diff --git a/src/HOL/Library/Lexord.thy b/src/HOL/Library/Lexord.thy
new file mode 100644
--- /dev/null
+++ b/src/HOL/Library/Lexord.thy
@@ -0,0 +1,208 @@
+section \<open>Lexicographic orderings\<close>
+
+theory Lexord
+  imports Main
+begin
+
+subsection \<open>The preorder case\<close>
+
+locale lex_preordering = preordering
+begin
+
+inductive lex_less :: \<open>'a list \<Rightarrow> 'a list \<Rightarrow> bool\<close>  (infix \<open>[\<^bold><]\<close> 50) 
+where
+  Nil: \<open>[] [\<^bold><] y # ys\<close>
+| Cons: \<open>x \<^bold>< y \<Longrightarrow> x # xs [\<^bold><] y # ys\<close>
+| Cons_eq: \<open>x \<^bold>\<le> y \<Longrightarrow> y \<^bold>\<le> x \<Longrightarrow> xs [\<^bold><] ys \<Longrightarrow> x # xs [\<^bold><] y # ys\<close>
+
+inductive lex_less_eq :: \<open>'a list \<Rightarrow> 'a list \<Rightarrow> bool\<close>  (infix \<open>[\<^bold>\<le>]\<close> 50)
+where
+  Nil: \<open>[] [\<^bold>\<le>] ys\<close>
+| Cons: \<open>x \<^bold>< y \<Longrightarrow> x # xs [\<^bold>\<le>] y # ys\<close>
+| Cons_eq: \<open>x \<^bold>\<le> y \<Longrightarrow> y \<^bold>\<le> x \<Longrightarrow> xs [\<^bold>\<le>] ys \<Longrightarrow> x # xs [\<^bold>\<le>] y # ys\<close>
+
+lemma lex_less_simps [simp]:
+  \<open>[] [\<^bold><] y # ys\<close>
+  \<open>\<not> xs [\<^bold><] []\<close>
+  \<open>x # xs [\<^bold><] y # ys \<longleftrightarrow> x \<^bold>< y \<or> x \<^bold>\<le> y \<and> y \<^bold>\<le> x \<and> xs [\<^bold><] ys\<close>
+  by (auto intro: lex_less.intros elim: lex_less.cases)
+
+lemma lex_less_eq_simps [simp]:
+  \<open>[] [\<^bold>\<le>] ys\<close>
+  \<open>\<not> x # xs [\<^bold>\<le>] []\<close>
+  \<open>x # xs [\<^bold>\<le>] y # ys \<longleftrightarrow> x \<^bold>< y \<or> x \<^bold>\<le> y \<and> y \<^bold>\<le> x \<and> xs [\<^bold>\<le>] ys\<close>
+  by (auto intro: lex_less_eq.intros elim: lex_less_eq.cases)
+
+lemma lex_less_code [code]:
+  \<open>[] [\<^bold><] y # ys \<longleftrightarrow> True\<close>
+  \<open>xs [\<^bold><] [] \<longleftrightarrow> False\<close>
+  \<open>x # xs [\<^bold><] y # ys \<longleftrightarrow> x \<^bold>< y \<or> x \<^bold>\<le> y \<and> y \<^bold>\<le> x \<and> xs [\<^bold><] ys\<close>
+  by simp_all
+
+lemma lex_less_eq_code [code]:
+  \<open>[] [\<^bold>\<le>] ys \<longleftrightarrow> True\<close>
+  \<open>x # xs [\<^bold>\<le>] [] \<longleftrightarrow> False\<close>
+  \<open>x # xs [\<^bold>\<le>] y # ys \<longleftrightarrow> x \<^bold>< y \<or> x \<^bold>\<le> y \<and> y \<^bold>\<le> x \<and> xs [\<^bold>\<le>] ys\<close>
+  by simp_all
+
+lemma preordering:
+  \<open>preordering ([\<^bold>\<le>]) ([\<^bold><])\<close>
+proof
+  fix xs ys zs
+  show \<open>xs [\<^bold>\<le>] xs\<close>
+    by (induction xs) (simp_all add: refl)
+  show \<open>xs [\<^bold>\<le>] zs\<close> if \<open>xs [\<^bold>\<le>] ys\<close> \<open>ys [\<^bold>\<le>] zs\<close>
+  using that proof (induction arbitrary: zs)
+    case (Nil ys)
+    then show ?case by simp
+  next
+    case (Cons x y xs ys)
+    then show ?case
+      by (cases zs) (auto dest: strict_trans strict_trans2)
+  next
+    case (Cons_eq x y xs ys)
+    then show ?case
+      by (cases zs) (auto dest: strict_trans1 intro: trans)
+  qed
+  show \<open>xs [\<^bold><] ys \<longleftrightarrow> xs [\<^bold>\<le>] ys \<and> \<not> ys [\<^bold>\<le>] xs\<close> (is \<open>?P \<longleftrightarrow> ?Q\<close>)
+  proof
+    assume ?P
+    then have \<open>xs [\<^bold>\<le>] ys\<close>
+      by induction simp_all
+    moreover have \<open>\<not> ys [\<^bold>\<le>] xs\<close>
+      using \<open>?P\<close>
+      by induction (simp_all, simp_all add: strict_iff_not asym)
+    ultimately show ?Q ..
+  next
+    assume ?Q
+    then have \<open>xs [\<^bold>\<le>] ys\<close> \<open>\<not> ys [\<^bold>\<le>] xs\<close>
+      by auto
+    then show ?P
+    proof induction
+      case (Nil ys)
+      then show ?case
+        by (cases ys) simp_all
+    next
+      case (Cons x y xs ys)
+      then show ?case
+        by simp
+    next
+      case (Cons_eq x y xs ys)
+      then show ?case
+        by simp
+    qed
+  qed
+qed
+
+interpretation lex: preordering \<open>([\<^bold>\<le>])\<close> \<open>([\<^bold><])\<close>
+  by (fact preordering)
+
+end
+
+
+subsection \<open>The order case\<close>
+
+locale lex_ordering = lex_preordering + ordering
+begin
+
+interpretation lex: preordering \<open>([\<^bold>\<le>])\<close> \<open>([\<^bold><])\<close>
+  by (fact preordering)
+
+lemma less_lex_Cons_iff [simp]:
+  \<open>x # xs [\<^bold><] y # ys \<longleftrightarrow> x \<^bold>< y \<or> x = y \<and> xs [\<^bold><] ys\<close>
+  by (auto intro: refl antisym)
+
+lemma less_eq_lex_Cons_iff [simp]:
+  \<open>x # xs [\<^bold>\<le>] y # ys \<longleftrightarrow> x \<^bold>< y \<or> x = y \<and> xs [\<^bold>\<le>] ys\<close>
+  by (auto intro: refl antisym)
+
+lemma ordering:
+  \<open>ordering ([\<^bold>\<le>]) ([\<^bold><])\<close>
+proof
+  fix xs ys
+  show *: \<open>xs = ys\<close> if \<open>xs [\<^bold>\<le>] ys\<close> \<open>ys [\<^bold>\<le>] xs\<close>
+  using that proof induction
+  case (Nil ys)
+    then show ?case by (cases ys) simp
+  next
+    case (Cons x y xs ys)
+    then show ?case by (auto dest: asym intro: antisym)
+      (simp add: strict_iff_not)
+  next
+    case (Cons_eq x y xs ys)
+    then show ?case by (auto intro: antisym)
+      (simp add: strict_iff_not)
+  qed
+  show \<open>xs [\<^bold><] ys \<longleftrightarrow> xs [\<^bold>\<le>] ys \<and> xs \<noteq> ys\<close>
+    by (auto simp add: lex.strict_iff_not dest: *)
+qed
+
+interpretation lex: ordering \<open>([\<^bold>\<le>])\<close> \<open>([\<^bold><])\<close>
+  by (fact ordering)
+
+end
+
+
+subsection \<open>Canonical instance\<close>
+
+instantiation list :: (preorder) preorder
+begin
+
+global_interpretation lex: lex_preordering \<open>(\<le>) :: 'a::preorder \<Rightarrow> 'a \<Rightarrow> bool\<close> \<open>(<) :: 'a \<Rightarrow> 'a \<Rightarrow> bool\<close>
+  defines less_eq_list = lex.lex_less_eq
+    and less_list = lex.lex_less ..
+
+instance
+  by (rule class.preorder.of_class.intro, rule preordering_preorderI, fact lex.preordering)
+
+end
+
+global_interpretation lex: lex_ordering \<open>(\<le>) :: 'a::order \<Rightarrow> 'a \<Rightarrow> bool\<close> \<open>(<) :: 'a \<Rightarrow> 'a \<Rightarrow> bool\<close>
+  rewrites \<open>lex_preordering.lex_less_eq (\<le>) (<) = ((\<le>) :: 'a list \<Rightarrow> 'a list \<Rightarrow> bool)\<close>
+    and \<open>lex_preordering.lex_less (\<le>) (<) = ((<) :: 'a list \<Rightarrow> 'a list \<Rightarrow> bool)\<close>
+proof -
+  interpret lex_ordering \<open>(\<le>) :: 'a \<Rightarrow> 'a \<Rightarrow> bool\<close> \<open>(<) :: 'a \<Rightarrow> 'a \<Rightarrow> bool\<close> ..
+  show \<open>lex_ordering ((\<le>)  :: 'a \<Rightarrow> 'a \<Rightarrow> bool) (<)\<close>
+    by (fact lex_ordering_axioms)
+  show \<open>lex_preordering.lex_less_eq (\<le>) (<) = (\<le>)\<close>
+    by (simp add: less_eq_list_def)
+  show \<open>lex_preordering.lex_less (\<le>) (<) = (<)\<close>
+    by (simp add: less_list_def)
+qed
+
+instance list :: (order) order
+  by (rule class.order.of_class.intro, rule ordering_orderI, fact lex.ordering)
+
+export_code \<open>(\<le>) :: _ list \<Rightarrow> _ list \<Rightarrow> bool\<close> \<open>(<) :: _ list \<Rightarrow> _ list \<Rightarrow> bool\<close> in Haskell
+
+
+subsection \<open>Non-canonical instance\<close>
+
+context comm_monoid_mult
+begin
+
+definition dvd_strict :: \<open>'a \<Rightarrow> 'a \<Rightarrow> bool\<close>
+  where \<open>dvd_strict a b \<longleftrightarrow> a dvd b \<and> \<not> b dvd a\<close>
+
+end
+
+global_interpretation dvd: lex_preordering \<open>(dvd) :: 'a::comm_monoid_mult \<Rightarrow> 'a \<Rightarrow> bool\<close> dvd_strict
+  defines lex_dvd = dvd.lex_less_eq
+    and lex_dvd_strict = dvd.lex_less
+  apply (rule lex_preordering.intro)
+  apply standard
+    apply (auto simp add: dvd_strict_def)
+  done
+
+print_theorems
+
+global_interpretation lex_dvd: preordering lex_dvd lex_dvd_strict
+  by (fact dvd.preordering)
+
+definition \<open>example = lex_dvd [(4::int), - 7, 8] [- 8, 13, 5]\<close>
+
+export_code example in Haskell
+
+value example
+
+end
diff --git a/src/HOL/Orderings.thy b/src/HOL/Orderings.thy
--- a/src/HOL/Orderings.thy
+++ b/src/HOL/Orderings.thy
@@ -1,1702 +1,1712 @@
 (*  Title:      HOL/Orderings.thy
     Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
 *)
 
 section \<open>Abstract orderings\<close>
 
 theory Orderings
 imports HOL
 keywords "print_orders" :: diag
 begin
 
 ML_file \<open>~~/src/Provers/order_procedure.ML\<close>
 ML_file \<open>~~/src/Provers/order_tac.ML\<close>
 
 subsection \<open>Abstract ordering\<close>
 
 locale partial_preordering =
   fixes less_eq :: \<open>'a \<Rightarrow> 'a \<Rightarrow> bool\<close> (infix \<open>\<^bold>\<le>\<close> 50)
   assumes refl: \<open>a \<^bold>\<le> a\<close> \<comment> \<open>not \<open>iff\<close>: makes problems due to multiple (dual) interpretations\<close>
     and trans: \<open>a \<^bold>\<le> b \<Longrightarrow> b \<^bold>\<le> c \<Longrightarrow> a \<^bold>\<le> c\<close>
 
 locale preordering = partial_preordering +
   fixes less :: \<open>'a \<Rightarrow> 'a \<Rightarrow> bool\<close> (infix \<open>\<^bold><\<close> 50)
   assumes strict_iff_not: \<open>a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> \<not> b \<^bold>\<le> a\<close>
 begin
 
 lemma strict_implies_order:
   \<open>a \<^bold>< b \<Longrightarrow> a \<^bold>\<le> b\<close>
   by (simp add: strict_iff_not)
 
 lemma irrefl: \<comment> \<open>not \<open>iff\<close>: makes problems due to multiple (dual) interpretations\<close>
   \<open>\<not> a \<^bold>< a\<close>
   by (simp add: strict_iff_not)
 
 lemma asym:
   \<open>a \<^bold>< b \<Longrightarrow> b \<^bold>< a \<Longrightarrow> False\<close>
   by (auto simp add: strict_iff_not)
 
 lemma strict_trans1:
   \<open>a \<^bold>\<le> b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c\<close>
   by (auto simp add: strict_iff_not intro: trans)
 
 lemma strict_trans2:
   \<open>a \<^bold>< b \<Longrightarrow> b \<^bold>\<le> c \<Longrightarrow> a \<^bold>< c\<close>
   by (auto simp add: strict_iff_not intro: trans)
 
 lemma strict_trans:
   \<open>a \<^bold>< b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c\<close>
   by (auto intro: strict_trans1 strict_implies_order)
 
 end
 
 lemma preordering_strictI: \<comment> \<open>Alternative introduction rule with bias towards strict order\<close>
   fixes less_eq (infix \<open>\<^bold>\<le>\<close> 50)
     and less (infix \<open>\<^bold><\<close> 50)
   assumes less_eq_less: \<open>\<And>a b. a \<^bold>\<le> b \<longleftrightarrow> a \<^bold>< b \<or> a = b\<close>
     assumes asym: \<open>\<And>a b. a \<^bold>< b \<Longrightarrow> \<not> b \<^bold>< a\<close>
   assumes irrefl: \<open>\<And>a. \<not> a \<^bold>< a\<close>
   assumes trans: \<open>\<And>a b c. a \<^bold>< b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c\<close>
   shows \<open>preordering (\<^bold>\<le>) (\<^bold><)\<close>
 proof
   fix a b
   show \<open>a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> \<not> b \<^bold>\<le> a\<close>
     by (auto simp add: less_eq_less asym irrefl)
 next
   fix a
   show \<open>a \<^bold>\<le> a\<close>
     by (auto simp add: less_eq_less)
 next
   fix a b c
   assume \<open>a \<^bold>\<le> b\<close> and \<open>b \<^bold>\<le> c\<close> then show \<open>a \<^bold>\<le> c\<close>
     by (auto simp add: less_eq_less intro: trans)
 qed
 
 lemma preordering_dualI:
   fixes less_eq (infix \<open>\<^bold>\<le>\<close> 50)
     and less (infix \<open>\<^bold><\<close> 50)
   assumes \<open>preordering (\<lambda>a b. b \<^bold>\<le> a) (\<lambda>a b. b \<^bold>< a)\<close>
   shows \<open>preordering (\<^bold>\<le>) (\<^bold><)\<close>
 proof -
   from assms interpret preordering \<open>\<lambda>a b. b \<^bold>\<le> a\<close> \<open>\<lambda>a b. b \<^bold>< a\<close> .
   show ?thesis
     by standard (auto simp: strict_iff_not refl intro: trans)
 qed
 
 locale ordering = partial_preordering +
   fixes less :: \<open>'a \<Rightarrow> 'a \<Rightarrow> bool\<close> (infix \<open>\<^bold><\<close> 50)
   assumes strict_iff_order: \<open>a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> a \<noteq> b\<close>
   assumes antisym: \<open>a \<^bold>\<le> b \<Longrightarrow> b \<^bold>\<le> a \<Longrightarrow> a = b\<close>
 begin
 
 sublocale preordering \<open>(\<^bold>\<le>)\<close> \<open>(\<^bold><)\<close>
 proof
   show \<open>a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> \<not> b \<^bold>\<le> a\<close> for a b
     by (auto simp add: strict_iff_order intro: antisym)
 qed
 
 lemma strict_implies_not_eq:
   \<open>a \<^bold>< b \<Longrightarrow> a \<noteq> b\<close>
   by (simp add: strict_iff_order)
 
 lemma not_eq_order_implies_strict:
   \<open>a \<noteq> b \<Longrightarrow> a \<^bold>\<le> b \<Longrightarrow> a \<^bold>< b\<close>
   by (simp add: strict_iff_order)
 
 lemma order_iff_strict:
   \<open>a \<^bold>\<le> b \<longleftrightarrow> a \<^bold>< b \<or> a = b\<close>
   by (auto simp add: strict_iff_order refl)
 
 lemma eq_iff: \<open>a = b \<longleftrightarrow> a \<^bold>\<le> b \<and> b \<^bold>\<le> a\<close>
   by (auto simp add: refl intro: antisym)
 
 end
 
 lemma ordering_strictI: \<comment> \<open>Alternative introduction rule with bias towards strict order\<close>
   fixes less_eq (infix \<open>\<^bold>\<le>\<close> 50)
     and less (infix \<open>\<^bold><\<close> 50)
   assumes less_eq_less: \<open>\<And>a b. a \<^bold>\<le> b \<longleftrightarrow> a \<^bold>< b \<or> a = b\<close>
     assumes asym: \<open>\<And>a b. a \<^bold>< b \<Longrightarrow> \<not> b \<^bold>< a\<close>
   assumes irrefl: \<open>\<And>a. \<not> a \<^bold>< a\<close>
   assumes trans: \<open>\<And>a b c. a \<^bold>< b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c\<close>
   shows \<open>ordering (\<^bold>\<le>) (\<^bold><)\<close>
 proof
   fix a b
   show \<open>a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> a \<noteq> b\<close>
     by (auto simp add: less_eq_less asym irrefl)
 next
   fix a
   show \<open>a \<^bold>\<le> a\<close>
     by (auto simp add: less_eq_less)
 next
   fix a b c
   assume \<open>a \<^bold>\<le> b\<close> and \<open>b \<^bold>\<le> c\<close> then show \<open>a \<^bold>\<le> c\<close>
     by (auto simp add: less_eq_less intro: trans)
 next
   fix a b
   assume \<open>a \<^bold>\<le> b\<close> and \<open>b \<^bold>\<le> a\<close> then show \<open>a = b\<close>
     by (auto simp add: less_eq_less asym)
 qed
 
 lemma ordering_dualI:
   fixes less_eq (infix \<open>\<^bold>\<le>\<close> 50)
     and less (infix \<open>\<^bold><\<close> 50)
   assumes \<open>ordering (\<lambda>a b. b \<^bold>\<le> a) (\<lambda>a b. b \<^bold>< a)\<close>
   shows \<open>ordering (\<^bold>\<le>) (\<^bold><)\<close>
 proof -
   from assms interpret ordering \<open>\<lambda>a b. b \<^bold>\<le> a\<close> \<open>\<lambda>a b. b \<^bold>< a\<close> .
   show ?thesis
     by standard (auto simp: strict_iff_order refl intro: antisym trans)
 qed
 
 locale ordering_top = ordering +
   fixes top :: \<open>'a\<close>  (\<open>\<^bold>\<top>\<close>)
   assumes extremum [simp]: \<open>a \<^bold>\<le> \<^bold>\<top>\<close>
 begin
 
 lemma extremum_uniqueI:
   \<open>\<^bold>\<top> \<^bold>\<le> a \<Longrightarrow> a = \<^bold>\<top>\<close>
   by (rule antisym) auto
 
 lemma extremum_unique:
   \<open>\<^bold>\<top> \<^bold>\<le> a \<longleftrightarrow> a = \<^bold>\<top>\<close>
   by (auto intro: antisym)
 
 lemma extremum_strict [simp]:
   \<open>\<not> (\<^bold>\<top> \<^bold>< a)\<close>
   using extremum [of a] by (auto simp add: order_iff_strict intro: asym irrefl)
 
 lemma not_eq_extremum:
   \<open>a \<noteq> \<^bold>\<top> \<longleftrightarrow> a \<^bold>< \<^bold>\<top>\<close>
   by (auto simp add: order_iff_strict intro: not_eq_order_implies_strict extremum)
 
 end
 
 
 subsection \<open>Syntactic orders\<close>
 
 class ord =
   fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
     and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
 begin
 
 notation
   less_eq  ("'(\<le>')") and
   less_eq  ("(_/ \<le> _)"  [51, 51] 50) and
   less  ("'(<')") and
   less  ("(_/ < _)"  [51, 51] 50)
 
 abbreviation (input)
   greater_eq  (infix "\<ge>" 50)
   where "x \<ge> y \<equiv> y \<le> x"
 
 abbreviation (input)
   greater  (infix ">" 50)
   where "x > y \<equiv> y < x"
 
 notation (ASCII)
   less_eq  ("'(<=')") and
   less_eq  ("(_/ <= _)" [51, 51] 50)
 
 notation (input)
   greater_eq  (infix ">=" 50)
 
 end
 
 
 subsection \<open>Quasi orders\<close>
 
 class preorder = ord +
   assumes less_le_not_le: "x < y \<longleftrightarrow> x \<le> y \<and> \<not> (y \<le> x)"
   and order_refl [iff]: "x \<le> x"
   and order_trans: "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
 begin
 
 sublocale order: preordering less_eq less + dual_order: preordering greater_eq greater
 proof -
   interpret preordering less_eq less
     by standard (auto intro: order_trans simp add: less_le_not_le)
   show \<open>preordering less_eq less\<close>
     by (fact preordering_axioms)
   then show \<open>preordering greater_eq greater\<close>
     by (rule preordering_dualI)
 qed
 
 text \<open>Reflexivity.\<close>
 
 lemma eq_refl: "x = y \<Longrightarrow> x \<le> y"
     \<comment> \<open>This form is useful with the classical reasoner.\<close>
 by (erule ssubst) (rule order_refl)
 
 lemma less_irrefl [iff]: "\<not> x < x"
 by (simp add: less_le_not_le)
 
 lemma less_imp_le: "x < y \<Longrightarrow> x \<le> y"
 by (simp add: less_le_not_le)
 
 
 text \<open>Asymmetry.\<close>
 
 lemma less_not_sym: "x < y \<Longrightarrow> \<not> (y < x)"
 by (simp add: less_le_not_le)
 
 lemma less_asym: "x < y \<Longrightarrow> (\<not> P \<Longrightarrow> y < x) \<Longrightarrow> P"
 by (drule less_not_sym, erule contrapos_np) simp
 
 
 text \<open>Transitivity.\<close>
 
 lemma less_trans: "x < y \<Longrightarrow> y < z \<Longrightarrow> x < z"
 by (auto simp add: less_le_not_le intro: order_trans)
 
 lemma le_less_trans: "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z"
 by (auto simp add: less_le_not_le intro: order_trans)
 
 lemma less_le_trans: "x < y \<Longrightarrow> y \<le> z \<Longrightarrow> x < z"
 by (auto simp add: less_le_not_le intro: order_trans)
 
 
 text \<open>Useful for simplification, but too risky to include by default.\<close>
 
 lemma less_imp_not_less: "x < y \<Longrightarrow> (\<not> y < x) \<longleftrightarrow> True"
 by (blast elim: less_asym)
 
 lemma less_imp_triv: "x < y \<Longrightarrow> (y < x \<longrightarrow> P) \<longleftrightarrow> True"
 by (blast elim: less_asym)
 
 
 text \<open>Transitivity rules for calculational reasoning\<close>
 
 lemma less_asym': "a < b \<Longrightarrow> b < a \<Longrightarrow> P"
 by (rule less_asym)
 
 
 text \<open>Dual order\<close>
 
 lemma dual_preorder:
   \<open>class.preorder (\<ge>) (>)\<close>
   by standard (auto simp add: less_le_not_le intro: order_trans)
 
 end
 
+lemma preordering_preorderI:
+  \<open>class.preorder (\<^bold>\<le>) (\<^bold><)\<close> if \<open>preordering (\<^bold>\<le>) (\<^bold><)\<close>
+    for less_eq (infix \<open>\<^bold>\<le>\<close> 50) and less (infix \<open>\<^bold><\<close> 50)
+proof -
+  from that interpret preordering \<open>(\<^bold>\<le>)\<close> \<open>(\<^bold><)\<close> .
+  show ?thesis
+    by standard (auto simp add: strict_iff_not refl intro: trans)
+qed
+
+
 
 subsection \<open>Partial orders\<close>
 
 class order = preorder +
   assumes order_antisym: "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y"
 begin
 
 lemma less_le: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> y"
   by (auto simp add: less_le_not_le intro: order_antisym)
 
 sublocale order: ordering less_eq less + dual_order: ordering greater_eq greater
 proof -
   interpret ordering less_eq less
     by standard (auto intro: order_antisym order_trans simp add: less_le)
   show "ordering less_eq less"
     by (fact ordering_axioms)
   then show "ordering greater_eq greater"
     by (rule ordering_dualI)
 qed
 
 print_theorems
 
 text \<open>Reflexivity.\<close>
 
 lemma le_less: "x \<le> y \<longleftrightarrow> x < y \<or> x = y"
     \<comment> \<open>NOT suitable for iff, since it can cause PROOF FAILED.\<close>
 by (fact order.order_iff_strict)
 
 lemma le_imp_less_or_eq: "x \<le> y \<Longrightarrow> x < y \<or> x = y"
 by (simp add: less_le)
 
 
 text \<open>Useful for simplification, but too risky to include by default.\<close>
 
 lemma less_imp_not_eq: "x < y \<Longrightarrow> (x = y) \<longleftrightarrow> False"
 by auto
 
 lemma less_imp_not_eq2: "x < y \<Longrightarrow> (y = x) \<longleftrightarrow> False"
 by auto
 
 
 text \<open>Transitivity rules for calculational reasoning\<close>
 
 lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<le> b \<Longrightarrow> a < b"
 by (fact order.not_eq_order_implies_strict)
 
 lemma le_neq_trans: "a \<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a < b"
 by (rule order.not_eq_order_implies_strict)
 
 
 text \<open>Asymmetry.\<close>
 
 lemma order_eq_iff: "x = y \<longleftrightarrow> x \<le> y \<and> y \<le> x"
   by (fact order.eq_iff)
 
 lemma antisym_conv: "y \<le> x \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y"
   by (simp add: order.eq_iff)
 
 lemma less_imp_neq: "x < y \<Longrightarrow> x \<noteq> y"
   by (fact order.strict_implies_not_eq)
 
 lemma antisym_conv1: "\<not> x < y \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y"
   by (simp add: local.le_less)
 
 lemma antisym_conv2: "x \<le> y \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
   by (simp add: local.less_le)
 
 lemma leD: "y \<le> x \<Longrightarrow> \<not> x < y"
   by (auto simp: less_le order.antisym)
 
 text \<open>Least value operator\<close>
 
 definition (in ord)
   Least :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "LEAST " 10) where
   "Least P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<le> y))"
 
 lemma Least_equality:
   assumes "P x"
     and "\<And>y. P y \<Longrightarrow> x \<le> y"
   shows "Least P = x"
 unfolding Least_def by (rule the_equality)
   (blast intro: assms order.antisym)+
 
 lemma LeastI2_order:
   assumes "P x"
     and "\<And>y. P y \<Longrightarrow> x \<le> y"
     and "\<And>x. P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> x \<le> y \<Longrightarrow> Q x"
   shows "Q (Least P)"
 unfolding Least_def by (rule theI2)
   (blast intro: assms order.antisym)+
 
 lemma Least_ex1:
   assumes   "\<exists>!x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<le> y)"
   shows     Least1I: "P (Least P)" and Least1_le: "P z \<Longrightarrow> Least P \<le> z"
   using     theI'[OF assms]
   unfolding Least_def
   by        auto
 
 text \<open>Greatest value operator\<close>
 
 definition Greatest :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "GREATEST " 10) where
 "Greatest P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<ge> y))"
 
 lemma GreatestI2_order:
   "\<lbrakk> P x;
     \<And>y. P y \<Longrightarrow> x \<ge> y;
     \<And>x. \<lbrakk> P x; \<forall>y. P y \<longrightarrow> x \<ge> y \<rbrakk> \<Longrightarrow> Q x \<rbrakk>
   \<Longrightarrow> Q (Greatest P)"
 unfolding Greatest_def
 by (rule theI2) (blast intro: order.antisym)+
 
 lemma Greatest_equality:
   "\<lbrakk> P x;  \<And>y. P y \<Longrightarrow> x \<ge> y \<rbrakk> \<Longrightarrow> Greatest P = x"
 unfolding Greatest_def
 by (rule the_equality) (blast intro: order.antisym)+
 
 end
 
 lemma ordering_orderI:
   fixes less_eq (infix "\<^bold>\<le>" 50)
     and less (infix "\<^bold><" 50)
   assumes "ordering less_eq less"
   shows "class.order less_eq less"
 proof -
   from assms interpret ordering less_eq less .
   show ?thesis
     by standard (auto intro: antisym trans simp add: refl strict_iff_order)
 qed
 
 lemma order_strictI:
-  fixes less (infix "\<sqsubset>" 50)
-    and less_eq (infix "\<sqsubseteq>" 50)
-  assumes "\<And>a b. a \<sqsubseteq> b \<longleftrightarrow> a \<sqsubset> b \<or> a = b"
-    assumes "\<And>a b. a \<sqsubset> b \<Longrightarrow> \<not> b \<sqsubset> a"
-  assumes "\<And>a. \<not> a \<sqsubset> a"
-  assumes "\<And>a b c. a \<sqsubset> b \<Longrightarrow> b \<sqsubset> c \<Longrightarrow> a \<sqsubset> c"
+  fixes less (infix "\<^bold><" 50)
+    and less_eq (infix "\<^bold>\<le>" 50)
+  assumes "\<And>a b. a \<^bold>\<le> b \<longleftrightarrow> a \<^bold>< b \<or> a = b"
+    assumes "\<And>a b. a \<^bold>< b \<Longrightarrow> \<not> b \<^bold>< a"
+  assumes "\<And>a. \<not> a \<^bold>< a"
+  assumes "\<And>a b c. a \<^bold>< b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c"
   shows "class.order less_eq less"
   by (rule ordering_orderI) (rule ordering_strictI, (fact assms)+)
 
 context order
 begin
 
 text \<open>Dual order\<close>
 
 lemma dual_order:
   "class.order (\<ge>) (>)"
   using dual_order.ordering_axioms by (rule ordering_orderI)
 
 end
 
 
 subsection \<open>Linear (total) orders\<close>
 
 class linorder = order +
   assumes linear: "x \<le> y \<or> y \<le> x"
 begin
 
 lemma less_linear: "x < y \<or> x = y \<or> y < x"
 unfolding less_le using less_le linear by blast
 
 lemma le_less_linear: "x \<le> y \<or> y < x"
 by (simp add: le_less less_linear)
 
 lemma le_cases [case_names le ge]:
   "(x \<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<le> x \<Longrightarrow> P) \<Longrightarrow> P"
 using linear by blast
 
 lemma (in linorder) le_cases3:
   "\<lbrakk>\<lbrakk>x \<le> y; y \<le> z\<rbrakk> \<Longrightarrow> P; \<lbrakk>y \<le> x; x \<le> z\<rbrakk> \<Longrightarrow> P; \<lbrakk>x \<le> z; z \<le> y\<rbrakk> \<Longrightarrow> P;
     \<lbrakk>z \<le> y; y \<le> x\<rbrakk> \<Longrightarrow> P; \<lbrakk>y \<le> z; z \<le> x\<rbrakk> \<Longrightarrow> P; \<lbrakk>z \<le> x; x \<le> y\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
 by (blast intro: le_cases)
 
 lemma linorder_cases [case_names less equal greater]:
   "(x < y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y < x \<Longrightarrow> P) \<Longrightarrow> P"
 using less_linear by blast
 
 lemma linorder_wlog[case_names le sym]:
   "(\<And>a b. a \<le> b \<Longrightarrow> P a b) \<Longrightarrow> (\<And>a b. P b a \<Longrightarrow> P a b) \<Longrightarrow> P a b"
   by (cases rule: le_cases[of a b]) blast+
 
 lemma not_less: "\<not> x < y \<longleftrightarrow> y \<le> x"
   unfolding less_le
   using linear by (blast intro: order.antisym)
 
 lemma not_less_iff_gr_or_eq: "\<not>(x < y) \<longleftrightarrow> (x > y \<or> x = y)"
   by (auto simp add:not_less le_less)
 
 lemma not_le: "\<not> x \<le> y \<longleftrightarrow> y < x"
   unfolding less_le
   using linear by (blast intro: order.antisym)
 
 lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x < y \<or> y < x"
 by (cut_tac x = x and y = y in less_linear, auto)
 
 lemma neqE: "x \<noteq> y \<Longrightarrow> (x < y \<Longrightarrow> R) \<Longrightarrow> (y < x \<Longrightarrow> R) \<Longrightarrow> R"
 by (simp add: neq_iff) blast
 
 lemma antisym_conv3: "\<not> y < x \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
 by (blast intro: order.antisym dest: not_less [THEN iffD1])
 
 lemma leI: "\<not> x < y \<Longrightarrow> y \<le> x"
 unfolding not_less .
 
 lemma not_le_imp_less: "\<not> y \<le> x \<Longrightarrow> x < y"
 unfolding not_le .
 
 lemma linorder_less_wlog[case_names less refl sym]:
      "\<lbrakk>\<And>a b. a < b \<Longrightarrow> P a b;  \<And>a. P a a;  \<And>a b. P b a \<Longrightarrow> P a b\<rbrakk> \<Longrightarrow> P a b"
   using antisym_conv3 by blast
 
 text \<open>Dual order\<close>
 
 lemma dual_linorder:
   "class.linorder (\<ge>) (>)"
 by (rule class.linorder.intro, rule dual_order) (unfold_locales, rule linear)
 
 end
 
 
 text \<open>Alternative introduction rule with bias towards strict order\<close>
 
 lemma linorder_strictI:
   fixes less_eq (infix "\<^bold>\<le>" 50)
     and less (infix "\<^bold><" 50)
   assumes "class.order less_eq less"
   assumes trichotomy: "\<And>a b. a \<^bold>< b \<or> a = b \<or> b \<^bold>< a"
   shows "class.linorder less_eq less"
 proof -
   interpret order less_eq less
     by (fact \<open>class.order less_eq less\<close>)
   show ?thesis
   proof
     fix a b
     show "a \<^bold>\<le> b \<or> b \<^bold>\<le> a"
       using trichotomy by (auto simp add: le_less)
   qed
 qed
 
 
 subsection \<open>Reasoning tools setup\<close>
 
 ML \<open>
 structure Logic_Signature : LOGIC_SIGNATURE = struct
   val mk_Trueprop = HOLogic.mk_Trueprop
   val dest_Trueprop = HOLogic.dest_Trueprop
   val Trueprop_conv = HOLogic.Trueprop_conv
   val Not = HOLogic.Not
   val conj = HOLogic.conj
   val disj = HOLogic.disj
   
   val notI = @{thm notI}
   val ccontr = @{thm ccontr}
   val conjI = @{thm conjI}  
   val conjE = @{thm conjE}
   val disjE = @{thm disjE}
 
   val not_not_conv = Conv.rewr_conv @{thm eq_reflection[OF not_not]}
   val de_Morgan_conj_conv = Conv.rewr_conv @{thm eq_reflection[OF de_Morgan_conj]}
   val de_Morgan_disj_conv = Conv.rewr_conv @{thm eq_reflection[OF de_Morgan_disj]}
   val conj_disj_distribL_conv = Conv.rewr_conv @{thm eq_reflection[OF conj_disj_distribL]}
   val conj_disj_distribR_conv = Conv.rewr_conv @{thm eq_reflection[OF conj_disj_distribR]}
 end
 
 structure HOL_Base_Order_Tac = Base_Order_Tac(
   structure Logic_Sig = Logic_Signature;
   (* Exclude types with specialised solvers. *)
   val excluded_types = [HOLogic.natT, HOLogic.intT, HOLogic.realT]
 )
 
 structure HOL_Order_Tac = Order_Tac(structure Base_Tac = HOL_Base_Order_Tac)
 
 fun print_orders ctxt0 =
   let
     val ctxt = Config.put show_sorts true ctxt0
     val orders = HOL_Order_Tac.Data.get (Context.Proof ctxt)
     fun pretty_term t = Pretty.block
       [Pretty.quote (Syntax.pretty_term ctxt t), Pretty.brk 1,
         Pretty.str "::", Pretty.brk 1,
         Pretty.quote (Syntax.pretty_typ ctxt (type_of t)), Pretty.brk 1]
     fun pretty_order ({kind = kind, ops = ops, ...}, _) =
       Pretty.block ([Pretty.str (@{make_string} kind), Pretty.str ":", Pretty.brk 1]
                     @ map pretty_term ops)
   in
     Pretty.writeln (Pretty.big_list "order structures:" (map pretty_order orders))
   end
 
 val _ =
   Outer_Syntax.command \<^command_keyword>\<open>print_orders\<close>
     "print order structures available to transitivity reasoner"
     (Scan.succeed (Toplevel.keep (print_orders o Toplevel.context_of)))
 
 \<close>
 
 method_setup order = \<open>
   Scan.succeed (fn ctxt => SIMPLE_METHOD' (HOL_Order_Tac.tac [] ctxt))
 \<close> "transitivity reasoner"
 
 
 text \<open>Declarations to set up transitivity reasoner of partial and linear orders.\<close>
 
 context order
 begin
 
 lemma nless_le: "(\<not> a < b) \<longleftrightarrow> (\<not> a \<le> b) \<or> a = b"
   using local.dual_order.order_iff_strict by blast
 
 local_setup \<open>
   HOL_Order_Tac.declare_order {
     ops = {eq = @{term \<open>(=) :: 'a \<Rightarrow> 'a \<Rightarrow> bool\<close>}, le = @{term \<open>(\<le>)\<close>}, lt = @{term \<open>(<)\<close>}},
     thms = {trans = @{thm order_trans}, refl = @{thm order_refl}, eqD1 = @{thm eq_refl},
             eqD2 = @{thm eq_refl[OF sym]}, antisym = @{thm order_antisym}, contr = @{thm notE}},
     conv_thms = {less_le = @{thm eq_reflection[OF less_le]},
                  nless_le = @{thm eq_reflection[OF nless_le]}}
   }
 \<close>
 
 end
 
 context linorder
 begin
 
 lemma nle_le: "(\<not> a \<le> b) \<longleftrightarrow> b \<le> a \<and> b \<noteq> a"
   using not_le less_le by simp
 
 local_setup \<open>
   HOL_Order_Tac.declare_linorder {
     ops = {eq = @{term \<open>(=) :: 'a \<Rightarrow> 'a \<Rightarrow> bool\<close>}, le = @{term \<open>(\<le>)\<close>}, lt = @{term \<open>(<)\<close>}},
     thms = {trans = @{thm order_trans}, refl = @{thm order_refl}, eqD1 = @{thm eq_refl},
             eqD2 = @{thm eq_refl[OF sym]}, antisym = @{thm order_antisym}, contr = @{thm notE}},
     conv_thms = {less_le = @{thm eq_reflection[OF less_le]},
                  nless_le = @{thm eq_reflection[OF not_less]},
                  nle_le = @{thm eq_reflection[OF nle_le]}}
   }
 \<close>
 
 end
 
 setup \<open>
   map_theory_simpset (fn ctxt0 => ctxt0 addSolver
     mk_solver "Transitivity" (fn ctxt => HOL_Order_Tac.tac (Simplifier.prems_of ctxt) ctxt))
 \<close>
 
 ML \<open>
 local
   fun prp t thm = Thm.prop_of thm = t;  (* FIXME proper aconv!? *)
 in
 
 fun antisym_le_simproc ctxt ct =
   (case Thm.term_of ct of
     (le as Const (_, T)) $ r $ s =>
      (let
         val prems = Simplifier.prems_of ctxt;
         val less = Const (\<^const_name>\<open>less\<close>, T);
         val t = HOLogic.mk_Trueprop(le $ s $ r);
       in
         (case find_first (prp t) prems of
           NONE =>
             let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s)) in
               (case find_first (prp t) prems of
                 NONE => NONE
               | SOME thm => SOME(mk_meta_eq(thm RS @{thm antisym_conv1})))
              end
          | SOME thm => SOME (mk_meta_eq (thm RS @{thm order_class.antisym_conv})))
       end handle THM _ => NONE)
   | _ => NONE);
 
 fun antisym_less_simproc ctxt ct =
   (case Thm.term_of ct of
     NotC $ ((less as Const(_,T)) $ r $ s) =>
      (let
        val prems = Simplifier.prems_of ctxt;
        val le = Const (\<^const_name>\<open>less_eq\<close>, T);
        val t = HOLogic.mk_Trueprop(le $ r $ s);
       in
         (case find_first (prp t) prems of
           NONE =>
             let val t = HOLogic.mk_Trueprop (NotC $ (less $ s $ r)) in
               (case find_first (prp t) prems of
                 NONE => NONE
               | SOME thm => SOME (mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3})))
             end
         | SOME thm => SOME (mk_meta_eq (thm RS @{thm antisym_conv2})))
       end handle THM _ => NONE)                           
   | _ => NONE);
 
 end;
 \<close>
 
 simproc_setup antisym_le ("(x::'a::order) \<le> y") = "K antisym_le_simproc"
 simproc_setup antisym_less ("\<not> (x::'a::linorder) < y") = "K antisym_less_simproc"
 
 
 subsection \<open>Bounded quantifiers\<close>
 
 syntax (ASCII)
   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)
 
   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)
 
   "_All_neq" :: "[idt, 'a, bool] => bool"    ("(3ALL _~=_./ _)"  [0, 0, 10] 10)
   "_Ex_neq" :: "[idt, 'a, bool] => bool"    ("(3EX _~=_./ _)"  [0, 0, 10] 10)
 
 syntax
   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
 
   "_All_greater" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
   "_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
   "_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
   "_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
 
   "_All_neq" :: "[idt, 'a, bool] => bool"    ("(3\<forall>_\<noteq>_./ _)"  [0, 0, 10] 10)
   "_Ex_neq" :: "[idt, 'a, bool] => bool"    ("(3\<exists>_\<noteq>_./ _)"  [0, 0, 10] 10)
 
 syntax (input)
   "_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
   "_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
   "_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
   "_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
   "_All_neq" :: "[idt, 'a, bool] => bool"    ("(3! _~=_./ _)"  [0, 0, 10] 10)
   "_Ex_neq" :: "[idt, 'a, bool] => bool"    ("(3? _~=_./ _)"  [0, 0, 10] 10)
 
 translations
   "\<forall>x<y. P" \<rightharpoonup> "\<forall>x. x < y \<longrightarrow> P"
   "\<exists>x<y. P" \<rightharpoonup> "\<exists>x. x < y \<and> P"
   "\<forall>x\<le>y. P" \<rightharpoonup> "\<forall>x. x \<le> y \<longrightarrow> P"
   "\<exists>x\<le>y. P" \<rightharpoonup> "\<exists>x. x \<le> y \<and> P"
   "\<forall>x>y. P" \<rightharpoonup> "\<forall>x. x > y \<longrightarrow> P"
   "\<exists>x>y. P" \<rightharpoonup> "\<exists>x. x > y \<and> P"
   "\<forall>x\<ge>y. P" \<rightharpoonup> "\<forall>x. x \<ge> y \<longrightarrow> P"
   "\<exists>x\<ge>y. P" \<rightharpoonup> "\<exists>x. x \<ge> y \<and> P"
   "\<forall>x\<noteq>y. P" \<rightharpoonup> "\<forall>x. x \<noteq> y \<longrightarrow> P"
   "\<exists>x\<noteq>y. P" \<rightharpoonup> "\<exists>x. x \<noteq> y \<and> P"
 
 print_translation \<open>
 let
   val All_binder = Mixfix.binder_name \<^const_syntax>\<open>All\<close>;
   val Ex_binder = Mixfix.binder_name \<^const_syntax>\<open>Ex\<close>;
   val impl = \<^const_syntax>\<open>HOL.implies\<close>;
   val conj = \<^const_syntax>\<open>HOL.conj\<close>;
   val less = \<^const_syntax>\<open>less\<close>;
   val less_eq = \<^const_syntax>\<open>less_eq\<close>;
 
   val trans =
    [((All_binder, impl, less),
     (\<^syntax_const>\<open>_All_less\<close>, \<^syntax_const>\<open>_All_greater\<close>)),
     ((All_binder, impl, less_eq),
     (\<^syntax_const>\<open>_All_less_eq\<close>, \<^syntax_const>\<open>_All_greater_eq\<close>)),
     ((Ex_binder, conj, less),
     (\<^syntax_const>\<open>_Ex_less\<close>, \<^syntax_const>\<open>_Ex_greater\<close>)),
     ((Ex_binder, conj, less_eq),
     (\<^syntax_const>\<open>_Ex_less_eq\<close>, \<^syntax_const>\<open>_Ex_greater_eq\<close>))];
 
   fun matches_bound v t =
     (case t of
       Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (v', _) => v = v'
     | _ => false);
   fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false);
   fun mk x c n P = Syntax.const c $ Syntax_Trans.mark_bound_body x $ n $ P;
 
   fun tr' q = (q, fn _ =>
     (fn [Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (v, T),
         Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
         (case AList.lookup (=) trans (q, c, d) of
           NONE => raise Match
         | SOME (l, g) =>
             if matches_bound v t andalso not (contains_var v u) then mk (v, T) l u P
             else if matches_bound v u andalso not (contains_var v t) then mk (v, T) g t P
             else raise Match)
       | _ => raise Match));
 in [tr' All_binder, tr' Ex_binder] end
 \<close>
 
 
 subsection \<open>Transitivity reasoning\<close>
 
 context ord
 begin
 
 lemma ord_le_eq_trans: "a \<le> b \<Longrightarrow> b = c \<Longrightarrow> a \<le> c"
   by (rule subst)
 
 lemma ord_eq_le_trans: "a = b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
   by (rule ssubst)
 
 lemma ord_less_eq_trans: "a < b \<Longrightarrow> b = c \<Longrightarrow> a < c"
   by (rule subst)
 
 lemma ord_eq_less_trans: "a = b \<Longrightarrow> b < c \<Longrightarrow> a < c"
   by (rule ssubst)
 
 end
 
 lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
   (!!x y. x < y ==> f x < f y) ==> f a < c"
 proof -
   assume r: "!!x y. x < y ==> f x < f y"
   assume "a < b" hence "f a < f b" by (rule r)
   also assume "f b < c"
   finally (less_trans) show ?thesis .
 qed
 
 lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
   (!!x y. x < y ==> f x < f y) ==> a < f c"
 proof -
   assume r: "!!x y. x < y ==> f x < f y"
   assume "a < f b"
   also assume "b < c" hence "f b < f c" by (rule r)
   finally (less_trans) show ?thesis .
 qed
 
 lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
   (!!x y. x <= y ==> f x <= f y) ==> f a < c"
 proof -
   assume r: "!!x y. x <= y ==> f x <= f y"
   assume "a <= b" hence "f a <= f b" by (rule r)
   also assume "f b < c"
   finally (le_less_trans) show ?thesis .
 qed
 
 lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
   (!!x y. x < y ==> f x < f y) ==> a < f c"
 proof -
   assume r: "!!x y. x < y ==> f x < f y"
   assume "a <= f b"
   also assume "b < c" hence "f b < f c" by (rule r)
   finally (le_less_trans) show ?thesis .
 qed
 
 lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
   (!!x y. x < y ==> f x < f y) ==> f a < c"
 proof -
   assume r: "!!x y. x < y ==> f x < f y"
   assume "a < b" hence "f a < f b" by (rule r)
   also assume "f b <= c"
   finally (less_le_trans) show ?thesis .
 qed
 
 lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
   (!!x y. x <= y ==> f x <= f y) ==> a < f c"
 proof -
   assume r: "!!x y. x <= y ==> f x <= f y"
   assume "a < f b"
   also assume "b <= c" hence "f b <= f c" by (rule r)
   finally (less_le_trans) show ?thesis .
 qed
 
 lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
   (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
 proof -
   assume r: "!!x y. x <= y ==> f x <= f y"
   assume "a <= f b"
   also assume "b <= c" hence "f b <= f c" by (rule r)
   finally (order_trans) show ?thesis .
 qed
 
 lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
   (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
 proof -
   assume r: "!!x y. x <= y ==> f x <= f y"
   assume "a <= b" hence "f a <= f b" by (rule r)
   also assume "f b <= c"
   finally (order_trans) show ?thesis .
 qed
 
 lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
   (!!x y. x <= y ==> f x <= f y) ==> f a <= c"
 proof -
   assume r: "!!x y. x <= y ==> f x <= f y"
   assume "a <= b" hence "f a <= f b" by (rule r)
   also assume "f b = c"
   finally (ord_le_eq_trans) show ?thesis .
 qed
 
 lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
   (!!x y. x <= y ==> f x <= f y) ==> a <= f c"
 proof -
   assume r: "!!x y. x <= y ==> f x <= f y"
   assume "a = f b"
   also assume "b <= c" hence "f b <= f c" by (rule r)
   finally (ord_eq_le_trans) show ?thesis .
 qed
 
 lemma ord_less_eq_subst: "a < b ==> f b = c ==>
   (!!x y. x < y ==> f x < f y) ==> f a < c"
 proof -
   assume r: "!!x y. x < y ==> f x < f y"
   assume "a < b" hence "f a < f b" by (rule r)
   also assume "f b = c"
   finally (ord_less_eq_trans) show ?thesis .
 qed
 
 lemma ord_eq_less_subst: "a = f b ==> b < c ==>
   (!!x y. x < y ==> f x < f y) ==> a < f c"
 proof -
   assume r: "!!x y. x < y ==> f x < f y"
   assume "a = f b"
   also assume "b < c" hence "f b < f c" by (rule r)
   finally (ord_eq_less_trans) show ?thesis .
 qed
 
 text \<open>
   Note that this list of rules is in reverse order of priorities.
 \<close>
 
 lemmas [trans] =
   order_less_subst2
   order_less_subst1
   order_le_less_subst2
   order_le_less_subst1
   order_less_le_subst2
   order_less_le_subst1
   order_subst2
   order_subst1
   ord_le_eq_subst
   ord_eq_le_subst
   ord_less_eq_subst
   ord_eq_less_subst
   forw_subst
   back_subst
   rev_mp
   mp
 
 lemmas (in order) [trans] =
   neq_le_trans
   le_neq_trans
 
 lemmas (in preorder) [trans] =
   less_trans
   less_asym'
   le_less_trans
   less_le_trans
   order_trans
 
 lemmas (in order) [trans] =
   order.antisym
 
 lemmas (in ord) [trans] =
   ord_le_eq_trans
   ord_eq_le_trans
   ord_less_eq_trans
   ord_eq_less_trans
 
 lemmas [trans] =
   trans
 
 lemmas order_trans_rules =
   order_less_subst2
   order_less_subst1
   order_le_less_subst2
   order_le_less_subst1
   order_less_le_subst2
   order_less_le_subst1
   order_subst2
   order_subst1
   ord_le_eq_subst
   ord_eq_le_subst
   ord_less_eq_subst
   ord_eq_less_subst
   forw_subst
   back_subst
   rev_mp
   mp
   neq_le_trans
   le_neq_trans
   less_trans
   less_asym'
   le_less_trans
   less_le_trans
   order_trans
   order.antisym
   ord_le_eq_trans
   ord_eq_le_trans
   ord_less_eq_trans
   ord_eq_less_trans
   trans
 
 text \<open>These support proving chains of decreasing inequalities
     a >= b >= c ... in Isar proofs.\<close>
 
 lemma xt1 [no_atp]:
   "a = b \<Longrightarrow> b > c \<Longrightarrow> a > c"
   "a > b \<Longrightarrow> b = c \<Longrightarrow> a > c"
   "a = b \<Longrightarrow> b \<ge> c \<Longrightarrow> a \<ge> c"
   "a \<ge> b \<Longrightarrow> b = c \<Longrightarrow> a \<ge> c"
   "(x::'a::order) \<ge> y \<Longrightarrow> y \<ge> x \<Longrightarrow> x = y"
   "(x::'a::order) \<ge> y \<Longrightarrow> y \<ge> z \<Longrightarrow> x \<ge> z"
   "(x::'a::order) > y \<Longrightarrow> y \<ge> z \<Longrightarrow> x > z"
   "(x::'a::order) \<ge> y \<Longrightarrow> y > z \<Longrightarrow> x > z"
   "(a::'a::order) > b \<Longrightarrow> b > a \<Longrightarrow> P"
   "(x::'a::order) > y \<Longrightarrow> y > z \<Longrightarrow> x > z"
   "(a::'a::order) \<ge> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a > b"
   "(a::'a::order) \<noteq> b \<Longrightarrow> a \<ge> b \<Longrightarrow> a > b"
   "a = f b \<Longrightarrow> b > c \<Longrightarrow> (\<And>x y. x > y \<Longrightarrow> f x > f y) \<Longrightarrow> a > f c"
   "a > b \<Longrightarrow> f b = c \<Longrightarrow> (\<And>x y. x > y \<Longrightarrow> f x > f y) \<Longrightarrow> f a > c"
   "a = f b \<Longrightarrow> b \<ge> c \<Longrightarrow> (\<And>x y. x \<ge> y \<Longrightarrow> f x \<ge> f y) \<Longrightarrow> a \<ge> f c"
   "a \<ge> b \<Longrightarrow> f b = c \<Longrightarrow> (\<And>x y. x \<ge> y \<Longrightarrow> f x \<ge> f y) \<Longrightarrow> f a \<ge> c"
   by auto
 
 lemma xt2 [no_atp]:
   "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"
 by (subgoal_tac "f b >= f c", force, force)
 
 lemma xt3 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==>
     (!!x y. x >= y ==> f x >= f y) ==> f a >= c"
 by (subgoal_tac "f a >= f b", force, force)
 
 lemma xt4 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) >= c ==>
   (!!x y. x >= y ==> f x >= f y) ==> a > f c"
 by (subgoal_tac "f b >= f c", force, force)
 
 lemma xt5 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) >= c==>
     (!!x y. x > y ==> f x > f y) ==> f a > c"
 by (subgoal_tac "f a > f b", force, force)
 
 lemma xt6 [no_atp]: "(a::'a::order) >= f b ==> b > c ==>
     (!!x y. x > y ==> f x > f y) ==> a > f c"
 by (subgoal_tac "f b > f c", force, force)
 
 lemma xt7 [no_atp]: "(a::'a::order) >= b ==> (f b::'b::order) > c ==>
     (!!x y. x >= y ==> f x >= f y) ==> f a > c"
 by (subgoal_tac "f a >= f b", force, force)
 
 lemma xt8 [no_atp]: "(a::'a::order) > f b ==> (b::'b::order) > c ==>
     (!!x y. x > y ==> f x > f y) ==> a > f c"
 by (subgoal_tac "f b > f c", force, force)
 
 lemma xt9 [no_atp]: "(a::'a::order) > b ==> (f b::'b::order) > c ==>
     (!!x y. x > y ==> f x > f y) ==> f a > c"
 by (subgoal_tac "f a > f b", force, force)
 
 lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
 
 (*
   Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands
   for the wrong thing in an Isar proof.
 
   The extra transitivity rules can be used as follows:
 
 lemma "(a::'a::order) > z"
 proof -
   have "a >= b" (is "_ >= ?rhs")
     sorry
   also have "?rhs >= c" (is "_ >= ?rhs")
     sorry
   also (xtrans) have "?rhs = d" (is "_ = ?rhs")
     sorry
   also (xtrans) have "?rhs >= e" (is "_ >= ?rhs")
     sorry
   also (xtrans) have "?rhs > f" (is "_ > ?rhs")
     sorry
   also (xtrans) have "?rhs > z"
     sorry
   finally (xtrans) show ?thesis .
 qed
 
   Alternatively, one can use "declare xtrans [trans]" and then
   leave out the "(xtrans)" above.
 *)
 
 
 subsection \<open>Monotonicity\<close>
 
 context order
 begin
 
 definition mono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool" where
   "mono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
 
 lemma monoI [intro?]:
   fixes f :: "'a \<Rightarrow> 'b::order"
   shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y) \<Longrightarrow> mono f"
   unfolding mono_def by iprover
 
 lemma monoD [dest?]:
   fixes f :: "'a \<Rightarrow> 'b::order"
   shows "mono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
   unfolding mono_def by iprover
 
 lemma monoE:
   fixes f :: "'a \<Rightarrow> 'b::order"
   assumes "mono f"
   assumes "x \<le> y"
   obtains "f x \<le> f y"
 proof
   from assms show "f x \<le> f y" by (simp add: mono_def)
 qed
 
 definition antimono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool" where
   "antimono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<ge> f y)"
 
 lemma antimonoI [intro?]:
   fixes f :: "'a \<Rightarrow> 'b::order"
   shows "(\<And>x y. x \<le> y \<Longrightarrow> f x \<ge> f y) \<Longrightarrow> antimono f"
   unfolding antimono_def by iprover
 
 lemma antimonoD [dest?]:
   fixes f :: "'a \<Rightarrow> 'b::order"
   shows "antimono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<ge> f y"
   unfolding antimono_def by iprover
 
 lemma antimonoE:
   fixes f :: "'a \<Rightarrow> 'b::order"
   assumes "antimono f"
   assumes "x \<le> y"
   obtains "f x \<ge> f y"
 proof
   from assms show "f x \<ge> f y" by (simp add: antimono_def)
 qed
 
 definition strict_mono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool" where
   "strict_mono f \<longleftrightarrow> (\<forall>x y. x < y \<longrightarrow> f x < f y)"
 
 lemma strict_monoI [intro?]:
   assumes "\<And>x y. x < y \<Longrightarrow> f x < f y"
   shows "strict_mono f"
   using assms unfolding strict_mono_def by auto
 
 lemma strict_monoD [dest?]:
   "strict_mono f \<Longrightarrow> x < y \<Longrightarrow> f x < f y"
   unfolding strict_mono_def by auto
 
 lemma strict_mono_mono [dest?]:
   assumes "strict_mono f"
   shows "mono f"
 proof (rule monoI)
   fix x y
   assume "x \<le> y"
   show "f x \<le> f y"
   proof (cases "x = y")
     case True then show ?thesis by simp
   next
     case False with \<open>x \<le> y\<close> have "x < y" by simp
     with assms strict_monoD have "f x < f y" by auto
     then show ?thesis by simp
 
   qed
 qed
 
 end
 
 context linorder
 begin
 
 lemma mono_invE:
   fixes f :: "'a \<Rightarrow> 'b::order"
   assumes "mono f"
   assumes "f x < f y"
   obtains "x \<le> y"
 proof
   show "x \<le> y"
   proof (rule ccontr)
     assume "\<not> x \<le> y"
     then have "y \<le> x" by simp
     with \<open>mono f\<close> obtain "f y \<le> f x" by (rule monoE)
     with \<open>f x < f y\<close> show False by simp
   qed
 qed
 
 lemma mono_strict_invE:
   fixes f :: "'a \<Rightarrow> 'b::order"
   assumes "mono f"
   assumes "f x < f y"
   obtains "x < y"
 proof
   show "x < y"
   proof (rule ccontr)
     assume "\<not> x < y"
     then have "y \<le> x" by simp
     with \<open>mono f\<close> obtain "f y \<le> f x" by (rule monoE)
     with \<open>f x < f y\<close> show False by simp
   qed
 qed
 
 lemma strict_mono_eq:
   assumes "strict_mono f"
   shows "f x = f y \<longleftrightarrow> x = y"
 proof
   assume "f x = f y"
   show "x = y" proof (cases x y rule: linorder_cases)
     case less with assms strict_monoD have "f x < f y" by auto
     with \<open>f x = f y\<close> show ?thesis by simp
   next
     case equal then show ?thesis .
   next
     case greater with assms strict_monoD have "f y < f x" by auto
     with \<open>f x = f y\<close> show ?thesis by simp
   qed
 qed simp
 
 lemma strict_mono_less_eq:
   assumes "strict_mono f"
   shows "f x \<le> f y \<longleftrightarrow> x \<le> y"
 proof
   assume "x \<le> y"
   with assms strict_mono_mono monoD show "f x \<le> f y" by auto
 next
   assume "f x \<le> f y"
   show "x \<le> y" proof (rule ccontr)
     assume "\<not> x \<le> y" then have "y < x" by simp
     with assms strict_monoD have "f y < f x" by auto
     with \<open>f x \<le> f y\<close> show False by simp
   qed
 qed
 
 lemma strict_mono_less:
   assumes "strict_mono f"
   shows "f x < f y \<longleftrightarrow> x < y"
   using assms
     by (auto simp add: less_le Orderings.less_le strict_mono_eq strict_mono_less_eq)
 
 end
 
 
 subsection \<open>min and max -- fundamental\<close>
 
 definition (in ord) min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
   "min a b = (if a \<le> b then a else b)"
 
 definition (in ord) max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
   "max a b = (if a \<le> b then b else a)"
 
 lemma min_absorb1: "x \<le> y \<Longrightarrow> min x y = x"
   by (simp add: min_def)
 
 lemma max_absorb2: "x \<le> y \<Longrightarrow> max x y = y"
   by (simp add: max_def)
 
 lemma min_absorb2: "(y::'a::order) \<le> x \<Longrightarrow> min x y = y"
   by (simp add:min_def)
 
 lemma max_absorb1: "(y::'a::order) \<le> x \<Longrightarrow> max x y = x"
   by (simp add: max_def)
 
 lemma max_min_same [simp]:
   fixes x y :: "'a :: linorder"
   shows "max x (min x y) = x" "max (min x y) x = x" "max (min x y) y = y" "max y (min x y) = y"
 by(auto simp add: max_def min_def)
 
 
 subsection \<open>(Unique) top and bottom elements\<close>
 
 class bot =
   fixes bot :: 'a ("\<bottom>")
 
 class order_bot = order + bot +
   assumes bot_least: "\<bottom> \<le> a"
 begin
 
 sublocale bot: ordering_top greater_eq greater bot
   by standard (fact bot_least)
 
 lemma le_bot:
   "a \<le> \<bottom> \<Longrightarrow> a = \<bottom>"
   by (fact bot.extremum_uniqueI)
 
 lemma bot_unique:
   "a \<le> \<bottom> \<longleftrightarrow> a = \<bottom>"
   by (fact bot.extremum_unique)
 
 lemma not_less_bot:
   "\<not> a < \<bottom>"
   by (fact bot.extremum_strict)
 
 lemma bot_less:
   "a \<noteq> \<bottom> \<longleftrightarrow> \<bottom> < a"
   by (fact bot.not_eq_extremum)
 
 lemma max_bot[simp]: "max bot x = x"
 by(simp add: max_def bot_unique)
 
 lemma max_bot2[simp]: "max x bot = x"
 by(simp add: max_def bot_unique)
 
 lemma min_bot[simp]: "min bot x = bot"
 by(simp add: min_def bot_unique)
 
 lemma min_bot2[simp]: "min x bot = bot"
 by(simp add: min_def bot_unique)
 
 end
 
 class top =
   fixes top :: 'a ("\<top>")
 
 class order_top = order + top +
   assumes top_greatest: "a \<le> \<top>"
 begin
 
 sublocale top: ordering_top less_eq less top
   by standard (fact top_greatest)
 
 lemma top_le:
   "\<top> \<le> a \<Longrightarrow> a = \<top>"
   by (fact top.extremum_uniqueI)
 
 lemma top_unique:
   "\<top> \<le> a \<longleftrightarrow> a = \<top>"
   by (fact top.extremum_unique)
 
 lemma not_top_less:
   "\<not> \<top> < a"
   by (fact top.extremum_strict)
 
 lemma less_top:
   "a \<noteq> \<top> \<longleftrightarrow> a < \<top>"
   by (fact top.not_eq_extremum)
 
 lemma max_top[simp]: "max top x = top"
 by(simp add: max_def top_unique)
 
 lemma max_top2[simp]: "max x top = top"
 by(simp add: max_def top_unique)
 
 lemma min_top[simp]: "min top x = x"
 by(simp add: min_def top_unique)
 
 lemma min_top2[simp]: "min x top = x"
 by(simp add: min_def top_unique)
 
 end
 
 
 subsection \<open>Dense orders\<close>
 
 class dense_order = order +
   assumes dense: "x < y \<Longrightarrow> (\<exists>z. x < z \<and> z < y)"
 
 class dense_linorder = linorder + dense_order
 begin
 
 lemma dense_le:
   fixes y z :: 'a
   assumes "\<And>x. x < y \<Longrightarrow> x \<le> z"
   shows "y \<le> z"
 proof (rule ccontr)
   assume "\<not> ?thesis"
   hence "z < y" by simp
   from dense[OF this]
   obtain x where "x < y" and "z < x" by safe
   moreover have "x \<le> z" using assms[OF \<open>x < y\<close>] .
   ultimately show False by auto
 qed
 
 lemma dense_le_bounded:
   fixes x y z :: 'a
   assumes "x < y"
   assumes *: "\<And>w. \<lbrakk> x < w ; w < y \<rbrakk> \<Longrightarrow> w \<le> z"
   shows "y \<le> z"
 proof (rule dense_le)
   fix w assume "w < y"
   from dense[OF \<open>x < y\<close>] obtain u where "x < u" "u < y" by safe
   from linear[of u w]
   show "w \<le> z"
   proof (rule disjE)
     assume "u \<le> w"
     from less_le_trans[OF \<open>x < u\<close> \<open>u \<le> w\<close>] \<open>w < y\<close>
     show "w \<le> z" by (rule *)
   next
     assume "w \<le> u"
     from \<open>w \<le> u\<close> *[OF \<open>x < u\<close> \<open>u < y\<close>]
     show "w \<le> z" by (rule order_trans)
   qed
 qed
 
 lemma dense_ge:
   fixes y z :: 'a
   assumes "\<And>x. z < x \<Longrightarrow> y \<le> x"
   shows "y \<le> z"
 proof (rule ccontr)
   assume "\<not> ?thesis"
   hence "z < y" by simp
   from dense[OF this]
   obtain x where "x < y" and "z < x" by safe
   moreover have "y \<le> x" using assms[OF \<open>z < x\<close>] .
   ultimately show False by auto
 qed
 
 lemma dense_ge_bounded:
   fixes x y z :: 'a
   assumes "z < x"
   assumes *: "\<And>w. \<lbrakk> z < w ; w < x \<rbrakk> \<Longrightarrow> y \<le> w"
   shows "y \<le> z"
 proof (rule dense_ge)
   fix w assume "z < w"
   from dense[OF \<open>z < x\<close>] obtain u where "z < u" "u < x" by safe
   from linear[of u w]
   show "y \<le> w"
   proof (rule disjE)
     assume "w \<le> u"
     from \<open>z < w\<close> le_less_trans[OF \<open>w \<le> u\<close> \<open>u < x\<close>]
     show "y \<le> w" by (rule *)
   next
     assume "u \<le> w"
     from *[OF \<open>z < u\<close> \<open>u < x\<close>] \<open>u \<le> w\<close>
     show "y \<le> w" by (rule order_trans)
   qed
 qed
 
 end
 
 class no_top = order +
   assumes gt_ex: "\<exists>y. x < y"
 
 class no_bot = order +
   assumes lt_ex: "\<exists>y. y < x"
 
 class unbounded_dense_linorder = dense_linorder + no_top + no_bot
 
 
 subsection \<open>Wellorders\<close>
 
 class wellorder = linorder +
   assumes less_induct [case_names less]: "(\<And>x. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P a"
 begin
 
 lemma wellorder_Least_lemma:
   fixes k :: 'a
   assumes "P k"
   shows LeastI: "P (LEAST x. P x)" and Least_le: "(LEAST x. P x) \<le> k"
 proof -
   have "P (LEAST x. P x) \<and> (LEAST x. P x) \<le> k"
   using assms proof (induct k rule: less_induct)
     case (less x) then have "P x" by simp
     show ?case proof (rule classical)
       assume assm: "\<not> (P (LEAST a. P a) \<and> (LEAST a. P a) \<le> x)"
       have "\<And>y. P y \<Longrightarrow> x \<le> y"
       proof (rule classical)
         fix y
         assume "P y" and "\<not> x \<le> y"
         with less have "P (LEAST a. P a)" and "(LEAST a. P a) \<le> y"
           by (auto simp add: not_le)
         with assm have "x < (LEAST a. P a)" and "(LEAST a. P a) \<le> y"
           by auto
         then show "x \<le> y" by auto
       qed
       with \<open>P x\<close> have Least: "(LEAST a. P a) = x"
         by (rule Least_equality)
       with \<open>P x\<close> show ?thesis by simp
     qed
   qed
   then show "P (LEAST x. P x)" and "(LEAST x. P x) \<le> k" by auto
 qed
 
 \<comment> \<open>The following 3 lemmas are due to Brian Huffman\<close>
 lemma LeastI_ex: "\<exists>x. P x \<Longrightarrow> P (Least P)"
   by (erule exE) (erule LeastI)
 
 lemma LeastI2:
   "P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)"
   by (blast intro: LeastI)
 
 lemma LeastI2_ex:
   "\<exists>a. P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)"
   by (blast intro: LeastI_ex)
 
 lemma LeastI2_wellorder:
   assumes "P a"
   and "\<And>a. \<lbrakk> P a; \<forall>b. P b \<longrightarrow> a \<le> b \<rbrakk> \<Longrightarrow> Q a"
   shows "Q (Least P)"
 proof (rule LeastI2_order)
   show "P (Least P)" using \<open>P a\<close> by (rule LeastI)
 next
   fix y assume "P y" thus "Least P \<le> y" by (rule Least_le)
 next
   fix x assume "P x" "\<forall>y. P y \<longrightarrow> x \<le> y" thus "Q x" by (rule assms(2))
 qed
 
 lemma LeastI2_wellorder_ex:
   assumes "\<exists>x. P x"
   and "\<And>a. \<lbrakk> P a; \<forall>b. P b \<longrightarrow> a \<le> b \<rbrakk> \<Longrightarrow> Q a"
   shows "Q (Least P)"
 using assms by clarify (blast intro!: LeastI2_wellorder)
 
 lemma not_less_Least: "k < (LEAST x. P x) \<Longrightarrow> \<not> P k"
 apply (simp add: not_le [symmetric])
 apply (erule contrapos_nn)
 apply (erule Least_le)
 done
 
 lemma exists_least_iff: "(\<exists>n. P n) \<longleftrightarrow> (\<exists>n. P n \<and> (\<forall>m < n. \<not> P m))" (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   assume ?rhs thus ?lhs by blast
 next
   assume H: ?lhs then obtain n where n: "P n" by blast
   let ?x = "Least P"
   { fix m assume m: "m < ?x"
     from not_less_Least[OF m] have "\<not> P m" . }
   with LeastI_ex[OF H] show ?rhs by blast
 qed
 
 end
 
 
 subsection \<open>Order on \<^typ>\<open>bool\<close>\<close>
 
 instantiation bool :: "{order_bot, order_top, linorder}"
 begin
 
 definition
   le_bool_def [simp]: "P \<le> Q \<longleftrightarrow> P \<longrightarrow> Q"
 
 definition
   [simp]: "(P::bool) < Q \<longleftrightarrow> \<not> P \<and> Q"
 
 definition
   [simp]: "\<bottom> \<longleftrightarrow> False"
 
 definition
   [simp]: "\<top> \<longleftrightarrow> True"
 
 instance proof
 qed auto
 
 end
 
 lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
   by simp
 
 lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
   by simp
 
 lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
   by simp
 
 lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
   by simp
 
 lemma bot_boolE: "\<bottom> \<Longrightarrow> P"
   by simp
 
 lemma top_boolI: \<top>
   by simp
 
 lemma [code]:
   "False \<le> b \<longleftrightarrow> True"
   "True \<le> b \<longleftrightarrow> b"
   "False < b \<longleftrightarrow> b"
   "True < b \<longleftrightarrow> False"
   by simp_all
 
 
 subsection \<open>Order on \<^typ>\<open>_ \<Rightarrow> _\<close>\<close>
 
 instantiation "fun" :: (type, ord) ord
 begin
 
 definition
   le_fun_def: "f \<le> g \<longleftrightarrow> (\<forall>x. f x \<le> g x)"
 
 definition
   "(f::'a \<Rightarrow> 'b) < g \<longleftrightarrow> f \<le> g \<and> \<not> (g \<le> f)"
 
 instance ..
 
 end
 
 instance "fun" :: (type, preorder) preorder proof
 qed (auto simp add: le_fun_def less_fun_def
   intro: order_trans order.antisym)
 
 instance "fun" :: (type, order) order proof
 qed (auto simp add: le_fun_def intro: order.antisym)
 
 instantiation "fun" :: (type, bot) bot
 begin
 
 definition
   "\<bottom> = (\<lambda>x. \<bottom>)"
 
 instance ..
 
 end
 
 instantiation "fun" :: (type, order_bot) order_bot
 begin
 
 lemma bot_apply [simp, code]:
   "\<bottom> x = \<bottom>"
   by (simp add: bot_fun_def)
 
 instance proof
 qed (simp add: le_fun_def)
 
 end
 
 instantiation "fun" :: (type, top) top
 begin
 
 definition
   [no_atp]: "\<top> = (\<lambda>x. \<top>)"
 
 instance ..
 
 end
 
 instantiation "fun" :: (type, order_top) order_top
 begin
 
 lemma top_apply [simp, code]:
   "\<top> x = \<top>"
   by (simp add: top_fun_def)
 
 instance proof
 qed (simp add: le_fun_def)
 
 end
 
 lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g"
   unfolding le_fun_def by simp
 
 lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P"
   unfolding le_fun_def by simp
 
 lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x"
   by (rule le_funE)
 
 lemma mono_compose: "mono Q \<Longrightarrow> mono (\<lambda>i x. Q i (f x))"
   unfolding mono_def le_fun_def by auto
 
 
 subsection \<open>Order on unary and binary predicates\<close>
 
 lemma predicate1I:
   assumes PQ: "\<And>x. P x \<Longrightarrow> Q x"
   shows "P \<le> Q"
   apply (rule le_funI)
   apply (rule le_boolI)
   apply (rule PQ)
   apply assumption
   done
 
 lemma predicate1D:
   "P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x"
   apply (erule le_funE)
   apply (erule le_boolE)
   apply assumption+
   done
 
 lemma rev_predicate1D:
   "P x \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x"
   by (rule predicate1D)
 
 lemma predicate2I:
   assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y"
   shows "P \<le> Q"
   apply (rule le_funI)+
   apply (rule le_boolI)
   apply (rule PQ)
   apply assumption
   done
 
 lemma predicate2D:
   "P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y"
   apply (erule le_funE)+
   apply (erule le_boolE)
   apply assumption+
   done
 
 lemma rev_predicate2D:
   "P x y \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x y"
   by (rule predicate2D)
 
 lemma bot1E [no_atp]: "\<bottom> x \<Longrightarrow> P"
   by (simp add: bot_fun_def)
 
 lemma bot2E: "\<bottom> x y \<Longrightarrow> P"
   by (simp add: bot_fun_def)
 
 lemma top1I: "\<top> x"
   by (simp add: top_fun_def)
 
 lemma top2I: "\<top> x y"
   by (simp add: top_fun_def)
 
 
 subsection \<open>Name duplicates\<close>
 
 lemmas antisym = order.antisym
 lemmas eq_iff = order.eq_iff
 
 lemmas order_eq_refl = preorder_class.eq_refl
 lemmas order_less_irrefl = preorder_class.less_irrefl
 lemmas order_less_imp_le = preorder_class.less_imp_le
 lemmas order_less_not_sym = preorder_class.less_not_sym
 lemmas order_less_asym = preorder_class.less_asym
 lemmas order_less_trans = preorder_class.less_trans
 lemmas order_le_less_trans = preorder_class.le_less_trans
 lemmas order_less_le_trans = preorder_class.less_le_trans
 lemmas order_less_imp_not_less = preorder_class.less_imp_not_less
 lemmas order_less_imp_triv = preorder_class.less_imp_triv
 lemmas order_less_asym' = preorder_class.less_asym'
 
 lemmas order_less_le = order_class.less_le
 lemmas order_le_less = order_class.le_less
 lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq
 lemmas order_less_imp_not_eq = order_class.less_imp_not_eq
 lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2
 lemmas order_neq_le_trans = order_class.neq_le_trans
 lemmas order_le_neq_trans = order_class.le_neq_trans
 lemmas order_eq_iff = order_class.order.eq_iff
 lemmas order_antisym_conv = order_class.antisym_conv
 
 lemmas linorder_linear = linorder_class.linear
 lemmas linorder_less_linear = linorder_class.less_linear
 lemmas linorder_le_less_linear = linorder_class.le_less_linear
 lemmas linorder_le_cases = linorder_class.le_cases
 lemmas linorder_not_less = linorder_class.not_less
 lemmas linorder_not_le = linorder_class.not_le
 lemmas linorder_neq_iff = linorder_class.neq_iff
 lemmas linorder_neqE = linorder_class.neqE
 
 end