(*  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_Id_Field:

assumes TOT: "Total r" and NID: "\<not> (r <= Id)"

shows "Field r = Field(r - Id)"

using mono_Field[of "r - Id" r] Diff_subset[of r Id]

proof(auto)

  have "r \<noteq> {}" using NID by fast

  then obtain b and c where "b \<noteq> c \<and> (b,c) \<in> r" using NID by fast

  hence 1: "b \<noteq> c \<and> {b,c} \<le> Field r" by (auto simp: Field_def)

  (*  *)

  fix a assume *: "a \<in> Field r"

  obtain d where 2: "d \<in> Field r" and 3: "d \<noteq> a"

  using * 1 by auto

  hence "(a,d) \<in> r \<or> (d,a) \<in> r" using * TOT

  by (simp add: total_on_def)

  thus "a \<in> Field(r - Id)" using 3 unfolding Field_def by blast

qed

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