src/HOL/Cardinals/Constructions_on_Wellorders.thy

make WWW_Find work again, now that its ML modules reside within a theory context (cf. bf5b45870110) -- patch by Rafal Kolanski;

(*  Title:      HOL/Cardinals/Constructions_on_Wellorders.thy
    Author:     Andrei Popescu, TU Muenchen
    Copyright   2012

Constructions on wellorders.
*)

header {* Constructions on Wellorders *}

theory Constructions_on_Wellorders
imports Constructions_on_Wellorders_Base Wellorder_Embedding
begin

declare
  ordLeq_Well_order_simp[simp]
  ordLess_Well_order_simp[simp]
  ordIso_Well_order_simp[simp]
  not_ordLeq_iff_ordLess[simp]
  not_ordLess_iff_ordLeq[simp]


subsection {* Restriction to a set  *}

lemma Restr_incr2:
"r <= r' \<Longrightarrow> Restr r A <= Restr r' A"
by blast

lemma Restr_incr:
"\<lbrakk>r \<le> r'; A \<le> A'\<rbrakk> \<Longrightarrow> Restr r A \<le> Restr r' A'"
by blast

lemma Restr_Int:
"Restr (Restr r A) B = Restr r (A Int B)"
by blast

lemma Restr_iff: "(a,b) : Restr r A = (a : A \<and> b : A \<and> (a,b) : r)"
by (auto simp add: Field_def)

lemma Restr_subset1: "Restr r A \<le> r"
by auto

lemma Restr_subset2: "Restr r A \<le> A \<times> A"
by auto

lemma wf_Restr:
"wf r \<Longrightarrow> wf(Restr r A)"
using wf_subset Restr_subset by blast

lemma Restr_incr1:
"A \<le> B \<Longrightarrow> Restr r A \<le> Restr r B"
by blast


subsection {* Order filters versus restrictions and embeddings  *}

lemma ofilter_Restr:
assumes WELL: "Well_order r" and
        OFA: "ofilter r A" and OFB: "ofilter r B" and SUB: "A \<le> B"
shows "ofilter (Restr r B) A"
proof-
  let ?rB = "Restr r B"
  have Well: "wo_rel r" unfolding wo_rel_def using WELL .
  hence Refl: "Refl r" by (auto simp add: wo_rel.REFL)
  hence Field: "Field ?rB = Field r Int B"
  using Refl_Field_Restr by blast
  have WellB: "wo_rel ?rB \<and> Well_order ?rB" using WELL
  by (auto simp add: Well_order_Restr wo_rel_def)
  (* Main proof *)
  show ?thesis
  proof(auto simp add: WellB wo_rel.ofilter_def)
    fix a assume "a \<in> A"
    hence "a \<in> Field r \<and> a \<in> B" using assms Well
    by (auto simp add: wo_rel.ofilter_def)
    with Field show "a \<in> Field(Restr r B)" by auto
  next
    fix a b assume *: "a \<in> A"  and "b \<in> under (Restr r B) a"
    hence "b \<in> under r a"
    using WELL OFB SUB ofilter_Restr_under[of r B a] by auto
    thus "b \<in> A" using * Well OFA by(auto simp add: wo_rel.ofilter_def)
  qed
qed

lemma ofilter_subset_iso:
assumes WELL: "Well_order r" and
        OFA: "ofilter r A" and OFB: "ofilter r B"
shows "(A = B) = iso (Restr r A) (Restr r B) id"
using assms
by (auto simp add: ofilter_subset_embedS_iso)


subsection {* Ordering the  well-orders by existence of embeddings *}

corollary ordLeq_refl_on: "refl_on {r. Well_order r} ordLeq"
using ordLeq_reflexive unfolding ordLeq_def refl_on_def
by blast

corollary ordLeq_trans: "trans ordLeq"
using trans_def[of ordLeq] ordLeq_transitive by blast

corollary ordLeq_preorder_on: "preorder_on {r. Well_order r} ordLeq"
by(auto simp add: preorder_on_def ordLeq_refl_on ordLeq_trans)

corollary ordIso_refl_on: "refl_on {r. Well_order r} ordIso"
using ordIso_reflexive unfolding refl_on_def ordIso_def
by blast

corollary ordIso_trans: "trans ordIso"
using trans_def[of ordIso] ordIso_transitive by blast

corollary ordIso_sym: "sym ordIso"
by (auto simp add: sym_def ordIso_symmetric)

corollary ordIso_equiv: "equiv {r. Well_order r} ordIso"
by (auto simp add:  equiv_def ordIso_sym ordIso_refl_on ordIso_trans)

lemma ordLess_irrefl: "irrefl ordLess"
by(unfold irrefl_def, auto simp add: ordLess_irreflexive)

lemma ordLess_or_ordIso:
assumes WELL: "Well_order r" and WELL': "Well_order r'"
shows "r <o r' \<or> r' <o r \<or> r =o r'"
unfolding ordLess_def ordIso_def
using assms embedS_or_iso[of r r'] by auto

corollary ordLeq_ordLess_Un_ordIso:
"ordLeq = ordLess \<union> ordIso"
by (auto simp add: ordLeq_iff_ordLess_or_ordIso)

lemma not_ordLeq_ordLess:
"r \<le>o r' \<Longrightarrow> \<not> r' <o r"
using not_ordLess_ordLeq by blast

lemma ordIso_or_ordLess:
assumes WELL: "Well_order r" and WELL': "Well_order r'"
shows "r =o r' \<or> r <o r' \<or> r' <o r"
using assms ordLess_or_ordLeq ordLeq_iff_ordLess_or_ordIso by blast

lemmas ord_trans = ordIso_transitive ordLeq_transitive ordLess_transitive
                   ordIso_ordLeq_trans ordLeq_ordIso_trans
                   ordIso_ordLess_trans ordLess_ordIso_trans
                   ordLess_ordLeq_trans ordLeq_ordLess_trans

lemma ofilter_ordLeq:
assumes "Well_order r" and "ofilter r A"
shows "Restr r A \<le>o r"
proof-
  have "A \<le> Field r" using assms by (auto simp add: wo_rel_def wo_rel.ofilter_def)
  thus ?thesis using assms
  by (simp add: ofilter_subset_ordLeq wo_rel.Field_ofilter
      wo_rel_def Restr_Field)
qed

corollary under_Restr_ordLeq:
"Well_order r \<Longrightarrow> Restr r (under r a) \<le>o r"
by (auto simp add: ofilter_ordLeq wo_rel.under_ofilter wo_rel_def)


subsection {* Copy via direct images  *}

lemma Id_dir_image: "dir_image Id f \<le> Id"
unfolding dir_image_def by auto

lemma Un_dir_image:
"dir_image (r1 \<union> r2) f = (dir_image r1 f) \<union> (dir_image r2 f)"
unfolding dir_image_def by auto

lemma Int_dir_image:
assumes "inj_on f (Field r1 \<union> Field r2)"
shows "dir_image (r1 Int r2) f = (dir_image r1 f) Int (dir_image r2 f)"
proof
  show "dir_image (r1 Int r2) f \<le> (dir_image r1 f) Int (dir_image r2 f)"
  using assms unfolding dir_image_def inj_on_def by auto
next
  show "(dir_image r1 f) Int (dir_image r2 f) \<le> dir_image (r1 Int r2) f"
  proof(clarify)
    fix a' b'
    assume "(a',b') \<in> dir_image r1 f" "(a',b') \<in> dir_image r2 f"
    then obtain a1 b1 a2 b2
    where 1: "a' = f a1 \<and> b' = f b1 \<and> a' = f a2 \<and> b' = f b2" and
          2: "(a1,b1) \<in> r1 \<and> (a2,b2) \<in> r2" and
          3: "{a1,b1} \<le> Field r1 \<and> {a2,b2} \<le> Field r2"
    unfolding dir_image_def Field_def by blast
    hence "a1 = a2 \<and> b1 = b2" using assms unfolding inj_on_def by auto
    hence "a' = f a1 \<and> b' = f b1 \<and> (a1,b1) \<in> r1 Int r2 \<and> (a2,b2) \<in> r1 Int r2"
    using 1 2 by auto
    thus "(a',b') \<in> dir_image (r1 \<inter> r2) f"
    unfolding dir_image_def by blast
  qed
qed


subsection {* Ordinal-like sum of two (disjoint) well-orders *}

text{* This is roughly obtained by ``concatenating" the two well-orders -- thus, all elements
of the first will be smaller than all elements of the second.  This construction
only makes sense if the fields of the two well-order relations are disjoint. *}

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'}"

abbreviation Osum2 :: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a rel" (infix "\<union>o" 60)
where "r \<union>o r' \<equiv> r Osum r'"

lemma Field_Osum: "Field(r Osum r') = Field r \<union> Field r'"
unfolding Osum_def Field_def by blast

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  (* Need first unfold Field_Osum, only then Osum_def *)
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_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 blast
    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 blast
    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_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 rel.Total_Id_Field by blast
      hence ?thesis unfolding Osum_def by auto
     }
     ultimately show ?thesis using * unfolding Osum_def by blast
   qed
  }
  thus ?thesis by(auto simp add: Osum_def)
qed


lemma wf_Int_Times:
assumes "A Int B = {}"
shows "wf(A \<times> B)"
proof(unfold wf_def, auto)
  fix P x
  assume *: "\<forall>x. (\<forall>y. y \<in> A \<and> x \<in> B \<longrightarrow> P y) \<longrightarrow> P x"
  moreover have "\<forall>y \<in> A. P y" using assms * by blast
  ultimately show "P x" using * by (case_tac "x \<in> B", auto)
qed

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_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 (auto simp add: 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 (auto simp add: 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

lemma Osum_embed:
assumes FLD: "Field r Int Field r' = {}" and
        WELL: "Well_order r" and WELL': "Well_order r'"
shows "embed r (r Osum r') id"
proof-
  have 1: "Well_order (r Osum r')"
  using assms by (auto simp add: Osum_Well_order)
  moreover
  have "compat r (r Osum r') id"
  unfolding compat_def Osum_def by auto
  moreover
  have "inj_on id (Field r)" by simp
  moreover
  have "ofilter (r Osum r') (Field r)"
  using 1 proof(auto simp add: wo_rel_def wo_rel.ofilter_def
                               Field_Osum rel.under_def)
    fix a b assume 2: "a \<in> Field r" and 3: "(b,a) \<in> r Osum r'"
    moreover
    {assume "(b,a) \<in> r'"
     hence "a \<in> Field r'" using Field_def[of r'] by blast
     hence False using 2 FLD by blast
    }
    moreover
    {assume "a \<in> Field r'"
     hence False using 2 FLD by blast
    }
    ultimately
    show "b \<in> Field r" by (auto simp add: Osum_def Field_def)
  qed
  ultimately show ?thesis
  using assms by (auto simp add: embed_iff_compat_inj_on_ofilter)
qed

corollary Osum_ordLeq:
assumes FLD: "Field r Int Field r' = {}" and
        WELL: "Well_order r" and WELL': "Well_order r'"
shows "r \<le>o r Osum r'"
using assms Osum_embed Osum_Well_order
unfolding ordLeq_def by blast

lemma Well_order_embed_copy:
assumes WELL: "well_order_on A r" and
        INJ: "inj_on f A" and SUB: "f ` A \<le> B"
shows "\<exists>r'. well_order_on B r' \<and> r \<le>o r'"
proof-
  have "bij_betw f A (f ` A)"
  using INJ inj_on_imp_bij_betw by blast
  then obtain r'' where "well_order_on (f ` A) r''" and 1: "r =o r''"
  using WELL  Well_order_iso_copy by blast
  hence 2: "Well_order r'' \<and> Field r'' = (f ` A)"
  using rel.well_order_on_Well_order by blast
  (*  *)
  let ?C = "B - (f ` A)"
  obtain r''' where "well_order_on ?C r'''"
  using well_order_on by blast
  hence 3: "Well_order r''' \<and> Field r''' = ?C"
  using rel.well_order_on_Well_order by blast
  (*  *)
  let ?r' = "r'' Osum r'''"
  have "Field r'' Int Field r''' = {}"
  using 2 3 by auto
  hence "r'' \<le>o ?r'" using Osum_ordLeq[of r'' r'''] 2 3 by blast
  hence 4: "r \<le>o ?r'" using 1 ordIso_ordLeq_trans by blast
  (*  *)
  hence "Well_order ?r'" unfolding ordLeq_def by auto
  moreover
  have "Field ?r' = B" using 2 3 SUB by (auto simp add: Field_Osum)
  ultimately show ?thesis using 4 by blast
qed


subsection {* The maxim among a finite set of ordinals  *}

text {* The correct phrasing would be ``a maxim of ...", as @{text "\<le>o"} is only a preorder. *}

definition isOmax :: "'a rel set \<Rightarrow> 'a rel \<Rightarrow> bool"
where
"isOmax  R r == r \<in> R \<and> (ALL r' : R. r' \<le>o r)"

definition omax :: "'a rel set \<Rightarrow> 'a rel"
where
"omax R == SOME r. isOmax R r"

lemma exists_isOmax:
assumes "finite R" and "R \<noteq> {}" and "\<forall> r \<in> R. Well_order r"
shows "\<exists> r. isOmax R r"
proof-
  have "finite R \<Longrightarrow> R \<noteq> {} \<longrightarrow> (\<forall> r \<in> R. Well_order r) \<longrightarrow> (\<exists> r. isOmax R r)"
  apply(erule finite_induct) apply(simp add: isOmax_def)
  proof(clarsimp)
    fix r :: "('a \<times> 'a) set" and R assume *: "finite R" and **: "r \<notin> R"
    and ***: "Well_order r" and ****: "\<forall>r\<in>R. Well_order r"
    and IH: "R \<noteq> {} \<longrightarrow> (\<exists>p. isOmax R p)"
    let ?R' = "insert r R"
    show "\<exists>p'. (isOmax ?R' p')"
    proof(cases "R = {}")
      assume Case1: "R = {}"
      thus ?thesis unfolding isOmax_def using ***
      by (simp add: ordLeq_reflexive)
    next
      assume Case2: "R \<noteq> {}"
      then obtain p where p: "isOmax R p" using IH by auto
      hence 1: "Well_order p" using **** unfolding isOmax_def by simp
      {assume Case21: "r \<le>o p"
       hence "isOmax ?R' p" using p unfolding isOmax_def by simp
       hence ?thesis by auto
      }
      moreover
      {assume Case22: "p \<le>o r"
       {fix r' assume "r' \<in> ?R'"
        moreover
        {assume "r' \<in> R"
         hence "r' \<le>o p" using p unfolding isOmax_def by simp
         hence "r' \<le>o r" using Case22 by(rule ordLeq_transitive)
        }
        moreover have "r \<le>o r" using *** by(rule ordLeq_reflexive)
        ultimately have "r' \<le>o r" by auto
       }
       hence "isOmax ?R' r" unfolding isOmax_def by simp
       hence ?thesis by auto
      }
      moreover have "r \<le>o p \<or> p \<le>o r"
      using 1 *** ordLeq_total by auto
      ultimately show ?thesis by blast
    qed
  qed
  thus ?thesis using assms by auto
qed

lemma omax_isOmax:
assumes "finite R" and "R \<noteq> {}" and "\<forall> r \<in> R. Well_order r"
shows "isOmax R (omax R)"
unfolding omax_def using assms
by(simp add: exists_isOmax someI_ex)

lemma omax_in:
assumes "finite R" and "R \<noteq> {}" and "\<forall> r \<in> R. Well_order r"
shows "omax R \<in> R"
using assms omax_isOmax unfolding isOmax_def by blast

lemma Well_order_omax:
assumes "finite R" and "R \<noteq> {}" and "\<forall>r\<in>R. Well_order r"
shows "Well_order (omax R)"
using assms apply - apply(drule omax_in) by auto

lemma omax_maxim:
assumes "finite R" and "\<forall> r \<in> R. Well_order r" and "r \<in> R"
shows "r \<le>o omax R"
using assms omax_isOmax unfolding isOmax_def by blast

lemma omax_ordLeq:
assumes "finite R" and "R \<noteq> {}" and *: "\<forall> r \<in> R. r \<le>o p"
shows "omax R \<le>o p"
proof-
  have "\<forall> r \<in> R. Well_order r" using * unfolding ordLeq_def by simp
  thus ?thesis using assms omax_in by auto
qed

lemma omax_ordLess:
assumes "finite R" and "R \<noteq> {}" and *: "\<forall> r \<in> R. r <o p"
shows "omax R <o p"
proof-
  have "\<forall> r \<in> R. Well_order r" using * unfolding ordLess_def by simp
  thus ?thesis using assms omax_in by auto
qed

lemma omax_ordLeq_elim:
assumes "finite R" and "\<forall> r \<in> R. Well_order r"
and "omax R \<le>o p" and "r \<in> R"
shows "r \<le>o p"
using assms omax_maxim[of R r] apply simp
using ordLeq_transitive by blast

lemma omax_ordLess_elim:
assumes "finite R" and "\<forall> r \<in> R. Well_order r"
and "omax R <o p" and "r \<in> R"
shows "r <o p"
using assms omax_maxim[of R r] apply simp
using ordLeq_ordLess_trans by blast

lemma ordLeq_omax:
assumes "finite R" and "\<forall> r \<in> R. Well_order r"
and "r \<in> R" and "p \<le>o r"
shows "p \<le>o omax R"
using assms omax_maxim[of R r] apply simp
using ordLeq_transitive by blast

lemma ordLess_omax:
assumes "finite R" and "\<forall> r \<in> R. Well_order r"
and "r \<in> R" and "p <o r"
shows "p <o omax R"
using assms omax_maxim[of R r] apply simp
using ordLess_ordLeq_trans by blast

lemma omax_ordLeq_mono:
assumes P: "finite P" and R: "finite R"
and NE_P: "P \<noteq> {}" and Well_R: "\<forall> r \<in> R. Well_order r"
and LEQ: "\<forall> p \<in> P. \<exists> r \<in> R. p \<le>o r"
shows "omax P \<le>o omax R"
proof-
  let ?mp = "omax P"  let ?mr = "omax R"
  {fix p assume "p : P"
   then obtain r where r: "r : R" and "p \<le>o r"
   using LEQ by blast
   moreover have "r <=o ?mr"
   using r R Well_R omax_maxim by blast
   ultimately have "p <=o ?mr"
   using ordLeq_transitive by blast
  }
  thus "?mp <=o ?mr"
  using NE_P P using omax_ordLeq by blast
qed

lemma omax_ordLess_mono:
assumes P: "finite P" and R: "finite R"
and NE_P: "P \<noteq> {}" and Well_R: "\<forall> r \<in> R. Well_order r"
and LEQ: "\<forall> p \<in> P. \<exists> r \<in> R. p <o r"
shows "omax P <o omax R"
proof-
  let ?mp = "omax P"  let ?mr = "omax R"
  {fix p assume "p : P"
   then obtain r where r: "r : R" and "p <o r"
   using LEQ by blast
   moreover have "r <=o ?mr"
   using r R Well_R omax_maxim by blast
   ultimately have "p <o ?mr"
   using ordLess_ordLeq_trans by blast
  }
  thus "?mp <o ?mr"
  using NE_P P omax_ordLess by blast
qed

end