theory Recur imports Main begin

 ML "Pretty.margin_default := 64"

 text{*

 @{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}

 *}

 text{*\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"wf r"}\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"wf r"} means that the relation @{term r} is wellfounded

 *}

 text{*

 @{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}

 *}

 end