(* Title: HOL/Algebra/Order.thy
Author: Clemens Ballarin, started 7 November 2003
Copyright: Clemens Ballarin
Most congruence rules by Stephan Hohe.
With additional contributions from Alasdair Armstrong and Simon Foster.
*)
theory Order
imports
"HOL-Library.FuncSet"
Congruence
begin
section \<open>Orders\<close>
subsection \<open>Partial Orders\<close>
record 'a gorder = "'a eq_object" +
le :: "['a, 'a] => bool" (infixl "\<sqsubseteq>\<index>" 50)
abbreviation inv_gorder :: "_ \<Rightarrow> 'a gorder" where
"inv_gorder L \<equiv>
\<lparr> carrier = carrier L,
eq = op .=\<^bsub>L\<^esub>,
le = (\<lambda> x y. y \<sqsubseteq>\<^bsub>L \<^esub>x) \<rparr>"
lemma inv_gorder_inv:
"inv_gorder (inv_gorder L) = L"
by simp
locale weak_partial_order = equivalence L for L (structure) +
assumes le_refl [intro, simp]:
"x \<in> carrier L ==> x \<sqsubseteq> x"
and weak_le_antisym [intro]:
"[| x \<sqsubseteq> y; y \<sqsubseteq> x; x \<in> carrier L; y \<in> carrier L |] ==> x .= y"
and le_trans [trans]:
"[| x \<sqsubseteq> y; y \<sqsubseteq> z; x \<in> carrier L; y \<in> carrier L; z \<in> carrier L |] ==> x \<sqsubseteq> z"
and le_cong:
"\<lbrakk> x .= y; z .= w; x \<in> carrier L; y \<in> carrier L; z \<in> carrier L; w \<in> carrier L \<rbrakk> \<Longrightarrow>
x \<sqsubseteq> z \<longleftrightarrow> y \<sqsubseteq> w"
definition
lless :: "[_, 'a, 'a] => bool" (infixl "\<sqsubset>\<index>" 50)
where "x \<sqsubset>\<^bsub>L\<^esub> y \<longleftrightarrow> x \<sqsubseteq>\<^bsub>L\<^esub> y \<and> x .\<noteq>\<^bsub>L\<^esub> y"
subsubsection \<open>The order relation\<close>
context weak_partial_order
begin
lemma le_cong_l [intro, trans]:
"\<lbrakk> x .= y; y \<sqsubseteq> z; x \<in> carrier L; y \<in> carrier L; z \<in> carrier L \<rbrakk> \<Longrightarrow> x \<sqsubseteq> z"
by (auto intro: le_cong [THEN iffD2])
lemma le_cong_r [intro, trans]:
"\<lbrakk> x \<sqsubseteq> y; y .= z; x \<in> carrier L; y \<in> carrier L; z \<in> carrier L \<rbrakk> \<Longrightarrow> x \<sqsubseteq> z"
by (auto intro: le_cong [THEN iffD1])
lemma weak_refl [intro, simp]: "\<lbrakk> x .= y; x \<in> carrier L; y \<in> carrier L \<rbrakk> \<Longrightarrow> x \<sqsubseteq> y"
by (simp add: le_cong_l)
end
lemma weak_llessI:
fixes R (structure)
assumes "x \<sqsubseteq> y" and "\<not>(x .= y)"
shows "x \<sqsubset> y"
using assms unfolding lless_def by simp
lemma lless_imp_le:
fixes R (structure)
assumes "x \<sqsubset> y"
shows "x \<sqsubseteq> y"
using assms unfolding lless_def by simp
lemma weak_lless_imp_not_eq:
fixes R (structure)
assumes "x \<sqsubset> y"
shows "\<not> (x .= y)"
using assms unfolding lless_def by simp
lemma weak_llessE:
fixes R (structure)
assumes p: "x \<sqsubset> y" and e: "\<lbrakk>x \<sqsubseteq> y; \<not> (x .= y)\<rbrakk> \<Longrightarrow> P"
shows "P"
using p by (blast dest: lless_imp_le weak_lless_imp_not_eq e)
lemma (in weak_partial_order) lless_cong_l [trans]:
assumes xx': "x .= x'"
and xy: "x' \<sqsubset> y"
and carr: "x \<in> carrier L" "x' \<in> carrier L" "y \<in> carrier L"
shows "x \<sqsubset> y"
using assms unfolding lless_def by (auto intro: trans sym)
lemma (in weak_partial_order) lless_cong_r [trans]:
assumes xy: "x \<sqsubset> y"
and yy': "y .= y'"
and carr: "x \<in> carrier L" "y \<in> carrier L" "y' \<in> carrier L"
shows "x \<sqsubset> y'"
using assms unfolding lless_def by (auto intro: trans sym) (*slow*)
lemma (in weak_partial_order) lless_antisym:
assumes "a \<in> carrier L" "b \<in> carrier L"
and "a \<sqsubset> b" "b \<sqsubset> a"
shows "P"
using assms
by (elim weak_llessE) auto
lemma (in weak_partial_order) lless_trans [trans]:
assumes "a \<sqsubset> b" "b \<sqsubset> c"
and carr[simp]: "a \<in> carrier L" "b \<in> carrier L" "c \<in> carrier L"
shows "a \<sqsubset> c"
using assms unfolding lless_def by (blast dest: le_trans intro: sym)
lemma weak_partial_order_subset:
assumes "weak_partial_order L" "A \<subseteq> carrier L"
shows "weak_partial_order (L\<lparr> carrier := A \<rparr>)"
proof -
interpret L: weak_partial_order L
by (simp add: assms)
interpret equivalence "(L\<lparr> carrier := A \<rparr>)"
by (simp add: L.equivalence_axioms assms(2) equivalence_subset)
show ?thesis
apply (unfold_locales, simp_all)
using assms(2) apply auto[1]
using assms(2) apply auto[1]
apply (meson L.le_trans assms(2) contra_subsetD)
apply (meson L.le_cong assms(2) subsetCE)
done
qed
subsubsection \<open>Upper and lower bounds of a set\<close>
definition
Upper :: "[_, 'a set] => 'a set"
where "Upper L A = {u. (\<forall>x. x \<in> A \<inter> carrier L \<longrightarrow> x \<sqsubseteq>\<^bsub>L\<^esub> u)} \<inter> carrier L"
definition
Lower :: "[_, 'a set] => 'a set"
where "Lower L A = {l. (\<forall>x. x \<in> A \<inter> carrier L \<longrightarrow> l \<sqsubseteq>\<^bsub>L\<^esub> x)} \<inter> carrier L"
lemma Upper_closed [intro!, simp]:
"Upper L A \<subseteq> carrier L"
by (unfold Upper_def) clarify
lemma Upper_memD [dest]:
fixes L (structure)
shows "[| u \<in> Upper L A; x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> u \<and> u \<in> carrier L"
by (unfold Upper_def) blast
lemma (in weak_partial_order) Upper_elemD [dest]:
"[| u .\<in> Upper L A; u \<in> carrier L; x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> u"
unfolding Upper_def elem_def
by (blast dest: sym)
lemma Upper_memI:
fixes L (structure)
shows "[| !! y. y \<in> A ==> y \<sqsubseteq> x; x \<in> carrier L |] ==> x \<in> Upper L A"
by (unfold Upper_def) blast
lemma (in weak_partial_order) Upper_elemI:
"[| !! y. y \<in> A ==> y \<sqsubseteq> x; x \<in> carrier L |] ==> x .\<in> Upper L A"
unfolding Upper_def by blast
lemma Upper_antimono:
"A \<subseteq> B ==> Upper L B \<subseteq> Upper L A"
by (unfold Upper_def) blast
lemma (in weak_partial_order) Upper_is_closed [simp]:
"A \<subseteq> carrier L ==> is_closed (Upper L A)"
by (rule is_closedI) (blast intro: Upper_memI)+
lemma (in weak_partial_order) Upper_mem_cong:
assumes a'carr: "a' \<in> carrier L" and Acarr: "A \<subseteq> carrier L"
and aa': "a .= a'"
and aelem: "a \<in> Upper L A"
shows "a' \<in> Upper L A"
proof (rule Upper_memI[OF _ a'carr])
fix y
assume yA: "y \<in> A"
hence "y \<sqsubseteq> a" by (intro Upper_memD[OF aelem, THEN conjunct1] Acarr)
also note aa'
finally
show "y \<sqsubseteq> a'"
by (simp add: a'carr subsetD[OF Acarr yA] subsetD[OF Upper_closed aelem])
qed
lemma (in weak_partial_order) Upper_cong:
assumes Acarr: "A \<subseteq> carrier L" and A'carr: "A' \<subseteq> carrier L"
and AA': "A {.=} A'"
shows "Upper L A = Upper L A'"
unfolding Upper_def
apply rule
apply (rule, clarsimp) defer 1
apply (rule, clarsimp) defer 1
proof -
fix x a'
assume carr: "x \<in> carrier L" "a' \<in> carrier L"
and a'A': "a' \<in> A'"
assume aLxCond[rule_format]: "\<forall>a. a \<in> A \<and> a \<in> carrier L \<longrightarrow> a \<sqsubseteq> x"
from AA' and a'A' have "\<exists>a\<in>A. a' .= a" by (rule set_eqD2)
from this obtain a
where aA: "a \<in> A"
and a'a: "a' .= a"
by auto
note [simp] = subsetD[OF Acarr aA] carr
note a'a
also have "a \<sqsubseteq> x" by (simp add: aLxCond aA)
finally show "a' \<sqsubseteq> x" by simp
next
fix x a
assume carr: "x \<in> carrier L" "a \<in> carrier L"
and aA: "a \<in> A"
assume a'LxCond[rule_format]: "\<forall>a'. a' \<in> A' \<and> a' \<in> carrier L \<longrightarrow> a' \<sqsubseteq> x"
from AA' and aA have "\<exists>a'\<in>A'. a .= a'" by (rule set_eqD1)
from this obtain a'
where a'A': "a' \<in> A'"
and aa': "a .= a'"
by auto
note [simp] = subsetD[OF A'carr a'A'] carr
note aa'
also have "a' \<sqsubseteq> x" by (simp add: a'LxCond a'A')
finally show "a \<sqsubseteq> x" by simp
qed
lemma Lower_closed [intro!, simp]:
"Lower L A \<subseteq> carrier L"
by (unfold Lower_def) clarify
lemma Lower_memD [dest]:
fixes L (structure)
shows "[| l \<in> Lower L A; x \<in> A; A \<subseteq> carrier L |] ==> l \<sqsubseteq> x \<and> l \<in> carrier L"
by (unfold Lower_def) blast
lemma Lower_memI:
fixes L (structure)
shows "[| !! y. y \<in> A ==> x \<sqsubseteq> y; x \<in> carrier L |] ==> x \<in> Lower L A"
by (unfold Lower_def) blast
lemma Lower_antimono:
"A \<subseteq> B ==> Lower L B \<subseteq> Lower L A"
by (unfold Lower_def) blast
lemma (in weak_partial_order) Lower_is_closed [simp]:
"A \<subseteq> carrier L \<Longrightarrow> is_closed (Lower L A)"
by (rule is_closedI) (blast intro: Lower_memI dest: sym)+
lemma (in weak_partial_order) Lower_mem_cong:
assumes a'carr: "a' \<in> carrier L" and Acarr: "A \<subseteq> carrier L"
and aa': "a .= a'"
and aelem: "a \<in> Lower L A"
shows "a' \<in> Lower L A"
using assms Lower_closed[of L A]
by (intro Lower_memI) (blast intro: le_cong_l[OF aa'[symmetric]])
lemma (in weak_partial_order) Lower_cong:
assumes Acarr: "A \<subseteq> carrier L" and A'carr: "A' \<subseteq> carrier L"
and AA': "A {.=} A'"
shows "Lower L A = Lower L A'"
unfolding Lower_def
apply rule
apply clarsimp defer 1
apply clarsimp defer 1
proof -
fix x a'
assume carr: "x \<in> carrier L" "a' \<in> carrier L"
and a'A': "a' \<in> A'"
assume "\<forall>a. a \<in> A \<and> a \<in> carrier L \<longrightarrow> x \<sqsubseteq> a"
hence aLxCond: "\<And>a. \<lbrakk>a \<in> A; a \<in> carrier L\<rbrakk> \<Longrightarrow> x \<sqsubseteq> a" by fast
from AA' and a'A' have "\<exists>a\<in>A. a' .= a" by (rule set_eqD2)
from this obtain a
where aA: "a \<in> A"
and a'a: "a' .= a"
by auto
from aA and subsetD[OF Acarr aA]
have "x \<sqsubseteq> a" by (rule aLxCond)
also note a'a[symmetric]
finally
show "x \<sqsubseteq> a'" by (simp add: carr subsetD[OF Acarr aA])
next
fix x a
assume carr: "x \<in> carrier L" "a \<in> carrier L"
and aA: "a \<in> A"
assume "\<forall>a'. a' \<in> A' \<and> a' \<in> carrier L \<longrightarrow> x \<sqsubseteq> a'"
hence a'LxCond: "\<And>a'. \<lbrakk>a' \<in> A'; a' \<in> carrier L\<rbrakk> \<Longrightarrow> x \<sqsubseteq> a'" by fast+
from AA' and aA have "\<exists>a'\<in>A'. a .= a'" by (rule set_eqD1)
from this obtain a'
where a'A': "a' \<in> A'"
and aa': "a .= a'"
by auto
from a'A' and subsetD[OF A'carr a'A']
have "x \<sqsubseteq> a'" by (rule a'LxCond)
also note aa'[symmetric]
finally show "x \<sqsubseteq> a" by (simp add: carr subsetD[OF A'carr a'A'])
qed
text \<open>Jacobson: Theorem 8.1\<close>
lemma Lower_empty [simp]:
"Lower L {} = carrier L"
by (unfold Lower_def) simp
lemma Upper_empty [simp]:
"Upper L {} = carrier L"
by (unfold Upper_def) simp
subsubsection \<open>Least and greatest, as predicate\<close>
definition
least :: "[_, 'a, 'a set] => bool"
where "least L l A \<longleftrightarrow> A \<subseteq> carrier L \<and> l \<in> A \<and> (\<forall>x\<in>A. l \<sqsubseteq>\<^bsub>L\<^esub> x)"
definition
greatest :: "[_, 'a, 'a set] => bool"
where "greatest L g A \<longleftrightarrow> A \<subseteq> carrier L \<and> g \<in> A \<and> (\<forall>x\<in>A. x \<sqsubseteq>\<^bsub>L\<^esub> g)"
text (in weak_partial_order) \<open>Could weaken these to @{term "l \<in> carrier L \<and> l
.\<in> A"} and @{term "g \<in> carrier L \<and> g .\<in> A"}.\<close>
lemma least_closed [intro, simp]:
"least L l A ==> l \<in> carrier L"
by (unfold least_def) fast
lemma least_mem:
"least L l A ==> l \<in> A"
by (unfold least_def) fast
lemma (in weak_partial_order) weak_least_unique:
"[| least L x A; least L y A |] ==> x .= y"
by (unfold least_def) blast
lemma least_le:
fixes L (structure)
shows "[| least L x A; a \<in> A |] ==> x \<sqsubseteq> a"
by (unfold least_def) fast
lemma (in weak_partial_order) least_cong:
"[| x .= x'; x \<in> carrier L; x' \<in> carrier L; is_closed A |] ==> least L x A = least L x' A"
by (unfold least_def) (auto dest: sym)
abbreviation is_lub :: "[_, 'a, 'a set] => bool"
where "is_lub L x A \<equiv> least L x (Upper L A)"
text (in weak_partial_order) \<open>@{const least} is not congruent in the second parameter for
@{term "A {.=} A'"}\<close>
lemma (in weak_partial_order) least_Upper_cong_l:
assumes "x .= x'"
and "x \<in> carrier L" "x' \<in> carrier L"
and "A \<subseteq> carrier L"
shows "least L x (Upper L A) = least L x' (Upper L A)"
apply (rule least_cong) using assms by auto
lemma (in weak_partial_order) least_Upper_cong_r:
assumes Acarrs: "A \<subseteq> carrier L" "A' \<subseteq> carrier L" (* unneccessary with current Upper? *)
and AA': "A {.=} A'"
shows "least L x (Upper L A) = least L x (Upper L A')"
apply (subgoal_tac "Upper L A = Upper L A'", simp)
by (rule Upper_cong) fact+
lemma least_UpperI:
fixes L (structure)
assumes above: "!! x. x \<in> A ==> x \<sqsubseteq> s"
and below: "!! y. y \<in> Upper L A ==> s \<sqsubseteq> y"
and L: "A \<subseteq> carrier L" "s \<in> carrier L"
shows "least L s (Upper L A)"
proof -
have "Upper L A \<subseteq> carrier L" by simp
moreover from above L have "s \<in> Upper L A" by (simp add: Upper_def)
moreover from below have "ALL x : Upper L A. s \<sqsubseteq> x" by fast
ultimately show ?thesis by (simp add: least_def)
qed
lemma least_Upper_above:
fixes L (structure)
shows "[| least L s (Upper L A); x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> s"
by (unfold least_def) blast
lemma greatest_closed [intro, simp]:
"greatest L l A ==> l \<in> carrier L"
by (unfold greatest_def) fast
lemma greatest_mem:
"greatest L l A ==> l \<in> A"
by (unfold greatest_def) fast
lemma (in weak_partial_order) weak_greatest_unique:
"[| greatest L x A; greatest L y A |] ==> x .= y"
by (unfold greatest_def) blast
lemma greatest_le:
fixes L (structure)
shows "[| greatest L x A; a \<in> A |] ==> a \<sqsubseteq> x"
by (unfold greatest_def) fast
lemma (in weak_partial_order) greatest_cong:
"[| x .= x'; x \<in> carrier L; x' \<in> carrier L; is_closed A |] ==>
greatest L x A = greatest L x' A"
by (unfold greatest_def) (auto dest: sym)
abbreviation is_glb :: "[_, 'a, 'a set] => bool"
where "is_glb L x A \<equiv> greatest L x (Lower L A)"
text (in weak_partial_order) \<open>@{const greatest} is not congruent in the second parameter for
@{term "A {.=} A'"} \<close>
lemma (in weak_partial_order) greatest_Lower_cong_l:
assumes "x .= x'"
and "x \<in> carrier L" "x' \<in> carrier L"
and "A \<subseteq> carrier L" (* unneccessary with current Lower *)
shows "greatest L x (Lower L A) = greatest L x' (Lower L A)"
apply (rule greatest_cong) using assms by auto
lemma (in weak_partial_order) greatest_Lower_cong_r:
assumes Acarrs: "A \<subseteq> carrier L" "A' \<subseteq> carrier L"
and AA': "A {.=} A'"
shows "greatest L x (Lower L A) = greatest L x (Lower L A')"
apply (subgoal_tac "Lower L A = Lower L A'", simp)
by (rule Lower_cong) fact+
lemma greatest_LowerI:
fixes L (structure)
assumes below: "!! x. x \<in> A ==> i \<sqsubseteq> x"
and above: "!! y. y \<in> Lower L A ==> y \<sqsubseteq> i"
and L: "A \<subseteq> carrier L" "i \<in> carrier L"
shows "greatest L i (Lower L A)"
proof -
have "Lower L A \<subseteq> carrier L" by simp
moreover from below L have "i \<in> Lower L A" by (simp add: Lower_def)
moreover from above have "ALL x : Lower L A. x \<sqsubseteq> i" by fast
ultimately show ?thesis by (simp add: greatest_def)
qed
lemma greatest_Lower_below:
fixes L (structure)
shows "[| greatest L i (Lower L A); x \<in> A; A \<subseteq> carrier L |] ==> i \<sqsubseteq> x"
by (unfold greatest_def) blast
lemma Lower_dual [simp]:
"Lower (inv_gorder L) A = Upper L A"
by (simp add:Upper_def Lower_def)
lemma Upper_dual [simp]:
"Upper (inv_gorder L) A = Lower L A"
by (simp add:Upper_def Lower_def)
lemma least_dual [simp]:
"least (inv_gorder L) x A = greatest L x A"
by (simp add:least_def greatest_def)
lemma greatest_dual [simp]:
"greatest (inv_gorder L) x A = least L x A"
by (simp add:least_def greatest_def)
lemma (in weak_partial_order) dual_weak_order:
"weak_partial_order (inv_gorder L)"
apply (unfold_locales)
apply (simp_all)
apply (metis sym)
apply (metis trans)
apply (metis weak_le_antisym)
apply (metis le_trans)
apply (metis le_cong_l le_cong_r sym)
done
lemma dual_weak_order_iff:
"weak_partial_order (inv_gorder A) \<longleftrightarrow> weak_partial_order A"
proof
assume "weak_partial_order (inv_gorder A)"
then interpret dpo: weak_partial_order "inv_gorder A"
rewrites "carrier (inv_gorder A) = carrier A"
and "le (inv_gorder A) = (\<lambda> x y. le A y x)"
and "eq (inv_gorder A) = eq A"
by (simp_all)
show "weak_partial_order A"
by (unfold_locales, auto intro: dpo.sym dpo.trans dpo.le_trans)
next
assume "weak_partial_order A"
thus "weak_partial_order (inv_gorder A)"
by (metis weak_partial_order.dual_weak_order)
qed
subsubsection \<open>Intervals\<close>
definition
at_least_at_most :: "('a, 'c) gorder_scheme \<Rightarrow> 'a => 'a => 'a set" ("(1\<lbrace>_.._\<rbrace>\<index>)")
where "\<lbrace>l..u\<rbrace>\<^bsub>A\<^esub> = {x \<in> carrier A. l \<sqsubseteq>\<^bsub>A\<^esub> x \<and> x \<sqsubseteq>\<^bsub>A\<^esub> u}"
context weak_partial_order
begin
lemma at_least_at_most_upper [dest]:
"x \<in> \<lbrace>a..b\<rbrace> \<Longrightarrow> x \<sqsubseteq> b"
by (simp add: at_least_at_most_def)
lemma at_least_at_most_lower [dest]:
"x \<in> \<lbrace>a..b\<rbrace> \<Longrightarrow> a \<sqsubseteq> x"
by (simp add: at_least_at_most_def)
lemma at_least_at_most_closed: "\<lbrace>a..b\<rbrace> \<subseteq> carrier L"
by (auto simp add: at_least_at_most_def)
lemma at_least_at_most_member [intro]:
"\<lbrakk> x \<in> carrier L; a \<sqsubseteq> x; x \<sqsubseteq> b \<rbrakk> \<Longrightarrow> x \<in> \<lbrace>a..b\<rbrace>"
by (simp add: at_least_at_most_def)
end
subsubsection \<open>Isotone functions\<close>
definition isotone :: "('a, 'c) gorder_scheme \<Rightarrow> ('b, 'd) gorder_scheme \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
where
"isotone A B f \<equiv>
weak_partial_order A \<and> weak_partial_order B \<and>
(\<forall>x\<in>carrier A. \<forall>y\<in>carrier A. x \<sqsubseteq>\<^bsub>A\<^esub> y \<longrightarrow> f x \<sqsubseteq>\<^bsub>B\<^esub> f y)"
lemma isotoneI [intro?]:
fixes f :: "'a \<Rightarrow> 'b"
assumes "weak_partial_order L1"
"weak_partial_order L2"
"(\<And>x y. \<lbrakk> x \<in> carrier L1; y \<in> carrier L1; x \<sqsubseteq>\<^bsub>L1\<^esub> y \<rbrakk>
\<Longrightarrow> f x \<sqsubseteq>\<^bsub>L2\<^esub> f y)"
shows "isotone L1 L2 f"
using assms by (auto simp add:isotone_def)
abbreviation Monotone :: "('a, 'b) gorder_scheme \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> bool" ("Mono\<index>")
where "Monotone L f \<equiv> isotone L L f"
lemma use_iso1:
"\<lbrakk>isotone A A f; x \<in> carrier A; y \<in> carrier A; x \<sqsubseteq>\<^bsub>A\<^esub> y\<rbrakk> \<Longrightarrow>
f x \<sqsubseteq>\<^bsub>A\<^esub> f y"
by (simp add: isotone_def)
lemma use_iso2:
"\<lbrakk>isotone A B f; x \<in> carrier A; y \<in> carrier A; x \<sqsubseteq>\<^bsub>A\<^esub> y\<rbrakk> \<Longrightarrow>
f x \<sqsubseteq>\<^bsub>B\<^esub> f y"
by (simp add: isotone_def)
lemma iso_compose:
"\<lbrakk>f \<in> carrier A \<rightarrow> carrier B; isotone A B f; g \<in> carrier B \<rightarrow> carrier C; isotone B C g\<rbrakk> \<Longrightarrow>
isotone A C (g \<circ> f)"
by (simp add: isotone_def, safe, metis Pi_iff)
lemma (in weak_partial_order) inv_isotone [simp]:
"isotone (inv_gorder A) (inv_gorder B) f = isotone A B f"
by (auto simp add:isotone_def dual_weak_order dual_weak_order_iff)
subsubsection \<open>Idempotent functions\<close>
definition idempotent ::
"('a, 'b) gorder_scheme \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> bool" ("Idem\<index>") where
"idempotent L f \<equiv> \<forall>x\<in>carrier L. f (f x) .=\<^bsub>L\<^esub> f x"
lemma (in weak_partial_order) idempotent:
"\<lbrakk> Idem f; x \<in> carrier L \<rbrakk> \<Longrightarrow> f (f x) .= f x"
by (auto simp add: idempotent_def)
subsubsection \<open>Order embeddings\<close>
definition order_emb :: "('a, 'c) gorder_scheme \<Rightarrow> ('b, 'd) gorder_scheme \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
where
"order_emb A B f \<equiv> weak_partial_order A
\<and> weak_partial_order B
\<and> (\<forall>x\<in>carrier A. \<forall>y\<in>carrier A. f x \<sqsubseteq>\<^bsub>B\<^esub> f y \<longleftrightarrow> x \<sqsubseteq>\<^bsub>A\<^esub> y )"
lemma order_emb_isotone: "order_emb A B f \<Longrightarrow> isotone A B f"
by (auto simp add: isotone_def order_emb_def)
subsubsection \<open>Commuting functions\<close>
definition commuting :: "('a, 'c) gorder_scheme \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> bool" where
"commuting A f g = (\<forall>x\<in>carrier A. (f \<circ> g) x .=\<^bsub>A\<^esub> (g \<circ> f) x)"
subsection \<open>Partial orders where \<open>eq\<close> is the Equality\<close>
locale partial_order = weak_partial_order +
assumes eq_is_equal: "op .= = op ="
begin
declare weak_le_antisym [rule del]
lemma le_antisym [intro]:
"[| x \<sqsubseteq> y; y \<sqsubseteq> x; x \<in> carrier L; y \<in> carrier L |] ==> x = y"
using weak_le_antisym unfolding eq_is_equal .
lemma lless_eq:
"x \<sqsubset> y \<longleftrightarrow> x \<sqsubseteq> y \<and> x \<noteq> y"
unfolding lless_def by (simp add: eq_is_equal)
lemma set_eq_is_eq: "A {.=} B \<longleftrightarrow> A = B"
by (auto simp add: set_eq_def elem_def eq_is_equal)
end
lemma (in partial_order) dual_order:
"partial_order (inv_gorder L)"
proof -
interpret dwo: weak_partial_order "inv_gorder L"
by (metis dual_weak_order)
show ?thesis
by (unfold_locales, simp add:eq_is_equal)
qed
lemma dual_order_iff:
"partial_order (inv_gorder A) \<longleftrightarrow> partial_order A"
proof
assume assm:"partial_order (inv_gorder A)"
then interpret po: partial_order "inv_gorder A"
rewrites "carrier (inv_gorder A) = carrier A"
and "le (inv_gorder A) = (\<lambda> x y. le A y x)"
and "eq (inv_gorder A) = eq A"
by (simp_all)
show "partial_order A"
apply (unfold_locales, simp_all)
apply (metis po.sym, metis po.trans)
apply (metis po.weak_le_antisym, metis po.le_trans)
apply (metis (full_types) po.eq_is_equal, metis po.eq_is_equal)
done
next
assume "partial_order A"
thus "partial_order (inv_gorder A)"
by (metis partial_order.dual_order)
qed
text \<open>Least and greatest, as predicate\<close>
lemma (in partial_order) least_unique:
"[| least L x A; least L y A |] ==> x = y"
using weak_least_unique unfolding eq_is_equal .
lemma (in partial_order) greatest_unique:
"[| greatest L x A; greatest L y A |] ==> x = y"
using weak_greatest_unique unfolding eq_is_equal .
subsection \<open>Bounded Orders\<close>
definition
top :: "_ => 'a" ("\<top>\<index>") where
"\<top>\<^bsub>L\<^esub> = (SOME x. greatest L x (carrier L))"
definition
bottom :: "_ => 'a" ("\<bottom>\<index>") where
"\<bottom>\<^bsub>L\<^esub> = (SOME x. least L x (carrier L))"
locale weak_partial_order_bottom = weak_partial_order L for L (structure) +
assumes bottom_exists: "\<exists> x. least L x (carrier L)"
begin
lemma bottom_least: "least L \<bottom> (carrier L)"
proof -
obtain x where "least L x (carrier L)"
by (metis bottom_exists)
thus ?thesis
by (auto intro:someI2 simp add: bottom_def)
qed
lemma bottom_closed [simp, intro]:
"\<bottom> \<in> carrier L"
by (metis bottom_least least_mem)
lemma bottom_lower [simp, intro]:
"x \<in> carrier L \<Longrightarrow> \<bottom> \<sqsubseteq> x"
by (metis bottom_least least_le)
end
locale weak_partial_order_top = weak_partial_order L for L (structure) +
assumes top_exists: "\<exists> x. greatest L x (carrier L)"
begin
lemma top_greatest: "greatest L \<top> (carrier L)"
proof -
obtain x where "greatest L x (carrier L)"
by (metis top_exists)
thus ?thesis
by (auto intro:someI2 simp add: top_def)
qed
lemma top_closed [simp, intro]:
"\<top> \<in> carrier L"
by (metis greatest_mem top_greatest)
lemma top_higher [simp, intro]:
"x \<in> carrier L \<Longrightarrow> x \<sqsubseteq> \<top>"
by (metis greatest_le top_greatest)
end
subsection \<open>Total Orders\<close>
locale weak_total_order = weak_partial_order +
assumes total: "\<lbrakk> x \<in> carrier L; y \<in> carrier L \<rbrakk> \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
text \<open>Introduction rule: the usual definition of total order\<close>
lemma (in weak_partial_order) weak_total_orderI:
assumes total: "!!x y. \<lbrakk> x \<in> carrier L; y \<in> carrier L \<rbrakk> \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
shows "weak_total_order L"
by unfold_locales (rule total)
subsection \<open>Total orders where \<open>eq\<close> is the Equality\<close>
locale total_order = partial_order +
assumes total_order_total: "\<lbrakk> x \<in> carrier L; y \<in> carrier L \<rbrakk> \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
sublocale total_order < weak?: weak_total_order
by unfold_locales (rule total_order_total)
text \<open>Introduction rule: the usual definition of total order\<close>
lemma (in partial_order) total_orderI:
assumes total: "!!x y. \<lbrakk> x \<in> carrier L; y \<in> carrier L \<rbrakk> \<Longrightarrow> x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
shows "total_order L"
by unfold_locales (rule total)
end