(*  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