src/HOL/Wellfounded.thy
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point-free characterization of well-foundedness
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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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*)
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header {*Well-founded Recursion*}
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theory Wellfounded
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imports Finite_Set Transitive_Closure
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uses ("Tools/Function/size.ML")
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begin
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subsection {* Basic Definitions *}
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constdefs
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  wf         :: "('a * 'a)set => bool"
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  "wf(r) == (!P. (!x. (!y. (y,x):r --> P(y)) --> P(x)) --> (!x. P(x)))"
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  wfP :: "('a => 'a => bool) => bool"
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  "wfP r == wf {(x, y). r x y}"
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  acyclic :: "('a*'a)set => bool"
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  "acyclic r == !x. (x,x) ~: r^+"
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abbreviation acyclicP :: "('a => 'a => bool) => bool" where
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  "acyclicP r == acyclic {(x, y). r x y}"
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lemma wfP_wf_eq [pred_set_conv]: "wfP (\<lambda>x y. (x, y) \<in> r) = wf r"
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  by (simp add: wfP_def)
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lemma wfUNIVI: 
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   "(!!P x. (ALL x. (ALL y. (y,x) : r --> P(y)) --> P(x)) ==> P(x)) ==> wf(r)"
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  unfolding wf_def by blast
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lemmas wfPUNIVI = wfUNIVI [to_pred]
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text{*Restriction to domain @{term A} and range @{term B}.  If @{term r} is
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    well-founded over their intersection, then @{term "wf r"}*}
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lemma wfI: 
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 "[| r \<subseteq> A <*> B; 
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     !!x P. [|\<forall>x. (\<forall>y. (y,x) : r --> P y) --> P x;  x : A; x : B |] ==> P x |]
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  ==>  wf r"
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  unfolding wf_def by blast
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lemma wf_induct: 
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    "[| wf(r);           
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        !!x.[| ALL y. (y,x): r --> P(y) |] ==> P(x)  
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     |]  ==>  P(a)"
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  unfolding wf_def by blast
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lemmas wfP_induct = wf_induct [to_pred]
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lemmas wf_induct_rule = wf_induct [rule_format, consumes 1, case_names less, induct set: wf]
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lemmas wfP_induct_rule = wf_induct_rule [to_pred, induct set: wfP]
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lemma wf_not_sym: "wf r ==> (a, x) : r ==> (x, a) ~: r"
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  by (induct a arbitrary: x set: wf) blast
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lemma wf_asym:
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  assumes "wf r" "(a, x) \<in> r"
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  obtains "(x, a) \<notin> r"
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  by (drule wf_not_sym[OF assms])
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lemma wf_not_refl [simp]: "wf r ==> (a, a) ~: r"
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  by (blast elim: wf_asym)
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lemma wf_irrefl: assumes "wf r" obtains "(a, a) \<notin> r"
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by (drule wf_not_refl[OF assms])
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lemma wf_wellorderI:
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  assumes wf: "wf {(x::'a::ord, y). x < y}"
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  assumes lin: "OFCLASS('a::ord, linorder_class)"
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  shows "OFCLASS('a::ord, wellorder_class)"
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using lin by (rule wellorder_class.intro)
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  (blast intro: wellorder_axioms.intro wf_induct_rule [OF wf])
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lemma (in wellorder) wf:
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  "wf {(x, y). x < y}"
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unfolding wf_def by (blast intro: less_induct)
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subsection {* Basic Results *}
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text {* Point-free characterization of well-foundedness *}
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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{*Minimal-element characterization of well-foundedness*}
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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{* Well-foundedness of transitive closure *}
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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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   149
    assume induct_step: "!!x. (!!y. (y, x) : r^+ ==> P y) ==> P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   150
    have "P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   151
    proof (rule induct_step)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   152
      fix y assume "(y, x) : r^+"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   153
      with `wf r` show "P y"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   154
      proof (induct x arbitrary: y)
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   155
        case (less x)
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   156
        note hyp = `\<And>x' y'. (x', x) : r ==> (y', x') : r^+ ==> P y'`
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   157
        from `(y, x) : r^+` show "P y"
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   158
        proof cases
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   159
          case base
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   160
          show "P y"
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   161
          proof (rule induct_step)
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   162
            fix y' assume "(y', y) : r^+"
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   163
            with `(y, x) : r` show "P y'" by (rule hyp [of y y'])
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   164
          qed
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   165
        next
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   166
          case step
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32704
diff changeset
   167
          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
   168
          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
   169
        qed
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   170
      qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   171
    qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   172
  } then show ?thesis unfolding wf_def by blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   173
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   174
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   175
lemmas wfP_trancl = wf_trancl [to_pred]
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_converse_trancl: "wf (r^-1) ==> wf ((r^+)^-1)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   178
  apply (subst trancl_converse [symmetric])
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   179
  apply (erule wf_trancl)
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
33216
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   182
text {* Well-foundedness of subsets *}
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   183
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   184
lemma wf_subset: "[| wf(r);  p<=r |] ==> wf(p)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   185
  apply (simp (no_asm_use) add: wf_eq_minimal)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   186
  apply fast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   187
  done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   188
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   189
lemmas wfP_subset = wf_subset [to_pred]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   190
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   191
text {* Well-foundedness of the empty relation *}
33216
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   192
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   193
lemma wf_empty [iff]: "wf {}"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   194
  by (simp add: wf_def)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   195
32205
49db434c157f explicit is better than implicit
haftmann
parents: 31775
diff changeset
   196
lemma wfP_empty [iff]:
49db434c157f explicit is better than implicit
haftmann
parents: 31775
diff changeset
   197
  "wfP (\<lambda>x y. False)"
49db434c157f explicit is better than implicit
haftmann
parents: 31775
diff changeset
   198
proof -
49db434c157f explicit is better than implicit
haftmann
parents: 31775
diff changeset
   199
  have "wfP bot" by (fact wf_empty [to_pred bot_empty_eq2])
49db434c157f explicit is better than implicit
haftmann
parents: 31775
diff changeset
   200
  then show ?thesis by (simp add: bot_fun_eq bot_bool_eq)
49db434c157f explicit is better than implicit
haftmann
parents: 31775
diff changeset
   201
qed
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   202
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   203
lemma wf_Int1: "wf r ==> wf (r Int r')"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   204
  apply (erule wf_subset)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   205
  apply (rule Int_lower1)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   206
  done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   207
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   208
lemma wf_Int2: "wf r ==> wf (r' Int r)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   209
  apply (erule wf_subset)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   210
  apply (rule Int_lower2)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   211
  done  
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   212
33216
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   213
text {* Exponentiation *}
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   214
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   215
lemma wf_exp:
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   216
  assumes "wf (R ^^ n)"
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   217
  shows "wf R"
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   218
proof (rule wfI_pf)
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   219
  fix A assume "A \<subseteq> R `` A"
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   220
  then have "A \<subseteq> (R ^^ n) `` A" by (induct n) force+
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   221
  with `wf (R ^^ n)`
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   222
  show "A = {}" by (rule wfE_pf)
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   223
qed
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   224
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   225
text {* Well-foundedness of insert *}
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   226
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   227
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
   228
apply (rule iffI)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   229
 apply (blast elim: wf_trancl [THEN wf_irrefl]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   230
              intro: rtrancl_into_trancl1 wf_subset 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   231
                     rtrancl_mono [THEN [2] rev_subsetD])
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   232
apply (simp add: wf_eq_minimal, safe)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   233
apply (rule allE, assumption, erule impE, blast) 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   234
apply (erule bexE)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   235
apply (rename_tac "a", case_tac "a = x")
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   236
 prefer 2
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   237
apply blast 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   238
apply (case_tac "y:Q")
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   239
 prefer 2 apply blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   240
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
   241
 apply assumption
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   242
apply (erule_tac V = "ALL Q. (EX x. x : Q) --> ?P Q" in thin_rl) 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   243
  --{*essential for speed*}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   244
txt{*Blast with new substOccur fails*}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   245
apply (fast intro: converse_rtrancl_into_rtrancl)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   246
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   247
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   248
text{*Well-foundedness of image*}
33216
7c61bc5d7310 point-free characterization of well-foundedness
krauss
parents: 33215
diff changeset
   249
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   250
lemma wf_prod_fun_image: "[| wf r; inj f |] ==> wf(prod_fun f f ` r)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   251
apply (simp only: wf_eq_minimal, clarify)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   252
apply (case_tac "EX p. f p : Q")
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   253
apply (erule_tac x = "{p. f p : Q}" in allE)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   254
apply (fast dest: inj_onD, blast)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   255
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   256
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   257
26976
cf147f69b3df rearranged subsections
krauss
parents: 26803
diff changeset
   258
subsection {* Well-Foundedness Results for Unions *}
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   259
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   260
lemma wf_union_compatible:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   261
  assumes "wf R" "wf S"
32235
8f9b8d14fc9f "more standard" argument order of relation composition (op O)
krauss
parents: 32205
diff changeset
   262
  assumes "R O S \<subseteq> R"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   263
  shows "wf (R \<union> S)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   264
proof (rule wfI_min)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   265
  fix x :: 'a and Q 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   266
  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
   267
  assume "x \<in> Q"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   268
  obtain a where "a \<in> ?Q'"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   269
    by (rule wfE_min [OF `wf R` `x \<in> Q`]) blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   270
  with `wf S`
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   271
  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
   272
  { 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   273
    fix y assume "(y, z) \<in> S"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   274
    then have "y \<notin> ?Q'" by (rule zmin)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   275
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   276
    have "y \<notin> Q"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   277
    proof 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   278
      assume "y \<in> Q"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   279
      with `y \<notin> ?Q'` 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   280
      obtain w where "(w, y) \<in> R" and "w \<in> Q" by auto
32235
8f9b8d14fc9f "more standard" argument order of relation composition (op O)
krauss
parents: 32205
diff changeset
   281
      from `(w, y) \<in> R` `(y, z) \<in> S` have "(w, z) \<in> R O S" by (rule rel_compI)
8f9b8d14fc9f "more standard" argument order of relation composition (op O)
krauss
parents: 32205
diff changeset
   282
      with `R O S \<subseteq> R` have "(w, z) \<in> R" ..
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   283
      with `z \<in> ?Q'` have "w \<notin> Q" by blast 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   284
      with `w \<in> Q` show False by contradiction
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   285
    qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   286
  }
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   287
  with `z \<in> ?Q'` show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<union> S \<longrightarrow> y \<notin> Q" by blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   288
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   289
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   290
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   291
text {* Well-foundedness of indexed union with disjoint domains and ranges *}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   292
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   293
lemma wf_UN: "[| ALL i:I. wf(r i);  
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   294
         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
   295
      |] ==> wf(UN i:I. r i)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   296
apply (simp only: wf_eq_minimal, clarify)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   297
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
   298
 prefer 2
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   299
 apply force 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   300
apply clarify
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   301
apply (drule bspec, assumption)  
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   302
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
   303
apply (blast elim!: allE)  
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   304
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   305
32263
8bc0fd4a23a0 explicit is better than implicit
haftmann
parents: 32244
diff changeset
   306
lemma wfP_SUP:
8bc0fd4a23a0 explicit is better than implicit
haftmann
parents: 32244
diff changeset
   307
  "\<forall>i. wfP (r i) \<Longrightarrow> \<forall>i j. r i \<noteq> r j \<longrightarrow> inf (DomainP (r i)) (RangeP (r j)) = bot \<Longrightarrow> wfP (SUPR UNIV r)"
32704
6f0a56d255f4 simplified proof
haftmann
parents: 32463
diff changeset
   308
  by (rule wf_UN [where I=UNIV and r="\<lambda>i. {(x, y). r i x y}", to_pred SUP_UN_eq2])
6f0a56d255f4 simplified proof
haftmann
parents: 32463
diff changeset
   309
    (simp_all add: Collect_def)
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
lemma wf_Union: 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   312
 "[| ALL r:R. wf r;  
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   313
     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
   314
  |] ==> wf(Union R)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   315
apply (simp add: Union_def)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   316
apply (blast intro: wf_UN)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   317
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   318
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   319
(*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
   320
             by case distinction.
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   321
  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
   322
     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
   323
     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
   324
     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
   325
     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
   326
  2. There is no such step.
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   327
     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
   328
     element of A as well.
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   329
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   330
*)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   331
lemma wf_Un:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   332
     "[| 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
   333
  using wf_union_compatible[of s r] 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   334
  by (auto simp: Un_ac)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   335
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   336
lemma wf_union_merge: 
32235
8f9b8d14fc9f "more standard" argument order of relation composition (op O)
krauss
parents: 32205
diff changeset
   337
  "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
   338
proof
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   339
  assume "wf ?A"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   340
  with wf_trancl have wfT: "wf (?A^+)" .
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   341
  moreover have "?B \<subseteq> ?A^+"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   342
    by (subst trancl_unfold, subst trancl_unfold) blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   343
  ultimately show "wf ?B" by (rule wf_subset)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   344
next
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   345
  assume "wf ?B"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   346
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   347
  show "wf ?A"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   348
  proof (rule wfI_min)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   349
    fix Q :: "'a set" and x 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   350
    assume "x \<in> Q"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   351
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   352
    with `wf ?B`
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   353
    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
   354
      by (erule wfE_min)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   355
    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
   356
      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
   357
      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
   358
      by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   359
    
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   360
    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
   361
    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
   362
      case True
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   363
      with `z \<in> Q` A3 show ?thesis by blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   364
    next
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   365
      case False 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   366
      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
   367
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   368
      have "\<forall>y. (y, z') \<in> ?A \<longrightarrow> y \<notin> Q"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   369
      proof (intro allI impI)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   370
        fix y assume "(y, z') \<in> ?A"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   371
        then show "y \<notin> Q"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   372
        proof
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   373
          assume "(y, z') \<in> R" 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   374
          then have "(y, z) \<in> R O R" using `(z', z) \<in> R` ..
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   375
          with A1 show "y \<notin> Q" .
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   376
        next
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   377
          assume "(y, z') \<in> S" 
32235
8f9b8d14fc9f "more standard" argument order of relation composition (op O)
krauss
parents: 32205
diff changeset
   378
          then have "(y, z) \<in> S O R" using  `(z', z) \<in> R` ..
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   379
          with A2 show "y \<notin> Q" .
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   380
        qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   381
      qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   382
      with `z' \<in> Q` show ?thesis ..
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   383
    qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   384
  qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   385
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   386
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   387
lemma wf_comp_self: "wf R = wf (R O R)"  -- {* special case *}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   388
  by (rule wf_union_merge [where S = "{}", simplified])
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   389
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   390
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   391
subsubsection {* acyclic *}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   392
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   393
lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   394
  by (simp add: acyclic_def)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   395
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   396
lemma wf_acyclic: "wf r ==> acyclic r"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   397
apply (simp add: acyclic_def)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   398
apply (blast elim: wf_trancl [THEN wf_irrefl])
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   399
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   400
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   401
lemmas wfP_acyclicP = wf_acyclic [to_pred]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   402
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   403
lemma acyclic_insert [iff]:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   404
     "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   405
apply (simp add: acyclic_def trancl_insert)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   406
apply (blast intro: rtrancl_trans)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   407
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   408
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   409
lemma acyclic_converse [iff]: "acyclic(r^-1) = acyclic r"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   410
by (simp add: acyclic_def trancl_converse)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   411
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   412
lemmas acyclicP_converse [iff] = acyclic_converse [to_pred]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   413
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   414
lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   415
apply (simp add: acyclic_def antisym_def)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   416
apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   417
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   418
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   419
(* Other direction:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   420
acyclic = no loops
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   421
antisym = only self loops
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   422
Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r - Id)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   423
==> antisym( r^* ) = acyclic(r - Id)";
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   424
*)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   425
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   426
lemma acyclic_subset: "[| acyclic s; r <= s |] ==> acyclic r"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   427
apply (simp add: acyclic_def)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   428
apply (blast intro: trancl_mono)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   429
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   430
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   431
text{* Wellfoundedness of finite acyclic relations*}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   432
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   433
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
   434
apply (erule finite_induct, blast)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   435
apply (simp (no_asm_simp) only: split_tupled_all)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   436
apply simp
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   437
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   438
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   439
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
   440
apply (erule finite_converse [THEN iffD2, THEN finite_acyclic_wf])
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   441
apply (erule acyclic_converse [THEN iffD2])
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   442
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   443
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   444
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
   445
by (blast intro: finite_acyclic_wf wf_acyclic)
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
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   448
subsection {* @{typ nat} is well-founded *}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   449
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   450
lemma less_nat_rel: "op < = (\<lambda>m n. n = Suc m)^++"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   451
proof (rule ext, rule ext, rule iffI)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   452
  fix n m :: nat
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   453
  assume "m < n"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   454
  then show "(\<lambda>m n. n = Suc m)^++ m n"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   455
  proof (induct n)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   456
    case 0 then show ?case by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   457
  next
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   458
    case (Suc n) then show ?case
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   459
      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
   460
  qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   461
next
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   462
  fix n m :: nat
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   463
  assume "(\<lambda>m n. n = Suc m)^++ m n"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   464
  then show "m < n"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   465
    by (induct n)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   466
      (simp_all add: less_Suc_eq_le reflexive le_less)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   467
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   468
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   469
definition
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   470
  pred_nat :: "(nat * nat) set" where
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   471
  "pred_nat = {(m, n). n = Suc m}"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   472
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   473
definition
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   474
  less_than :: "(nat * nat) set" where
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   475
  "less_than = pred_nat^+"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   476
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   477
lemma less_eq: "(m, n) \<in> pred_nat^+ \<longleftrightarrow> m < n"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   478
  unfolding less_nat_rel pred_nat_def trancl_def by simp
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   479
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   480
lemma pred_nat_trancl_eq_le:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   481
  "(m, n) \<in> pred_nat^* \<longleftrightarrow> m \<le> n"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   482
  unfolding less_eq rtrancl_eq_or_trancl by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   483
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   484
lemma wf_pred_nat: "wf pred_nat"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   485
  apply (unfold wf_def pred_nat_def, clarify)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   486
  apply (induct_tac x, blast+)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   487
  done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   488
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   489
lemma wf_less_than [iff]: "wf less_than"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   490
  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
   491
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   492
lemma trans_less_than [iff]: "trans less_than"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   493
  by (simp add: less_than_def trans_trancl)
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_than_iff [iff]: "((x,y): less_than) = (x<y)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   496
  by (simp add: less_than_def less_eq)
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 wf_less: "wf {(x, y::nat). x < y}"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   499
  using wf_less_than by (simp add: less_than_def less_eq [symmetric])
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   500
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
subsection {* Accessible Part *}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   503
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   504
text {*
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   505
 Inductive definition of the accessible part @{term "acc r"} of a
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   506
 relation; see also \cite{paulin-tlca}.
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   507
*}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   508
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   509
inductive_set
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   510
  acc :: "('a * 'a) set => 'a set"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   511
  for r :: "('a * 'a) set"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   512
  where
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   513
    accI: "(!!y. (y, x) : r ==> y : acc r) ==> x : acc r"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   514
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   515
abbreviation
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   516
  termip :: "('a => 'a => bool) => 'a => bool" where
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   517
  "termip r == accp (r\<inverse>\<inverse>)"
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
abbreviation
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   520
  termi :: "('a * 'a) set => 'a set" where
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   521
  "termi r == acc (r\<inverse>)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   522
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   523
lemmas accpI = accp.accI
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   524
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   525
text {* Induction rules *}
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
theorem accp_induct:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   528
  assumes major: "accp r a"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   529
  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
   530
  shows "P a"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   531
  apply (rule major [THEN accp.induct])
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   532
  apply (rule hyp)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   533
   apply (rule accp.accI)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   534
   apply fast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   535
  apply fast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   536
  done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   537
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   538
theorems accp_induct_rule = accp_induct [rule_format, induct set: accp]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   539
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   540
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
   541
  apply (erule accp.cases)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   542
  apply fast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   543
  done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   544
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   545
lemma not_accp_down:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   546
  assumes na: "\<not> accp R x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   547
  obtains z where "R z x" and "\<not> accp R z"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   548
proof -
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   549
  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
   550
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   551
  show thesis
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   552
  proof (cases "\<forall>z. R z x \<longrightarrow> accp R z")
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   553
    case True
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   554
    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
   555
    hence "accp R x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   556
      by (rule accp.accI)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   557
    with na show thesis ..
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   558
  next
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   559
    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
   560
      by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   561
    with a show thesis .
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   562
  qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   563
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   564
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   565
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
   566
  apply (erule rtranclp_induct)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   567
   apply blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   568
  apply (blast dest: accp_downward)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   569
  done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   570
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   571
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
   572
  apply (blast dest: accp_downwards_aux)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   573
  done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   574
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   575
theorem accp_wfPI: "\<forall>x. accp r x ==> wfP r"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   576
  apply (rule wfPUNIVI)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   577
  apply (induct_tac P x rule: accp_induct)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   578
   apply blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   579
  apply blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   580
  done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   581
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   582
theorem accp_wfPD: "wfP r ==> accp r x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   583
  apply (erule wfP_induct_rule)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   584
  apply (rule accp.accI)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   585
  apply blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   586
  done
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
theorem wfP_accp_iff: "wfP r = (\<forall>x. accp r x)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   589
  apply (blast intro: accp_wfPI dest: accp_wfPD)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   590
  done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   591
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   592
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   593
text {* Smaller relations have bigger accessible parts: *}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   594
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   595
lemma accp_subset:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   596
  assumes sub: "R1 \<le> R2"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   597
  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
   598
proof (rule predicate1I)
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   599
  fix x assume "accp R2 x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   600
  then show "accp R1 x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   601
  proof (induct x)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   602
    fix x
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   603
    assume ih: "\<And>y. R2 y x \<Longrightarrow> accp R1 y"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   604
    with sub show "accp R1 x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   605
      by (blast intro: accp.accI)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   606
  qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   607
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   608
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   609
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   610
text {* This is a generalized induction theorem that works on
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   611
  subsets of the accessible part. *}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   612
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   613
lemma accp_subset_induct:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   614
  assumes subset: "D \<le> accp R"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   615
    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
   616
    and "D x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   617
    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
   618
  shows "P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   619
proof -
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   620
  from subset and `D x`
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   621
  have "accp R x" ..
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   622
  then show "P x" using `D x`
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   623
  proof (induct x)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   624
    fix x
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   625
    assume "D x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   626
      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
   627
    with dcl and istep show "P x" by blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   628
  qed
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
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
text {* Set versions of the above theorems *}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   633
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   634
lemmas acc_induct = accp_induct [to_set]
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
lemmas acc_induct_rule = acc_induct [rule_format, induct set: acc]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   637
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   638
lemmas acc_downward = accp_downward [to_set]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   639
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   640
lemmas not_acc_down = not_accp_down [to_set]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   641
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   642
lemmas acc_downwards_aux = accp_downwards_aux [to_set]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   643
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   644
lemmas acc_downwards = accp_downwards [to_set]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   645
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   646
lemmas acc_wfI = accp_wfPI [to_set]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   647
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   648
lemmas acc_wfD = accp_wfPD [to_set]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   649
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   650
lemmas wf_acc_iff = wfP_accp_iff [to_set]
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   651
26803
0af0f674845d - Explicitely passed pred_subset_eq and pred_equals_eq as an argument to the
berghofe
parents: 26748
diff changeset
   652
lemmas acc_subset = accp_subset [to_set pred_subset_eq]
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   653
26803
0af0f674845d - Explicitely passed pred_subset_eq and pred_equals_eq as an argument to the
berghofe
parents: 26748
diff changeset
   654
lemmas acc_subset_induct = accp_subset_induct [to_set pred_subset_eq]
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   655
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
subsection {* Tools for building wellfounded relations *}
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
text {* Inverse Image *}
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
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
   662
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
   663
apply clarify
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   664
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
   665
prefer 2 apply (blast del: allE)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   666
apply (erule allE)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   667
apply (erule (1) notE impE)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   668
apply blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   669
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   670
31775
2b04504fcb69 uniformly capitialized names for subdirectories
haftmann
parents: 30989
diff changeset
   671
text {* Measure Datatypes into @{typ nat} *}
26748
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
definition measure :: "('a => nat) => ('a * 'a)set"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   674
where "measure == inv_image less_than"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   675
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   676
lemma in_measure[simp]: "((x,y) : measure f) = (f x < f y)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   677
  by (simp add:measure_def)
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
lemma wf_measure [iff]: "wf (measure f)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   680
apply (unfold measure_def)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   681
apply (rule wf_less_than [THEN wf_inv_image])
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   682
done
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
text{* Lexicographic combinations *}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   685
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   686
definition
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   687
 lex_prod  :: "[('a*'a)set, ('b*'b)set] => (('a*'b)*('a*'b))set"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   688
               (infixr "<*lex*>" 80)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   689
where
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   690
    "ra <*lex*> rb == {((a,b),(a',b')). (a,a') : ra | a=a' & (b,b') : rb}"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   691
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   692
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
   693
apply (unfold wf_def lex_prod_def) 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   694
apply (rule allI, rule impI)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   695
apply (simp (no_asm_use) only: split_paired_All)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   696
apply (drule spec, erule mp) 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   697
apply (rule allI, rule impI)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   698
apply (drule spec, erule mp, blast) 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   699
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   700
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   701
lemma in_lex_prod[simp]: 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   702
  "(((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
   703
  by (auto simp:lex_prod_def)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   704
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   705
text{* @{term "op <*lex*>"} preserves transitivity *}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   706
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   707
lemma trans_lex_prod [intro!]: 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   708
    "[| trans R1; trans R2 |] ==> trans (R1 <*lex*> R2)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   709
by (unfold trans_def lex_prod_def, blast) 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   710
31775
2b04504fcb69 uniformly capitialized names for subdirectories
haftmann
parents: 30989
diff changeset
   711
text {* lexicographic combinations with measure Datatypes *}
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   712
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   713
definition 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   714
  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
   715
where
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   716
  "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
   717
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   718
lemma wf_mlex: "wf R \<Longrightarrow> wf (f <*mlex*> R)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   719
unfolding mlex_prod_def
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   720
by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   721
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   722
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
   723
unfolding mlex_prod_def by simp
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   724
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   725
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
   726
unfolding mlex_prod_def by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   727
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   728
text {* proper subset relation on finite sets *}
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
definition finite_psubset  :: "('a set * 'a set) set"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   731
where "finite_psubset == {(A,B). A < B & finite B}"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   732
28260
703046c93ffe wf_finite_psubset[simp], in_finite_psubset[simp]
krauss
parents: 27823
diff changeset
   733
lemma wf_finite_psubset[simp]: "wf(finite_psubset)"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   734
apply (unfold finite_psubset_def)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   735
apply (rule wf_measure [THEN wf_subset])
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   736
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
   737
apply (fast elim!: psubset_card_mono)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   738
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   739
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   740
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
   741
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
   742
28260
703046c93ffe wf_finite_psubset[simp], in_finite_psubset[simp]
krauss
parents: 27823
diff changeset
   743
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
   744
unfolding finite_psubset_def by auto
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   745
28735
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   746
text {* max- and min-extension of order to finite sets *}
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   747
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   748
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
   749
for R :: "('a \<times> 'a) set"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   750
where
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   751
  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
   752
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   753
lemma max_ext_wf:
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   754
  assumes wf: "wf r"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   755
  shows "wf (max_ext r)"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   756
proof (rule acc_wfI, intro allI)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   757
  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
   758
  proof cases
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   759
    assume "finite M"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   760
    thus ?thesis
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   761
    proof (induct M)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   762
      show "{} \<in> ?W"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   763
        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
   764
    next
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   765
      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
   766
      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
   767
      proof (induct arbitrary: M)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   768
        fix M a
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   769
        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
   770
        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
   771
        {
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   772
          fix N M :: "'a set"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   773
          assume "finite N" "finite M"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   774
          then
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   775
          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
   776
            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
   777
        }
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   778
        note add_less = this
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   779
        
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   780
        show "insert a M \<in> ?W"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   781
        proof (rule accI)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   782
          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
   783
          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
   784
            by (auto elim!: max_ext.cases)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   785
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   786
          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
   787
          let ?N2 = "{ n \<in> N. (n, a) \<notin> r }"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   788
          have N: "?N1 \<union> ?N2 = N" by (rule set_ext) auto
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   789
          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
   790
          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
   791
          
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   792
          have "?N2 \<in> ?W"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   793
          proof cases
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   794
            assume [simp]: "M = {}"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   795
            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
   796
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   797
            from asm1 have "?N2 = {}" by auto
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   798
            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
   799
          next
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   800
            assume "M \<noteq> {}"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   801
            have N2: "(?N2, M) \<in> max_ext r" 
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   802
              by (rule max_extI[OF _ _ `M \<noteq> {}`]) (insert asm1, auto intro: finites)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   803
            
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   804
            with `M \<in> ?W` show "?N2 \<in> ?W" by (rule acc_downward)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   805
          qed
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   806
          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
   807
            by (rule add_less) simp
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   808
          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
   809
        qed
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   810
      qed
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   811
    qed
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   812
  next
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   813
    assume [simp]: "\<not> finite M"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   814
    show ?thesis
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   815
      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
   816
  qed
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   817
qed
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   818
29125
d41182a8135c method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents: 28845
diff changeset
   819
lemma max_ext_additive: 
d41182a8135c method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents: 28845
diff changeset
   820
 "(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
   821
  (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
   822
by (force elim!: max_ext.cases)
d41182a8135c method "sizechange" proves termination of functions; added more infrastructure for termination proofs
krauss
parents: 28845
diff changeset
   823
28735
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   824
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   825
definition
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   826
  min_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
   827
where
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   828
  [code del]: "min_ext r = {(X, Y) | X Y. X \<noteq> {} \<and> (\<forall>y \<in> Y. (\<exists>x \<in> X. (x, y) \<in> r))}"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   829
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   830
lemma min_ext_wf:
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   831
  assumes "wf r"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   832
  shows "wf (min_ext r)"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   833
proof (rule wfI_min)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   834
  fix Q :: "'a set set"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   835
  fix x
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   836
  assume nonempty: "x \<in> Q"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   837
  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
   838
  proof cases
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   839
    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
   840
  next
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   841
    assume "Q \<noteq> {{}}"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   842
    with nonempty
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   843
    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
   844
    then have eU: "e \<in> \<Union>Q" by auto
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   845
    with `wf r` 
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   846
    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
   847
      by (erule wfE_min)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   848
    from z obtain m where "m \<in> Q" "z \<in> m" by auto
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   849
    from `m \<in> Q`
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   850
    show ?thesis
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   851
    proof (rule, intro bexI allI impI)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   852
      fix n
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   853
      assume smaller: "(n, m) \<in> min_ext r"
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   854
      with `z \<in> m` obtain y where y: "y \<in> n" "(y, z) \<in> r" by (auto simp: min_ext_def)
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   855
      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
   856
    qed      
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   857
  qed
bed31381e6b6 min_ext/max_ext lifting wellfounded relations on finite sets. Preserves wf
krauss
parents: 28562
diff changeset
   858
qed
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   859
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   860
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   861
subsection{*Weakly decreasing sequences (w.r.t. some well-founded order) 
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   862
   stabilize.*}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   863
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   864
text{*This material does not appear to be used any longer.*}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   865
28845
cdfc8ef54a99 removed lemmas called lemma1 and lemma2
krauss
parents: 28735
diff changeset
   866
lemma sequence_trans: "[| ALL i. (f (Suc i), f i) : r^* |] ==> (f (i+k), f i) : r^*"
cdfc8ef54a99 removed lemmas called lemma1 and lemma2
krauss
parents: 28735
diff changeset
   867
by (induct k) (auto intro: rtrancl_trans)
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   868
28845
cdfc8ef54a99 removed lemmas called lemma1 and lemma2
krauss
parents: 28735
diff changeset
   869
lemma wf_weak_decr_stable: 
cdfc8ef54a99 removed lemmas called lemma1 and lemma2
krauss
parents: 28735
diff changeset
   870
  assumes as: "ALL i. (f (Suc i), f i) : r^*" "wf (r^+)"
cdfc8ef54a99 removed lemmas called lemma1 and lemma2
krauss
parents: 28735
diff changeset
   871
  shows "EX i. ALL k. f (i+k) = f i"
cdfc8ef54a99 removed lemmas called lemma1 and lemma2
krauss
parents: 28735
diff changeset
   872
proof -
cdfc8ef54a99 removed lemmas called lemma1 and lemma2
krauss
parents: 28735
diff changeset
   873
  have lem: "!!x. [| ALL i. (f (Suc i), f i) : r^*; wf (r^+) |]  
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   874
      ==> ALL m. f m = x --> (EX i. ALL k. f (m+i+k) = f (m+i))"
28845
cdfc8ef54a99 removed lemmas called lemma1 and lemma2
krauss
parents: 28735
diff changeset
   875
  apply (erule wf_induct, clarify)
cdfc8ef54a99 removed lemmas called lemma1 and lemma2
krauss
parents: 28735
diff changeset
   876
  apply (case_tac "EX j. (f (m+j), f m) : r^+")
cdfc8ef54a99 removed lemmas called lemma1 and lemma2
krauss
parents: 28735
diff changeset
   877
   apply clarify
cdfc8ef54a99 removed lemmas called lemma1 and lemma2
krauss
parents: 28735
diff changeset
   878
   apply (subgoal_tac "EX i. ALL k. f ((m+j) +i+k) = f ( (m+j) +i) ")
cdfc8ef54a99 removed lemmas called lemma1 and lemma2
krauss
parents: 28735
diff changeset
   879
    apply clarify
cdfc8ef54a99 removed lemmas called lemma1 and lemma2
krauss
parents: 28735
diff changeset
   880
    apply (rule_tac x = "j+i" in exI)
cdfc8ef54a99 removed lemmas called lemma1 and lemma2
krauss
parents: 28735
diff changeset
   881
    apply (simp add: add_ac, blast)
cdfc8ef54a99 removed lemmas called lemma1 and lemma2
krauss
parents: 28735
diff changeset
   882
  apply (rule_tac x = 0 in exI, clarsimp)
cdfc8ef54a99 removed lemmas called lemma1 and lemma2
krauss
parents: 28735
diff changeset
   883
  apply (drule_tac i = m and k = k in sequence_trans)
cdfc8ef54a99 removed lemmas called lemma1 and lemma2
krauss
parents: 28735
diff changeset
   884
  apply (blast elim: rtranclE dest: rtrancl_into_trancl1)
cdfc8ef54a99 removed lemmas called lemma1 and lemma2
krauss
parents: 28735
diff changeset
   885
  done
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   886
28845
cdfc8ef54a99 removed lemmas called lemma1 and lemma2
krauss
parents: 28735
diff changeset
   887
  from lem[OF as, THEN spec, of 0, simplified] 
cdfc8ef54a99 removed lemmas called lemma1 and lemma2
krauss
parents: 28735
diff changeset
   888
  show ?thesis by auto
cdfc8ef54a99 removed lemmas called lemma1 and lemma2
krauss
parents: 28735
diff changeset
   889
qed
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   890
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   891
(* special case of the theorem above: <= *)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   892
lemma weak_decr_stable:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   893
     "ALL i. f (Suc i) <= ((f i)::nat) ==> EX i. ALL k. f (i+k) = f i"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   894
apply (rule_tac r = pred_nat in wf_weak_decr_stable)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   895
apply (simp add: pred_nat_trancl_eq_le)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   896
apply (intro wf_trancl wf_pred_nat)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   897
done
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   898
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   899
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   900
subsection {* size of a datatype value *}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   901
31775
2b04504fcb69 uniformly capitialized names for subdirectories
haftmann
parents: 30989
diff changeset
   902
use "Tools/Function/size.ML"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   903
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   904
setup Size.setup
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   905
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28524
diff changeset
   906
lemma size_bool [code]:
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 26976
diff changeset
   907
  "size (b\<Colon>bool) = 0" by (cases b) auto
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 26976
diff changeset
   908
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28524
diff changeset
   909
lemma nat_size [simp, code]: "size (n\<Colon>nat) = n"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   910
  by (induct n) simp_all
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   911
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 26976
diff changeset
   912
declare "prod.size" [noatp]
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   913
30430
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 29609
diff changeset
   914
lemma [code]:
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 29609
diff changeset
   915
  "size (P :: 'a Predicate.pred) = 0" by (cases P) simp
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 29609
diff changeset
   916
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 29609
diff changeset
   917
lemma [code]:
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 29609
diff changeset
   918
  "pred_size f P = 0" by (cases P) simp
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 29609
diff changeset
   919
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
diff changeset
   920
end