--- a/src/HOL/Cardinals/Order_Relation_More_FP.thy Fri Jan 17 09:52:19 2014 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,204 +0,0 @@
-(* Title: HOL/Cardinals/Order_Relation_More_FP.thy
- Author: Andrei Popescu, TU Muenchen
- Copyright 2012
-
-Basics on order-like relations (FP).
-*)
-
-header {* Basics on Order-Like Relations (FP) *}
-
-theory Order_Relation_More_FP
-imports 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. *}
-
-
-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 (auto 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 auto
-
-
-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 rel \<Rightarrow> 'a \<Rightarrow> 'a set"
-where "under r a \<equiv> {b. (b,a) \<in> r}"
-
-definition underS::"'a rel \<Rightarrow> 'a \<Rightarrow> 'a set"
-where "underS r a \<equiv> {b. b \<noteq> a \<and> (b,a) \<in> r}"
-
-definition Under::"'a rel \<Rightarrow> 'a set \<Rightarrow> 'a set"
-where "Under r A \<equiv> {b \<in> Field r. \<forall>a \<in> A. (b,a) \<in> r}"
-
-definition UnderS::"'a rel \<Rightarrow> 'a set \<Rightarrow> 'a set"
-where "UnderS r A \<equiv> {b \<in> Field r. \<forall>a \<in> A. b \<noteq> a \<and> (b,a) \<in> r}"
-
-definition above::"'a rel \<Rightarrow> 'a \<Rightarrow> 'a set"
-where "above r a \<equiv> {b. (a,b) \<in> r}"
-
-definition aboveS::"'a rel \<Rightarrow> 'a \<Rightarrow> 'a set"
-where "aboveS r a \<equiv> {b. b \<noteq> a \<and> (a,b) \<in> r}"
-
-definition Above::"'a rel \<Rightarrow> 'a set \<Rightarrow> 'a set"
-where "Above r A \<equiv> {b \<in> Field r. \<forall>a \<in> A. (a,b) \<in> r}"
-
-definition AboveS::"'a rel \<Rightarrow> 'a set \<Rightarrow> 'a set"
-where "AboveS r 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 r a \<le> under r a"
-by(auto simp add: underS_def under_def)
-
-
-lemma underS_notIn: "a \<notin> underS r a"
-by(simp add: underS_def)
-
-
-lemma Refl_under_in: "\<lbrakk>Refl r; a \<in> Field r\<rbrakk> \<Longrightarrow> a \<in> under r a"
-by(simp add: refl_on_def under_def)
-
-
-lemma AboveS_disjoint: "A Int (AboveS r A) = {}"
-by(auto simp add: AboveS_def)
-
-lemma in_AboveS_underS: "a \<in> Field r \<Longrightarrow> a \<in> AboveS r (underS r a)"
-by(auto simp add: AboveS_def underS_def)
-
-lemma Refl_under_underS:
- assumes "Refl r" "a \<in> Field r"
- shows "under r a = underS r 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 r a = {}"
-by (auto simp: Field_def underS_def)
-
-lemma under_Field: "under r a \<le> Field r"
-by(unfold under_def Field_def, auto)
-
-lemma underS_Field: "underS r a \<le> Field r"
-by(unfold underS_def Field_def, auto)
-
-lemma underS_Field2:
-"a \<in> Field r \<Longrightarrow> underS r a < Field r"
-using underS_notIn underS_Field by fast
-
-lemma underS_Field3:
-"Field r \<noteq> {} \<Longrightarrow> underS r a < Field r"
-by(cases "a \<in> Field r", simp add: underS_Field2, auto simp add: underS_empty)
-
-lemma AboveS_Field: "AboveS r 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 r a \<le> under r 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 r a \<le> underS r 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 r a \<le> underS r b) = ((a,b) \<in> r)"
-proof
- assume "(a,b) \<in> r"
- thus "underS r a \<le> underS r b" using LO
- by (simp add: order_on_defs underS_incr)
-next
- assume *: "underS r a \<le> underS r 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 r a" unfolding underS_def by blast
- hence "b \<in> underS r 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
-
-end