author popescua Mon, 27 May 2013 20:09:20 +0200 changeset 52184 d6627b50b131 parent 52183 667961fa6a60 child 52190 c87b7f26e2c7
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Order_Union.thy	Mon May 27 20:09:20 2013 +0200
@@ -0,0 +1,378 @@
+(*  Title:      HOL/Library/Order_Union.thy
+    Author:     Andrei Popescu, TU Muenchen
+
+Subset of Constructions_on_Wellorders that provides the ordinal sum but does
+not rely on the ~/HOL/Library/Zorn.thy.
+*)
+
+
+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'}"
+
+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 \<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
+```