(*  Title:      HOL/Orderings.thy

     Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson

 *)

 header {* Abstract orderings *}

 theory Orderings

 imports HOL

 uses

   "~~/src/Provers/order.ML"

   "~~/src/Provers/quasi.ML"  (* FIXME unused? *)

 begin

 subsection {* Syntactic orders *}

 class ord =

   fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"

     and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"

 begin

 notation

   less_eq  ("op <=") and

   less_eq  ("(_/ <= _)" [51, 51] 50) and

   less  ("op <") and

   less  ("(_/ < _)"  [51, 51] 50)

 notation (xsymbols)

   less_eq  ("op \<le>") and

   less_eq  ("(_/ \<le> _)"  [51, 51] 50)

 notation (HTML output)

   less_eq  ("op \<le>") and

   less_eq  ("(_/ \<le> _)"  [51, 51] 50)

 abbreviation (input)

   greater_eq  (infix ">=" 50) where

   "x >= y \<equiv> y <= x"

 notation (input)

   greater_eq  (infix "\<ge>" 50)

 abbreviation (input)

   greater  (infix ">" 50) where

   "x > y \<equiv> y < x"

 end

 subsection {* Quasi orders *}

 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

 text {* Reflexivity. *}

 lemma eq_refl: "x = y \<Longrightarrow> x \<le> y"

     -- {* This form is useful with the classical reasoner. *}

 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"

 unfolding less_le_not_le by blast

 text {* Asymmetry. *}

 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 {* Transitivity. *}

 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 {* Useful for simplification, but too risky to include by default. *}

 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 {* Transitivity rules for calculational reasoning *}

 lemma less_asym': "a < b \<Longrightarrow> b < a \<Longrightarrow> P"

 by (rule less_asym)

 text {* Dual order *}

 lemma dual_preorder:

   "class.preorder (op \<ge>) (op >)"

 proof qed (auto simp add: less_le_not_le intro: order_trans)

 end

 subsection {* Partial orders *}

 class order = preorder +

   assumes antisym: "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y"

 begin

 text {* Reflexivity. *}

 lemma less_le: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> y"

 by (auto simp add: less_le_not_le intro: antisym)

 lemma le_less: "x \<le> y \<longleftrightarrow> x < y \<or> x = y"

     -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}

 by (simp add: less_le) blast

 lemma le_imp_less_or_eq: "x \<le> y \<Longrightarrow> x < y \<or> x = y"

 unfolding less_le by blast

 text {* Useful for simplification, but too risky to include by default. *}

 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 {* Transitivity rules for calculational reasoning *}

 lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<le> b \<Longrightarrow> a < b"

 by (simp add: less_le)

 lemma le_neq_trans: "a \<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a < b"

 by (simp add: less_le)

 text {* Asymmetry. *}

 lemma eq_iff: "x = y \<longleftrightarrow> x \<le> y \<and> y \<le> x"

 by (blast intro: antisym)

 lemma antisym_conv: "y \<le> x \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y"

 by (blast intro: antisym)

 lemma less_imp_neq: "x < y \<Longrightarrow> x \<noteq> y"

 by (erule contrapos_pn, erule subst, rule less_irrefl)

 text {* Least value operator *}

 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 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 antisym)+

 text {* Dual order *}

 lemma dual_order:

   "class.order (op \<ge>) (op >)"

 by (intro_locales, rule dual_preorder) (unfold_locales, rule antisym)

 end

 subsection {* Linear (total) orders *}

 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 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 not_less: "\<not> x < y \<longleftrightarrow> y \<le> x"

 apply (simp add: less_le)

 using linear apply (blast intro: antisym)

 done

 lemma not_less_iff_gr_or_eq:

  "\<not>(x < y) \<longleftrightarrow> (x > y | x = y)"

 apply(simp add:not_less le_less)

 apply blast

 done

 lemma not_le: "\<not> x \<le> y \<longleftrightarrow> y < x"

 apply (simp add: less_le)

 using linear apply (blast intro: antisym)

 done

 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_conv1: "\<not> x < y \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y"

 by (blast intro: antisym dest: not_less [THEN iffD1])

 lemma antisym_conv2: "x \<le> y \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"

 by (blast intro: antisym dest: not_less [THEN iffD1])

 lemma antisym_conv3: "\<not> y < x \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"

 by (blast intro: antisym dest: not_less [THEN iffD1])

 lemma leI: "\<not> x < y \<Longrightarrow> y \<le> x"

 unfolding not_less .

 lemma leD: "y \<le> x \<Longrightarrow> \<not> x < y"

 unfolding not_less .

 (*FIXME inappropriate name (or delete altogether)*)

 lemma not_leE: "\<not> y \<le> x \<Longrightarrow> x < y"

 unfolding not_le .

 text {* Dual order *}

 lemma dual_linorder:

   "class.linorder (op \<ge>) (op >)"

 by (rule class.linorder.intro, rule dual_order) (unfold_locales, rule linear)

 text {* min/max *}

 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_le_iff_disj:

   "min x y \<le> z \<longleftrightarrow> x \<le> z \<or> y \<le> z"

 unfolding min_def using linear by (auto intro: order_trans)

 lemma le_max_iff_disj:

   "z \<le> max x y \<longleftrightarrow> z \<le> x \<or> z \<le> y"

 unfolding max_def using linear by (auto intro: order_trans)

 lemma min_less_iff_disj:

   "min x y < z \<longleftrightarrow> x < z \<or> y < z"

 unfolding min_def le_less using less_linear by (auto intro: less_trans)

 lemma less_max_iff_disj:

   "z < max x y \<longleftrightarrow> z < x \<or> z < y"

 unfolding max_def le_less using less_linear by (auto intro: less_trans)

 lemma min_less_iff_conj [simp]:

   "z < min x y \<longleftrightarrow> z < x \<and> z < y"

 unfolding min_def le_less using less_linear by (auto intro: less_trans)

 lemma max_less_iff_conj [simp]:

   "max x y < z \<longleftrightarrow> x < z \<and> y < z"

 unfolding max_def le_less using less_linear by (auto intro: less_trans)

 lemma split_min [no_atp]:

   "P (min i j) \<longleftrightarrow> (i \<le> j \<longrightarrow> P i) \<and> (\<not> i \<le> j \<longrightarrow> P j)"

 by (simp add: min_def)

 lemma split_max [no_atp]:

   "P (max i j) \<longleftrightarrow> (i \<le> j \<longrightarrow> P j) \<and> (\<not> i \<le> j \<longrightarrow> P i)"

 by (simp add: max_def)

 end

 subsection {* Reasoning tools setup *}

 ML {*

 signature ORDERS =

 sig

   val print_structures: Proof.context -> unit

   val setup: theory -> theory

   val order_tac: Proof.context -> thm list -> int -> tactic

 end;

 structure Orders: ORDERS =

 struct

 (** Theory and context data **)

 fun struct_eq ((s1: string, ts1), (s2, ts2)) =

   (s1 = s2) andalso eq_list (op aconv) (ts1, ts2);

 structure Data = Generic_Data

 (

   type T = ((string * term list) * Order_Tac.less_arith) list;

     (* Order structures:

        identifier of the structure, list of operations and record of theorems

        needed to set up the transitivity reasoner,

        identifier and operations identify the structure uniquely. *)

   val empty = [];

   val extend = I;

   fun merge data = AList.join struct_eq (K fst) data;

 );

 fun print_structures ctxt =

   let

     val structs = 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))];

     fun pretty_struct ((s, ts), _) = Pretty.block

       [Pretty.str s, Pretty.str ":", Pretty.brk 1,

        Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))];

   in

     Pretty.writeln (Pretty.big_list "Order structures:" (map pretty_struct structs))

   end;

 (** Method **)

 fun struct_tac ((s, [eq, le, less]), thms) ctxt prems =

   let

     fun decomp thy (@{const Trueprop} $ t) =

       let

         fun excluded t =

           (* exclude numeric types: linear arithmetic subsumes transitivity *)

           let val T = type_of t

           in

             T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT

           end;

         fun rel (bin_op $ t1 $ t2) =

               if excluded t1 then NONE

               else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2)

               else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2)

               else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2)

               else NONE

           | rel _ = NONE;

         fun dec (Const (@{const_name Not}, _) $ t) = (case rel t

               of NONE => NONE

                | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2))

           | dec x = rel x;

       in dec t end

       | decomp thy _ = NONE;

   in

     case s of

       "order" => Order_Tac.partial_tac decomp thms ctxt prems

     | "linorder" => Order_Tac.linear_tac decomp thms ctxt prems

     | _ => error ("Unknown kind of order `" ^ s ^ "' encountered in transitivity reasoner.")

   end

 fun order_tac ctxt prems =

   FIRST' (map (fn s => CHANGED o struct_tac s ctxt prems) (Data.get (Context.Proof ctxt)));

 (** Attribute **)

 fun add_struct_thm s tag =

   Thm.declaration_attribute

     (fn thm => Data.map (AList.map_default struct_eq (s, Order_Tac.empty TrueI) (Order_Tac.update tag thm)));

 fun del_struct s =

   Thm.declaration_attribute

     (fn _ => Data.map (AList.delete struct_eq s));

 val attrib_setup =

   Attrib.setup @{binding order}

     (Scan.lift ((Args.add -- Args.name >> (fn (_, s) => SOME s) || Args.del >> K NONE) --|

       Args.colon (* FIXME || Scan.succeed true *) ) -- Scan.lift Args.name --

       Scan.repeat Args.term

       >> (fn ((SOME tag, n), ts) => add_struct_thm (n, ts) tag

            | ((NONE, n), ts) => del_struct (n, ts)))

     "theorems controlling transitivity reasoner";

 (** Diagnostic command **)

 val _ =

   Outer_Syntax.improper_command "print_orders"

     "print order structures available to transitivity reasoner" Keyword.diag

     (Scan.succeed (Toplevel.no_timing o Toplevel.unknown_context o

         Toplevel.keep (print_structures o Toplevel.context_of)));

 (** Setup **)

 val setup =

   Method.setup @{binding order} (Scan.succeed (fn ctxt => SIMPLE_METHOD' (order_tac ctxt [])))

     "transitivity reasoner" #>

   attrib_setup;

 end;

 *}

 setup Orders.setup

 text {* Declarations to set up transitivity reasoner of partial and linear orders. *}

 context order

 begin

 (* The type constraint on @{term op =} below is necessary since the operation

    is not a parameter of the locale. *)

 declare less_irrefl [THEN notE, order add less_reflE: order "op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool" "op <=" "op <"]

 declare order_refl  [order add le_refl: order "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare less_imp_le [order add less_imp_le: order "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare antisym [order add eqI: order "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare eq_refl [order add eqD1: order "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare sym [THEN eq_refl, order add eqD2: order "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare less_trans [order add less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare less_le_trans [order add less_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare le_less_trans [order add le_less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare order_trans [order add le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare le_neq_trans [order add le_neq_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare neq_le_trans [order add neq_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare less_imp_neq [order add less_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare not_sym [order add not_sym: order "op = :: 'a => 'a => bool" "op <=" "op <"]

 end

 context linorder

 begin

 declare [[order del: order "op = :: 'a => 'a => bool" "op <=" "op <"]]

 declare less_irrefl [THEN notE, order add less_reflE: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare order_refl [order add le_refl: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare less_imp_le [order add less_imp_le: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare not_less [THEN iffD2, order add not_lessI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare not_le [THEN iffD2, order add not_leI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare not_less [THEN iffD1, order add not_lessD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare not_le [THEN iffD1, order add not_leD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare antisym [order add eqI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare eq_refl [order add eqD1: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare sym [THEN eq_refl, order add eqD2: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare less_trans [order add less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare less_le_trans [order add less_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare le_less_trans [order add le_less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare order_trans [order add le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare le_neq_trans [order add le_neq_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare neq_le_trans [order add neq_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare less_imp_neq [order add less_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare eq_neq_eq_imp_neq [order add eq_neq_eq_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]

 declare not_sym [order add not_sym: linorder "op = :: 'a => 'a => bool" "op <=" "op <"]

 end

 setup {*

 let

 fun prp t thm = Thm.prop_of thm = t;  (* FIXME aconv!? *)

 fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) =

   let val prems = Simplifier.prems_of ss;

       val less = Const (@{const_name less}, 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 linorder_class.antisym_conv1}))

          end

      | SOME thm => SOME(mk_meta_eq(thm RS @{thm order_class.antisym_conv}))

   end

   handle THM _ => NONE;

 fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) =

   let val prems = Simplifier.prems_of ss;

       val le = Const (@{const_name less_eq}, 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 linorder_class.antisym_conv2}))

   end

   handle THM _ => NONE;

 fun add_simprocs procs thy =

   Simplifier.map_simpset_global (fn ss => ss

     addsimprocs (map (fn (name, raw_ts, proc) =>

       Simplifier.simproc_global thy name raw_ts proc) procs)) thy;

 fun add_solver name tac =

   Simplifier.map_simpset_global (fn ss => ss addSolver

     mk_solver name (fn ss => tac (Simplifier.the_context ss) (Simplifier.prems_of ss)));

 in

   add_simprocs [

        ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le),

        ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less)

      ]

   #> add_solver "Transitivity" Orders.order_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). *)

 end

 *}

 subsection {* Bounded quantifiers *}

 syntax

   "_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)

 syntax (xsymbols)

   "_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)

 syntax (HOL)

   "_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)

 syntax (HTML output)

   "_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)

 translations

   "ALL x<y. P"   =>  "ALL x. x < y \<longrightarrow> P"

   "EX x<y. P"    =>  "EX x. x < y \<and> P"

   "ALL x<=y. P"  =>  "ALL x. x <= y \<longrightarrow> P"

   "EX x<=y. P"   =>  "EX x. x <= y \<and> P"

   "ALL x>y. P"   =>  "ALL x. x > y \<longrightarrow> P"

   "EX x>y. P"    =>  "EX x. x > y \<and> P"

   "ALL x>=y. P"  =>  "ALL x. x >= y \<longrightarrow> P"

   "EX x>=y. P"   =>  "EX x. x >= y \<and> P"

 print_translation {*

 let

   val All_binder = Mixfix.binder_name @{const_syntax All};

   val Ex_binder = Mixfix.binder_name @{const_syntax Ex};

   val impl = @{const_syntax HOL.implies};

   val conj = @{const_syntax HOL.conj};

   val less = @{const_syntax less};

   val less_eq = @{const_syntax less_eq};

   val trans =

    [((All_binder, impl, less),

     (@{syntax_const "_All_less"}, @{syntax_const "_All_greater"})),

     ((All_binder, impl, less_eq),

     (@{syntax_const "_All_less_eq"}, @{syntax_const "_All_greater_eq"})),

     ((Ex_binder, conj, less),

     (@{syntax_const "_Ex_less"}, @{syntax_const "_Ex_greater"})),

     ((Ex_binder, conj, less_eq),

     (@{syntax_const "_Ex_less_eq"}, @{syntax_const "_Ex_greater_eq"}))];

   fun matches_bound v t =

     (case t of

       Const (@{syntax_const "_bound"}, _) $ Free (v', _) => v = v'

     | _ => false);

   fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false);

   fun mk v c n P = Syntax.const c $ Syntax_Trans.mark_bound v $ n $ P;

   fun tr' q = (q,

     fn [Const (@{syntax_const "_bound"}, _) $ Free (v, _),

         Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>

         (case AList.lookup (op =) 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 l u P

             else if matches_bound v u andalso not (contains_var v t) then mk v g t P

             else raise Match)

      | _ => raise Match);

 in [tr' All_binder, tr' Ex_binder] end

 *}

 subsection {* Transitivity reasoning *}

 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 {*

   Note that this list of rules is in reverse order of priorities.

 *}

 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] =

   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

   antisym

   ord_le_eq_trans

   ord_eq_le_trans

   ord_less_eq_trans

   ord_eq_less_trans

   trans

 text {* These support proving chains of decreasing inequalities

     a >= b >= c ... in Isar proofs. *}

 lemma xt1 [no_atp]:

   "a = b ==> b > c ==> a > c"

   "a > b ==> b = c ==> a > c"

   "a = b ==> b >= c ==> a >= c"

   "a >= b ==> b = c ==> a >= c"

   "(x::'a::order) >= y ==> y >= x ==> x = y"

   "(x::'a::order) >= y ==> y >= z ==> x >= z"

   "(x::'a::order) > y ==> y >= z ==> x > z"

   "(x::'a::order) >= y ==> y > z ==> x > z"

   "(a::'a::order) > b ==> b > a ==> P"

   "(x::'a::order) > y ==> y > z ==> x > z"

   "(a::'a::order) >= b ==> a ~= b ==> a > b"

   "(a::'a::order) ~= b ==> a >= b ==> a > b"

   "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" 

   "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c"

   "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c"

   "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= 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 [no_atp]

 (* 

   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 {* Monotonicity, least value operator and min/max *}

 context order

 begin

 definition mono :: "('a \<Rightarrow> 'b\<Colon>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\<Colon>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\<Colon>order"

   shows "mono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"

   unfolding mono_def by iprover

 definition strict_mono :: "('a \<Rightarrow> 'b\<Colon>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 `x \<le> y` 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 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 `f x = f y` 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 `f x = f y` 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 `f x \<le> f y` 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)

 lemma min_of_mono:

   fixes f :: "'a \<Rightarrow> 'b\<Colon>linorder"

   shows "mono f \<Longrightarrow> min (f m) (f n) = f (min m n)"

   by (auto simp: mono_def Orderings.min_def min_def intro: Orderings.antisym)

 lemma max_of_mono:

   fixes f :: "'a \<Rightarrow> 'b\<Colon>linorder"

   shows "mono f \<Longrightarrow> max (f m) (f n) = f (max m n)"

   by (auto simp: mono_def Orderings.max_def max_def intro: Orderings.antisym)

 end

 lemma min_absorb1: "x \<le> y \<Longrightarrow> min x y = x"

 by (simp add: min_def)

 lemma max_absorb2: "x \<le> y ==> max x y = y"

 by (simp add: max_def)

 lemma min_absorb2: "(y\<Colon>'a\<Colon>order) \<le> x \<Longrightarrow> min x y = y"

 by (simp add:min_def)

 lemma max_absorb1: "(y\<Colon>'a\<Colon>order) \<le> x \<Longrightarrow> max x y = x"

 by (simp add: max_def)

 subsection {* (Unique) top and bottom elements *}

 class bot = order +

   fixes bot :: 'a ("\<bottom>")

   assumes bot_least [simp]: "\<bottom> \<le> a"

 begin

 lemma le_bot:

   "a \<le> \<bottom> \<Longrightarrow> a = \<bottom>"

   by (auto intro: antisym)

 lemma bot_unique:

   "a \<le> \<bottom> \<longleftrightarrow> a = \<bottom>"

   by (auto intro: antisym)

 lemma not_less_bot [simp]:

   "\<not> (a < \<bottom>)"

   using bot_least [of a] by (auto simp: le_less)

 lemma bot_less:

   "a \<noteq> \<bottom> \<longleftrightarrow> \<bottom> < a"

   by (auto simp add: less_le_not_le intro!: antisym)

 end

 class top = order +

   fixes top :: 'a ("\<top>")

   assumes top_greatest [simp]: "a \<le> \<top>"

 begin

 lemma top_le:

   "\<top> \<le> a \<Longrightarrow> a = \<top>"

   by (rule antisym) auto

 lemma top_unique:

   "\<top> \<le> a \<longleftrightarrow> a = \<top>"

   by (auto intro: antisym)

 lemma not_top_less [simp]: "\<not> (\<top> < a)"

   using top_greatest [of a] by (auto simp: le_less)

 lemma less_top:

   "a \<noteq> \<top> \<longleftrightarrow> a < \<top>"

   by (auto simp add: less_le_not_le intro!: antisym)

 end

 subsection {* Dense orders *}

 class dense_linorder = linorder + 

   assumes gt_ex: "\<exists>y. x < y" 

   and lt_ex: "\<exists>y. y < x"

   and dense: "x < y \<Longrightarrow> (\<exists>z. x < z \<and> z < y)"

 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 `x < y`] .

   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 `x < y`] 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 `x < u` `u \<le> w`] `w < y`

     show "w \<le> z" by (rule *)

   next

     assume "w \<le> u"

     from `w \<le> u` *[OF `x < u` `u < y`]

     show "w \<le> z" by (rule order_trans)

   qed

 qed

 end

 subsection {* Wellorders *}

 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 `P x` have Least: "(LEAST a. P a) = x"

         by (rule Least_equality)

       with `P x` show ?thesis by simp

     qed

   qed

   then show "P (LEAST x. P x)" and "(LEAST x. P x) \<le> k" by auto

 qed

 -- "The following 3 lemmas are due to Brian Huffman"

 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 `P a` 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 not_less_Least: "k < (LEAST x. P x) \<Longrightarrow> \<not> P k"

 apply (simp (no_asm_use) add: not_le [symmetric])

 apply (erule contrapos_nn)

 apply (erule Least_le)

 done

 end

 subsection {* Order on @{typ bool} *}

 instantiation bool :: "{bot, top, linorder}"

 begin

 definition

   le_bool_def [simp]: "P \<le> Q \<longleftrightarrow> P \<longrightarrow> Q"

 definition

   [simp]: "(P\<Colon>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 {* Order on @{typ "_ \<Rightarrow> _"} *}

 instantiation "fun" :: (type, ord) ord

 begin

 definition

   le_fun_def: "f \<le> g \<longleftrightarrow> (\<forall>x. f x \<le> g x)"

 definition

   "(f\<Colon>'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 antisym)

 instance "fun" :: (type, order) order proof

 qed (auto simp add: le_fun_def intro: antisym)

 instantiation "fun" :: (type, bot) bot

 begin

 definition

   "\<bottom> = (\<lambda>x. \<bottom>)"

 lemma bot_apply [simp] (* CANDIDATE [code] *):

   "\<bottom> x = \<bottom>"

   by (simp add: bot_fun_def)

 instance proof

 qed (simp add: le_fun_def bot_apply)

 end

 instantiation "fun" :: (type, top) top

 begin

 definition

   [no_atp]: "\<top> = (\<lambda>x. \<top>)"

 declare top_fun_def_raw [no_atp]

 lemma top_apply [simp] (* CANDIDATE [code] *):

   "\<top> x = \<top>"

   by (simp add: top_fun_def)

 instance proof

 qed (simp add: le_fun_def top_apply)

 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"

   unfolding le_fun_def by simp

 subsection {* Order on unary and binary predicates *}

 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 {* Name duplicates *}

 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_antisym = order_class.antisym

 lemmas order_eq_iff = order_class.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

 lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1

 lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2

 lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3

 end