(*  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_Base" 

 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 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_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 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_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

 end