(* Title: HOL/Wellfounded.thy
Author: Tobias Nipkow
Author: Lawrence C Paulson
Author: Konrad Slind
Author: Alexander Krauss
Author: Andrei Popescu, TU Muenchen
Author: Martin Desharnais, MPI-INF Saarbruecken
*)
section \<open>Well-founded Recursion\<close>
theory Wellfounded
imports Transitive_Closure
begin
subsection \<open>Basic Definitions\<close>
definition wf_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool" where
"wf_on A r \<longleftrightarrow> (\<forall>P. (\<forall>x \<in> A. (\<forall>y \<in> A. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x) \<longrightarrow> (\<forall>x \<in> A. P x))"
abbreviation wf :: "('a \<times> 'a) set \<Rightarrow> bool" where
"wf \<equiv> wf_on UNIV"
definition wfp_on :: "'a set \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
"wfp_on A R \<longleftrightarrow> (\<forall>P. (\<forall>x \<in> A. (\<forall>y \<in> A. R y x \<longrightarrow> P y) \<longrightarrow> P x) \<longrightarrow> (\<forall>x \<in> A. P x))"
abbreviation wfP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
"wfP \<equiv> wfp_on UNIV"
alias wfp = wfP
text \<open>We keep old name \<^const>\<open>wfP\<close> for backward compatibility, but offer new name \<^const>\<open>wfp\<close> to be
consistent with similar predicates, e.g., \<^const>\<open>asymp\<close>, \<^const>\<open>transp\<close>, \<^const>\<open>totalp\<close>.\<close>
subsection \<open>Equivalence of Definitions\<close>
lemma wfp_on_wf_on_eq[pred_set_conv]: "wfp_on A (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> wf_on A r"
by (simp add: wfp_on_def wf_on_def)
lemma wf_def: "wf r \<longleftrightarrow> (\<forall>P. (\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x) \<longrightarrow> (\<forall>x. P x))"
unfolding wf_on_def by simp
lemma wfP_def: "wfP r \<longleftrightarrow> wf {(x, y). r x y}"
unfolding wf_def wfp_on_def by simp
lemma wfP_wf_eq: "wfP (\<lambda>x y. (x, y) \<in> r) = wf r"
using wfp_on_wf_on_eq .
subsection \<open>Induction Principles\<close>
lemma wf_on_induct[consumes 1, case_names in_set less, induct set: wf_on]:
assumes "wf_on A r" and "x \<in> A" and "\<And>x. x \<in> A \<Longrightarrow> (\<And>y. y \<in> A \<Longrightarrow> (y, x) \<in> r \<Longrightarrow> P y) \<Longrightarrow> P x"
shows "P x"
using assms(2,3) by (auto intro: \<open>wf_on A r\<close>[unfolded wf_on_def, rule_format])
lemma wfp_on_induct[consumes 1, case_names in_set less, induct pred: wfp_on]:
assumes "wfp_on A r" and "x \<in> A" and "\<And>x. x \<in> A \<Longrightarrow> (\<And>y. y \<in> A \<Longrightarrow> r y x \<Longrightarrow> P y) \<Longrightarrow> P x"
shows "P x"
using assms by (fact wf_on_induct[to_pred])
lemma wf_induct:
assumes "wf r"
and "\<And>x. \<forall>y. (y, x) \<in> r \<longrightarrow> P y \<Longrightarrow> P x"
shows "P a"
using assms by (auto intro: wf_on_induct[of UNIV])
lemmas wfP_induct = wf_induct [to_pred]
lemmas wf_induct_rule = wf_induct [rule_format, consumes 1, case_names less, induct set: wf]
lemmas wfP_induct_rule = wf_induct_rule [to_pred, induct set: wfP]
lemma wf_on_iff_wf: "wf_on A r \<longleftrightarrow> wf {(x, y) \<in> r. x \<in> A \<and> y \<in> A}"
proof (rule iffI)
assume wf: "wf_on A r"
show "wf {(x, y) \<in> r. x \<in> A \<and> y \<in> A}"
unfolding wf_def
proof (intro allI impI ballI)
fix P x
assume IH: "\<forall>x. (\<forall>y. (y, x) \<in> {(x, y). (x, y) \<in> r \<and> x \<in> A \<and> y \<in> A} \<longrightarrow> P y) \<longrightarrow> P x"
show "P x"
proof (cases "x \<in> A")
case True
show ?thesis
using wf
proof (induction x rule: wf_on_induct)
case in_set
thus ?case
using True .
next
case (less x)
thus ?case
by (auto intro: IH[rule_format])
qed
next
case False
then show ?thesis
by (auto intro: IH[rule_format])
qed
qed
next
assume wf: "wf {(x, y). (x, y) \<in> r \<and> x \<in> A \<and> y \<in> A}"
show "wf_on A r"
unfolding wf_on_def
proof (intro allI impI ballI)
fix P x
assume IH: "\<forall>x\<in>A. (\<forall>y\<in>A. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x" and "x \<in> A"
show "P x"
using wf \<open>x \<in> A\<close>
proof (induction x rule: wf_on_induct)
case in_set
show ?case
by simp
next
case (less y)
hence "\<And>z. (z, y) \<in> r \<Longrightarrow> z \<in> A \<Longrightarrow> P z"
by simp
thus ?case
using IH[rule_format, OF \<open>y \<in> A\<close>] by simp
qed
qed
qed
subsection \<open>Introduction Rules\<close>
lemma wfUNIVI: "(\<And>P x. (\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x) \<Longrightarrow> P x) \<Longrightarrow> wf r"
unfolding wf_def by blast
lemmas wfPUNIVI = wfUNIVI [to_pred]
text \<open>Restriction to domain \<open>A\<close> and range \<open>B\<close>.
If \<open>r\<close> is well-founded over their intersection, then \<open>wf r\<close>.\<close>
lemma wfI:
assumes "r \<subseteq> A \<times> B"
and "\<And>x P. \<lbrakk>\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x; x \<in> A; x \<in> B\<rbrakk> \<Longrightarrow> P x"
shows "wf r"
using assms unfolding wf_def by blast
subsection \<open>Ordering Properties\<close>
lemma wf_not_sym: "wf r \<Longrightarrow> (a, x) \<in> r \<Longrightarrow> (x, a) \<notin> r"
by (induct a arbitrary: x set: wf) blast
lemma wf_asym:
assumes "wf r" "(a, x) \<in> r"
obtains "(x, a) \<notin> r"
by (drule wf_not_sym[OF assms])
lemma wf_imp_asym: "wf r \<Longrightarrow> asym r"
by (auto intro: asymI elim: wf_asym)
lemma wfP_imp_asymp: "wfP r \<Longrightarrow> asymp r"
by (rule wf_imp_asym[to_pred])
lemma wf_not_refl [simp]: "wf r \<Longrightarrow> (a, a) \<notin> r"
by (blast elim: wf_asym)
lemma wf_irrefl:
assumes "wf r"
obtains "(a, a) \<notin> r"
by (drule wf_not_refl[OF assms])
lemma wf_imp_irrefl:
assumes "wf r" shows "irrefl r"
using wf_irrefl [OF assms] by (auto simp add: irrefl_def)
lemma wfP_imp_irreflp: "wfP r \<Longrightarrow> irreflp r"
by (rule wf_imp_irrefl[to_pred])
lemma wf_wellorderI:
assumes wf: "wf {(x::'a::ord, y). x < y}"
and lin: "OFCLASS('a::ord, linorder_class)"
shows "OFCLASS('a::ord, wellorder_class)"
apply (rule wellorder_class.intro [OF lin])
apply (simp add: wellorder_class.intro class.wellorder_axioms.intro wf_induct_rule [OF wf])
done
lemma (in wellorder) wf: "wf {(x, y). x < y}"
unfolding wf_def by (blast intro: less_induct)
lemma (in wellorder) wfP_less[simp]: "wfP (<)"
by (simp add: wf wfP_def)
lemma (in wellorder) wfp_on_less[simp]: "wfp_on A (<)"
unfolding wfp_on_def
proof (intro allI impI ballI)
fix P x
assume hyps: "\<forall>x\<in>A. (\<forall>y\<in>A. y < x \<longrightarrow> P y) \<longrightarrow> P x"
show "x \<in> A \<Longrightarrow> P x"
proof (induction x rule: less_induct)
case (less x)
show ?case
proof (rule hyps[rule_format])
show "x \<in> A"
using \<open>x \<in> A\<close> .
next
show "\<And>y. y \<in> A \<Longrightarrow> y < x \<Longrightarrow> P y"
using less.IH .
qed
qed
qed
subsection \<open>Basic Results\<close>
text \<open>Point-free characterization of well-foundedness\<close>
lemma wf_onE_pf:
assumes wf: "wf_on A r" and "B \<subseteq> A" and "B \<subseteq> r `` B"
shows "B = {}"
proof -
have "x \<notin> B" if "x \<in> A" for x
using wf
proof (induction x rule: wf_on_induct)
case in_set
show ?case
using that .
next
case (less x)
have "x \<notin> r `` B"
using less.IH \<open>B \<subseteq> A\<close> by blast
thus ?case
using \<open>B \<subseteq> r `` B\<close> by blast
qed
with \<open>B \<subseteq> A\<close> show ?thesis
by blast
qed
lemma wfE_pf: "wf R \<Longrightarrow> A \<subseteq> R `` A \<Longrightarrow> A = {}"
using wf_onE_pf[of UNIV, simplified] .
lemma wf_onI_pf:
assumes "\<And>B. B \<subseteq> A \<Longrightarrow> B \<subseteq> R `` B \<Longrightarrow> B = {}"
shows "wf_on A R"
unfolding wf_on_def
proof (intro allI impI ballI)
fix P :: "'a \<Rightarrow> bool" and x :: 'a
let ?B = "{x \<in> A. \<not> P x}"
assume "\<forall>x\<in>A. (\<forall>y\<in>A. (y, x) \<in> R \<longrightarrow> P y) \<longrightarrow> P x"
hence "?B \<subseteq> R `` ?B" by blast
hence "{x \<in> A. \<not> P x} = {}"
using assms(1)[of ?B] by simp
moreover assume "x \<in> A"
ultimately show "P x"
by simp
qed
lemma wfI_pf: "(\<And>A. A \<subseteq> R `` A \<Longrightarrow> A = {}) \<Longrightarrow> wf R"
using wf_onI_pf[of UNIV, simplified] .
subsubsection \<open>Minimal-element characterization of well-foundedness\<close>
lemma wf_on_iff_ex_minimal: "wf_on A R \<longleftrightarrow> (\<forall>B \<subseteq> A. B \<noteq> {} \<longrightarrow> (\<exists>z \<in> B. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> B))"
proof (intro iffI allI impI)
fix B
assume "wf_on A R" and "B \<subseteq> A" and "B \<noteq> {}"
show "\<exists>z \<in> B. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> B"
using wf_onE_pf[OF \<open>wf_on A R\<close> \<open>B \<subseteq> A\<close>] \<open>B \<noteq> {}\<close> by blast
next
assume ex_min: "\<forall>B\<subseteq>A. B \<noteq> {} \<longrightarrow> (\<exists>z\<in>B. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> B)"
show "wf_on A R "
proof (rule wf_onI_pf)
fix B
assume "B \<subseteq> A" and "B \<subseteq> R `` B"
have False if "B \<noteq> {}"
using ex_min[rule_format, OF \<open>B \<subseteq> A\<close> \<open>B \<noteq> {}\<close>]
using \<open>B \<subseteq> R `` B\<close> by blast
thus "B = {}"
by blast
qed
qed
lemma wf_iff_ex_minimal: "wf R \<longleftrightarrow> (\<forall>B. B \<noteq> {} \<longrightarrow> (\<exists>z \<in> B. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> B))"
using wf_on_iff_ex_minimal[of UNIV, simplified] .
lemma wfp_on_iff_ex_minimal: "wfp_on A R \<longleftrightarrow> (\<forall>B \<subseteq> A. B \<noteq> {} \<longrightarrow> (\<exists>z \<in> B. \<forall>y. R y z \<longrightarrow> y \<notin> B))"
using wf_on_iff_ex_minimal[of A, to_pred] by simp
lemma wfp_iff_ex_minimal: "wfp R \<longleftrightarrow> (\<forall>B. B \<noteq> {} \<longrightarrow> (\<exists>z \<in> B. \<forall>y. R y z \<longrightarrow> y \<notin> B))"
using wfp_on_iff_ex_minimal[of UNIV, simplified] .
lemma wfE_min:
assumes wf: "wf R" and Q: "x \<in> Q"
obtains z where "z \<in> Q" "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
using Q wfE_pf[OF wf, of Q] by blast
lemma wfE_min':
"wf R \<Longrightarrow> Q \<noteq> {} \<Longrightarrow> (\<And>z. z \<in> Q \<Longrightarrow> (\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q) \<Longrightarrow> thesis) \<Longrightarrow> thesis"
using wfE_min[of R _ Q] by blast
lemma wfI_min:
assumes a: "\<And>x Q. x \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q"
shows "wf R"
proof (rule wfI_pf)
fix A
assume b: "A \<subseteq> R `` A"
have False if "x \<in> A" for x
using a[OF that] b by blast
then show "A = {}" by blast
qed
lemma wf_eq_minimal: "wf r \<longleftrightarrow> (\<forall>Q x. x \<in> Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q))"
unfolding wf_iff_ex_minimal by blast
lemmas wfP_eq_minimal = wf_eq_minimal [to_pred]
subsubsection \<open>Antimonotonicity\<close>
lemma wf_on_antimono_strong:
assumes "wf_on B r" and "A \<subseteq> B" and "(\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> (x, y) \<in> q \<Longrightarrow> (x, y) \<in> r)"
shows "wf_on A q"
unfolding wf_on_iff_ex_minimal
proof (intro allI impI)
fix AA assume "AA \<subseteq> A" and "AA \<noteq> {}"
hence "\<exists>z\<in>AA. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> AA"
using \<open>wf_on B r\<close> \<open>A \<subseteq> B\<close>
by (simp add: wf_on_iff_ex_minimal)
then show "\<exists>z\<in>AA. \<forall>y. (y, z) \<in> q \<longrightarrow> y \<notin> AA"
using \<open>AA \<subseteq> A\<close> assms(3) by blast
qed
lemma wfp_on_antimono_strong:
"wfp_on B R \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> Q x y \<Longrightarrow> R x y) \<Longrightarrow> wfp_on A Q"
using wf_on_antimono_strong[of B _ A, to_pred] .
lemma wf_on_antimono: "A \<subseteq> B \<Longrightarrow> q \<subseteq> r \<Longrightarrow> wf_on B r \<le> wf_on A q"
using wf_on_antimono_strong[of B r A q] by auto
lemma wfp_on_antimono: "A \<subseteq> B \<Longrightarrow> Q \<le> R \<Longrightarrow> wfp_on B R \<le> wfp_on A Q"
using wfp_on_antimono_strong[of B R A Q] by auto
lemma wf_on_subset: "wf_on B r \<Longrightarrow> A \<subseteq> B \<Longrightarrow> wf_on A r"
using wf_on_antimono_strong .
lemma wfp_on_subset: "wfp_on B R \<Longrightarrow> A \<subseteq> B \<Longrightarrow> wfp_on A R"
using wfp_on_antimono_strong .
subsubsection \<open>Well-foundedness of transitive closure\<close>
lemma wf_trancl:
assumes "wf r"
shows "wf (r\<^sup>+)"
proof -
have "P x" if induct_step: "\<And>x. (\<And>y. (y, x) \<in> r\<^sup>+ \<Longrightarrow> P y) \<Longrightarrow> P x" for P x
proof (rule induct_step)
show "P y" if "(y, x) \<in> r\<^sup>+" for y
using \<open>wf r\<close> and that
proof (induct x arbitrary: y)
case (less x)
note hyp = \<open>\<And>x' y'. (x', x) \<in> r \<Longrightarrow> (y', x') \<in> r\<^sup>+ \<Longrightarrow> P y'\<close>
from \<open>(y, x) \<in> r\<^sup>+\<close> show "P y"
proof cases
case base
show "P y"
proof (rule induct_step)
fix y'
assume "(y', y) \<in> r\<^sup>+"
with \<open>(y, x) \<in> r\<close> show "P y'"
by (rule hyp [of y y'])
qed
next
case step
then obtain x' where "(x', x) \<in> r" and "(y, x') \<in> r\<^sup>+"
by simp
then show "P y" by (rule hyp [of x' y])
qed
qed
qed
then show ?thesis unfolding wf_def by blast
qed
lemmas wfP_trancl = wf_trancl [to_pred]
lemma wf_converse_trancl: "wf (r\<inverse>) \<Longrightarrow> wf ((r\<^sup>+)\<inverse>)"
apply (subst trancl_converse [symmetric])
apply (erule wf_trancl)
done
text \<open>Well-foundedness of subsets\<close>
lemma wf_subset: "wf r \<Longrightarrow> p \<subseteq> r \<Longrightarrow> wf p"
by (simp add: wf_eq_minimal) fast
lemmas wfP_subset = wf_subset [to_pred]
text \<open>Well-foundedness of the empty relation\<close>
lemma wf_empty [iff]: "wf {}"
by (simp add: wf_def)
lemma wfP_empty [iff]: "wfP (\<lambda>x y. False)"
proof -
have "wfP bot"
by (fact wf_empty[to_pred bot_empty_eq2])
then show ?thesis
by (simp add: bot_fun_def)
qed
lemma wf_Int1: "wf r \<Longrightarrow> wf (r \<inter> r')"
by (erule wf_subset) (rule Int_lower1)
lemma wf_Int2: "wf r \<Longrightarrow> wf (r' \<inter> r)"
by (erule wf_subset) (rule Int_lower2)
text \<open>Exponentiation.\<close>
lemma wf_exp:
assumes "wf (R ^^ n)"
shows "wf R"
proof (rule wfI_pf)
fix A assume "A \<subseteq> R `` A"
then have "A \<subseteq> (R ^^ n) `` A"
by (induct n) force+
with \<open>wf (R ^^ n)\<close> show "A = {}"
by (rule wfE_pf)
qed
text \<open>Well-foundedness of \<open>insert\<close>.\<close>
lemma wf_insert [iff]: "wf (insert (y,x) r) \<longleftrightarrow> wf r \<and> (x,y) \<notin> r\<^sup>*" (is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
by (blast elim: wf_trancl [THEN wf_irrefl]
intro: rtrancl_into_trancl1 wf_subset rtrancl_mono [THEN subsetD])
next
assume R: ?rhs
then have R': "Q \<noteq> {} \<Longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q)" for Q
by (auto simp: wf_eq_minimal)
show ?lhs
unfolding wf_eq_minimal
proof clarify
fix Q :: "'a set" and q
assume "q \<in> Q"
then obtain a where "a \<in> Q" and a: "\<And>y. (y, a) \<in> r \<Longrightarrow> y \<notin> Q"
using R by (auto simp: wf_eq_minimal)
show "\<exists>z\<in>Q. \<forall>y'. (y', z) \<in> insert (y, x) r \<longrightarrow> y' \<notin> Q"
proof (cases "a=x")
case True
show ?thesis
proof (cases "y \<in> Q")
case True
then obtain z where "z \<in> Q" "(z, y) \<in> r\<^sup>*"
"\<And>z'. (z', z) \<in> r \<longrightarrow> z' \<in> Q \<longrightarrow> (z', y) \<notin> r\<^sup>*"
using R' [of "{z \<in> Q. (z,y) \<in> r\<^sup>*}"] by auto
then have "\<forall>y'. (y', z) \<in> insert (y, x) r \<longrightarrow> y' \<notin> Q"
using R by(blast intro: rtrancl_trans)+
then show ?thesis
by (rule bexI) fact
next
case False
then show ?thesis
using a \<open>a \<in> Q\<close> by blast
qed
next
case False
with a \<open>a \<in> Q\<close> show ?thesis
by blast
qed
qed
qed
subsubsection \<open>Well-foundedness of image\<close>
lemma wf_map_prod_image_Dom_Ran:
fixes r:: "('a \<times> 'a) set"
and f:: "'a \<Rightarrow> 'b"
assumes wf_r: "wf r"
and inj: "\<And> a a'. a \<in> Domain r \<Longrightarrow> a' \<in> Range r \<Longrightarrow> f a = f a' \<Longrightarrow> a = a'"
shows "wf (map_prod f f ` r)"
proof (unfold wf_eq_minimal, clarify)
fix B :: "'b set" and b::"'b"
assume "b \<in> B"
define A where "A = f -` B \<inter> Domain r"
show "\<exists>z\<in>B. \<forall>y. (y, z) \<in> map_prod f f ` r \<longrightarrow> y \<notin> B"
proof (cases "A = {}")
case False
then obtain a0 where "a0 \<in> A" and "\<forall>a. (a, a0) \<in> r \<longrightarrow> a \<notin> A"
using wfE_min[OF wf_r] by auto
thus ?thesis
using inj unfolding A_def
by (intro bexI[of _ "f a0"]) auto
qed (use \<open>b \<in> B\<close> in \<open>unfold A_def, auto\<close>)
qed
lemma wf_map_prod_image: "wf r \<Longrightarrow> inj f \<Longrightarrow> wf (map_prod f f ` r)"
by(rule wf_map_prod_image_Dom_Ran) (auto dest: inj_onD)
lemma wfp_on_image: "wfp_on (f ` A) R \<longleftrightarrow> wfp_on A (\<lambda>a b. R (f a) (f b))"
proof (rule iffI)
assume hyp: "wfp_on (f ` A) R"
show "wfp_on A (\<lambda>a b. R (f a) (f b))"
unfolding wfp_on_iff_ex_minimal
proof (intro allI impI)
fix B
assume "B \<subseteq> A" and "B \<noteq> {}"
hence "f ` B \<subseteq> f ` A" and "f ` B \<noteq> {}"
unfolding atomize_conj image_is_empty
using image_mono by iprover
hence "\<exists>z\<in>f ` B. \<forall>y. R y z \<longrightarrow> y \<notin> f ` B"
using hyp[unfolded wfp_on_iff_ex_minimal, rule_format] by iprover
then obtain fz where "fz \<in> f ` B" and fz_max: "\<forall>y. R y fz \<longrightarrow> y \<notin> f ` B" ..
obtain z where "z \<in> B" and "fz = f z"
using \<open>fz \<in> f ` B\<close> unfolding image_iff ..
show "\<exists>z\<in>B. \<forall>y. R (f y) (f z) \<longrightarrow> y \<notin> B"
proof (intro bexI allI impI)
show "z \<in> B"
using \<open>z \<in> B\<close> .
next
fix y assume "R (f y) (f z)"
hence "f y \<notin> f ` B"
using fz_max \<open>fz = f z\<close> by iprover
thus "y \<notin> B"
by (rule contrapos_nn) (rule imageI)
qed
qed
next
assume hyp: "wfp_on A (\<lambda>a b. R (f a) (f b))"
show "wfp_on (f ` A) R"
unfolding wfp_on_iff_ex_minimal
proof (intro allI impI)
fix fA
assume "fA \<subseteq> f ` A" and "fA \<noteq> {}"
then obtain A' where "A' \<subseteq> A" and "A' \<noteq> {}" and "fA = f ` A'"
by (auto simp only: subset_image_iff)
obtain z where "z \<in> A'" and z_max: "\<forall>y. R (f y) (f z) \<longrightarrow> y \<notin> A'"
using hyp[unfolded wfp_on_iff_ex_minimal, rule_format, OF \<open>A' \<subseteq> A\<close> \<open>A' \<noteq> {}\<close>] by blast
show "\<exists>z\<in>fA. \<forall>y. R y z \<longrightarrow> y \<notin> fA"
proof (intro bexI allI impI)
show "f z \<in> fA"
unfolding \<open>fA = f ` A'\<close>
using imageI[OF \<open>z \<in> A'\<close>] .
next
show "\<And>y. R y (f z) \<Longrightarrow> y \<notin> fA"
unfolding \<open>fA = f ` A'\<close>
using z_max by auto
qed
qed
qed
subsection \<open>Well-Foundedness Results for Unions\<close>
lemma wf_union_compatible:
assumes "wf R" "wf S"
assumes "R O S \<subseteq> R"
shows "wf (R \<union> S)"
proof (rule wfI_min)
fix x :: 'a and Q
let ?Q' = "{x \<in> Q. \<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> Q}"
assume "x \<in> Q"
obtain a where "a \<in> ?Q'"
by (rule wfE_min [OF \<open>wf R\<close> \<open>x \<in> Q\<close>]) blast
with \<open>wf S\<close> obtain z where "z \<in> ?Q'" and zmin: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> ?Q'"
by (erule wfE_min)
have "y \<notin> Q" if "(y, z) \<in> S" for y
proof
from that have "y \<notin> ?Q'" by (rule zmin)
assume "y \<in> Q"
with \<open>y \<notin> ?Q'\<close> obtain w where "(w, y) \<in> R" and "w \<in> Q" by auto
from \<open>(w, y) \<in> R\<close> \<open>(y, z) \<in> S\<close> have "(w, z) \<in> R O S" by (rule relcompI)
with \<open>R O S \<subseteq> R\<close> have "(w, z) \<in> R" ..
with \<open>z \<in> ?Q'\<close> have "w \<notin> Q" by blast
with \<open>w \<in> Q\<close> show False by contradiction
qed
with \<open>z \<in> ?Q'\<close> show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<union> S \<longrightarrow> y \<notin> Q" by blast
qed
text \<open>Well-foundedness of indexed union with disjoint domains and ranges.\<close>
lemma wf_UN:
assumes r: "\<And>i. i \<in> I \<Longrightarrow> wf (r i)"
and disj: "\<And>i j. \<lbrakk>i \<in> I; j \<in> I; r i \<noteq> r j\<rbrakk> \<Longrightarrow> Domain (r i) \<inter> Range (r j) = {}"
shows "wf (\<Union>i\<in>I. r i)"
unfolding wf_eq_minimal
proof clarify
fix A and a :: "'b"
assume "a \<in> A"
show "\<exists>z\<in>A. \<forall>y. (y, z) \<in> \<Union>(r ` I) \<longrightarrow> y \<notin> A"
proof (cases "\<exists>i\<in>I. \<exists>a\<in>A. \<exists>b\<in>A. (b, a) \<in> r i")
case True
then obtain i b c where ibc: "i \<in> I" "b \<in> A" "c \<in> A" "(c,b) \<in> r i"
by blast
have ri: "\<And>Q. Q \<noteq> {} \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> r i \<longrightarrow> y \<notin> Q"
using r [OF \<open>i \<in> I\<close>] unfolding wf_eq_minimal by auto
show ?thesis
using ri [of "{a. a \<in> A \<and> (\<exists>b\<in>A. (b, a) \<in> r i) }"] ibc disj
by blast
next
case False
with \<open>a \<in> A\<close> show ?thesis
by blast
qed
qed
lemma wfP_SUP:
"\<forall>i. wfP (r i) \<Longrightarrow> \<forall>i j. r i \<noteq> r j \<longrightarrow> inf (Domainp (r i)) (Rangep (r j)) = bot \<Longrightarrow>
wfP (\<Squnion>(range r))"
by (rule wf_UN[to_pred]) simp_all
lemma wf_Union:
assumes "\<forall>r\<in>R. wf r"
and "\<forall>r\<in>R. \<forall>s\<in>R. r \<noteq> s \<longrightarrow> Domain r \<inter> Range s = {}"
shows "wf (\<Union>R)"
using assms wf_UN[of R "\<lambda>i. i"] by simp
text \<open>
Intuition: We find an \<open>R \<union> S\<close>-min element of a nonempty subset \<open>A\<close> by case distinction.
\<^enum> There is a step \<open>a \<midarrow>R\<rightarrow> b\<close> with \<open>a, b \<in> A\<close>.
Pick an \<open>R\<close>-min element \<open>z\<close> of the (nonempty) set \<open>{a\<in>A | \<exists>b\<in>A. a \<midarrow>R\<rightarrow> b}\<close>.
By definition, there is \<open>z' \<in> A\<close> s.t. \<open>z \<midarrow>R\<rightarrow> z'\<close>. Because \<open>z\<close> is \<open>R\<close>-min in the
subset, \<open>z'\<close> must be \<open>R\<close>-min in \<open>A\<close>. Because \<open>z'\<close> has an \<open>R\<close>-predecessor, it cannot
have an \<open>S\<close>-successor and is thus \<open>S\<close>-min in \<open>A\<close> as well.
\<^enum> There is no such step.
Pick an \<open>S\<close>-min element of \<open>A\<close>. In this case it must be an \<open>R\<close>-min
element of \<open>A\<close> as well.
\<close>
lemma wf_Un: "wf r \<Longrightarrow> wf s \<Longrightarrow> Domain r \<inter> Range s = {} \<Longrightarrow> wf (r \<union> s)"
using wf_union_compatible[of s r]
by (auto simp: Un_ac)
lemma wf_union_merge: "wf (R \<union> S) = wf (R O R \<union> S O R \<union> S)"
(is "wf ?A = wf ?B")
proof
assume "wf ?A"
with wf_trancl have wfT: "wf (?A\<^sup>+)" .
moreover have "?B \<subseteq> ?A\<^sup>+"
by (subst trancl_unfold, subst trancl_unfold) blast
ultimately show "wf ?B" by (rule wf_subset)
next
assume "wf ?B"
show "wf ?A"
proof (rule wfI_min)
fix Q :: "'a set" and x
assume "x \<in> Q"
with \<open>wf ?B\<close> obtain z where "z \<in> Q" and "\<And>y. (y, z) \<in> ?B \<Longrightarrow> y \<notin> Q"
by (erule wfE_min)
then have 1: "\<And>y. (y, z) \<in> R O R \<Longrightarrow> y \<notin> Q"
and 2: "\<And>y. (y, z) \<in> S O R \<Longrightarrow> y \<notin> Q"
and 3: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> Q"
by auto
show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> ?A \<longrightarrow> y \<notin> Q"
proof (cases "\<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q")
case True
with \<open>z \<in> Q\<close> 3 show ?thesis by blast
next
case False
then obtain z' where "z'\<in>Q" "(z', z) \<in> R" by blast
have "\<forall>y. (y, z') \<in> ?A \<longrightarrow> y \<notin> Q"
proof (intro allI impI)
fix y assume "(y, z') \<in> ?A"
then show "y \<notin> Q"
proof
assume "(y, z') \<in> R"
then have "(y, z) \<in> R O R" using \<open>(z', z) \<in> R\<close> ..
with 1 show "y \<notin> Q" .
next
assume "(y, z') \<in> S"
then have "(y, z) \<in> S O R" using \<open>(z', z) \<in> R\<close> ..
with 2 show "y \<notin> Q" .
qed
qed
with \<open>z' \<in> Q\<close> show ?thesis ..
qed
qed
qed
lemma wf_comp_self: "wf R \<longleftrightarrow> wf (R O R)" \<comment> \<open>special case\<close>
by (rule wf_union_merge [where S = "{}", simplified])
subsection \<open>Well-Foundedness of Composition\<close>
text \<open>Bachmair and Dershowitz 1986, Lemma 2. [Provided by Tjark Weber]\<close>
lemma qc_wf_relto_iff:
assumes "R O S \<subseteq> (R \<union> S)\<^sup>* O R" \<comment> \<open>R quasi-commutes over S\<close>
shows "wf (S\<^sup>* O R O S\<^sup>*) \<longleftrightarrow> wf R"
(is "wf ?S \<longleftrightarrow> _")
proof
show "wf R" if "wf ?S"
proof -
have "R \<subseteq> ?S" by auto
with wf_subset [of ?S] that show "wf R"
by auto
qed
next
show "wf ?S" if "wf R"
proof (rule wfI_pf)
fix A
assume A: "A \<subseteq> ?S `` A"
let ?X = "(R \<union> S)\<^sup>* `` A"
have *: "R O (R \<union> S)\<^sup>* \<subseteq> (R \<union> S)\<^sup>* O R"
proof -
have "(x, z) \<in> (R \<union> S)\<^sup>* O R" if "(y, z) \<in> (R \<union> S)\<^sup>*" and "(x, y) \<in> R" for x y z
using that
proof (induct y z)
case rtrancl_refl
then show ?case by auto
next
case (rtrancl_into_rtrancl a b c)
then have "(x, c) \<in> ((R \<union> S)\<^sup>* O (R \<union> S)\<^sup>*) O R"
using assms by blast
then show ?case by simp
qed
then show ?thesis by auto
qed
then have "R O S\<^sup>* \<subseteq> (R \<union> S)\<^sup>* O R"
using rtrancl_Un_subset by blast
then have "?S \<subseteq> (R \<union> S)\<^sup>* O (R \<union> S)\<^sup>* O R"
by (simp add: relcomp_mono rtrancl_mono)
also have "\<dots> = (R \<union> S)\<^sup>* O R"
by (simp add: O_assoc[symmetric])
finally have "?S O (R \<union> S)\<^sup>* \<subseteq> (R \<union> S)\<^sup>* O R O (R \<union> S)\<^sup>*"
by (simp add: O_assoc[symmetric] relcomp_mono)
also have "\<dots> \<subseteq> (R \<union> S)\<^sup>* O (R \<union> S)\<^sup>* O R"
using * by (simp add: relcomp_mono)
finally have "?S O (R \<union> S)\<^sup>* \<subseteq> (R \<union> S)\<^sup>* O R"
by (simp add: O_assoc[symmetric])
then have "(?S O (R \<union> S)\<^sup>*) `` A \<subseteq> ((R \<union> S)\<^sup>* O R) `` A"
by (simp add: Image_mono)
moreover have "?X \<subseteq> (?S O (R \<union> S)\<^sup>*) `` A"
using A by (auto simp: relcomp_Image)
ultimately have "?X \<subseteq> R `` ?X"
by (auto simp: relcomp_Image)
then have "?X = {}"
using \<open>wf R\<close> by (simp add: wfE_pf)
moreover have "A \<subseteq> ?X" by auto
ultimately show "A = {}" by simp
qed
qed
corollary wf_relcomp_compatible:
assumes "wf R" and "R O S \<subseteq> S O R"
shows "wf (S O R)"
proof -
have "R O S \<subseteq> (R \<union> S)\<^sup>* O R"
using assms by blast
then have "wf (S\<^sup>* O R O S\<^sup>*)"
by (simp add: assms qc_wf_relto_iff)
then show ?thesis
by (rule Wellfounded.wf_subset) blast
qed
subsection \<open>Acyclic relations\<close>
lemma wf_acyclic: "wf r \<Longrightarrow> acyclic r"
by (simp add: acyclic_def) (blast elim: wf_trancl [THEN wf_irrefl])
lemmas wfP_acyclicP = wf_acyclic [to_pred]
subsubsection \<open>Wellfoundedness of finite acyclic relations\<close>
lemma finite_acyclic_wf:
assumes "finite r" "acyclic r" shows "wf r"
using assms
proof (induction r rule: finite_induct)
case (insert x r)
then show ?case
by (cases x) simp
qed simp
lemma finite_acyclic_wf_converse: "finite r \<Longrightarrow> acyclic r \<Longrightarrow> wf (r\<inverse>)"
apply (erule finite_converse [THEN iffD2, THEN finite_acyclic_wf])
apply (erule acyclic_converse [THEN iffD2])
done
text \<open>
Observe that the converse of an irreflexive, transitive,
and finite relation is again well-founded. Thus, we may
employ it for well-founded induction.
\<close>
lemma wf_converse:
assumes "irrefl r" and "trans r" and "finite r"
shows "wf (r\<inverse>)"
proof -
have "acyclic r"
using \<open>irrefl r\<close> and \<open>trans r\<close>
by (simp add: irrefl_def acyclic_irrefl)
with \<open>finite r\<close> show ?thesis
by (rule finite_acyclic_wf_converse)
qed
lemma wf_iff_acyclic_if_finite: "finite r \<Longrightarrow> wf r = acyclic r"
by (blast intro: finite_acyclic_wf wf_acyclic)
subsection \<open>\<^typ>\<open>nat\<close> is well-founded\<close>
lemma less_nat_rel: "(<) = (\<lambda>m n. n = Suc m)\<^sup>+\<^sup>+"
proof (rule ext, rule ext, rule iffI)
fix n m :: nat
show "(\<lambda>m n. n = Suc m)\<^sup>+\<^sup>+ m n" if "m < n"
using that
proof (induct n)
case 0
then show ?case by auto
next
case (Suc n)
then show ?case
by (auto simp add: less_Suc_eq_le le_less intro: tranclp.trancl_into_trancl)
qed
show "m < n" if "(\<lambda>m n. n = Suc m)\<^sup>+\<^sup>+ m n"
using that by (induct n) (simp_all add: less_Suc_eq_le reflexive le_less)
qed
definition pred_nat :: "(nat \<times> nat) set"
where "pred_nat = {(m, n). n = Suc m}"
definition less_than :: "(nat \<times> nat) set"
where "less_than = pred_nat\<^sup>+"
lemma less_eq: "(m, n) \<in> pred_nat\<^sup>+ \<longleftrightarrow> m < n"
unfolding less_nat_rel pred_nat_def trancl_def by simp
lemma pred_nat_trancl_eq_le: "(m, n) \<in> pred_nat\<^sup>* \<longleftrightarrow> m \<le> n"
unfolding less_eq rtrancl_eq_or_trancl by auto
lemma wf_pred_nat: "wf pred_nat"
unfolding wf_def
proof clarify
fix P x
assume "\<forall>x'. (\<forall>y. (y, x') \<in> pred_nat \<longrightarrow> P y) \<longrightarrow> P x'"
then show "P x"
unfolding pred_nat_def by (induction x) blast+
qed
lemma wf_less_than [iff]: "wf less_than"
by (simp add: less_than_def wf_pred_nat [THEN wf_trancl])
lemma trans_less_than [iff]: "trans less_than"
by (simp add: less_than_def)
lemma less_than_iff [iff]: "((x,y) \<in> less_than) = (x<y)"
by (simp add: less_than_def less_eq)
lemma irrefl_less_than: "irrefl less_than"
using irrefl_def by blast
lemma asym_less_than: "asym less_than"
by (rule asymI) simp
lemma total_less_than: "total less_than" and total_on_less_than [simp]: "total_on A less_than"
using total_on_def by force+
lemma wf_less: "wf {(x, y::nat). x < y}"
by (rule Wellfounded.wellorder_class.wf)
subsection \<open>Accessible Part\<close>
text \<open>
Inductive definition of the accessible part \<open>acc r\<close> of a
relation; see also \<^cite>\<open>"paulin-tlca"\<close>.
\<close>
inductive_set acc :: "('a \<times> 'a) set \<Rightarrow> 'a set" for r :: "('a \<times> 'a) set"
where accI: "(\<And>y. (y, x) \<in> r \<Longrightarrow> y \<in> acc r) \<Longrightarrow> x \<in> acc r"
abbreviation termip :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
where "termip r \<equiv> accp (r\<inverse>\<inverse>)"
abbreviation termi :: "('a \<times> 'a) set \<Rightarrow> 'a set"
where "termi r \<equiv> acc (r\<inverse>)"
lemmas accpI = accp.accI
lemma accp_eq_acc [code]: "accp r = (\<lambda>x. x \<in> Wellfounded.acc {(x, y). r x y})"
by (simp add: acc_def)
text \<open>Induction rules\<close>
theorem accp_induct:
assumes major: "accp r a"
assumes hyp: "\<And>x. accp r x \<Longrightarrow> \<forall>y. r y x \<longrightarrow> P y \<Longrightarrow> P x"
shows "P a"
apply (rule major [THEN accp.induct])
apply (rule hyp)
apply (rule accp.accI)
apply auto
done
lemmas accp_induct_rule = accp_induct [rule_format, induct set: accp]
theorem accp_downward: "accp r b \<Longrightarrow> r a b \<Longrightarrow> accp r a"
by (cases rule: accp.cases)
lemma not_accp_down:
assumes na: "\<not> accp R x"
obtains z where "R z x" and "\<not> accp R z"
proof -
assume a: "\<And>z. R z x \<Longrightarrow> \<not> accp R z \<Longrightarrow> thesis"
show thesis
proof (cases "\<forall>z. R z x \<longrightarrow> accp R z")
case True
then have "\<And>z. R z x \<Longrightarrow> accp R z" by auto
then have "accp R x" by (rule accp.accI)
with na show thesis ..
next
case False then obtain z where "R z x" and "\<not> accp R z"
by auto
with a show thesis .
qed
qed
lemma accp_downwards_aux: "r\<^sup>*\<^sup>* b a \<Longrightarrow> accp r a \<longrightarrow> accp r b"
by (erule rtranclp_induct) (blast dest: accp_downward)+
theorem accp_downwards: "accp r a \<Longrightarrow> r\<^sup>*\<^sup>* b a \<Longrightarrow> accp r b"
by (blast dest: accp_downwards_aux)
theorem accp_wfPI: "\<forall>x. accp r x \<Longrightarrow> wfP r"
proof (rule wfPUNIVI)
fix P x
assume "\<forall>x. accp r x" "\<forall>x. (\<forall>y. r y x \<longrightarrow> P y) \<longrightarrow> P x"
then show "P x"
using accp_induct[where P = P] by blast
qed
theorem accp_wfPD: "wfP r \<Longrightarrow> accp r x"
apply (erule wfP_induct_rule)
apply (rule accp.accI)
apply blast
done
theorem wfp_iff_accp: "wfp r = (\<forall>x. accp r x)"
by (blast intro: accp_wfPI dest: accp_wfPD)
text \<open>Smaller relations have bigger accessible parts:\<close>
lemma accp_subset:
assumes "R1 \<le> R2"
shows "accp R2 \<le> accp R1"
proof (rule predicate1I)
fix x
assume "accp R2 x"
then show "accp R1 x"
proof (induct x)
fix x
assume "\<And>y. R2 y x \<Longrightarrow> accp R1 y"
with assms show "accp R1 x"
by (blast intro: accp.accI)
qed
qed
text \<open>This is a generalized induction theorem that works on
subsets of the accessible part.\<close>
lemma accp_subset_induct:
assumes subset: "D \<le> accp R"
and dcl: "\<And>x z. D x \<Longrightarrow> R z x \<Longrightarrow> D z"
and "D x"
and istep: "\<And>x. D x \<Longrightarrow> (\<And>z. R z x \<Longrightarrow> P z) \<Longrightarrow> P x"
shows "P x"
proof -
from subset and \<open>D x\<close>
have "accp R x" ..
then show "P x" using \<open>D x\<close>
proof (induct x)
fix x
assume "D x" and "\<And>y. R y x \<Longrightarrow> D y \<Longrightarrow> P y"
with dcl and istep show "P x" by blast
qed
qed
text \<open>Set versions of the above theorems\<close>
lemmas acc_induct = accp_induct [to_set]
lemmas acc_induct_rule = acc_induct [rule_format, induct set: acc]
lemmas acc_downward = accp_downward [to_set]
lemmas not_acc_down = not_accp_down [to_set]
lemmas acc_downwards_aux = accp_downwards_aux [to_set]
lemmas acc_downwards = accp_downwards [to_set]
lemmas acc_wfI = accp_wfPI [to_set]
lemmas acc_wfD = accp_wfPD [to_set]
lemmas wf_iff_acc = wfp_iff_accp [to_set]
lemmas acc_subset = accp_subset [to_set]
lemmas acc_subset_induct = accp_subset_induct [to_set]
subsection \<open>Tools for building wellfounded relations\<close>
text \<open>Inverse Image\<close>
lemma wf_inv_image [simp,intro!]:
fixes f :: "'a \<Rightarrow> 'b"
assumes "wf r"
shows "wf (inv_image r f)"
proof -
have "\<And>x P. x \<in> P \<Longrightarrow> \<exists>z\<in>P. \<forall>y. (f y, f z) \<in> r \<longrightarrow> y \<notin> P"
proof -
fix P and x::'a
assume "x \<in> P"
then obtain w where w: "w \<in> {w. \<exists>x::'a. x \<in> P \<and> f x = w}"
by auto
have *: "\<And>Q u. u \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q"
using assms by (auto simp add: wf_eq_minimal)
show "\<exists>z\<in>P. \<forall>y. (f y, f z) \<in> r \<longrightarrow> y \<notin> P"
using * [OF w] by auto
qed
then show ?thesis
by (clarsimp simp: inv_image_def wf_eq_minimal)
qed
lemma wfp_on_inv_imagep:
assumes wf: "wfp_on (f ` A) R"
shows "wfp_on A (inv_imagep R f)"
unfolding wfp_on_iff_ex_minimal
proof (intro allI impI)
fix B assume "B \<subseteq> A" and "B \<noteq> {}"
hence "\<exists>z\<in>f ` B. \<forall>y. R y z \<longrightarrow> y \<notin> f ` B"
using wf[unfolded wfp_on_iff_ex_minimal, rule_format, of "f ` B"] by blast
thus "\<exists>z\<in>B. \<forall>y. inv_imagep R f y z \<longrightarrow> y \<notin> B"
unfolding inv_imagep_def
by auto
qed
subsubsection \<open>Conversion to a known well-founded relation\<close>
lemma wfp_on_if_convertible_to_wfp_on:
assumes
wf: "wfp_on (f ` A) Q" and
convertible: "(\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> R x y \<Longrightarrow> Q (f x) (f y))"
shows "wfp_on A R"
unfolding wfp_on_iff_ex_minimal
proof (intro allI impI)
fix B assume "B \<subseteq> A" and "B \<noteq> {}"
moreover from wf have "wfp_on A (inv_imagep Q f)"
by (rule wfp_on_inv_imagep)
ultimately obtain y where "y \<in> B" and "\<And>z. Q (f z) (f y) \<Longrightarrow> z \<notin> B"
unfolding wfp_on_iff_ex_minimal in_inv_imagep
by blast
thus "\<exists>z \<in> B. \<forall>y. R y z \<longrightarrow> y \<notin> B"
using \<open>B \<subseteq> A\<close> convertible by blast
qed
lemma wf_on_if_convertible_to_wf_on: "wf_on (f ` A) Q \<Longrightarrow> (\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (f x, f y) \<in> Q) \<Longrightarrow> wf_on A R"
using wfp_on_if_convertible_to_wfp_on[to_set] .
lemma wf_if_convertible_to_wf:
fixes r :: "'a rel" and s :: "'b rel" and f :: "'a \<Rightarrow> 'b"
assumes "wf s" and convertible: "\<And>x y. (x, y) \<in> r \<Longrightarrow> (f x, f y) \<in> s"
shows "wf r"
proof (rule wf_on_if_convertible_to_wf_on)
show "wf_on (range f) s"
using wf_on_subset[OF \<open>wf s\<close> subset_UNIV] .
next
show "\<And>x y. (x, y) \<in> r \<Longrightarrow> (f x, f y) \<in> s"
using convertible .
qed
lemma wfp_if_convertible_to_wfp: "wfP S \<Longrightarrow> (\<And>x y. R x y \<Longrightarrow> S (f x) (f y)) \<Longrightarrow> wfP R"
using wf_if_convertible_to_wf[to_pred, of S R f] by simp
text \<open>Converting to @{typ nat} is a very common special case that might be found more easily by
Sledgehammer.\<close>
lemma wfp_if_convertible_to_nat:
fixes f :: "_ \<Rightarrow> nat"
shows "(\<And>x y. R x y \<Longrightarrow> f x < f y) \<Longrightarrow> wfP R"
by (rule wfp_if_convertible_to_wfp[of "(<) :: nat \<Rightarrow> nat \<Rightarrow> bool", simplified])
subsubsection \<open>Measure functions into \<^typ>\<open>nat\<close>\<close>
definition measure :: "('a \<Rightarrow> nat) \<Rightarrow> ('a \<times> 'a) set"
where "measure = inv_image less_than"
lemma in_measure[simp, code_unfold]: "(x, y) \<in> measure f \<longleftrightarrow> f x < f y"
by (simp add:measure_def)
lemma wf_measure [iff]: "wf (measure f)"
unfolding measure_def by (rule wf_less_than [THEN wf_inv_image])
lemma wf_if_measure: "(\<And>x. P x \<Longrightarrow> f(g x) < f x) \<Longrightarrow> wf {(y,x). P x \<and> y = g x}"
for f :: "'a \<Rightarrow> nat"
using wf_measure[of f] unfolding measure_def inv_image_def less_than_def less_eq
by (rule wf_subset) auto
subsubsection \<open>Lexicographic combinations\<close>
definition lex_prod :: "('a \<times>'a) set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> (('a \<times> 'b) \<times> ('a \<times> 'b)) set"
(infixr "<*lex*>" 80)
where "ra <*lex*> rb = {((a, b), (a', b')). (a, a') \<in> ra \<or> a = a' \<and> (b, b') \<in> rb}"
lemma in_lex_prod[simp]: "((a, b), (a', b')) \<in> r <*lex*> s \<longleftrightarrow> (a, a') \<in> r \<or> a = a' \<and> (b, b') \<in> s"
by (auto simp:lex_prod_def)
lemma wf_lex_prod [intro!]:
assumes "wf ra" "wf rb"
shows "wf (ra <*lex*> rb)"
proof (rule wfI)
fix z :: "'a \<times> 'b" and P
assume * [rule_format]: "\<forall>u. (\<forall>v. (v, u) \<in> ra <*lex*> rb \<longrightarrow> P v) \<longrightarrow> P u"
obtain x y where zeq: "z = (x,y)"
by fastforce
have "P(x,y)" using \<open>wf ra\<close>
proof (induction x arbitrary: y rule: wf_induct_rule)
case (less x)
note lessx = less
show ?case using \<open>wf rb\<close> less
proof (induction y rule: wf_induct_rule)
case (less y)
show ?case
by (force intro: * less.IH lessx)
qed
qed
then show "P z"
by (simp add: zeq)
qed auto
lemma refl_lex_prod[simp]: "refl r\<^sub>B \<Longrightarrow> refl (r\<^sub>A <*lex*> r\<^sub>B)"
by (auto intro!: reflI dest: refl_onD)
lemma irrefl_on_lex_prod[simp]:
"irrefl_on A r\<^sub>A \<Longrightarrow> irrefl_on B r\<^sub>B \<Longrightarrow> irrefl_on (A \<times> B) (r\<^sub>A <*lex*> r\<^sub>B)"
by (auto intro!: irrefl_onI dest: irrefl_onD)
lemma irrefl_lex_prod[simp]: "irrefl r\<^sub>A \<Longrightarrow> irrefl r\<^sub>B \<Longrightarrow> irrefl (r\<^sub>A <*lex*> r\<^sub>B)"
by (rule irrefl_on_lex_prod[of UNIV _ UNIV, unfolded UNIV_Times_UNIV])
lemma sym_on_lex_prod[simp]:
"sym_on A r\<^sub>A \<Longrightarrow> sym_on B r\<^sub>B \<Longrightarrow> sym_on (A \<times> B) (r\<^sub>A <*lex*> r\<^sub>B)"
by (auto intro!: sym_onI dest: sym_onD)
lemma sym_lex_prod[simp]:
"sym r\<^sub>A \<Longrightarrow> sym r\<^sub>B \<Longrightarrow> sym (r\<^sub>A <*lex*> r\<^sub>B)"
by (rule sym_on_lex_prod[of UNIV _ UNIV, unfolded UNIV_Times_UNIV])
lemma asym_on_lex_prod[simp]:
"asym_on A r\<^sub>A \<Longrightarrow> asym_on B r\<^sub>B \<Longrightarrow> asym_on (A \<times> B) (r\<^sub>A <*lex*> r\<^sub>B)"
by (auto intro!: asym_onI dest: asym_onD)
lemma asym_lex_prod[simp]:
"asym r\<^sub>A \<Longrightarrow> asym r\<^sub>B \<Longrightarrow> asym (r\<^sub>A <*lex*> r\<^sub>B)"
by (rule asym_on_lex_prod[of UNIV _ UNIV, unfolded UNIV_Times_UNIV])
lemma trans_on_lex_prod[simp]:
assumes "trans_on A r\<^sub>A" and "trans_on B r\<^sub>B"
shows "trans_on (A \<times> B) (r\<^sub>A <*lex*> r\<^sub>B)"
proof (rule trans_onI)
fix x y z
show "x \<in> A \<times> B \<Longrightarrow> y \<in> A \<times> B \<Longrightarrow> z \<in> A \<times> B \<Longrightarrow>
(x, y) \<in> r\<^sub>A <*lex*> r\<^sub>B \<Longrightarrow> (y, z) \<in> r\<^sub>A <*lex*> r\<^sub>B \<Longrightarrow> (x, z) \<in> r\<^sub>A <*lex*> r\<^sub>B"
using trans_onD[OF \<open>trans_on A r\<^sub>A\<close>, of "fst x" "fst y" "fst z"]
using trans_onD[OF \<open>trans_on B r\<^sub>B\<close>, of "snd x" "snd y" "snd z"]
by auto
qed
lemma trans_lex_prod [simp,intro!]: "trans r\<^sub>A \<Longrightarrow> trans r\<^sub>B \<Longrightarrow> trans (r\<^sub>A <*lex*> r\<^sub>B)"
by (rule trans_on_lex_prod[of UNIV _ UNIV, unfolded UNIV_Times_UNIV])
lemma total_on_lex_prod[simp]:
"total_on A r\<^sub>A \<Longrightarrow> total_on B r\<^sub>B \<Longrightarrow> total_on (A \<times> B) (r\<^sub>A <*lex*> r\<^sub>B)"
by (auto simp: total_on_def)
lemma total_lex_prod[simp]: "total r\<^sub>A \<Longrightarrow> total r\<^sub>B \<Longrightarrow> total (r\<^sub>A <*lex*> r\<^sub>B)"
by (rule total_on_lex_prod[of UNIV _ UNIV, unfolded UNIV_Times_UNIV])
text \<open>lexicographic combinations with measure functions\<close>
definition mlex_prod :: "('a \<Rightarrow> nat) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" (infixr "<*mlex*>" 80)
where "f <*mlex*> R = inv_image (less_than <*lex*> R) (\<lambda>x. (f x, x))"
lemma
wf_mlex: "wf R \<Longrightarrow> wf (f <*mlex*> R)" and
mlex_less: "f x < f y \<Longrightarrow> (x, y) \<in> f <*mlex*> R" and
mlex_leq: "f x \<le> f y \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (x, y) \<in> f <*mlex*> R" and
mlex_iff: "(x, y) \<in> f <*mlex*> R \<longleftrightarrow> f x < f y \<or> f x = f y \<and> (x, y) \<in> R"
by (auto simp: mlex_prod_def)
text \<open>Proper subset relation on finite sets.\<close>
definition finite_psubset :: "('a set \<times> 'a set) set"
where "finite_psubset = {(A, B). A \<subset> B \<and> finite B}"
lemma wf_finite_psubset[simp]: "wf finite_psubset"
apply (unfold finite_psubset_def)
apply (rule wf_measure [THEN wf_subset])
apply (simp add: measure_def inv_image_def less_than_def less_eq)
apply (fast elim!: psubset_card_mono)
done
lemma trans_finite_psubset: "trans finite_psubset"
by (auto simp: finite_psubset_def less_le trans_def)
lemma in_finite_psubset[simp]: "(A, B) \<in> finite_psubset \<longleftrightarrow> A \<subset> B \<and> finite B"
unfolding finite_psubset_def by auto
text \<open>max- and min-extension of order to finite sets\<close>
inductive_set max_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set"
for R :: "('a \<times> 'a) set"
where max_extI[intro]:
"finite X \<Longrightarrow> finite Y \<Longrightarrow> Y \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> \<exists>y\<in>Y. (x, y) \<in> R) \<Longrightarrow> (X, Y) \<in> max_ext R"
lemma max_ext_wf:
assumes wf: "wf r"
shows "wf (max_ext r)"
proof (rule acc_wfI, intro allI)
show "M \<in> acc (max_ext r)" (is "_ \<in> ?W") for M
proof (induct M rule: infinite_finite_induct)
case empty
show ?case
by (rule accI) (auto elim: max_ext.cases)
next
case (insert a M)
from wf \<open>M \<in> ?W\<close> \<open>finite M\<close> show "insert a M \<in> ?W"
proof (induct arbitrary: M)
fix M a
assume "M \<in> ?W"
assume [intro]: "finite M"
assume hyp: "\<And>b M. (b, a) \<in> r \<Longrightarrow> M \<in> ?W \<Longrightarrow> finite M \<Longrightarrow> insert b M \<in> ?W"
have add_less: "M \<in> ?W \<Longrightarrow> (\<And>y. y \<in> N \<Longrightarrow> (y, a) \<in> r) \<Longrightarrow> N \<union> M \<in> ?W"
if "finite N" "finite M" for N M :: "'a set"
using that by (induct N arbitrary: M) (auto simp: hyp)
show "insert a M \<in> ?W"
proof (rule accI)
fix N
assume Nless: "(N, insert a M) \<in> max_ext r"
then have *: "\<And>x. x \<in> N \<Longrightarrow> (x, a) \<in> r \<or> (\<exists>y \<in> M. (x, y) \<in> r)"
by (auto elim!: max_ext.cases)
let ?N1 = "{n \<in> N. (n, a) \<in> r}"
let ?N2 = "{n \<in> N. (n, a) \<notin> r}"
have N: "?N1 \<union> ?N2 = N" by (rule set_eqI) auto
from Nless have "finite N" by (auto elim: max_ext.cases)
then have finites: "finite ?N1" "finite ?N2" by auto
have "?N2 \<in> ?W"
proof (cases "M = {}")
case [simp]: True
have Mw: "{} \<in> ?W" by (rule accI) (auto elim: max_ext.cases)
from * have "?N2 = {}" by auto
with Mw show "?N2 \<in> ?W" by (simp only:)
next
case False
from * finites have N2: "(?N2, M) \<in> max_ext r"
using max_extI[OF _ _ \<open>M \<noteq> {}\<close>, where ?X = ?N2] by auto
with \<open>M \<in> ?W\<close> show "?N2 \<in> ?W" by (rule acc_downward)
qed
with finites have "?N1 \<union> ?N2 \<in> ?W"
by (rule add_less) simp
then show "N \<in> ?W" by (simp only: N)
qed
qed
next
case infinite
show ?case
by (rule accI) (auto elim: max_ext.cases simp: infinite)
qed
qed
lemma max_ext_additive: "(A, B) \<in> max_ext R \<Longrightarrow> (C, D) \<in> max_ext R \<Longrightarrow> (A \<union> C, B \<union> D) \<in> max_ext R"
by (force elim!: max_ext.cases)
definition min_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set"
where "min_ext r = {(X, Y) | X Y. X \<noteq> {} \<and> (\<forall>y \<in> Y. (\<exists>x \<in> X. (x, y) \<in> r))}"
lemma min_ext_wf:
assumes "wf r"
shows "wf (min_ext r)"
proof (rule wfI_min)
show "\<exists>m \<in> Q. (\<forall>n. (n, m) \<in> min_ext r \<longrightarrow> n \<notin> Q)" if nonempty: "x \<in> Q"
for Q :: "'a set set" and x
proof (cases "Q = {{}}")
case True
then show ?thesis by (simp add: min_ext_def)
next
case False
with nonempty obtain e x where "x \<in> Q" "e \<in> x" by force
then have eU: "e \<in> \<Union>Q" by auto
with \<open>wf r\<close>
obtain z where z: "z \<in> \<Union>Q" "\<And>y. (y, z) \<in> r \<Longrightarrow> y \<notin> \<Union>Q"
by (erule wfE_min)
from z obtain m where "m \<in> Q" "z \<in> m" by auto
from \<open>m \<in> Q\<close> show ?thesis
proof (intro rev_bexI allI impI)
fix n
assume smaller: "(n, m) \<in> min_ext r"
with \<open>z \<in> m\<close> obtain y where "y \<in> n" "(y, z) \<in> r"
by (auto simp: min_ext_def)
with z(2) show "n \<notin> Q" by auto
qed
qed
qed
subsubsection \<open>Bounded increase must terminate\<close>
lemma wf_bounded_measure:
fixes ub :: "'a \<Rightarrow> nat"
and f :: "'a \<Rightarrow> nat"
assumes "\<And>a b. (b, a) \<in> r \<Longrightarrow> ub b \<le> ub a \<and> ub a \<ge> f b \<and> f b > f a"
shows "wf r"
by (rule wf_subset[OF wf_measure[of "\<lambda>a. ub a - f a"]]) (auto dest: assms)
lemma wf_bounded_set:
fixes ub :: "'a \<Rightarrow> 'b set"
and f :: "'a \<Rightarrow> 'b set"
assumes "\<And>a b. (b,a) \<in> r \<Longrightarrow> finite (ub a) \<and> ub b \<subseteq> ub a \<and> ub a \<supseteq> f b \<and> f b \<supset> f a"
shows "wf r"
apply (rule wf_bounded_measure[of r "\<lambda>a. card (ub a)" "\<lambda>a. card (f a)"])
apply (drule assms)
apply (blast intro: card_mono finite_subset psubset_card_mono dest: psubset_eq[THEN iffD2])
done
lemma finite_subset_wf:
assumes "finite A"
shows "wf {(X, Y). X \<subset> Y \<and> Y \<subseteq> A}"
by (rule wf_subset[OF wf_finite_psubset[unfolded finite_psubset_def]])
(auto intro: finite_subset[OF _ assms])
hide_const (open) acc accp
subsection \<open>Code Generation Setup\<close>
text \<open>Code equations with \<^const>\<open>wf\<close> or \<^const>\<open>wfp\<close> on the left-hand side are not supported by the
code generation module because of the \<^const>\<open>UNIV\<close> hidden behind the abbreviations. To sidestep this
problem, we provide the following wrapper definitions and use @{attribute code_abbrev} to register
the definitions with the pre- and post-processors of the code generator.\<close>
definition wf_code :: "('a \<times> 'a) set \<Rightarrow> bool" where
[code_abbrev]: "wf_code r \<longleftrightarrow> wf r"
definition wfp_code :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
[code_abbrev]: "wfp_code R \<longleftrightarrow> wfp R"
end