

text \<open>
  \begin{matharray}{rcl}
    @{command_def (HOL) "partial_function"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
    @{attribute_def (HOL) "partial_function_mono"} & : & @{text attribute} \\
  \end{matharray}

  @{rail \<open>
    @@{command (HOL) partial_function} @{syntax target}?
      '(' @{syntax nameref} ')' @{syntax "fixes"} \<newline>
      @'where' @{syntax thmdecl}? @{syntax prop}
  \<close>}

  \begin{description}

  \item @{command (HOL) "partial_function"}~@{text "(mode)"} defines
  recursive functions based on fixpoints in complete partial
  orders. No termination proof is required from the user or
  constructed internally. Instead, the possibility of non-termination
  is modelled explicitly in the result type, which contains an
  explicit bottom element.

  Pattern matching and mutual recursion are currently not supported.
  Thus, the specification consists of a single function described by a
  single recursive equation.

  There are no fixed syntactic restrictions on the body of the
  function, but the induced functional must be provably monotonic
  wrt.\ the underlying order.  The 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 @{text mode} argument specifies the mode of operation
  of the command, which directly corresponds to a complete partial
  order on the result type. By default, the following modes are
  defined:

  \begin{description}

  \item @{text option} defines functions that map into the @{type
  option} type. Here, the value @{term None} is used to model a
  non-terminating computation. Monotonicity requires that if @{term
  None} is returned by a recursive call, then the overall result must
  also be @{term None}. This is best achieved through the use of the
  monadic operator @{const "Option.bind"}.

  \item @{text tailrec} defines functions with an arbitrary result
  type and uses the slightly degenerated partial order where @{term
  "undefined"} is the bottom element.  Now, monotonicity requires that
  if @{term undefined} is returned by a recursive call, then the
  overall result must also be @{term undefined}. In practice, this is
  only satisfied when each recursive call is a tail call, whose result
  is directly returned. Thus, this mode of operation allows the
  definition of arbitrary tail-recursive functions.

  \end{description}

  Experienced users may define new modes by instantiating the locale
  @{const "partial_function_definitions"} appropriately.

  \item @{attribute (HOL) partial_function_mono} declares rules for
  use in the internal monotonicity proofs of partial function
  definitions.

  \end{description}

\<close>

subsection \<open>Old-style recursive function definitions (TFL)\<close>

text \<open>
  \begin{matharray}{rcl}
    @{command_def (HOL) "recdef"} & : & @{text "theory \<rightarrow> theory)"} \\
    @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
  \end{matharray}

  The old TFL commands @{command (HOL) "recdef"} and @{command (HOL)
  "recdef_tc"} for defining recursive are mostly obsolete; @{command
  (HOL) "function"} or @{command (HOL) "fun"} should be used instead.

  @{rail \<open>
    @@{command (HOL) recdef} ('(' @'permissive' ')')? \<newline>
      @{syntax name} @{syntax term} (@{syntax prop} +) hints?
    ;
    recdeftc @{syntax thmdecl}? tc
    ;
    hints: '(' @'hints' ( recdefmod * ) ')'
    ;
    recdefmod: (('recdef_simp' | 'recdef_cong' | 'recdef_wf')
      (() | 'add' | 'del') ':' @{syntax thmrefs}) | @{syntax clasimpmod}
    ;
    tc: @{syntax nameref} ('(' @{syntax nat} ')')?
  \<close>}

  \begin{description}

  \item @{command (HOL) "recdef"} defines general well-founded
  recursive functions (using the TFL package), see also
  @{cite "isabelle-HOL"}.  The ``@{text "(permissive)"}'' option tells
  TFL to recover from failed proof attempts, returning unfinished
  results.  The @{text recdef_simp}, @{text recdef_cong}, and @{text
  recdef_wf} hints refer to auxiliary rules to be used in the internal
  automated proof process of TFL.  Additional @{syntax clasimpmod}
  declarations may be given to tune the context of the Simplifier
  (cf.\ \secref{sec:simplifier}) and Classical reasoner (cf.\
  \secref{sec:classical}).

  \item @{command (HOL) "recdef_tc"}~@{text "c (i)"} recommences the
  proof for leftover termination condition number @{text i} (default
  1) as generated by a @{command (HOL) "recdef"} definition of
  constant @{text c}.

  Note that in most cases, @{command (HOL) "recdef"} is able to finish
  its internal proofs without manual intervention.

  \end{description}

  \medskip Hints for @{command (HOL) "recdef"} may be also declared
  globally, using the following attributes.

  \begin{matharray}{rcl}
    @{attribute_def (HOL) recdef_simp} & : & @{text attribute} \\
    @{attribute_def (HOL) recdef_cong} & : & @{text attribute} \\
    @{attribute_def (HOL) recdef_wf} & : & @{text attribute} \\
  \end{matharray}

  @{rail \<open>
    (@@{attribute (HOL) recdef_simp} | @@{attribute (HOL) recdef_cong} |
      @@{attribute (HOL) recdef_wf}) (() | 'add' | 'del')
  \<close>}
\<close>

section \<open>Old-style datatypes \label{sec:hol-datatype}\<close>

text \<open>
  \begin{matharray}{rcl}
    @{command_def (HOL) "old_datatype"} & : & @{text "theory \<rightarrow> theory"} \\
    @{command_def (HOL) "old_rep_datatype"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
  \end{matharray}

  @{rail \<open>
    @@{command (HOL) old_datatype} (spec + @'and')
    ;
    @@{command (HOL) old_rep_datatype} ('(' (@{syntax name} +) ')')? (@{syntax term} +)
    ;

    spec: @{syntax typespec_sorts} @{syntax mixfix}? '=' (cons + '|')
    ;
    cons: @{syntax name} (@{syntax type} * ) @{syntax mixfix}?
  \<close>}

  \begin{description}

  \item @{command (HOL) "old_datatype"} defines old-style inductive
  datatypes in HOL.

  \item @{command (HOL) "old_rep_datatype"} represents existing types as
  old-style datatypes.

  \end{description}

  These commands are mostly obsolete; @{command (HOL) "datatype"}
  should be used instead.

  See @{cite "isabelle-HOL"} for more details on datatypes, but beware of
  the old-style theory syntax being used there!  Apart from proper
  proof methods for case-analysis and induction, there are also
  emulations of ML tactics @{method (HOL) case_tac} and @{method (HOL)
  induct_tac} available, see \secref{sec:hol-induct-tac}; these admit
  to refer directly to the internal structure of subgoals (including
  internally bound parameters).
\<close>

subsubsection \<open>Examples\<close>

text \<open>We define a type of finite sequences, with slightly different
  names than the existing @{typ "'a list"} that is already in @{theory
  Main}:\<close>

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

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{fixed} and \emph{schematic} records
  at the level of terms and types.  The notation is as follows:

  \begin{center}
  \begin{tabular}{l|l|l}
    & record terms & record types \\ \hline
    fixed & @{text "\<lparr>x = a, y = b\<rparr>"} & @{text "\<lparr>x :: A, y :: B\<rparr>"} \\
    schematic & @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} &
      @{text "\<lparr>x :: A, y :: B, \<dots> :: M\<rparr>"} \\
  \end{tabular}
  \end{center}

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

  A fixed record @{text "\<lparr>x = a, y = b\<rparr>"} has field @{text x} of value
  @{text a} and field @{text y} of value @{text b}.  The corresponding
  type is @{text "\<lparr>x :: A, y :: B\<rparr>"}, assuming that @{text "a :: A"}
  and @{text "b :: B"}.

  A record scheme like @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} contains fields
  @{text x} and @{text y} as before, but also possibly further fields
  as indicated by the ``@{text "\<dots>"}'' notation (which is actually part
  of the syntax).  The improper field ``@{text "\<dots>"}'' of a record
  scheme is called the \emph{more part}.  Logically it is just a free
  variable, which is occasionally referred to as ``row variable'' in
  the literature.  The more part of a record scheme may be
  instantiated by zero or more further components.  For example, the
  previous scheme may get instantiated to @{text "\<lparr>x = a, y = b, z =
  c, \<dots> = m'\<rparr>"}, where @{text m'} refers to a different more part.
  Fixed records are special instances of record schemes, where
  ``@{text "\<dots>"}'' is properly terminated by the @{text "() :: unit"}
  element.  In fact, @{text "\<lparr>x = a, y = b\<rparr>"} is just an abbreviation
  for @{text "\<lparr>x = a, y = b, \<dots> = ()\<rparr>"}.

  \medskip Two key observations make extensible records in a simply
  typed language like HOL work out:

  \begin{enumerate}

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

  \item field names are externalized, they cannot be accessed within
  the logic as first-class values.

  \end{enumerate}

  \medskip In Isabelle/HOL record types have to be defined explicitly,
  fixing their field names and types, and their (optional) parent
  record.  Afterwards, records may be formed using above syntax, while
  obeying the canonical order of fields as given by their declaration.
  The record package provides several standard operations like
  selectors and updates.  The common setup for various generic proof
  tools enable succinct reasoning patterns.  See also the Isabelle/HOL
  tutorial @{cite "isabelle-hol-book"} for further instructions on using
  records in practice.
\<close>

subsection \<open>Record specifications\<close>

text \<open>
  \begin{matharray}{rcl}
    @{command_def (HOL) "record"} & : & @{text "theory \<rightarrow> theory"} \\
  \end{matharray}

  @{rail \<open>
    @@{command (HOL) record} @{syntax typespec_sorts} '=' \<newline>
      (@{syntax type} '+')? (constdecl +)
    ;
    constdecl: @{syntax name} '::' @{syntax type} @{syntax mixfix}?
  \<close>}

  \begin{description}

  \item @{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1
  \<dots> c\<^sub>n :: \<sigma>\<^sub>n"} defines extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"},
  derived from the optional parent record @{text "\<tau>"} by adding new
  field components @{text "c\<^sub>i :: \<sigma>\<^sub>i"} etc.

  The type variables of @{text "\<tau>"} and @{text "\<sigma>\<^sub>i"} need to be
  covered by the (distinct) parameters @{text "\<alpha>\<^sub>1, \<dots>,
  \<alpha>\<^sub>m"}.  Type constructor @{text t} has to be new, while @{text
  \<tau>} needs to specify an instance of an existing record type.  At
  least one new field @{text "c\<^sub>i"} has to be specified.
  Basically, field names need to belong to a unique record.  This is
  not a real restriction in practice, since fields are qualified by
  the record name internally.

  The parent record specification @{text \<tau>} is optional; if omitted
  @{text t} becomes a root record.  The hierarchy of all records
  declared within a theory context forms a forest structure, i.e.\ a
  set of trees starting with a root record each.  There is no way to
  merge multiple parent records!

  For convenience, @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} is made a
  type abbreviation for the fixed record type @{text "\<lparr>c\<^sub>1 ::
  \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<rparr>"}, likewise is @{text
  "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m, \<zeta>) t_scheme"} made an abbreviation for
  @{text "\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n, \<dots> ::
  \<zeta>\<rparr>"}.

  \end{description}
\<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 ``@{text more}''.  There are also
  cumulative record constructor functions.  To simplify the
  presentation below, we assume for now that @{text "(\<alpha>\<^sub>1, \<dots>,
  \<alpha>\<^sub>m) t"} is a root record with fields @{text "c\<^sub>1 ::
  \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n"}.

  \medskip \textbf{Selectors} and \textbf{updates} are available for
  any field (including ``@{text more}''):

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

  There is special syntax for application of updates: @{text "r\<lparr>x :=
  a\<rparr>"} abbreviates term @{text "x_update a r"}.  Further notation for
  repeated updates is also available: @{text "r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<lparr>z :=
  c\<rparr>"} may be written @{text "r\<lparr>x := a, y := b, z := c\<rparr>"}.  Note that
  because of postfix notation the order of fields shown here is
  reverse than in the actual term.  Since repeated updates are just
  function applications, fields may be freely permuted in @{text "\<lparr>x
  := a, y := b, z := c\<rparr>"}, as far as logical equality is concerned.
  Thus commutativity of independent updates can be proven within the
  logic for any two fields, but not as a general theorem.

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

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

  \medskip We now reconsider the case of non-root records, which are
  derived of some parent.  In general, the latter may depend on
  another parent as well, resulting in a list of \emph{ancestor
  records}.  Appending the lists of fields of all ancestors results in
  a certain field prefix.  The record package automatically takes care
  of this by lifting operations over this context of ancestor fields.
  Assuming that @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} has ancestor
  fields @{text "b\<^sub>1 :: \<rho>\<^sub>1, \<dots>, b\<^sub>k :: \<rho>\<^sub>k"},
  the above record operations will get the following types:

  \medskip
  \begin{tabular}{lll}
    @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
    @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow>
      \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow>
      \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
    @{text "t.make"} & @{text "::"} & @{text "\<rho>\<^sub>1 \<Rightarrow> \<dots> \<rho>\<^sub>k \<Rightarrow> \<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow>
      \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
  \end{tabular}
  \medskip

  \noindent Some further operations address the extension aspect of a
  derived record scheme specifically: @{text "t.fields"} produces a
  record fragment consisting of exactly the new fields introduced here
  (the result may serve as a more part elsewhere); @{text "t.extend"}
  takes a fixed record and adds a given more part; @{text
  "t.truncate"} restricts a record scheme to a fixed record.

  \medskip
  \begin{tabular}{lll}
    @{text "t.fields"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
    @{text "t.extend"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr> \<Rightarrow>
      \<zeta> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
    @{text "t.truncate"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
  \end{tabular}
  \medskip

  \noindent Note that @{text "t.make"} and @{text "t.fields"} coincide
  for root records.
\<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
  @{text t} is a record type as specified above.

  \begin{enumerate}

  \item Standard conversions for selectors or updates applied to
  record constructor terms are made part of the default Simplifier
  context; thus proofs by reduction of basic operations merely require
  the @{method simp} method without further arguments.  These rules
  are available as @{text "t.simps"}, too.

  \item Selectors applied to updated records are automatically reduced
  by an internal simplification procedure, which is also part of the
  standard Simplifier setup.

  \item Inject equations of a form analogous to @{prop "(x, y) = (x',
  y') \<equiv> x = x' \<and> y = y'"} are declared to the Simplifier and Classical
  Reasoner as @{attribute iff} rules.  These rules are available as
  @{text "t.iffs"}.

  \item The introduction rule for record equality analogous to @{text
  "x r = x r' \<Longrightarrow> y r = y r' \<dots> \<Longrightarrow> r = r'"} is declared to the Simplifier,
  and as the basic rule context as ``@{attribute intro}@{text "?"}''.
  The rule is called @{text "t.equality"}.

  \item Representations of arbitrary record expressions as canonical
  constructor terms are provided both in @{method cases} and @{method
  induct} format (cf.\ the generic proof methods of the same name,
  \secref{sec:cases-induct}).  Several variations are available, for
  fixed records, record schemes, more parts etc.

  The generic proof methods are sufficiently smart to pick the most
  sensible rule according to the type of the indicated record
  expression: users just need to apply something like ``@{text "(cases
  r)"}'' to a certain proof problem.

  \item The derived record operations @{text "t.make"}, @{text
  "t.fields"}, @{text "t.extend"}, @{text "t.truncate"} are \emph{not}
  treated automatically, but usually need to be expanded by hand,
  using the collective fact @{text "t.defs"}.

  \end{enumerate}
\<close>

subsubsection \<open>Examples\<close>

text \<open>See @{file "~~/src/HOL/ex/Records.thy"}, for example.\<close>

section \<open>Typedef axiomatization \label{sec:hol-typedef}\<close>

text \<open>
  \begin{matharray}{rcl}
    @{command_def (HOL) "typedef"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
  \end{matharray}

  A Gordon/HOL-style type definition is a certain axiom scheme that
  identifies a new type with a subset of an existing type.  More
  precisely, the new type is defined by exhibiting an existing type
  @{text \<tau>}, a set @{text "A :: \<tau> set"}, and a theorem that proves
  @{prop "\<exists>x. x \<in> A"}.  Thus @{text A} is a non-empty subset of @{text
  \<tau>}, and the new type denotes this subset.  New functions are
  postulated that establish an isomorphism between the new type and
  the subset.  In general, the type @{text \<tau>} may involve type
  variables @{text "\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n"} which means that the type definition
  produces a type constructor @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t"} depending on
  those type arguments.

  The axiomatization can be considered a ``definition'' in the sense of the
  particular set-theoretic interpretation of HOL @{cite pitts93}, where the
  universe of types is required to be downwards-closed wrt.\ arbitrary
  non-empty subsets. Thus genuinely new types introduced by @{command
  "typedef"} stay within the range of HOL models by construction.

  In contrast, the command @{command_ref type_synonym} from Isabelle/Pure
  merely introduces syntactic abbreviations, without any logical
  significance. Thus it is more faithful to the idea of a genuine type
  definition, but less powerful in practice.

  @{rail \<open>
    @@{command (HOL) typedef} abs_type '=' rep_set
    ;
    abs_type: @{syntax typespec_sorts} @{syntax mixfix}?
    ;
    rep_set: @{syntax term} (@'morphisms' @{syntax name} @{syntax name})?
  \<close>}

  \begin{description}

  \item @{command (HOL) "typedef"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = A"} produces an
  axiomatization (\secref{sec:axiomatizations}) for a type definition in the
  background theory of the current context, depending on a non-emptiness
  result of the set @{text A} that needs to be proven here. The set @{text
  A} may contain type variables @{text "\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n"} as specified on the
  LHS, but no term variables.

  Even though a local theory specification, the newly introduced type
  constructor cannot depend on parameters or assumptions of the
  context: this is structurally impossible in HOL.  In contrast, the
  non-emptiness proof may use local assumptions in unusual situations,
  which could result in different interpretations in target contexts:
  the meaning of the bijection between the representing set @{text A}
  and the new type @{text t} may then change in different application
  contexts.

  For @{command (HOL) "typedef"}~@{text "t = A"} the newly introduced
  type @{text t} 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 @{text Rep_t} and its inverse @{text
  Abs_t}: An explicit @{keyword (HOL) "morphisms"} specification
  allows to provide alternative names.

  The core axiomatization uses the locale predicate @{const
  type_definition} as defined in Isabelle/HOL.  Various basic
  consequences of that are instantiated accordingly, re-using the
  locale facts with names derived from the new type constructor.  Thus
  the generic @{thm type_definition.Rep} is turned into the specific
  @{text "Rep_t"}, for example.

  Theorems @{thm type_definition.Rep}, @{thm
  type_definition.Rep_inverse}, and @{thm type_definition.Abs_inverse}
  provide the most basic characterization as a corresponding
  injection/surjection pair (in both directions).  The derived rules
  @{thm type_definition.Rep_inject} and @{thm
  type_definition.Abs_inject} provide a more convenient version of
  injectivity, suitable for automated proof tools (e.g.\ in
  declarations involving @{attribute simp} or @{attribute iff}).
  Furthermore, the rules @{thm type_definition.Rep_cases}~/ @{thm
  type_definition.Rep_induct}, and @{thm type_definition.Abs_cases}~/
  @{thm type_definition.Abs_induct} provide alternative views on
  surjectivity.  These rules are already declared as set or type rules
  for the generic @{method cases} and @{method induct} methods,
  respectively.

  \end{description}
\<close>

subsubsection \<open>Examples\<close>

text \<open>Type definitions permit the introduction of abstract data
  types in a safe way, namely by providing models based on already
  existing types.  Given some abstract axiomatic description @{text P}
  of a type, this involves two steps:

  \begin{enumerate}

  \item Find an appropriate type @{text \<tau>} and subset @{text A} which
  has the desired properties @{text P}, and make a type definition
  based on this representation.

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

  \end{enumerate}

  You can later forget about the representation and work solely in
  terms of the abstract properties @{text P}.

  \medskip The following trivial example pulls a three-element type
  into existence within the formal logical environment of HOL.\<close>

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)

text \<open>Note that such trivial constructions are better done with
  derived specification mechanisms such as @{command datatype}:\<close>

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

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"} & : & @{text "local_theory \<rightarrow> proof(prove)"}
  \end{matharray}

  @{rail \<open>
    @@{command (HOL) functor} (@{syntax name} ':')? @{syntax term}
  \<close>}

  \begin{description}

  \item @{command (HOL) "functor"}~@{text "prefix: m"} allows to
  prove and register properties about the functorial structure of type
  constructors.  These properties then can be used by other packages
  to deal with those type constructors in certain type constructions.
  Characteristic theorems are noted in the current local theory.  By
  default, they are prefixed with the base name of the type
  constructor, an explicit prefix can be given alternatively.

  The given term @{text "m"} is considered as \emph{mapper} for the
  corresponding type constructor and must conform to the following
  type pattern:

  \begin{matharray}{lll}
    @{text "m"} & @{text "::"} &
      @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>k \<Rightarrow> (\<^vec>\<alpha>\<^sub>n) t \<Rightarrow> (\<^vec>\<beta>\<^sub>n) t"} \\
  \end{matharray}

  \noindent where @{text t} is the type constructor, @{text
  "\<^vec>\<alpha>\<^sub>n"} and @{text "\<^vec>\<beta>\<^sub>n"} are distinct
  type variables free in the local theory and @{text "\<sigma>\<^sub>1"},
  \ldots, @{text "\<sigma>\<^sub>k"} is a subsequence of @{text "\<alpha>\<^sub>1 \<Rightarrow>
  \<beta>\<^sub>1"}, @{text "\<beta>\<^sub>1 \<Rightarrow> \<alpha>\<^sub>1"}, \ldots,
  @{text "\<alpha>\<^sub>n \<Rightarrow> \<beta>\<^sub>n"}, @{text "\<beta>\<^sub>n \<Rightarrow>
  \<alpha>\<^sub>n"}.

  \end{description}
\<close>

section \<open>Quotient types\<close>

text \<open>
  \begin{matharray}{rcl}
    @{command_def (HOL) "quotient_type"} & : & @{text "local_theory \<rightarrow> proof(prove)"}\\
    @{command_def (HOL) "quotient_definition"} & : & @{text "local_theory \<rightarrow> proof(prove)"}\\
    @{command_def (HOL) "print_quotmapsQ3"} & : & @{text "context \<rightarrow>"}\\
    @{command_def (HOL) "print_quotientsQ3"} & : & @{text "context \<rightarrow>"}\\
    @{command_def (HOL) "print_quotconsts"} & : & @{text "context \<rightarrow>"}\\
    @{method_def (HOL) "lifting"} & : & @{text method} \\
    @{method_def (HOL) "lifting_setup"} & : & @{text method} \\
    @{method_def (HOL) "descending"} & : & @{text method} \\
    @{method_def (HOL) "descending_setup"} & : & @{text method} \\
    @{method_def (HOL) "partiality_descending"} & : & @{text method} \\
    @{method_def (HOL) "partiality_descending_setup"} & : & @{text method} \\
    @{method_def (HOL) "regularize"} & : & @{text method} \\
    @{method_def (HOL) "injection"} & : & @{text method} \\
    @{method_def (HOL) "cleaning"} & : & @{text method} \\
    @{attribute_def (HOL) "quot_thm"} & : & @{text attribute} \\
    @{attribute_def (HOL) "quot_lifted"} & : & @{text attribute} \\
    @{attribute_def (HOL) "quot_respect"} & : & @{text attribute} \\
    @{attribute_def (HOL) "quot_preserve"} & : & @{text attribute} \\
  \end{matharray}

  The quotient package defines a new quotient type given a raw type
  and a partial equivalence relation. The package also historically
  includes automation for transporting definitions and theorems.
  But most of this automation was superseded by the Lifting and Transfer
  packages. The user should consider using these two new packages for
  lifting definitions and transporting theorems.

  @{rail \<open>
    @@{command (HOL) quotient_type} (spec)
    ;
    spec: @{syntax typespec} @{syntax mixfix}? '=' \<newline>
     @{syntax type} '/' ('partial' ':')? @{syntax term} \<newline>
     (@'morphisms' @{syntax name} @{syntax name})? (@'parametric' @{syntax thmref})?
  \<close>}

  @{rail \<open>
    @@{command (HOL) quotient_definition} constdecl? @{syntax thmdecl}? \<newline>
    @{syntax term} 'is' @{syntax term}
    ;
    constdecl: @{syntax name} ('::' @{syntax type})? @{syntax mixfix}?
  \<close>}

  @{rail \<open>
    @@{method (HOL) lifting} @{syntax thmrefs}?
    ;
    @@{method (HOL) lifting_setup} @{syntax thmrefs}?
  \<close>}

  \begin{description}

  \item @{command (HOL) "quotient_type"} defines a new quotient type @{text \<tau>}. The
  injection from a quotient type to a raw type is called @{text
  rep_\<tau>}, its inverse @{text abs_\<tau>} 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 @{text equivp}), unless the
  user specifies explicitly @{text partial} in which case the
  obligation is @{text part_equivp}.  A quotient defined with @{text
  partial} 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.

  \end{description}

  The most of the rest of the package was superseded by the Lifting and Transfer
  packages. The user should consider using these two new packages for
  lifting definitions and transporting theorems.

  \begin{description}

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

  \item @{command (HOL) "print_quotmapsQ3"} prints quotient map
  functions.

  \item @{command (HOL) "print_quotientsQ3"} prints quotients.

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

  \item @{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.

  \item @{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.

  \item @{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.

  \item @{method (HOL) "regularize"} applies the regularization
    heuristics to the current subgoal.

  \item @{method (HOL) "injection"} applies the injection heuristics
    to the current goal using the stored quotient respectfulness
    theorems.

  \item @{method (HOL) "cleaning"} applies the injection cleaning
    heuristics to the current subgoal using the stored quotient
    preservation theorems.

  \item @{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.

  \item @{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.

  \item @{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 "Quotient3 R Abs Rep \<Longrightarrow> Quotient3
    (rel_map R) (map Abs) (map Rep)"}. Quotient extension theorems
    are stored in a database and are used all the steps of lifting
    theorems.

  \end{description}
\<close>

section \<open>Definition by specification \label{sec:hol-specification}\<close>

text \<open>
  \begin{matharray}{rcl}
    @{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
  \end{matharray}

  @{rail \<open>
    @@{command (HOL) specification} '(' (decl +) ')' \<newline>
      (@{syntax thmdecl}? @{syntax prop} +)
    ;
    decl: (@{syntax name} ':')? @{syntax term} ('(' @'overloaded' ')')?
  \<close>}

  \begin{description}

  \item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a
  goal stating the existence of terms with the properties specified to
  hold for the constants given in @{text decls}.  After finishing the
  proof, the theory will be augmented with definitions for the given
  constants, as well as with theorems stating the properties for these
  constants.

  @{text decl} declares a constant to be defined by the
  specification given.  The definition for the constant @{text c} is
  bound to the name @{text c_def} unless a theorem name is given in
  the declaration.  Overloaded constants should be declared as such.

  \end{description}
\<close>

section \<open>Adhoc overloading of constants\<close>

text \<open>
  \begin{tabular}{rcll}
  @{command_def "adhoc_overloading"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
  @{command_def "no_adhoc_overloading"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
  @{attribute_def "show_variants"} & : & @{text "attribute"} & default @{text false} \\
  \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 "~~/src/HOL/ex/Adhoc_Overloading_Examples.thy"} and
  @{file "~~/src/HOL/Library/Monad_Syntax.thy"}.

  @{rail \<open>
    (@@{command adhoc_overloading} | @@{command no_adhoc_overloading})
      (@{syntax nameref} (@{syntax term} + ) + @'and')
  \<close>}

  \begin{description}

  \item @{command "adhoc_overloading"}~@{text "c v\<^sub>1 ... v\<^sub>n"}
  associates variants with an existing constant.

  \item @{command "no_adhoc_overloading"} is similar to
  @{command "adhoc_overloading"}, but removes the specified variants
  from the present context.

  \item @{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.

  \end{description}
\<close>

chapter \<open>Proof tools\<close>

section \<open>Adhoc tuples\<close>

text \<open>
  \begin{matharray}{rcl}
    @{attribute_def (HOL) split_format}@{text "\<^sup>*"} & : & @{text attribute} \\
  \end{matharray}

  @{rail \<open>
    @@{attribute (HOL) split_format} ('(' 'complete' ')')?
  \<close>}

  \begin{description}

  \item @{attribute (HOL) split_format}\ @{text "(complete)"} causes
  arguments in function applications to be represented canonically
  according to their tuple type structure.

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

  \end{description}
\<close>

section \<open>Transfer package\<close>

text \<open>
  \begin{matharray}{rcl}
    @{method_def (HOL) "transfer"} & : & @{text method} \\
    @{method_def (HOL) "transfer'"} & : & @{text method} \\
    @{method_def (HOL) "transfer_prover"} & : & @{text method} \\
    @{attribute_def (HOL) "Transfer.transferred"} & : & @{text attribute} \\
    @{attribute_def (HOL) "untransferred"} & : & @{text attribute} \\
    @{attribute_def (HOL) "transfer_rule"} & : & @{text attribute} \\
    @{attribute_def (HOL) "transfer_domain_rule"} & : & @{text attribute} \\
    @{attribute_def (HOL) "relator_eq"} & : & @{text attribute} \\
    @{attribute_def (HOL) "relator_domain"} & : & @{text attribute} \\
  \end{matharray}

  \begin{description}

  \item @{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.
    @{text "transfer fixing: x y z"}.

  \item @{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.

  \item @{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.

  \item @{attribute (HOL) "untransferred"} proves the same equivalent theorem
     as @{method (HOL) "transfer"} internally does.

  \item @{attribute (HOL) Transfer.transferred} works in the opposite
    direction than @{method (HOL) "transfer'"}. E.g., given the transfer
    relation @{text "ZN x n \<equiv> (x = int n)"}, corresponding transfer rules and the theorem
    @{text "\<forall>x::int \<in> {0..}. x < x + 1"}, the attribute would prove
    @{text "\<forall>n::nat. n < n + 1"}. The attribute is still in experimental
    phase of development.

  \item @{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:

    @{text "((A ===> B) ===> list_all2 A ===> list_all2 B) map map"}\\
    @{text "(cr_int ===> cr_int ===> cr_int) (\<lambda>(x,y) (u,v). (x+u, y+v)) plus"}

    Lemmas involving predicates on relations can also be registered
    using the same attribute. For example:

    @{text "bi_unique A \<Longrightarrow> (list_all2 A ===> op =) distinct distinct"}\\
    @{text "\<lbrakk>bi_unique A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (rel_prod A B)"}

    Preservation of predicates on relations (@{text "bi_unique, bi_total,
    right_unique, right_total, left_unique, left_total"}) with the respect to a relator
    is proved automatically if the involved type is BNF
    @{cite "isabelle-datatypes"} without dead variables.

  \item @{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 @{text "ZN x n \<equiv> (x = int n)"},
    one can register the following transfer domain rule:
    @{text "Domainp ZN = (\<lambda>x. x \<ge> 0)"}. The rules allow the package to produce
    more readable transferred goals, e.g., when quantifiers are transferred.

  \item @{attribute (HOL) relator_eq} attribute collects identity laws
    for relators of various type constructors, e.g. @{term "rel_set
    (op =) = (op =)"}. The @{method (HOL) transfer} method uses these
    lemmas to infer transfer rules for non-polymorphic constants on
    the fly. For examples see @{file
    "~~/src/HOL/Lifting_Set.thy"} or @{file "~~/src/HOL/Lifting.thy"}.
    This property is proved automatically if the involved type is BNF without dead variables.

  \item @{attribute_def (HOL) "relator_domain"} attribute collects rules
    describing domains of relators by predicators. E.g.,
    @{term "Domainp (rel_set T) = (\<lambda>A. Ball A (Domainp T))"}. This allows the package
    to lift transfer domain rules through type constructors. For examples see @{file
    "~~/src/HOL/Lifting_Set.thy"} or @{file "~~/src/HOL/Lifting.thy"}.
    This property is proved automatically if the involved type is BNF without dead variables.

  \end{description}

  Theoretical background can be found in @{cite "Huffman-Kuncar:2013:lifting_transfer"}.
\<close>

section \<open>Lifting package\<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 (Abs and Rep) with the raw term. It also proves an appropriate
  transfer rule for the 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"} & : & @{text "local_theory \<rightarrow> local_theory"}\\
    @{command_def (HOL) "lift_definition"} & : & @{text "local_theory \<rightarrow> proof(prove)"}\\
    @{command_def (HOL) "lifting_forget"} & : & @{text "local_theory \<rightarrow> local_theory"}\\
    @{command_def (HOL) "lifting_update"} & : & @{text "local_theory \<rightarrow> local_theory"}\\
    @{command_def (HOL) "print_quot_maps"} & : & @{text "context \<rightarrow>"}\\
    @{command_def (HOL) "print_quotients"} & : & @{text "context \<rightarrow>"}\\
    @{attribute_def (HOL) "quot_map"} & : & @{text attribute} \\
    @{attribute_def (HOL) "relator_eq_onp"} & : & @{text attribute} \\
    @{attribute_def (HOL) "relator_mono"} & : & @{text attribute} \\
    @{attribute_def (HOL) "relator_distr"} & : & @{text attribute} \\
    @{attribute_def (HOL) "quot_del"} & : & @{text attribute} \\
    @{attribute_def (HOL) "lifting_restore"} & : & @{text attribute} \\
  \end{matharray}

  @{rail \<open>
    @@{command (HOL) setup_lifting} ('(' 'no_code' ')')? \<newline>
      @{syntax thmref} @{syntax thmref}? (@'parametric' @{syntax thmref})?;
  \<close>}

  @{rail \<open>
    @@{command (HOL) lift_definition} @{syntax name} '::' @{syntax type}  @{syntax mixfix}? \<newline>
      'is' @{syntax term} (@'parametric' (@{syntax thmref}+))?;
  \<close>}

  @{rail \<open>
    @@{command (HOL) lifting_forget} @{syntax nameref};
  \<close>}

  @{rail \<open>
    @@{command (HOL) lifting_update} @{syntax nameref};
  \<close>}

  @{rail \<open>
    @@{attribute (HOL) lifting_restore} @{syntax thmref} (@{syntax thmref} @{syntax thmref})?;
  \<close>}

  \begin{description}

  \item @{command (HOL) "setup_lifting"} Sets up the Lifting package
    to work with a user-defined type.
    The command supports two modes. 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 "Quotient R Abs Rep T"}. 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 "reflp R"}, 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 R 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 @{text Quotient} theorem manually for
    new types, as special commands exist to automate the process.

    When a new subtype is defined by @{command (HOL) typedef}, @{command (HOL) "lift_definition"}
    can be used in its
    second mode, where only the type_definition theorem @{text "type_definition Rep Abs A"}
    is used as an argument of the command. The command internally proves the corresponding
    Quotient 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}.

    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.
    If the option @{text "no_code"} is specified, the Lifting package does not set up the code
    generator and as a consequence no code equations involving an abstract type are registered
    by @{command (HOL) "lift_definition"}.

  \item @{command (HOL) "lift_definition"} @{text "f :: \<tau>"} @{keyword (HOL) "is"} @{text t}
    Defines a new function @{text f} with an abstract type @{text \<tau>}
    in terms of a corresponding operation @{text t} on a
    representation type. More formally, if @{text "t :: \<sigma>"}, then
    the command builds a term @{text "F"} as a corresponding combination of abstraction
    and representation functions such that @{text "F :: \<sigma> \<Rightarrow> \<tau>" } and
    defines @{text f} is as @{text "f \<equiv> F t"}.
    The term @{text t} does not have to be necessarily a constant but it can be any term.

    The command opens a proof environment and the user must discharge
    a respectfulness proof obligation. For a type copy, i.e., a typedef with @{text
    UNIV}, the obligation is discharged automatically. The proof goal is
    presented in a user-friendly, readable form. A respectfulness
    theorem in the standard format @{text f.rsp} and a transfer rule
    @{text f.transfer} for the Transfer package are generated by the
    package.

    The user can specify a parametricity theorems for @{text t} after the keyword
    @{keyword "parametric"}, which allows the command
    to generate parametric transfer rules for @{text f}.

    For each constant defined through trivial quotients (type copies or
    subtypes) @{text f.rep_eq} is generated. The equation is a code certificate
    that defines @{text f} using the representation function.

    For each constant @{text f.abs_eq} is generated. The equation is unconditional
    for total quotients. The equation defines @{text f} using
    the abstraction function.

    Integration with [@{attribute code} abstract]: For subtypes (e.g.,
    corresponding to a datatype invariant, such as dlist), @{command
    (HOL) "lift_definition"} uses a code certificate theorem
    @{text f.rep_eq} as a code equation.

    Integration with [@{attribute code} equation]: For total quotients, @{command
    (HOL) "lift_definition"} uses @{text f.abs_eq} as a code equation.

  \item @{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 @{text "\<tau>"} by
    @{command_def (HOL) "lift_definition"}, the package defines a new bundle
    that is called @{text "\<tau>.lifting"}. 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"} @{text "\<tau>.lifting"}.

    The command @{command (HOL) "lifting_forget"} @{text "\<tau>.lifting"} deletes set-up of the Lifting
    package
    for @{text \<tau>} and deletes all the transfer rules that were introduced
    by @{command (HOL) "lift_definition"} using @{text \<tau>} 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"}).

  \item @{command (HOL) "print_quot_maps"} prints stored quotient map
    theorems.

  \item @{command (HOL) "print_quotients"} prints stored quotient
    theorems.

  \item @{attribute (HOL) quot_map} registers a quotient map
    theorem, a theorem showing how to "lift" quotients over type constructors.
    E.g., @{term "Quotient R Abs Rep T \<Longrightarrow>
    Quotient (rel_set R) (image Abs) (image Rep) (rel_set T)"}.
    For examples see @{file
    "~~/src/HOL/Lifting_Set.thy"} or @{file "~~/src/HOL/Lifting.thy"}.
    This property is proved automatically if the involved type is BNF without dead variables.

  \item @{attribute (HOL) relator_eq_onp} registers a theorem that
    shows that a relator applied to an equality restricted by a predicate @{term P} (i.e., @{term
    "eq_onp P"}) is equal
    to a predicator applied to the @{term P}. The combinator @{const eq_onp} is used for
    internal encoding of proper subtypes. Such theorems allows the package to hide @{text
    eq_onp} from a user in a user-readable form of a
    respectfulness theorem. For examples see @{file
    "~~/src/HOL/Lifting_Set.thy"} or @{file "~~/src/HOL/Lifting.thy"}.
    This property is proved automatically if the involved type is BNF without dead variables.

  \item @{attribute (HOL) "relator_mono"} registers a property describing a monotonicity of a relator.
    E.g., @{term "A \<le> B \<Longrightarrow> rel_set A \<le> rel_set B"}.
    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
    "~~/src/HOL/Lifting_Set.thy"} or @{file "~~/src/HOL/Lifting.thy"}.
    This property is proved automatically if the involved type is BNF without dead variables.

  \item @{attribute (HOL) "relator_distr"} registers a property describing a distributivity
    of the relation composition and a relator. E.g.,
    @{text "rel_set R \<circ>\<circ> rel_set S = rel_set (R \<circ>\<circ> S)"}.
    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
    "~~/src/HOL/Lifting_Set.thy"} or @{file "~~/src/HOL/Lifting.thy"}.
    This property is proved automatically if the involved type is BNF without dead variables.

  \item @{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.

  \item @{attribute (HOL) lifting_restore} @{text "Quotient_thm pcr_def pcr_cr_eq_thm"}
    registers the Quotient theorem @{text Quotient_thm} in the Lifting infrastructure
    and thus sets up lifting for an abstract type @{text \<tau>} (that is defined by @{text Quotient_thm}).
    Optional theorems @{text pcr_def} and @{text pcr_cr_eq_thm} can be specified to register
    the parametrized
    correspondence relation for @{text \<tau>}. E.g., for @{text "'a dlist"}, @{text pcr_def} is
    @{text "pcr_dlist A \<equiv> list_all2 A \<circ>\<circ> cr_dlist"} and @{text pcr_cr_eq_thm} is
    @{text "pcr_dlist op= = op="}.
    This attribute is rather used for low-level
    manipulation with set-up of the Lifting package because using of the bundle @{text \<tau>.lifting}
    together with the commands @{command (HOL) lifting_forget} and @{command (HOL) lifting_update} is
    preferred for normal usage.

  \item 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.

  \end{description}
\<close>

section \<open>Coercive subtyping\<close>

text \<open>
  \begin{matharray}{rcl}
    @{attribute_def (HOL) coercion} & : & @{text attribute} \\
    @{attribute_def (HOL) coercion_enabled} & : & @{text attribute} \\
    @{attribute_def (HOL) coercion_map} & : & @{text attribute} \\
  \end{matharray}

  Coercive subtyping allows the user to omit explicit type
  conversions, also called \emph{coercions}.  Type inference will add
  them as necessary when parsing a term. See
  @{cite "traytel-berghofer-nipkow-2011"} for details.

  @{rail \<open>
    @@{attribute (HOL) coercion} (@{syntax term})?
    ;
    @@{attribute (HOL) coercion_map} (@{syntax term})?
  \<close>}

  \begin{description}

  \item @{attribute (HOL) "coercion"}~@{text "f"} registers a new
  coercion function @{text "f :: \<sigma>\<^sub>1 \<Rightarrow> \<sigma>\<^sub>2"} where @{text "\<sigma>\<^sub>1"} and
  @{text "\<sigma>\<^sub>2"} 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.

  \item @{attribute (HOL) "coercion_map"}~@{text "map"} registers a
  new map function to lift coercions through type constructors. The
  function @{text "map"} must conform to the following type pattern

  \begin{matharray}{lll}
    @{text "map"} & @{text "::"} &
      @{text "f\<^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"} \\
  \end{matharray}

  where @{text "t"} is a type constructor and @{text "f\<^sub>i"} is of type
  @{text "\<alpha>\<^sub>i \<Rightarrow> \<beta>\<^sub>i"} or @{text "\<beta>\<^sub>i \<Rightarrow> \<alpha>\<^sub>i"}.  Registering a map function
  overwrites any existing map function for this particular type
  constructor.

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

  \end{description}
\<close>

section \<open>Arithmetic proof support\<close>

text \<open>
  \begin{matharray}{rcl}
    @{method_def (HOL) arith} & : & @{text method} \\
    @{attribute_def (HOL) arith} & : & @{text attribute} \\
    @{attribute_def (HOL) arith_split} & : & @{text attribute} \\
  \end{matharray}

  \begin{description}

  \item @{method (HOL) arith} decides linear arithmetic problems (on
  types @{text nat}, @{text int}, @{text real}).  Any current facts
  are inserted into the goal before running the procedure.

  \item @{attribute (HOL) arith} declares facts that are supplied to
  the arithmetic provers implicitly.

  \item @{attribute (HOL) arith_split} attribute declares case split
  rules to be expanded before @{method (HOL) arith} is invoked.

  \end{description}

  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} & : & @{text method} \\
  \end{matharray}

  @{rail \<open>
    @@{method (HOL) iprover} (@{syntax rulemod} *)
  \<close>}

  \begin{description}

  \item @{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
  ``@{text "!"}'' indicator refers to ``safe'' rules, which may be
  applied aggressively (without considering back-tracking later).
  Rules declared with ``@{text "?"}'' are ignored in proof search (the
  single-step @{method (Pure) rule} method still observes these).  An
  explicit weight annotation may be given as well; otherwise the
  number of rule premises will be taken into account here.

  \end{description}
\<close>

section \<open>Model Elimination and Resolution\<close>

text \<open>
  \begin{matharray}{rcl}
    @{method_def (HOL) "meson"} & : & @{text method} \\
    @{method_def (HOL) "metis"} & : & @{text method} \\
  \end{matharray}

  @{rail \<open>
    @@{method (HOL) meson} @{syntax thmrefs}?
    ;
    @@{method (HOL) metis}
      ('(' ('partial_types' | 'full_types' | 'no_types' | @{syntax name}) ')')?
      @{syntax thmrefs}?
  \<close>}

  \begin{description}

  \item @{method (HOL) meson} implements Loveland's model elimination
  procedure @{cite "loveland-78"}.  See @{file
  "~~/src/HOL/ex/Meson_Test.thy"} for examples.

  \item @{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 @{file "~~/src/HOL/Metis_Examples"} contains several small
  theories developed to a large extent using @{method (HOL) metis}.

  \end{description}
\<close>

section \<open>Algebraic reasoning via Gr\"obner bases\<close>

text \<open>
  \begin{matharray}{rcl}
    @{method_def (HOL) "algebra"} & : & @{text method} \\
    @{attribute_def (HOL) algebra} & : & @{text attribute} \\
  \end{matharray}

  @{rail \<open>
    @@{method (HOL) algebra}
      ('add' ':' @{syntax thmrefs})?
      ('del' ':' @{syntax thmrefs})?
    ;
    @@{attribute (HOL) algebra} (() | 'add' | 'del')
  \<close>}

  \begin{description}

  \item @{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:

  \begin{enumerate}

  \item 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 @{text "p = 0"} or inequations
  @{text "q \<noteq> 0"} anywhere within a universal problem statement.

  \item 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 @{text "e\<^sub>1, \<dots>, e\<^sub>n"} and the @{text "p\<^sub>i\<^sub>j"} are multivariate
  polynomials only in the variables mentioned as arguments.

  \end{enumerate}

  The proof method is preceded by a simplification step, which may be
  modified by using the form @{text "(algebra add: ths\<^sub>1 del: ths\<^sub>2)"}.
  This acts like declarations for the Simplifier
  (\secref{sec:simplifier}) on a private simpset for this tool.

  \item @{attribute algebra} (as attribute) manages the default
  collection of pre-simplification rules of the above proof method.

  \end{description}
\<close>

subsubsection \<open>Example\<close>

text \<open>The subsequent example is from geometry: collinearity is
  invariant by rotation.\<close>

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)

text \<open>
 See also @{file "~~/src/HOL/ex/Groebner_Examples.thy"}.
\<close>

section \<open>Coherent Logic\<close>

text \<open>
  \begin{matharray}{rcl}
    @{method_def (HOL) "coherent"} & : & @{text method} \\
  \end{matharray}

  @{rail \<open>
    @@{method (HOL) coherent} @{syntax thmrefs}?
  \<close>}

  \begin{description}

  \item @{method (HOL) coherent} solves problems of \emph{Coherent
  Logic} @{cite "Bezem-Coquand:2005"}, which covers applications in
  confluence theory, lattice theory and projective geometry.  See
  @{file "~~/src/HOL/ex/Coherent.thy"} for some examples.

  \end{description}
\<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"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
    @{command_def (HOL) "try"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
    @{command_def (HOL) "try0"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
    @{command_def (HOL) "sledgehammer"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
    @{command_def (HOL) "sledgehammer_params"} & : & @{text "theory \<rightarrow> theory"}
  \end{matharray}

  @{rail \<open>
    @@{command (HOL) try}
    ;

    @@{command (HOL) try0} ( ( ( 'simp' | 'intro' | 'elim' | 'dest' ) ':' @{syntax thmrefs} ) + ) ?
      @{syntax nat}?
    ;

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

    @@{command (HOL) sledgehammer_params} ( ( '[' args ']' ) ? )
    ;
    args: ( @{syntax name} '=' value + ',' )
    ;
    facts: '(' ( ( ( ( 'add' | 'del' ) ':' ) ? @{syntax thmrefs} ) + ) ? ')'
  \<close>} % FIXME check args "value"

  \begin{description}

  \item @{command (HOL) "solve_direct"} checks whether the current
  subgoals can be solved directly by an existing theorem. Duplicate
  lemmas can be detected in this way.

  \item @{command (HOL) "try0"} attempts to prove a subgoal
  using a combination of standard proof methods (@{method auto},
  @{method simp}, @{method blast}, etc.).  Additional facts supplied
  via @{text "simp:"}, @{text "intro:"}, @{text "elim:"}, and @{text
  "dest:"} are passed to the appropriate proof methods.

  \item @{command (HOL) "try"} attempts to prove or disprove a subgoal
  using a combination of provers and disprovers (@{command (HOL)
  "solve_direct"}, @{command (HOL) "quickcheck"}, @{command (HOL)
  "try0"}, @{command (HOL) "sledgehammer"}, @{command (HOL)
  "nitpick"}).

  \item @{command (HOL) "sledgehammer"} attempts to prove a subgoal
  using external automatic provers (resolution provers and SMT
  solvers). See the Sledgehammer manual @{cite "isabelle-sledgehammer"}
  for details.

  \item @{command (HOL) "sledgehammer_params"} changes @{command (HOL)
  "sledgehammer"} configuration options persistently.

  \end{description}
\<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"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
    @{command_def (HOL) "values"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
    @{command_def (HOL) "quickcheck"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
    @{command_def (HOL) "nitpick"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
    @{command_def (HOL) "quickcheck_params"} & : & @{text "theory \<rightarrow> theory"} \\
    @{command_def (HOL) "nitpick_params"} & : & @{text "theory \<rightarrow> theory"} \\
    @{command_def (HOL) "quickcheck_generator"} & : & @{text "theory \<rightarrow> theory"} \\
    @{command_def (HOL) "find_unused_assms"} & : & @{text "context \<rightarrow>"}
  \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 nameref} \<newline>
      'operations:' ( @{syntax term} +)
    ;

    @@{command (HOL) find_unused_assms} @{syntax name}?
    ;
    modes: '(' (@{syntax name} +) ')'
    ;
    args: ( @{syntax name} '=' value + ',' )
  \<close>} % FIXME check "value"

  \begin{description}

  \item @{command (HOL) "value"}~@{text t} evaluates and prints a
  term; optionally @{text modes} can be specified, which are appended
  to the current print mode; see \secref{sec:print-modes}.
  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 @{text simp} for fully symbolic evaluation
  using the simplifier, @{text nbe} for \emph{normalization by
  evaluation} and \emph{code} for code generation in SML.

  \item @{command (HOL) "values"}~@{text t} enumerates a set
  comprehension by evaluation and prints its values up to the given
  number of solutions; optionally @{text modes} can be specified,
  which are appended to the current print mode; see
  \secref{sec:print-modes}.

  \item @{command (HOL) "quickcheck"} tests the current goal for
  counterexamples using a series of assignments for its free
  variables; by default the first subgoal is tested, an other can be
  selected explicitly using an optional goal index.  Assignments can
  be chosen exhausting the search space 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:

    \begin{description}

    \item[@{text tester}] specifies which testing approach to apply.
    There are three testers, @{text exhaustive}, @{text random}, and
    @{text narrowing}.  An unknown configuration option is treated as
    an argument to tester, making @{text "tester ="} optional.  When
    multiple testers are given, these are applied in parallel.  If no
    tester is specified, quickcheck uses the testers that are set
    active, i.e., configurations @{attribute
    quickcheck_exhaustive_active}, @{attribute
    quickcheck_random_active}, @{attribute
    quickcheck_narrowing_active} are set to true.

    \item[@{text size}] specifies the maximum size of the search space
    for assignment values.

    \item[@{text genuine_only}] sets quickcheck only to return genuine
    counterexample, but not potentially spurious counterexamples due
    to underspecified functions.

    \item[@{text abort_potential}] 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 @{text genuine_only} option
    must be set to false.

    \item[@{text eval}] 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.

    \item[@{text iterations}] sets how many sets of assignments are
    generated for each particular size.

    \item[@{text no_assms}] specifies whether assumptions in
    structured proofs should be ignored.

    \item[@{text locale}] 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
    @{text interpret} and @{text expand}. 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.

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

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

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

    \item[@{text use_subtype}] if set quickcheck automatically lifts
    conjectures to registered subtypes if possible, and tests the
    lifted conjecture.

    \item[@{text quiet}] if set quickcheck does not output anything
    while testing.

    \item[@{text verbose}] if set quickcheck informs about the current
    size and cardinality while testing.

    \item[@{text expect}] can be used to check if the user's
    expectation was met (@{text no_expectation}, @{text
    no_counterexample}, or @{text counterexample}).

    \end{description}

  These option can be given within square brackets.

  Using the following type classes, the testers generate values and convert
  them back into Isabelle terms for displaying counterexamples.
    \begin{description}
    \item[@{text exhaustive}] The parameters of the type classes @{class exhaustive}
      and @{class full_exhaustive} implement the testing. They take a
      testing function as a parameter, which takes a value of type @{typ "'a"}
      and optionally produces a counterexample, and a size parameter for the test values.
      In @{class full_exhaustive}, the testing function parameter additionally
      expects a lazy term reconstruction in the type @{typ Code_Evaluation.term}
      of the tested value.

      The canonical implementation for @{text exhaustive} testers calls the given
      testing function on all values up to the given size and stops as soon
      as a counterexample is found.

    \item[@{text random}] The operation @{const Quickcheck_Random.random}
      of the type class @{class random} generates a pseudo-random
      value of the given size and a lazy term reconstruction of the value
      in the type @{typ Code_Evaluation.term}. A pseudo-randomness generator
      is defined in theory @{theory Random}.

    \item[@{text narrowing}] implements Haskell's Lazy Smallcheck @{cite "runciman-naylor-lindblad"}
      using the type classes @{class narrowing} and @{class partial_term_of}.
      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 narrowing}.
      Narrowing views every value as a sum-of-products which is expressed using the operations
      @{const Quickcheck_Narrowing.cons} (embedding a value),
      @{const Quickcheck_Narrowing.apply} (product) and @{const Quickcheck_Narrowing.sum} (sum).
      The refinement should enable further evaluation of the goal.

      For example, @{const narrowing} for the list type @{typ "'a :: narrowing list"}
      can be recursively defined as
      @{term "Quickcheck_Narrowing.sum (Quickcheck_Narrowing.cons [])
                (Quickcheck_Narrowing.apply
                  (Quickcheck_Narrowing.apply
                    (Quickcheck_Narrowing.cons (op #))
                    narrowing)
                  narrowing)"}.
      If a symbolic variable of type @{typ "_ list"} is evaluated, it is replaced by (i)~the empty
      list @{term "[]"} and (ii)~by a non-empty list whose head and tail can then be recursively
      refined if needed.

      To reconstruct counterexamples, the operation @{const partial_term_of} transforms
      @{text narrowing}'s deep representation of terms to the type @{typ Code_Evaluation.term}.
      The deep representation models symbolic variables as
      @{const Quickcheck_Narrowing.Narrowing_variable}, which are normally converted to
      @{const Code_Evaluation.Free}, and refined values as
      @{term "Quickcheck_Narrowing.Narrowing_constructor i args"}, where @{term "i :: integer"}
      denotes the index in the sum of refinements. In the above example for lists,
      @{term "0"} corresponds to @{term "[]"} and @{term "1"}
      to @{term "op #"}.

      The command @{command (HOL) "code_datatype"} sets up @{const partial_term_of}
      such that the @{term "i"}-th refinement is interpreted as the @{term "i"}-th constructor,
      but it does not ensures consistency with @{const narrowing}.
    \end{description}

  \item @{command (HOL) "quickcheck_params"} changes @{command (HOL)
  "quickcheck"} configuration options persistently.

  \item @{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.

  \item @{command (HOL) "nitpick"} tests the current goal for
  counterexamples using a reduction to first-order relational
  logic. See the Nitpick manual @{cite "isabelle-nitpick"} for details.

  \item @{command (HOL) "nitpick_params"} changes @{command (HOL)
  "nitpick"} configuration options persistently.

  \item @{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.

  \end{description}
\<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}@{text "\<^sup>*"} & : & @{text method} \\
    @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\
    @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\
    @{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
  \end{matharray}

  @{rail \<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}+) (@'for' (@{syntax name}+))?
    ;
    @@{command (HOL) inductive_cases} (@{syntax thmdecl}? (@{syntax prop}+) + @'and')
    ;
    rule: 'rule' ':' @{syntax thmref}
  \<close>}

  \begin{description}

  \item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit
  to reason about inductive types.  Rules are selected according to
  the declarations by the @{attribute cases} and @{attribute induct}
  attributes, cf.\ \secref{sec:cases-induct}.  The @{command (HOL)
  datatype} package already takes care of this.

  These unstructured tactics feature both goal addressing and dynamic
  instantiation.  Note that named rule cases are \emph{not} provided
  as would be by the proper @{method cases} and @{method induct} proof
  methods (see \secref{sec:cases-induct}).  Unlike the @{method
  induct} method, @{method induct_tac} does not handle structured rule
  statements, only the compact object-logic conclusion of the subgoal
  being addressed.

  \item @{method (HOL) ind_cases} and @{command (HOL)
  "inductive_cases"} provide an interface to the internal @{ML_text
  mk_cases} operation.  Rules are simplified in an unrestricted
  forward manner.

  While @{method (HOL) ind_cases} is a proof method to apply the
  result immediately as elimination rules, @{command (HOL)
  "inductive_cases"} provides case split theorems at the theory level
  for later use.  The @{keyword "for"} argument of the @{method (HOL)
  ind_cases} method allows to specify a list of variables that should
  be generalized before applying the resulting rule.

  \end{description}
\<close>

chapter \<open>Executable code\<close>

text \<open>For validation purposes, it is often useful to \emph{execute}
  specifications.  In principle, execution could be simulated by
  Isabelle's inference kernel, i.e. by a combination of resolution and
  simplification.  Unfortunately, this approach is rather inefficient.
  A more efficient way of executing specifications is to translate
  them into a functional programming language such as ML.

  Isabelle provides 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{selection} of code theorems,
  \emph{translation} into an abstract executable view and
  \emph{serialization} to a specific \emph{target language}.
  Inductive specifications can be executed using the predicate
  compiler which operates within HOL.  See @{cite "isabelle-codegen"} for
  an introduction.

  \begin{matharray}{rcl}
    @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
    @{attribute_def (HOL) code} & : & @{text attribute} \\
    @{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\
    @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
    @{attribute_def (HOL) code_unfold} & : & @{text attribute} \\
    @{attribute_def (HOL) code_post} & : & @{text attribute} \\
    @{attribute_def (HOL) code_abbrev} & : & @{text attribute} \\
    @{command_def (HOL) "print_codeproc"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
    @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
    @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
    @{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\
    @{command_def (HOL) "code_printing"} & : & @{text "theory \<rightarrow> theory"} \\
    @{command_def (HOL) "code_identifier"} & : & @{text "theory \<rightarrow> theory"} \\
    @{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\
    @{command_def (HOL) "code_reflect"} & : & @{text "theory \<rightarrow> theory"} \\
    @{command_def (HOL) "code_pred"} & : & @{text "theory \<rightarrow> proof(prove)"}
  \end{matharray}

  @{rail \<open>
    @@{command (HOL) export_code} ( @'open' ) ? ( constexpr + ) \<newline>
       ( ( @'in' target ( @'module_name' @{syntax string} ) ? \<newline>
        ( @'file' @{syntax string} ) ? ( '(' args ')' ) ?) + ) ?
    ;

    const: @{syntax term}
    ;

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

    typeconstructor: @{syntax nameref}
    ;

    class: @{syntax nameref}
    ;

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

    @@{attribute (HOL) code} ( 'del' | 'equation' | 'abstype' | 'abstract'
      | 'drop:' ( const + ) | 'abort:' ( const + ) )?
    ;

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

    @@{attribute (HOL) code_unfold} ( 'del' ) ?
    ;

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

    @@{attribute (HOL) code_abbrev}
    ;

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

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

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

    symbol_const: ( @'constant' const )
    ;

    symbol_typeconstructor: ( @'type_constructor' typeconstructor )
    ;

    symbol_class: ( @'type_class' class )
    ;

    symbol_class_relation: ( @'class_relation' class ( '<' | '\<subseteq>' ) class )
    ;

    symbol_class_instance: ( @'class_instance' typeconstructor @'::' 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_typeconstructor: symbol_typeconstructor ( '\<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 ')' ( @{syntax string} ( @'attach' ( const + ) ) ? ) ? + @'and' )
    ;

    @@{command (HOL) code_printing} ( ( printing_const | printing_typeconstructor
      | printing_class | printing_class_relation | printing_class_instance
      | printing_module ) + '|' )
    ;

    @@{command (HOL) code_identifier} ( ( symbol_const | symbol_typeconstructor
      | 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>}

  \begin{description}

  \item @{command (HOL) "export_code"} generates code for a given list
  of constants in the specified target language(s).  If no
  serialization instruction is given, only abstract code is generated
  internally.

  Constants may be specified by giving them literally, referring to
  all executable constants within a certain theory by giving @{text
  "name._"}, or referring to \emph{all} executable constants currently
  available by giving @{text "_"}.

  By default, exported identifiers are minimized per module.  This
  can be suppressed by prepending @{keyword "open"} before the list
  of contants.

  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{all} code is
  placed in this module.

  For \emph{SML}, \emph{OCaml} and \emph{Scala} the file specification
  refers to a single file; for \emph{Haskell}, it refers to a whole
  directory, where code is generated in multiple files reflecting the
  module hierarchy.  Omitting the file specification denotes standard
  output.

  Serializers take an optional list of arguments in parentheses.
  For \emph{Haskell} a module name prefix may be given using the
  ``@{text "root:"}'' argument; ``@{text string_classes}'' adds a
  ``@{verbatim "deriving (Read, Show)"}'' clause to each appropriate
  datatype declaration.

  \item @{attribute (HOL) code} declare code equations for code
  generation.  Variant @{text "code equation"} declares a conventional
  equation as code equation.  Variants @{text "code abstype"} and
  @{text "code abstract"} declare abstract datatype certificates or
  code equations on abstract datatype representations respectively.
  Vanilla @{text "code"} falls back to @{text "code equation"}
  or @{text "code abstype"} depending on the syntactic shape
  of the underlying equation.  Variant @{text "code del"}
  deselects a code equation for code generation.

  Variants @{text "code drop:"} and @{text "code abort:"} take
  a list of constant as arguments and drop all code equations declared
  for them.  In the case of {text abort}, these constants then are
  are not required to have a definition by means of code equations;
  if needed these are implemented by program abort (exception) instead.

  Usually packages introducing code equations provide a reasonable
  default setup for selection.

  \item @{command (HOL) "code_datatype"} specifies a constructor set
  for a logical type.

  \item @{command (HOL) "print_codesetup"} gives an overview on
  selected code equations and code generator datatypes.

  \item @{attribute (HOL) code_unfold} declares (or with option
  ``@{text "del"}'' removes) theorems which during preprocessing
  are applied as rewrite rules to any code equation or evaluation
  input.

  \item @{attribute (HOL) code_post} declares (or with option ``@{text
  "del"}'' removes) theorems which are applied as rewrite rules to any
  result of an evaluation.

  \item @{attribute (HOL) code_abbrev} declares (or with option ``@{text
  "del"}'' 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.

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

  \item @{command (HOL) "code_thms"} prints a list of theorems
  representing the corresponding program containing all given
  constants after preprocessing.

  \item @{command (HOL) "code_deps"} visualizes dependencies of
  theorems representing the corresponding program containing all given
  constants after preprocessing.

  \item @{command (HOL) "code_reserved"} declares a list of names as
  reserved for a given target, preventing it to be shadowed by any
  generated code.

  \item @{command (HOL) "code_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.

  \item @{command (HOL) "code_monad"} provides an auxiliary mechanism
  to generate monadic code for Haskell.

  \item @{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{Warning:} 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.

  \item @{command (HOL) "code_reflect"} without a ``@{text "file"}''
  argument compiles code into the system runtime environment and
  modifies the code generator setup that future invocations of system
  runtime code generation referring to one of the ``@{text
  "datatypes"}'' or ``@{text "functions"}'' entities use these
  precompiled entities.  With a ``@{text "file"}'' argument, the
  corresponding code is generated into that specified file without
  modifying the code generator setup.

  \item @{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.

  \end{description}
\<close>

end