src/HOL/Wellfounded.thy
author haftmann
Tue, 13 Oct 2015 09:21:15 +0200
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(*  Title:      HOL/Wellfounded.thy
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    Author:     Tobias Nipkow
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    Author:     Lawrence C Paulson
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    Author:     Konrad Slind
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    Author:     Alexander Krauss
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    Author:     Andrei Popescu, TU Muenchen
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*)
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section \<open>Well-founded Recursion\<close>
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theory Wellfounded
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imports Transitive_Closure
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begin
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subsection \<open>Basic Definitions\<close>
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definition wf :: "('a * 'a) set => bool" where
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  "wf r \<longleftrightarrow> (!P. (!x. (!y. (y,x):r --> P(y)) --> P(x)) --> (!x. P(x)))"
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definition wfP :: "('a => 'a => bool) => bool" where
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  "wfP r \<longleftrightarrow> wf {(x, y). r x y}"
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lemma wfP_wf_eq [pred_set_conv]: "wfP (\<lambda>x y. (x, y) \<in> r) = wf r"
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  by (simp add: wfP_def)
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lemma wfUNIVI: 
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   "(!!P x. (ALL x. (ALL y. (y,x) : r --> P(y)) --> P(x)) ==> P(x)) ==> wf(r)"
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  unfolding wf_def by blast
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lemmas wfPUNIVI = wfUNIVI [to_pred]
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text\<open>Restriction to domain @{term A} and range @{term B}.  If @{term r} is
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    well-founded over their intersection, then @{term "wf r"}\<close>
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lemma wfI: 
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 "[| r \<subseteq> A <*> B; 
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     !!x P. [|\<forall>x. (\<forall>y. (y,x) : r --> P y) --> P x;  x : A; x : B |] ==> P x |]
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  ==>  wf r"
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  unfolding wf_def by blast
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lemma wf_induct: 
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    "[| wf(r);           
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        !!x.[| ALL y. (y,x): r --> P(y) |] ==> P(x)  
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     |]  ==>  P(a)"
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  unfolding wf_def by blast
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lemmas wfP_induct = wf_induct [to_pred]
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lemmas wf_induct_rule = wf_induct [rule_format, consumes 1, case_names less, induct set: wf]
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lemmas wfP_induct_rule = wf_induct_rule [to_pred, induct set: wfP]
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lemma wf_not_sym: "wf r ==> (a, x) : r ==> (x, a) ~: r"
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  by (induct a arbitrary: x set: wf) blast
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lemma wf_asym:
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  assumes "wf r" "(a, x) \<in> r"
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  obtains "(x, a) \<notin> r"
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  by (drule wf_not_sym[OF assms])
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lemma wf_not_refl [simp]: "wf r ==> (a, a) ~: r"
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  by (blast elim: wf_asym)
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lemma wf_irrefl: assumes "wf r" obtains "(a, a) \<notin> r"
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by (drule wf_not_refl[OF assms])
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lemma wf_wellorderI:
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  assumes wf: "wf {(x::'a::ord, y). x < y}"
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  assumes lin: "OFCLASS('a::ord, linorder_class)"
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  shows "OFCLASS('a::ord, wellorder_class)"
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using lin by (rule wellorder_class.intro)
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  (rule class.wellorder_axioms.intro, rule wf_induct_rule [OF wf], simp)
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lemma (in wellorder) wf:
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  "wf {(x, y). x < y}"
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unfolding wf_def by (blast intro: less_induct)
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subsection \<open>Basic Results\<close>
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text \<open>Point-free characterization of well-foundedness\<close>
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lemma wfE_pf:
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  assumes wf: "wf R"
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  assumes a: "A \<subseteq> R `` A"
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  shows "A = {}"
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proof -
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  { fix x
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    from wf have "x \<notin> A"
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    proof induct
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      fix x assume "\<And>y. (y, x) \<in> R \<Longrightarrow> y \<notin> A"
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      then have "x \<notin> R `` A" by blast
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      with a show "x \<notin> A" by blast
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    qed
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  } thus ?thesis by auto
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qed
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lemma wfI_pf:
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  assumes a: "\<And>A. A \<subseteq> R `` A \<Longrightarrow> A = {}"
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  shows "wf R"
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proof (rule wfUNIVI)
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  fix P :: "'a \<Rightarrow> bool" and x
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  let ?A = "{x. \<not> P x}"
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  assume "\<forall>x. (\<forall>y. (y, x) \<in> R \<longrightarrow> P y) \<longrightarrow> P x"
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  then have "?A \<subseteq> R `` ?A" by blast
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  with a show "P x" by blast
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qed
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text\<open>Minimal-element characterization of well-foundedness\<close>
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lemma wfE_min:
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  assumes wf: "wf R" and Q: "x \<in> Q"
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  obtains z where "z \<in> Q" "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
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  using Q wfE_pf[OF wf, of Q] by blast
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lemma wfI_min:
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  assumes a: "\<And>x Q. x \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q"
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  shows "wf R"
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proof (rule wfI_pf)
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  fix A assume b: "A \<subseteq> R `` A"
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  { fix x assume "x \<in> A"
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    from a[OF this] b have "False" by blast
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  }
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  thus "A = {}" by blast
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qed
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lemma wf_eq_minimal: "wf r = (\<forall>Q x. x\<in>Q --> (\<exists>z\<in>Q. \<forall>y. (y,z)\<in>r --> y\<notin>Q))"
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apply auto
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apply (erule wfE_min, assumption, blast)
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apply (rule wfI_min, auto)
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done
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lemmas wfP_eq_minimal = wf_eq_minimal [to_pred]
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text\<open>Well-foundedness of transitive closure\<close>
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lemma wf_trancl:
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  assumes "wf r"
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  shows "wf (r^+)"
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proof -
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  {
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    fix P and x
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    assume induct_step: "!!x. (!!y. (y, x) : r^+ ==> P y) ==> P x"
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    have "P x"
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    proof (rule induct_step)
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      fix y assume "(y, x) : r^+"
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      with \<open>wf r\<close> show "P y"
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      proof (induct x arbitrary: y)
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wenzelm
parents: 32704
diff changeset
   148
        case (less x)
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   149
        note hyp = \<open>\<And>x' y'. (x', x) : r ==> (y', x') : r^+ ==> P y'\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   150
        from \<open>(y, x) : r^+\<close> show "P y"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   151
        proof cases
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   152
          case base
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   153
          show "P y"
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   154
          proof (rule induct_step)
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   155
            fix y' assume "(y', y) : r^+"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   156
            with \<open>(y, x) : r\<close> show "P y'" by (rule hyp [of y y'])
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   157
          qed
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   158
        next
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   159
          case step
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   160
          then obtain x' where "(x', x) : r" and "(y, x') : r^+" by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   161
          then show "P y" by (rule hyp [of x' y])
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   162
        qed
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   163
      qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   164
    qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   165
  } then show ?thesis unfolding wf_def by blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   166
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   167
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   168
lemmas wfP_trancl = wf_trancl [to_pred]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   169
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   170
lemma wf_converse_trancl: "wf (r^-1) ==> wf ((r^+)^-1)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   171
  apply (subst trancl_converse [symmetric])
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   172
  apply (erule wf_trancl)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   173
  done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   174
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   175
text \<open>Well-foundedness of subsets\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   176
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   177
lemma wf_subset: "[| wf(r);  p<=r |] ==> wf(p)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   178
  apply (simp (no_asm_use) add: wf_eq_minimal)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   179
  apply fast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   180
  done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   181
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   182
lemmas wfP_subset = wf_subset [to_pred]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   183
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   184
text \<open>Well-foundedness of the empty relation\<close>
33216
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   185
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   186
lemma wf_empty [iff]: "wf {}"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   187
  by (simp add: wf_def)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   188
32205
49db434c157f explicit is better than implicit
haftmann
parents: 31775
diff changeset
   189
lemma wfP_empty [iff]:
49db434c157f explicit is better than implicit
haftmann
parents: 31775
diff changeset
   190
  "wfP (\<lambda>x y. False)"
49db434c157f explicit is better than implicit
haftmann
parents: 31775
diff changeset
   191
proof -
49db434c157f explicit is better than implicit
haftmann
parents: 31775
diff changeset
   192
  have "wfP bot" by (fact wf_empty [to_pred bot_empty_eq2])
44921
58eef4843641 tuned proofs
huffman
parents: 44144
diff changeset
   193
  then show ?thesis by (simp add: bot_fun_def)
32205
49db434c157f explicit is better than implicit
haftmann
parents: 31775
diff changeset
   194
qed
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   195
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   196
lemma wf_Int1: "wf r ==> wf (r Int r')"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   197
  apply (erule wf_subset)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   198
  apply (rule Int_lower1)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   199
  done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   200
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   201
lemma wf_Int2: "wf r ==> wf (r' Int r)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   202
  apply (erule wf_subset)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   203
  apply (rule Int_lower2)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   204
  done  
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   205
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   206
text \<open>Exponentiation\<close>
33216
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   207
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   208
lemma wf_exp:
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   209
  assumes "wf (R ^^ n)"
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   210
  shows "wf R"
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   211
proof (rule wfI_pf)
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   212
  fix A assume "A \<subseteq> R `` A"
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   213
  then have "A \<subseteq> (R ^^ n) `` A" by (induct n) force+
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   214
  with \<open>wf (R ^^ n)\<close>
33216
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   215
  show "A = {}" by (rule wfE_pf)
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   216
qed
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   217
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   218
text \<open>Well-foundedness of insert\<close>
33216
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   219
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   220
lemma wf_insert [iff]: "wf(insert (y,x) r) = (wf(r) & (x,y) ~: r^*)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   221
apply (rule iffI)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   222
 apply (blast elim: wf_trancl [THEN wf_irrefl]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   223
              intro: rtrancl_into_trancl1 wf_subset 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   224
                     rtrancl_mono [THEN [2] rev_subsetD])
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   225
apply (simp add: wf_eq_minimal, safe)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   226
apply (rule allE, assumption, erule impE, blast) 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   227
apply (erule bexE)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   228
apply (rename_tac "a", case_tac "a = x")
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   229
 prefer 2
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   230
apply blast 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   231
apply (case_tac "y:Q")
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   232
 prefer 2 apply blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   233
apply (rule_tac x = "{z. z:Q & (z,y) : r^*}" in allE)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   234
 apply assumption
59807
22bc39064290 prefer local fixes;
wenzelm
parents: 58889
diff changeset
   235
apply (erule_tac V = "ALL Q. (EX x. x : Q) --> P Q" for P in thin_rl) 
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   236
  --\<open>essential for speed\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   237
txt\<open>Blast with new substOccur fails\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   238
apply (fast intro: converse_rtrancl_into_rtrancl)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   239
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   240
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   241
text\<open>Well-foundedness of image\<close>
33216
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   242
55932
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55027
diff changeset
   243
lemma wf_map_prod_image: "[| wf r; inj f |] ==> wf (map_prod f f ` r)"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   244
apply (simp only: wf_eq_minimal, clarify)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   245
apply (case_tac "EX p. f p : Q")
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   246
apply (erule_tac x = "{p. f p : Q}" in allE)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   247
apply (fast dest: inj_onD, blast)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   248
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   249
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   250
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   251
subsection \<open>Well-Foundedness Results for Unions\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   252
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   253
lemma wf_union_compatible:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   254
  assumes "wf R" "wf S"
32235
8f9b8d14fc9f "more standard" argument order of relation composition (op O)
krauss
parents: 32205
diff changeset
   255
  assumes "R O S \<subseteq> R"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   256
  shows "wf (R \<union> S)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   257
proof (rule wfI_min)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   258
  fix x :: 'a and Q 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   259
  let ?Q' = "{x \<in> Q. \<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> Q}"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   260
  assume "x \<in> Q"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   261
  obtain a where "a \<in> ?Q'"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   262
    by (rule wfE_min [OF \<open>wf R\<close> \<open>x \<in> Q\<close>]) blast
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   263
  with \<open>wf S\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   264
  obtain z where "z \<in> ?Q'" and zmin: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> ?Q'" by (erule wfE_min)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   265
  { 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   266
    fix y assume "(y, z) \<in> S"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   267
    then have "y \<notin> ?Q'" by (rule zmin)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   268
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   269
    have "y \<notin> Q"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   270
    proof 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   271
      assume "y \<in> Q"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   272
      with \<open>y \<notin> ?Q'\<close> 
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   273
      obtain w where "(w, y) \<in> R" and "w \<in> Q" by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   274
      from \<open>(w, y) \<in> R\<close> \<open>(y, z) \<in> S\<close> have "(w, z) \<in> R O S" by (rule relcompI)
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   275
      with \<open>R O S \<subseteq> R\<close> have "(w, z) \<in> R" ..
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   276
      with \<open>z \<in> ?Q'\<close> have "w \<notin> Q" by blast 
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   277
      with \<open>w \<in> Q\<close> show False by contradiction
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   278
    qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   279
  }
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   280
  with \<open>z \<in> ?Q'\<close> show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<union> S \<longrightarrow> y \<notin> Q" by blast
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   281
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   282
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   283
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   284
text \<open>Well-foundedness of indexed union with disjoint domains and ranges\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   285
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   286
lemma wf_UN: "[| ALL i:I. wf(r i);  
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   287
         ALL i:I. ALL j:I. r i ~= r j --> Domain(r i) Int Range(r j) = {}  
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   288
      |] ==> wf(UN i:I. r i)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   289
apply (simp only: wf_eq_minimal, clarify)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   290
apply (rename_tac A a, case_tac "EX i:I. EX a:A. EX b:A. (b,a) : r i")
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   291
 prefer 2
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   292
 apply force 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   293
apply clarify
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   294
apply (drule bspec, assumption)  
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   295
apply (erule_tac x="{a. a:A & (EX b:A. (b,a) : r i) }" in allE)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   296
apply (blast elim!: allE)  
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   297
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   298
32263
8bc0fd4a23a0 explicit is better than implicit
haftmann
parents: 32244
diff changeset
   299
lemma wfP_SUP:
56218
1c3f1f2431f9 elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents: 56166
diff changeset
   300
  "\<forall>i. wfP (r i) \<Longrightarrow> \<forall>i j. r i \<noteq> r j \<longrightarrow> inf (DomainP (r i)) (RangeP (r j)) = bot \<Longrightarrow> wfP (SUPREMUM UNIV r)"
46883
eec472dae593 tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents: 46882
diff changeset
   301
  apply (rule wf_UN[to_pred])
46882
6242b4bc05bc tuned simpset
noschinl
parents: 46664
diff changeset
   302
  apply simp_all
45970
b6d0cff57d96 adjusted to set/pred distinction by means of type constructor `set`
haftmann
parents: 45139
diff changeset
   303
  done
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   304
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   305
lemma wf_Union: 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   306
 "[| ALL r:R. wf r;  
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   307
     ALL r:R. ALL s:R. r ~= s --> Domain r Int Range s = {}  
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   308
  |] ==> wf(Union R)"
56166
9a241bc276cd normalising simp rules for compound operators
haftmann
parents: 55932
diff changeset
   309
  using wf_UN[of R "\<lambda>i. i"] by simp
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   310
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   311
(*Intuition: we find an (R u S)-min element of a nonempty subset A
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   312
             by case distinction.
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   313
  1. There is a step a -R-> b with a,b : A.
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   314
     Pick an R-min element z of the (nonempty) set {a:A | EX b:A. a -R-> b}.
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   315
     By definition, there is z':A s.t. z -R-> z'. Because z is R-min in the
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   316
     subset, z' must be R-min in A. Because z' has an R-predecessor, it cannot
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   317
     have an S-successor and is thus S-min in A as well.
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   318
  2. There is no such step.
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   319
     Pick an S-min element of A. In this case it must be an R-min
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   320
     element of A as well.
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   321
*)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   322
lemma wf_Un:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   323
     "[| wf r; wf s; Domain r Int Range s = {} |] ==> wf(r Un s)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   324
  using wf_union_compatible[of s r] 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   325
  by (auto simp: Un_ac)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   326
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   327
lemma wf_union_merge: 
32235
8f9b8d14fc9f "more standard" argument order of relation composition (op O)
krauss
parents: 32205
diff changeset
   328
  "wf (R \<union> S) = wf (R O R \<union> S O R \<union> S)" (is "wf ?A = wf ?B")
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   329
proof
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   330
  assume "wf ?A"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   331
  with wf_trancl have wfT: "wf (?A^+)" .
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   332
  moreover have "?B \<subseteq> ?A^+"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   333
    by (subst trancl_unfold, subst trancl_unfold) blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   334
  ultimately show "wf ?B" by (rule wf_subset)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   335
next
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   336
  assume "wf ?B"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   337
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   338
  show "wf ?A"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   339
  proof (rule wfI_min)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   340
    fix Q :: "'a set" and x 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   341
    assume "x \<in> Q"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   342
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   343
    with \<open>wf ?B\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   344
    obtain z where "z \<in> Q" and "\<And>y. (y, z) \<in> ?B \<Longrightarrow> y \<notin> Q" 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   345
      by (erule wfE_min)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   346
    then have A1: "\<And>y. (y, z) \<in> R O R \<Longrightarrow> y \<notin> Q"
32235
8f9b8d14fc9f "more standard" argument order of relation composition (op O)
krauss
parents: 32205
diff changeset
   347
      and A2: "\<And>y. (y, z) \<in> S O R \<Longrightarrow> y \<notin> Q"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   348
      and A3: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> Q"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   349
      by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   350
    
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   351
    show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> ?A \<longrightarrow> y \<notin> Q"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   352
    proof (cases "\<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q")
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   353
      case True
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   354
      with \<open>z \<in> Q\<close> A3 show ?thesis by blast
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   355
    next
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   356
      case False 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   357
      then obtain z' where "z'\<in>Q" "(z', z) \<in> R" by blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   358
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   359
      have "\<forall>y. (y, z') \<in> ?A \<longrightarrow> y \<notin> Q"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   360
      proof (intro allI impI)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   361
        fix y assume "(y, z') \<in> ?A"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   362
        then show "y \<notin> Q"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   363
        proof
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   364
          assume "(y, z') \<in> R" 
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   365
          then have "(y, z) \<in> R O R" using \<open>(z', z) \<in> R\<close> ..
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   366
          with A1 show "y \<notin> Q" .
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   367
        next
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   368
          assume "(y, z') \<in> S" 
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   369
          then have "(y, z) \<in> S O R" using  \<open>(z', z) \<in> R\<close> ..
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   370
          with A2 show "y \<notin> Q" .
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   371
        qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   372
      qed
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   373
      with \<open>z' \<in> Q\<close> show ?thesis ..
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   374
    qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   375
  qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   376
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   377
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   378
lemma wf_comp_self: "wf R = wf (R O R)"  -- \<open>special case\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   379
  by (rule wf_union_merge [where S = "{}", simplified])
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   380
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   381
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   382
subsection \<open>Well-Foundedness of Composition\<close>
60148
f0fc2378a479 new lemma
nipkow
parents: 59807
diff changeset
   383
60493
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   384
text \<open>Bachmair and Dershowitz 1986, Lemma 2. [Provided by Tjark Weber]\<close>
60148
f0fc2378a479 new lemma
nipkow
parents: 59807
diff changeset
   385
60493
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   386
lemma qc_wf_relto_iff:
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   387
  assumes "R O S \<subseteq> (R \<union> S)\<^sup>* O R" -- \<open>R quasi-commutes over S\<close>
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   388
  shows "wf (S\<^sup>* O R O S\<^sup>*) \<longleftrightarrow> wf R" (is "wf ?S \<longleftrightarrow> _")
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   389
proof
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   390
  assume "wf ?S"
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   391
  moreover have "R \<subseteq> ?S" by auto
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   392
  ultimately show "wf R" using wf_subset by auto
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   393
next
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   394
  assume "wf R"
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   395
  show "wf ?S"
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   396
  proof (rule wfI_pf)
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   397
    fix A assume A: "A \<subseteq> ?S `` A"
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   398
    let ?X = "(R \<union> S)\<^sup>* `` A"
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   399
    have *: "R O (R \<union> S)\<^sup>* \<subseteq> (R \<union> S)\<^sup>* O R"
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   400
      proof -
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   401
        { fix x y z assume "(y, z) \<in> (R \<union> S)\<^sup>*" and "(x, y) \<in> R"
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   402
          then have "(x, z) \<in> (R \<union> S)\<^sup>* O R"
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   403
          proof (induct y z)
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   404
            case rtrancl_refl then show ?case by auto
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   405
          next
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   406
            case (rtrancl_into_rtrancl a b c)
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   407
            then have "(x, c) \<in> ((R \<union> S)\<^sup>* O (R \<union> S)\<^sup>*) O R" using assms by blast
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   408
            then show ?case by simp
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   409
          qed }
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   410
        then show ?thesis by auto
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   411
      qed
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   412
    then have "R O S\<^sup>* \<subseteq> (R \<union> S)\<^sup>* O R" using rtrancl_Un_subset by blast
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   413
    then have "?S \<subseteq> (R \<union> S)\<^sup>* O (R \<union> S)\<^sup>* O R" by (simp add: relcomp_mono rtrancl_mono)
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   414
    also have "\<dots> = (R \<union> S)\<^sup>* O R" by (simp add: O_assoc[symmetric])
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   415
    finally have "?S O (R \<union> S)\<^sup>* \<subseteq> (R \<union> S)\<^sup>* O R O (R \<union> S)\<^sup>*" by (simp add: O_assoc[symmetric] relcomp_mono)
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   416
    also have "\<dots> \<subseteq> (R \<union> S)\<^sup>* O (R \<union> S)\<^sup>* O R" using * by (simp add: relcomp_mono)
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   417
    finally have "?S O (R \<union> S)\<^sup>* \<subseteq> (R \<union> S)\<^sup>* O R" by (simp add: O_assoc[symmetric])
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   418
    then have "(?S O (R \<union> S)\<^sup>*) `` A \<subseteq> ((R \<union> S)\<^sup>* O R) `` A" by (simp add: Image_mono)
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   419
    moreover have "?X \<subseteq> (?S O (R \<union> S)\<^sup>*) `` A" using A by (auto simp: relcomp_Image)
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   420
    ultimately have "?X \<subseteq> R `` ?X" by (auto simp: relcomp_Image)
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   421
    then have "?X = {}" using \<open>wf R\<close> by (simp add: wfE_pf)
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   422
    moreover have "A \<subseteq> ?X" by auto
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   423
    ultimately show "A = {}" by simp
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   424
  qed
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   425
qed
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   426
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   427
corollary wf_relcomp_compatible:
60148
f0fc2378a479 new lemma
nipkow
parents: 59807
diff changeset
   428
  assumes "wf R" and "R O S \<subseteq> S O R"
f0fc2378a479 new lemma
nipkow
parents: 59807
diff changeset
   429
  shows "wf (S O R)"
60493
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   430
proof -
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   431
  have "R O S \<subseteq> (R \<union> S)\<^sup>* O R"
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   432
    using assms by blast
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   433
  then have "wf (S\<^sup>* O R O S\<^sup>*)"
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   434
    by (simp add: assms qc_wf_relto_iff)
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   435
  then show ?thesis
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   436
    by (rule Wellfounded.wf_subset) blast
60148
f0fc2378a479 new lemma
nipkow
parents: 59807
diff changeset
   437
qed
f0fc2378a479 new lemma
nipkow
parents: 59807
diff changeset
   438
f0fc2378a479 new lemma
nipkow
parents: 59807
diff changeset
   439
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   440
subsection \<open>Acyclic relations\<close>
33217
ab979f6e99f4 authentic constants; moved "acyclic" further down
krauss
parents: 33216
diff changeset
   441
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   442
lemma wf_acyclic: "wf r ==> acyclic r"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   443
apply (simp add: acyclic_def)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   444
apply (blast elim: wf_trancl [THEN wf_irrefl])
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   445
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   446
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   447
lemmas wfP_acyclicP = wf_acyclic [to_pred]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   448
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   449
text\<open>Wellfoundedness of finite acyclic relations\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   450
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   451
lemma finite_acyclic_wf [rule_format]: "finite r ==> acyclic r --> wf r"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   452
apply (erule finite_induct, blast)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   453
apply (simp (no_asm_simp) only: split_tupled_all)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   454
apply simp
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   455
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   456
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   457
lemma finite_acyclic_wf_converse: "[|finite r; acyclic r|] ==> wf (r^-1)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   458
apply (erule finite_converse [THEN iffD2, THEN finite_acyclic_wf])
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   459
apply (erule acyclic_converse [THEN iffD2])
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   460
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   461
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   462
lemma wf_iff_acyclic_if_finite: "finite r ==> wf r = acyclic r"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   463
by (blast intro: finite_acyclic_wf wf_acyclic)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   464
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   465
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   466
subsection \<open>@{typ nat} is well-founded\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   467
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   468
lemma less_nat_rel: "op < = (\<lambda>m n. n = Suc m)^++"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   469
proof (rule ext, rule ext, rule iffI)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   470
  fix n m :: nat
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   471
  assume "m < n"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   472
  then show "(\<lambda>m n. n = Suc m)^++ m n"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   473
  proof (induct n)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   474
    case 0 then show ?case by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   475
  next
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   476
    case (Suc n) then show ?case
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   477
      by (auto simp add: less_Suc_eq_le le_less intro: tranclp.trancl_into_trancl)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   478
  qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   479
next
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   480
  fix n m :: nat
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   481
  assume "(\<lambda>m n. n = Suc m)^++ m n"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   482
  then show "m < n"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   483
    by (induct n)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   484
      (simp_all add: less_Suc_eq_le reflexive le_less)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   485
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   486
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   487
definition
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   488
  pred_nat :: "(nat * nat) set" where
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   489
  "pred_nat = {(m, n). n = Suc m}"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   490
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   491
definition
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   492
  less_than :: "(nat * nat) set" where
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   493
  "less_than = pred_nat^+"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   494
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   495
lemma less_eq: "(m, n) \<in> pred_nat^+ \<longleftrightarrow> m < n"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   496
  unfolding less_nat_rel pred_nat_def trancl_def by simp
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   497
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   498
lemma pred_nat_trancl_eq_le:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   499
  "(m, n) \<in> pred_nat^* \<longleftrightarrow> m \<le> n"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   500
  unfolding less_eq rtrancl_eq_or_trancl by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   501
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   502
lemma wf_pred_nat: "wf pred_nat"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   503
  apply (unfold wf_def pred_nat_def, clarify)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   504
  apply (induct_tac x, blast+)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   505
  done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   506
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   507
lemma wf_less_than [iff]: "wf less_than"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   508
  by (simp add: less_than_def wf_pred_nat [THEN wf_trancl])
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   509
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   510
lemma trans_less_than [iff]: "trans less_than"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 33217
diff changeset
   511
  by (simp add: less_than_def)
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   512
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   513
lemma less_than_iff [iff]: "((x,y): less_than) = (x<y)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   514
  by (simp add: less_than_def less_eq)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   515
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   516
lemma wf_less: "wf {(x, y::nat). x < y}"
60493
866f41a869e6 New WF theorem by Tjark Weber. Replaced the proof of the subsequent theorem.
paulson <lp15@cam.ac.uk>
parents: 60148
diff changeset
   517
  by (rule Wellfounded.wellorder_class.wf)
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   518
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   519
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   520
subsection \<open>Accessible Part\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   521
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   522
text \<open>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   523
 Inductive definition of the accessible part @{term "acc r"} of a
58623
2db1df2c8467 more bibtex entries;
wenzelm
parents: 56643
diff changeset
   524
 relation; see also @{cite "paulin-tlca"}.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   525
\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   526
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   527
inductive_set
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   528
  acc :: "('a * 'a) set => 'a set"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   529
  for r :: "('a * 'a) set"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   530
  where
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   531
    accI: "(!!y. (y, x) : r ==> y : acc r) ==> x : acc r"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   532
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   533
abbreviation
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   534
  termip :: "('a => 'a => bool) => 'a => bool" where
45137
6e422d180de8 modernized definitions
haftmann
parents: 45012
diff changeset
   535
  "termip r \<equiv> accp (r\<inverse>\<inverse>)"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   536
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   537
abbreviation
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   538
  termi :: "('a * 'a) set => 'a set" where
45137
6e422d180de8 modernized definitions
haftmann
parents: 45012
diff changeset
   539
  "termi r \<equiv> acc (r\<inverse>)"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   540
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   541
lemmas accpI = accp.accI
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   542
54295
45a5523d4a63 qualifed popular user space names
haftmann
parents: 49945
diff changeset
   543
lemma accp_eq_acc [code]:
45a5523d4a63 qualifed popular user space names
haftmann
parents: 49945
diff changeset
   544
  "accp r = (\<lambda>x. x \<in> Wellfounded.acc {(x, y). r x y})"
45a5523d4a63 qualifed popular user space names
haftmann
parents: 49945
diff changeset
   545
  by (simp add: acc_def)
45a5523d4a63 qualifed popular user space names
haftmann
parents: 49945
diff changeset
   546
45a5523d4a63 qualifed popular user space names
haftmann
parents: 49945
diff changeset
   547
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   548
text \<open>Induction rules\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   549
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   550
theorem accp_induct:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   551
  assumes major: "accp r a"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   552
  assumes hyp: "!!x. accp r x ==> \<forall>y. r y x --> P y ==> P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   553
  shows "P a"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   554
  apply (rule major [THEN accp.induct])
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   555
  apply (rule hyp)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   556
   apply (rule accp.accI)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   557
   apply fast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   558
  apply fast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   559
  done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   560
61337
4645502c3c64 fewer aliases for toplevel theorem statements;
wenzelm
parents: 60758
diff changeset
   561
lemmas accp_induct_rule = accp_induct [rule_format, induct set: accp]
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   562
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   563
theorem accp_downward: "accp r b ==> r a b ==> accp r a"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   564
  apply (erule accp.cases)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   565
  apply fast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   566
  done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   567
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   568
lemma not_accp_down:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   569
  assumes na: "\<not> accp R x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   570
  obtains z where "R z x" and "\<not> accp R z"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   571
proof -
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   572
  assume a: "\<And>z. \<lbrakk>R z x; \<not> accp R z\<rbrakk> \<Longrightarrow> thesis"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   573
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   574
  show thesis
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   575
  proof (cases "\<forall>z. R z x \<longrightarrow> accp R z")
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   576
    case True
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   577
    hence "\<And>z. R z x \<Longrightarrow> accp R z" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   578
    hence "accp R x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   579
      by (rule accp.accI)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   580
    with na show thesis ..
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   581
  next
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   582
    case False then obtain z where "R z x" and "\<not> accp R z"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   583
      by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   584
    with a show thesis .
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   585
  qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   586
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   587
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   588
lemma accp_downwards_aux: "r\<^sup>*\<^sup>* b a ==> accp r a --> accp r b"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   589
  apply (erule rtranclp_induct)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   590
   apply blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   591
  apply (blast dest: accp_downward)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   592
  done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   593
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   594
theorem accp_downwards: "accp r a ==> r\<^sup>*\<^sup>* b a ==> accp r b"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   595
  apply (blast dest: accp_downwards_aux)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   596
  done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   597
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   598
theorem accp_wfPI: "\<forall>x. accp r x ==> wfP r"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   599
  apply (rule wfPUNIVI)
44921
58eef4843641 tuned proofs
huffman
parents: 44144
diff changeset
   600
  apply (rule_tac P=P in accp_induct)
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   601
   apply blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   602
  apply blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   603
  done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   604
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   605
theorem accp_wfPD: "wfP r ==> accp r x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   606
  apply (erule wfP_induct_rule)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   607
  apply (rule accp.accI)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   608
  apply blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   609
  done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   610
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   611
theorem wfP_accp_iff: "wfP r = (\<forall>x. accp r x)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   612
  apply (blast intro: accp_wfPI dest: accp_wfPD)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   613
  done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   614
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   615
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   616
text \<open>Smaller relations have bigger accessible parts:\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   617
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   618
lemma accp_subset:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   619
  assumes sub: "R1 \<le> R2"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   620
  shows "accp R2 \<le> accp R1"
26803
0af0f674845d - Explicitely passed pred_subset_eq and pred_equals_eq as an argument to the
berghofe
parents: 26748
diff changeset
   621
proof (rule predicate1I)
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   622
  fix x assume "accp R2 x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   623
  then show "accp R1 x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   624
  proof (induct x)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   625
    fix x
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   626
    assume ih: "\<And>y. R2 y x \<Longrightarrow> accp R1 y"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   627
    with sub show "accp R1 x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   628
      by (blast intro: accp.accI)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   629
  qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   630
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   631
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   632
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   633
text \<open>This is a generalized induction theorem that works on
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   634
  subsets of the accessible part.\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   635
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   636
lemma accp_subset_induct:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   637
  assumes subset: "D \<le> accp R"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   638
    and dcl: "\<And>x z. \<lbrakk>D x; R z x\<rbrakk> \<Longrightarrow> D z"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   639
    and "D x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   640
    and istep: "\<And>x. \<lbrakk>D x; (\<And>z. R z x \<Longrightarrow> P z)\<rbrakk> \<Longrightarrow> P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   641
  shows "P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   642
proof -
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   643
  from subset and \<open>D x\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   644
  have "accp R x" ..
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   645
  then show "P x" using \<open>D x\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   646
  proof (induct x)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   647
    fix x
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   648
    assume "D x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   649
      and "\<And>y. R y x \<Longrightarrow> D y \<Longrightarrow> P y"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   650
    with dcl and istep show "P x" by blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   651
  qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   652
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   653
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   654
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   655
text \<open>Set versions of the above theorems\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   656
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   657
lemmas acc_induct = accp_induct [to_set]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   658
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   659
lemmas acc_induct_rule = acc_induct [rule_format, induct set: acc]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   660
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   661
lemmas acc_downward = accp_downward [to_set]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   662
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   663
lemmas not_acc_down = not_accp_down [to_set]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   664
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   665
lemmas acc_downwards_aux = accp_downwards_aux [to_set]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   666
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   667
lemmas acc_downwards = accp_downwards [to_set]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   668
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   669
lemmas acc_wfI = accp_wfPI [to_set]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   670
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   671
lemmas acc_wfD = accp_wfPD [to_set]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   672
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   673
lemmas wf_acc_iff = wfP_accp_iff [to_set]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   674
46177
adac34829e10 pred_subset_eq and SUP_UN_eq2 are now standard pred_set_conv rules
berghofe
parents: 45970
diff changeset
   675
lemmas acc_subset = accp_subset [to_set]
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   676
46177
adac34829e10 pred_subset_eq and SUP_UN_eq2 are now standard pred_set_conv rules
berghofe
parents: 45970
diff changeset
   677
lemmas acc_subset_induct = accp_subset_induct [to_set]
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   678
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   679
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   680
subsection \<open>Tools for building wellfounded relations\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   681
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   682
text \<open>Inverse Image\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   683
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   684
lemma wf_inv_image [simp,intro!]: "wf(r) ==> wf(inv_image r (f::'a=>'b))"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   685
apply (simp (no_asm_use) add: inv_image_def wf_eq_minimal)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   686
apply clarify
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   687
apply (subgoal_tac "EX (w::'b) . w : {w. EX (x::'a) . x: Q & (f x = w) }")
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   688
prefer 2 apply (blast del: allE)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   689
apply (erule allE)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   690
apply (erule (1) notE impE)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   691
apply blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   692
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   693
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   694
text \<open>Measure functions into @{typ nat}\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   695
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   696
definition measure :: "('a => nat) => ('a * 'a)set"
45137
6e422d180de8 modernized definitions
haftmann
parents: 45012
diff changeset
   697
where "measure = inv_image less_than"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   698
46356
48fcca8965f4 adding code_unfold to make measure executable
bulwahn
parents: 46349
diff changeset
   699
lemma in_measure[simp, code_unfold]: "((x,y) : measure f) = (f x < f y)"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   700
  by (simp add:measure_def)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   701
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   702
lemma wf_measure [iff]: "wf (measure f)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   703
apply (unfold measure_def)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   704
apply (rule wf_less_than [THEN wf_inv_image])
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   705
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   706
41720
f749155883d7 added termination lemmas
nipkow
parents: 41075
diff changeset
   707
lemma wf_if_measure: fixes f :: "'a \<Rightarrow> nat"
f749155883d7 added termination lemmas
nipkow
parents: 41075
diff changeset
   708
shows "(!!x. P x \<Longrightarrow> f(g x) < f x) \<Longrightarrow> wf {(y,x). P x \<and> y = g x}"
f749155883d7 added termination lemmas
nipkow
parents: 41075
diff changeset
   709
apply(insert wf_measure[of f])
f749155883d7 added termination lemmas
nipkow
parents: 41075
diff changeset
   710
apply(simp only: measure_def inv_image_def less_than_def less_eq)
f749155883d7 added termination lemmas
nipkow
parents: 41075
diff changeset
   711
apply(erule wf_subset)
f749155883d7 added termination lemmas
nipkow
parents: 41075
diff changeset
   712
apply auto
f749155883d7 added termination lemmas
nipkow
parents: 41075
diff changeset
   713
done
f749155883d7 added termination lemmas
nipkow
parents: 41075
diff changeset
   714
f749155883d7 added termination lemmas
nipkow
parents: 41075
diff changeset
   715
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   716
text\<open>Lexicographic combinations\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   717
37767
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 37407
diff changeset
   718
definition lex_prod :: "('a \<times>'a) set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> (('a \<times> 'b) \<times> ('a \<times> 'b)) set" (infixr "<*lex*>" 80) where
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 37407
diff changeset
   719
  "ra <*lex*> rb = {((a, b), (a', b')). (a, a') \<in> ra \<or> a = a' \<and> (b, b') \<in> rb}"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   720
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   721
lemma wf_lex_prod [intro!]: "[| wf(ra); wf(rb) |] ==> wf(ra <*lex*> rb)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   722
apply (unfold wf_def lex_prod_def) 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   723
apply (rule allI, rule impI)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   724
apply (simp (no_asm_use) only: split_paired_All)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   725
apply (drule spec, erule mp) 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   726
apply (rule allI, rule impI)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   727
apply (drule spec, erule mp, blast) 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   728
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   729
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   730
lemma in_lex_prod[simp]: 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   731
  "(((a,b),(a',b')): r <*lex*> s) = ((a,a'): r \<or> (a = a' \<and> (b, b') : s))"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   732
  by (auto simp:lex_prod_def)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   733
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   734
text\<open>@{term "op <*lex*>"} preserves transitivity\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   735
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   736
lemma trans_lex_prod [intro!]: 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   737
    "[| trans R1; trans R2 |] ==> trans (R1 <*lex*> R2)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   738
by (unfold trans_def lex_prod_def, blast) 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   739
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   740
text \<open>lexicographic combinations with measure functions\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   741
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   742
definition 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   743
  mlex_prod :: "('a \<Rightarrow> nat) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" (infixr "<*mlex*>" 80)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   744
where
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   745
  "f <*mlex*> R = inv_image (less_than <*lex*> R) (%x. (f x, x))"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   746
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   747
lemma wf_mlex: "wf R \<Longrightarrow> wf (f <*mlex*> R)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   748
unfolding mlex_prod_def
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   749
by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   750
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   751
lemma mlex_less: "f x < f y \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   752
unfolding mlex_prod_def by simp
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   753
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   754
lemma mlex_leq: "f x \<le> f y \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   755
unfolding mlex_prod_def by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   756
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   757
text \<open>proper subset relation on finite sets\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   758
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   759
definition finite_psubset  :: "('a set * 'a set) set"
45137
6e422d180de8 modernized definitions
haftmann
parents: 45012
diff changeset
   760
where "finite_psubset = {(A,B). A < B & finite B}"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   761
28260
703046c93ffe wf_finite_psubset[simp], in_finite_psubset[simp]
krauss
parents: 27823
diff changeset
   762
lemma wf_finite_psubset[simp]: "wf(finite_psubset)"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   763
apply (unfold finite_psubset_def)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   764
apply (rule wf_measure [THEN wf_subset])
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   765
apply (simp add: measure_def inv_image_def less_than_def less_eq)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   766
apply (fast elim!: psubset_card_mono)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   767
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   768
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   769
lemma trans_finite_psubset: "trans finite_psubset"
26803
0af0f674845d - Explicitely passed pred_subset_eq and pred_equals_eq as an argument to the
berghofe
parents: 26748
diff changeset
   770
by (simp add: finite_psubset_def less_le trans_def, blast)
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   771
28260
703046c93ffe wf_finite_psubset[simp], in_finite_psubset[simp]
krauss
parents: 27823
diff changeset
   772
lemma in_finite_psubset[simp]: "(A, B) \<in> finite_psubset = (A < B & finite B)"
703046c93ffe wf_finite_psubset[simp], in_finite_psubset[simp]
krauss
parents: 27823
diff changeset
   773
unfolding finite_psubset_def by auto
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   774
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   775
text \<open>max- and min-extension of order to finite sets\<close>
28735
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   776
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   777
inductive_set max_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set" 
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   778
for R :: "('a \<times> 'a) set"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   779
where
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   780
  max_extI[intro]: "finite X \<Longrightarrow> finite Y \<Longrightarrow> Y \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> \<exists>y\<in>Y. (x, y) \<in> R) \<Longrightarrow> (X, Y) \<in> max_ext R"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   781
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   782
lemma max_ext_wf:
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   783
  assumes wf: "wf r"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   784
  shows "wf (max_ext r)"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   785
proof (rule acc_wfI, intro allI)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   786
  fix M show "M \<in> acc (max_ext r)" (is "_ \<in> ?W")
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   787
  proof cases
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   788
    assume "finite M"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   789
    thus ?thesis
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   790
    proof (induct M)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   791
      show "{} \<in> ?W"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   792
        by (rule accI) (auto elim: max_ext.cases)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   793
    next
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   794
      fix M a assume "M \<in> ?W" "finite M"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   795
      with wf show "insert a M \<in> ?W"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   796
      proof (induct arbitrary: M)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   797
        fix M a
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   798
        assume "M \<in> ?W"  and  [intro]: "finite M"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   799
        assume hyp: "\<And>b M. (b, a) \<in> r \<Longrightarrow> M \<in> ?W \<Longrightarrow> finite M \<Longrightarrow> insert b M \<in> ?W"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   800
        {
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   801
          fix N M :: "'a set"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   802
          assume "finite N" "finite M"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   803
          then
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   804
          have "\<lbrakk>M \<in> ?W ; (\<And>y. y \<in> N \<Longrightarrow> (y, a) \<in> r)\<rbrakk> \<Longrightarrow>  N \<union> M \<in> ?W"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   805
            by (induct N arbitrary: M) (auto simp: hyp)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   806
        }
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   807
        note add_less = this
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   808
        
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   809
        show "insert a M \<in> ?W"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   810
        proof (rule accI)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   811
          fix N assume Nless: "(N, insert a M) \<in> max_ext r"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   812
          hence asm1: "\<And>x. x \<in> N \<Longrightarrow> (x, a) \<in> r \<or> (\<exists>y \<in> M. (x, y) \<in> r)"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   813
            by (auto elim!: max_ext.cases)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   814
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   815
          let ?N1 = "{ n \<in> N. (n, a) \<in> r }"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   816
          let ?N2 = "{ n \<in> N. (n, a) \<notin> r }"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 37767
diff changeset
   817
          have N: "?N1 \<union> ?N2 = N" by (rule set_eqI) auto
28735
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   818
          from Nless have "finite N" by (auto elim: max_ext.cases)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   819
          then have finites: "finite ?N1" "finite ?N2" by auto
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   820
          
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   821
          have "?N2 \<in> ?W"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   822
          proof cases
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   823
            assume [simp]: "M = {}"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   824
            have Mw: "{} \<in> ?W" by (rule accI) (auto elim: max_ext.cases)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   825
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   826
            from asm1 have "?N2 = {}" by auto
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   827
            with Mw show "?N2 \<in> ?W" by (simp only:)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   828
          next
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   829
            assume "M \<noteq> {}"
49945
fb696ff1f086 adjusting proofs
bulwahn
parents: 48891
diff changeset
   830
            from asm1 finites have N2: "(?N2, M) \<in> max_ext r" 
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   831
              by (rule_tac max_extI[OF _ _ \<open>M \<noteq> {}\<close>]) auto
49945
fb696ff1f086 adjusting proofs
bulwahn
parents: 48891
diff changeset
   832
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   833
            with \<open>M \<in> ?W\<close> show "?N2 \<in> ?W" by (rule acc_downward)
28735
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   834
          qed
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   835
          with finites have "?N1 \<union> ?N2 \<in> ?W" 
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   836
            by (rule add_less) simp
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   837
          then show "N \<in> ?W" by (simp only: N)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   838
        qed
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   839
      qed
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   840
    qed
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   841
  next
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   842
    assume [simp]: "\<not> finite M"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   843
    show ?thesis
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   844
      by (rule accI) (auto elim: max_ext.cases)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   845
  qed
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   846
qed
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   847
29125
d41182a8135c method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents: 28845
diff changeset
   848
lemma max_ext_additive: 
d41182a8135c method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents: 28845
diff changeset
   849
 "(A, B) \<in> max_ext R \<Longrightarrow> (C, D) \<in> max_ext R \<Longrightarrow>
d41182a8135c method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents: 28845
diff changeset
   850
  (A \<union> C, B \<union> D) \<in> max_ext R"
d41182a8135c method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents: 28845
diff changeset
   851
by (force elim!: max_ext.cases)
d41182a8135c method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents: 28845
diff changeset
   852
28735
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   853
37767
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 37407
diff changeset
   854
definition min_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set"  where
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 37407
diff changeset
   855
  "min_ext r = {(X, Y) | X Y. X \<noteq> {} \<and> (\<forall>y \<in> Y. (\<exists>x \<in> X. (x, y) \<in> r))}"
28735
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   856
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   857
lemma min_ext_wf:
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   858
  assumes "wf r"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   859
  shows "wf (min_ext r)"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   860
proof (rule wfI_min)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   861
  fix Q :: "'a set set"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   862
  fix x
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   863
  assume nonempty: "x \<in> Q"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   864
  show "\<exists>m \<in> Q. (\<forall> n. (n, m) \<in> min_ext r \<longrightarrow> n \<notin> Q)"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   865
  proof cases
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   866
    assume "Q = {{}}" thus ?thesis by (simp add: min_ext_def)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   867
  next
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   868
    assume "Q \<noteq> {{}}"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   869
    with nonempty
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   870
    obtain e x where "x \<in> Q" "e \<in> x" by force
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   871
    then have eU: "e \<in> \<Union>Q" by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   872
    with \<open>wf r\<close> 
28735
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   873
    obtain z where z: "z \<in> \<Union>Q" "\<And>y. (y, z) \<in> r \<Longrightarrow> y \<notin> \<Union>Q" 
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   874
      by (erule wfE_min)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   875
    from z obtain m where "m \<in> Q" "z \<in> m" by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   876
    from \<open>m \<in> Q\<close>
28735
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   877
    show ?thesis
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   878
    proof (rule, intro bexI allI impI)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   879
      fix n
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   880
      assume smaller: "(n, m) \<in> min_ext r"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   881
      with \<open>z \<in> m\<close> obtain y where y: "y \<in> n" "(y, z) \<in> r" by (auto simp: min_ext_def)
28735
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   882
      then show "n \<notin> Q" using z(2) by auto
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   883
    qed      
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   884
  qed
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   885
qed
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   886
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60493
diff changeset
   887
text\<open>Bounded increase must terminate:\<close>
43137
32b888e1a170 new lemmas
nipkow
parents: 41720
diff changeset
   888
32b888e1a170 new lemmas
nipkow
parents: 41720
diff changeset
   889
lemma wf_bounded_measure:
32b888e1a170 new lemmas
nipkow
parents: 41720
diff changeset
   890
fixes ub :: "'a \<Rightarrow> nat" and f :: "'a \<Rightarrow> nat"
43140
504d72a39638 tuned lemmas
nipkow
parents: 43137
diff changeset
   891
assumes "!!a b. (b,a) : r \<Longrightarrow> ub b \<le> ub a & ub a \<ge> f b & f b > f a"
43137
32b888e1a170 new lemmas
nipkow
parents: 41720
diff changeset
   892
shows "wf r"
32b888e1a170 new lemmas
nipkow
parents: 41720
diff changeset
   893
apply(rule wf_subset[OF wf_measure[of "%a. ub a - f a"]])
32b888e1a170 new lemmas
nipkow
parents: 41720
diff changeset
   894
apply (auto dest: assms)
32b888e1a170 new lemmas
nipkow
parents: 41720
diff changeset
   895
done
32b888e1a170 new lemmas
nipkow
parents: 41720
diff changeset
   896
32b888e1a170 new lemmas
nipkow
parents: 41720
diff changeset
   897
lemma wf_bounded_set:
32b888e1a170 new lemmas
nipkow
parents: 41720
diff changeset
   898
fixes ub :: "'a \<Rightarrow> 'b set" and f :: "'a \<Rightarrow> 'b set"
32b888e1a170 new lemmas
nipkow
parents: 41720
diff changeset
   899
assumes "!!a b. (b,a) : r \<Longrightarrow>
43140
504d72a39638 tuned lemmas
nipkow
parents: 43137
diff changeset
   900
  finite(ub a) & ub b \<subseteq> ub a & ub a \<supseteq> f b & f b \<supset> f a"
43137
32b888e1a170 new lemmas
nipkow
parents: 41720
diff changeset
   901
shows "wf r"
32b888e1a170 new lemmas
nipkow
parents: 41720
diff changeset
   902
apply(rule wf_bounded_measure[of r "%a. card(ub a)" "%a. card(f a)"])
32b888e1a170 new lemmas
nipkow
parents: 41720
diff changeset
   903
apply(drule assms)
43140
504d72a39638 tuned lemmas
nipkow
parents: 43137
diff changeset
   904
apply (blast intro: card_mono finite_subset psubset_card_mono dest: psubset_eq[THEN iffD2])
43137
32b888e1a170 new lemmas
nipkow
parents: 41720
diff changeset
   905
done
32b888e1a170 new lemmas
nipkow
parents: 41720
diff changeset
   906
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   907
54295
45a5523d4a63 qualifed popular user space names
haftmann
parents: 49945
diff changeset
   908
hide_const (open) acc accp
45a5523d4a63 qualifed popular user space names
haftmann
parents: 49945
diff changeset
   909
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   910
end