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(* Title: HOL/Library/Order_Union.thy
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Author: Andrei Popescu, TU Muenchen
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The ordinal-like sum of two orders with disjoint fields
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*)
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header {* Order Union *}
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theory Order_Union
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imports "~~/src/HOL/Cardinals/Wellfounded_More_Base"
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begin
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definition Osum :: "'a rel \<Rightarrow> 'a rel \<Rightarrow> 'a rel" (infix "Osum" 60) where
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"r Osum r' = r \<union> r' \<union> {(a, a'). a \<in> Field r \<and> a' \<in> Field r'}"
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notation Osum (infix "\<union>o" 60)
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lemma Field_Osum: "Field (r \<union>o r') = Field r \<union> Field r'"
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unfolding Osum_def Field_def by blast
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lemma Osum_wf:
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assumes FLD: "Field r Int Field r' = {}" and
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WF: "wf r" and WF': "wf r'"
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shows "wf (r Osum r')"
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unfolding wf_eq_minimal2 unfolding Field_Osum
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proof(intro allI impI, elim conjE)
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fix A assume *: "A \<subseteq> Field r \<union> Field r'" and **: "A \<noteq> {}"
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obtain B where B_def: "B = A Int Field r" by blast
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show "\<exists>a\<in>A. \<forall>a'\<in>A. (a', a) \<notin> r \<union>o r'"
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proof(cases "B = {}")
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assume Case1: "B \<noteq> {}"
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hence "B \<noteq> {} \<and> B \<le> Field r" using B_def by auto
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then obtain a where 1: "a \<in> B" and 2: "\<forall>a1 \<in> B. (a1,a) \<notin> r"
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using WF unfolding wf_eq_minimal2 by blast
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hence 3: "a \<in> Field r \<and> a \<notin> Field r'" using B_def FLD by auto
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(* *)
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have "\<forall>a1 \<in> A. (a1,a) \<notin> r Osum r'"
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proof(intro ballI)
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fix a1 assume **: "a1 \<in> A"
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{assume Case11: "a1 \<in> Field r"
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hence "(a1,a) \<notin> r" using B_def ** 2 by auto
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moreover
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have "(a1,a) \<notin> r'" using 3 by (auto simp add: Field_def)
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ultimately have "(a1,a) \<notin> r Osum r'"
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using 3 unfolding Osum_def by auto
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}
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moreover
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{assume Case12: "a1 \<notin> Field r"
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hence "(a1,a) \<notin> r" unfolding Field_def by auto
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moreover
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have "(a1,a) \<notin> r'" using 3 unfolding Field_def by auto
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ultimately have "(a1,a) \<notin> r Osum r'"
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using 3 unfolding Osum_def by auto
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}
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ultimately show "(a1,a) \<notin> r Osum r'" by blast
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qed
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thus ?thesis using 1 B_def by auto
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next
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assume Case2: "B = {}"
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hence 1: "A \<noteq> {} \<and> A \<le> Field r'" using * ** B_def by auto
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then obtain a' where 2: "a' \<in> A" and 3: "\<forall>a1' \<in> A. (a1',a') \<notin> r'"
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using WF' unfolding wf_eq_minimal2 by blast
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hence 4: "a' \<in> Field r' \<and> a' \<notin> Field r" using 1 FLD by blast
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(* *)
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have "\<forall>a1' \<in> A. (a1',a') \<notin> r Osum r'"
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proof(unfold Osum_def, auto simp add: 3)
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fix a1' assume "(a1', a') \<in> r"
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thus False using 4 unfolding Field_def by blast
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next
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fix a1' assume "a1' \<in> A" and "a1' \<in> Field r"
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thus False using Case2 B_def by auto
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qed
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thus ?thesis using 2 by blast
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qed
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qed
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lemma Osum_Refl:
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assumes FLD: "Field r Int Field r' = {}" and
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REFL: "Refl r" and REFL': "Refl r'"
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shows "Refl (r Osum r')"
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using assms
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unfolding refl_on_def Field_Osum unfolding Osum_def by blast
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lemma Osum_trans:
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assumes FLD: "Field r Int Field r' = {}" and
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TRANS: "trans r" and TRANS': "trans r'"
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shows "trans (r Osum r')"
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proof(unfold trans_def, auto)
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fix x y z assume *: "(x, y) \<in> r \<union>o r'" and **: "(y, z) \<in> r \<union>o r'"
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show "(x, z) \<in> r \<union>o r'"
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proof-
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{assume Case1: "(x,y) \<in> r"
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hence 1: "x \<in> Field r \<and> y \<in> Field r" unfolding Field_def by auto
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have ?thesis
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proof-
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{assume Case11: "(y,z) \<in> r"
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hence "(x,z) \<in> r" using Case1 TRANS trans_def[of r] by blast
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hence ?thesis unfolding Osum_def by auto
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}
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moreover
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{assume Case12: "(y,z) \<in> r'"
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hence "y \<in> Field r'" unfolding Field_def by auto
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hence False using FLD 1 by auto
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}
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moreover
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{assume Case13: "z \<in> Field r'"
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hence ?thesis using 1 unfolding Osum_def by auto
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}
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ultimately show ?thesis using ** unfolding Osum_def by blast
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qed
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}
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moreover
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{assume Case2: "(x,y) \<in> r'"
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hence 2: "x \<in> Field r' \<and> y \<in> Field r'" unfolding Field_def by auto
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have ?thesis
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proof-
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{assume Case21: "(y,z) \<in> r"
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hence "y \<in> Field r" unfolding Field_def by auto
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hence False using FLD 2 by auto
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}
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moreover
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{assume Case22: "(y,z) \<in> r'"
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hence "(x,z) \<in> r'" using Case2 TRANS' trans_def[of r'] by blast
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hence ?thesis unfolding Osum_def by auto
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}
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moreover
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{assume Case23: "y \<in> Field r"
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hence False using FLD 2 by auto
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}
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ultimately show ?thesis using ** unfolding Osum_def by blast
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qed
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}
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moreover
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{assume Case3: "x \<in> Field r \<and> y \<in> Field r'"
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have ?thesis
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proof-
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{assume Case31: "(y,z) \<in> r"
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hence "y \<in> Field r" unfolding Field_def by auto
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hence False using FLD Case3 by auto
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}
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moreover
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{assume Case32: "(y,z) \<in> r'"
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hence "z \<in> Field r'" unfolding Field_def by blast
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hence ?thesis unfolding Osum_def using Case3 by auto
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}
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moreover
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{assume Case33: "y \<in> Field r"
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hence False using FLD Case3 by auto
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}
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ultimately show ?thesis using ** unfolding Osum_def by blast
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qed
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}
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ultimately show ?thesis using * unfolding Osum_def by blast
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qed
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qed
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lemma Osum_Preorder:
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"\<lbrakk>Field r Int Field r' = {}; Preorder r; Preorder r'\<rbrakk> \<Longrightarrow> Preorder (r Osum r')"
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unfolding preorder_on_def using Osum_Refl Osum_trans by blast
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lemma Osum_antisym:
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assumes FLD: "Field r Int Field r' = {}" and
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AN: "antisym r" and AN': "antisym r'"
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shows "antisym (r Osum r')"
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proof(unfold antisym_def, auto)
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fix x y assume *: "(x, y) \<in> r \<union>o r'" and **: "(y, x) \<in> r \<union>o r'"
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show "x = y"
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proof-
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{assume Case1: "(x,y) \<in> r"
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hence 1: "x \<in> Field r \<and> y \<in> Field r" unfolding Field_def by auto
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have ?thesis
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proof-
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have "(y,x) \<in> r \<Longrightarrow> ?thesis"
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using Case1 AN antisym_def[of r] by blast
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moreover
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{assume "(y,x) \<in> r'"
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hence "y \<in> Field r'" unfolding Field_def by auto
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hence False using FLD 1 by auto
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}
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moreover
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have "x \<in> Field r' \<Longrightarrow> False" using FLD 1 by auto
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ultimately show ?thesis using ** unfolding Osum_def by blast
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qed
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}
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moreover
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{assume Case2: "(x,y) \<in> r'"
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hence 2: "x \<in> Field r' \<and> y \<in> Field r'" unfolding Field_def by auto
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have ?thesis
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proof-
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{assume "(y,x) \<in> r"
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hence "y \<in> Field r" unfolding Field_def by auto
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hence False using FLD 2 by auto
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}
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moreover
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have "(y,x) \<in> r' \<Longrightarrow> ?thesis"
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using Case2 AN' antisym_def[of r'] by blast
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moreover
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{assume "y \<in> Field r"
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hence False using FLD 2 by auto
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}
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ultimately show ?thesis using ** unfolding Osum_def by blast
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qed
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}
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moreover
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{assume Case3: "x \<in> Field r \<and> y \<in> Field r'"
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have ?thesis
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proof-
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{assume "(y,x) \<in> r"
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hence "y \<in> Field r" unfolding Field_def by auto
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hence False using FLD Case3 by auto
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}
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moreover
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{assume Case32: "(y,x) \<in> r'"
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hence "x \<in> Field r'" unfolding Field_def by blast
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hence False using FLD Case3 by auto
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}
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moreover
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have "\<not> y \<in> Field r" using FLD Case3 by auto
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ultimately show ?thesis using ** unfolding Osum_def by blast
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qed
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}
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ultimately show ?thesis using * unfolding Osum_def by blast
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qed
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qed
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lemma Osum_Partial_order:
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"\<lbrakk>Field r Int Field r' = {}; Partial_order r; Partial_order r'\<rbrakk> \<Longrightarrow>
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Partial_order (r Osum r')"
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unfolding partial_order_on_def using Osum_Preorder Osum_antisym by blast
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lemma Osum_Total:
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assumes FLD: "Field r Int Field r' = {}" and
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TOT: "Total r" and TOT': "Total r'"
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shows "Total (r Osum r')"
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using assms
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unfolding total_on_def Field_Osum unfolding Osum_def by blast
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lemma Osum_Linear_order:
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"\<lbrakk>Field r Int Field r' = {}; Linear_order r; Linear_order r'\<rbrakk> \<Longrightarrow>
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Linear_order (r Osum r')"
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unfolding linear_order_on_def using Osum_Partial_order Osum_Total by blast
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lemma Osum_minus_Id1:
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assumes "r \<le> Id"
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shows "(r Osum r') - Id \<le> (r' - Id) \<union> (Field r \<times> Field r')"
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proof-
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let ?Left = "(r Osum r') - Id"
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let ?Right = "(r' - Id) \<union> (Field r \<times> Field r')"
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{fix a::'a and b assume *: "(a,b) \<notin> Id"
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{assume "(a,b) \<in> r"
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with * have False using assms by auto
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}
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moreover
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{assume "(a,b) \<in> r'"
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with * have "(a,b) \<in> r' - Id" by auto
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}
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ultimately
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have "(a,b) \<in> ?Left \<Longrightarrow> (a,b) \<in> ?Right"
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unfolding Osum_def by auto
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}
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thus ?thesis by auto
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qed
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lemma Osum_minus_Id2:
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assumes "r' \<le> Id"
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shows "(r Osum r') - Id \<le> (r - Id) \<union> (Field r \<times> Field r')"
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proof-
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let ?Left = "(r Osum r') - Id"
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let ?Right = "(r - Id) \<union> (Field r \<times> Field r')"
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{fix a::'a and b assume *: "(a,b) \<notin> Id"
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{assume "(a,b) \<in> r'"
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with * have False using assms by auto
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}
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moreover
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{assume "(a,b) \<in> r"
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with * have "(a,b) \<in> r - Id" by auto
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}
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ultimately
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have "(a,b) \<in> ?Left \<Longrightarrow> (a,b) \<in> ?Right"
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unfolding Osum_def by auto
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}
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thus ?thesis by auto
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qed
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lemma Osum_minus_Id:
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assumes TOT: "Total r" and TOT': "Total r'" and
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NID: "\<not> (r \<le> Id)" and NID': "\<not> (r' \<le> Id)"
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shows "(r Osum r') - Id \<le> (r - Id) Osum (r' - Id)"
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proof-
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{fix a a' assume *: "(a,a') \<in> (r Osum r')" and **: "a \<noteq> a'"
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have "(a,a') \<in> (r - Id) Osum (r' - Id)"
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proof-
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{assume "(a,a') \<in> r \<or> (a,a') \<in> r'"
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with ** have ?thesis unfolding Osum_def by auto
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}
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moreover
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{assume "a \<in> Field r \<and> a' \<in> Field r'"
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hence "a \<in> Field(r - Id) \<and> a' \<in> Field (r' - Id)"
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using assms Total_Id_Field by blast
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hence ?thesis unfolding Osum_def by auto
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}
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ultimately show ?thesis using * unfolding Osum_def by blast
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qed
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}
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thus ?thesis by(auto simp add: Osum_def)
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qed
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lemma wf_Int_Times:
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assumes "A Int B = {}"
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shows "wf(A \<times> B)"
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proof(unfold wf_def, auto)
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fix P x
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assume *: "\<forall>x. (\<forall>y. y \<in> A \<and> x \<in> B \<longrightarrow> P y) \<longrightarrow> P x"
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moreover have "\<forall>y \<in> A. P y" using assms * by blast
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ultimately show "P x" using * by (case_tac "x \<in> B", auto)
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qed
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lemma Osum_wf_Id:
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assumes TOT: "Total r" and TOT': "Total r'" and
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FLD: "Field r Int Field r' = {}" and
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WF: "wf(r - Id)" and WF': "wf(r' - Id)"
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shows "wf ((r Osum r') - Id)"
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proof(cases "r \<le> Id \<or> r' \<le> Id")
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assume Case1: "\<not>(r \<le> Id \<or> r' \<le> Id)"
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have "Field(r - Id) Int Field(r' - Id) = {}"
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using FLD mono_Field[of "r - Id" r] mono_Field[of "r' - Id" r']
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Diff_subset[of r Id] Diff_subset[of r' Id] by blast
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thus ?thesis
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using Case1 Osum_minus_Id[of r r'] assms Osum_wf[of "r - Id" "r' - Id"]
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wf_subset[of "(r - Id) \<union>o (r' - Id)" "(r Osum r') - Id"] by auto
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next
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have 1: "wf(Field r \<times> Field r')"
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using FLD by (auto simp add: wf_Int_Times)
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assume Case2: "r \<le> Id \<or> r' \<le> Id"
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moreover
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{assume Case21: "r \<le> Id"
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hence "(r Osum r') - Id \<le> (r' - Id) \<union> (Field r \<times> Field r')"
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using Osum_minus_Id1[of r r'] by simp
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moreover
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{have "Domain(Field r \<times> Field r') Int Range(r' - Id) = {}"
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using FLD unfolding Field_def by blast
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hence "wf((r' - Id) \<union> (Field r \<times> Field r'))"
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using 1 WF' wf_Un[of "Field r \<times> Field r'" "r' - Id"]
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by (auto simp add: Un_commute)
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}
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ultimately have ?thesis by (auto simp add: wf_subset)
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}
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moreover
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{assume Case22: "r' \<le> Id"
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hence "(r Osum r') - Id \<le> (r - Id) \<union> (Field r \<times> Field r')"
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using Osum_minus_Id2[of r' r] by simp
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|
352 |
moreover
|
|
353 |
{have "Range(Field r \<times> Field r') Int Domain(r - Id) = {}"
|
|
354 |
using FLD unfolding Field_def by blast
|
|
355 |
hence "wf((r - Id) \<union> (Field r \<times> Field r'))"
|
|
356 |
using 1 WF wf_Un[of "r - Id" "Field r \<times> Field r'"]
|
|
357 |
by (auto simp add: Un_commute)
|
|
358 |
}
|
|
359 |
ultimately have ?thesis by (auto simp add: wf_subset)
|
|
360 |
}
|
|
361 |
ultimately show ?thesis by blast
|
|
362 |
qed
|
|
363 |
|
|
364 |
lemma Osum_Well_order:
|
|
365 |
assumes FLD: "Field r Int Field r' = {}" and
|
|
366 |
WELL: "Well_order r" and WELL': "Well_order r'"
|
|
367 |
shows "Well_order (r Osum r')"
|
|
368 |
proof-
|
|
369 |
have "Total r \<and> Total r'" using WELL WELL'
|
|
370 |
by (auto simp add: order_on_defs)
|
|
371 |
thus ?thesis using assms unfolding well_order_on_def
|
|
372 |
using Osum_Linear_order Osum_wf_Id by blast
|
|
373 |
qed
|
|
374 |
|
|
375 |
end
|
|
376 |
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