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-
-This section describes advanced features of inductive definitions. 
-The premises of introduction rules may contain universal quantifiers and
-monotonic functions.  Theorems may be proved by rule inversion.
-
-\subsection{Universal quantifiers in introduction rules}
-\label{sec:gterm-datatype}
-
-As a running example, this section develops the theory of \textbf{ground
-terms}: terms constructed from constant and function 
-symbols but not variables. To simplify matters further, we regard a
-constant as a function applied to the null argument  list.  Let us declare a
-datatype \isa{gterm} for the type of ground  terms. It is a type constructor
-whose argument is a type of  function symbols. 
-\begin{isabelle}
-\isacommand{datatype}\ 'f\ gterm\ =\ Apply\ 'f\ "'f\ gterm\ list"
-\end{isabelle}
-To try it out, we declare a datatype of some integer operations: 
-integer constants, the unary minus operator and the addition 
-operator. 
-\begin{isabelle}
-\isacommand{datatype}\ integer_op\ =\ Number\ int\ |\ UnaryMinus\ |\ Plus
-\end{isabelle}
-Now the type \isa{integer\_op gterm} denotes the ground 
-terms built over those symbols.
-
-The type constructor \texttt{gterm} can be generalized to a function 
-over sets.  It returns 
-the set of ground terms that can be formed over a set \isa{F} of function symbols. For
-example,  we could consider the set of ground terms formed from the finite 
-set {\isa{\{Number 2, UnaryMinus, Plus\}}}.
-
-This concept is inductive. If we have a list \isa{args} of ground terms 
-over~\isa{F} and a function symbol \isa{f} in \isa{F}, then we 
-can apply \isa{f} to  \isa{args} to obtain another ground term. 
-The only difficulty is that the argument list may be of any length. Hitherto, 
-each rule in an inductive definition referred to the inductively 
-defined set a fixed number of times, typically once or twice. 
-A universal quantifier in the premise of the introduction rule 
-expresses that every element of \isa{args} belongs
-to our inductively defined set: is a ground term 
-over~\isa{F}.  The function {\isa{set}} denotes the set of elements in a given 
-list. 
-\begin{isabelle}
-\isacommand{consts}\ gterms\ ::\ "'f\ set\ \isasymRightarrow \ 'f\ gterm\ set"\isanewline
-\isacommand{inductive}\ "gterms\ F"\isanewline
-\isakeyword{intros}\isanewline
-step[intro!]:\ "\isasymlbrakk \isasymforall t\ \isasymin \ set\ args.\ t\ \isasymin \ gterms\ F;\ \ f\ \isasymin \ F\isasymrbrakk \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \isasymLongrightarrow \ (Apply\ f\ args)\ \isasymin \ gterms\
-F"
-\end{isabelle}
-
-To demonstrate a proof from this definition, let us 
-show that the function \isa{gterms}
-is \textbf{monotonic}.  We shall need this concept shortly.
-\begin{isabelle}
-\isacommand{lemma}\ "F\isasymsubseteq G\ \isasymLongrightarrow \ gterms\ F\ \isasymsubseteq \ gterms\ G"\isanewline
-\isacommand{apply}\ clarify\isanewline
-\isacommand{apply}\ (erule\ gterms.induct)\isanewline
-\isacommand{apply}\ blast\isanewline
-\isacommand{done}
-\end{isabelle}
-Intuitively, this theorem says that
-enlarging the set of function symbols enlarges the set of ground 
-terms. The proof is a trivial rule induction.
-First we use the \isa{clarify} method to assume the existence of an element of
-\isa{terms~F}.  (We could have used \isa{intro subsetI}.)  We then
-apply rule induction. Here is the resulting subgoal: 
-\begin{isabelle}
-1.\ \isasymAnd x\ f\ args.\isanewline
-\ \ \ \ \ \ \isasymlbrakk F\ \isasymsubseteq \ G;\ \isasymforall t\isasymin set\ args.\ t\ \isasymin \ gterms\ F\ \isasymand \ t\ \isasymin \ gterms\ G;\ f\ \isasymin \ F\isasymrbrakk \isanewline
-\ \ \ \ \ \ \isasymLongrightarrow \ Apply\ f\ args\ \isasymin \ gterms\ G%
-\end{isabelle}
-%
-The assumptions state that \isa{f} belongs 
-to~\isa{F}, which is included in~\isa{G}, and that every element of the list \isa{args} is
-a ground term over~\isa{G}.  The \isa{blast} method finds this chain of reasoning easily.  
-
-\textit{Remark}: why do we call this function \isa{gterms} instead 
-of {\isa{gterm}}? Isabelle maintains separate name spaces for types 
-and constants, so there is no danger of confusion. However, name 
-clashes could arise in the theorems that Isabelle generates. 
-Our choice of names keeps {\isa{gterms.induct}} separate from {\isa{gterm.induct}}.
-
-\subsection{Rule inversion}\label{sec:rule-inversion}
-
-Case analysis on an inductive definition is called \textbf{rule inversion}. 
-It is frequently used in proofs about operational semantics.  It can be
-highly effective when it is applied automatically.  Let us look at how rule
-inversion is done in Isabelle.
-
-Recall that \isa{even} is the minimal set closed under these two rules:
-\begin{isabelle}
-0\ \isasymin \ even\isanewline
-n\ \isasymin \ even\ \isasymLongrightarrow \ (Suc\ (Suc\ n))\ \isasymin
-\ even
-\end{isabelle}
-Minimality means that \isa{even} contains only the elements that these
-rules force it to contain.  If we are told that \isa{a}
-belongs to
-\isa{even} then there are only two possibilities.  Either \isa{a} is \isa{0}
-or else \isa{a} has the form \isa{Suc(Suc~n)}, for an arbitrary \isa{n}
-that belongs to
-\isa{even}.  That is the gist of the \isa{cases} rule, which Isabelle proves
-for us when it accepts an inductive definition:
-\begin{isabelle}
-\isasymlbrakk a\ \isasymin \ even;\isanewline
-\ a\ =\ 0\ \isasymLongrightarrow \ P;\isanewline
-\ \isasymAnd n.\ \isasymlbrakk a\ =\ Suc(Suc\ n);\ n\ \isasymin \
-even\isasymrbrakk \ \isasymLongrightarrow \ P\isasymrbrakk \
-\isasymLongrightarrow \ P
-\rulename{even.cases}
-\end{isabelle}
-
-This general rule is less useful than instances of it for
-specific patterns.  For example, if \isa{a} has the form
-\isa{Suc(Suc~n)} then the first case becomes irrelevant, while the second
-case tells us that \isa{n} belongs to \isa{even}.  Isabelle will generate
-this instance for us:
-\begin{isabelle}
-\isacommand{inductive\_cases}\ Suc_Suc_cases\ [elim!]:
-\ "Suc(Suc\ n)\ \isasymin \ even"
-\end{isabelle}
-The \isacommand{inductive\_cases} command generates an instance of the
-\isa{cases} rule for the supplied pattern and gives it the supplied name:
-%
-\begin{isabelle}
-\isasymlbrakk Suc\ (Suc\ n)\ \isasymin \ even;\ n\ \isasymin \ even\
-\isasymLongrightarrow \ P\isasymrbrakk \ \isasymLongrightarrow \ P%
-\rulename{Suc_Suc_cases}
-\end{isabelle}
-%
-Applying this as an elimination rule yields one case where \isa{even.cases}
-would yield two.  Rule inversion works well when the conclusions of the
-introduction rules involve datatype constructors like \isa{Suc} and \isa{\#}
-(list `cons'); freeness reasoning discards all but one or two cases.
-
-In the \isacommand{inductive\_cases} command we supplied an
-attribute, \isa{elim!}, indicating that this elimination rule can be applied
-aggressively.  The original
-\isa{cases} rule would loop if used in that manner because the
-pattern~\isa{a} matches everything.
-
-The rule \isa{Suc_Suc_cases} is equivalent to the following implication:
-\begin{isabelle}
-Suc (Suc\ n)\ \isasymin \ even\ \isasymLongrightarrow \ n\ \isasymin \
-even
-\end{isabelle}
-%
-In \S\ref{sec:gen-rule-induction} we devoted some effort to proving precisely
-this result.  Yet we could have obtained it by a one-line declaration. 
-This example also justifies the terminology \textbf{rule inversion}: the new
-rule inverts the introduction rule \isa{even.step}.
-
-For one-off applications of rule inversion, use the \isa{ind_cases} method. 
-Here is an example:
-\begin{isabelle}
-\isacommand{apply}\ (ind_cases\ "Suc(Suc\ n)\ \isasymin \ even")
-\end{isabelle}
-The specified instance of the \isa{cases} rule is generated, applied, and
-discarded.
-
-\medskip
-Let us try rule inversion on our ground terms example:
-\begin{isabelle}
-\isacommand{inductive\_cases}\ gterm_Apply_elim\ [elim!]:\ "Apply\ f\ args\
-\isasymin\ gterms\ F"
-\end{isabelle}
-%
-Here is the result.  No cases are discarded (there was only one to begin
-with) but the rule applies specifically to the pattern \isa{Apply\ f\ args}.
-It can be applied repeatedly as an elimination rule without looping, so we
-have given the
-\isa{elim!}\ attribute. 
-\begin{isabelle}
-\isasymlbrakk Apply\ f\ args\ \isasymin \ gterms\ F;\isanewline
-\ \isasymlbrakk
-\isasymforall t\isasymin set\ args.\ t\ \isasymin \ gterms\ F;\ f\ \isasymin
-\ F\isasymrbrakk \ \isasymLongrightarrow \ P\isasymrbrakk\isanewline
-\isasymLongrightarrow \ P%
-\rulename{gterm_Apply_elim}
-\end{isabelle}
-
-This rule replaces an assumption about \isa{Apply\ f\ args} by 
-assumptions about \isa{f} and~\isa{args}.  Here is a proof in which this
-inference is essential.  We show that if \isa{t} is a ground term over both
-of the sets
-\isa{F} and~\isa{G} then it is also a ground term over their intersection,
-\isa{F\isasyminter G}.
-\begin{isabelle}
-\isacommand{lemma}\ gterms_IntI\ [rule_format]:\isanewline
-\ \ \ \ \ "t\ \isasymin \ gterms\ F\ \isasymLongrightarrow \ t\ \isasymin \ gterms\ G\ \isasymlongrightarrow \ t\ \isasymin \ gterms\ (F\isasyminter G)"\isanewline
-\isacommand{apply}\ (erule\ gterms.induct)\isanewline
-\isacommand{apply}\ blast\isanewline
-\isacommand{done}
-\end{isabelle}
-%
-The proof begins with rule induction over the definition of
-\isa{gterms}, which leaves a single subgoal:  
-\begin{isabelle}
-1.\ \isasymAnd args\ f.\isanewline
-\ \ \ \ \ \ \isasymlbrakk \isasymforall t\isasymin set\ args.\ t\ \isasymin \ gterms\ F\ \isasymand\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (t\ \isasymin \ gterms\ G\ \isasymlongrightarrow \ t\ \isasymin \ gterms\ (F\ \isasyminter \ G));\isanewline
-\ \ \ \ \ \ \ f\ \isasymin \ F\isasymrbrakk \isanewline
-\ \ \ \ \ \ \isasymLongrightarrow \ Apply\ f\ args\ \isasymin \ gterms\ G\ \isasymlongrightarrow \ Apply\ f\ args\ \isasymin \ gterms\ (F\ \isasyminter \ G)
-\end{isabelle}
-%
-The induction hypothesis states that every element of \isa{args} belongs to 
-\isa{gterms\ (F\ \isasyminter \ G)} --- provided it already belongs to
-\isa{gterms\ G}.  How do we meet that condition?  
-
-By assuming (as we may) the formula \isa{Apply\ f\ args\ \isasymin \ gterms\
-G}.  Rule inversion, in the form of \isa{gterm_Apply_elim}, infers that every
-element of \isa{args} belongs to 
-\isa{gterms~G}.  It also yields \isa{f\ \isasymin \ G}, which will allow us
-to conclude \isa{f\ \isasymin \ F\ \isasyminter \ G}.  All of this reasoning
-is done by \isa{blast}.
-
-\medskip
-
-To summarize, every inductive definition produces a \isa{cases} rule.  The
-\isacommand{inductive\_cases} command stores an instance of the \isa{cases}
-rule for a given pattern.  Within a proof, the \isa{ind_cases} method
-applies an instance of the \isa{cases}
-rule.
-
-
-\subsection{Continuing the `ground terms' example}
-
-Call a term \textbf{well-formed} if each symbol occurring in it has 
-the correct number of arguments. To formalize this concept, we 
-introduce a function mapping each symbol to its arity, a natural 
-number. 
-
-Let us define the set of well-formed ground terms. 
-Suppose we are given a function called \isa{arity}, specifying the arities to be used.
-In the inductive step, we have a list \isa{args} of such terms and a function 
-symbol~\isa{f}. If the length of the list matches the function's arity 
-then applying \isa{f} to \isa{args} yields a well-formed term. 
-\begin{isabelle}
-\isacommand{consts}\ well_formed_gterm\ ::\ "('f\ \isasymRightarrow \ nat)\ \isasymRightarrow \ 'f\ gterm\ set"\isanewline
-\isacommand{inductive}\ "well_formed_gterm\ arity"\isanewline
-\isakeyword{intros}\isanewline
-step[intro!]:\ "\isasymlbrakk \isasymforall t\ \isasymin \ set\ args.\ t\ \isasymin \ well_formed_gterm\ arity;\ \ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ length\ args\ =\ arity\ f\isasymrbrakk \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \isasymLongrightarrow \ (Apply\ f\ args)\ \isasymin \ well_formed_gterm\
-arity"
-\end{isabelle}
-%
-The inductive definition neatly captures the reasoning above.
-It is just an elaboration of the previous one, consisting of a single 
-introduction rule. The universal quantification over the
-\isa{set} of arguments expresses that all of them are well-formed.
-
-\subsection{Alternative definition using a monotonic function}
-
-An inductive definition may refer to the inductively defined 
-set through an arbitrary monotonic function.  To demonstrate this
-powerful feature, let us
-change the  inductive definition above, replacing the
-quantifier by a use of the function \isa{lists}. This
-function, from the Isabelle library, is analogous to the
-function \isa{gterms} declared above. If \isa{A} is a set then
-{\isa{lists A}} is the set of lists whose elements belong to
-\isa{A}.  
-
-In the inductive definition of well-formed terms, examine the one
-introduction rule.  The first premise states that \isa{args} belongs to
-the \isa{lists} of well-formed terms.  This formulation is more
-direct, if more obscure, than using a universal quantifier.
-\begin{isabelle}
-\isacommand{consts}\ well_formed_gterm'\ ::\ "('f\ \isasymRightarrow \ nat)\ \isasymRightarrow \ 'f\ gterm\ set"\isanewline
-\isacommand{inductive}\ "well_formed_gterm'\ arity"\isanewline
-\isakeyword{intros}\isanewline
-step[intro!]:\ "\isasymlbrakk args\ \isasymin \ lists\ (well_formed_gterm'\ arity);\ \ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ length\ args\ =\ arity\ f\isasymrbrakk \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \isasymLongrightarrow \ (Apply\ f\ args)\ \isasymin \ well_formed_gterm'\ arity"\isanewline
-\isakeyword{monos}\ lists_mono
-\end{isabelle}
-
-We must cite the theorem \isa{lists_mono} in order to justify 
-using the function \isa{lists}. 
-\begin{isabelle}
-A\ \isasymsubseteq\ B\ \isasymLongrightarrow \ lists\ A\ \isasymsubseteq
-\ lists\ B\rulename{lists_mono}
-\end{isabelle}
-%
-Why must the function be monotonic?  An inductive definition describes
-an iterative construction: each element of the set is constructed by a
-finite number of introduction rule applications.  For example, the
-elements of \isa{even} are constructed by finitely many applications of
-the rules 
-\begin{isabelle}
-0\ \isasymin \ even\isanewline
-n\ \isasymin \ even\ \isasymLongrightarrow \ (Suc\ (Suc\ n))\ \isasymin
-\ even
-\end{isabelle}
-All references to a set in its
-inductive definition must be positive.  Applications of an
-introduction rule cannot invalidate previous applications, allowing the
-construction process to converge.
-The following pair of rules do not constitute an inductive definition:
-\begin{isabelle}
-0\ \isasymin \ even\isanewline
-n\ \isasymnotin \ even\ \isasymLongrightarrow \ (Suc\ n)\ \isasymin
-\ even
-\end{isabelle}
-%
-Showing that 4 is even using these rules requires showing that 3 is not
-even.  It is far from trivial to show that this set of rules
-characterizes the even numbers.  
-
-Even with its use of the function \isa{lists}, the premise of our
-introduction rule is positive:
-\begin{isabelle}
-args\ \isasymin \ lists\ (well_formed_gterm'\ arity)
-\end{isabelle}
-To apply the rule we construct a list \isa{args} of previously
-constructed well-formed terms.  We obtain a
-new term, \isa{Apply\ f\ args}.  Because \isa{lists} is monotonic,
-applications of the rule remain valid as new terms are constructed.
-Further lists of well-formed
-terms become available and none are taken away.
-
-
-\subsection{A proof of equivalence}
-
-We naturally hope that these two inductive definitions of `well-formed' 
-coincide.  The equality can be proved by separate inclusions in 
-each direction.  Each is a trivial rule induction. 
-\begin{isabelle}
-\isacommand{lemma}\ "well_formed_gterm\ arity\ \isasymsubseteq \ well_formed_gterm'\ arity"\isanewline
-\isacommand{apply}\ clarify\isanewline
-\isacommand{apply}\ (erule\ well_formed_gterm.induct)\isanewline
-\isacommand{apply}\ auto\isanewline
-\isacommand{done}
-\end{isabelle}
-
-The \isa{clarify} method gives
-us an element of \isa{well_formed_gterm\ arity} on which to perform 
-induction.  The resulting subgoal can be proved automatically:
-\begin{isabelle}
-{\isadigit{1}}{\isachardot}\ {\isasymAnd}x\ args\ f{\isachardot}\isanewline
-\ \ \ \ \ \ {\isasymlbrakk}{\isasymforall}t{\isasymin}set\ args{\isachardot}\isanewline
-\ \ \ \ \ \ \ \ \ \ t\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm\ arity\ {\isasymand}\ t\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity{\isacharsemicolon}\isanewline
-\ \ \ \ \ \ \ length\ args\ {\isacharequal}\ arity\ f{\isasymrbrakk}\isanewline
-\ \ \ \ \ \ {\isasymLongrightarrow}\ Apply\ f\ args\ {\isasymin}\ well{\isacharunderscore}formed{\isacharunderscore}gterm{\isacharprime}\ arity%
-\end{isabelle}
-%
-This proof resembles the one given in
-\S\ref{sec:gterm-datatype} above, especially in the form of the
-induction hypothesis.  Next, we consider the opposite inclusion:
-\begin{isabelle}
-\isacommand{lemma}\ "well_formed_gterm'\ arity\ \isasymsubseteq \ well_formed_gterm\ arity"\isanewline
-\isacommand{apply}\ clarify\isanewline
-\isacommand{apply}\ (erule\ well_formed_gterm'.induct)\isanewline
-\isacommand{apply}\ auto\isanewline
-\isacommand{done}
-\end{isabelle}
-%
-The proof script is identical, but the subgoal after applying induction may
-be surprising:
-\begin{isabelle}
-1.\ \isasymAnd x\ args\ f.\isanewline
-\ \ \ \ \ \ \isasymlbrakk args\ \isasymin \ lists\ (well_formed_gterm'\ arity\ \isasyminter\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacharbraceleft u.\ u\ \isasymin \ well_formed_gterm\ arity\isacharbraceright );\isanewline
-\ \ \ \ \ \ \ length\ args\ =\ arity\ f\isasymrbrakk \isanewline
-\ \ \ \ \ \ \isasymLongrightarrow \ Apply\ f\ args\ \isasymin \ well_formed_gterm\ arity%
-\end{isabelle}
-The induction hypothesis contains an application of \isa{lists}.  Using a
-monotonic function in the inductive definition always has this effect.  The
-subgoal may look uninviting, but fortunately a useful rewrite rule concerning
-\isa{lists} is available:
-\begin{isabelle}
-lists\ (A\ \isasyminter \ B)\ =\ lists\ A\ \isasyminter \ lists\ B
-\rulename{lists_Int_eq}
-\end{isabelle}
-%
-Thanks to this default simplification rule, the induction hypothesis 
-is quickly replaced by its two parts:
-\begin{isabelle}
-\ \ \ \ \ \ \ args\ \isasymin \ lists\ (well_formed_gterm'\ arity)\isanewline
-\ \ \ \ \ \ \ args\ \isasymin \ lists\ (well_formed_gterm\ arity)
-\end{isabelle}
-Invoking the rule \isa{well_formed_gterm.step} completes the proof.  The
-call to
-\isa{auto} does all this work.
-
-This example is typical of how monotonic functions can be used.  In
-particular, a rewrite rule analogous to \isa{lists_Int_eq} holds in most
-cases.  Monotonicity implies one direction of this set equality; we have
-this theorem:
-\begin{isabelle}
-mono\ f\ \isasymLongrightarrow \ f\ (A\ \isasyminter \ B)\ \isasymsubseteq \
-f\ A\ \isasyminter \ f\ B%
-\rulename{mono_Int}
-\end{isabelle}
-
-
-To summarize: a universal quantifier in an introduction rule 
-lets it refer to any number of instances of 
-the inductively defined set.  A monotonic function in an introduction 
-rule lets it refer to constructions over the inductively defined 
-set.  Each element of an inductively defined set is created 
-through finitely many applications of the introduction rules.  So each rule
-must be positive, and never can it refer to \textit{infinitely} many
-previous instances of the inductively defined set. 
-
-
-
-\begin{exercise}
-Prove this theorem, one direction of which was proved in
-\S\ref{sec:rule-inversion} above.
-\begin{isabelle}
-\isacommand{lemma}\ gterms_Int_eq\ [simp]:\ "gterms\ (F\isasyminter G)\ =\
-gterms\ F\ \isasyminter \ gterms\ G"\isanewline
-\end{isabelle}
-\end{exercise}
-
-
-\begin{exercise}
-A function mapping function symbols to their 
-types is called a \textbf{signature}.  Given a type 
-ranging over type symbols, we can represent a function's type by a
-list of argument types paired with the result type. 
-Complete this inductive definition:
-\begin{isabelle}
-\isacommand{consts}\ well_typed_gterm::\ "('f\ \isasymRightarrow \ 't\ list\ *\ 't)\ \isasymRightarrow \ ('f\ gterm\ *\ 't)set"\isanewline
-\isacommand{inductive}\ "well_typed_gterm\ sig"\isanewline
-\end{isabelle}
-\end{exercise}