src/HOL/Wellfounded.thy
author wenzelm
Sat Jul 18 22:58:50 2015 +0200 (2015-07-18)
changeset 60758 d8d85a8172b5
parent 60493 866f41a869e6
child 61337 4645502c3c64
permissions -rw-r--r--
isabelle update_cartouches;
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(*  Title:      HOL/Wellfounded.thy
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    Author:     Tobias Nipkow
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    Author:     Lawrence C Paulson
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    Author:     Konrad Slind
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    Author:     Alexander Krauss
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    Author:     Andrei Popescu, TU Muenchen
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*)
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section \<open>Well-founded Recursion\<close>
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theory Wellfounded
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imports Transitive_Closure
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begin
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subsection \<open>Basic Definitions\<close>
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definition wf :: "('a * 'a) set => bool" where
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  "wf r \<longleftrightarrow> (!P. (!x. (!y. (y,x):r --> P(y)) --> P(x)) --> (!x. P(x)))"
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definition wfP :: "('a => 'a => bool) => bool" where
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  "wfP r \<longleftrightarrow> wf {(x, y). r x y}"
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lemma wfP_wf_eq [pred_set_conv]: "wfP (\<lambda>x y. (x, y) \<in> r) = wf r"
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  by (simp add: wfP_def)
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lemma wfUNIVI: 
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   "(!!P x. (ALL x. (ALL y. (y,x) : r --> P(y)) --> P(x)) ==> P(x)) ==> wf(r)"
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  unfolding wf_def by blast
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lemmas wfPUNIVI = wfUNIVI [to_pred]
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text\<open>Restriction to domain @{term A} and range @{term B}.  If @{term r} is
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    well-founded over their intersection, then @{term "wf r"}\<close>
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lemma wfI: 
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 "[| r \<subseteq> A <*> B; 
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     !!x P. [|\<forall>x. (\<forall>y. (y,x) : r --> P y) --> P x;  x : A; x : B |] ==> P x |]
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  ==>  wf r"
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  unfolding wf_def by blast
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lemma wf_induct: 
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    "[| wf(r);           
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        !!x.[| ALL y. (y,x): r --> P(y) |] ==> P(x)  
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     |]  ==>  P(a)"
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  unfolding wf_def by blast
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lemmas wfP_induct = wf_induct [to_pred]
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lemmas wf_induct_rule = wf_induct [rule_format, consumes 1, case_names less, induct set: wf]
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lemmas wfP_induct_rule = wf_induct_rule [to_pred, induct set: wfP]
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lemma wf_not_sym: "wf r ==> (a, x) : r ==> (x, a) ~: r"
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  by (induct a arbitrary: x set: wf) blast
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lemma wf_asym:
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  assumes "wf r" "(a, x) \<in> r"
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  obtains "(x, a) \<notin> r"
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  by (drule wf_not_sym[OF assms])
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lemma wf_not_refl [simp]: "wf r ==> (a, a) ~: r"
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  by (blast elim: wf_asym)
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lemma wf_irrefl: assumes "wf r" obtains "(a, a) \<notin> r"
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by (drule wf_not_refl[OF assms])
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lemma wf_wellorderI:
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  assumes wf: "wf {(x::'a::ord, y). x < y}"
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  assumes lin: "OFCLASS('a::ord, linorder_class)"
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  shows "OFCLASS('a::ord, wellorder_class)"
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using lin by (rule wellorder_class.intro)
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  (blast intro: class.wellorder_axioms.intro wf_induct_rule [OF wf])
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lemma (in wellorder) wf:
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  "wf {(x, y). x < y}"
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unfolding wf_def by (blast intro: less_induct)
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subsection \<open>Basic Results\<close>
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text \<open>Point-free characterization of well-foundedness\<close>
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lemma wfE_pf:
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  assumes wf: "wf R"
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  assumes a: "A \<subseteq> R `` A"
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  shows "A = {}"
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proof -
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  { fix x
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    from wf have "x \<notin> A"
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    proof induct
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      fix x assume "\<And>y. (y, x) \<in> R \<Longrightarrow> y \<notin> A"
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      then have "x \<notin> R `` A" by blast
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      with a show "x \<notin> A" by blast
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    qed
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  } thus ?thesis by auto
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qed
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lemma wfI_pf:
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  assumes a: "\<And>A. A \<subseteq> R `` A \<Longrightarrow> A = {}"
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  shows "wf R"
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proof (rule wfUNIVI)
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  fix P :: "'a \<Rightarrow> bool" and x
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  let ?A = "{x. \<not> P x}"
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  assume "\<forall>x. (\<forall>y. (y, x) \<in> R \<longrightarrow> P y) \<longrightarrow> P x"
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  then have "?A \<subseteq> R `` ?A" by blast
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  with a show "P x" by blast
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qed
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text\<open>Minimal-element characterization of well-foundedness\<close>
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lemma wfE_min:
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  assumes wf: "wf R" and Q: "x \<in> Q"
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  obtains z where "z \<in> Q" "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
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  using Q wfE_pf[OF wf, of Q] by blast
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lemma wfI_min:
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  assumes a: "\<And>x Q. x \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q"
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  shows "wf R"
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proof (rule wfI_pf)
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  fix A assume b: "A \<subseteq> R `` A"
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  { fix x assume "x \<in> A"
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    from a[OF this] b have "False" by blast
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  }
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  thus "A = {}" by blast
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qed
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lemma wf_eq_minimal: "wf r = (\<forall>Q x. x\<in>Q --> (\<exists>z\<in>Q. \<forall>y. (y,z)\<in>r --> y\<notin>Q))"
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apply auto
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apply (erule wfE_min, assumption, blast)
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apply (rule wfI_min, auto)
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done
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lemmas wfP_eq_minimal = wf_eq_minimal [to_pred]
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text\<open>Well-foundedness of transitive closure\<close>
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lemma wf_trancl:
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  assumes "wf r"
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  shows "wf (r^+)"
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proof -
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  {
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    fix P and x
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    assume induct_step: "!!x. (!!y. (y, x) : r^+ ==> P y) ==> P x"
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    have "P x"
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    proof (rule induct_step)
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      fix y assume "(y, x) : r^+"
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      with \<open>wf r\<close> show "P y"
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      proof (induct x arbitrary: y)
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        case (less x)
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        note hyp = \<open>\<And>x' y'. (x', x) : r ==> (y', x') : r^+ ==> P y'\<close>
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        from \<open>(y, x) : r^+\<close> show "P y"
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        proof cases
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          case base
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          show "P y"
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          proof (rule induct_step)
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            fix y' assume "(y', y) : r^+"
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            with \<open>(y, x) : r\<close> show "P y'" by (rule hyp [of y y'])
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          qed
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        next
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          case step
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          then obtain x' where "(x', x) : r" and "(y, x') : r^+" by simp
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          then show "P y" by (rule hyp [of x' y])
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        qed
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      qed
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    qed
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  } then show ?thesis unfolding wf_def by blast
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qed
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lemmas wfP_trancl = wf_trancl [to_pred]
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lemma wf_converse_trancl: "wf (r^-1) ==> wf ((r^+)^-1)"
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  apply (subst trancl_converse [symmetric])
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  apply (erule wf_trancl)
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  done
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text \<open>Well-foundedness of subsets\<close>
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lemma wf_subset: "[| wf(r);  p<=r |] ==> wf(p)"
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  apply (simp (no_asm_use) add: wf_eq_minimal)
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  apply fast
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  done
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lemmas wfP_subset = wf_subset [to_pred]
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text \<open>Well-foundedness of the empty relation\<close>
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lemma wf_empty [iff]: "wf {}"
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  by (simp add: wf_def)
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lemma wfP_empty [iff]:
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  "wfP (\<lambda>x y. False)"
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proof -
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  have "wfP bot" by (fact wf_empty [to_pred bot_empty_eq2])
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  then show ?thesis by (simp add: bot_fun_def)
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qed
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lemma wf_Int1: "wf r ==> wf (r Int r')"
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  apply (erule wf_subset)
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  apply (rule Int_lower1)
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  done
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lemma wf_Int2: "wf r ==> wf (r' Int r)"
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  apply (erule wf_subset)
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  apply (rule Int_lower2)
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  done  
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text \<open>Exponentiation\<close>
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lemma wf_exp:
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  assumes "wf (R ^^ n)"
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  shows "wf R"
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proof (rule wfI_pf)
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  fix A assume "A \<subseteq> R `` A"
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  then have "A \<subseteq> (R ^^ n) `` A" by (induct n) force+
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  with \<open>wf (R ^^ n)\<close>
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  show "A = {}" by (rule wfE_pf)
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qed
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text \<open>Well-foundedness of insert\<close>
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lemma wf_insert [iff]: "wf(insert (y,x) r) = (wf(r) & (x,y) ~: r^*)"
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apply (rule iffI)
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 apply (blast elim: wf_trancl [THEN wf_irrefl]
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              intro: rtrancl_into_trancl1 wf_subset 
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                     rtrancl_mono [THEN [2] rev_subsetD])
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apply (simp add: wf_eq_minimal, safe)
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apply (rule allE, assumption, erule impE, blast) 
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apply (erule bexE)
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apply (rename_tac "a", case_tac "a = x")
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 prefer 2
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apply blast 
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apply (case_tac "y:Q")
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 prefer 2 apply blast
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apply (rule_tac x = "{z. z:Q & (z,y) : r^*}" in allE)
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 apply assumption
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apply (erule_tac V = "ALL Q. (EX x. x : Q) --> P Q" for P in thin_rl) 
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  --\<open>essential for speed\<close>
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txt\<open>Blast with new substOccur fails\<close>
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apply (fast intro: converse_rtrancl_into_rtrancl)
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done
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text\<open>Well-foundedness of image\<close>
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lemma wf_map_prod_image: "[| wf r; inj f |] ==> wf (map_prod f f ` r)"
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apply (simp only: wf_eq_minimal, clarify)
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apply (case_tac "EX p. f p : Q")
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apply (erule_tac x = "{p. f p : Q}" in allE)
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apply (fast dest: inj_onD, blast)
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done
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subsection \<open>Well-Foundedness Results for Unions\<close>
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lemma wf_union_compatible:
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  assumes "wf R" "wf S"
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  assumes "R O S \<subseteq> R"
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  shows "wf (R \<union> S)"
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proof (rule wfI_min)
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  fix x :: 'a and Q 
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  let ?Q' = "{x \<in> Q. \<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> Q}"
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  assume "x \<in> Q"
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  obtain a where "a \<in> ?Q'"
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    by (rule wfE_min [OF \<open>wf R\<close> \<open>x \<in> Q\<close>]) blast
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  with \<open>wf S\<close>
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  obtain z where "z \<in> ?Q'" and zmin: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> ?Q'" by (erule wfE_min)
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  { 
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    fix y assume "(y, z) \<in> S"
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    then have "y \<notin> ?Q'" by (rule zmin)
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    have "y \<notin> Q"
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    proof 
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      assume "y \<in> Q"
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      with \<open>y \<notin> ?Q'\<close> 
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      obtain w where "(w, y) \<in> R" and "w \<in> Q" by auto
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      from \<open>(w, y) \<in> R\<close> \<open>(y, z) \<in> S\<close> have "(w, z) \<in> R O S" by (rule relcompI)
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      with \<open>R O S \<subseteq> R\<close> have "(w, z) \<in> R" ..
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      with \<open>z \<in> ?Q'\<close> have "w \<notin> Q" by blast 
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      with \<open>w \<in> Q\<close> show False by contradiction
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    qed
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  }
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  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
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qed
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text \<open>Well-foundedness of indexed union with disjoint domains and ranges\<close>
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lemma wf_UN: "[| ALL i:I. wf(r i);  
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         ALL i:I. ALL j:I. r i ~= r j --> Domain(r i) Int Range(r j) = {}  
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      |] ==> wf(UN i:I. r i)"
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apply (simp only: wf_eq_minimal, clarify)
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apply (rename_tac A a, case_tac "EX i:I. EX a:A. EX b:A. (b,a) : r i")
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 prefer 2
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 apply force 
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apply clarify
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apply (drule bspec, assumption)  
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apply (erule_tac x="{a. a:A & (EX b:A. (b,a) : r i) }" in allE)
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apply (blast elim!: allE)  
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done
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lemma wfP_SUP:
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  "\<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 (SUPREMUM UNIV r)"
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  apply (rule wf_UN[to_pred])
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  apply simp_all
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  done
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lemma wf_Union: 
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 "[| ALL r:R. wf r;  
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     ALL r:R. ALL s:R. r ~= s --> Domain r Int Range s = {}  
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  |] ==> wf(Union R)"
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  using wf_UN[of R "\<lambda>i. i"] by simp
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(*Intuition: we find an (R u S)-min element of a nonempty subset A
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             by case distinction.
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  1. There is a step a -R-> b with a,b : A.
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     Pick an R-min element z of the (nonempty) set {a:A | EX b:A. a -R-> b}.
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     By definition, there is z':A s.t. z -R-> z'. Because z is R-min in the
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     subset, z' must be R-min in A. Because z' has an R-predecessor, it cannot
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     have an S-successor and is thus S-min in A as well.
krauss@26748
   318
  2. There is no such step.
krauss@26748
   319
     Pick an S-min element of A. In this case it must be an R-min
krauss@26748
   320
     element of A as well.
krauss@26748
   321
*)
krauss@26748
   322
lemma wf_Un:
krauss@26748
   323
     "[| wf r; wf s; Domain r Int Range s = {} |] ==> wf(r Un s)"
krauss@26748
   324
  using wf_union_compatible[of s r] 
krauss@26748
   325
  by (auto simp: Un_ac)
krauss@26748
   326
krauss@26748
   327
lemma wf_union_merge: 
krauss@32235
   328
  "wf (R \<union> S) = wf (R O R \<union> S O R \<union> S)" (is "wf ?A = wf ?B")
krauss@26748
   329
proof
krauss@26748
   330
  assume "wf ?A"
krauss@26748
   331
  with wf_trancl have wfT: "wf (?A^+)" .
krauss@26748
   332
  moreover have "?B \<subseteq> ?A^+"
krauss@26748
   333
    by (subst trancl_unfold, subst trancl_unfold) blast
krauss@26748
   334
  ultimately show "wf ?B" by (rule wf_subset)
krauss@26748
   335
next
krauss@26748
   336
  assume "wf ?B"
krauss@26748
   337
krauss@26748
   338
  show "wf ?A"
krauss@26748
   339
  proof (rule wfI_min)
krauss@26748
   340
    fix Q :: "'a set" and x 
krauss@26748
   341
    assume "x \<in> Q"
krauss@26748
   342
wenzelm@60758
   343
    with \<open>wf ?B\<close>
krauss@26748
   344
    obtain z where "z \<in> Q" and "\<And>y. (y, z) \<in> ?B \<Longrightarrow> y \<notin> Q" 
krauss@26748
   345
      by (erule wfE_min)
krauss@26748
   346
    then have A1: "\<And>y. (y, z) \<in> R O R \<Longrightarrow> y \<notin> Q"
krauss@32235
   347
      and A2: "\<And>y. (y, z) \<in> S O R \<Longrightarrow> y \<notin> Q"
krauss@26748
   348
      and A3: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> Q"
krauss@26748
   349
      by auto
krauss@26748
   350
    
krauss@26748
   351
    show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> ?A \<longrightarrow> y \<notin> Q"
krauss@26748
   352
    proof (cases "\<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q")
krauss@26748
   353
      case True
wenzelm@60758
   354
      with \<open>z \<in> Q\<close> A3 show ?thesis by blast
krauss@26748
   355
    next
krauss@26748
   356
      case False 
krauss@26748
   357
      then obtain z' where "z'\<in>Q" "(z', z) \<in> R" by blast
krauss@26748
   358
krauss@26748
   359
      have "\<forall>y. (y, z') \<in> ?A \<longrightarrow> y \<notin> Q"
krauss@26748
   360
      proof (intro allI impI)
krauss@26748
   361
        fix y assume "(y, z') \<in> ?A"
krauss@26748
   362
        then show "y \<notin> Q"
krauss@26748
   363
        proof
krauss@26748
   364
          assume "(y, z') \<in> R" 
wenzelm@60758
   365
          then have "(y, z) \<in> R O R" using \<open>(z', z) \<in> R\<close> ..
krauss@26748
   366
          with A1 show "y \<notin> Q" .
krauss@26748
   367
        next
krauss@26748
   368
          assume "(y, z') \<in> S" 
wenzelm@60758
   369
          then have "(y, z) \<in> S O R" using  \<open>(z', z) \<in> R\<close> ..
krauss@26748
   370
          with A2 show "y \<notin> Q" .
krauss@26748
   371
        qed
krauss@26748
   372
      qed
wenzelm@60758
   373
      with \<open>z' \<in> Q\<close> show ?thesis ..
krauss@26748
   374
    qed
krauss@26748
   375
  qed
krauss@26748
   376
qed
krauss@26748
   377
wenzelm@60758
   378
lemma wf_comp_self: "wf R = wf (R O R)"  -- \<open>special case\<close>
krauss@26748
   379
  by (rule wf_union_merge [where S = "{}", simplified])
krauss@26748
   380
krauss@26748
   381
wenzelm@60758
   382
subsection \<open>Well-Foundedness of Composition\<close>
nipkow@60148
   383
lp15@60493
   384
text \<open>Bachmair and Dershowitz 1986, Lemma 2. [Provided by Tjark Weber]\<close>
nipkow@60148
   385
lp15@60493
   386
lemma qc_wf_relto_iff:
lp15@60493
   387
  assumes "R O S \<subseteq> (R \<union> S)\<^sup>* O R" -- \<open>R quasi-commutes over S\<close>
lp15@60493
   388
  shows "wf (S\<^sup>* O R O S\<^sup>*) \<longleftrightarrow> wf R" (is "wf ?S \<longleftrightarrow> _")
lp15@60493
   389
proof
lp15@60493
   390
  assume "wf ?S"
lp15@60493
   391
  moreover have "R \<subseteq> ?S" by auto
lp15@60493
   392
  ultimately show "wf R" using wf_subset by auto
lp15@60493
   393
next
lp15@60493
   394
  assume "wf R"
lp15@60493
   395
  show "wf ?S"
lp15@60493
   396
  proof (rule wfI_pf)
lp15@60493
   397
    fix A assume A: "A \<subseteq> ?S `` A"
lp15@60493
   398
    let ?X = "(R \<union> S)\<^sup>* `` A"
lp15@60493
   399
    have *: "R O (R \<union> S)\<^sup>* \<subseteq> (R \<union> S)\<^sup>* O R"
lp15@60493
   400
      proof -
lp15@60493
   401
        { fix x y z assume "(y, z) \<in> (R \<union> S)\<^sup>*" and "(x, y) \<in> R"
lp15@60493
   402
          then have "(x, z) \<in> (R \<union> S)\<^sup>* O R"
lp15@60493
   403
          proof (induct y z)
lp15@60493
   404
            case rtrancl_refl then show ?case by auto
lp15@60493
   405
          next
lp15@60493
   406
            case (rtrancl_into_rtrancl a b c)
lp15@60493
   407
            then have "(x, c) \<in> ((R \<union> S)\<^sup>* O (R \<union> S)\<^sup>*) O R" using assms by blast
lp15@60493
   408
            then show ?case by simp
lp15@60493
   409
          qed }
lp15@60493
   410
        then show ?thesis by auto
lp15@60493
   411
      qed
lp15@60493
   412
    then have "R O S\<^sup>* \<subseteq> (R \<union> S)\<^sup>* O R" using rtrancl_Un_subset by blast
lp15@60493
   413
    then have "?S \<subseteq> (R \<union> S)\<^sup>* O (R \<union> S)\<^sup>* O R" by (simp add: relcomp_mono rtrancl_mono)
lp15@60493
   414
    also have "\<dots> = (R \<union> S)\<^sup>* O R" by (simp add: O_assoc[symmetric])
lp15@60493
   415
    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)
lp15@60493
   416
    also have "\<dots> \<subseteq> (R \<union> S)\<^sup>* O (R \<union> S)\<^sup>* O R" using * by (simp add: relcomp_mono)
lp15@60493
   417
    finally have "?S O (R \<union> S)\<^sup>* \<subseteq> (R \<union> S)\<^sup>* O R" by (simp add: O_assoc[symmetric])
lp15@60493
   418
    then have "(?S O (R \<union> S)\<^sup>*) `` A \<subseteq> ((R \<union> S)\<^sup>* O R) `` A" by (simp add: Image_mono)
lp15@60493
   419
    moreover have "?X \<subseteq> (?S O (R \<union> S)\<^sup>*) `` A" using A by (auto simp: relcomp_Image)
lp15@60493
   420
    ultimately have "?X \<subseteq> R `` ?X" by (auto simp: relcomp_Image)
lp15@60493
   421
    then have "?X = {}" using \<open>wf R\<close> by (simp add: wfE_pf)
lp15@60493
   422
    moreover have "A \<subseteq> ?X" by auto
lp15@60493
   423
    ultimately show "A = {}" by simp
lp15@60493
   424
  qed
lp15@60493
   425
qed
lp15@60493
   426
lp15@60493
   427
corollary wf_relcomp_compatible:
nipkow@60148
   428
  assumes "wf R" and "R O S \<subseteq> S O R"
nipkow@60148
   429
  shows "wf (S O R)"
lp15@60493
   430
proof -
lp15@60493
   431
  have "R O S \<subseteq> (R \<union> S)\<^sup>* O R"
lp15@60493
   432
    using assms by blast
lp15@60493
   433
  then have "wf (S\<^sup>* O R O S\<^sup>*)"
lp15@60493
   434
    by (simp add: assms qc_wf_relto_iff)
lp15@60493
   435
  then show ?thesis
lp15@60493
   436
    by (rule Wellfounded.wf_subset) blast
nipkow@60148
   437
qed
nipkow@60148
   438
nipkow@60148
   439
wenzelm@60758
   440
subsection \<open>Acyclic relations\<close>
krauss@33217
   441
krauss@26748
   442
lemma wf_acyclic: "wf r ==> acyclic r"
krauss@26748
   443
apply (simp add: acyclic_def)
krauss@26748
   444
apply (blast elim: wf_trancl [THEN wf_irrefl])
krauss@26748
   445
done
krauss@26748
   446
krauss@26748
   447
lemmas wfP_acyclicP = wf_acyclic [to_pred]
krauss@26748
   448
wenzelm@60758
   449
text\<open>Wellfoundedness of finite acyclic relations\<close>
krauss@26748
   450
krauss@26748
   451
lemma finite_acyclic_wf [rule_format]: "finite r ==> acyclic r --> wf r"
krauss@26748
   452
apply (erule finite_induct, blast)
krauss@26748
   453
apply (simp (no_asm_simp) only: split_tupled_all)
krauss@26748
   454
apply simp
krauss@26748
   455
done
krauss@26748
   456
krauss@26748
   457
lemma finite_acyclic_wf_converse: "[|finite r; acyclic r|] ==> wf (r^-1)"
krauss@26748
   458
apply (erule finite_converse [THEN iffD2, THEN finite_acyclic_wf])
krauss@26748
   459
apply (erule acyclic_converse [THEN iffD2])
krauss@26748
   460
done
krauss@26748
   461
krauss@26748
   462
lemma wf_iff_acyclic_if_finite: "finite r ==> wf r = acyclic r"
krauss@26748
   463
by (blast intro: finite_acyclic_wf wf_acyclic)
krauss@26748
   464
krauss@26748
   465
wenzelm@60758
   466
subsection \<open>@{typ nat} is well-founded\<close>
krauss@26748
   467
krauss@26748
   468
lemma less_nat_rel: "op < = (\<lambda>m n. n = Suc m)^++"
krauss@26748
   469
proof (rule ext, rule ext, rule iffI)
krauss@26748
   470
  fix n m :: nat
krauss@26748
   471
  assume "m < n"
krauss@26748
   472
  then show "(\<lambda>m n. n = Suc m)^++ m n"
krauss@26748
   473
  proof (induct n)
krauss@26748
   474
    case 0 then show ?case by auto
krauss@26748
   475
  next
krauss@26748
   476
    case (Suc n) then show ?case
krauss@26748
   477
      by (auto simp add: less_Suc_eq_le le_less intro: tranclp.trancl_into_trancl)
krauss@26748
   478
  qed
krauss@26748
   479
next
krauss@26748
   480
  fix n m :: nat
krauss@26748
   481
  assume "(\<lambda>m n. n = Suc m)^++ m n"
krauss@26748
   482
  then show "m < n"
krauss@26748
   483
    by (induct n)
krauss@26748
   484
      (simp_all add: less_Suc_eq_le reflexive le_less)
krauss@26748
   485
qed
krauss@26748
   486
krauss@26748
   487
definition
krauss@26748
   488
  pred_nat :: "(nat * nat) set" where
krauss@26748
   489
  "pred_nat = {(m, n). n = Suc m}"
krauss@26748
   490
krauss@26748
   491
definition
krauss@26748
   492
  less_than :: "(nat * nat) set" where
krauss@26748
   493
  "less_than = pred_nat^+"
krauss@26748
   494
krauss@26748
   495
lemma less_eq: "(m, n) \<in> pred_nat^+ \<longleftrightarrow> m < n"
krauss@26748
   496
  unfolding less_nat_rel pred_nat_def trancl_def by simp
krauss@26748
   497
krauss@26748
   498
lemma pred_nat_trancl_eq_le:
krauss@26748
   499
  "(m, n) \<in> pred_nat^* \<longleftrightarrow> m \<le> n"
krauss@26748
   500
  unfolding less_eq rtrancl_eq_or_trancl by auto
krauss@26748
   501
krauss@26748
   502
lemma wf_pred_nat: "wf pred_nat"
krauss@26748
   503
  apply (unfold wf_def pred_nat_def, clarify)
krauss@26748
   504
  apply (induct_tac x, blast+)
krauss@26748
   505
  done
krauss@26748
   506
krauss@26748
   507
lemma wf_less_than [iff]: "wf less_than"
krauss@26748
   508
  by (simp add: less_than_def wf_pred_nat [THEN wf_trancl])
krauss@26748
   509
krauss@26748
   510
lemma trans_less_than [iff]: "trans less_than"
huffman@35216
   511
  by (simp add: less_than_def)
krauss@26748
   512
krauss@26748
   513
lemma less_than_iff [iff]: "((x,y): less_than) = (x<y)"
krauss@26748
   514
  by (simp add: less_than_def less_eq)
krauss@26748
   515
krauss@26748
   516
lemma wf_less: "wf {(x, y::nat). x < y}"
lp15@60493
   517
  by (rule Wellfounded.wellorder_class.wf)
krauss@26748
   518
krauss@26748
   519
wenzelm@60758
   520
subsection \<open>Accessible Part\<close>
krauss@26748
   521
wenzelm@60758
   522
text \<open>
krauss@26748
   523
 Inductive definition of the accessible part @{term "acc r"} of a
wenzelm@58623
   524
 relation; see also @{cite "paulin-tlca"}.
wenzelm@60758
   525
\<close>
krauss@26748
   526
krauss@26748
   527
inductive_set
krauss@26748
   528
  acc :: "('a * 'a) set => 'a set"
krauss@26748
   529
  for r :: "('a * 'a) set"
krauss@26748
   530
  where
krauss@26748
   531
    accI: "(!!y. (y, x) : r ==> y : acc r) ==> x : acc r"
krauss@26748
   532
krauss@26748
   533
abbreviation
krauss@26748
   534
  termip :: "('a => 'a => bool) => 'a => bool" where
haftmann@45137
   535
  "termip r \<equiv> accp (r\<inverse>\<inverse>)"
krauss@26748
   536
krauss@26748
   537
abbreviation
krauss@26748
   538
  termi :: "('a * 'a) set => 'a set" where
haftmann@45137
   539
  "termi r \<equiv> acc (r\<inverse>)"
krauss@26748
   540
krauss@26748
   541
lemmas accpI = accp.accI
krauss@26748
   542
haftmann@54295
   543
lemma accp_eq_acc [code]:
haftmann@54295
   544
  "accp r = (\<lambda>x. x \<in> Wellfounded.acc {(x, y). r x y})"
haftmann@54295
   545
  by (simp add: acc_def)
haftmann@54295
   546
haftmann@54295
   547
wenzelm@60758
   548
text \<open>Induction rules\<close>
krauss@26748
   549
krauss@26748
   550
theorem accp_induct:
krauss@26748
   551
  assumes major: "accp r a"
krauss@26748
   552
  assumes hyp: "!!x. accp r x ==> \<forall>y. r y x --> P y ==> P x"
krauss@26748
   553
  shows "P a"
krauss@26748
   554
  apply (rule major [THEN accp.induct])
krauss@26748
   555
  apply (rule hyp)
krauss@26748
   556
   apply (rule accp.accI)
krauss@26748
   557
   apply fast
krauss@26748
   558
  apply fast
krauss@26748
   559
  done
krauss@26748
   560
krauss@26748
   561
theorems accp_induct_rule = accp_induct [rule_format, induct set: accp]
krauss@26748
   562
krauss@26748
   563
theorem accp_downward: "accp r b ==> r a b ==> accp r a"
krauss@26748
   564
  apply (erule accp.cases)
krauss@26748
   565
  apply fast
krauss@26748
   566
  done
krauss@26748
   567
krauss@26748
   568
lemma not_accp_down:
krauss@26748
   569
  assumes na: "\<not> accp R x"
krauss@26748
   570
  obtains z where "R z x" and "\<not> accp R z"
krauss@26748
   571
proof -
krauss@26748
   572
  assume a: "\<And>z. \<lbrakk>R z x; \<not> accp R z\<rbrakk> \<Longrightarrow> thesis"
krauss@26748
   573
krauss@26748
   574
  show thesis
krauss@26748
   575
  proof (cases "\<forall>z. R z x \<longrightarrow> accp R z")
krauss@26748
   576
    case True
krauss@26748
   577
    hence "\<And>z. R z x \<Longrightarrow> accp R z" by auto
krauss@26748
   578
    hence "accp R x"
krauss@26748
   579
      by (rule accp.accI)
krauss@26748
   580
    with na show thesis ..
krauss@26748
   581
  next
krauss@26748
   582
    case False then obtain z where "R z x" and "\<not> accp R z"
krauss@26748
   583
      by auto
krauss@26748
   584
    with a show thesis .
krauss@26748
   585
  qed
krauss@26748
   586
qed
krauss@26748
   587
krauss@26748
   588
lemma accp_downwards_aux: "r\<^sup>*\<^sup>* b a ==> accp r a --> accp r b"
krauss@26748
   589
  apply (erule rtranclp_induct)
krauss@26748
   590
   apply blast
krauss@26748
   591
  apply (blast dest: accp_downward)
krauss@26748
   592
  done
krauss@26748
   593
krauss@26748
   594
theorem accp_downwards: "accp r a ==> r\<^sup>*\<^sup>* b a ==> accp r b"
krauss@26748
   595
  apply (blast dest: accp_downwards_aux)
krauss@26748
   596
  done
krauss@26748
   597
krauss@26748
   598
theorem accp_wfPI: "\<forall>x. accp r x ==> wfP r"
krauss@26748
   599
  apply (rule wfPUNIVI)
huffman@44921
   600
  apply (rule_tac P=P in accp_induct)
krauss@26748
   601
   apply blast
krauss@26748
   602
  apply blast
krauss@26748
   603
  done
krauss@26748
   604
krauss@26748
   605
theorem accp_wfPD: "wfP r ==> accp r x"
krauss@26748
   606
  apply (erule wfP_induct_rule)
krauss@26748
   607
  apply (rule accp.accI)
krauss@26748
   608
  apply blast
krauss@26748
   609
  done
krauss@26748
   610
krauss@26748
   611
theorem wfP_accp_iff: "wfP r = (\<forall>x. accp r x)"
krauss@26748
   612
  apply (blast intro: accp_wfPI dest: accp_wfPD)
krauss@26748
   613
  done
krauss@26748
   614
krauss@26748
   615
wenzelm@60758
   616
text \<open>Smaller relations have bigger accessible parts:\<close>
krauss@26748
   617
krauss@26748
   618
lemma accp_subset:
krauss@26748
   619
  assumes sub: "R1 \<le> R2"
krauss@26748
   620
  shows "accp R2 \<le> accp R1"
berghofe@26803
   621
proof (rule predicate1I)
krauss@26748
   622
  fix x assume "accp R2 x"
krauss@26748
   623
  then show "accp R1 x"
krauss@26748
   624
  proof (induct x)
krauss@26748
   625
    fix x
krauss@26748
   626
    assume ih: "\<And>y. R2 y x \<Longrightarrow> accp R1 y"
krauss@26748
   627
    with sub show "accp R1 x"
krauss@26748
   628
      by (blast intro: accp.accI)
krauss@26748
   629
  qed
krauss@26748
   630
qed
krauss@26748
   631
krauss@26748
   632
wenzelm@60758
   633
text \<open>This is a generalized induction theorem that works on
wenzelm@60758
   634
  subsets of the accessible part.\<close>
krauss@26748
   635
krauss@26748
   636
lemma accp_subset_induct:
krauss@26748
   637
  assumes subset: "D \<le> accp R"
krauss@26748
   638
    and dcl: "\<And>x z. \<lbrakk>D x; R z x\<rbrakk> \<Longrightarrow> D z"
krauss@26748
   639
    and "D x"
krauss@26748
   640
    and istep: "\<And>x. \<lbrakk>D x; (\<And>z. R z x \<Longrightarrow> P z)\<rbrakk> \<Longrightarrow> P x"
krauss@26748
   641
  shows "P x"
krauss@26748
   642
proof -
wenzelm@60758
   643
  from subset and \<open>D x\<close>
krauss@26748
   644
  have "accp R x" ..
wenzelm@60758
   645
  then show "P x" using \<open>D x\<close>
krauss@26748
   646
  proof (induct x)
krauss@26748
   647
    fix x
krauss@26748
   648
    assume "D x"
krauss@26748
   649
      and "\<And>y. R y x \<Longrightarrow> D y \<Longrightarrow> P y"
krauss@26748
   650
    with dcl and istep show "P x" by blast
krauss@26748
   651
  qed
krauss@26748
   652
qed
krauss@26748
   653
krauss@26748
   654
wenzelm@60758
   655
text \<open>Set versions of the above theorems\<close>
krauss@26748
   656
krauss@26748
   657
lemmas acc_induct = accp_induct [to_set]
krauss@26748
   658
krauss@26748
   659
lemmas acc_induct_rule = acc_induct [rule_format, induct set: acc]
krauss@26748
   660
krauss@26748
   661
lemmas acc_downward = accp_downward [to_set]
krauss@26748
   662
krauss@26748
   663
lemmas not_acc_down = not_accp_down [to_set]
krauss@26748
   664
krauss@26748
   665
lemmas acc_downwards_aux = accp_downwards_aux [to_set]
krauss@26748
   666
krauss@26748
   667
lemmas acc_downwards = accp_downwards [to_set]
krauss@26748
   668
krauss@26748
   669
lemmas acc_wfI = accp_wfPI [to_set]
krauss@26748
   670
krauss@26748
   671
lemmas acc_wfD = accp_wfPD [to_set]
krauss@26748
   672
krauss@26748
   673
lemmas wf_acc_iff = wfP_accp_iff [to_set]
krauss@26748
   674
berghofe@46177
   675
lemmas acc_subset = accp_subset [to_set]
krauss@26748
   676
berghofe@46177
   677
lemmas acc_subset_induct = accp_subset_induct [to_set]
krauss@26748
   678
krauss@26748
   679
wenzelm@60758
   680
subsection \<open>Tools for building wellfounded relations\<close>
krauss@26748
   681
wenzelm@60758
   682
text \<open>Inverse Image\<close>
krauss@26748
   683
krauss@26748
   684
lemma wf_inv_image [simp,intro!]: "wf(r) ==> wf(inv_image r (f::'a=>'b))"
krauss@26748
   685
apply (simp (no_asm_use) add: inv_image_def wf_eq_minimal)
krauss@26748
   686
apply clarify
krauss@26748
   687
apply (subgoal_tac "EX (w::'b) . w : {w. EX (x::'a) . x: Q & (f x = w) }")
krauss@26748
   688
prefer 2 apply (blast del: allE)
krauss@26748
   689
apply (erule allE)
krauss@26748
   690
apply (erule (1) notE impE)
krauss@26748
   691
apply blast
krauss@26748
   692
done
krauss@26748
   693
wenzelm@60758
   694
text \<open>Measure functions into @{typ nat}\<close>
krauss@26748
   695
krauss@26748
   696
definition measure :: "('a => nat) => ('a * 'a)set"
haftmann@45137
   697
where "measure = inv_image less_than"
krauss@26748
   698
bulwahn@46356
   699
lemma in_measure[simp, code_unfold]: "((x,y) : measure f) = (f x < f y)"
krauss@26748
   700
  by (simp add:measure_def)
krauss@26748
   701
krauss@26748
   702
lemma wf_measure [iff]: "wf (measure f)"
krauss@26748
   703
apply (unfold measure_def)
krauss@26748
   704
apply (rule wf_less_than [THEN wf_inv_image])
krauss@26748
   705
done
krauss@26748
   706
nipkow@41720
   707
lemma wf_if_measure: fixes f :: "'a \<Rightarrow> nat"
nipkow@41720
   708
shows "(!!x. P x \<Longrightarrow> f(g x) < f x) \<Longrightarrow> wf {(y,x). P x \<and> y = g x}"
nipkow@41720
   709
apply(insert wf_measure[of f])
nipkow@41720
   710
apply(simp only: measure_def inv_image_def less_than_def less_eq)
nipkow@41720
   711
apply(erule wf_subset)
nipkow@41720
   712
apply auto
nipkow@41720
   713
done
nipkow@41720
   714
nipkow@41720
   715
wenzelm@60758
   716
text\<open>Lexicographic combinations\<close>
krauss@26748
   717
haftmann@37767
   718
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
haftmann@37767
   719
  "ra <*lex*> rb = {((a, b), (a', b')). (a, a') \<in> ra \<or> a = a' \<and> (b, b') \<in> rb}"
krauss@26748
   720
krauss@26748
   721
lemma wf_lex_prod [intro!]: "[| wf(ra); wf(rb) |] ==> wf(ra <*lex*> rb)"
krauss@26748
   722
apply (unfold wf_def lex_prod_def) 
krauss@26748
   723
apply (rule allI, rule impI)
krauss@26748
   724
apply (simp (no_asm_use) only: split_paired_All)
krauss@26748
   725
apply (drule spec, erule mp) 
krauss@26748
   726
apply (rule allI, rule impI)
krauss@26748
   727
apply (drule spec, erule mp, blast) 
krauss@26748
   728
done
krauss@26748
   729
krauss@26748
   730
lemma in_lex_prod[simp]: 
krauss@26748
   731
  "(((a,b),(a',b')): r <*lex*> s) = ((a,a'): r \<or> (a = a' \<and> (b, b') : s))"
krauss@26748
   732
  by (auto simp:lex_prod_def)
krauss@26748
   733
wenzelm@60758
   734
text\<open>@{term "op <*lex*>"} preserves transitivity\<close>
krauss@26748
   735
krauss@26748
   736
lemma trans_lex_prod [intro!]: 
krauss@26748
   737
    "[| trans R1; trans R2 |] ==> trans (R1 <*lex*> R2)"
krauss@26748
   738
by (unfold trans_def lex_prod_def, blast) 
krauss@26748
   739
wenzelm@60758
   740
text \<open>lexicographic combinations with measure functions\<close>
krauss@26748
   741
krauss@26748
   742
definition 
krauss@26748
   743
  mlex_prod :: "('a \<Rightarrow> nat) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" (infixr "<*mlex*>" 80)
krauss@26748
   744
where
krauss@26748
   745
  "f <*mlex*> R = inv_image (less_than <*lex*> R) (%x. (f x, x))"
krauss@26748
   746
krauss@26748
   747
lemma wf_mlex: "wf R \<Longrightarrow> wf (f <*mlex*> R)"
krauss@26748
   748
unfolding mlex_prod_def
krauss@26748
   749
by auto
krauss@26748
   750
krauss@26748
   751
lemma mlex_less: "f x < f y \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
krauss@26748
   752
unfolding mlex_prod_def by simp
krauss@26748
   753
krauss@26748
   754
lemma mlex_leq: "f x \<le> f y \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
krauss@26748
   755
unfolding mlex_prod_def by auto
krauss@26748
   756
wenzelm@60758
   757
text \<open>proper subset relation on finite sets\<close>
krauss@26748
   758
krauss@26748
   759
definition finite_psubset  :: "('a set * 'a set) set"
haftmann@45137
   760
where "finite_psubset = {(A,B). A < B & finite B}"
krauss@26748
   761
krauss@28260
   762
lemma wf_finite_psubset[simp]: "wf(finite_psubset)"
krauss@26748
   763
apply (unfold finite_psubset_def)
krauss@26748
   764
apply (rule wf_measure [THEN wf_subset])
krauss@26748
   765
apply (simp add: measure_def inv_image_def less_than_def less_eq)
krauss@26748
   766
apply (fast elim!: psubset_card_mono)
krauss@26748
   767
done
krauss@26748
   768
krauss@26748
   769
lemma trans_finite_psubset: "trans finite_psubset"
berghofe@26803
   770
by (simp add: finite_psubset_def less_le trans_def, blast)
krauss@26748
   771
krauss@28260
   772
lemma in_finite_psubset[simp]: "(A, B) \<in> finite_psubset = (A < B & finite B)"
krauss@28260
   773
unfolding finite_psubset_def by auto
krauss@26748
   774
wenzelm@60758
   775
text \<open>max- and min-extension of order to finite sets\<close>
krauss@28735
   776
krauss@28735
   777
inductive_set max_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set" 
krauss@28735
   778
for R :: "('a \<times> 'a) set"
krauss@28735
   779
where
krauss@28735
   780
  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"
krauss@28735
   781
krauss@28735
   782
lemma max_ext_wf:
krauss@28735
   783
  assumes wf: "wf r"
krauss@28735
   784
  shows "wf (max_ext r)"
krauss@28735
   785
proof (rule acc_wfI, intro allI)
krauss@28735
   786
  fix M show "M \<in> acc (max_ext r)" (is "_ \<in> ?W")
krauss@28735
   787
  proof cases
krauss@28735
   788
    assume "finite M"
krauss@28735
   789
    thus ?thesis
krauss@28735
   790
    proof (induct M)
krauss@28735
   791
      show "{} \<in> ?W"
krauss@28735
   792
        by (rule accI) (auto elim: max_ext.cases)
krauss@28735
   793
    next
krauss@28735
   794
      fix M a assume "M \<in> ?W" "finite M"
krauss@28735
   795
      with wf show "insert a M \<in> ?W"
krauss@28735
   796
      proof (induct arbitrary: M)
krauss@28735
   797
        fix M a
krauss@28735
   798
        assume "M \<in> ?W"  and  [intro]: "finite M"
krauss@28735
   799
        assume hyp: "\<And>b M. (b, a) \<in> r \<Longrightarrow> M \<in> ?W \<Longrightarrow> finite M \<Longrightarrow> insert b M \<in> ?W"
krauss@28735
   800
        {
krauss@28735
   801
          fix N M :: "'a set"
krauss@28735
   802
          assume "finite N" "finite M"
krauss@28735
   803
          then
krauss@28735
   804
          have "\<lbrakk>M \<in> ?W ; (\<And>y. y \<in> N \<Longrightarrow> (y, a) \<in> r)\<rbrakk> \<Longrightarrow>  N \<union> M \<in> ?W"
krauss@28735
   805
            by (induct N arbitrary: M) (auto simp: hyp)
krauss@28735
   806
        }
krauss@28735
   807
        note add_less = this
krauss@28735
   808
        
krauss@28735
   809
        show "insert a M \<in> ?W"
krauss@28735
   810
        proof (rule accI)
krauss@28735
   811
          fix N assume Nless: "(N, insert a M) \<in> max_ext r"
krauss@28735
   812
          hence asm1: "\<And>x. x \<in> N \<Longrightarrow> (x, a) \<in> r \<or> (\<exists>y \<in> M. (x, y) \<in> r)"
krauss@28735
   813
            by (auto elim!: max_ext.cases)
krauss@28735
   814
krauss@28735
   815
          let ?N1 = "{ n \<in> N. (n, a) \<in> r }"
krauss@28735
   816
          let ?N2 = "{ n \<in> N. (n, a) \<notin> r }"
nipkow@39302
   817
          have N: "?N1 \<union> ?N2 = N" by (rule set_eqI) auto
krauss@28735
   818
          from Nless have "finite N" by (auto elim: max_ext.cases)
krauss@28735
   819
          then have finites: "finite ?N1" "finite ?N2" by auto
krauss@28735
   820
          
krauss@28735
   821
          have "?N2 \<in> ?W"
krauss@28735
   822
          proof cases
krauss@28735
   823
            assume [simp]: "M = {}"
krauss@28735
   824
            have Mw: "{} \<in> ?W" by (rule accI) (auto elim: max_ext.cases)
krauss@28735
   825
krauss@28735
   826
            from asm1 have "?N2 = {}" by auto
krauss@28735
   827
            with Mw show "?N2 \<in> ?W" by (simp only:)
krauss@28735
   828
          next
krauss@28735
   829
            assume "M \<noteq> {}"
bulwahn@49945
   830
            from asm1 finites have N2: "(?N2, M) \<in> max_ext r" 
wenzelm@60758
   831
              by (rule_tac max_extI[OF _ _ \<open>M \<noteq> {}\<close>]) auto
bulwahn@49945
   832
wenzelm@60758
   833
            with \<open>M \<in> ?W\<close> show "?N2 \<in> ?W" by (rule acc_downward)
krauss@28735
   834
          qed
krauss@28735
   835
          with finites have "?N1 \<union> ?N2 \<in> ?W" 
krauss@28735
   836
            by (rule add_less) simp
krauss@28735
   837
          then show "N \<in> ?W" by (simp only: N)
krauss@28735
   838
        qed
krauss@28735
   839
      qed
krauss@28735
   840
    qed
krauss@28735
   841
  next
krauss@28735
   842
    assume [simp]: "\<not> finite M"
krauss@28735
   843
    show ?thesis
krauss@28735
   844
      by (rule accI) (auto elim: max_ext.cases)
krauss@28735
   845
  qed
krauss@28735
   846
qed
krauss@28735
   847
krauss@29125
   848
lemma max_ext_additive: 
krauss@29125
   849
 "(A, B) \<in> max_ext R \<Longrightarrow> (C, D) \<in> max_ext R \<Longrightarrow>
krauss@29125
   850
  (A \<union> C, B \<union> D) \<in> max_ext R"
krauss@29125
   851
by (force elim!: max_ext.cases)
krauss@29125
   852
krauss@28735
   853
haftmann@37767
   854
definition min_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set"  where
haftmann@37767
   855
  "min_ext r = {(X, Y) | X Y. X \<noteq> {} \<and> (\<forall>y \<in> Y. (\<exists>x \<in> X. (x, y) \<in> r))}"
krauss@28735
   856
krauss@28735
   857
lemma min_ext_wf:
krauss@28735
   858
  assumes "wf r"
krauss@28735
   859
  shows "wf (min_ext r)"
krauss@28735
   860
proof (rule wfI_min)
krauss@28735
   861
  fix Q :: "'a set set"
krauss@28735
   862
  fix x
krauss@28735
   863
  assume nonempty: "x \<in> Q"
krauss@28735
   864
  show "\<exists>m \<in> Q. (\<forall> n. (n, m) \<in> min_ext r \<longrightarrow> n \<notin> Q)"
krauss@28735
   865
  proof cases
krauss@28735
   866
    assume "Q = {{}}" thus ?thesis by (simp add: min_ext_def)
krauss@28735
   867
  next
krauss@28735
   868
    assume "Q \<noteq> {{}}"
krauss@28735
   869
    with nonempty
krauss@28735
   870
    obtain e x where "x \<in> Q" "e \<in> x" by force
krauss@28735
   871
    then have eU: "e \<in> \<Union>Q" by auto
wenzelm@60758
   872
    with \<open>wf r\<close> 
krauss@28735
   873
    obtain z where z: "z \<in> \<Union>Q" "\<And>y. (y, z) \<in> r \<Longrightarrow> y \<notin> \<Union>Q" 
krauss@28735
   874
      by (erule wfE_min)
krauss@28735
   875
    from z obtain m where "m \<in> Q" "z \<in> m" by auto
wenzelm@60758
   876
    from \<open>m \<in> Q\<close>
krauss@28735
   877
    show ?thesis
krauss@28735
   878
    proof (rule, intro bexI allI impI)
krauss@28735
   879
      fix n
krauss@28735
   880
      assume smaller: "(n, m) \<in> min_ext r"
wenzelm@60758
   881
      with \<open>z \<in> m\<close> obtain y where y: "y \<in> n" "(y, z) \<in> r" by (auto simp: min_ext_def)
krauss@28735
   882
      then show "n \<notin> Q" using z(2) by auto
krauss@28735
   883
    qed      
krauss@28735
   884
  qed
krauss@28735
   885
qed
krauss@26748
   886
wenzelm@60758
   887
text\<open>Bounded increase must terminate:\<close>
nipkow@43137
   888
nipkow@43137
   889
lemma wf_bounded_measure:
nipkow@43137
   890
fixes ub :: "'a \<Rightarrow> nat" and f :: "'a \<Rightarrow> nat"
nipkow@43140
   891
assumes "!!a b. (b,a) : r \<Longrightarrow> ub b \<le> ub a & ub a \<ge> f b & f b > f a"
nipkow@43137
   892
shows "wf r"
nipkow@43137
   893
apply(rule wf_subset[OF wf_measure[of "%a. ub a - f a"]])
nipkow@43137
   894
apply (auto dest: assms)
nipkow@43137
   895
done
nipkow@43137
   896
nipkow@43137
   897
lemma wf_bounded_set:
nipkow@43137
   898
fixes ub :: "'a \<Rightarrow> 'b set" and f :: "'a \<Rightarrow> 'b set"
nipkow@43137
   899
assumes "!!a b. (b,a) : r \<Longrightarrow>
nipkow@43140
   900
  finite(ub a) & ub b \<subseteq> ub a & ub a \<supseteq> f b & f b \<supset> f a"
nipkow@43137
   901
shows "wf r"
nipkow@43137
   902
apply(rule wf_bounded_measure[of r "%a. card(ub a)" "%a. card(f a)"])
nipkow@43137
   903
apply(drule assms)
nipkow@43140
   904
apply (blast intro: card_mono finite_subset psubset_card_mono dest: psubset_eq[THEN iffD2])
nipkow@43137
   905
done
nipkow@43137
   906
krauss@26748
   907
haftmann@54295
   908
hide_const (open) acc accp
haftmann@54295
   909
krauss@26748
   910
end