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author | popescua |

Mon, 27 May 2013 20:09:20 +0200 | |

changeset 52184 | d6627b50b131 |

parent 52183 | 667961fa6a60 |

child 52190 | c87b7f26e2c7 |

added Ordered_Union

--- /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. +*) + +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'}" + +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 +