theory Recur imports Main begin
text\<open>
@{thm[display] mono_def[no_vars]}
\rulename{mono_def}
@{thm[display] monoI[no_vars]}
\rulename{monoI}
@{thm[display] monoD[no_vars]}
\rulename{monoD}
@{thm[display] lfp_unfold[no_vars]}
\rulename{lfp_unfold}
@{thm[display] lfp_induct[no_vars]}
\rulename{lfp_induct}
@{thm[display] gfp_unfold[no_vars]}
\rulename{gfp_unfold}
@{thm[display] coinduct[no_vars]}
\rulename{coinduct}
\<close>
text\<open>\noindent
A relation $<$ is
\bfindex{wellfounded} if it has no infinite descending chain $\cdots <
a@2 < a@1 < a@0$. Clearly, a function definition is total iff the set
of all pairs $(r,l)$, where $l$ is the argument on the left-hand side
of an equation and $r$ the argument of some recursive call on the
corresponding right-hand side, induces a wellfounded relation.
The HOL library formalizes
some of the theory of wellfounded relations. For example
\<^prop>\<open>wf r\<close>\index{*wf|bold} means that relation @{term[show_types]"r::('a*'a)set"} is
wellfounded.
Finally we should mention that HOL already provides the mother of all
inductions, \textbf{wellfounded
induction}\indexbold{induction!wellfounded}\index{wellfounded
induction|see{induction, wellfounded}} (@{thm[source]wf_induct}):
@{thm[display]wf_induct[no_vars]}
where \<^term>\<open>wf r\<close> means that the relation \<^term>\<open>r\<close> is wellfounded
\<close>
text\<open>
@{thm[display] wf_induct[no_vars]}
\rulename{wf_induct}
@{thm[display] less_than_iff[no_vars]}
\rulename{less_than_iff}
@{thm[display] inv_image_def[no_vars]}
\rulename{inv_image_def}
@{thm[display] measure_def[no_vars]}
\rulename{measure_def}
@{thm[display] wf_less_than[no_vars]}
\rulename{wf_less_than}
@{thm[display] wf_inv_image[no_vars]}
\rulename{wf_inv_image}
@{thm[display] wf_measure[no_vars]}
\rulename{wf_measure}
@{thm[display] lex_prod_def[no_vars]}
\rulename{lex_prod_def}
@{thm[display] wf_lex_prod[no_vars]}
\rulename{wf_lex_prod}
\<close>
end