author | blanchet |
Wed, 24 Sep 2014 15:45:55 +0200 | |
changeset 58425 | 246985c6b20b |
parent 58127 | b7cab82f488e |
child 58889 | 5b7a9633cfa8 |
permissions | -rw-r--r-- |
54545
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effectively reverted d25fc4c0ff62, to avoid quasi-cyclic dependencies with HOL-Cardinals and minimize BNF dependencies
blanchet
parents:
54482
diff
changeset
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(* Title: HOL/Cardinals/Order_Union.thy |
52184 | 2 |
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 Order_Relation |
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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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compile (importing 'Metis' or 'Main' would have been an alternative)
blanchet
parents:
54545
diff
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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 |
36 |
(* *) |
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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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55021
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compile (importing 'Metis' or 'Main' would have been an alternative)
blanchet
parents:
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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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58127
b7cab82f488e
renamed '(BNF_)Constructions_on_Wellorders' to '(BNF_)Wellorder_Constructions'
blanchet
parents:
55027
diff
changeset
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using assms |
52184 | 82 |
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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||
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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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54482 | 302 |
ultimately show ?thesis using * unfolding Osum_def by fast |
52184 | 303 |
qed |
304 |
} |
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thus ?thesis by(auto simp add: Osum_def) |
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qed |
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307 |
||
308 |
lemma wf_Int_Times: |
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309 |
assumes "A Int B = {}" |
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310 |
shows "wf(A \<times> B)" |
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54482 | 311 |
unfolding wf_def using assms by blast |
52184 | 312 |
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313 |
lemma Osum_wf_Id: |
|
314 |
assumes TOT: "Total r" and TOT': "Total r'" and |
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315 |
FLD: "Field r Int Field r' = {}" and |
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316 |
WF: "wf(r - Id)" and WF': "wf(r' - Id)" |
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317 |
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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320 |
have "Field(r - Id) Int Field(r' - Id) = {}" |
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321 |
using FLD mono_Field[of "r - Id" r] mono_Field[of "r' - Id" r'] |
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322 |
Diff_subset[of r Id] Diff_subset[of r' Id] by blast |
|
323 |
thus ?thesis |
|
324 |
using Case1 Osum_minus_Id[of r r'] assms Osum_wf[of "r - Id" "r' - Id"] |
|
325 |
wf_subset[of "(r - Id) \<union>o (r' - Id)" "(r Osum r') - Id"] by auto |
|
326 |
next |
|
327 |
have 1: "wf(Field r \<times> Field r')" |
|
328 |
using FLD by (auto simp add: wf_Int_Times) |
|
329 |
assume Case2: "r \<le> Id \<or> r' \<le> Id" |
|
330 |
moreover |
|
331 |
{assume Case21: "r \<le> Id" |
|
332 |
hence "(r Osum r') - Id \<le> (r' - Id) \<union> (Field r \<times> Field r')" |
|
333 |
using Osum_minus_Id1[of r r'] by simp |
|
334 |
moreover |
|
335 |
{have "Domain(Field r \<times> Field r') Int Range(r' - Id) = {}" |
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336 |
using FLD unfolding Field_def by blast |
|
337 |
hence "wf((r' - Id) \<union> (Field r \<times> Field r'))" |
|
338 |
using 1 WF' wf_Un[of "Field r \<times> Field r'" "r' - Id"] |
|
339 |
by (auto simp add: Un_commute) |
|
340 |
} |
|
55021
e40090032de9
compile (importing 'Metis' or 'Main' would have been an alternative)
blanchet
parents:
54545
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changeset
|
341 |
ultimately have ?thesis using wf_subset by blast |
52184 | 342 |
} |
343 |
moreover |
|
344 |
{assume Case22: "r' \<le> Id" |
|
345 |
hence "(r Osum r') - Id \<le> (r - Id) \<union> (Field r \<times> Field r')" |
|
346 |
using Osum_minus_Id2[of r' r] by simp |
|
347 |
moreover |
|
348 |
{have "Range(Field r \<times> Field r') Int Domain(r - Id) = {}" |
|
349 |
using FLD unfolding Field_def by blast |
|
350 |
hence "wf((r - Id) \<union> (Field r \<times> Field r'))" |
|
351 |
using 1 WF wf_Un[of "r - Id" "Field r \<times> Field r'"] |
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352 |
by (auto simp add: Un_commute) |
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353 |
} |
|
55021
e40090032de9
compile (importing 'Metis' or 'Main' would have been an alternative)
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parents:
54545
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changeset
|
354 |
ultimately have ?thesis using wf_subset by blast |
52184 | 355 |
} |
356 |
ultimately show ?thesis by blast |
|
357 |
qed |
|
358 |
||
359 |
lemma Osum_Well_order: |
|
360 |
assumes FLD: "Field r Int Field r' = {}" and |
|
361 |
WELL: "Well_order r" and WELL': "Well_order r'" |
|
362 |
shows "Well_order (r Osum r')" |
|
363 |
proof- |
|
364 |
have "Total r \<and> Total r'" using WELL WELL' |
|
365 |
by (auto simp add: order_on_defs) |
|
366 |
thus ?thesis using assms unfolding well_order_on_def |
|
367 |
using Osum_Linear_order Osum_wf_Id by blast |
|
368 |
qed |
|
369 |
||
370 |
end |