(* Title: HOL/Lattice/Bounds.thy
ID: $Id$
Author: Markus Wenzel, TU Muenchen
*)
header {* Bounds *}
theory Bounds = Orders:
subsection {* Infimum and supremum *}
text {*
Given a partial order, we define infimum (greatest lower bound) and
supremum (least upper bound) wrt.\ @{text \<sqsubseteq>} for two and for any
number of elements.
*}
constdefs
is_inf :: "'a::partial_order \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
"is_inf x y inf \<equiv> inf \<sqsubseteq> x \<and> inf \<sqsubseteq> y \<and> (\<forall>z. z \<sqsubseteq> x \<and> z \<sqsubseteq> y \<longrightarrow> z \<sqsubseteq> inf)"
is_sup :: "'a::partial_order \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
"is_sup x y sup \<equiv> x \<sqsubseteq> sup \<and> y \<sqsubseteq> sup \<and> (\<forall>z. x \<sqsubseteq> z \<and> y \<sqsubseteq> z \<longrightarrow> sup \<sqsubseteq> z)"
is_Inf :: "'a::partial_order set \<Rightarrow> 'a \<Rightarrow> bool"
"is_Inf A inf \<equiv> (\<forall>x \<in> A. inf \<sqsubseteq> x) \<and> (\<forall>z. (\<forall>x \<in> A. z \<sqsubseteq> x) \<longrightarrow> z \<sqsubseteq> inf)"
is_Sup :: "'a::partial_order set \<Rightarrow> 'a \<Rightarrow> bool"
"is_Sup A sup \<equiv> (\<forall>x \<in> A. x \<sqsubseteq> sup) \<and> (\<forall>z. (\<forall>x \<in> A. x \<sqsubseteq> z) \<longrightarrow> sup \<sqsubseteq> z)"
text {*
These definitions entail the following basic properties of boundary
elements.
*}
lemma is_infI [intro?]: "inf \<sqsubseteq> x \<Longrightarrow> inf \<sqsubseteq> y \<Longrightarrow>
(\<And>z. z \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> y \<Longrightarrow> z \<sqsubseteq> inf) \<Longrightarrow> is_inf x y inf"
by (unfold is_inf_def) blast
lemma is_inf_greatest [elim?]:
"is_inf x y inf \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> y \<Longrightarrow> z \<sqsubseteq> inf"
by (unfold is_inf_def) blast
lemma is_inf_lower [elim?]:
"is_inf x y inf \<Longrightarrow> (inf \<sqsubseteq> x \<Longrightarrow> inf \<sqsubseteq> y \<Longrightarrow> C) \<Longrightarrow> C"
by (unfold is_inf_def) blast
lemma is_supI [intro?]: "x \<sqsubseteq> sup \<Longrightarrow> y \<sqsubseteq> sup \<Longrightarrow>
(\<And>z. x \<sqsubseteq> z \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> sup \<sqsubseteq> z) \<Longrightarrow> is_sup x y sup"
by (unfold is_sup_def) blast
lemma is_sup_least [elim?]:
"is_sup x y sup \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> sup \<sqsubseteq> z"
by (unfold is_sup_def) blast
lemma is_sup_upper [elim?]:
"is_sup x y sup \<Longrightarrow> (x \<sqsubseteq> sup \<Longrightarrow> y \<sqsubseteq> sup \<Longrightarrow> C) \<Longrightarrow> C"
by (unfold is_sup_def) blast
lemma is_InfI [intro?]: "(\<And>x. x \<in> A \<Longrightarrow> inf \<sqsubseteq> x) \<Longrightarrow>
(\<And>z. (\<forall>x \<in> A. z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> inf) \<Longrightarrow> is_Inf A inf"
by (unfold is_Inf_def) blast
lemma is_Inf_greatest [elim?]:
"is_Inf A inf \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> inf"
by (unfold is_Inf_def) blast
lemma is_Inf_lower [dest?]:
"is_Inf A inf \<Longrightarrow> x \<in> A \<Longrightarrow> inf \<sqsubseteq> x"
by (unfold is_Inf_def) blast
lemma is_SupI [intro?]: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> sup) \<Longrightarrow>
(\<And>z. (\<forall>x \<in> A. x \<sqsubseteq> z) \<Longrightarrow> sup \<sqsubseteq> z) \<Longrightarrow> is_Sup A sup"
by (unfold is_Sup_def) blast
lemma is_Sup_least [elim?]:
"is_Sup A sup \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> sup \<sqsubseteq> z"
by (unfold is_Sup_def) blast
lemma is_Sup_upper [dest?]:
"is_Sup A sup \<Longrightarrow> x \<in> A \<Longrightarrow> x \<sqsubseteq> sup"
by (unfold is_Sup_def) blast
subsection {* Duality *}
text {*
Infimum and supremum are dual to each other.
*}
theorem dual_inf [iff?]:
"is_inf (dual x) (dual y) (dual sup) = is_sup x y sup"
by (simp add: is_inf_def is_sup_def dual_all [symmetric] dual_leq)
theorem dual_sup [iff?]:
"is_sup (dual x) (dual y) (dual inf) = is_inf x y inf"
by (simp add: is_inf_def is_sup_def dual_all [symmetric] dual_leq)
theorem dual_Inf [iff?]:
"is_Inf (dual `` A) (dual sup) = is_Sup A sup"
by (simp add: is_Inf_def is_Sup_def dual_all [symmetric] dual_leq)
theorem dual_Sup [iff?]:
"is_Sup (dual `` A) (dual inf) = is_Inf A inf"
by (simp add: is_Inf_def is_Sup_def dual_all [symmetric] dual_leq)
subsection {* Uniqueness *}
text {*
Infima and suprema on partial orders are unique; this is mainly due
to anti-symmetry of the underlying relation.
*}
theorem is_inf_uniq: "is_inf x y inf \<Longrightarrow> is_inf x y inf' \<Longrightarrow> inf = inf'"
proof -
assume inf: "is_inf x y inf"
assume inf': "is_inf x y inf'"
show ?thesis
proof (rule leq_antisym)
from inf' show "inf \<sqsubseteq> inf'"
proof (rule is_inf_greatest)
from inf show "inf \<sqsubseteq> x" ..
from inf show "inf \<sqsubseteq> y" ..
qed
from inf show "inf' \<sqsubseteq> inf"
proof (rule is_inf_greatest)
from inf' show "inf' \<sqsubseteq> x" ..
from inf' show "inf' \<sqsubseteq> y" ..
qed
qed
qed
theorem is_sup_uniq: "is_sup x y sup \<Longrightarrow> is_sup x y sup' \<Longrightarrow> sup = sup'"
proof -
assume sup: "is_sup x y sup" and sup': "is_sup x y sup'"
have "dual sup = dual sup'"
proof (rule is_inf_uniq)
from sup show "is_inf (dual x) (dual y) (dual sup)" ..
from sup' show "is_inf (dual x) (dual y) (dual sup')" ..
qed
thus "sup = sup'" ..
qed
theorem is_Inf_uniq: "is_Inf A inf \<Longrightarrow> is_Inf A inf' \<Longrightarrow> inf = inf'"
proof -
assume inf: "is_Inf A inf"
assume inf': "is_Inf A inf'"
show ?thesis
proof (rule leq_antisym)
from inf' show "inf \<sqsubseteq> inf'"
proof (rule is_Inf_greatest)
fix x assume "x \<in> A"
from inf show "inf \<sqsubseteq> x" ..
qed
from inf show "inf' \<sqsubseteq> inf"
proof (rule is_Inf_greatest)
fix x assume "x \<in> A"
from inf' show "inf' \<sqsubseteq> x" ..
qed
qed
qed
theorem is_Sup_uniq: "is_Sup A sup \<Longrightarrow> is_Sup A sup' \<Longrightarrow> sup = sup'"
proof -
assume sup: "is_Sup A sup" and sup': "is_Sup A sup'"
have "dual sup = dual sup'"
proof (rule is_Inf_uniq)
from sup show "is_Inf (dual `` A) (dual sup)" ..
from sup' show "is_Inf (dual `` A) (dual sup')" ..
qed
thus "sup = sup'" ..
qed
subsection {* Related elements *}
text {*
The binary bound of related elements is either one of the argument.
*}
theorem is_inf_related [elim?]: "x \<sqsubseteq> y \<Longrightarrow> is_inf x y x"
proof -
assume "x \<sqsubseteq> y"
show ?thesis
proof
show "x \<sqsubseteq> x" ..
show "x \<sqsubseteq> y" .
fix z assume "z \<sqsubseteq> x" and "z \<sqsubseteq> y" show "z \<sqsubseteq> x" .
qed
qed
theorem is_sup_related [elim?]: "x \<sqsubseteq> y \<Longrightarrow> is_sup x y y"
proof -
assume "x \<sqsubseteq> y"
show ?thesis
proof
show "x \<sqsubseteq> y" .
show "y \<sqsubseteq> y" ..
fix z assume "x \<sqsubseteq> z" and "y \<sqsubseteq> z"
show "y \<sqsubseteq> z" .
qed
qed
subsection {* General versus binary bounds \label{sec:gen-bin-bounds} *}
text {*
General bounds of two-element sets coincide with binary bounds.
*}
theorem is_Inf_binary: "is_Inf {x, y} inf = is_inf x y inf"
proof -
let ?A = "{x, y}"
show ?thesis
proof
assume is_Inf: "is_Inf ?A inf"
show "is_inf x y inf"
proof
have "x \<in> ?A" by simp
with is_Inf show "inf \<sqsubseteq> x" ..
have "y \<in> ?A" by simp
with is_Inf show "inf \<sqsubseteq> y" ..
fix z assume zx: "z \<sqsubseteq> x" and zy: "z \<sqsubseteq> y"
from is_Inf show "z \<sqsubseteq> inf"
proof (rule is_Inf_greatest)
fix a assume "a \<in> ?A"
hence "a = x \<or> a = y" by blast
thus "z \<sqsubseteq> a"
proof
assume "a = x"
with zx show ?thesis by simp
next
assume "a = y"
with zy show ?thesis by simp
qed
qed
qed
next
assume is_inf: "is_inf x y inf"
show "is_Inf {x, y} inf"
proof
fix a assume "a \<in> ?A"
hence "a = x \<or> a = y" by blast
thus "inf \<sqsubseteq> a"
proof
assume "a = x"
also from is_inf have "inf \<sqsubseteq> x" ..
finally show ?thesis .
next
assume "a = y"
also from is_inf have "inf \<sqsubseteq> y" ..
finally show ?thesis .
qed
next
fix z assume z: "\<forall>a \<in> ?A. z \<sqsubseteq> a"
from is_inf show "z \<sqsubseteq> inf"
proof (rule is_inf_greatest)
from z show "z \<sqsubseteq> x" by blast
from z show "z \<sqsubseteq> y" by blast
qed
qed
qed
qed
theorem is_Sup_binary: "is_Sup {x, y} sup = is_sup x y sup"
proof -
have "is_Sup {x, y} sup = is_Inf (dual `` {x, y}) (dual sup)"
by (simp only: dual_Inf)
also have "dual `` {x, y} = {dual x, dual y}"
by simp
also have "is_Inf \<dots> (dual sup) = is_inf (dual x) (dual y) (dual sup)"
by (rule is_Inf_binary)
also have "\<dots> = is_sup x y sup"
by (simp only: dual_inf)
finally show ?thesis .
qed
subsection {* Connecting general bounds \label{sec:connect-bounds} *}
text {*
Either kind of general bounds is sufficient to express the other.
The least upper bound (supremum) is the same as the the greatest
lower bound of the set of all upper bounds; the dual statements
holds as well; the dual statement holds as well.
*}
theorem Inf_Sup: "is_Inf {b. \<forall>a \<in> A. a \<sqsubseteq> b} sup \<Longrightarrow> is_Sup A sup"
proof -
let ?B = "{b. \<forall>a \<in> A. a \<sqsubseteq> b}"
assume is_Inf: "is_Inf ?B sup"
show "is_Sup A sup"
proof
fix x assume x: "x \<in> A"
from is_Inf show "x \<sqsubseteq> sup"
proof (rule is_Inf_greatest)
fix y assume "y \<in> ?B"
hence "\<forall>a \<in> A. a \<sqsubseteq> y" ..
from this x show "x \<sqsubseteq> y" ..
qed
next
fix z assume "\<forall>x \<in> A. x \<sqsubseteq> z"
hence "z \<in> ?B" ..
with is_Inf show "sup \<sqsubseteq> z" ..
qed
qed
theorem Sup_Inf: "is_Sup {b. \<forall>a \<in> A. b \<sqsubseteq> a} inf \<Longrightarrow> is_Inf A inf"
proof -
assume "is_Sup {b. \<forall>a \<in> A. b \<sqsubseteq> a} inf"
hence "is_Inf (dual `` {b. \<forall>a \<in> A. dual a \<sqsubseteq> dual b}) (dual inf)"
by (simp only: dual_Inf dual_leq)
also have "dual `` {b. \<forall>a \<in> A. dual a \<sqsubseteq> dual b} = {b'. \<forall>a' \<in> dual `` A. a' \<sqsubseteq> b'}"
by (auto iff: dual_ball dual_Collect) (* FIXME !? *)
finally have "is_Inf \<dots> (dual inf)" .
hence "is_Sup (dual `` A) (dual inf)"
by (rule Inf_Sup)
thus ?thesis ..
qed
end