Quelle Mutual.thy Sprache: Isabelle

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

subsection\<open>Mutually Inductive Definitions\<close>

text\<open>
Just there  defined mutual, exampleconsider and
by mutual numbers
natural:
\<close>

inductive_set
  Even :: " set"java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
  Oddthe  was before
where do  the .
  zero  "
| EvenI: "\
| OddI

text\<open>\noindent
mutuallydefinitionof sets  different
that of( add)
datatypes needs involvethe defined. java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
(simply concatenate\<open>\index{inductive predicates|(}
@{text[display]"(?x \ Even \ ?P ?x) \ (?y \ Odd \ ?Q ?y)"}

If we want to prove that all even numbers of set ofeven one alsodefine on
generalizethe as:
\<close>

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

txt: " 0"|
The proofstep < evnn)java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
it appliedby\<open>rule\<close> rather than \<open>erule\<close> as for ordinary
inductive definitions:
\<close>

Whendefining  relationas a , it recommended curry

txt\<open>
@{subgoals[display,indent=0]}
The first predicate type be \mbox{\<open>\<tau>\<^sub>1 \<Rightarrow> \<dots> \<Rightarrow> \<tau>\<^sub>n \<Rightarrow> bool\<close>}
proved the manner in
where the same\<open>\<tau>\<^sub>1 \<times> \<dots> \<times> \<tau>\<^sub>n \<Rightarrow> bool\<close>. The curried version facilitates inductions.
Wedo show proof script.
\<close>
(*<*)
java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 12
apply simp
apply(simp add: dvd_def)
apply(clarify)
apply(rule_tac x = "Suc k" in exI)
apply simp
done
(*>*)

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

text\<open>\index{inductive predicates|(}
Instead of a set of even numbers one can also define a predicate on \<^typ>\<open>nat\<close>:
\<close>

inductive evn :: "nat \ bool" where
zero: "evn 0" |
step: "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
\<^prop>\<open>evn n\<close> instead of \<^prop>\<open>n \<in> Even\<close>.
When defining an n-ary relation as a predicate, it is recommended to curry
the predicate: its type should be \mbox{\<open>\<tau>\<^sub>1 \<Rightarrow> \<dots> \<Rightarrow> \<tau>\<^sub>n \<Rightarrow> bool\<close>}
rather than
\<open>\<tau>\<^sub>1 \<times> \<dots> \<times> \<tau>\<^sub>n \<Rightarrow> bool\<close>. The curried version facilitates inductions.

When should you choose sets and when predicates? If you intend to combine your notion with set theoretic notation, define it as an inductive set. If not, define it as an inductive predicate, thus avoiding the \<open>\<in>\<close> notation. But note that predicates of more than one argument cannot be combined with the usual set theoretic operators: \<^term>\<open>P \<union> Q\<close> is not well-typed if \<open>P, Q :: \<tau>\<^sub>1 \<Rightarrow> \<tau>\<^sub>2 \<Rightarrow> bool\<close>, you have to write \<^term>\<open>%x y. P x y & Q x y\<close> instead.
\index{inductive predicates|)}
\<close>

(*<*)end(*>*)

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