src/HOL/Wellfounded.thy
author krauss
Wed Sep 17 15:59:23 2008 +0200 (2008-09-17)
changeset 28260 703046c93ffe
parent 27823 52971512d1a2
child 28524 644b62cf678f
permissions -rw-r--r--
wf_finite_psubset[simp], in_finite_psubset[simp]
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(*  ID:         $Id$
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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 Nat
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uses ("Tools/function_package/size.ML")
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begin
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subsection {* Basic Definitions *}
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inductive
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  wfrec_rel :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b => bool"
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  for R :: "('a * 'a) set"
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  and F :: "('a => 'b) => 'a => 'b"
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where
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  wfrecI: "ALL z. (z, x) : R --> wfrec_rel R F z (g z) ==>
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            wfrec_rel R F x (F g x)"
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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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  cut        :: "('a => 'b) => ('a * 'a)set => 'a => 'a => 'b"
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  "cut f r x == (%y. if (y,x):r then f y else arbitrary)"
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  adm_wf :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => bool"
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  "adm_wf R F == ALL f g x.
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     (ALL z. (z, x) : R --> f z = g z) --> F f x = F g x"
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  wfrec :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b"
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  [code func del]: "wfrec R F == %x. THE y. wfrec_rel R (%f x. F (cut f R x) x) x y"
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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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lemmas wfP_empty [iff] =
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  wf_empty [to_pred bot_empty_eq2, simplified bot_fun_eq bot_bool_eq]
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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 "S O R \<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> S O R" by (rule rel_compI)
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      with `S O R \<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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lemmas wfP_SUP = wf_UN [where I=UNIV and r="\<lambda>i. {(x, y). r i x y}",
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  to_pred SUP_UN_eq2 bot_empty_eq pred_equals_eq, simplified, standard]
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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.
krauss@26748
   301
  2. There is no such step.
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   302
     Pick an S-min element of A. In this case it must be an R-min
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   303
     element of A as well.
krauss@26748
   304
krauss@26748
   305
*)
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   306
lemma wf_Un:
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   307
     "[| wf r; wf s; Domain r Int Range s = {} |] ==> wf(r Un s)"
krauss@26748
   308
  using wf_union_compatible[of s r] 
krauss@26748
   309
  by (auto simp: Un_ac)
krauss@26748
   310
krauss@26748
   311
lemma wf_union_merge: 
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   312
  "wf (R \<union> S) = wf (R O R \<union> R O S \<union> S)" (is "wf ?A = wf ?B")
krauss@26748
   313
proof
krauss@26748
   314
  assume "wf ?A"
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   315
  with wf_trancl have wfT: "wf (?A^+)" .
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   316
  moreover have "?B \<subseteq> ?A^+"
krauss@26748
   317
    by (subst trancl_unfold, subst trancl_unfold) blast
krauss@26748
   318
  ultimately show "wf ?B" by (rule wf_subset)
krauss@26748
   319
next
krauss@26748
   320
  assume "wf ?B"
krauss@26748
   321
krauss@26748
   322
  show "wf ?A"
krauss@26748
   323
  proof (rule wfI_min)
krauss@26748
   324
    fix Q :: "'a set" and x 
krauss@26748
   325
    assume "x \<in> Q"
krauss@26748
   326
krauss@26748
   327
    with `wf ?B`
krauss@26748
   328
    obtain z where "z \<in> Q" and "\<And>y. (y, z) \<in> ?B \<Longrightarrow> y \<notin> Q" 
krauss@26748
   329
      by (erule wfE_min)
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   330
    then have A1: "\<And>y. (y, z) \<in> R O R \<Longrightarrow> y \<notin> Q"
krauss@26748
   331
      and A2: "\<And>y. (y, z) \<in> R O S \<Longrightarrow> y \<notin> Q"
krauss@26748
   332
      and A3: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> Q"
krauss@26748
   333
      by auto
krauss@26748
   334
    
krauss@26748
   335
    show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> ?A \<longrightarrow> y \<notin> Q"
krauss@26748
   336
    proof (cases "\<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q")
krauss@26748
   337
      case True
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   338
      with `z \<in> Q` A3 show ?thesis by blast
krauss@26748
   339
    next
krauss@26748
   340
      case False 
krauss@26748
   341
      then obtain z' where "z'\<in>Q" "(z', z) \<in> R" by blast
krauss@26748
   342
krauss@26748
   343
      have "\<forall>y. (y, z') \<in> ?A \<longrightarrow> y \<notin> Q"
krauss@26748
   344
      proof (intro allI impI)
krauss@26748
   345
        fix y assume "(y, z') \<in> ?A"
krauss@26748
   346
        then show "y \<notin> Q"
krauss@26748
   347
        proof
krauss@26748
   348
          assume "(y, z') \<in> R" 
krauss@26748
   349
          then have "(y, z) \<in> R O R" using `(z', z) \<in> R` ..
krauss@26748
   350
          with A1 show "y \<notin> Q" .
krauss@26748
   351
        next
krauss@26748
   352
          assume "(y, z') \<in> S" 
krauss@26748
   353
          then have "(y, z) \<in> R O S" using  `(z', z) \<in> R` ..
krauss@26748
   354
          with A2 show "y \<notin> Q" .
krauss@26748
   355
        qed
krauss@26748
   356
      qed
krauss@26748
   357
      with `z' \<in> Q` show ?thesis ..
krauss@26748
   358
    qed
krauss@26748
   359
  qed
krauss@26748
   360
qed
krauss@26748
   361
krauss@26748
   362
lemma wf_comp_self: "wf R = wf (R O R)"  -- {* special case *}
krauss@26748
   363
  by (rule wf_union_merge [where S = "{}", simplified])
krauss@26748
   364
krauss@26748
   365
krauss@26748
   366
subsubsection {* acyclic *}
krauss@26748
   367
krauss@26748
   368
lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r"
krauss@26748
   369
  by (simp add: acyclic_def)
krauss@26748
   370
krauss@26748
   371
lemma wf_acyclic: "wf r ==> acyclic r"
krauss@26748
   372
apply (simp add: acyclic_def)
krauss@26748
   373
apply (blast elim: wf_trancl [THEN wf_irrefl])
krauss@26748
   374
done
krauss@26748
   375
krauss@26748
   376
lemmas wfP_acyclicP = wf_acyclic [to_pred]
krauss@26748
   377
krauss@26748
   378
lemma acyclic_insert [iff]:
krauss@26748
   379
     "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"
krauss@26748
   380
apply (simp add: acyclic_def trancl_insert)
krauss@26748
   381
apply (blast intro: rtrancl_trans)
krauss@26748
   382
done
krauss@26748
   383
krauss@26748
   384
lemma acyclic_converse [iff]: "acyclic(r^-1) = acyclic r"
krauss@26748
   385
by (simp add: acyclic_def trancl_converse)
krauss@26748
   386
krauss@26748
   387
lemmas acyclicP_converse [iff] = acyclic_converse [to_pred]
krauss@26748
   388
krauss@26748
   389
lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)"
krauss@26748
   390
apply (simp add: acyclic_def antisym_def)
krauss@26748
   391
apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
krauss@26748
   392
done
krauss@26748
   393
krauss@26748
   394
(* Other direction:
krauss@26748
   395
acyclic = no loops
krauss@26748
   396
antisym = only self loops
krauss@26748
   397
Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r - Id)
krauss@26748
   398
==> antisym( r^* ) = acyclic(r - Id)";
krauss@26748
   399
*)
krauss@26748
   400
krauss@26748
   401
lemma acyclic_subset: "[| acyclic s; r <= s |] ==> acyclic r"
krauss@26748
   402
apply (simp add: acyclic_def)
krauss@26748
   403
apply (blast intro: trancl_mono)
krauss@26748
   404
done
krauss@26748
   405
krauss@26748
   406
text{* Wellfoundedness of finite acyclic relations*}
krauss@26748
   407
krauss@26748
   408
lemma finite_acyclic_wf [rule_format]: "finite r ==> acyclic r --> wf r"
krauss@26748
   409
apply (erule finite_induct, blast)
krauss@26748
   410
apply (simp (no_asm_simp) only: split_tupled_all)
krauss@26748
   411
apply simp
krauss@26748
   412
done
krauss@26748
   413
krauss@26748
   414
lemma finite_acyclic_wf_converse: "[|finite r; acyclic r|] ==> wf (r^-1)"
krauss@26748
   415
apply (erule finite_converse [THEN iffD2, THEN finite_acyclic_wf])
krauss@26748
   416
apply (erule acyclic_converse [THEN iffD2])
krauss@26748
   417
done
krauss@26748
   418
krauss@26748
   419
lemma wf_iff_acyclic_if_finite: "finite r ==> wf r = acyclic r"
krauss@26748
   420
by (blast intro: finite_acyclic_wf wf_acyclic)
krauss@26748
   421
krauss@26748
   422
krauss@26748
   423
subsection{*Well-Founded Recursion*}
krauss@26748
   424
krauss@26748
   425
text{*cut*}
krauss@26748
   426
krauss@26748
   427
lemma cuts_eq: "(cut f r x = cut g r x) = (ALL y. (y,x):r --> f(y)=g(y))"
krauss@26748
   428
by (simp add: expand_fun_eq cut_def)
krauss@26748
   429
krauss@26748
   430
lemma cut_apply: "(x,a):r ==> (cut f r a)(x) = f(x)"
krauss@26748
   431
by (simp add: cut_def)
krauss@26748
   432
krauss@26748
   433
text{*Inductive characterization of wfrec combinator; for details see:  
krauss@26748
   434
John Harrison, "Inductive definitions: automation and application"*}
krauss@26748
   435
krauss@26748
   436
lemma wfrec_unique: "[| adm_wf R F; wf R |] ==> EX! y. wfrec_rel R F x y"
krauss@26748
   437
apply (simp add: adm_wf_def)
krauss@26748
   438
apply (erule_tac a=x in wf_induct) 
krauss@26748
   439
apply (rule ex1I)
krauss@26748
   440
apply (rule_tac g = "%x. THE y. wfrec_rel R F x y" in wfrec_rel.wfrecI)
krauss@26748
   441
apply (fast dest!: theI')
krauss@26748
   442
apply (erule wfrec_rel.cases, simp)
krauss@26748
   443
apply (erule allE, erule allE, erule allE, erule mp)
krauss@26748
   444
apply (fast intro: the_equality [symmetric])
krauss@26748
   445
done
krauss@26748
   446
krauss@26748
   447
lemma adm_lemma: "adm_wf R (%f x. F (cut f R x) x)"
krauss@26748
   448
apply (simp add: adm_wf_def)
krauss@26748
   449
apply (intro strip)
krauss@26748
   450
apply (rule cuts_eq [THEN iffD2, THEN subst], assumption)
krauss@26748
   451
apply (rule refl)
krauss@26748
   452
done
krauss@26748
   453
krauss@26748
   454
lemma wfrec: "wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a"
krauss@26748
   455
apply (simp add: wfrec_def)
krauss@26748
   456
apply (rule adm_lemma [THEN wfrec_unique, THEN the1_equality], assumption)
krauss@26748
   457
apply (rule wfrec_rel.wfrecI)
krauss@26748
   458
apply (intro strip)
krauss@26748
   459
apply (erule adm_lemma [THEN wfrec_unique, THEN theI'])
krauss@26748
   460
done
krauss@26748
   461
krauss@26748
   462
subsection {* Code generator setup *}
krauss@26748
   463
krauss@26748
   464
consts_code
krauss@26748
   465
  "wfrec"   ("\<module>wfrec?")
krauss@26748
   466
attach {*
krauss@26748
   467
fun wfrec f x = f (wfrec f) x;
krauss@26748
   468
*}
krauss@26748
   469
krauss@26748
   470
krauss@26748
   471
subsection {* @{typ nat} is well-founded *}
krauss@26748
   472
krauss@26748
   473
lemma less_nat_rel: "op < = (\<lambda>m n. n = Suc m)^++"
krauss@26748
   474
proof (rule ext, rule ext, rule iffI)
krauss@26748
   475
  fix n m :: nat
krauss@26748
   476
  assume "m < n"
krauss@26748
   477
  then show "(\<lambda>m n. n = Suc m)^++ m n"
krauss@26748
   478
  proof (induct n)
krauss@26748
   479
    case 0 then show ?case by auto
krauss@26748
   480
  next
krauss@26748
   481
    case (Suc n) then show ?case
krauss@26748
   482
      by (auto simp add: less_Suc_eq_le le_less intro: tranclp.trancl_into_trancl)
krauss@26748
   483
  qed
krauss@26748
   484
next
krauss@26748
   485
  fix n m :: nat
krauss@26748
   486
  assume "(\<lambda>m n. n = Suc m)^++ m n"
krauss@26748
   487
  then show "m < n"
krauss@26748
   488
    by (induct n)
krauss@26748
   489
      (simp_all add: less_Suc_eq_le reflexive le_less)
krauss@26748
   490
qed
krauss@26748
   491
krauss@26748
   492
definition
krauss@26748
   493
  pred_nat :: "(nat * nat) set" where
krauss@26748
   494
  "pred_nat = {(m, n). n = Suc m}"
krauss@26748
   495
krauss@26748
   496
definition
krauss@26748
   497
  less_than :: "(nat * nat) set" where
krauss@26748
   498
  "less_than = pred_nat^+"
krauss@26748
   499
krauss@26748
   500
lemma less_eq: "(m, n) \<in> pred_nat^+ \<longleftrightarrow> m < n"
krauss@26748
   501
  unfolding less_nat_rel pred_nat_def trancl_def by simp
krauss@26748
   502
krauss@26748
   503
lemma pred_nat_trancl_eq_le:
krauss@26748
   504
  "(m, n) \<in> pred_nat^* \<longleftrightarrow> m \<le> n"
krauss@26748
   505
  unfolding less_eq rtrancl_eq_or_trancl by auto
krauss@26748
   506
krauss@26748
   507
lemma wf_pred_nat: "wf pred_nat"
krauss@26748
   508
  apply (unfold wf_def pred_nat_def, clarify)
krauss@26748
   509
  apply (induct_tac x, blast+)
krauss@26748
   510
  done
krauss@26748
   511
krauss@26748
   512
lemma wf_less_than [iff]: "wf less_than"
krauss@26748
   513
  by (simp add: less_than_def wf_pred_nat [THEN wf_trancl])
krauss@26748
   514
krauss@26748
   515
lemma trans_less_than [iff]: "trans less_than"
krauss@26748
   516
  by (simp add: less_than_def trans_trancl)
krauss@26748
   517
krauss@26748
   518
lemma less_than_iff [iff]: "((x,y): less_than) = (x<y)"
krauss@26748
   519
  by (simp add: less_than_def less_eq)
krauss@26748
   520
krauss@26748
   521
lemma wf_less: "wf {(x, y::nat). x < y}"
krauss@26748
   522
  using wf_less_than by (simp add: less_than_def less_eq [symmetric])
krauss@26748
   523
krauss@26748
   524
krauss@26748
   525
subsection {* Accessible Part *}
krauss@26748
   526
krauss@26748
   527
text {*
krauss@26748
   528
 Inductive definition of the accessible part @{term "acc r"} of a
krauss@26748
   529
 relation; see also \cite{paulin-tlca}.
krauss@26748
   530
*}
krauss@26748
   531
krauss@26748
   532
inductive_set
krauss@26748
   533
  acc :: "('a * 'a) set => 'a set"
krauss@26748
   534
  for r :: "('a * 'a) set"
krauss@26748
   535
  where
krauss@26748
   536
    accI: "(!!y. (y, x) : r ==> y : acc r) ==> x : acc r"
krauss@26748
   537
krauss@26748
   538
abbreviation
krauss@26748
   539
  termip :: "('a => 'a => bool) => 'a => bool" where
krauss@26748
   540
  "termip r == accp (r\<inverse>\<inverse>)"
krauss@26748
   541
krauss@26748
   542
abbreviation
krauss@26748
   543
  termi :: "('a * 'a) set => 'a set" where
krauss@26748
   544
  "termi r == acc (r\<inverse>)"
krauss@26748
   545
krauss@26748
   546
lemmas accpI = accp.accI
krauss@26748
   547
krauss@26748
   548
text {* Induction rules *}
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)
krauss@26748
   600
  apply (induct_tac P x rule: 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
krauss@26748
   616
text {* Smaller relations have bigger accessible parts: *}
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
krauss@26748
   633
text {* This is a generalized induction theorem that works on
krauss@26748
   634
  subsets of the accessible part. *}
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 -
krauss@26748
   643
  from subset and `D x`
krauss@26748
   644
  have "accp R x" ..
krauss@26748
   645
  then show "P x" using `D x`
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
krauss@26748
   655
text {* Set versions of the above theorems *}
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@26803
   675
lemmas acc_subset = accp_subset [to_set pred_subset_eq]
krauss@26748
   676
berghofe@26803
   677
lemmas acc_subset_induct = accp_subset_induct [to_set pred_subset_eq]
krauss@26748
   678
krauss@26748
   679
krauss@26748
   680
subsection {* Tools for building wellfounded relations *}
krauss@26748
   681
krauss@26748
   682
text {* Inverse Image *}
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
krauss@26748
   694
lemma in_inv_image[simp]: "((x,y) : inv_image r f) = ((f x, f y) : r)"
krauss@26748
   695
  by (auto simp:inv_image_def)
krauss@26748
   696
krauss@26748
   697
text {* Measure functions into @{typ nat} *}
krauss@26748
   698
krauss@26748
   699
definition measure :: "('a => nat) => ('a * 'a)set"
krauss@26748
   700
where "measure == inv_image less_than"
krauss@26748
   701
krauss@26748
   702
lemma in_measure[simp]: "((x,y) : measure f) = (f x < f y)"
krauss@26748
   703
  by (simp add:measure_def)
krauss@26748
   704
krauss@26748
   705
lemma wf_measure [iff]: "wf (measure f)"
krauss@26748
   706
apply (unfold measure_def)
krauss@26748
   707
apply (rule wf_less_than [THEN wf_inv_image])
krauss@26748
   708
done
krauss@26748
   709
krauss@26748
   710
text{* Lexicographic combinations *}
krauss@26748
   711
krauss@26748
   712
definition
krauss@26748
   713
 lex_prod  :: "[('a*'a)set, ('b*'b)set] => (('a*'b)*('a*'b))set"
krauss@26748
   714
               (infixr "<*lex*>" 80)
krauss@26748
   715
where
krauss@26748
   716
    "ra <*lex*> rb == {((a,b),(a',b')). (a,a') : ra | a=a' & (b,b') : rb}"
krauss@26748
   717
krauss@26748
   718
lemma wf_lex_prod [intro!]: "[| wf(ra); wf(rb) |] ==> wf(ra <*lex*> rb)"
krauss@26748
   719
apply (unfold wf_def lex_prod_def) 
krauss@26748
   720
apply (rule allI, rule impI)
krauss@26748
   721
apply (simp (no_asm_use) only: split_paired_All)
krauss@26748
   722
apply (drule spec, erule mp) 
krauss@26748
   723
apply (rule allI, rule impI)
krauss@26748
   724
apply (drule spec, erule mp, blast) 
krauss@26748
   725
done
krauss@26748
   726
krauss@26748
   727
lemma in_lex_prod[simp]: 
krauss@26748
   728
  "(((a,b),(a',b')): r <*lex*> s) = ((a,a'): r \<or> (a = a' \<and> (b, b') : s))"
krauss@26748
   729
  by (auto simp:lex_prod_def)
krauss@26748
   730
krauss@26748
   731
text{* @{term "op <*lex*>"} preserves transitivity *}
krauss@26748
   732
krauss@26748
   733
lemma trans_lex_prod [intro!]: 
krauss@26748
   734
    "[| trans R1; trans R2 |] ==> trans (R1 <*lex*> R2)"
krauss@26748
   735
by (unfold trans_def lex_prod_def, blast) 
krauss@26748
   736
krauss@26748
   737
text {* lexicographic combinations with measure functions *}
krauss@26748
   738
krauss@26748
   739
definition 
krauss@26748
   740
  mlex_prod :: "('a \<Rightarrow> nat) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" (infixr "<*mlex*>" 80)
krauss@26748
   741
where
krauss@26748
   742
  "f <*mlex*> R = inv_image (less_than <*lex*> R) (%x. (f x, x))"
krauss@26748
   743
krauss@26748
   744
lemma wf_mlex: "wf R \<Longrightarrow> wf (f <*mlex*> R)"
krauss@26748
   745
unfolding mlex_prod_def
krauss@26748
   746
by auto
krauss@26748
   747
krauss@26748
   748
lemma mlex_less: "f x < f y \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
krauss@26748
   749
unfolding mlex_prod_def by simp
krauss@26748
   750
krauss@26748
   751
lemma mlex_leq: "f x \<le> f y \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
krauss@26748
   752
unfolding mlex_prod_def by auto
krauss@26748
   753
krauss@26748
   754
text {* proper subset relation on finite sets *}
krauss@26748
   755
krauss@26748
   756
definition finite_psubset  :: "('a set * 'a set) set"
krauss@26748
   757
where "finite_psubset == {(A,B). A < B & finite B}"
krauss@26748
   758
krauss@28260
   759
lemma wf_finite_psubset[simp]: "wf(finite_psubset)"
krauss@26748
   760
apply (unfold finite_psubset_def)
krauss@26748
   761
apply (rule wf_measure [THEN wf_subset])
krauss@26748
   762
apply (simp add: measure_def inv_image_def less_than_def less_eq)
krauss@26748
   763
apply (fast elim!: psubset_card_mono)
krauss@26748
   764
done
krauss@26748
   765
krauss@26748
   766
lemma trans_finite_psubset: "trans finite_psubset"
berghofe@26803
   767
by (simp add: finite_psubset_def less_le trans_def, blast)
krauss@26748
   768
krauss@28260
   769
lemma in_finite_psubset[simp]: "(A, B) \<in> finite_psubset = (A < B & finite B)"
krauss@28260
   770
unfolding finite_psubset_def by auto
krauss@26748
   771
krauss@26748
   772
krauss@26748
   773
text {*Wellfoundedness of @{text same_fst}*}
krauss@26748
   774
krauss@26748
   775
definition
krauss@26748
   776
 same_fst :: "('a => bool) => ('a => ('b * 'b)set) => (('a*'b)*('a*'b))set"
krauss@26748
   777
where
krauss@26748
   778
    "same_fst P R == {((x',y'),(x,y)) . x'=x & P x & (y',y) : R x}"
krauss@26748
   779
   --{*For @{text rec_def} declarations where the first n parameters
krauss@26748
   780
       stay unchanged in the recursive call. 
krauss@26748
   781
       See @{text "Library/While_Combinator.thy"} for an application.*}
krauss@26748
   782
krauss@26748
   783
lemma same_fstI [intro!]:
krauss@26748
   784
     "[| P x; (y',y) : R x |] ==> ((x,y'),(x,y)) : same_fst P R"
krauss@26748
   785
by (simp add: same_fst_def)
krauss@26748
   786
krauss@26748
   787
lemma wf_same_fst:
krauss@26748
   788
  assumes prem: "(!!x. P x ==> wf(R x))"
krauss@26748
   789
  shows "wf(same_fst P R)"
krauss@26748
   790
apply (simp cong del: imp_cong add: wf_def same_fst_def)
krauss@26748
   791
apply (intro strip)
krauss@26748
   792
apply (rename_tac a b)
krauss@26748
   793
apply (case_tac "wf (R a)")
krauss@26748
   794
 apply (erule_tac a = b in wf_induct, blast)
krauss@26748
   795
apply (blast intro: prem)
krauss@26748
   796
done
krauss@26748
   797
krauss@26748
   798
krauss@26748
   799
subsection{*Weakly decreasing sequences (w.r.t. some well-founded order) 
krauss@26748
   800
   stabilize.*}
krauss@26748
   801
krauss@26748
   802
text{*This material does not appear to be used any longer.*}
krauss@26748
   803
krauss@26748
   804
lemma lemma1: "[| ALL i. (f (Suc i), f i) : r^* |] ==> (f (i+k), f i) : r^*"
krauss@26748
   805
apply (induct_tac "k", simp_all)
krauss@26748
   806
apply (blast intro: rtrancl_trans)
krauss@26748
   807
done
krauss@26748
   808
krauss@26748
   809
lemma lemma2: "[| ALL i. (f (Suc i), f i) : r^*; wf (r^+) |]  
krauss@26748
   810
      ==> ALL m. f m = x --> (EX i. ALL k. f (m+i+k) = f (m+i))"
krauss@26748
   811
apply (erule wf_induct, clarify)
krauss@26748
   812
apply (case_tac "EX j. (f (m+j), f m) : r^+")
krauss@26748
   813
 apply clarify
krauss@26748
   814
 apply (subgoal_tac "EX i. ALL k. f ((m+j) +i+k) = f ( (m+j) +i) ")
krauss@26748
   815
  apply clarify
krauss@26748
   816
  apply (rule_tac x = "j+i" in exI)
krauss@26748
   817
  apply (simp add: add_ac, blast)
krauss@26748
   818
apply (rule_tac x = 0 in exI, clarsimp)
krauss@26748
   819
apply (drule_tac i = m and k = k in lemma1)
krauss@26748
   820
apply (blast elim: rtranclE dest: rtrancl_into_trancl1)
krauss@26748
   821
done
krauss@26748
   822
krauss@26748
   823
lemma wf_weak_decr_stable: "[| ALL i. (f (Suc i), f i) : r^*; wf (r^+) |]  
krauss@26748
   824
      ==> EX i. ALL k. f (i+k) = f i"
krauss@26748
   825
apply (drule_tac x = 0 in lemma2 [THEN spec], auto)
krauss@26748
   826
done
krauss@26748
   827
krauss@26748
   828
(* special case of the theorem above: <= *)
krauss@26748
   829
lemma weak_decr_stable:
krauss@26748
   830
     "ALL i. f (Suc i) <= ((f i)::nat) ==> EX i. ALL k. f (i+k) = f i"
krauss@26748
   831
apply (rule_tac r = pred_nat in wf_weak_decr_stable)
krauss@26748
   832
apply (simp add: pred_nat_trancl_eq_le)
krauss@26748
   833
apply (intro wf_trancl wf_pred_nat)
krauss@26748
   834
done
krauss@26748
   835
krauss@26748
   836
krauss@26748
   837
subsection {* size of a datatype value *}
krauss@26748
   838
krauss@26748
   839
use "Tools/function_package/size.ML"
krauss@26748
   840
krauss@26748
   841
setup Size.setup
krauss@26748
   842
haftmann@27823
   843
lemma size_bool [code func]:
haftmann@27823
   844
  "size (b\<Colon>bool) = 0" by (cases b) auto
haftmann@27823
   845
krauss@26748
   846
lemma nat_size [simp, code func]: "size (n\<Colon>nat) = n"
krauss@26748
   847
  by (induct n) simp_all
krauss@26748
   848
haftmann@27823
   849
declare "prod.size" [noatp]
krauss@26748
   850
krauss@26748
   851
end