--- a/src/HOL/Cardinals/Constructions_on_Wellorders.thy Tue May 28 19:59:54 2013 +0200
+++ b/src/HOL/Cardinals/Constructions_on_Wellorders.thy Tue May 28 20:00:29 2013 +0200
@@ -188,376 +188,7 @@
qed
qed
-
-subsection {* Ordinal-like sum of two (disjoint) well-orders *}
-
-text{* This is roughly obtained by ``concatenating" the two well-orders -- thus, all elements
-of the first will be smaller than all elements of the second. This construction
-only makes sense if the fields of the two well-order relations are disjoint. *}
-
-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 Osum r') = Field r \<union> Field r'"
-unfolding Osum_def Field_def by blast
-
-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 (* Need first unfold Field_Osum, only then Osum_def *)
-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_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 metis
- 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 metis
- 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_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 mem_Sigma_iff, intro impI allI)
- 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") blast+
-qed
-
-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_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
+(* More facts on ordinal sum: *)
lemma Osum_embed:
assumes FLD: "Field r Int Field r' = {}" and