src/HOL/Algebra/Order.thy

   "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"
+      "x \<in> carrier L \<Longrightarrow> x \<sqsubseteq> x"
     and weak_le_antisym [intro]:
-      "[| x \<sqsubseteq> y; y \<sqsubseteq> x; x \<in> carrier L; y \<in> carrier L |] ==> x .= y"
+      "\<lbrakk>x \<sqsubseteq> y; y \<sqsubseteq> x; x \<in> carrier L; y \<in> carrier L\<rbrakk> \<Longrightarrow> 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"
+      "\<lbrakk>x \<sqsubseteq> y; y \<sqsubseteq> z; x \<in> carrier L; y \<in> carrier L; z \<in> carrier L\<rbrakk> \<Longrightarrow> 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>
+      "\<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"
+  "\<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"
+  "\<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"
+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:

 
 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 "\<forall>x \<in> 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 "\<forall>x \<in> 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 (in weak_partial_order) equivalence_dual: "equivalence (inv_gorder L)"
+  by (rule equivalence.intro) (auto simp: intro: sym trans)
+
+lemma  (in weak_partial_order) dual_weak_order: "weak_partial_order (inv_gorder L)"
+  by intro_locales (auto simp add: weak_partial_order_axioms_def le_cong intro: equivalence_dual le_trans)
+
+lemma (in weak_partial_order) dual_eq_iff [simp]: "A {.=}\<^bsub>inv_gorder L\<^esub> A' \<longleftrightarrow> A {.=} A'"
+  by (auto simp: set_eq_def elem_def)
 
 lemma dual_weak_order_iff:
   "weak_partial_order (inv_gorder A) \<longleftrightarrow> weak_partial_order A"
 proof
   assume "weak_partial_order (inv_gorder A)"

   assume "weak_partial_order A"
   thus "weak_partial_order (inv_gorder A)"
     by (metis weak_partial_order.dual_weak_order)
 qed
 
+lemma Upper_closed [iff]:
+  "Upper L A \<subseteq> carrier L"
+  by (unfold Upper_def) clarify
+
+lemma Upper_memD [dest]:
+  fixes L (structure)
+  shows "\<lbrakk>u \<in> Upper L A; x \<in> A; A \<subseteq> carrier L\<rbrakk> \<Longrightarrow> x \<sqsubseteq> u \<and> u \<in> carrier L"
+  by (unfold Upper_def) blast
+
+lemma (in weak_partial_order) Upper_elemD [dest]:
+  "\<lbrakk>u .\<in> Upper L A; u \<in> carrier L; x \<in> A; A \<subseteq> carrier L\<rbrakk> \<Longrightarrow> x \<sqsubseteq> u"
+  unfolding Upper_def elem_def
+  by (blast dest: sym)
+
+lemma Upper_memI:
+  fixes L (structure)
+  shows "\<lbrakk>!! y. y \<in> A \<Longrightarrow> y \<sqsubseteq> x; x \<in> carrier L\<rbrakk> \<Longrightarrow> x \<in> Upper L A"
+  by (unfold Upper_def) blast
+
+lemma (in weak_partial_order) Upper_elemI:
+  "\<lbrakk>!! y. y \<in> A \<Longrightarrow> y \<sqsubseteq> x; x \<in> carrier L\<rbrakk> \<Longrightarrow> x .\<in> Upper L A"
+  unfolding Upper_def by blast
+
+lemma Upper_antimono:
+  "A \<subseteq> B \<Longrightarrow> 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 \<Longrightarrow> is_closed (Upper L A)"
+  by (rule is_closedI) (blast intro: Upper_memI)+
+
+lemma (in weak_partial_order) Upper_mem_cong:
+  assumes  "a' \<in> carrier L" "A \<subseteq> carrier L" "a .= a'" "a \<in> Upper L A"
+  shows "a' \<in> Upper L A"
+  by (metis assms Upper_closed Upper_is_closed closure_of_eq complete_classes)
+
+lemma (in weak_partial_order) Upper_semi_cong:
+  assumes "A \<subseteq> carrier L" "A {.=} A'"
+  shows "Upper L A \<subseteq> Upper L A'"
+  unfolding Upper_def
+   by clarsimp (meson assms equivalence.refl equivalence_axioms le_cong set_eqD2 subset_eq)
+
+lemma (in weak_partial_order) Upper_cong:
+  assumes "A \<subseteq> carrier L" "A' \<subseteq> carrier L" "A {.=} A'"
+  shows "Upper L A = Upper L A'"
+  using assms by (simp add: Upper_semi_cong set_eq_sym subset_antisym)
+
+lemma Lower_closed [intro!, simp]:
+  "Lower L A \<subseteq> carrier L"
+  by (unfold Lower_def) clarify
+
+lemma Lower_memD [dest]:
+  fixes L (structure)
+  shows "\<lbrakk>l \<in> Lower L A; x \<in> A; A \<subseteq> carrier L\<rbrakk> \<Longrightarrow> l \<sqsubseteq> x \<and> l \<in> carrier L"
+  by (unfold Lower_def) blast
+
+lemma Lower_memI:
+  fixes L (structure)
+  shows "\<lbrakk>!! y. y \<in> A \<Longrightarrow> x \<sqsubseteq> y; x \<in> carrier L\<rbrakk> \<Longrightarrow> x \<in> Lower L A"
+  by (unfold Lower_def) blast
+
+lemma Lower_antimono:
+  "A \<subseteq> B \<Longrightarrow> 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' \<in> carrier L"  "A \<subseteq> carrier L" "a .= a'" "a \<in> Lower L A"
+  shows "a' \<in> Lower L A"
+  by (meson assms Lower_closed Lower_is_closed is_closed_eq subsetCE)
+
+lemma (in weak_partial_order) Lower_cong:
+  assumes "A \<subseteq> carrier L" "A' \<subseteq> carrier L" "A {.=} A'"
+  shows "Lower L A = Lower L A'"
+  unfolding Upper_dual [symmetric]
+  by (rule weak_partial_order.Upper_cong [OF dual_weak_order]) (simp_all add: assms)
+
+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_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 least_closed [intro, simp]:
+  "least L l A \<Longrightarrow> l \<in> carrier L"
+  by (unfold least_def) fast
+
+lemma least_mem:
+  "least L l A \<Longrightarrow> l \<in> A"
+  by (unfold least_def) fast
+
+lemma (in weak_partial_order) weak_least_unique:
+  "\<lbrakk>least L x A; least L y A\<rbrakk> \<Longrightarrow> x .= y"
+  by (unfold least_def) blast
+
+lemma least_le:
+  fixes L (structure)
+  shows "\<lbrakk>least L x A; a \<in> A\<rbrakk> \<Longrightarrow> x \<sqsubseteq> a"
+  by (unfold least_def) fast
+
+lemma (in weak_partial_order) least_cong:
+  "\<lbrakk>x .= x'; x \<in> carrier L; x' \<in> carrier L; is_closed A\<rbrakk> \<Longrightarrow> least L x A = least L x' A"
+  unfolding least_def
+  by (meson is_closed_eq is_closed_eq_rev le_cong local.refl subset_iff)
+
+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 "A \<subseteq> carrier L" "A' \<subseteq> carrier L" "A {.=} A'"
+  shows "least L x (Upper L A) = least L x (Upper L A')"
+  using Upper_cong assms by auto
+
+lemma least_UpperI:
+  fixes L (structure)
+  assumes above: "!! x. x \<in> A \<Longrightarrow> x \<sqsubseteq> s"
+    and below: "!! y. y \<in> Upper L A \<Longrightarrow> 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 "\<forall>x \<in> 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 "\<lbrakk>least L s (Upper L A); x \<in> A; A \<subseteq> carrier L\<rbrakk> \<Longrightarrow> x \<sqsubseteq> s"
+  by (unfold least_def) blast
+
+lemma greatest_closed [intro, simp]:
+  "greatest L l A \<Longrightarrow> l \<in> carrier L"
+  by (unfold greatest_def) fast
+
+lemma greatest_mem:
+  "greatest L l A \<Longrightarrow> l \<in> A"
+  by (unfold greatest_def) fast
+
+lemma (in weak_partial_order) weak_greatest_unique:
+  "\<lbrakk>greatest L x A; greatest L y A\<rbrakk> \<Longrightarrow> x .= y"
+  by (unfold greatest_def) blast
+
+lemma greatest_le:
+  fixes L (structure)
+  shows "\<lbrakk>greatest L x A; a \<in> A\<rbrakk> \<Longrightarrow> a \<sqsubseteq> x"
+  by (unfold greatest_def) fast
+
+lemma (in weak_partial_order) greatest_cong:
+  "\<lbrakk>x .= x'; x \<in> carrier L; x' \<in> carrier L; is_closed A\<rbrakk> \<Longrightarrow>
+  greatest L x A = greatest L x' A"
+  unfolding greatest_def
+  by (meson is_closed_eq_rev le_cong_r local.sym subset_eq)
+
+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"
+  shows "greatest L x (Lower L A) = greatest L x' (Lower L A)"
+proof -
+  have "\<forall>A. is_closed (Lower L (A \<inter> carrier L))"
+    by simp
+  then show ?thesis
+    by (simp add: Lower_def assms greatest_cong)
+qed
+
+lemma (in weak_partial_order) greatest_Lower_cong_r:
+  assumes "A \<subseteq> carrier L" "A' \<subseteq> carrier L" "A {.=} A'"
+  shows "greatest L x (Lower L A) = greatest L x (Lower L A')"
+  using Lower_cong assms by auto
+
+lemma greatest_LowerI:
+  fixes L (structure)
+  assumes below: "!! x. x \<in> A \<Longrightarrow> i \<sqsubseteq> x"
+    and above: "!! y. y \<in> Lower L A \<Longrightarrow> 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 "\<forall>x \<in> 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 "\<lbrakk>greatest L i (Lower L A); x \<in> A; A \<subseteq> carrier L\<rbrakk> \<Longrightarrow> i \<sqsubseteq> x"
+  by (unfold greatest_def) blast
+
 
 subsubsection \<open>Intervals\<close>
 
 definition
   at_least_at_most :: "('a, 'c) gorder_scheme \<Rightarrow> 'a => 'a => 'a set" ("(1\<lbrace>_.._\<rbrace>\<index>)")

 
   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>"
+    "\<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
 
 

 
 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> 
+          "(\<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>")

 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"
+  "\<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>
 

 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"
+  "\<lbrakk>x \<sqsubseteq> y; y \<sqsubseteq> x; x \<in> carrier L; y \<in> carrier L\<rbrakk> \<Longrightarrow> 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)

   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 (unfold_locales, simp_all add: po.sym)
+    apply (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"

 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"
+  "\<lbrakk>least L x A; least L y A\<rbrakk> \<Longrightarrow> 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"
+  "\<lbrakk>greatest L x A; greatest L y A\<rbrakk> \<Longrightarrow> x = y"
   using weak_greatest_unique unfolding eq_is_equal .
 
 
 subsection \<open>Bounded Orders\<close>
 

 
 
 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"
+  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"
+  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"
+  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"
+  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