(* Title: HOL/Wfrec.thy
Author: Tobias Nipkow
Author: Lawrence C Paulson
Author: Konrad Slind
*)
section \<open>Well-Founded Recursion Combinator\<close>
theory Wfrec
imports Wellfounded
begin
inductive wfrec_rel :: "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" for R F
where wfrecI: "(\<And>z. (z, x) \<in> R \<Longrightarrow> wfrec_rel R F z (g z)) \<Longrightarrow> wfrec_rel R F x (F g x)"
definition cut :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b"
where "cut f R x = (\<lambda>y. if (y, x) \<in> R then f y else undefined)"
definition adm_wf :: "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)) \<Rightarrow> bool"
where "adm_wf R F \<longleftrightarrow> (\<forall>f g x. (\<forall>z. (z, x) \<in> R \<longrightarrow> f z = g z) \<longrightarrow> F f x = F g x)"
definition wfrec :: "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)) \<Rightarrow> ('a \<Rightarrow> 'b)"
where "wfrec R F = (\<lambda>x. THE y. wfrec_rel R (\<lambda>f x. F (cut f R x) x) x y)"
lemma cuts_eq: "(cut f R x = cut g R x) \<longleftrightarrow> (\<forall>y. (y, x) \<in> R \<longrightarrow> f y = g y)"
by (simp add: fun_eq_iff cut_def)
lemma cut_apply: "(x, a) \<in> R \<Longrightarrow> cut f R a x = f x"
by (simp add: cut_def)
text \<open>
Inductive characterization of \<open>wfrec\<close> combinator; for details see:
John Harrison, "Inductive definitions: automation and application".
\<close>
lemma theI_unique: "\<exists>!x. P x \<Longrightarrow> P x \<longleftrightarrow> x = The P"
by (auto intro: the_equality[symmetric] theI)
lemma wfrec_unique:
assumes "adm_wf R F" "wf R"
shows "\<exists>!y. wfrec_rel R F x y"
using \<open>wf R\<close>
proof induct
define f where "f y = (THE z. wfrec_rel R F y z)" for y
case (less x)
then have "\<And>y z. (y, x) \<in> R \<Longrightarrow> wfrec_rel R F y z \<longleftrightarrow> z = f y"
unfolding f_def by (rule theI_unique)
with \<open>adm_wf R F\<close> show ?case
by (subst wfrec_rel.simps) (auto simp: adm_wf_def)
qed
lemma adm_lemma: "adm_wf R (\<lambda>f x. F (cut f R x) x)"
by (auto simp: adm_wf_def intro!: arg_cong[where f="\<lambda>x. F x y" for y] cuts_eq[THEN iffD2])
lemma wfrec: "wf R \<Longrightarrow> wfrec R F a = F (cut (wfrec R F) R a) a"
apply (simp add: wfrec_def)
apply (rule adm_lemma [THEN wfrec_unique, THEN the1_equality])
apply assumption
apply (rule wfrec_rel.wfrecI)
apply (erule adm_lemma [THEN wfrec_unique, THEN theI'])
done
text \<open>This form avoids giant explosions in proofs. NOTE USE OF \<open>\<equiv>\<close>.\<close>
lemma def_wfrec: "f \<equiv> wfrec R F \<Longrightarrow> wf R \<Longrightarrow> f a = F (cut f R a) a"
by (auto intro: wfrec)
subsubsection \<open>Well-founded recursion via genuine fixpoints\<close>
lemma wfrec_fixpoint:
assumes wf: "wf R"
and adm: "adm_wf R F"
shows "wfrec R F = F (wfrec R F)"
proof (rule ext)
fix x
have "wfrec R F x = F (cut (wfrec R F) R x) x"
using wfrec[of R F] wf by simp
also
have "\<And>y. (y, x) \<in> R \<Longrightarrow> cut (wfrec R F) R x y = wfrec R F y"
by (auto simp add: cut_apply)
then have "F (cut (wfrec R F) R x) x = F (wfrec R F) x"
using adm adm_wf_def[of R F] by auto
finally show "wfrec R F x = F (wfrec R F) x" .
qed
subsection \<open>Wellfoundedness of \<open>same_fst\<close>\<close>
definition same_fst :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> ('b \<times> 'b) set) \<Rightarrow> (('a \<times> 'b) \<times> ('a \<times> 'b)) set"
where "same_fst P R = {((x', y'), (x, y)) . x' = x \<and> P x \<and> (y',y) \<in> R x}"
\<comment> \<open>For \<^const>\<open>wfrec\<close> declarations where the first n parameters
stay unchanged in the recursive call.\<close>
lemma same_fstI [intro!]: "P x \<Longrightarrow> (y', y) \<in> R x \<Longrightarrow> ((x, y'), (x, y)) \<in> same_fst P R"
by (simp add: same_fst_def)
lemma wf_same_fst:
assumes prem: "\<And>x. P x \<Longrightarrow> wf (R x)"
shows "wf (same_fst P R)"
apply (simp cong del: imp_cong add: wf_def same_fst_def)
apply (intro strip)
apply (rename_tac a b)
apply (case_tac "wf (R a)")
apply (erule_tac a = b in wf_induct)
apply blast
apply (blast intro: prem)
done
end