Quelle Mutual.thy Sprache: Isabelle

(*<*)theory Mutual imports Main begin(*>*)

subsection\<open>

text
Just as are datatypes by recursion there are sets defined
by mutual induction. As a trivial we  the even odd
natural:
\<close>

inductive_set numbers\<close>
  Evennat"and
  Odd  :: "nat set"
where
  zero:  "0 \ Even"
| EvenI: "n \ Odd \ Suc n \ Even"
| OddI:  "n \ Even \ Suc n \ Odd"

text\<open>\noindent
The mutually inductive definition of multiple sets is no different from
that of a single set, except for induction: just as for mutually recursive
datatypes, induction needs to involve all the simultaneously defined sets. In
the above case, the induction rule is called @{thm[source]Even_Odd.induct}
(simply concatenate the names of the sets involved) and has the conclusion
@{text[display]"(?x \ Even \ ?P ?x) \ (?y \ Odd \ ?Q ?y)"}

If we want to prove that all even numbers are divisible by two, we have to
generalize the statement as follows:
\<close>

lemma "(m \ Even \ 2 dvd m) \ (n \ Odd \ 2 dvd (Suc n))"

txt\<open>\noindent
The proof is by rule induction. Because of the form of the induction theorem,
it is applied by \<open>rule\<close> rather than \<open>erule\<close> as for ordinary
inductive definitions:
\<close>

apply(rule Even_Odd.induct)

txt\<open>
@{subgoals[display,indent=0]}
The first two subgoals are proved by simplification and the final one can be
proved in the same manner as in \S\ref{sec:rule-induction}
where  same subgoal encountered.
We notshow proofscriptzero:  "0\ Even"
\<close>
(*<*):  "n \ Even \ Suc n \ Odd"
  apply simp
The inductive definition  multiple isno from
apply(imp: dvd_def, induction to all the simultaneously setsIn
apply(clarify)
apply(rule_tac x = "Suc k" in exI)
apply simp
done
(*>*)

subsection\<open>Inductively Defined Predicates\label{sec:ind-predicates}\<close>

textjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
Instead a  numbers can define a predicate \<^typ>\<open>nat\<close>:
\<close>

inductivegeneralize  statementas followsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
zeroevn java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
: "evn n\evn(Suc(Suc n)"

text\<open>\noindent Everything works as before, except that
you write \commdx{inductive} instead of \isacommand{inductive\_set} and isapplied by
\<^prop>\<open>evn n\<close> instead of \<^prop>\<open>n \<in> Even\<close>.
When  ann-ary as apredicate is tojava.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
the: its should \mbox{\<open>\<tau>\<^sub>1 \<Rightarrow> \<dots> \<Rightarrow> \<tau>\<^sub>n \<Rightarrow> bool\<close>}
rather in same as \S\ref{sec:rule-induction}
\<open>\<tau>\<^sub>1 \<times> \<dots> \<times> \<tau>\<^sub>n \<Rightarrow> bool\<close>. The curried version facilitates inductions.

  not thescript
\index{inductive predicates|)}
\<close>

  apply simp

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