diff -r cfb21e03fe2a -r d64a4ef26edb src/HOL/Cardinals/Order_Relation_More_Base.thy
--- a/src/HOL/Cardinals/Order_Relation_More_Base.thy	Thu Dec 05 17:52:12 2013 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,286 +0,0 @@
-(*  Title:      HOL/Cardinals/Order_Relation_More_Base.thy
-    Author:     Andrei Popescu, TU Muenchen
-    Copyright   2012
-
-Basics on order-like relations (base).
-*)
-
-header {* Basics on Order-Like Relations (Base) *}
-
-theory Order_Relation_More_Base
-imports "~~/src/HOL/Library/Order_Relation"
-begin
-
-
-text{* In this section, we develop basic concepts and results pertaining
-to order-like relations, i.e., to reflexive and/or transitive and/or symmetric and/or
-total relations.  The development is placed on top of the definitions
-from the theory @{text "Order_Relation"}.  We also
-further define upper and lower bounds operators. *}
-
-
-locale rel = fixes r :: "'a rel"
-
-text{* The following context encompasses all this section, except
-for its last subsection. In other words, for the rest of this section except its last
-subsection, we consider a fixed relation @{text "r"}. *}
-
-context rel
-begin
-
-
-subsection {* Auxiliaries *}
-
-
-lemma refl_on_domain:
-"\<lbrakk>refl_on A r; (a,b) : r\<rbrakk> \<Longrightarrow> a \<in> A \<and> b \<in> A"
-by(auto simp add: refl_on_def)
-
-
-corollary well_order_on_domain:
-"\<lbrakk>well_order_on A r; (a,b) \<in> r\<rbrakk> \<Longrightarrow> a \<in> A \<and> b \<in> A"
-by(simp add: refl_on_domain order_on_defs)
-
-
-lemma well_order_on_Field:
-"well_order_on A r \<Longrightarrow> A = Field r"
-by(auto simp add: refl_on_def Field_def order_on_defs)
-
-
-lemma well_order_on_Well_order:
-"well_order_on A r \<Longrightarrow> A = Field r \<and> Well_order r"
-using well_order_on_Field by simp
-
-
-lemma Total_subset_Id:
-assumes TOT: "Total r" and SUB: "r \<le> Id"
-shows "r = {} \<or> (\<exists>a. r = {(a,a)})"
-proof-
-  {assume "r \<noteq> {}"
-   then obtain a b where 1: "(a,b) \<in> r" by fast
-   hence "a = b" using SUB by blast
-   hence 2: "(a,a) \<in> r" using 1 by simp
-   {fix c d assume "(c,d) \<in> r"
-    hence "{a,c,d} \<le> Field r" using 1 unfolding Field_def by blast
-    hence "((a,c) \<in> r \<or> (c,a) \<in> r \<or> a = c) \<and>
-           ((a,d) \<in> r \<or> (d,a) \<in> r \<or> a = d)"
-    using TOT unfolding total_on_def by blast
-    hence "a = c \<and> a = d" using SUB by blast
-   }
-   hence "r \<le> {(a,a)}" by auto
-   with 2 have "\<exists>a. r = {(a,a)}" by blast
-  }
-  thus ?thesis by blast
-qed
-
-
-lemma Linear_order_in_diff_Id:
-assumes LI: "Linear_order r" and
-        IN1: "a \<in> Field r" and IN2: "b \<in> Field r"
-shows "((a,b) \<in> r) = ((b,a) \<notin> r - Id)"
-using assms unfolding order_on_defs total_on_def antisym_def Id_def refl_on_def by force
-
-
-subsection {* The upper and lower bounds operators  *}
-
-
-text{* Here we define upper (``above") and lower (``below") bounds operators.
-We think of @{text "r"} as a {\em non-strict} relation.  The suffix ``S"
-at the names of some operators indicates that the bounds are strict -- e.g.,
-@{text "underS a"} is the set of all strict lower bounds of @{text "a"} (w.r.t. @{text "r"}).
-Capitalization of the first letter in the name reminds that the operator acts on sets, rather
-than on individual elements. *}
-
-definition under::"'a \<Rightarrow> 'a set"
-where "under a \<equiv> {b. (b,a) \<in> r}"
-
-definition underS::"'a \<Rightarrow> 'a set"
-where "underS a \<equiv> {b. b \<noteq> a \<and> (b,a) \<in> r}"
-
-definition Under::"'a set \<Rightarrow> 'a set"
-where "Under A \<equiv> {b \<in> Field r. \<forall>a \<in> A. (b,a) \<in> r}"
-
-definition UnderS::"'a set \<Rightarrow> 'a set"
-where "UnderS A \<equiv> {b \<in> Field r. \<forall>a \<in> A. b \<noteq> a \<and> (b,a) \<in> r}"
-
-definition above::"'a \<Rightarrow> 'a set"
-where "above a \<equiv> {b. (a,b) \<in> r}"
-
-definition aboveS::"'a \<Rightarrow> 'a set"
-where "aboveS a \<equiv> {b. b \<noteq> a \<and> (a,b) \<in> r}"
-
-definition Above::"'a set \<Rightarrow> 'a set"
-where "Above A \<equiv> {b \<in> Field r. \<forall>a \<in> A. (a,b) \<in> r}"
-
-definition AboveS::"'a set \<Rightarrow> 'a set"
-where "AboveS A \<equiv> {b \<in> Field r. \<forall>a \<in> A. b \<noteq> a \<and> (a,b) \<in> r}"
-(*  *)
-
-text{* Note:  In the definitions of @{text "Above[S]"} and @{text "Under[S]"},
-  we bounded comprehension by @{text "Field r"} in order to properly cover
-  the case of @{text "A"} being empty. *}
-
-
-lemma UnderS_subset_Under: "UnderS A \<le> Under A"
-by(auto simp add: UnderS_def Under_def)
-
-
-lemma underS_subset_under: "underS a \<le> under a"
-by(auto simp add: underS_def under_def)
-
-
-lemma underS_notIn: "a \<notin> underS a"
-by(simp add: underS_def)
-
-
-lemma Refl_under_in: "\<lbrakk>Refl r; a \<in> Field r\<rbrakk> \<Longrightarrow> a \<in> under a"
-by(simp add: refl_on_def under_def)
-
-
-lemma AboveS_disjoint: "A Int (AboveS A) = {}"
-by(auto simp add: AboveS_def)
-
-
-lemma in_AboveS_underS: "a \<in> Field r \<Longrightarrow> a \<in> AboveS (underS a)"
-by(auto simp add: AboveS_def underS_def)
-
-
-lemma Refl_under_underS:
-assumes "Refl r" "a \<in> Field r"
-shows "under a = underS a \<union> {a}"
-unfolding under_def underS_def
-using assms refl_on_def[of _ r] by fastforce
-
-
-lemma underS_empty: "a \<notin> Field r \<Longrightarrow> underS a = {}"
-by (auto simp: Field_def underS_def)
-
-
-lemma under_Field: "under a \<le> Field r"
-by(unfold under_def Field_def, auto)
-
-
-lemma underS_Field: "underS a \<le> Field r"
-by(unfold underS_def Field_def, auto)
-
-
-lemma underS_Field2:
-"a \<in> Field r \<Longrightarrow> underS a < Field r"
-using assms underS_notIn underS_Field by blast
-
-
-lemma underS_Field3:
-"Field r \<noteq> {} \<Longrightarrow> underS a < Field r"
-by(cases "a \<in> Field r", simp add: underS_Field2, auto simp add: underS_empty)
-
-
-lemma Under_Field: "Under A \<le> Field r"
-by(unfold Under_def Field_def, auto)
-
-
-lemma UnderS_Field: "UnderS A \<le> Field r"
-by(unfold UnderS_def Field_def, auto)
-
-
-lemma AboveS_Field: "AboveS A \<le> Field r"
-by(unfold AboveS_def Field_def, auto)
-
-
-lemma under_incr:
-assumes TRANS: "trans r" and REL: "(a,b) \<in> r"
-shows "under a \<le> under b"
-proof(unfold under_def, auto)
-  fix x assume "(x,a) \<in> r"
-  with REL TRANS trans_def[of r]
-  show "(x,b) \<in> r" by blast
-qed
-
-
-lemma underS_incr:
-assumes TRANS: "trans r" and ANTISYM: "antisym r" and
-        REL: "(a,b) \<in> r"
-shows "underS a \<le> underS b"
-proof(unfold underS_def, auto)
-  assume *: "b \<noteq> a" and **: "(b,a) \<in> r"
-  with ANTISYM antisym_def[of r] REL
-  show False by blast
-next
-  fix x assume "x \<noteq> a" "(x,a) \<in> r"
-  with REL TRANS trans_def[of r]
-  show "(x,b) \<in> r" by blast
-qed
-
-
-lemma underS_incl_iff:
-assumes LO: "Linear_order r" and
-        INa: "a \<in> Field r" and INb: "b \<in> Field r"
-shows "(underS a \<le> underS b) = ((a,b) \<in> r)"
-proof
-  assume "(a,b) \<in> r"
-  thus "underS a \<le> underS b" using LO
-  by (simp add: order_on_defs underS_incr)
-next
-  assume *: "underS a \<le> underS b"
-  {assume "a = b"
-   hence "(a,b) \<in> r" using assms
-   by (simp add: order_on_defs refl_on_def)
-  }
-  moreover
-  {assume "a \<noteq> b \<and> (b,a) \<in> r"
-   hence "b \<in> underS a" unfolding underS_def by blast
-   hence "b \<in> underS b" using * by blast
-   hence False by (simp add: underS_notIn)
-  }
-  ultimately
-  show "(a,b) \<in> r" using assms
-  order_on_defs[of "Field r" r] total_on_def[of "Field r" r] by blast
-qed
-
-
-lemma under_Under_trans:
-assumes TRANS: "trans r" and
-        IN1: "a \<in> under b" and IN2: "b \<in> Under C"
-shows "a \<in> Under C"
-proof-
-  have "(a,b) \<in> r \<and> (\<forall>c \<in> C. (b,c) \<in> r)"
-  using IN1 IN2 under_def Under_def by blast
-  hence "\<forall>c \<in> C. (a,c) \<in> r"
-  using TRANS trans_def[of r] by blast
-  moreover
-  have "a \<in> Field r" using IN1 unfolding Field_def under_def by blast
-  ultimately
-  show ?thesis unfolding Under_def by blast
-qed
-
-
-lemma under_UnderS_trans:
-assumes TRANS: "trans r" and ANTISYM: "antisym r" and
-        IN1: "a \<in> under b" and IN2: "b \<in> UnderS C"
-shows "a \<in> UnderS C"
-proof-
-  from IN2 have "b \<in> Under C"
-  using UnderS_subset_Under[of C] by blast
-  with assms under_Under_trans
-  have "a \<in> Under C" by blast
-  (*  *)
-  moreover
-  have "a \<notin> C"
-  proof
-    assume *: "a \<in> C"
-    have 1: "(a,b) \<in> r"
-    using IN1 under_def[of b] by auto
-    have "\<forall>c \<in> C. b \<noteq> c \<and> (b,c) \<in> r"
-    using IN2 UnderS_def[of C] by blast
-    with * have "b \<noteq> a \<and> (b,a) \<in> r" by blast
-    with 1 ANTISYM antisym_def[of r]
-    show False by blast
-  qed
-  (*  *)
-  ultimately
-  show ?thesis unfolding UnderS_def Under_def by fast
-qed
-
-
-end  (* context rel *)
-
-end