# HG changeset patch
# User blanchet
# Date 1384985646 -3600
# Node ID 483131676087620895373bcdd5b465c8747effb2
# Parent  7d23f8e501d4bcdab690a7f949023700a16d0c49
effectively reverted d25fc4c0ff62, to avoid quasi-cyclic dependencies with HOL-Cardinals and minimize BNF dependencies

diff -r 7d23f8e501d4 -r 483131676087 src/HOL/Cardinals/Cardinals.thy
--- a/src/HOL/Cardinals/Cardinals.thy	Wed Nov 20 21:28:58 2013 +0100
+++ b/src/HOL/Cardinals/Cardinals.thy	Wed Nov 20 23:14:06 2013 +0100
@@ -9,7 +9,7 @@
 header {* Theory of Ordinals and Cardinals  *}
 
 theory Cardinals
-imports Cardinal_Order_Relation Cardinal_Arithmetic
+imports Cardinal_Order_Relation Cardinal_Arithmetic Wellorder_Extension
 begin
 
 end
diff -r 7d23f8e501d4 -r 483131676087 src/HOL/Cardinals/Constructions_on_Wellorders.thy
--- a/src/HOL/Cardinals/Constructions_on_Wellorders.thy	Wed Nov 20 21:28:58 2013 +0100
+++ b/src/HOL/Cardinals/Constructions_on_Wellorders.thy	Wed Nov 20 23:14:06 2013 +0100
@@ -8,7 +8,7 @@
 header {* Constructions on Wellorders *}
 
 theory Constructions_on_Wellorders
-imports Constructions_on_Wellorders_FP Wellorder_Embedding
+imports Constructions_on_Wellorders_FP Wellorder_Embedding Order_Union
 begin
 
 declare
diff -r 7d23f8e501d4 -r 483131676087 src/HOL/Cardinals/Order_Union.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Cardinals/Order_Union.thy	Wed Nov 20 23:14:06 2013 +0100
@@ -0,0 +1,370 @@
+(*  Title:      HOL/Cardinals/Order_Union.thy
+    Author:     Andrei Popescu, TU Muenchen
+
+The ordinal-like sum of two orders with disjoint fields
+*)
+
+header {* Order Union *}
+
+theory Order_Union
+imports Wellfounded_More_FP
+begin
+
+definition Osum :: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a rel"  (infix "Osum" 60) where
+  "r Osum r' = r \<union> r' \<union> {(a, a'). a \<in> Field r \<and> a' \<in> Field r'}"
+
+notation Osum  (infix "\<union>o" 60)
+
+lemma Field_Osum: "Field (r \<union>o r') = Field r \<union> Field r'"
+  unfolding Osum_def Field_def by blast
+
+lemma Osum_wf:
+assumes FLD: "Field r Int Field r' = {}" and
+        WF: "wf r" and WF': "wf r'"
+shows "wf (r Osum r')"
+unfolding wf_eq_minimal2 unfolding Field_Osum
+proof(intro allI impI, elim conjE)
+  fix A assume *: "A \<subseteq> Field r \<union> Field r'" and **: "A \<noteq> {}"
+  obtain B where B_def: "B = A Int Field r" by blast
+  show "\<exists>a\<in>A. \<forall>a'\<in>A. (a', a) \<notin> r \<union>o r'"
+  proof(cases "B = {}")
+    assume Case1: "B \<noteq> {}"
+    hence "B \<noteq> {} \<and> B \<le> Field r" using B_def by auto
+    then obtain a where 1: "a \<in> B" and 2: "\<forall>a1 \<in> B. (a1,a) \<notin> r"
+    using WF unfolding wf_eq_minimal2 by metis
+    hence 3: "a \<in> Field r \<and> a \<notin> Field r'" using B_def FLD by auto
+    (*  *)
+    have "\<forall>a1 \<in> A. (a1,a) \<notin> r Osum r'"
+    proof(intro ballI)
+      fix a1 assume **: "a1 \<in> A"
+      {assume Case11: "a1 \<in> Field r"
+       hence "(a1,a) \<notin> r" using B_def ** 2 by auto
+       moreover
+       have "(a1,a) \<notin> r'" using 3 by (auto simp add: Field_def)
+       ultimately have "(a1,a) \<notin> r Osum r'"
+       using 3 unfolding Osum_def by auto
+      }
+      moreover
+      {assume Case12: "a1 \<notin> Field r"
+       hence "(a1,a) \<notin> r" unfolding Field_def by auto
+       moreover
+       have "(a1,a) \<notin> r'" using 3 unfolding Field_def by auto
+       ultimately have "(a1,a) \<notin> r Osum r'"
+       using 3 unfolding Osum_def by auto
+      }
+      ultimately show "(a1,a) \<notin> r Osum r'" by blast
+    qed
+    thus ?thesis using 1 B_def by auto
+  next
+    assume Case2: "B = {}"
+    hence 1: "A \<noteq> {} \<and> A \<le> Field r'" using * ** B_def by auto
+    then obtain a' where 2: "a' \<in> A" and 3: "\<forall>a1' \<in> A. (a1',a') \<notin> r'"
+    using WF' unfolding wf_eq_minimal2 by metis
+    hence 4: "a' \<in> Field r' \<and> a' \<notin> Field r" using 1 FLD by blast
+    (*  *)
+    have "\<forall>a1' \<in> A. (a1',a') \<notin> r Osum r'"
+    proof(unfold Osum_def, auto simp add: 3)
+      fix a1' assume "(a1', a') \<in> r"
+      thus False using 4 unfolding Field_def by blast
+    next
+      fix a1' assume "a1' \<in> A" and "a1' \<in> Field r"
+      thus False using Case2 B_def by auto
+    qed
+    thus ?thesis using 2 by blast
+  qed
+qed
+
+lemma Osum_Refl:
+assumes FLD: "Field r Int Field r' = {}" and
+        REFL: "Refl r" and REFL': "Refl r'"
+shows "Refl (r Osum r')"
+using assms 
+unfolding refl_on_def Field_Osum unfolding Osum_def by blast
+
+lemma Osum_trans:
+assumes FLD: "Field r Int Field r' = {}" and
+        TRANS: "trans r" and TRANS': "trans r'"
+shows "trans (r Osum r')"
+proof(unfold trans_def, auto)
+  fix x y z assume *: "(x, y) \<in> r \<union>o r'" and **: "(y, z) \<in> r \<union>o r'"
+  show  "(x, z) \<in> r \<union>o r'"
+  proof-
+    {assume Case1: "(x,y) \<in> r"
+     hence 1: "x \<in> Field r \<and> y \<in> Field r" unfolding Field_def by auto
+     have ?thesis
+     proof-
+       {assume Case11: "(y,z) \<in> r"
+        hence "(x,z) \<in> r" using Case1 TRANS trans_def[of r] by blast
+        hence ?thesis unfolding Osum_def by auto
+       }
+       moreover
+       {assume Case12: "(y,z) \<in> r'"
+        hence "y \<in> Field r'" unfolding Field_def by auto
+        hence False using FLD 1 by auto
+       }
+       moreover
+       {assume Case13: "z \<in> Field r'"
+        hence ?thesis using 1 unfolding Osum_def by auto
+       }
+       ultimately show ?thesis using ** unfolding Osum_def by blast
+     qed
+    }
+    moreover
+    {assume Case2: "(x,y) \<in> r'"
+     hence 2: "x \<in> Field r' \<and> y \<in> Field r'" unfolding Field_def by auto
+     have ?thesis
+     proof-
+       {assume Case21: "(y,z) \<in> r"
+        hence "y \<in> Field r" unfolding Field_def by auto
+        hence False using FLD 2 by auto
+       }
+       moreover
+       {assume Case22: "(y,z) \<in> r'"
+        hence "(x,z) \<in> r'" using Case2 TRANS' trans_def[of r'] by blast
+        hence ?thesis unfolding Osum_def by auto
+       }
+       moreover
+       {assume Case23: "y \<in> Field r"
+        hence False using FLD 2 by auto
+       }
+       ultimately show ?thesis using ** unfolding Osum_def by blast
+     qed
+    }
+    moreover
+    {assume Case3: "x \<in> Field r \<and> y \<in> Field r'"
+     have ?thesis
+     proof-
+       {assume Case31: "(y,z) \<in> r"
+        hence "y \<in> Field r" unfolding Field_def by auto
+        hence False using FLD Case3 by auto
+       }
+       moreover
+       {assume Case32: "(y,z) \<in> r'"
+        hence "z \<in> Field r'" unfolding Field_def by blast
+        hence ?thesis unfolding Osum_def using Case3 by auto
+       }
+       moreover
+       {assume Case33: "y \<in> Field r"
+        hence False using FLD Case3 by auto
+       }
+       ultimately show ?thesis using ** unfolding Osum_def by blast
+     qed
+    }
+    ultimately show ?thesis using * unfolding Osum_def by blast
+  qed
+qed
+
+lemma Osum_Preorder:
+"\<lbrakk>Field r Int Field r' = {}; Preorder r; Preorder r'\<rbrakk> \<Longrightarrow> Preorder (r Osum r')"
+unfolding preorder_on_def using Osum_Refl Osum_trans by blast
+
+lemma Osum_antisym:
+assumes FLD: "Field r Int Field r' = {}" and
+        AN: "antisym r" and AN': "antisym r'"
+shows "antisym (r Osum r')"
+proof(unfold antisym_def, auto)
+  fix x y assume *: "(x, y) \<in> r \<union>o r'" and **: "(y, x) \<in> r \<union>o r'"
+  show  "x = y"
+  proof-
+    {assume Case1: "(x,y) \<in> r"
+     hence 1: "x \<in> Field r \<and> y \<in> Field r" unfolding Field_def by auto
+     have ?thesis
+     proof-
+       have "(y,x) \<in> r \<Longrightarrow> ?thesis"
+       using Case1 AN antisym_def[of r] by blast
+       moreover
+       {assume "(y,x) \<in> r'"
+        hence "y \<in> Field r'" unfolding Field_def by auto
+        hence False using FLD 1 by auto
+       }
+       moreover
+       have "x \<in> Field r' \<Longrightarrow> False" using FLD 1 by auto
+       ultimately show ?thesis using ** unfolding Osum_def by blast
+     qed
+    }
+    moreover
+    {assume Case2: "(x,y) \<in> r'"
+     hence 2: "x \<in> Field r' \<and> y \<in> Field r'" unfolding Field_def by auto
+     have ?thesis
+     proof-
+       {assume "(y,x) \<in> r"
+        hence "y \<in> Field r" unfolding Field_def by auto
+        hence False using FLD 2 by auto
+       }
+       moreover
+       have "(y,x) \<in> r' \<Longrightarrow> ?thesis"
+       using Case2 AN' antisym_def[of r'] by blast
+       moreover
+       {assume "y \<in> Field r"
+        hence False using FLD 2 by auto
+       }
+       ultimately show ?thesis using ** unfolding Osum_def by blast
+     qed
+    }
+    moreover
+    {assume Case3: "x \<in> Field r \<and> y \<in> Field r'"
+     have ?thesis
+     proof-
+       {assume "(y,x) \<in> r"
+        hence "y \<in> Field r" unfolding Field_def by auto
+        hence False using FLD Case3 by auto
+       }
+       moreover
+       {assume Case32: "(y,x) \<in> r'"
+        hence "x \<in> Field r'" unfolding Field_def by blast
+        hence False using FLD Case3 by auto
+       }
+       moreover
+       have "\<not> y \<in> Field r" using FLD Case3 by auto
+       ultimately show ?thesis using ** unfolding Osum_def by blast
+     qed
+    }
+    ultimately show ?thesis using * unfolding Osum_def by blast
+  qed
+qed
+
+lemma Osum_Partial_order:
+"\<lbrakk>Field r Int Field r' = {}; Partial_order r; Partial_order r'\<rbrakk> \<Longrightarrow>
+ Partial_order (r Osum r')"
+unfolding partial_order_on_def using Osum_Preorder Osum_antisym by blast
+
+lemma Osum_Total:
+assumes FLD: "Field r Int Field r' = {}" and
+        TOT: "Total r" and TOT': "Total r'"
+shows "Total (r Osum r')"
+using assms
+unfolding total_on_def  Field_Osum unfolding Osum_def by blast
+
+lemma Osum_Linear_order:
+"\<lbrakk>Field r Int Field r' = {}; Linear_order r; Linear_order r'\<rbrakk> \<Longrightarrow>
+ Linear_order (r Osum r')"
+unfolding linear_order_on_def using Osum_Partial_order Osum_Total by blast
+
+lemma Osum_minus_Id1:
+assumes "r \<le> Id"
+shows "(r Osum r') - Id \<le> (r' - Id) \<union> (Field r \<times> Field r')"
+proof-
+  let ?Left = "(r Osum r') - Id"
+  let ?Right = "(r' - Id) \<union> (Field r \<times> Field r')"
+  {fix a::'a and b assume *: "(a,b) \<notin> Id"
+   {assume "(a,b) \<in> r"
+    with * have False using assms by auto
+   }
+   moreover
+   {assume "(a,b) \<in> r'"
+    with * have "(a,b) \<in> r' - Id" by auto
+   }
+   ultimately
+   have "(a,b) \<in> ?Left \<Longrightarrow> (a,b) \<in> ?Right"
+   unfolding Osum_def by auto
+  }
+  thus ?thesis by auto
+qed
+
+lemma Osum_minus_Id2:
+assumes "r' \<le> Id"
+shows "(r Osum r') - Id \<le> (r - Id) \<union> (Field r \<times> Field r')"
+proof-
+  let ?Left = "(r Osum r') - Id"
+  let ?Right = "(r - Id) \<union> (Field r \<times> Field r')"
+  {fix a::'a and b assume *: "(a,b) \<notin> Id"
+   {assume "(a,b) \<in> r'"
+    with * have False using assms by auto
+   }
+   moreover
+   {assume "(a,b) \<in> r"
+    with * have "(a,b) \<in> r - Id" by auto
+   }
+   ultimately
+   have "(a,b) \<in> ?Left \<Longrightarrow> (a,b) \<in> ?Right"
+   unfolding Osum_def by auto
+  }
+  thus ?thesis by auto
+qed
+
+lemma Osum_minus_Id:
+assumes TOT: "Total r" and TOT': "Total r'" and
+        NID: "\<not> (r \<le> Id)" and NID': "\<not> (r' \<le> Id)"
+shows "(r Osum r') - Id \<le> (r - Id) Osum (r' - Id)"
+proof-
+  {fix a a' assume *: "(a,a') \<in> (r Osum r')" and **: "a \<noteq> a'"
+   have "(a,a') \<in> (r - Id) Osum (r' - Id)"
+   proof-
+     {assume "(a,a') \<in> r \<or> (a,a') \<in> r'"
+      with ** have ?thesis unfolding Osum_def by auto
+     }
+     moreover
+     {assume "a \<in> Field r \<and> a' \<in> Field r'"
+      hence "a \<in> Field(r - Id) \<and> a' \<in> Field (r' - Id)"
+      using assms Total_Id_Field by blast
+      hence ?thesis unfolding Osum_def by auto
+     }
+     ultimately show ?thesis using * unfolding Osum_def by fast
+   qed
+  }
+  thus ?thesis by(auto simp add: Osum_def)
+qed
+
+lemma wf_Int_Times:
+assumes "A Int B = {}"
+shows "wf(A \<times> B)"
+unfolding wf_def using assms by blast
+
+lemma Osum_wf_Id:
+assumes TOT: "Total r" and TOT': "Total r'" and
+        FLD: "Field r Int Field r' = {}" and
+        WF: "wf(r - Id)" and WF': "wf(r' - Id)"
+shows "wf ((r Osum r') - Id)"
+proof(cases "r \<le> Id \<or> r' \<le> Id")
+  assume Case1: "\<not>(r \<le> Id \<or> r' \<le> Id)"
+  have "Field(r - Id) Int Field(r' - Id) = {}"
+  using FLD mono_Field[of "r - Id" r]  mono_Field[of "r' - Id" r']
+            Diff_subset[of r Id] Diff_subset[of r' Id] by blast
+  thus ?thesis
+  using Case1 Osum_minus_Id[of r r'] assms Osum_wf[of "r - Id" "r' - Id"]
+        wf_subset[of "(r - Id) \<union>o (r' - Id)" "(r Osum r') - Id"] by auto
+next
+  have 1: "wf(Field r \<times> Field r')"
+  using FLD by (auto simp add: wf_Int_Times)
+  assume Case2: "r \<le> Id \<or> r' \<le> Id"
+  moreover
+  {assume Case21: "r \<le> Id"
+   hence "(r Osum r') - Id \<le> (r' - Id) \<union> (Field r \<times> Field r')"
+   using Osum_minus_Id1[of r r'] by simp
+   moreover
+   {have "Domain(Field r \<times> Field r') Int Range(r' - Id) = {}"
+    using FLD unfolding Field_def by blast
+    hence "wf((r' - Id) \<union> (Field r \<times> Field r'))"
+    using 1 WF' wf_Un[of "Field r \<times> Field r'" "r' - Id"]
+    by (auto simp add: Un_commute)
+   }
+   ultimately have ?thesis by (metis wf_subset)
+  }
+  moreover
+  {assume Case22: "r' \<le> Id"
+   hence "(r Osum r') - Id \<le> (r - Id) \<union> (Field r \<times> Field r')"
+   using Osum_minus_Id2[of r' r] by simp
+   moreover
+   {have "Range(Field r \<times> Field r') Int Domain(r - Id) = {}"
+    using FLD unfolding Field_def by blast
+    hence "wf((r - Id) \<union> (Field r \<times> Field r'))"
+    using 1 WF wf_Un[of "r - Id" "Field r \<times> Field r'"]
+    by (auto simp add: Un_commute)
+   }
+   ultimately have ?thesis by (metis wf_subset)
+  }
+  ultimately show ?thesis by blast
+qed
+
+lemma Osum_Well_order:
+assumes FLD: "Field r Int Field r' = {}" and
+        WELL: "Well_order r" and WELL': "Well_order r'"
+shows "Well_order (r Osum r')"
+proof-
+  have "Total r \<and> Total r'" using WELL WELL'
+  by (auto simp add: order_on_defs)
+  thus ?thesis using assms unfolding well_order_on_def
+  using Osum_Linear_order Osum_wf_Id by blast
+qed
+
+end
diff -r 7d23f8e501d4 -r 483131676087 src/HOL/Cardinals/Wellorder_Extension.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Cardinals/Wellorder_Extension.thy	Wed Nov 20 23:14:06 2013 +0100
@@ -0,0 +1,213 @@
+(*  Title:      HOL/Cardinals/Wellorder_Extension.thy
+    Author:     Christian Sternagel, JAIST
+*)
+
+header {* Extending Well-founded Relations to Wellorders *}
+
+theory Wellorder_Extension
+imports "~~/src/HOL/Library/Zorn" Order_Union
+begin
+
+subsection {* Extending Well-founded Relations to Wellorders *}
+
+text {*A \emph{downset} (also lower set, decreasing set, initial segment, or
+downward closed set) is closed w.r.t.\ smaller elements.*}
+definition downset_on where
+  "downset_on A r = (\<forall>x y. (x, y) \<in> r \<and> y \<in> A \<longrightarrow> x \<in> A)"
+
+(*
+text {*Connection to order filters of the @{theory Cardinals} theory.*}
+lemma (in wo_rel) ofilter_downset_on_conv:
+  "ofilter A \<longleftrightarrow> downset_on A r \<and> A \<subseteq> Field r"
+  by (auto simp: downset_on_def ofilter_def under_def)
+*)
+
+lemma downset_onI:
+  "(\<And>x y. (x, y) \<in> r \<Longrightarrow> y \<in> A \<Longrightarrow> x \<in> A) \<Longrightarrow> downset_on A r"
+  by (auto simp: downset_on_def)
+
+lemma downset_onD:
+  "downset_on A r \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> y \<in> A \<Longrightarrow> x \<in> A"
+  unfolding downset_on_def by blast
+
+text {*Extensions of relations w.r.t.\ a given set.*}
+definition extension_on where
+  "extension_on A r s = (\<forall>x\<in>A. \<forall>y\<in>A. (x, y) \<in> s \<longrightarrow> (x, y) \<in> r)"
+
+lemma extension_onI:
+  "(\<And>x y. \<lbrakk>x \<in> A; y \<in> A; (x, y) \<in> s\<rbrakk> \<Longrightarrow> (x, y) \<in> r) \<Longrightarrow> extension_on A r s"
+  by (auto simp: extension_on_def)
+
+lemma extension_onD:
+  "extension_on A r s \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> (x, y) \<in> s \<Longrightarrow> (x, y) \<in> r"
+  by (auto simp: extension_on_def)
+
+lemma downset_on_Union:
+  assumes "\<And>r. r \<in> R \<Longrightarrow> downset_on (Field r) p"
+  shows "downset_on (Field (\<Union>R)) p"
+  using assms by (auto intro: downset_onI dest: downset_onD)
+
+lemma chain_subset_extension_on_Union:
+  assumes "chain\<^sub>\<subseteq> R" and "\<And>r. r \<in> R \<Longrightarrow> extension_on (Field r) r p"
+  shows "extension_on (Field (\<Union>R)) (\<Union>R) p"
+  using assms
+  by (simp add: chain_subset_def extension_on_def)
+     (metis (no_types) mono_Field set_mp)
+
+lemma downset_on_empty [simp]: "downset_on {} p"
+  by (auto simp: downset_on_def)
+
+lemma extension_on_empty [simp]: "extension_on {} p q"
+  by (auto simp: extension_on_def)
+
+text {*Every well-founded relation can be extended to a wellorder.*}
+theorem well_order_extension:
+  assumes "wf p"
+  shows "\<exists>w. p \<subseteq> w \<and> Well_order w"
+proof -
+  let ?K = "{r. Well_order r \<and> downset_on (Field r) p \<and> extension_on (Field r) r p}"
+  def I \<equiv> "init_seg_of \<inter> ?K \<times> ?K"
+  have I_init: "I \<subseteq> init_seg_of" by (simp add: I_def)
+  then have subch: "\<And>R. R \<in> Chains I \<Longrightarrow> chain\<^sub>\<subseteq> R"
+    by (auto simp: init_seg_of_def chain_subset_def Chains_def)
+  have Chains_wo: "\<And>R r. R \<in> Chains I \<Longrightarrow> r \<in> R \<Longrightarrow>
+      Well_order r \<and> downset_on (Field r) p \<and> extension_on (Field r) r p"
+    by (simp add: Chains_def I_def) blast
+  have FI: "Field I = ?K" by (auto simp: I_def init_seg_of_def Field_def)
+  then have 0: "Partial_order I"
+    by (auto simp: partial_order_on_def preorder_on_def antisym_def antisym_init_seg_of refl_on_def
+      trans_def I_def elim: trans_init_seg_of)
+  { fix R assume "R \<in> Chains I"
+    then have Ris: "R \<in> Chains init_seg_of" using mono_Chains [OF I_init] by blast
+    have subch: "chain\<^sub>\<subseteq> R" using `R \<in> Chains I` I_init
+      by (auto simp: init_seg_of_def chain_subset_def Chains_def)
+    have "\<forall>r\<in>R. Refl r" and "\<forall>r\<in>R. trans r" and "\<forall>r\<in>R. antisym r" and
+      "\<forall>r\<in>R. Total r" and "\<forall>r\<in>R. wf (r - Id)" and
+      "\<And>r. r \<in> R \<Longrightarrow> downset_on (Field r) p" and
+      "\<And>r. r \<in> R \<Longrightarrow> extension_on (Field r) r p"
+      using Chains_wo [OF `R \<in> Chains I`] by (simp_all add: order_on_defs)
+    have "Refl (\<Union>R)" using `\<forall>r\<in>R. Refl r`  unfolding refl_on_def by fastforce
+    moreover have "trans (\<Union>R)"
+      by (rule chain_subset_trans_Union [OF subch `\<forall>r\<in>R. trans r`])
+    moreover have "antisym (\<Union>R)"
+      by (rule chain_subset_antisym_Union [OF subch `\<forall>r\<in>R. antisym r`])
+    moreover have "Total (\<Union>R)"
+      by (rule chain_subset_Total_Union [OF subch `\<forall>r\<in>R. Total r`])
+    moreover have "wf ((\<Union>R) - Id)"
+    proof -
+      have "(\<Union>R) - Id = \<Union>{r - Id | r. r \<in> R}" by blast
+      with `\<forall>r\<in>R. wf (r - Id)` wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]]
+      show ?thesis by fastforce
+    qed
+    ultimately have "Well_order (\<Union>R)" by (simp add: order_on_defs)
+    moreover have "\<forall>r\<in>R. r initial_segment_of \<Union>R" using Ris
+      by (simp add: Chains_init_seg_of_Union)
+    moreover have "downset_on (Field (\<Union>R)) p"
+      by (rule downset_on_Union [OF `\<And>r. r \<in> R \<Longrightarrow> downset_on (Field r) p`])
+    moreover have "extension_on (Field (\<Union>R)) (\<Union>R) p"
+      by (rule chain_subset_extension_on_Union [OF subch `\<And>r. r \<in> R \<Longrightarrow> extension_on (Field r) r p`])
+    ultimately have "\<Union>R \<in> ?K \<and> (\<forall>r\<in>R. (r,\<Union>R) \<in> I)"
+      using mono_Chains [OF I_init] and `R \<in> Chains I`
+      by (simp (no_asm) add: I_def del: Field_Union) (metis Chains_wo)
+  }
+  then have 1: "\<forall>R\<in>Chains I. \<exists>u\<in>Field I. \<forall>r\<in>R. (r, u) \<in> I" by (subst FI) blast
+  txt {*Zorn's Lemma yields a maximal wellorder m.*}
+  from Zorns_po_lemma [OF 0 1] obtain m :: "('a \<times> 'a) set"
+    where "Well_order m" and "downset_on (Field m) p" and "extension_on (Field m) m p" and
+    max: "\<forall>r. Well_order r \<and> downset_on (Field r) p \<and> extension_on (Field r) r p \<and>
+      (m, r) \<in> I \<longrightarrow> r = m"
+    by (auto simp: FI)
+  have "Field p \<subseteq> Field m"
+  proof (rule ccontr)
+    let ?Q = "Field p - Field m"
+    assume "\<not> (Field p \<subseteq> Field m)"
+    with assms [unfolded wf_eq_minimal, THEN spec, of ?Q]
+      obtain x where "x \<in> Field p" and "x \<notin> Field m" and
+      min: "\<forall>y. (y, x) \<in> p \<longrightarrow> y \<notin> ?Q" by blast
+    txt {*Add @{term x} as topmost element to @{term m}.*}
+    let ?s = "{(y, x) | y. y \<in> Field m}"
+    let ?m = "insert (x, x) m \<union> ?s"
+    have Fm: "Field ?m = insert x (Field m)" by (auto simp: Field_def)
+    have "Refl m" and "trans m" and "antisym m" and "Total m" and "wf (m - Id)"
+      using `Well_order m` by (simp_all add: order_on_defs)
+    txt {*We show that the extension is a wellorder.*}
+    have "Refl ?m" using `Refl m` Fm by (auto simp: refl_on_def)
+    moreover have "trans ?m" using `trans m` `x \<notin> Field m`
+      unfolding trans_def Field_def Domain_unfold Domain_converse [symmetric] by blast
+    moreover have "antisym ?m" using `antisym m` `x \<notin> Field m`
+      unfolding antisym_def Field_def Domain_unfold Domain_converse [symmetric] by blast
+    moreover have "Total ?m" using `Total m` Fm by (auto simp: Relation.total_on_def)
+    moreover have "wf (?m - Id)"
+    proof -
+      have "wf ?s" using `x \<notin> Field m`
+        by (simp add: wf_eq_minimal Field_def Domain_unfold Domain_converse [symmetric]) metis
+      thus ?thesis using `wf (m - Id)` `x \<notin> Field m`
+        wf_subset [OF `wf ?s` Diff_subset]
+        by (fastforce intro!: wf_Un simp add: Un_Diff Field_def)
+    qed
+    ultimately have "Well_order ?m" by (simp add: order_on_defs)
+    moreover have "extension_on (Field ?m) ?m p"
+      using `extension_on (Field m) m p` `downset_on (Field m) p`
+      by (subst Fm) (auto simp: extension_on_def dest: downset_onD)
+    moreover have "downset_on (Field ?m) p"
+      apply (subst Fm)
+      using `downset_on (Field m) p` and min
+      unfolding downset_on_def Field_def by blast
+    moreover have "(m, ?m) \<in> I"
+      using `Well_order m` and `Well_order ?m` and
+      `downset_on (Field m) p` and `downset_on (Field ?m) p` and
+      `extension_on (Field m) m p` and `extension_on (Field ?m) ?m p` and
+      `Refl m` and `x \<notin> Field m`
+      by (auto simp: I_def init_seg_of_def refl_on_def)
+    ultimately
+    --{*This contradicts maximality of m:*}
+    show False using max and `x \<notin> Field m` unfolding Field_def by blast
+  qed
+  have "p \<subseteq> m"
+    using `Field p \<subseteq> Field m` and `extension_on (Field m) m p`
+    unfolding Field_def extension_on_def by auto fast
+  with `Well_order m` show ?thesis by blast
+qed
+
+text {*Every well-founded relation can be extended to a total wellorder.*}
+corollary total_well_order_extension:
+  assumes "wf p"
+  shows "\<exists>w. p \<subseteq> w \<and> Well_order w \<and> Field w = UNIV"
+proof -
+  from well_order_extension [OF assms] obtain w
+    where "p \<subseteq> w" and wo: "Well_order w" by blast
+  let ?A = "UNIV - Field w"
+  from well_order_on [of ?A] obtain w' where wo': "well_order_on ?A w'" ..
+  have [simp]: "Field w' = ?A" using rel.well_order_on_Well_order [OF wo'] by simp
+  have *: "Field w \<inter> Field w' = {}" by simp
+  let ?w = "w \<union>o w'"
+  have "p \<subseteq> ?w" using `p \<subseteq> w` by (auto simp: Osum_def)
+  moreover have "Well_order ?w" using Osum_Well_order [OF * wo] and wo' by simp
+  moreover have "Field ?w = UNIV" by (simp add: Field_Osum)
+  ultimately show ?thesis by blast
+qed
+
+corollary well_order_on_extension:
+  assumes "wf p" and "Field p \<subseteq> A"
+  shows "\<exists>w. p \<subseteq> w \<and> well_order_on A w"
+proof -
+  from total_well_order_extension [OF `wf p`] obtain r
+    where "p \<subseteq> r" and wo: "Well_order r" and univ: "Field r = UNIV" by blast
+  let ?r = "{(x, y). x \<in> A \<and> y \<in> A \<and> (x, y) \<in> r}"
+  from `p \<subseteq> r` have "p \<subseteq> ?r" using `Field p \<subseteq> A` by (auto simp: Field_def)
+  have 1: "Field ?r = A" using wo univ
+    by (fastforce simp: Field_def order_on_defs refl_on_def)
+  have "Refl r" "trans r" "antisym r" "Total r" "wf (r - Id)"
+    using `Well_order r` by (simp_all add: order_on_defs)
+  have "refl_on A ?r" using `Refl r` by (auto simp: refl_on_def univ)
+  moreover have "trans ?r" using `trans r`
+    unfolding trans_def by blast
+  moreover have "antisym ?r" using `antisym r`
+    unfolding antisym_def by blast
+  moreover have "total_on A ?r" using `Total r` by (simp add: total_on_def univ)
+  moreover have "wf (?r - Id)" by (rule wf_subset [OF `wf(r - Id)`]) blast
+  ultimately have "well_order_on A ?r" by (simp add: order_on_defs)
+  with `p \<subseteq> ?r` show ?thesis by blast
+qed
+
+end
diff -r 7d23f8e501d4 -r 483131676087 src/HOL/Library/Library.thy
--- a/src/HOL/Library/Library.thy	Wed Nov 20 21:28:58 2013 +0100
+++ b/src/HOL/Library/Library.thy	Wed Nov 20 23:14:06 2013 +0100
@@ -41,7 +41,6 @@
   Numeral_Type
   OptionalSugar
   Option_ord
-  Order_Union
   Parallel
   Permutation
   Permutations
@@ -66,7 +65,6 @@
   Transitive_Closure_Table
   Wfrec
   While_Combinator
-  Zorn
 begin
 end
 (*>*)
diff -r 7d23f8e501d4 -r 483131676087 src/HOL/Library/Order_Union.thy
--- a/src/HOL/Library/Order_Union.thy	Wed Nov 20 21:28:58 2013 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,371 +0,0 @@
-(*  Title:      HOL/Library/Order_Union.thy
-    Author:     Andrei Popescu, TU Muenchen
-
-The ordinal-like sum of two orders with disjoint fields
-*)
-
-header {* Order Union *}
-
-theory Order_Union
-imports "~~/src/HOL/Cardinals/Wellfounded_More_FP" 
-begin
-
-definition Osum :: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a rel"  (infix "Osum" 60) where
-  "r Osum r' = r \<union> r' \<union> {(a, a'). a \<in> Field r \<and> a' \<in> Field r'}"
-
-notation Osum  (infix "\<union>o" 60)
-
-lemma Field_Osum: "Field (r \<union>o r') = Field r \<union> Field r'"
-  unfolding Osum_def Field_def by blast
-
-lemma Osum_wf:
-assumes FLD: "Field r Int Field r' = {}" and
-        WF: "wf r" and WF': "wf r'"
-shows "wf (r Osum r')"
-unfolding wf_eq_minimal2 unfolding Field_Osum
-proof(intro allI impI, elim conjE)
-  fix A assume *: "A \<subseteq> Field r \<union> Field r'" and **: "A \<noteq> {}"
-  obtain B where B_def: "B = A Int Field r" by blast
-  show "\<exists>a\<in>A. \<forall>a'\<in>A. (a', a) \<notin> r \<union>o r'"
-  proof(cases "B = {}")
-    assume Case1: "B \<noteq> {}"
-    hence "B \<noteq> {} \<and> B \<le> Field r" using B_def by auto
-    then obtain a where 1: "a \<in> B" and 2: "\<forall>a1 \<in> B. (a1,a) \<notin> r"
-    using WF unfolding wf_eq_minimal2 by metis
-    hence 3: "a \<in> Field r \<and> a \<notin> Field r'" using B_def FLD by auto
-    (*  *)
-    have "\<forall>a1 \<in> A. (a1,a) \<notin> r Osum r'"
-    proof(intro ballI)
-      fix a1 assume **: "a1 \<in> A"
-      {assume Case11: "a1 \<in> Field r"
-       hence "(a1,a) \<notin> r" using B_def ** 2 by auto
-       moreover
-       have "(a1,a) \<notin> r'" using 3 by (auto simp add: Field_def)
-       ultimately have "(a1,a) \<notin> r Osum r'"
-       using 3 unfolding Osum_def by auto
-      }
-      moreover
-      {assume Case12: "a1 \<notin> Field r"
-       hence "(a1,a) \<notin> r" unfolding Field_def by auto
-       moreover
-       have "(a1,a) \<notin> r'" using 3 unfolding Field_def by auto
-       ultimately have "(a1,a) \<notin> r Osum r'"
-       using 3 unfolding Osum_def by auto
-      }
-      ultimately show "(a1,a) \<notin> r Osum r'" by blast
-    qed
-    thus ?thesis using 1 B_def by auto
-  next
-    assume Case2: "B = {}"
-    hence 1: "A \<noteq> {} \<and> A \<le> Field r'" using * ** B_def by auto
-    then obtain a' where 2: "a' \<in> A" and 3: "\<forall>a1' \<in> A. (a1',a') \<notin> r'"
-    using WF' unfolding wf_eq_minimal2 by metis
-    hence 4: "a' \<in> Field r' \<and> a' \<notin> Field r" using 1 FLD by blast
-    (*  *)
-    have "\<forall>a1' \<in> A. (a1',a') \<notin> r Osum r'"
-    proof(unfold Osum_def, auto simp add: 3)
-      fix a1' assume "(a1', a') \<in> r"
-      thus False using 4 unfolding Field_def by blast
-    next
-      fix a1' assume "a1' \<in> A" and "a1' \<in> Field r"
-      thus False using Case2 B_def by auto
-    qed
-    thus ?thesis using 2 by blast
-  qed
-qed
-
-lemma Osum_Refl:
-assumes FLD: "Field r Int Field r' = {}" and
-        REFL: "Refl r" and REFL': "Refl r'"
-shows "Refl (r Osum r')"
-using assms 
-unfolding refl_on_def Field_Osum unfolding Osum_def by blast
-
-lemma Osum_trans:
-assumes FLD: "Field r Int Field r' = {}" and
-        TRANS: "trans r" and TRANS': "trans r'"
-shows "trans (r Osum r')"
-proof(unfold trans_def, auto)
-  fix x y z assume *: "(x, y) \<in> r \<union>o r'" and **: "(y, z) \<in> r \<union>o r'"
-  show  "(x, z) \<in> r \<union>o r'"
-  proof-
-    {assume Case1: "(x,y) \<in> r"
-     hence 1: "x \<in> Field r \<and> y \<in> Field r" unfolding Field_def by auto
-     have ?thesis
-     proof-
-       {assume Case11: "(y,z) \<in> r"
-        hence "(x,z) \<in> r" using Case1 TRANS trans_def[of r] by blast
-        hence ?thesis unfolding Osum_def by auto
-       }
-       moreover
-       {assume Case12: "(y,z) \<in> r'"
-        hence "y \<in> Field r'" unfolding Field_def by auto
-        hence False using FLD 1 by auto
-       }
-       moreover
-       {assume Case13: "z \<in> Field r'"
-        hence ?thesis using 1 unfolding Osum_def by auto
-       }
-       ultimately show ?thesis using ** unfolding Osum_def by blast
-     qed
-    }
-    moreover
-    {assume Case2: "(x,y) \<in> r'"
-     hence 2: "x \<in> Field r' \<and> y \<in> Field r'" unfolding Field_def by auto
-     have ?thesis
-     proof-
-       {assume Case21: "(y,z) \<in> r"
-        hence "y \<in> Field r" unfolding Field_def by auto
-        hence False using FLD 2 by auto
-       }
-       moreover
-       {assume Case22: "(y,z) \<in> r'"
-        hence "(x,z) \<in> r'" using Case2 TRANS' trans_def[of r'] by blast
-        hence ?thesis unfolding Osum_def by auto
-       }
-       moreover
-       {assume Case23: "y \<in> Field r"
-        hence False using FLD 2 by auto
-       }
-       ultimately show ?thesis using ** unfolding Osum_def by blast
-     qed
-    }
-    moreover
-    {assume Case3: "x \<in> Field r \<and> y \<in> Field r'"
-     have ?thesis
-     proof-
-       {assume Case31: "(y,z) \<in> r"
-        hence "y \<in> Field r" unfolding Field_def by auto
-        hence False using FLD Case3 by auto
-       }
-       moreover
-       {assume Case32: "(y,z) \<in> r'"
-        hence "z \<in> Field r'" unfolding Field_def by blast
-        hence ?thesis unfolding Osum_def using Case3 by auto
-       }
-       moreover
-       {assume Case33: "y \<in> Field r"
-        hence False using FLD Case3 by auto
-       }
-       ultimately show ?thesis using ** unfolding Osum_def by blast
-     qed
-    }
-    ultimately show ?thesis using * unfolding Osum_def by blast
-  qed
-qed
-
-lemma Osum_Preorder:
-"\<lbrakk>Field r Int Field r' = {}; Preorder r; Preorder r'\<rbrakk> \<Longrightarrow> Preorder (r Osum r')"
-unfolding preorder_on_def using Osum_Refl Osum_trans by blast
-
-lemma Osum_antisym:
-assumes FLD: "Field r Int Field r' = {}" and
-        AN: "antisym r" and AN': "antisym r'"
-shows "antisym (r Osum r')"
-proof(unfold antisym_def, auto)
-  fix x y assume *: "(x, y) \<in> r \<union>o r'" and **: "(y, x) \<in> r \<union>o r'"
-  show  "x = y"
-  proof-
-    {assume Case1: "(x,y) \<in> r"
-     hence 1: "x \<in> Field r \<and> y \<in> Field r" unfolding Field_def by auto
-     have ?thesis
-     proof-
-       have "(y,x) \<in> r \<Longrightarrow> ?thesis"
-       using Case1 AN antisym_def[of r] by blast
-       moreover
-       {assume "(y,x) \<in> r'"
-        hence "y \<in> Field r'" unfolding Field_def by auto
-        hence False using FLD 1 by auto
-       }
-       moreover
-       have "x \<in> Field r' \<Longrightarrow> False" using FLD 1 by auto
-       ultimately show ?thesis using ** unfolding Osum_def by blast
-     qed
-    }
-    moreover
-    {assume Case2: "(x,y) \<in> r'"
-     hence 2: "x \<in> Field r' \<and> y \<in> Field r'" unfolding Field_def by auto
-     have ?thesis
-     proof-
-       {assume "(y,x) \<in> r"
-        hence "y \<in> Field r" unfolding Field_def by auto
-        hence False using FLD 2 by auto
-       }
-       moreover
-       have "(y,x) \<in> r' \<Longrightarrow> ?thesis"
-       using Case2 AN' antisym_def[of r'] by blast
-       moreover
-       {assume "y \<in> Field r"
-        hence False using FLD 2 by auto
-       }
-       ultimately show ?thesis using ** unfolding Osum_def by blast
-     qed
-    }
-    moreover
-    {assume Case3: "x \<in> Field r \<and> y \<in> Field r'"
-     have ?thesis
-     proof-
-       {assume "(y,x) \<in> r"
-        hence "y \<in> Field r" unfolding Field_def by auto
-        hence False using FLD Case3 by auto
-       }
-       moreover
-       {assume Case32: "(y,x) \<in> r'"
-        hence "x \<in> Field r'" unfolding Field_def by blast
-        hence False using FLD Case3 by auto
-       }
-       moreover
-       have "\<not> y \<in> Field r" using FLD Case3 by auto
-       ultimately show ?thesis using ** unfolding Osum_def by blast
-     qed
-    }
-    ultimately show ?thesis using * unfolding Osum_def by blast
-  qed
-qed
-
-lemma Osum_Partial_order:
-"\<lbrakk>Field r Int Field r' = {}; Partial_order r; Partial_order r'\<rbrakk> \<Longrightarrow>
- Partial_order (r Osum r')"
-unfolding partial_order_on_def using Osum_Preorder Osum_antisym by blast
-
-lemma Osum_Total:
-assumes FLD: "Field r Int Field r' = {}" and
-        TOT: "Total r" and TOT': "Total r'"
-shows "Total (r Osum r')"
-using assms
-unfolding total_on_def  Field_Osum unfolding Osum_def by blast
-
-lemma Osum_Linear_order:
-"\<lbrakk>Field r Int Field r' = {}; Linear_order r; Linear_order r'\<rbrakk> \<Longrightarrow>
- Linear_order (r Osum r')"
-unfolding linear_order_on_def using Osum_Partial_order Osum_Total by blast
-
-lemma Osum_minus_Id1:
-assumes "r \<le> Id"
-shows "(r Osum r') - Id \<le> (r' - Id) \<union> (Field r \<times> Field r')"
-proof-
-  let ?Left = "(r Osum r') - Id"
-  let ?Right = "(r' - Id) \<union> (Field r \<times> Field r')"
-  {fix a::'a and b assume *: "(a,b) \<notin> Id"
-   {assume "(a,b) \<in> r"
-    with * have False using assms by auto
-   }
-   moreover
-   {assume "(a,b) \<in> r'"
-    with * have "(a,b) \<in> r' - Id" by auto
-   }
-   ultimately
-   have "(a,b) \<in> ?Left \<Longrightarrow> (a,b) \<in> ?Right"
-   unfolding Osum_def by auto
-  }
-  thus ?thesis by auto
-qed
-
-lemma Osum_minus_Id2:
-assumes "r' \<le> Id"
-shows "(r Osum r') - Id \<le> (r - Id) \<union> (Field r \<times> Field r')"
-proof-
-  let ?Left = "(r Osum r') - Id"
-  let ?Right = "(r - Id) \<union> (Field r \<times> Field r')"
-  {fix a::'a and b assume *: "(a,b) \<notin> Id"
-   {assume "(a,b) \<in> r'"
-    with * have False using assms by auto
-   }
-   moreover
-   {assume "(a,b) \<in> r"
-    with * have "(a,b) \<in> r - Id" by auto
-   }
-   ultimately
-   have "(a,b) \<in> ?Left \<Longrightarrow> (a,b) \<in> ?Right"
-   unfolding Osum_def by auto
-  }
-  thus ?thesis by auto
-qed
-
-lemma Osum_minus_Id:
-assumes TOT: "Total r" and TOT': "Total r'" and
-        NID: "\<not> (r \<le> Id)" and NID': "\<not> (r' \<le> Id)"
-shows "(r Osum r') - Id \<le> (r - Id) Osum (r' - Id)"
-proof-
-  {fix a a' assume *: "(a,a') \<in> (r Osum r')" and **: "a \<noteq> a'"
-   have "(a,a') \<in> (r - Id) Osum (r' - Id)"
-   proof-
-     {assume "(a,a') \<in> r \<or> (a,a') \<in> r'"
-      with ** have ?thesis unfolding Osum_def by auto
-     }
-     moreover
-     {assume "a \<in> Field r \<and> a' \<in> Field r'"
-      hence "a \<in> Field(r - Id) \<and> a' \<in> Field (r' - Id)"
-      using assms Total_Id_Field by blast
-      hence ?thesis unfolding Osum_def by auto
-     }
-     ultimately show ?thesis using * unfolding Osum_def by fast
-   qed
-  }
-  thus ?thesis by(auto simp add: Osum_def)
-qed
-
-lemma wf_Int_Times:
-assumes "A Int B = {}"
-shows "wf(A \<times> B)"
-unfolding wf_def using assms by blast
-
-lemma Osum_wf_Id:
-assumes TOT: "Total r" and TOT': "Total r'" and
-        FLD: "Field r Int Field r' = {}" and
-        WF: "wf(r - Id)" and WF': "wf(r' - Id)"
-shows "wf ((r Osum r') - Id)"
-proof(cases "r \<le> Id \<or> r' \<le> Id")
-  assume Case1: "\<not>(r \<le> Id \<or> r' \<le> Id)"
-  have "Field(r - Id) Int Field(r' - Id) = {}"
-  using FLD mono_Field[of "r - Id" r]  mono_Field[of "r' - Id" r']
-            Diff_subset[of r Id] Diff_subset[of r' Id] by blast
-  thus ?thesis
-  using Case1 Osum_minus_Id[of r r'] assms Osum_wf[of "r - Id" "r' - Id"]
-        wf_subset[of "(r - Id) \<union>o (r' - Id)" "(r Osum r') - Id"] by auto
-next
-  have 1: "wf(Field r \<times> Field r')"
-  using FLD by (auto simp add: wf_Int_Times)
-  assume Case2: "r \<le> Id \<or> r' \<le> Id"
-  moreover
-  {assume Case21: "r \<le> Id"
-   hence "(r Osum r') - Id \<le> (r' - Id) \<union> (Field r \<times> Field r')"
-   using Osum_minus_Id1[of r r'] by simp
-   moreover
-   {have "Domain(Field r \<times> Field r') Int Range(r' - Id) = {}"
-    using FLD unfolding Field_def by blast
-    hence "wf((r' - Id) \<union> (Field r \<times> Field r'))"
-    using 1 WF' wf_Un[of "Field r \<times> Field r'" "r' - Id"]
-    by (auto simp add: Un_commute)
-   }
-   ultimately have ?thesis by (metis wf_subset)
-  }
-  moreover
-  {assume Case22: "r' \<le> Id"
-   hence "(r Osum r') - Id \<le> (r - Id) \<union> (Field r \<times> Field r')"
-   using Osum_minus_Id2[of r' r] by simp
-   moreover
-   {have "Range(Field r \<times> Field r') Int Domain(r - Id) = {}"
-    using FLD unfolding Field_def by blast
-    hence "wf((r - Id) \<union> (Field r \<times> Field r'))"
-    using 1 WF wf_Un[of "r - Id" "Field r \<times> Field r'"]
-    by (auto simp add: Un_commute)
-   }
-   ultimately have ?thesis by (metis wf_subset)
-  }
-  ultimately show ?thesis by blast
-qed
-
-lemma Osum_Well_order:
-assumes FLD: "Field r Int Field r' = {}" and
-        WELL: "Well_order r" and WELL': "Well_order r'"
-shows "Well_order (r Osum r')"
-proof-
-  have "Total r \<and> Total r'" using WELL WELL'
-  by (auto simp add: order_on_defs)
-  thus ?thesis using assms unfolding well_order_on_def
-  using Osum_Linear_order Osum_wf_Id by blast
-qed
-
-end
-
diff -r 7d23f8e501d4 -r 483131676087 src/HOL/Library/Zorn.thy
--- a/src/HOL/Library/Zorn.thy	Wed Nov 20 21:28:58 2013 +0100
+++ b/src/HOL/Library/Zorn.thy	Wed Nov 20 23:14:06 2013 +0100
@@ -5,13 +5,12 @@
 
 Zorn's Lemma (ported from Larry Paulson's Zorn.thy in ZF).
 The well-ordering theorem.
-The extension of any well-founded relation to a well-order. 
 *)
 
 header {* Zorn's Lemma *}
 
 theory Zorn
-imports Order_Union
+imports Order_Relation
 begin
 
 subsection {* Zorn's Lemma for the Subset Relation *}
@@ -710,207 +709,4 @@
   with 1 show ?thesis by auto
 qed
 
-subsection {* Extending Well-founded Relations to Well-Orders *}
-
-text {*A \emph{downset} (also lower set, decreasing set, initial segment, or
-downward closed set) is closed w.r.t.\ smaller elements.*}
-definition downset_on where
-  "downset_on A r = (\<forall>x y. (x, y) \<in> r \<and> y \<in> A \<longrightarrow> x \<in> A)"
-
-(*
-text {*Connection to order filters of the @{theory Cardinals} theory.*}
-lemma (in wo_rel) ofilter_downset_on_conv:
-  "ofilter A \<longleftrightarrow> downset_on A r \<and> A \<subseteq> Field r"
-  by (auto simp: downset_on_def ofilter_def under_def)
-*)
-
-lemma downset_onI:
-  "(\<And>x y. (x, y) \<in> r \<Longrightarrow> y \<in> A \<Longrightarrow> x \<in> A) \<Longrightarrow> downset_on A r"
-  by (auto simp: downset_on_def)
-
-lemma downset_onD:
-  "downset_on A r \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> y \<in> A \<Longrightarrow> x \<in> A"
-  unfolding downset_on_def by blast
-
-text {*Extensions of relations w.r.t.\ a given set.*}
-definition extension_on where
-  "extension_on A r s = (\<forall>x\<in>A. \<forall>y\<in>A. (x, y) \<in> s \<longrightarrow> (x, y) \<in> r)"
-
-lemma extension_onI:
-  "(\<And>x y. \<lbrakk>x \<in> A; y \<in> A; (x, y) \<in> s\<rbrakk> \<Longrightarrow> (x, y) \<in> r) \<Longrightarrow> extension_on A r s"
-  by (auto simp: extension_on_def)
-
-lemma extension_onD:
-  "extension_on A r s \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> (x, y) \<in> s \<Longrightarrow> (x, y) \<in> r"
-  by (auto simp: extension_on_def)
-
-lemma downset_on_Union:
-  assumes "\<And>r. r \<in> R \<Longrightarrow> downset_on (Field r) p"
-  shows "downset_on (Field (\<Union>R)) p"
-  using assms by (auto intro: downset_onI dest: downset_onD)
-
-lemma chain_subset_extension_on_Union:
-  assumes "chain\<^sub>\<subseteq> R" and "\<And>r. r \<in> R \<Longrightarrow> extension_on (Field r) r p"
-  shows "extension_on (Field (\<Union>R)) (\<Union>R) p"
-  using assms
-  by (simp add: chain_subset_def extension_on_def)
-     (metis (no_types) mono_Field set_mp)
-
-lemma downset_on_empty [simp]: "downset_on {} p"
-  by (auto simp: downset_on_def)
-
-lemma extension_on_empty [simp]: "extension_on {} p q"
-  by (auto simp: extension_on_def)
-
-text {*Every well-founded relation can be extended to a well-order.*}
-theorem well_order_extension:
-  assumes "wf p"
-  shows "\<exists>w. p \<subseteq> w \<and> Well_order w"
-proof -
-  let ?K = "{r. Well_order r \<and> downset_on (Field r) p \<and> extension_on (Field r) r p}"
-  def I \<equiv> "init_seg_of \<inter> ?K \<times> ?K"
-  have I_init: "I \<subseteq> init_seg_of" by (simp add: I_def)
-  then have subch: "\<And>R. R \<in> Chains I \<Longrightarrow> chain\<^sub>\<subseteq> R"
-    by (auto simp: init_seg_of_def chain_subset_def Chains_def)
-  have Chains_wo: "\<And>R r. R \<in> Chains I \<Longrightarrow> r \<in> R \<Longrightarrow>
-      Well_order r \<and> downset_on (Field r) p \<and> extension_on (Field r) r p"
-    by (simp add: Chains_def I_def) blast
-  have FI: "Field I = ?K" by (auto simp: I_def init_seg_of_def Field_def)
-  then have 0: "Partial_order I"
-    by (auto simp: partial_order_on_def preorder_on_def antisym_def antisym_init_seg_of refl_on_def
-      trans_def I_def elim: trans_init_seg_of)
-  { fix R assume "R \<in> Chains I"
-    then have Ris: "R \<in> Chains init_seg_of" using mono_Chains [OF I_init] by blast
-    have subch: "chain\<^sub>\<subseteq> R" using `R \<in> Chains I` I_init
-      by (auto simp: init_seg_of_def chain_subset_def Chains_def)
-    have "\<forall>r\<in>R. Refl r" and "\<forall>r\<in>R. trans r" and "\<forall>r\<in>R. antisym r" and
-      "\<forall>r\<in>R. Total r" and "\<forall>r\<in>R. wf (r - Id)" and
-      "\<And>r. r \<in> R \<Longrightarrow> downset_on (Field r) p" and
-      "\<And>r. r \<in> R \<Longrightarrow> extension_on (Field r) r p"
-      using Chains_wo [OF `R \<in> Chains I`] by (simp_all add: order_on_defs)
-    have "Refl (\<Union>R)" using `\<forall>r\<in>R. Refl r`  unfolding refl_on_def by fastforce
-    moreover have "trans (\<Union>R)"
-      by (rule chain_subset_trans_Union [OF subch `\<forall>r\<in>R. trans r`])
-    moreover have "antisym (\<Union>R)"
-      by (rule chain_subset_antisym_Union [OF subch `\<forall>r\<in>R. antisym r`])
-    moreover have "Total (\<Union>R)"
-      by (rule chain_subset_Total_Union [OF subch `\<forall>r\<in>R. Total r`])
-    moreover have "wf ((\<Union>R) - Id)"
-    proof -
-      have "(\<Union>R) - Id = \<Union>{r - Id | r. r \<in> R}" by blast
-      with `\<forall>r\<in>R. wf (r - Id)` wf_Union_wf_init_segs [OF Chains_inits_DiffI [OF Ris]]
-      show ?thesis by fastforce
-    qed
-    ultimately have "Well_order (\<Union>R)" by (simp add: order_on_defs)
-    moreover have "\<forall>r\<in>R. r initial_segment_of \<Union>R" using Ris
-      by (simp add: Chains_init_seg_of_Union)
-    moreover have "downset_on (Field (\<Union>R)) p"
-      by (rule downset_on_Union [OF `\<And>r. r \<in> R \<Longrightarrow> downset_on (Field r) p`])
-    moreover have "extension_on (Field (\<Union>R)) (\<Union>R) p"
-      by (rule chain_subset_extension_on_Union [OF subch `\<And>r. r \<in> R \<Longrightarrow> extension_on (Field r) r p`])
-    ultimately have "\<Union>R \<in> ?K \<and> (\<forall>r\<in>R. (r,\<Union>R) \<in> I)"
-      using mono_Chains [OF I_init] and `R \<in> Chains I`
-      by (simp (no_asm) add: I_def del: Field_Union) (metis Chains_wo)
-  }
-  then have 1: "\<forall>R\<in>Chains I. \<exists>u\<in>Field I. \<forall>r\<in>R. (r, u) \<in> I" by (subst FI) blast
-  txt {*Zorn's Lemma yields a maximal well-order m.*}
-  from Zorns_po_lemma [OF 0 1] obtain m :: "('a \<times> 'a) set"
-    where "Well_order m" and "downset_on (Field m) p" and "extension_on (Field m) m p" and
-    max: "\<forall>r. Well_order r \<and> downset_on (Field r) p \<and> extension_on (Field r) r p \<and>
-      (m, r) \<in> I \<longrightarrow> r = m"
-    by (auto simp: FI)
-  have "Field p \<subseteq> Field m"
-  proof (rule ccontr)
-    let ?Q = "Field p - Field m"
-    assume "\<not> (Field p \<subseteq> Field m)"
-    with assms [unfolded wf_eq_minimal, THEN spec, of ?Q]
-      obtain x where "x \<in> Field p" and "x \<notin> Field m" and
-      min: "\<forall>y. (y, x) \<in> p \<longrightarrow> y \<notin> ?Q" by blast
-    txt {*Add @{term x} as topmost element to @{term m}.*}
-    let ?s = "{(y, x) | y. y \<in> Field m}"
-    let ?m = "insert (x, x) m \<union> ?s"
-    have Fm: "Field ?m = insert x (Field m)" by (auto simp: Field_def)
-    have "Refl m" and "trans m" and "antisym m" and "Total m" and "wf (m - Id)"
-      using `Well_order m` by (simp_all add: order_on_defs)
-    txt {*We show that the extension is a well-order.*}
-    have "Refl ?m" using `Refl m` Fm by (auto simp: refl_on_def)
-    moreover have "trans ?m" using `trans m` `x \<notin> Field m`
-      unfolding trans_def Field_def Domain_unfold Domain_converse [symmetric] by blast
-    moreover have "antisym ?m" using `antisym m` `x \<notin> Field m`
-      unfolding antisym_def Field_def Domain_unfold Domain_converse [symmetric] by blast
-    moreover have "Total ?m" using `Total m` Fm by (auto simp: Relation.total_on_def)
-    moreover have "wf (?m - Id)"
-    proof -
-      have "wf ?s" using `x \<notin> Field m`
-        by (simp add: wf_eq_minimal Field_def Domain_unfold Domain_converse [symmetric]) metis
-      thus ?thesis using `wf (m - Id)` `x \<notin> Field m`
-        wf_subset [OF `wf ?s` Diff_subset]
-        by (fastforce intro!: wf_Un simp add: Un_Diff Field_def)
-    qed
-    ultimately have "Well_order ?m" by (simp add: order_on_defs)
-    moreover have "extension_on (Field ?m) ?m p"
-      using `extension_on (Field m) m p` `downset_on (Field m) p`
-      by (subst Fm) (auto simp: extension_on_def dest: downset_onD)
-    moreover have "downset_on (Field ?m) p"
-      apply (subst Fm)
-      using `downset_on (Field m) p` and min
-      unfolding downset_on_def Field_def by blast
-    moreover have "(m, ?m) \<in> I"
-      using `Well_order m` and `Well_order ?m` and
-      `downset_on (Field m) p` and `downset_on (Field ?m) p` and
-      `extension_on (Field m) m p` and `extension_on (Field ?m) ?m p` and
-      `Refl m` and `x \<notin> Field m`
-      by (auto simp: I_def init_seg_of_def refl_on_def)
-    ultimately
-    --{*This contradicts maximality of m:*}
-    show False using max and `x \<notin> Field m` unfolding Field_def by blast
-  qed
-  have "p \<subseteq> m"
-    using `Field p \<subseteq> Field m` and `extension_on (Field m) m p`
-    unfolding Field_def extension_on_def by auto fast
-  with `Well_order m` show ?thesis by blast
-qed
-
-text {*Every well-founded relation can be extended to a total well-order.*}
-corollary total_well_order_extension:
-  assumes "wf p"
-  shows "\<exists>w. p \<subseteq> w \<and> Well_order w \<and> Field w = UNIV"
-proof -
-  from well_order_extension [OF assms] obtain w
-    where "p \<subseteq> w" and wo: "Well_order w" by blast
-  let ?A = "UNIV - Field w"
-  from well_order_on [of ?A] obtain w' where wo': "well_order_on ?A w'" ..
-  have [simp]: "Field w' = ?A" using rel.well_order_on_Well_order [OF wo'] by simp
-  have *: "Field w \<inter> Field w' = {}" by simp
-  let ?w = "w \<union>o w'"
-  have "p \<subseteq> ?w" using `p \<subseteq> w` by (auto simp: Osum_def)
-  moreover have "Well_order ?w" using Osum_Well_order [OF * wo] and wo' by simp
-  moreover have "Field ?w = UNIV" by (simp add: Field_Osum)
-  ultimately show ?thesis by blast
-qed
-
-corollary well_order_on_extension:
-  assumes "wf p" and "Field p \<subseteq> A"
-  shows "\<exists>w. p \<subseteq> w \<and> well_order_on A w"
-proof -
-  from total_well_order_extension [OF `wf p`] obtain r
-    where "p \<subseteq> r" and wo: "Well_order r" and univ: "Field r = UNIV" by blast
-  let ?r = "{(x, y). x \<in> A \<and> y \<in> A \<and> (x, y) \<in> r}"
-  from `p \<subseteq> r` have "p \<subseteq> ?r" using `Field p \<subseteq> A` by (auto simp: Field_def)
-  have 1: "Field ?r = A" using wo univ
-    by (fastforce simp: Field_def order_on_defs refl_on_def)
-  have "Refl r" "trans r" "antisym r" "Total r" "wf (r - Id)"
-    using `Well_order r` by (simp_all add: order_on_defs)
-  have "refl_on A ?r" using `Refl r` by (auto simp: refl_on_def univ)
-  moreover have "trans ?r" using `trans r`
-    unfolding trans_def by blast
-  moreover have "antisym ?r" using `antisym r`
-    unfolding antisym_def by blast
-  moreover have "total_on A ?r" using `Total r` by (simp add: total_on_def univ)
-  moreover have "wf (?r - Id)" by (rule wf_subset [OF `wf(r - Id)`]) blast
-  ultimately have "well_order_on A ?r" by (simp add: order_on_defs)
-  with `p \<subseteq> ?r` show ?thesis by blast
-qed
-
 end
-