src/HOL/Wellfounded.thy
author haftmann
Wed Sep 23 16:32:53 2009 +0200 (2009-09-23)
changeset 32704 6f0a56d255f4
parent 32463 3a0a65ca2261
child 32960 69916a850301
permissions -rw-r--r--
simplified proof
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(*  Author:     Tobias Nipkow
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Author:     Konrad Slind, Alexander Krauss
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    Copyright   1992-2008  University of Cambridge and TU Muenchen
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*)
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header {*Well-founded Recursion*}
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theory Wellfounded
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imports Finite_Set Transitive_Closure
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uses ("Tools/Function/size.ML")
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begin
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subsection {* Basic Definitions *}
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constdefs
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  wf         :: "('a * 'a)set => bool"
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  "wf(r) == (!P. (!x. (!y. (y,x):r --> P(y)) --> P(x)) --> (!x. P(x)))"
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  wfP :: "('a => 'a => bool) => bool"
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  "wfP r == wf {(x, y). r x y}"
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  acyclic :: "('a*'a)set => bool"
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  "acyclic r == !x. (x,x) ~: r^+"
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abbreviation acyclicP :: "('a => 'a => bool) => bool" where
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  "acyclicP r == acyclic {(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{*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"}*}
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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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(* [| wf r;  ~Z ==> (a,x) : r;  (x,a) ~: r ==> Z |] ==> Z *)
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lemmas wf_asym = wf_not_sym [elim_format]
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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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(* [| wf r;  (a,a) ~: r ==> PROP W |] ==> PROP W *)
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lemmas wf_irrefl = wf_not_refl [elim_format]
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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: 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 {* Basic Results *}
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text{*transitive closure of a well-founded relation is well-founded! *}
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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 `wf r` 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 = `\<And>x' y'. (x', x) : r ==> (y', x') : r^+ ==> P y'`
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	from `(y, x) : r^+` 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 `(y, x) : r` 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{*Minimal-element characterization of well-foundedness*}
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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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proof (intro iffI strip)
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  fix Q :: "'a set" and x
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  assume "wf r" and "x \<in> Q"
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  then show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q"
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    unfolding wf_def
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    by (blast dest: spec [of _ "%x. x\<in>Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y,z) \<in> r \<longrightarrow> y\<notin>Q)"]) 
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next
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  assume 1: "\<forall>Q x. x \<in> Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q)"
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  show "wf r"
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  proof (rule wfUNIVI)
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    fix P :: "'a \<Rightarrow> bool" and x
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    assume 2: "\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x"
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    let ?Q = "{x. \<not> P x}"
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    have "x \<in> ?Q \<longrightarrow> (\<exists>z \<in> ?Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> ?Q)"
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      by (rule 1 [THEN spec, THEN spec])
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    then have "\<not> P x \<longrightarrow> (\<exists>z. \<not> P z \<and> (\<forall>y. (y, z) \<in> r \<longrightarrow> P y))" by simp
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    with 2 have "\<not> P x \<longrightarrow> (\<exists>z. \<not> P z \<and> P z)" by fast
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    then show "P x" by simp
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  qed
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qed
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lemma wfE_min: 
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  assumes "wf R" "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 assms unfolding wf_eq_minimal by blast
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lemma wfI_min:
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  "(\<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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  \<Longrightarrow> wf R"
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  unfolding wf_eq_minimal by blast
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lemmas wfP_eq_minimal = wf_eq_minimal [to_pred]
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text {* Well-foundedness of subsets *}
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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 {* Well-foundedness of the empty relation *}
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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_eq bot_bool_eq)
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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{*Well-foundedness of insert*}
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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" in thin_rl) 
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  --{*essential for speed*}
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txt{*Blast with new substOccur fails*}
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apply (fast intro: converse_rtrancl_into_rtrancl)
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done
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text{*Well-foundedness of image*}
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lemma wf_prod_fun_image: "[| wf r; inj f |] ==> wf(prod_fun 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 {* Well-Foundedness Results for Unions *}
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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 `wf R` `x \<in> Q`]) blast
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  with `wf S`
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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 `y \<notin> ?Q'` 
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      obtain w where "(w, y) \<in> R" and "w \<in> Q" by auto
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      from `(w, y) \<in> R` `(y, z) \<in> S` have "(w, z) \<in> R O S" by (rule rel_compI)
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      with `R O S \<subseteq> R` have "(w, z) \<in> R" ..
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      with `z \<in> ?Q'` have "w \<notin> Q" by blast 
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      with `w \<in> Q` show False by contradiction
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    qed
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  }
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  with `z \<in> ?Q'` 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 {* Well-foundedness of indexed union with disjoint domains and ranges *}
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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 (SUPR UNIV r)"
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  by (rule wf_UN [where I=UNIV and r="\<lambda>i. {(x, y). r i x y}", to_pred SUP_UN_eq2])
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    (simp_all add: Collect_def)
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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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apply (simp add: Union_def)
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apply (blast intro: wf_UN)
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done
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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.
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  2. There is no such step.
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     Pick an S-min element of A. In this case it must be an R-min
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     element of A as well.
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*)
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lemma wf_Un:
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     "[| wf r; wf s; Domain r Int Range s = {} |] ==> wf(r Un s)"
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  using wf_union_compatible[of s r] 
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  by (auto simp: Un_ac)
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lemma wf_union_merge: 
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  "wf (R \<union> S) = wf (R O R \<union> S O R \<union> S)" (is "wf ?A = wf ?B")
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proof
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  assume "wf ?A"
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  with wf_trancl have wfT: "wf (?A^+)" .
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  moreover have "?B \<subseteq> ?A^+"
krauss@26748
   304
    by (subst trancl_unfold, subst trancl_unfold) blast
krauss@26748
   305
  ultimately show "wf ?B" by (rule wf_subset)
krauss@26748
   306
next
krauss@26748
   307
  assume "wf ?B"
krauss@26748
   308
krauss@26748
   309
  show "wf ?A"
krauss@26748
   310
  proof (rule wfI_min)
krauss@26748
   311
    fix Q :: "'a set" and x 
krauss@26748
   312
    assume "x \<in> Q"
krauss@26748
   313
krauss@26748
   314
    with `wf ?B`
krauss@26748
   315
    obtain z where "z \<in> Q" and "\<And>y. (y, z) \<in> ?B \<Longrightarrow> y \<notin> Q" 
krauss@26748
   316
      by (erule wfE_min)
krauss@26748
   317
    then have A1: "\<And>y. (y, z) \<in> R O R \<Longrightarrow> y \<notin> Q"
krauss@32235
   318
      and A2: "\<And>y. (y, z) \<in> S O R \<Longrightarrow> y \<notin> Q"
krauss@26748
   319
      and A3: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> Q"
krauss@26748
   320
      by auto
krauss@26748
   321
    
krauss@26748
   322
    show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> ?A \<longrightarrow> y \<notin> Q"
krauss@26748
   323
    proof (cases "\<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q")
krauss@26748
   324
      case True
krauss@26748
   325
      with `z \<in> Q` A3 show ?thesis by blast
krauss@26748
   326
    next
krauss@26748
   327
      case False 
krauss@26748
   328
      then obtain z' where "z'\<in>Q" "(z', z) \<in> R" by blast
krauss@26748
   329
krauss@26748
   330
      have "\<forall>y. (y, z') \<in> ?A \<longrightarrow> y \<notin> Q"
krauss@26748
   331
      proof (intro allI impI)
krauss@26748
   332
        fix y assume "(y, z') \<in> ?A"
krauss@26748
   333
        then show "y \<notin> Q"
krauss@26748
   334
        proof
krauss@26748
   335
          assume "(y, z') \<in> R" 
krauss@26748
   336
          then have "(y, z) \<in> R O R" using `(z', z) \<in> R` ..
krauss@26748
   337
          with A1 show "y \<notin> Q" .
krauss@26748
   338
        next
krauss@26748
   339
          assume "(y, z') \<in> S" 
krauss@32235
   340
          then have "(y, z) \<in> S O R" using  `(z', z) \<in> R` ..
krauss@26748
   341
          with A2 show "y \<notin> Q" .
krauss@26748
   342
        qed
krauss@26748
   343
      qed
krauss@26748
   344
      with `z' \<in> Q` show ?thesis ..
krauss@26748
   345
    qed
krauss@26748
   346
  qed
krauss@26748
   347
qed
krauss@26748
   348
krauss@26748
   349
lemma wf_comp_self: "wf R = wf (R O R)"  -- {* special case *}
krauss@26748
   350
  by (rule wf_union_merge [where S = "{}", simplified])
krauss@26748
   351
krauss@26748
   352
krauss@26748
   353
subsubsection {* acyclic *}
krauss@26748
   354
krauss@26748
   355
lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r"
krauss@26748
   356
  by (simp add: acyclic_def)
krauss@26748
   357
krauss@26748
   358
lemma wf_acyclic: "wf r ==> acyclic r"
krauss@26748
   359
apply (simp add: acyclic_def)
krauss@26748
   360
apply (blast elim: wf_trancl [THEN wf_irrefl])
krauss@26748
   361
done
krauss@26748
   362
krauss@26748
   363
lemmas wfP_acyclicP = wf_acyclic [to_pred]
krauss@26748
   364
krauss@26748
   365
lemma acyclic_insert [iff]:
krauss@26748
   366
     "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"
krauss@26748
   367
apply (simp add: acyclic_def trancl_insert)
krauss@26748
   368
apply (blast intro: rtrancl_trans)
krauss@26748
   369
done
krauss@26748
   370
krauss@26748
   371
lemma acyclic_converse [iff]: "acyclic(r^-1) = acyclic r"
krauss@26748
   372
by (simp add: acyclic_def trancl_converse)
krauss@26748
   373
krauss@26748
   374
lemmas acyclicP_converse [iff] = acyclic_converse [to_pred]
krauss@26748
   375
krauss@26748
   376
lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)"
krauss@26748
   377
apply (simp add: acyclic_def antisym_def)
krauss@26748
   378
apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
krauss@26748
   379
done
krauss@26748
   380
krauss@26748
   381
(* Other direction:
krauss@26748
   382
acyclic = no loops
krauss@26748
   383
antisym = only self loops
krauss@26748
   384
Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r - Id)
krauss@26748
   385
==> antisym( r^* ) = acyclic(r - Id)";
krauss@26748
   386
*)
krauss@26748
   387
krauss@26748
   388
lemma acyclic_subset: "[| acyclic s; r <= s |] ==> acyclic r"
krauss@26748
   389
apply (simp add: acyclic_def)
krauss@26748
   390
apply (blast intro: trancl_mono)
krauss@26748
   391
done
krauss@26748
   392
krauss@26748
   393
text{* Wellfoundedness of finite acyclic relations*}
krauss@26748
   394
krauss@26748
   395
lemma finite_acyclic_wf [rule_format]: "finite r ==> acyclic r --> wf r"
krauss@26748
   396
apply (erule finite_induct, blast)
krauss@26748
   397
apply (simp (no_asm_simp) only: split_tupled_all)
krauss@26748
   398
apply simp
krauss@26748
   399
done
krauss@26748
   400
krauss@26748
   401
lemma finite_acyclic_wf_converse: "[|finite r; acyclic r|] ==> wf (r^-1)"
krauss@26748
   402
apply (erule finite_converse [THEN iffD2, THEN finite_acyclic_wf])
krauss@26748
   403
apply (erule acyclic_converse [THEN iffD2])
krauss@26748
   404
done
krauss@26748
   405
krauss@26748
   406
lemma wf_iff_acyclic_if_finite: "finite r ==> wf r = acyclic r"
krauss@26748
   407
by (blast intro: finite_acyclic_wf wf_acyclic)
krauss@26748
   408
krauss@26748
   409
krauss@26748
   410
subsection {* @{typ nat} is well-founded *}
krauss@26748
   411
krauss@26748
   412
lemma less_nat_rel: "op < = (\<lambda>m n. n = Suc m)^++"
krauss@26748
   413
proof (rule ext, rule ext, rule iffI)
krauss@26748
   414
  fix n m :: nat
krauss@26748
   415
  assume "m < n"
krauss@26748
   416
  then show "(\<lambda>m n. n = Suc m)^++ m n"
krauss@26748
   417
  proof (induct n)
krauss@26748
   418
    case 0 then show ?case by auto
krauss@26748
   419
  next
krauss@26748
   420
    case (Suc n) then show ?case
krauss@26748
   421
      by (auto simp add: less_Suc_eq_le le_less intro: tranclp.trancl_into_trancl)
krauss@26748
   422
  qed
krauss@26748
   423
next
krauss@26748
   424
  fix n m :: nat
krauss@26748
   425
  assume "(\<lambda>m n. n = Suc m)^++ m n"
krauss@26748
   426
  then show "m < n"
krauss@26748
   427
    by (induct n)
krauss@26748
   428
      (simp_all add: less_Suc_eq_le reflexive le_less)
krauss@26748
   429
qed
krauss@26748
   430
krauss@26748
   431
definition
krauss@26748
   432
  pred_nat :: "(nat * nat) set" where
krauss@26748
   433
  "pred_nat = {(m, n). n = Suc m}"
krauss@26748
   434
krauss@26748
   435
definition
krauss@26748
   436
  less_than :: "(nat * nat) set" where
krauss@26748
   437
  "less_than = pred_nat^+"
krauss@26748
   438
krauss@26748
   439
lemma less_eq: "(m, n) \<in> pred_nat^+ \<longleftrightarrow> m < n"
krauss@26748
   440
  unfolding less_nat_rel pred_nat_def trancl_def by simp
krauss@26748
   441
krauss@26748
   442
lemma pred_nat_trancl_eq_le:
krauss@26748
   443
  "(m, n) \<in> pred_nat^* \<longleftrightarrow> m \<le> n"
krauss@26748
   444
  unfolding less_eq rtrancl_eq_or_trancl by auto
krauss@26748
   445
krauss@26748
   446
lemma wf_pred_nat: "wf pred_nat"
krauss@26748
   447
  apply (unfold wf_def pred_nat_def, clarify)
krauss@26748
   448
  apply (induct_tac x, blast+)
krauss@26748
   449
  done
krauss@26748
   450
krauss@26748
   451
lemma wf_less_than [iff]: "wf less_than"
krauss@26748
   452
  by (simp add: less_than_def wf_pred_nat [THEN wf_trancl])
krauss@26748
   453
krauss@26748
   454
lemma trans_less_than [iff]: "trans less_than"
krauss@26748
   455
  by (simp add: less_than_def trans_trancl)
krauss@26748
   456
krauss@26748
   457
lemma less_than_iff [iff]: "((x,y): less_than) = (x<y)"
krauss@26748
   458
  by (simp add: less_than_def less_eq)
krauss@26748
   459
krauss@26748
   460
lemma wf_less: "wf {(x, y::nat). x < y}"
krauss@26748
   461
  using wf_less_than by (simp add: less_than_def less_eq [symmetric])
krauss@26748
   462
krauss@26748
   463
krauss@26748
   464
subsection {* Accessible Part *}
krauss@26748
   465
krauss@26748
   466
text {*
krauss@26748
   467
 Inductive definition of the accessible part @{term "acc r"} of a
krauss@26748
   468
 relation; see also \cite{paulin-tlca}.
krauss@26748
   469
*}
krauss@26748
   470
krauss@26748
   471
inductive_set
krauss@26748
   472
  acc :: "('a * 'a) set => 'a set"
krauss@26748
   473
  for r :: "('a * 'a) set"
krauss@26748
   474
  where
krauss@26748
   475
    accI: "(!!y. (y, x) : r ==> y : acc r) ==> x : acc r"
krauss@26748
   476
krauss@26748
   477
abbreviation
krauss@26748
   478
  termip :: "('a => 'a => bool) => 'a => bool" where
krauss@26748
   479
  "termip r == accp (r\<inverse>\<inverse>)"
krauss@26748
   480
krauss@26748
   481
abbreviation
krauss@26748
   482
  termi :: "('a * 'a) set => 'a set" where
krauss@26748
   483
  "termi r == acc (r\<inverse>)"
krauss@26748
   484
krauss@26748
   485
lemmas accpI = accp.accI
krauss@26748
   486
krauss@26748
   487
text {* Induction rules *}
krauss@26748
   488
krauss@26748
   489
theorem accp_induct:
krauss@26748
   490
  assumes major: "accp r a"
krauss@26748
   491
  assumes hyp: "!!x. accp r x ==> \<forall>y. r y x --> P y ==> P x"
krauss@26748
   492
  shows "P a"
krauss@26748
   493
  apply (rule major [THEN accp.induct])
krauss@26748
   494
  apply (rule hyp)
krauss@26748
   495
   apply (rule accp.accI)
krauss@26748
   496
   apply fast
krauss@26748
   497
  apply fast
krauss@26748
   498
  done
krauss@26748
   499
krauss@26748
   500
theorems accp_induct_rule = accp_induct [rule_format, induct set: accp]
krauss@26748
   501
krauss@26748
   502
theorem accp_downward: "accp r b ==> r a b ==> accp r a"
krauss@26748
   503
  apply (erule accp.cases)
krauss@26748
   504
  apply fast
krauss@26748
   505
  done
krauss@26748
   506
krauss@26748
   507
lemma not_accp_down:
krauss@26748
   508
  assumes na: "\<not> accp R x"
krauss@26748
   509
  obtains z where "R z x" and "\<not> accp R z"
krauss@26748
   510
proof -
krauss@26748
   511
  assume a: "\<And>z. \<lbrakk>R z x; \<not> accp R z\<rbrakk> \<Longrightarrow> thesis"
krauss@26748
   512
krauss@26748
   513
  show thesis
krauss@26748
   514
  proof (cases "\<forall>z. R z x \<longrightarrow> accp R z")
krauss@26748
   515
    case True
krauss@26748
   516
    hence "\<And>z. R z x \<Longrightarrow> accp R z" by auto
krauss@26748
   517
    hence "accp R x"
krauss@26748
   518
      by (rule accp.accI)
krauss@26748
   519
    with na show thesis ..
krauss@26748
   520
  next
krauss@26748
   521
    case False then obtain z where "R z x" and "\<not> accp R z"
krauss@26748
   522
      by auto
krauss@26748
   523
    with a show thesis .
krauss@26748
   524
  qed
krauss@26748
   525
qed
krauss@26748
   526
krauss@26748
   527
lemma accp_downwards_aux: "r\<^sup>*\<^sup>* b a ==> accp r a --> accp r b"
krauss@26748
   528
  apply (erule rtranclp_induct)
krauss@26748
   529
   apply blast
krauss@26748
   530
  apply (blast dest: accp_downward)
krauss@26748
   531
  done
krauss@26748
   532
krauss@26748
   533
theorem accp_downwards: "accp r a ==> r\<^sup>*\<^sup>* b a ==> accp r b"
krauss@26748
   534
  apply (blast dest: accp_downwards_aux)
krauss@26748
   535
  done
krauss@26748
   536
krauss@26748
   537
theorem accp_wfPI: "\<forall>x. accp r x ==> wfP r"
krauss@26748
   538
  apply (rule wfPUNIVI)
krauss@26748
   539
  apply (induct_tac P x rule: accp_induct)
krauss@26748
   540
   apply blast
krauss@26748
   541
  apply blast
krauss@26748
   542
  done
krauss@26748
   543
krauss@26748
   544
theorem accp_wfPD: "wfP r ==> accp r x"
krauss@26748
   545
  apply (erule wfP_induct_rule)
krauss@26748
   546
  apply (rule accp.accI)
krauss@26748
   547
  apply blast
krauss@26748
   548
  done
krauss@26748
   549
krauss@26748
   550
theorem wfP_accp_iff: "wfP r = (\<forall>x. accp r x)"
krauss@26748
   551
  apply (blast intro: accp_wfPI dest: accp_wfPD)
krauss@26748
   552
  done
krauss@26748
   553
krauss@26748
   554
krauss@26748
   555
text {* Smaller relations have bigger accessible parts: *}
krauss@26748
   556
krauss@26748
   557
lemma accp_subset:
krauss@26748
   558
  assumes sub: "R1 \<le> R2"
krauss@26748
   559
  shows "accp R2 \<le> accp R1"
berghofe@26803
   560
proof (rule predicate1I)
krauss@26748
   561
  fix x assume "accp R2 x"
krauss@26748
   562
  then show "accp R1 x"
krauss@26748
   563
  proof (induct x)
krauss@26748
   564
    fix x
krauss@26748
   565
    assume ih: "\<And>y. R2 y x \<Longrightarrow> accp R1 y"
krauss@26748
   566
    with sub show "accp R1 x"
krauss@26748
   567
      by (blast intro: accp.accI)
krauss@26748
   568
  qed
krauss@26748
   569
qed
krauss@26748
   570
krauss@26748
   571
krauss@26748
   572
text {* This is a generalized induction theorem that works on
krauss@26748
   573
  subsets of the accessible part. *}
krauss@26748
   574
krauss@26748
   575
lemma accp_subset_induct:
krauss@26748
   576
  assumes subset: "D \<le> accp R"
krauss@26748
   577
    and dcl: "\<And>x z. \<lbrakk>D x; R z x\<rbrakk> \<Longrightarrow> D z"
krauss@26748
   578
    and "D x"
krauss@26748
   579
    and istep: "\<And>x. \<lbrakk>D x; (\<And>z. R z x \<Longrightarrow> P z)\<rbrakk> \<Longrightarrow> P x"
krauss@26748
   580
  shows "P x"
krauss@26748
   581
proof -
krauss@26748
   582
  from subset and `D x`
krauss@26748
   583
  have "accp R x" ..
krauss@26748
   584
  then show "P x" using `D x`
krauss@26748
   585
  proof (induct x)
krauss@26748
   586
    fix x
krauss@26748
   587
    assume "D x"
krauss@26748
   588
      and "\<And>y. R y x \<Longrightarrow> D y \<Longrightarrow> P y"
krauss@26748
   589
    with dcl and istep show "P x" by blast
krauss@26748
   590
  qed
krauss@26748
   591
qed
krauss@26748
   592
krauss@26748
   593
krauss@26748
   594
text {* Set versions of the above theorems *}
krauss@26748
   595
krauss@26748
   596
lemmas acc_induct = accp_induct [to_set]
krauss@26748
   597
krauss@26748
   598
lemmas acc_induct_rule = acc_induct [rule_format, induct set: acc]
krauss@26748
   599
krauss@26748
   600
lemmas acc_downward = accp_downward [to_set]
krauss@26748
   601
krauss@26748
   602
lemmas not_acc_down = not_accp_down [to_set]
krauss@26748
   603
krauss@26748
   604
lemmas acc_downwards_aux = accp_downwards_aux [to_set]
krauss@26748
   605
krauss@26748
   606
lemmas acc_downwards = accp_downwards [to_set]
krauss@26748
   607
krauss@26748
   608
lemmas acc_wfI = accp_wfPI [to_set]
krauss@26748
   609
krauss@26748
   610
lemmas acc_wfD = accp_wfPD [to_set]
krauss@26748
   611
krauss@26748
   612
lemmas wf_acc_iff = wfP_accp_iff [to_set]
krauss@26748
   613
berghofe@26803
   614
lemmas acc_subset = accp_subset [to_set pred_subset_eq]
krauss@26748
   615
berghofe@26803
   616
lemmas acc_subset_induct = accp_subset_induct [to_set pred_subset_eq]
krauss@26748
   617
krauss@26748
   618
krauss@26748
   619
subsection {* Tools for building wellfounded relations *}
krauss@26748
   620
krauss@26748
   621
text {* Inverse Image *}
krauss@26748
   622
krauss@26748
   623
lemma wf_inv_image [simp,intro!]: "wf(r) ==> wf(inv_image r (f::'a=>'b))"
krauss@26748
   624
apply (simp (no_asm_use) add: inv_image_def wf_eq_minimal)
krauss@26748
   625
apply clarify
krauss@26748
   626
apply (subgoal_tac "EX (w::'b) . w : {w. EX (x::'a) . x: Q & (f x = w) }")
krauss@26748
   627
prefer 2 apply (blast del: allE)
krauss@26748
   628
apply (erule allE)
krauss@26748
   629
apply (erule (1) notE impE)
krauss@26748
   630
apply blast
krauss@26748
   631
done
krauss@26748
   632
haftmann@31775
   633
text {* Measure Datatypes into @{typ nat} *}
krauss@26748
   634
krauss@26748
   635
definition measure :: "('a => nat) => ('a * 'a)set"
krauss@26748
   636
where "measure == inv_image less_than"
krauss@26748
   637
krauss@26748
   638
lemma in_measure[simp]: "((x,y) : measure f) = (f x < f y)"
krauss@26748
   639
  by (simp add:measure_def)
krauss@26748
   640
krauss@26748
   641
lemma wf_measure [iff]: "wf (measure f)"
krauss@26748
   642
apply (unfold measure_def)
krauss@26748
   643
apply (rule wf_less_than [THEN wf_inv_image])
krauss@26748
   644
done
krauss@26748
   645
krauss@26748
   646
text{* Lexicographic combinations *}
krauss@26748
   647
krauss@26748
   648
definition
krauss@26748
   649
 lex_prod  :: "[('a*'a)set, ('b*'b)set] => (('a*'b)*('a*'b))set"
krauss@26748
   650
               (infixr "<*lex*>" 80)
krauss@26748
   651
where
krauss@26748
   652
    "ra <*lex*> rb == {((a,b),(a',b')). (a,a') : ra | a=a' & (b,b') : rb}"
krauss@26748
   653
krauss@26748
   654
lemma wf_lex_prod [intro!]: "[| wf(ra); wf(rb) |] ==> wf(ra <*lex*> rb)"
krauss@26748
   655
apply (unfold wf_def lex_prod_def) 
krauss@26748
   656
apply (rule allI, rule impI)
krauss@26748
   657
apply (simp (no_asm_use) only: split_paired_All)
krauss@26748
   658
apply (drule spec, erule mp) 
krauss@26748
   659
apply (rule allI, rule impI)
krauss@26748
   660
apply (drule spec, erule mp, blast) 
krauss@26748
   661
done
krauss@26748
   662
krauss@26748
   663
lemma in_lex_prod[simp]: 
krauss@26748
   664
  "(((a,b),(a',b')): r <*lex*> s) = ((a,a'): r \<or> (a = a' \<and> (b, b') : s))"
krauss@26748
   665
  by (auto simp:lex_prod_def)
krauss@26748
   666
krauss@26748
   667
text{* @{term "op <*lex*>"} preserves transitivity *}
krauss@26748
   668
krauss@26748
   669
lemma trans_lex_prod [intro!]: 
krauss@26748
   670
    "[| trans R1; trans R2 |] ==> trans (R1 <*lex*> R2)"
krauss@26748
   671
by (unfold trans_def lex_prod_def, blast) 
krauss@26748
   672
haftmann@31775
   673
text {* lexicographic combinations with measure Datatypes *}
krauss@26748
   674
krauss@26748
   675
definition 
krauss@26748
   676
  mlex_prod :: "('a \<Rightarrow> nat) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" (infixr "<*mlex*>" 80)
krauss@26748
   677
where
krauss@26748
   678
  "f <*mlex*> R = inv_image (less_than <*lex*> R) (%x. (f x, x))"
krauss@26748
   679
krauss@26748
   680
lemma wf_mlex: "wf R \<Longrightarrow> wf (f <*mlex*> R)"
krauss@26748
   681
unfolding mlex_prod_def
krauss@26748
   682
by auto
krauss@26748
   683
krauss@26748
   684
lemma mlex_less: "f x < f y \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
krauss@26748
   685
unfolding mlex_prod_def by simp
krauss@26748
   686
krauss@26748
   687
lemma mlex_leq: "f x \<le> f y \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
krauss@26748
   688
unfolding mlex_prod_def by auto
krauss@26748
   689
krauss@26748
   690
text {* proper subset relation on finite sets *}
krauss@26748
   691
krauss@26748
   692
definition finite_psubset  :: "('a set * 'a set) set"
krauss@26748
   693
where "finite_psubset == {(A,B). A < B & finite B}"
krauss@26748
   694
krauss@28260
   695
lemma wf_finite_psubset[simp]: "wf(finite_psubset)"
krauss@26748
   696
apply (unfold finite_psubset_def)
krauss@26748
   697
apply (rule wf_measure [THEN wf_subset])
krauss@26748
   698
apply (simp add: measure_def inv_image_def less_than_def less_eq)
krauss@26748
   699
apply (fast elim!: psubset_card_mono)
krauss@26748
   700
done
krauss@26748
   701
krauss@26748
   702
lemma trans_finite_psubset: "trans finite_psubset"
berghofe@26803
   703
by (simp add: finite_psubset_def less_le trans_def, blast)
krauss@26748
   704
krauss@28260
   705
lemma in_finite_psubset[simp]: "(A, B) \<in> finite_psubset = (A < B & finite B)"
krauss@28260
   706
unfolding finite_psubset_def by auto
krauss@26748
   707
krauss@28735
   708
text {* max- and min-extension of order to finite sets *}
krauss@28735
   709
krauss@28735
   710
inductive_set max_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set" 
krauss@28735
   711
for R :: "('a \<times> 'a) set"
krauss@28735
   712
where
krauss@28735
   713
  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
   714
krauss@28735
   715
lemma max_ext_wf:
krauss@28735
   716
  assumes wf: "wf r"
krauss@28735
   717
  shows "wf (max_ext r)"
krauss@28735
   718
proof (rule acc_wfI, intro allI)
krauss@28735
   719
  fix M show "M \<in> acc (max_ext r)" (is "_ \<in> ?W")
krauss@28735
   720
  proof cases
krauss@28735
   721
    assume "finite M"
krauss@28735
   722
    thus ?thesis
krauss@28735
   723
    proof (induct M)
krauss@28735
   724
      show "{} \<in> ?W"
krauss@28735
   725
        by (rule accI) (auto elim: max_ext.cases)
krauss@28735
   726
    next
krauss@28735
   727
      fix M a assume "M \<in> ?W" "finite M"
krauss@28735
   728
      with wf show "insert a M \<in> ?W"
krauss@28735
   729
      proof (induct arbitrary: M)
krauss@28735
   730
        fix M a
krauss@28735
   731
        assume "M \<in> ?W"  and  [intro]: "finite M"
krauss@28735
   732
        assume hyp: "\<And>b M. (b, a) \<in> r \<Longrightarrow> M \<in> ?W \<Longrightarrow> finite M \<Longrightarrow> insert b M \<in> ?W"
krauss@28735
   733
        {
krauss@28735
   734
          fix N M :: "'a set"
krauss@28735
   735
          assume "finite N" "finite M"
krauss@28735
   736
          then
krauss@28735
   737
          have "\<lbrakk>M \<in> ?W ; (\<And>y. y \<in> N \<Longrightarrow> (y, a) \<in> r)\<rbrakk> \<Longrightarrow>  N \<union> M \<in> ?W"
krauss@28735
   738
            by (induct N arbitrary: M) (auto simp: hyp)
krauss@28735
   739
        }
krauss@28735
   740
        note add_less = this
krauss@28735
   741
        
krauss@28735
   742
        show "insert a M \<in> ?W"
krauss@28735
   743
        proof (rule accI)
krauss@28735
   744
          fix N assume Nless: "(N, insert a M) \<in> max_ext r"
krauss@28735
   745
          hence asm1: "\<And>x. x \<in> N \<Longrightarrow> (x, a) \<in> r \<or> (\<exists>y \<in> M. (x, y) \<in> r)"
krauss@28735
   746
            by (auto elim!: max_ext.cases)
krauss@28735
   747
krauss@28735
   748
          let ?N1 = "{ n \<in> N. (n, a) \<in> r }"
krauss@28735
   749
          let ?N2 = "{ n \<in> N. (n, a) \<notin> r }"
krauss@28735
   750
          have N: "?N1 \<union> ?N2 = N" by (rule set_ext) auto
krauss@28735
   751
          from Nless have "finite N" by (auto elim: max_ext.cases)
krauss@28735
   752
          then have finites: "finite ?N1" "finite ?N2" by auto
krauss@28735
   753
          
krauss@28735
   754
          have "?N2 \<in> ?W"
krauss@28735
   755
          proof cases
krauss@28735
   756
            assume [simp]: "M = {}"
krauss@28735
   757
            have Mw: "{} \<in> ?W" by (rule accI) (auto elim: max_ext.cases)
krauss@28735
   758
krauss@28735
   759
            from asm1 have "?N2 = {}" by auto
krauss@28735
   760
            with Mw show "?N2 \<in> ?W" by (simp only:)
krauss@28735
   761
          next
krauss@28735
   762
            assume "M \<noteq> {}"
krauss@28735
   763
            have N2: "(?N2, M) \<in> max_ext r" 
krauss@28735
   764
              by (rule max_extI[OF _ _ `M \<noteq> {}`]) (insert asm1, auto intro: finites)
krauss@28735
   765
            
krauss@28735
   766
            with `M \<in> ?W` show "?N2 \<in> ?W" by (rule acc_downward)
krauss@28735
   767
          qed
krauss@28735
   768
          with finites have "?N1 \<union> ?N2 \<in> ?W" 
krauss@28735
   769
            by (rule add_less) simp
krauss@28735
   770
          then show "N \<in> ?W" by (simp only: N)
krauss@28735
   771
        qed
krauss@28735
   772
      qed
krauss@28735
   773
    qed
krauss@28735
   774
  next
krauss@28735
   775
    assume [simp]: "\<not> finite M"
krauss@28735
   776
    show ?thesis
krauss@28735
   777
      by (rule accI) (auto elim: max_ext.cases)
krauss@28735
   778
  qed
krauss@28735
   779
qed
krauss@28735
   780
krauss@29125
   781
lemma max_ext_additive: 
krauss@29125
   782
 "(A, B) \<in> max_ext R \<Longrightarrow> (C, D) \<in> max_ext R \<Longrightarrow>
krauss@29125
   783
  (A \<union> C, B \<union> D) \<in> max_ext R"
krauss@29125
   784
by (force elim!: max_ext.cases)
krauss@29125
   785
krauss@28735
   786
krauss@28735
   787
definition
krauss@28735
   788
  min_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set" 
krauss@28735
   789
where
krauss@28735
   790
  [code del]: "min_ext r = {(X, Y) | X Y. X \<noteq> {} \<and> (\<forall>y \<in> Y. (\<exists>x \<in> X. (x, y) \<in> r))}"
krauss@28735
   791
krauss@28735
   792
lemma min_ext_wf:
krauss@28735
   793
  assumes "wf r"
krauss@28735
   794
  shows "wf (min_ext r)"
krauss@28735
   795
proof (rule wfI_min)
krauss@28735
   796
  fix Q :: "'a set set"
krauss@28735
   797
  fix x
krauss@28735
   798
  assume nonempty: "x \<in> Q"
krauss@28735
   799
  show "\<exists>m \<in> Q. (\<forall> n. (n, m) \<in> min_ext r \<longrightarrow> n \<notin> Q)"
krauss@28735
   800
  proof cases
krauss@28735
   801
    assume "Q = {{}}" thus ?thesis by (simp add: min_ext_def)
krauss@28735
   802
  next
krauss@28735
   803
    assume "Q \<noteq> {{}}"
krauss@28735
   804
    with nonempty
krauss@28735
   805
    obtain e x where "x \<in> Q" "e \<in> x" by force
krauss@28735
   806
    then have eU: "e \<in> \<Union>Q" by auto
krauss@28735
   807
    with `wf r` 
krauss@28735
   808
    obtain z where z: "z \<in> \<Union>Q" "\<And>y. (y, z) \<in> r \<Longrightarrow> y \<notin> \<Union>Q" 
krauss@28735
   809
      by (erule wfE_min)
krauss@28735
   810
    from z obtain m where "m \<in> Q" "z \<in> m" by auto
krauss@28735
   811
    from `m \<in> Q`
krauss@28735
   812
    show ?thesis
krauss@28735
   813
    proof (rule, intro bexI allI impI)
krauss@28735
   814
      fix n
krauss@28735
   815
      assume smaller: "(n, m) \<in> min_ext r"
krauss@28735
   816
      with `z \<in> m` obtain y where y: "y \<in> n" "(y, z) \<in> r" by (auto simp: min_ext_def)
krauss@28735
   817
      then show "n \<notin> Q" using z(2) by auto
krauss@28735
   818
    qed      
krauss@28735
   819
  qed
krauss@28735
   820
qed
krauss@26748
   821
krauss@26748
   822
krauss@26748
   823
subsection{*Weakly decreasing sequences (w.r.t. some well-founded order) 
krauss@26748
   824
   stabilize.*}
krauss@26748
   825
krauss@26748
   826
text{*This material does not appear to be used any longer.*}
krauss@26748
   827
krauss@28845
   828
lemma sequence_trans: "[| ALL i. (f (Suc i), f i) : r^* |] ==> (f (i+k), f i) : r^*"
krauss@28845
   829
by (induct k) (auto intro: rtrancl_trans)
krauss@26748
   830
krauss@28845
   831
lemma wf_weak_decr_stable: 
krauss@28845
   832
  assumes as: "ALL i. (f (Suc i), f i) : r^*" "wf (r^+)"
krauss@28845
   833
  shows "EX i. ALL k. f (i+k) = f i"
krauss@28845
   834
proof -
krauss@28845
   835
  have lem: "!!x. [| ALL i. (f (Suc i), f i) : r^*; wf (r^+) |]  
krauss@26748
   836
      ==> ALL m. f m = x --> (EX i. ALL k. f (m+i+k) = f (m+i))"
krauss@28845
   837
  apply (erule wf_induct, clarify)
krauss@28845
   838
  apply (case_tac "EX j. (f (m+j), f m) : r^+")
krauss@28845
   839
   apply clarify
krauss@28845
   840
   apply (subgoal_tac "EX i. ALL k. f ((m+j) +i+k) = f ( (m+j) +i) ")
krauss@28845
   841
    apply clarify
krauss@28845
   842
    apply (rule_tac x = "j+i" in exI)
krauss@28845
   843
    apply (simp add: add_ac, blast)
krauss@28845
   844
  apply (rule_tac x = 0 in exI, clarsimp)
krauss@28845
   845
  apply (drule_tac i = m and k = k in sequence_trans)
krauss@28845
   846
  apply (blast elim: rtranclE dest: rtrancl_into_trancl1)
krauss@28845
   847
  done
krauss@26748
   848
krauss@28845
   849
  from lem[OF as, THEN spec, of 0, simplified] 
krauss@28845
   850
  show ?thesis by auto
krauss@28845
   851
qed
krauss@26748
   852
krauss@26748
   853
(* special case of the theorem above: <= *)
krauss@26748
   854
lemma weak_decr_stable:
krauss@26748
   855
     "ALL i. f (Suc i) <= ((f i)::nat) ==> EX i. ALL k. f (i+k) = f i"
krauss@26748
   856
apply (rule_tac r = pred_nat in wf_weak_decr_stable)
krauss@26748
   857
apply (simp add: pred_nat_trancl_eq_le)
krauss@26748
   858
apply (intro wf_trancl wf_pred_nat)
krauss@26748
   859
done
krauss@26748
   860
krauss@26748
   861
krauss@26748
   862
subsection {* size of a datatype value *}
krauss@26748
   863
haftmann@31775
   864
use "Tools/Function/size.ML"
krauss@26748
   865
krauss@26748
   866
setup Size.setup
krauss@26748
   867
haftmann@28562
   868
lemma size_bool [code]:
haftmann@27823
   869
  "size (b\<Colon>bool) = 0" by (cases b) auto
haftmann@27823
   870
haftmann@28562
   871
lemma nat_size [simp, code]: "size (n\<Colon>nat) = n"
krauss@26748
   872
  by (induct n) simp_all
krauss@26748
   873
haftmann@27823
   874
declare "prod.size" [noatp]
krauss@26748
   875
haftmann@30430
   876
lemma [code]:
haftmann@30430
   877
  "size (P :: 'a Predicate.pred) = 0" by (cases P) simp
haftmann@30430
   878
haftmann@30430
   879
lemma [code]:
haftmann@30430
   880
  "pred_size f P = 0" by (cases P) simp
haftmann@30430
   881
krauss@26748
   882
end