src/HOL/Cardinals/Wellorder_Embedding.thy

+(*  Title:      HOL/Cardinals/Wellorder_Embedding.thy
+    Author:     Andrei Popescu, TU Muenchen
+    Copyright   2012
+
+Well-order embeddings.
+*)
+
+header {* Well-Order Embeddings *}
+
+theory Wellorder_Embedding
+imports Wellorder_Embedding_Base Fun_More Wellorder_Relation
+begin
+
+
+subsection {* Auxiliaries *}
+
+lemma UNION_bij_betw_ofilter:
+assumes WELL: "Well_order r" and
+        OF: "\<And> i. i \<in> I \<Longrightarrow> ofilter r (A i)" and
+       BIJ: "\<And> i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)"
+shows "bij_betw f (\<Union> i \<in> I. A i) (\<Union> i \<in> I. A' i)"
+proof-
+  have "wo_rel r" using WELL by (simp add: wo_rel_def)
+  hence "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i"
+  using wo_rel.ofilter_linord[of r] OF by blast
+  with WELL BIJ show ?thesis
+  by (auto simp add: bij_betw_UNION_chain)
+qed
+
+
+subsection {* (Well-order) embeddings, strict embeddings, isomorphisms and order-compatible
+functions *}
+
+lemma embed_halfcong:
+assumes EQ: "\<And> a. a \<in> Field r \<Longrightarrow> f a = g a" and
+        EMB: "embed r r' f"
+shows "embed r r' g"
+proof(unfold embed_def, auto)
+  fix a assume *: "a \<in> Field r"
+  hence "bij_betw f (under r a) (under r' (f a))"
+  using EMB unfolding embed_def by simp
+  moreover
+  {have "under r a \<le> Field r"
+   by (auto simp add: rel.under_Field)
+   hence "\<And> b. b \<in> under r a \<Longrightarrow> f b = g b"
+   using EQ by blast
+  }
+  moreover have "f a = g a" using * EQ by auto
+  ultimately show "bij_betw g (under r a) (under r' (g a))"
+  using bij_betw_cong[of "under r a" f g "under r' (f a)"] by auto
+qed
+
+lemma embed_cong[fundef_cong]:
+assumes "\<And> a. a \<in> Field r \<Longrightarrow> f a = g a"
+shows "embed r r' f = embed r r' g"
+using assms embed_halfcong[of r f g r']
+            embed_halfcong[of r g f r'] by auto
+
+lemma embedS_cong[fundef_cong]:
+assumes "\<And> a. a \<in> Field r \<Longrightarrow> f a = g a"
+shows "embedS r r' f = embedS r r' g"
+unfolding embedS_def using assms
+embed_cong[of r f g r'] bij_betw_cong[of "Field r" f g "Field r'"] by blast
+
+lemma iso_cong[fundef_cong]:
+assumes "\<And> a. a \<in> Field r \<Longrightarrow> f a = g a"
+shows "iso r r' f = iso r r' g"
+unfolding iso_def using assms
+embed_cong[of r f g r'] bij_betw_cong[of "Field r" f g "Field r'"] by blast
+
+lemma id_compat: "compat r r id"
+by(auto simp add: id_def compat_def)
+
+lemma comp_compat:
+"\<lbrakk>compat r r' f; compat r' r'' f'\<rbrakk> \<Longrightarrow> compat r r'' (f' o f)"
+by(auto simp add: comp_def compat_def)
+
+corollary one_set_greater:
+"(\<exists>f::'a \<Rightarrow> 'a'. f ` A \<le> A' \<and> inj_on f A) \<or> (\<exists>g::'a' \<Rightarrow> 'a. g ` A' \<le> A \<and> inj_on g A')"
+proof-
+  obtain r where "well_order_on A r" by (fastforce simp add: well_order_on)
+  hence 1: "A = Field r \<and> Well_order r"
+  using rel.well_order_on_Well_order by auto
+  obtain r' where 2: "well_order_on A' r'" by (fastforce simp add: well_order_on)
+  hence 2: "A' = Field r' \<and> Well_order r'"
+  using rel.well_order_on_Well_order by auto
+  hence "(\<exists>f. embed r r' f) \<or> (\<exists>g. embed r' r g)"
+  using 1 2 by (auto simp add: wellorders_totally_ordered)
+  moreover
+  {fix f assume "embed r r' f"
+   hence "f`A \<le> A' \<and> inj_on f A"
+   using 1 2 by (auto simp add: embed_Field embed_inj_on)
+  }
+  moreover
+  {fix g assume "embed r' r g"
+   hence "g`A' \<le> A \<and> inj_on g A'"
+   using 1 2 by (auto simp add: embed_Field embed_inj_on)
+  }
+  ultimately show ?thesis by blast
+qed
+
+corollary one_type_greater:
+"(\<exists>f::'a \<Rightarrow> 'a'. inj f) \<or> (\<exists>g::'a' \<Rightarrow> 'a. inj g)"
+using one_set_greater[of UNIV UNIV] by auto
+
+
+subsection {* Uniqueness of embeddings  *}
+
+lemma comp_embedS:
+assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
+        and  EMB: "embedS r r' f" and EMB': "embedS r' r'' f'"
+shows "embedS r r'' (f' o f)"
+proof-
+  have "embed r' r'' f'" using EMB' unfolding embedS_def by simp
+  thus ?thesis using assms by (auto simp add: embedS_comp_embed)
+qed
+
+lemma iso_iff4:
+assumes WELL: "Well_order r"  and WELL': "Well_order r'"
+shows "iso r r' f = (embed r r' f \<and> embed r' r (inv_into (Field r) f))"
+using assms embed_bothWays_iso
+by(unfold iso_def, auto simp add: inv_into_Field_embed_bij_betw)
+
+lemma embed_embedS_iso:
+"embed r r' f = (embedS r r' f \<or> iso r r' f)"
+unfolding embedS_def iso_def by blast
+
+lemma not_embedS_iso:
+"\<not> (embedS r r' f \<and> iso r r' f)"
+unfolding embedS_def iso_def by blast
+
+lemma embed_embedS_iff_not_iso:
+assumes "embed r r' f"
+shows "embedS r r' f = (\<not> iso r r' f)"
+using assms unfolding embedS_def iso_def by blast
+
+lemma iso_inv_into:
+assumes WELL: "Well_order r" and ISO: "iso r r' f"
+shows "iso r' r (inv_into (Field r) f)"
+using assms unfolding iso_def
+using bij_betw_inv_into inv_into_Field_embed_bij_betw by blast
+
+lemma embedS_or_iso:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "(\<exists>g. embedS r r' g) \<or> (\<exists>h. embedS r' r h) \<or> (\<exists>f. iso r r' f)"
+proof-
+  {fix f assume *: "embed r r' f"
+   {assume "bij_betw f (Field r) (Field r')"
+    hence ?thesis using * by (auto simp add: iso_def)
+   }
+   moreover
+   {assume "\<not> bij_betw f (Field r) (Field r')"
+    hence ?thesis using * by (auto simp add: embedS_def)
+   }
+   ultimately have ?thesis by auto
+  }
+  moreover
+  {fix f assume *: "embed r' r f"
+   {assume "bij_betw f (Field r') (Field r)"
+    hence "iso r' r f" using * by (auto simp add: iso_def)
+    hence "iso r r' (inv_into (Field r') f)"
+    using WELL' by (auto simp add: iso_inv_into)
+    hence ?thesis by blast
+   }
+   moreover
+   {assume "\<not> bij_betw f (Field r') (Field r)"
+    hence ?thesis using * by (auto simp add: embedS_def)
+   }
+   ultimately have ?thesis by auto
+  }
+  ultimately show ?thesis using WELL WELL'
+  wellorders_totally_ordered[of r r'] by blast
+qed
+
+end