# HG changeset patch
# User nipkow
# Date 1108057872 -3600
# Node ID 2ef571f80a552f57fe40401c6937dc97728360f9
# Parent  617996110388f4d1c26b7718da418f7c6b0319e7
Moved oderings from HOL into the new Orderings.thy

diff -r 617996110388 -r 2ef571f80a55 src/HOL/HOL.ML
--- a/src/HOL/HOL.ML	Thu Feb 10 17:09:15 2005 +0100
+++ b/src/HOL/HOL.ML	Thu Feb 10 18:51:12 2005 +0100
@@ -1,58 +1,5 @@
-
 (* legacy ML bindings *)
 
-val Least_def = thm "Least_def";
-val Least_equality = thm "Least_equality";
-val mono_def = thm "mono_def";
-val monoI = thm "monoI";
-val monoD = thm "monoD";
-val min_def = thm "min_def";
-val min_of_mono = thm "min_of_mono";
-val max_def = thm "max_def";
-val max_of_mono = thm "max_of_mono";
-val min_leastL = thm "min_leastL";
-val max_leastL = thm "max_leastL";
-val min_leastR = thm "min_leastR";
-val max_leastR = thm "max_leastR";
-val order_eq_refl = thm "order_eq_refl";
-val order_less_irrefl = thm "order_less_irrefl";
-val order_le_less = thm "order_le_less";
-val order_le_imp_less_or_eq = thm "order_le_imp_less_or_eq";
-val order_less_imp_le = thm "order_less_imp_le";
-val order_less_not_sym = thm "order_less_not_sym";
-val order_less_asym = thm "order_less_asym";
-val order_less_trans = thm "order_less_trans";
-val order_le_less_trans = thm "order_le_less_trans";
-val order_less_le_trans = thm "order_less_le_trans";
-val order_less_imp_not_less = thm "order_less_imp_not_less";
-val order_less_imp_triv = thm "order_less_imp_triv";
-val order_less_imp_not_eq = thm "order_less_imp_not_eq";
-val order_less_imp_not_eq2 = thm "order_less_imp_not_eq2";
-val linorder_less_linear = thm "linorder_less_linear";
-val linorder_cases = thm "linorder_cases";
-val linorder_not_less = thm "linorder_not_less";
-val linorder_not_le = thm "linorder_not_le";
-val linorder_neq_iff = thm "linorder_neq_iff";
-val linorder_neqE = thm "linorder_neqE";
-val min_same = thm "min_same";
-val max_same = thm "max_same";
-val le_max_iff_disj = thm "le_max_iff_disj";
-val le_maxI1 = thm "le_maxI1";
-val le_maxI2 = thm "le_maxI2";
-val less_max_iff_disj = thm "less_max_iff_disj";
-val max_le_iff_conj = thm "max_le_iff_conj";
-val max_less_iff_conj = thm "max_less_iff_conj";
-val le_min_iff_conj = thm "le_min_iff_conj";
-val min_less_iff_conj = thm "min_less_iff_conj";
-val min_le_iff_disj = thm "min_le_iff_disj";
-val min_less_iff_disj = thm "min_less_iff_disj";
-val split_min = thm "split_min";
-val split_max = thm "split_max";
-val order_refl = thm "order_refl";
-val order_trans = thm "order_trans";
-val order_antisym = thm "order_antisym";
-val order_less_le = thm "order_less_le";
-val linorder_linear = thm "linorder_linear";
 val choice_eq = thm "choice_eq";
 
 structure HOL =
diff -r 617996110388 -r 2ef571f80a55 src/HOL/HOL.thy
--- a/src/HOL/HOL.thy	Thu Feb 10 17:09:15 2005 +0100
+++ b/src/HOL/HOL.thy	Thu Feb 10 18:51:12 2005 +0100
@@ -7,8 +7,7 @@
 
 theory HOL
 imports CPure
-files ("cladata.ML") ("blastdata.ML") ("simpdata.ML") ("antisym_setup.ML")
-      ("eqrule_HOL_data.ML")
+files ("cladata.ML") ("blastdata.ML") ("simpdata.ML") ("eqrule_HOL_data.ML")
       ("~~/src/Provers/eqsubst.ML")
 begin
 
@@ -248,6 +247,14 @@
 apply assumption+
 done
 
+text {* For calculational reasoning: *}
+
+lemma forw_subst: "a = b ==> P b ==> P a"
+  by (rule ssubst)
+
+lemma back_subst: "P a ==> a = b ==> P b"
+  by (rule subst)
+
 
 subsection {*Congruence rules for application*}
 
@@ -1224,563 +1231,5 @@
 setup InductMethod.setup
 
 
-subsection {* Order signatures and orders *}
-
-axclass
-  ord < type
-
-syntax
-  "op <"        :: "['a::ord, 'a] => bool"             ("op <")
-  "op <="       :: "['a::ord, 'a] => bool"             ("op <=")
-
-global
-
-consts
-  "op <"        :: "['a::ord, 'a] => bool"             ("(_/ < _)"  [50, 51] 50)
-  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ <= _)" [50, 51] 50)
-
-local
-
-syntax (xsymbols)
-  "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
-  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
-
-syntax (HTML output)
-  "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
-  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
-
-text{* Syntactic sugar: *}
-
-consts
-  "_gt" :: "'a::ord => 'a => bool"             (infixl ">" 50)
-  "_ge" :: "'a::ord => 'a => bool"             (infixl ">=" 50)
-translations
-  "x > y"  => "y < x"
-  "x >= y" => "y <= x"
-
-syntax (xsymbols)
-  "_ge"       :: "'a::ord => 'a => bool"             (infixl "\<ge>" 50)
-
-syntax (HTML output)
-  "_ge"       :: "['a::ord, 'a] => bool"             (infixl "\<ge>" 50)
-
-
-subsubsection {* Monotonicity *}
-
-locale mono =
-  fixes f
-  assumes mono: "A <= B ==> f A <= f B"
-
-lemmas monoI [intro?] = mono.intro
-  and monoD [dest?] = mono.mono
-
-constdefs
-  min :: "['a::ord, 'a] => 'a"
-  "min a b == (if a <= b then a else b)"
-  max :: "['a::ord, 'a] => 'a"
-  "max a b == (if a <= b then b else a)"
-
-lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
-  by (simp add: min_def)
-
-lemma min_of_mono:
-    "ALL x y. (f x <= f y) = (x <= y) ==> min (f m) (f n) = f (min m n)"
-  by (simp add: min_def)
-
-lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
-  by (simp add: max_def)
-
-lemma max_of_mono:
-    "ALL x y. (f x <= f y) = (x <= y) ==> max (f m) (f n) = f (max m n)"
-  by (simp add: max_def)
-
-
-subsubsection "Orders"
-
-axclass order < ord
-  order_refl [iff]: "x <= x"
-  order_trans: "x <= y ==> y <= z ==> x <= z"
-  order_antisym: "x <= y ==> y <= x ==> x = y"
-  order_less_le: "(x < y) = (x <= y & x ~= y)"
-
-
-text {* Reflexivity. *}
-
-lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y"
-    -- {* This form is useful with the classical reasoner. *}
-  apply (erule ssubst)
-  apply (rule order_refl)
-  done
-
-lemma order_less_irrefl [iff]: "~ x < (x::'a::order)"
-  by (simp add: order_less_le)
-
-lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
-    -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
-  apply (simp add: order_less_le, blast)
-  done
-
-lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
-
-lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
-  by (simp add: order_less_le)
-
-
-text {* Asymmetry. *}
-
-lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"
-  by (simp add: order_less_le order_antisym)
-
-lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P"
-  apply (drule order_less_not_sym)
-  apply (erule contrapos_np, simp)
-  done
-
-lemma order_eq_iff: "!!x::'a::order. (x = y) = (x \<le> y & y \<le> x)"
-by (blast intro: order_antisym)
-
-lemma order_antisym_conv: "(y::'a::order) <= x ==> (x <= y) = (x = y)"
-by(blast intro:order_antisym)
-
-text {* Transitivity. *}
-
-lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z"
-  apply (simp add: order_less_le)
-  apply (blast intro: order_trans order_antisym)
-  done
-
-lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
-  apply (simp add: order_less_le)
-  apply (blast intro: order_trans order_antisym)
-  done
-
-lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
-  apply (simp add: order_less_le)
-  apply (blast intro: order_trans order_antisym)
-  done
-
-
-text {* Useful for simplification, but too risky to include by default. *}
-
-lemma order_less_imp_not_less: "(x::'a::order) < y ==>  (~ y < x) = True"
-  by (blast elim: order_less_asym)
-
-lemma order_less_imp_triv: "(x::'a::order) < y ==>  (y < x --> P) = True"
-  by (blast elim: order_less_asym)
-
-lemma order_less_imp_not_eq: "(x::'a::order) < y ==>  (x = y) = False"
-  by auto
-
-lemma order_less_imp_not_eq2: "(x::'a::order) < y ==>  (y = x) = False"
-  by auto
-
-
-text {* Other operators. *}
-
-lemma min_leastR: "(!!x::'a::order. least <= x) ==> min x least = least"
-  apply (simp add: min_def)
-  apply (blast intro: order_antisym)
-  done
-
-lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x"
-  apply (simp add: max_def)
-  apply (blast intro: order_antisym)
-  done
-
-
-subsubsection {* Least value operator *}
-
-constdefs
-  Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
-  "Least P == THE x. P x & (ALL y. P y --> x <= y)"
-    -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
-
-lemma LeastI2:
-  "[| P (x::'a::order);
-      !!y. P y ==> x <= y;
-      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
-   ==> Q (Least P)"
-  apply (unfold Least_def)
-  apply (rule theI2)
-    apply (blast intro: order_antisym)+
-  done
-
-lemma Least_equality:
-    "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
-  apply (simp add: Least_def)
-  apply (rule the_equality)
-  apply (auto intro!: order_antisym)
-  done
-
-
-subsubsection "Linear / total orders"
-
-axclass linorder < order
-  linorder_linear: "x <= y | y <= x"
-
-lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x"
-  apply (simp add: order_less_le)
-  apply (insert linorder_linear, blast)
-  done
-
-lemma linorder_le_less_linear: "!!x::'a::linorder. x\<le>y | y<x"
-  by (simp add: order_le_less linorder_less_linear)
-
-lemma linorder_le_cases [case_names le ge]:
-    "((x::'a::linorder) \<le> y ==> P) ==> (y \<le> x ==> P) ==> P"
-  by (insert linorder_linear, blast)
-
-lemma linorder_cases [case_names less equal greater]:
-    "((x::'a::linorder) < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P"
-  by (insert linorder_less_linear, blast)
-
-lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)"
-  apply (simp add: order_less_le)
-  apply (insert linorder_linear)
-  apply (blast intro: order_antisym)
-  done
-
-lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
-  apply (simp add: order_less_le)
-  apply (insert linorder_linear)
-  apply (blast intro: order_antisym)
-  done
-
-lemma linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)"
-by (cut_tac x = x and y = y in linorder_less_linear, auto)
-
-lemma linorder_neqE: "x ~= (y::'a::linorder) ==> (x < y ==> R) ==> (y < x ==> R) ==> R"
-by (simp add: linorder_neq_iff, blast)
-
-lemma linorder_antisym_conv1: "~ (x::'a::linorder) < y ==> (x <= y) = (x = y)"
-by(blast intro:order_antisym dest:linorder_not_less[THEN iffD1])
-
-lemma linorder_antisym_conv2: "(x::'a::linorder) <= y ==> (~ x < y) = (x = y)"
-by(blast intro:order_antisym dest:linorder_not_less[THEN iffD1])
-
-lemma linorder_antisym_conv3: "~ (y::'a::linorder) < x ==> (~ x < y) = (x = y)"
-by(blast intro:order_antisym dest:linorder_not_less[THEN iffD1])
-
-use "antisym_setup.ML";
-setup antisym_setup
-
-subsubsection "Min and max on (linear) orders"
-
-lemma min_same [simp]: "min (x::'a::order) x = x"
-  by (simp add: min_def)
-
-lemma max_same [simp]: "max (x::'a::order) x = x"
-  by (simp add: max_def)
-
-lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
-  apply (simp add: max_def)
-  apply (insert linorder_linear)
-  apply (blast intro: order_trans)
-  done
-
-lemma le_maxI1: "(x::'a::linorder) <= max x y"
-  by (simp add: le_max_iff_disj)
-
-lemma le_maxI2: "(y::'a::linorder) <= max x y"
-    -- {* CANNOT use with @{text "[intro!]"} because blast will give PROOF FAILED. *}
-  by (simp add: le_max_iff_disj)
-
-lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
-  apply (simp add: max_def order_le_less)
-  apply (insert linorder_less_linear)
-  apply (blast intro: order_less_trans)
-  done
-
-lemma max_le_iff_conj [simp]:
-    "!!z::'a::linorder. (max x y <= z) = (x <= z & y <= z)"
-  apply (simp add: max_def)
-  apply (insert linorder_linear)
-  apply (blast intro: order_trans)
-  done
-
-lemma max_less_iff_conj [simp]:
-    "!!z::'a::linorder. (max x y < z) = (x < z & y < z)"
-  apply (simp add: order_le_less max_def)
-  apply (insert linorder_less_linear)
-  apply (blast intro: order_less_trans)
-  done
-
-lemma le_min_iff_conj [simp]:
-    "!!z::'a::linorder. (z <= min x y) = (z <= x & z <= y)"
-    -- {* @{text "[iff]"} screws up a @{text blast} in MiniML *}
-  apply (simp add: min_def)
-  apply (insert linorder_linear)
-  apply (blast intro: order_trans)
-  done
-
-lemma min_less_iff_conj [simp]:
-    "!!z::'a::linorder. (z < min x y) = (z < x & z < y)"
-  apply (simp add: order_le_less min_def)
-  apply (insert linorder_less_linear)
-  apply (blast intro: order_less_trans)
-  done
-
-lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)"
-  apply (simp add: min_def)
-  apply (insert linorder_linear)
-  apply (blast intro: order_trans)
-  done
-
-lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)"
-  apply (simp add: min_def order_le_less)
-  apply (insert linorder_less_linear)
-  apply (blast intro: order_less_trans)
-  done
-
-lemma max_assoc: "!!x::'a::linorder. max (max x y) z = max x (max y z)"
-apply(simp add:max_def)
-apply(rule conjI)
-apply(blast intro:order_trans)
-apply(simp add:linorder_not_le)
-apply(blast dest: order_less_trans order_le_less_trans)
-done
-
-lemma max_commute: "!!x::'a::linorder. max x y = max y x"
-apply(simp add:max_def)
-apply(simp add:linorder_not_le)
-apply(blast dest: order_less_trans)
-done
-
-lemmas max_ac = max_assoc max_commute
-                mk_left_commute[of max,OF max_assoc max_commute]
-
-lemma min_assoc: "!!x::'a::linorder. min (min x y) z = min x (min y z)"
-apply(simp add:min_def)
-apply(rule conjI)
-apply(blast intro:order_trans)
-apply(simp add:linorder_not_le)
-apply(blast dest: order_less_trans order_le_less_trans)
-done
-
-lemma min_commute: "!!x::'a::linorder. min x y = min y x"
-apply(simp add:min_def)
-apply(simp add:linorder_not_le)
-apply(blast dest: order_less_trans)
-done
-
-lemmas min_ac = min_assoc min_commute
-                mk_left_commute[of min,OF min_assoc min_commute]
-
-lemma split_min:
-    "P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"
-  by (simp add: min_def)
-
-lemma split_max:
-    "P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"
-  by (simp add: max_def)
-
-
-subsubsection {* Transitivity rules for calculational reasoning *}
-
-
-lemma order_neq_le_trans: "a ~= b ==> (a::'a::order) <= b ==> a < b"
-  by (simp add: order_less_le)
-
-lemma order_le_neq_trans: "(a::'a::order) <= b ==> a ~= b ==> a < b"
-  by (simp add: order_less_le)
-
-lemma order_less_asym': "(a::'a::order) < b ==> b < a ==> P"
-  by (rule order_less_asym)
-
-
-subsubsection {* Setup of transitivity reasoner as Solver *}
-
-lemma less_imp_neq: "[| (x::'a::order) < y |] ==> x ~= y"
-  by (erule contrapos_pn, erule subst, rule order_less_irrefl)
-
-lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
-  by (erule subst, erule ssubst, assumption)
-
-ML_setup {*
-
-(* The setting up of Quasi_Tac serves as a demo.  Since there is no
-   class for quasi orders, the tactics Quasi_Tac.trans_tac and
-   Quasi_Tac.quasi_tac are not of much use. *)
-
-fun decomp_gen sort sign (Trueprop $ t) =
-  let fun of_sort t = Sign.of_sort sign (type_of t, sort)
-  fun dec (Const ("Not", _) $ t) = (
-	  case dec t of
-	    None => None
-	  | Some (t1, rel, t2) => Some (t1, "~" ^ rel, t2))
-	| dec (Const ("op =",  _) $ t1 $ t2) =
-	    if of_sort t1
-	    then Some (t1, "=", t2)
-	    else None
-	| dec (Const ("op <=",  _) $ t1 $ t2) =
-	    if of_sort t1
-	    then Some (t1, "<=", t2)
-	    else None
-	| dec (Const ("op <",  _) $ t1 $ t2) =
-	    if of_sort t1
-	    then Some (t1, "<", t2)
-	    else None
-	| dec _ = None
-  in dec t end;
-
-structure Quasi_Tac = Quasi_Tac_Fun (
-  struct
-    val le_trans = thm "order_trans";
-    val le_refl = thm "order_refl";
-    val eqD1 = thm "order_eq_refl";
-    val eqD2 = thm "sym" RS thm "order_eq_refl";
-    val less_reflE = thm "order_less_irrefl" RS thm "notE";
-    val less_imp_le = thm "order_less_imp_le";
-    val le_neq_trans = thm "order_le_neq_trans";
-    val neq_le_trans = thm "order_neq_le_trans";
-    val less_imp_neq = thm "less_imp_neq";
-    val decomp_trans = decomp_gen ["HOL.order"];
-    val decomp_quasi = decomp_gen ["HOL.order"];
-
-  end);  (* struct *)
-
-structure Order_Tac = Order_Tac_Fun (
-  struct
-    val less_reflE = thm "order_less_irrefl" RS thm "notE";
-    val le_refl = thm "order_refl";
-    val less_imp_le = thm "order_less_imp_le";
-    val not_lessI = thm "linorder_not_less" RS thm "iffD2";
-    val not_leI = thm "linorder_not_le" RS thm "iffD2";
-    val not_lessD = thm "linorder_not_less" RS thm "iffD1";
-    val not_leD = thm "linorder_not_le" RS thm "iffD1";
-    val eqI = thm "order_antisym";
-    val eqD1 = thm "order_eq_refl";
-    val eqD2 = thm "sym" RS thm "order_eq_refl";
-    val less_trans = thm "order_less_trans";
-    val less_le_trans = thm "order_less_le_trans";
-    val le_less_trans = thm "order_le_less_trans";
-    val le_trans = thm "order_trans";
-    val le_neq_trans = thm "order_le_neq_trans";
-    val neq_le_trans = thm "order_neq_le_trans";
-    val less_imp_neq = thm "less_imp_neq";
-    val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq";
-    val decomp_part = decomp_gen ["HOL.order"];
-    val decomp_lin = decomp_gen ["HOL.linorder"];
-
-  end);  (* struct *)
-
-simpset_ref() := simpset ()
-    addSolver (mk_solver "Trans_linear" (fn _ => Order_Tac.linear_tac))
-    addSolver (mk_solver "Trans_partial" (fn _ => Order_Tac.partial_tac));
-  (* Adding the transitivity reasoners also as safe solvers showed a slight
-     speed up, but the reasoning strength appears to be not higher (at least
-     no breaking of additional proofs in the entire HOL distribution, as
-     of 5 March 2004, was observed). *)
-*}
-
-(* Optional setup of methods *)
-
-(*
-method_setup trans_partial =
-  {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (Order_Tac.partial_tac)) *}
-  {* transitivity reasoner for partial orders *}	
-method_setup trans_linear =
-  {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (Order_Tac.linear_tac)) *}
-  {* transitivity reasoner for linear orders *}
-*)
-
-(*
-declare order.order_refl [simp del] order_less_irrefl [simp del]
-
-can currently not be removed, abel_cancel relies on it.
-*)
-
-subsubsection "Bounded quantifiers"
-
-syntax
-  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3ALL _<_./ _)"  [0, 0, 10] 10)
-  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3EX _<_./ _)"  [0, 0, 10] 10)
-  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3ALL _<=_./ _)" [0, 0, 10] 10)
-  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3EX _<=_./ _)" [0, 0, 10] 10)
-
-  "_gtAll" :: "[idt, 'a, bool] => bool"   ("(3ALL _>_./ _)"  [0, 0, 10] 10)
-  "_gtEx"  :: "[idt, 'a, bool] => bool"   ("(3EX _>_./ _)"  [0, 0, 10] 10)
-  "_geAll"   :: "[idt, 'a, bool] => bool"   ("(3ALL _>=_./ _)" [0, 0, 10] 10)
-  "_geEx"    :: "[idt, 'a, bool] => bool"   ("(3EX _>=_./ _)" [0, 0, 10] 10)
-
-syntax (xsymbols)
-  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
-  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
-  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
-  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
-
-  "_gtAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
-  "_gtEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
-  "_geAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
-  "_geEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
-
-syntax (HOL)
-  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
-  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
-  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
-  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
-
-syntax (HTML output)
-  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
-  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
-  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
-  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
-
-  "_gtAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
-  "_gtEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
-  "_geAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
-  "_geEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
-
-translations
- "ALL x<y. P"   =>  "ALL x. x < y --> P"
- "EX x<y. P"    =>  "EX x. x < y  & P"
- "ALL x<=y. P"  =>  "ALL x. x <= y --> P"
- "EX x<=y. P"   =>  "EX x. x <= y & P"
- "ALL x>y. P"   =>  "ALL x. x > y --> P"
- "EX x>y. P"    =>  "EX x. x > y  & P"
- "ALL x>=y. P"  =>  "ALL x. x >= y --> P"
- "EX x>=y. P"   =>  "EX x. x >= y & P"
-
-print_translation {*
-let
-  fun mk v v' q n P =
-    if v=v' andalso not(v  mem (map fst (Term.add_frees([],n))))
-    then Syntax.const q $ Syntax.mark_bound v' $ n $ P else raise Match;
-  fun all_tr' [Const ("_bound",_) $ Free (v,_),
-               Const("op -->",_) $ (Const ("op <",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] =
-    mk v v' "_lessAll" n P
-
-  | all_tr' [Const ("_bound",_) $ Free (v,_),
-               Const("op -->",_) $ (Const ("op <=",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] =
-    mk v v' "_leAll" n P
-
-  | all_tr' [Const ("_bound",_) $ Free (v,_),
-               Const("op -->",_) $ (Const ("op <",_) $ n $ (Const ("_bound",_) $ Free (v',_))) $ P] =
-    mk v v' "_gtAll" n P
-
-  | all_tr' [Const ("_bound",_) $ Free (v,_),
-               Const("op -->",_) $ (Const ("op <=",_) $ n $ (Const ("_bound",_) $ Free (v',_))) $ P] =
-    mk v v' "_geAll" n P;
-
-  fun ex_tr' [Const ("_bound",_) $ Free (v,_),
-               Const("op &",_) $ (Const ("op <",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] =
-    mk v v' "_lessEx" n P
-
-  | ex_tr' [Const ("_bound",_) $ Free (v,_),
-               Const("op &",_) $ (Const ("op <=",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] =
-    mk v v' "_leEx" n P
-
-  | ex_tr' [Const ("_bound",_) $ Free (v,_),
-               Const("op &",_) $ (Const ("op <",_) $ n $ (Const ("_bound",_) $ Free (v',_))) $ P] =
-    mk v v' "_gtEx" n P
-
-  | ex_tr' [Const ("_bound",_) $ Free (v,_),
-               Const("op &",_) $ (Const ("op <=",_) $ n $ (Const ("_bound",_) $ Free (v',_))) $ P] =
-    mk v v' "_geEx" n P
-in
-[("ALL ", all_tr'), ("EX ", ex_tr')]
-end
-*}
-
 end
 
diff -r 617996110388 -r 2ef571f80a55 src/HOL/LOrder.thy
--- a/src/HOL/LOrder.thy	Thu Feb 10 17:09:15 2005 +0100
+++ b/src/HOL/LOrder.thy	Thu Feb 10 18:51:12 2005 +0100
@@ -6,7 +6,7 @@
 header {* Lattice Orders *}
 
 theory LOrder
-imports HOL
+imports Orderings
 begin
 
 text {*
diff -r 617996110388 -r 2ef571f80a55 src/HOL/Lattice_Locales.thy
--- a/src/HOL/Lattice_Locales.thy	Thu Feb 10 17:09:15 2005 +0100
+++ b/src/HOL/Lattice_Locales.thy	Thu Feb 10 18:51:12 2005 +0100
@@ -6,7 +6,7 @@
 header {* Lattices via Locales *}
 
 theory Lattice_Locales
-imports Set
+imports HOL
 begin
 
 subsection{* Lattices *}
@@ -64,6 +64,16 @@
 lemma (in upper_semilattice) sup_absorb: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
 by(blast intro: antisym sup_ge2 sup_greatest refl)
 
+
+lemma (in lower_semilattice) below_inf_conv[simp]:
+ "x \<sqsubseteq> y \<sqinter> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"
+by(blast intro: antisym inf_le1 inf_le2 inf_least refl trans)
+
+lemma (in upper_semilattice) above_sup_conv[simp]:
+ "x \<squnion> y \<sqsubseteq> z = (x \<sqsubseteq> z \<and> y \<sqsubseteq> z)"
+by(blast intro: antisym sup_ge1 sup_ge2 sup_greatest refl trans)
+
+
 text{* Towards distributivity: if you have one of them, you have them all. *}
 
 lemma (in lattice) distrib_imp1:
@@ -97,7 +107,7 @@
   have "x \<sqinter> (y \<sqinter> z) = (y \<sqinter> z) \<sqinter> x" by (simp only: inf_commute)
   also have "... = y \<sqinter> (z \<sqinter> x)" by (simp only: inf_assoc)
   also have "z \<sqinter> x = x \<sqinter> z" by (simp only: inf_commute)
-  finally show ?thesis .
+  finally(back_subst) show ?thesis .
 qed
 
 lemma (in upper_semilattice) sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
@@ -105,7 +115,7 @@
   have "x \<squnion> (y \<squnion> z) = (y \<squnion> z) \<squnion> x" by (simp only: sup_commute)
   also have "... = y \<squnion> (z \<squnion> x)" by (simp only: sup_assoc)
   also have "z \<squnion> x = x \<squnion> z" by (simp only: sup_commute)
-  finally show ?thesis .
+  finally(back_subst) show ?thesis .
 qed
 
 lemma (in lower_semilattice) inf_left_idem: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
diff -r 617996110388 -r 2ef571f80a55 src/HOL/Orderings.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Orderings.ML	Thu Feb 10 18:51:12 2005 +0100
@@ -0,0 +1,54 @@
+(* legacy ML bindings *)
+
+val Least_def = thm "Least_def";
+val Least_equality = thm "Least_equality";
+val mono_def = thm "mono_def";
+val monoI = thm "monoI";
+val monoD = thm "monoD";
+val min_def = thm "min_def";
+val min_of_mono = thm "min_of_mono";
+val max_def = thm "max_def";
+val max_of_mono = thm "max_of_mono";
+val min_leastL = thm "min_leastL";
+val max_leastL = thm "max_leastL";
+val min_leastR = thm "min_leastR";
+val max_leastR = thm "max_leastR";
+val order_eq_refl = thm "order_eq_refl";
+val order_less_irrefl = thm "order_less_irrefl";
+val order_le_less = thm "order_le_less";
+val order_le_imp_less_or_eq = thm "order_le_imp_less_or_eq";
+val order_less_imp_le = thm "order_less_imp_le";
+val order_less_not_sym = thm "order_less_not_sym";
+val order_less_asym = thm "order_less_asym";
+val order_less_trans = thm "order_less_trans";
+val order_le_less_trans = thm "order_le_less_trans";
+val order_less_le_trans = thm "order_less_le_trans";
+val order_less_imp_not_less = thm "order_less_imp_not_less";
+val order_less_imp_triv = thm "order_less_imp_triv";
+val order_less_imp_not_eq = thm "order_less_imp_not_eq";
+val order_less_imp_not_eq2 = thm "order_less_imp_not_eq2";
+val linorder_less_linear = thm "linorder_less_linear";
+val linorder_cases = thm "linorder_cases";
+val linorder_not_less = thm "linorder_not_less";
+val linorder_not_le = thm "linorder_not_le";
+val linorder_neq_iff = thm "linorder_neq_iff";
+val linorder_neqE = thm "linorder_neqE";
+val min_same = thm "min_same";
+val max_same = thm "max_same";
+val le_max_iff_disj = thm "le_max_iff_disj";
+val le_maxI1 = thm "le_maxI1";
+val le_maxI2 = thm "le_maxI2";
+val less_max_iff_disj = thm "less_max_iff_disj";
+val max_le_iff_conj = thm "max_le_iff_conj";
+val max_less_iff_conj = thm "max_less_iff_conj";
+val le_min_iff_conj = thm "le_min_iff_conj";
+val min_less_iff_conj = thm "min_less_iff_conj";
+val min_le_iff_disj = thm "min_le_iff_disj";
+val min_less_iff_disj = thm "min_less_iff_disj";
+val split_min = thm "split_min";
+val split_max = thm "split_max";
+val order_refl = thm "order_refl";
+val order_trans = thm "order_trans";
+val order_antisym = thm "order_antisym";
+val order_less_le = thm "order_less_le";
+val linorder_linear = thm "linorder_linear";
diff -r 617996110388 -r 2ef571f80a55 src/HOL/Orderings.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Orderings.thy	Thu Feb 10 18:51:12 2005 +0100
@@ -0,0 +1,612 @@
+(*  Title:      HOL/Orderings.thy
+    ID:         $Id$
+    Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
+
+FIXME: derive more of the min/max laws generically via semilattices
+*)
+
+header {* Type classes for $\le$ *}
+
+theory Orderings
+imports Lattice_Locales
+files ("antisym_setup.ML")
+begin
+
+subsection {* Order signatures and orders *}
+
+axclass
+  ord < type
+
+syntax
+  "op <"        :: "['a::ord, 'a] => bool"             ("op <")
+  "op <="       :: "['a::ord, 'a] => bool"             ("op <=")
+
+global
+
+consts
+  "op <"        :: "['a::ord, 'a] => bool"             ("(_/ < _)"  [50, 51] 50)
+  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ <= _)" [50, 51] 50)
+
+local
+
+syntax (xsymbols)
+  "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
+  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
+
+syntax (HTML output)
+  "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
+  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
+
+text{* Syntactic sugar: *}
+
+consts
+  "_gt" :: "'a::ord => 'a => bool"             (infixl ">" 50)
+  "_ge" :: "'a::ord => 'a => bool"             (infixl ">=" 50)
+translations
+  "x > y"  => "y < x"
+  "x >= y" => "y <= x"
+
+syntax (xsymbols)
+  "_ge"       :: "'a::ord => 'a => bool"             (infixl "\<ge>" 50)
+
+syntax (HTML output)
+  "_ge"       :: "['a::ord, 'a] => bool"             (infixl "\<ge>" 50)
+
+
+subsection {* Monotonicity *}
+
+locale mono =
+  fixes f
+  assumes mono: "A <= B ==> f A <= f B"
+
+lemmas monoI [intro?] = mono.intro
+  and monoD [dest?] = mono.mono
+
+constdefs
+  min :: "['a::ord, 'a] => 'a"
+  "min a b == (if a <= b then a else b)"
+  max :: "['a::ord, 'a] => 'a"
+  "max a b == (if a <= b then b else a)"
+
+lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
+  by (simp add: min_def)
+
+lemma min_of_mono:
+    "ALL x y. (f x <= f y) = (x <= y) ==> min (f m) (f n) = f (min m n)"
+  by (simp add: min_def)
+
+lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
+  by (simp add: max_def)
+
+lemma max_of_mono:
+    "ALL x y. (f x <= f y) = (x <= y) ==> max (f m) (f n) = f (max m n)"
+  by (simp add: max_def)
+
+
+subsection "Orders"
+
+axclass order < ord
+  order_refl [iff]: "x <= x"
+  order_trans: "x <= y ==> y <= z ==> x <= z"
+  order_antisym: "x <= y ==> y <= x ==> x = y"
+  order_less_le: "(x < y) = (x <= y & x ~= y)"
+
+text{* Connection to locale: *}
+
+lemma partial_order_order:
+ "partial_order (op \<le> :: 'a::order \<Rightarrow> 'a \<Rightarrow> bool)"
+apply(rule partial_order.intro)
+apply(rule order_refl, erule (1) order_trans, erule (1) order_antisym)
+done
+
+text {* Reflexivity. *}
+
+lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y"
+    -- {* This form is useful with the classical reasoner. *}
+  apply (erule ssubst)
+  apply (rule order_refl)
+  done
+
+lemma order_less_irrefl [iff]: "~ x < (x::'a::order)"
+  by (simp add: order_less_le)
+
+lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
+    -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
+  apply (simp add: order_less_le, blast)
+  done
+
+lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
+
+lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
+  by (simp add: order_less_le)
+
+
+text {* Asymmetry. *}
+
+lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"
+  by (simp add: order_less_le order_antisym)
+
+lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P"
+  apply (drule order_less_not_sym)
+  apply (erule contrapos_np, simp)
+  done
+
+lemma order_eq_iff: "!!x::'a::order. (x = y) = (x \<le> y & y \<le> x)"
+by (blast intro: order_antisym)
+
+lemma order_antisym_conv: "(y::'a::order) <= x ==> (x <= y) = (x = y)"
+by(blast intro:order_antisym)
+
+text {* Transitivity. *}
+
+lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z"
+  apply (simp add: order_less_le)
+  apply (blast intro: order_trans order_antisym)
+  done
+
+lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
+  apply (simp add: order_less_le)
+  apply (blast intro: order_trans order_antisym)
+  done
+
+lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
+  apply (simp add: order_less_le)
+  apply (blast intro: order_trans order_antisym)
+  done
+
+
+text {* Useful for simplification, but too risky to include by default. *}
+
+lemma order_less_imp_not_less: "(x::'a::order) < y ==>  (~ y < x) = True"
+  by (blast elim: order_less_asym)
+
+lemma order_less_imp_triv: "(x::'a::order) < y ==>  (y < x --> P) = True"
+  by (blast elim: order_less_asym)
+
+lemma order_less_imp_not_eq: "(x::'a::order) < y ==>  (x = y) = False"
+  by auto
+
+lemma order_less_imp_not_eq2: "(x::'a::order) < y ==>  (y = x) = False"
+  by auto
+
+
+text {* Other operators. *}
+
+lemma min_leastR: "(!!x::'a::order. least <= x) ==> min x least = least"
+  apply (simp add: min_def)
+  apply (blast intro: order_antisym)
+  done
+
+lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x"
+  apply (simp add: max_def)
+  apply (blast intro: order_antisym)
+  done
+
+
+subsection {* Transitivity rules for calculational reasoning *}
+
+
+lemma order_neq_le_trans: "a ~= b ==> (a::'a::order) <= b ==> a < b"
+  by (simp add: order_less_le)
+
+lemma order_le_neq_trans: "(a::'a::order) <= b ==> a ~= b ==> a < b"
+  by (simp add: order_less_le)
+
+lemma order_less_asym': "(a::'a::order) < b ==> b < a ==> P"
+  by (rule order_less_asym)
+
+
+subsection {* Least value operator *}
+
+constdefs
+  Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
+  "Least P == THE x. P x & (ALL y. P y --> x <= y)"
+    -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
+
+lemma LeastI2:
+  "[| P (x::'a::order);
+      !!y. P y ==> x <= y;
+      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
+   ==> Q (Least P)"
+  apply (unfold Least_def)
+  apply (rule theI2)
+    apply (blast intro: order_antisym)+
+  done
+
+lemma Least_equality:
+    "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
+  apply (simp add: Least_def)
+  apply (rule the_equality)
+  apply (auto intro!: order_antisym)
+  done
+
+
+subsection "Linear / total orders"
+
+axclass linorder < order
+  linorder_linear: "x <= y | y <= x"
+
+lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x"
+  apply (simp add: order_less_le)
+  apply (insert linorder_linear, blast)
+  done
+
+lemma linorder_le_less_linear: "!!x::'a::linorder. x\<le>y | y<x"
+  by (simp add: order_le_less linorder_less_linear)
+
+lemma linorder_le_cases [case_names le ge]:
+    "((x::'a::linorder) \<le> y ==> P) ==> (y \<le> x ==> P) ==> P"
+  by (insert linorder_linear, blast)
+
+lemma linorder_cases [case_names less equal greater]:
+    "((x::'a::linorder) < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P"
+  by (insert linorder_less_linear, blast)
+
+lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)"
+  apply (simp add: order_less_le)
+  apply (insert linorder_linear)
+  apply (blast intro: order_antisym)
+  done
+
+lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
+  apply (simp add: order_less_le)
+  apply (insert linorder_linear)
+  apply (blast intro: order_antisym)
+  done
+
+lemma linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)"
+by (cut_tac x = x and y = y in linorder_less_linear, auto)
+
+lemma linorder_neqE: "x ~= (y::'a::linorder) ==> (x < y ==> R) ==> (y < x ==> R) ==> R"
+by (simp add: linorder_neq_iff, blast)
+
+lemma linorder_antisym_conv1: "~ (x::'a::linorder) < y ==> (x <= y) = (x = y)"
+by(blast intro:order_antisym dest:linorder_not_less[THEN iffD1])
+
+lemma linorder_antisym_conv2: "(x::'a::linorder) <= y ==> (~ x < y) = (x = y)"
+by(blast intro:order_antisym dest:linorder_not_less[THEN iffD1])
+
+lemma linorder_antisym_conv3: "~ (y::'a::linorder) < x ==> (~ x < y) = (x = y)"
+by(blast intro:order_antisym dest:linorder_not_less[THEN iffD1])
+
+use "antisym_setup.ML";
+setup antisym_setup
+
+subsection {* Setup of transitivity reasoner as Solver *}
+
+lemma less_imp_neq: "[| (x::'a::order) < y |] ==> x ~= y"
+  by (erule contrapos_pn, erule subst, rule order_less_irrefl)
+
+lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
+  by (erule subst, erule ssubst, assumption)
+
+ML_setup {*
+
+(* The setting up of Quasi_Tac serves as a demo.  Since there is no
+   class for quasi orders, the tactics Quasi_Tac.trans_tac and
+   Quasi_Tac.quasi_tac are not of much use. *)
+
+fun decomp_gen sort sign (Trueprop $ t) =
+  let fun of_sort t = Sign.of_sort sign (type_of t, sort)
+  fun dec (Const ("Not", _) $ t) = (
+	  case dec t of
+	    None => None
+	  | Some (t1, rel, t2) => Some (t1, "~" ^ rel, t2))
+	| dec (Const ("op =",  _) $ t1 $ t2) =
+	    if of_sort t1
+	    then Some (t1, "=", t2)
+	    else None
+	| dec (Const ("op <=",  _) $ t1 $ t2) =
+	    if of_sort t1
+	    then Some (t1, "<=", t2)
+	    else None
+	| dec (Const ("op <",  _) $ t1 $ t2) =
+	    if of_sort t1
+	    then Some (t1, "<", t2)
+	    else None
+	| dec _ = None
+  in dec t end;
+
+structure Quasi_Tac = Quasi_Tac_Fun (
+  struct
+    val le_trans = thm "order_trans";
+    val le_refl = thm "order_refl";
+    val eqD1 = thm "order_eq_refl";
+    val eqD2 = thm "sym" RS thm "order_eq_refl";
+    val less_reflE = thm "order_less_irrefl" RS thm "notE";
+    val less_imp_le = thm "order_less_imp_le";
+    val le_neq_trans = thm "order_le_neq_trans";
+    val neq_le_trans = thm "order_neq_le_trans";
+    val less_imp_neq = thm "less_imp_neq";
+    val decomp_trans = decomp_gen ["Orderings.order"];
+    val decomp_quasi = decomp_gen ["Orderings.order"];
+
+  end);  (* struct *)
+
+structure Order_Tac = Order_Tac_Fun (
+  struct
+    val less_reflE = thm "order_less_irrefl" RS thm "notE";
+    val le_refl = thm "order_refl";
+    val less_imp_le = thm "order_less_imp_le";
+    val not_lessI = thm "linorder_not_less" RS thm "iffD2";
+    val not_leI = thm "linorder_not_le" RS thm "iffD2";
+    val not_lessD = thm "linorder_not_less" RS thm "iffD1";
+    val not_leD = thm "linorder_not_le" RS thm "iffD1";
+    val eqI = thm "order_antisym";
+    val eqD1 = thm "order_eq_refl";
+    val eqD2 = thm "sym" RS thm "order_eq_refl";
+    val less_trans = thm "order_less_trans";
+    val less_le_trans = thm "order_less_le_trans";
+    val le_less_trans = thm "order_le_less_trans";
+    val le_trans = thm "order_trans";
+    val le_neq_trans = thm "order_le_neq_trans";
+    val neq_le_trans = thm "order_neq_le_trans";
+    val less_imp_neq = thm "less_imp_neq";
+    val eq_neq_eq_imp_neq = thm "eq_neq_eq_imp_neq";
+    val decomp_part = decomp_gen ["Orderings.order"];
+    val decomp_lin = decomp_gen ["Orderings.linorder"];
+
+  end);  (* struct *)
+
+simpset_ref() := simpset ()
+    addSolver (mk_solver "Trans_linear" (fn _ => Order_Tac.linear_tac))
+    addSolver (mk_solver "Trans_partial" (fn _ => Order_Tac.partial_tac));
+  (* Adding the transitivity reasoners also as safe solvers showed a slight
+     speed up, but the reasoning strength appears to be not higher (at least
+     no breaking of additional proofs in the entire HOL distribution, as
+     of 5 March 2004, was observed). *)
+*}
+
+(* Optional setup of methods *)
+
+(*
+method_setup trans_partial =
+  {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (Order_Tac.partial_tac)) *}
+  {* transitivity reasoner for partial orders *}	
+method_setup trans_linear =
+  {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (Order_Tac.linear_tac)) *}
+  {* transitivity reasoner for linear orders *}
+*)
+
+(*
+declare order.order_refl [simp del] order_less_irrefl [simp del]
+
+can currently not be removed, abel_cancel relies on it.
+*)
+
+
+subsection "Min and max on (linear) orders"
+
+lemma min_same [simp]: "min (x::'a::order) x = x"
+  by (simp add: min_def)
+
+lemma max_same [simp]: "max (x::'a::order) x = x"
+  by (simp add: max_def)
+
+text{* Instantiate locales: *}
+
+lemma lower_semilattice_lin_min:
+  "lower_semilattice(op \<le>) (min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
+apply(rule lower_semilattice.intro)
+apply(rule partial_order_order)
+apply(rule lower_semilattice_axioms.intro)
+apply(simp add:min_def linorder_not_le order_less_imp_le)
+apply(simp add:min_def linorder_not_le order_less_imp_le)
+apply(simp add:min_def linorder_not_le order_less_imp_le)
+done
+
+lemma upper_semilattice_lin_max:
+  "upper_semilattice(op \<le>) (max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a)"
+apply(rule upper_semilattice.intro)
+apply(rule partial_order_order)
+apply(rule upper_semilattice_axioms.intro)
+apply(simp add: max_def linorder_not_le order_less_imp_le)
+apply(simp add: max_def linorder_not_le order_less_imp_le)
+apply(simp add: max_def linorder_not_le order_less_imp_le)
+done
+
+lemma lattice_min_max: "lattice (op \<le>) (min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a) max"
+apply(rule lattice.intro)
+apply(rule partial_order_order)
+apply(rule lower_semilattice.axioms[OF lower_semilattice_lin_min])
+apply(rule upper_semilattice.axioms[OF upper_semilattice_lin_max])
+done
+
+lemma distrib_lattice_min_max:
+ "distrib_lattice (op \<le>) (min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a) max"
+apply(rule distrib_lattice.intro)
+apply(rule partial_order_order)
+apply(rule lower_semilattice.axioms[OF lower_semilattice_lin_min])
+apply(rule upper_semilattice.axioms[OF upper_semilattice_lin_max])
+apply(rule distrib_lattice_axioms.intro)
+apply(rule_tac x=x and y=y in linorder_le_cases)
+apply(rule_tac x=x and y=z in linorder_le_cases)
+apply(rule_tac x=y and y=z in linorder_le_cases)
+apply(simp add:min_def max_def)
+apply(simp add:min_def max_def)
+apply(rule_tac x=y and y=z in linorder_le_cases)
+apply(simp add:min_def max_def)
+apply(simp add:min_def max_def)
+apply(rule_tac x=x and y=z in linorder_le_cases)
+apply(rule_tac x=y and y=z in linorder_le_cases)
+apply(simp add:min_def max_def)
+apply(simp add:min_def max_def)
+apply(rule_tac x=y and y=z in linorder_le_cases)
+apply(simp add:min_def max_def)
+apply(simp add:min_def max_def)
+done
+
+lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
+  apply(simp add:max_def)
+  apply (insert linorder_linear)
+  apply (blast intro: order_trans)
+  done
+
+lemma le_maxI1: "(x::'a::linorder) <= max x y"
+by(rule upper_semilattice.sup_ge1[OF upper_semilattice_lin_max])
+
+lemma le_maxI2: "(y::'a::linorder) <= max x y"
+    -- {* CANNOT use with @{text "[intro!]"} because blast will give PROOF FAILED. *}
+by(rule upper_semilattice.sup_ge2[OF upper_semilattice_lin_max])
+
+lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
+  apply (simp add: max_def order_le_less)
+  apply (insert linorder_less_linear)
+  apply (blast intro: order_less_trans)
+  done
+
+lemma max_le_iff_conj [simp]:
+    "!!z::'a::linorder. (max x y <= z) = (x <= z & y <= z)"
+by (rule upper_semilattice.above_sup_conv[OF upper_semilattice_lin_max])
+
+lemma max_less_iff_conj [simp]:
+    "!!z::'a::linorder. (max x y < z) = (x < z & y < z)"
+  apply (simp add: order_le_less max_def)
+  apply (insert linorder_less_linear)
+  apply (blast intro: order_less_trans)
+  done
+
+lemma le_min_iff_conj [simp]:
+    "!!z::'a::linorder. (z <= min x y) = (z <= x & z <= y)"
+    -- {* @{text "[iff]"} screws up a @{text blast} in MiniML *}
+by (rule lower_semilattice.below_inf_conv[OF lower_semilattice_lin_min])
+
+lemma min_less_iff_conj [simp]:
+    "!!z::'a::linorder. (z < min x y) = (z < x & z < y)"
+  apply (simp add: order_le_less min_def)
+  apply (insert linorder_less_linear)
+  apply (blast intro: order_less_trans)
+  done
+
+lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)"
+  apply (simp add: min_def)
+  apply (insert linorder_linear)
+  apply (blast intro: order_trans)
+  done
+
+lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)"
+  apply (simp add: min_def order_le_less)
+  apply (insert linorder_less_linear)
+  apply (blast intro: order_less_trans)
+  done
+
+lemma max_assoc: "!!x::'a::linorder. max (max x y) z = max x (max y z)"
+by (rule upper_semilattice.sup_assoc[OF upper_semilattice_lin_max])
+
+lemma max_commute: "!!x::'a::linorder. max x y = max y x"
+by (rule upper_semilattice.sup_commute[OF upper_semilattice_lin_max])
+
+lemmas max_ac = max_assoc max_commute
+                mk_left_commute[of max,OF max_assoc max_commute]
+
+lemma min_assoc: "!!x::'a::linorder. min (min x y) z = min x (min y z)"
+by (rule lower_semilattice.inf_assoc[OF lower_semilattice_lin_min])
+
+lemma min_commute: "!!x::'a::linorder. min x y = min y x"
+by (rule lower_semilattice.inf_commute[OF lower_semilattice_lin_min])
+
+lemmas min_ac = min_assoc min_commute
+                mk_left_commute[of min,OF min_assoc min_commute]
+
+lemma split_min:
+    "P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"
+  by (simp add: min_def)
+
+lemma split_max:
+    "P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"
+  by (simp add: max_def)
+
+
+subsection "Bounded quantifiers"
+
+syntax
+  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3ALL _<_./ _)"  [0, 0, 10] 10)
+  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3EX _<_./ _)"  [0, 0, 10] 10)
+  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3ALL _<=_./ _)" [0, 0, 10] 10)
+  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3EX _<=_./ _)" [0, 0, 10] 10)
+
+  "_gtAll" :: "[idt, 'a, bool] => bool"   ("(3ALL _>_./ _)"  [0, 0, 10] 10)
+  "_gtEx"  :: "[idt, 'a, bool] => bool"   ("(3EX _>_./ _)"  [0, 0, 10] 10)
+  "_geAll"   :: "[idt, 'a, bool] => bool"   ("(3ALL _>=_./ _)" [0, 0, 10] 10)
+  "_geEx"    :: "[idt, 'a, bool] => bool"   ("(3EX _>=_./ _)" [0, 0, 10] 10)
+
+syntax (xsymbols)
+  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
+  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
+  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
+  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
+
+  "_gtAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
+  "_gtEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
+  "_geAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
+  "_geEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
+
+syntax (HOL)
+  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
+  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
+  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
+  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
+
+syntax (HTML output)
+  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
+  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
+  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
+  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
+
+  "_gtAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_>_./ _)"  [0, 0, 10] 10)
+  "_gtEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_>_./ _)"  [0, 0, 10] 10)
+  "_geAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
+  "_geEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
+
+translations
+ "ALL x<y. P"   =>  "ALL x. x < y --> P"
+ "EX x<y. P"    =>  "EX x. x < y  & P"
+ "ALL x<=y. P"  =>  "ALL x. x <= y --> P"
+ "EX x<=y. P"   =>  "EX x. x <= y & P"
+ "ALL x>y. P"   =>  "ALL x. x > y --> P"
+ "EX x>y. P"    =>  "EX x. x > y  & P"
+ "ALL x>=y. P"  =>  "ALL x. x >= y --> P"
+ "EX x>=y. P"   =>  "EX x. x >= y & P"
+
+print_translation {*
+let
+  fun mk v v' q n P =
+    if v=v' andalso not(v  mem (map fst (Term.add_frees([],n))))
+    then Syntax.const q $ Syntax.mark_bound v' $ n $ P else raise Match;
+  fun all_tr' [Const ("_bound",_) $ Free (v,_),
+               Const("op -->",_) $ (Const ("op <",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] =
+    mk v v' "_lessAll" n P
+
+  | all_tr' [Const ("_bound",_) $ Free (v,_),
+               Const("op -->",_) $ (Const ("op <=",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] =
+    mk v v' "_leAll" n P
+
+  | all_tr' [Const ("_bound",_) $ Free (v,_),
+               Const("op -->",_) $ (Const ("op <",_) $ n $ (Const ("_bound",_) $ Free (v',_))) $ P] =
+    mk v v' "_gtAll" n P
+
+  | all_tr' [Const ("_bound",_) $ Free (v,_),
+               Const("op -->",_) $ (Const ("op <=",_) $ n $ (Const ("_bound",_) $ Free (v',_))) $ P] =
+    mk v v' "_geAll" n P;
+
+  fun ex_tr' [Const ("_bound",_) $ Free (v,_),
+               Const("op &",_) $ (Const ("op <",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] =
+    mk v v' "_lessEx" n P
+
+  | ex_tr' [Const ("_bound",_) $ Free (v,_),
+               Const("op &",_) $ (Const ("op <=",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] =
+    mk v v' "_leEx" n P
+
+  | ex_tr' [Const ("_bound",_) $ Free (v,_),
+               Const("op &",_) $ (Const ("op <",_) $ n $ (Const ("_bound",_) $ Free (v',_))) $ P] =
+    mk v v' "_gtEx" n P
+
+  | ex_tr' [Const ("_bound",_) $ Free (v,_),
+               Const("op &",_) $ (Const ("op <=",_) $ n $ (Const ("_bound",_) $ Free (v',_))) $ P] =
+    mk v v' "_geEx" n P
+in
+[("ALL ", all_tr'), ("EX ", ex_tr')]
+end
+*}
+
+end
diff -r 617996110388 -r 2ef571f80a55 src/HOL/Set.thy
--- a/src/HOL/Set.thy	Thu Feb 10 17:09:15 2005 +0100
+++ b/src/HOL/Set.thy	Thu Feb 10 18:51:12 2005 +0100
@@ -6,7 +6,7 @@
 header {* Set theory for higher-order logic *}
 
 theory Set
-imports HOL
+imports Orderings
 begin
 
 text {* A set in HOL is simply a predicate. *}
@@ -2057,12 +2057,6 @@
 
 subsection {* Transitivity rules for calculational reasoning *}
 
-lemma forw_subst: "a = b ==> P b ==> P a"
-  by (rule ssubst)
-
-lemma back_subst: "P a ==> a = b ==> P b"
-  by (rule subst)
-
 lemma set_rev_mp: "x:A ==> A \<subseteq> B ==> x:B"
   by (rule subsetD)