# Theory Orderings

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theory Orderings
imports Lattice_Locales
uses (antisym_setup.ML) [Orderings.ML]
begin

    (*  Title:      HOL/Orderings.thy
        ID:         $Id: Orderings.thy,v 1.14 2005/08/03 12:48:36 avigad Exp $
        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
    uses ("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 ≤")
      "op <="       :: "['a::ord, 'a] => bool"             ("(_/ ≤ _)"  [50, 51] 50)

    syntax (HTML output)
      "op <="       :: "['a::ord, 'a] => bool"             ("op ≤")
      "op <="       :: "['a::ord, 'a] => bool"             ("(_/ ≤ _)"  [50, 51] 50)

    text{* Syntactic sugar: *}

    syntax
      "_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 "≥" 50)

    syntax (HTML output)
      "_ge"       :: "['a::ord, 'a] => bool"             (infixl "≥" 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: *}

    interpretation order:
      partial_order["op ≤ :: 'a::order => 'a => 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 ≤ y & y ≤ 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_order:
      "[| P (x::'a::order);
          !!y. P y ==> x <= y;
          !!x. [| P x; ALL y. P y --> x ≤ 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≤y | y<x"
      by (simp add: order_le_less linorder_less_linear)

    lemma linorder_le_cases [case_names le ge]:
        "((x::'a::linorder) ≤ y ==> P) ==> (y ≤ 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])

    text{*Replacing the old Nat.leI*}
    lemma leI: "~ x < y ==> y <= (x::'a::linorder)"
      by (simp only: linorder_not_less)

    lemma leD: "y <= (x::'a::linorder) ==> ~ x < y"
      by (simp only: linorder_not_less)

    (*FIXME inappropriate name (or delete altogether)*)
    lemma not_leE: "~ y <= (x::'a::linorder) ==> x < y"
      by (simp only: linorder_not_le)

    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 = let val T = type_of t in
            (* exclude numeric types: linear arithmetic subsumes transitivity *)
            T <> HOLogic.natT andalso T <> HOLogic.intT andalso
            T <> HOLogic.realT andalso Sign.of_sort sign (T, sort) end
      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 not_sym = thm "not_sym";
        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"

    text{* Instantiate locales: *}

    interpretation min_max:
      lower_semilattice["op ≤" "min :: 'a::linorder => 'a => 'a"]
    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

    interpretation min_max:
      upper_semilattice["op ≤" "max :: 'a::linorder => 'a => 'a"]
    apply -
    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

    interpretation min_max:
      lattice["op ≤" "min :: 'a::linorder => 'a => 'a" "max"]
    .

    interpretation min_max:
      distrib_lattice["op ≤" "min :: 'a::linorder => 'a => 'a" "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

    lemmas le_maxI1 = min_max.sup_ge1
    lemmas le_maxI2 = min_max.sup_ge2

    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_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 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

    lemmas max_ac = min_max.sup_assoc min_max.sup_commute
                   mk_left_commute[of max,OF min_max.sup_assoc min_max.sup_commute]

    lemmas min_ac = min_max.inf_assoc min_max.inf_commute
                   mk_left_commute[of min,OF min_max.inf_assoc min_max.inf_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∀_<_./ _)"  [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)

      "_gtAll" :: "[idt, 'a, bool] => bool"   ("(3∀_>_./ _)"  [0, 0, 10] 10)
      "_gtEx"  :: "[idt, 'a, bool] => bool"   ("(3∃_>_./ _)"  [0, 0, 10] 10)
      "_geAll"   :: "[idt, 'a, bool] => bool"   ("(3∀_≥_./ _)" [0, 0, 10] 10)
      "_geEx"    :: "[idt, 'a, bool] => bool"   ("(3∃_≥_./ _)" [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∀_<_./ _)"  [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)

      "_gtAll" :: "[idt, 'a, bool] => bool"   ("(3∀_>_./ _)"  [0, 0, 10] 10)
      "_gtEx"  :: "[idt, 'a, bool] => bool"   ("(3∃_>_./ _)"  [0, 0, 10] 10)
      "_geAll"   :: "[idt, 'a, bool] => bool"   ("(3∀_≥_./ _)" [0, 0, 10] 10)
      "_geEx"    :: "[idt, 'a, bool] => bool"   ("(3∃_≥_./ _)" [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
    *}

    subsection {* Extra transitivity rules *}

    text {* These support proving chains of decreasing inequalities
        a >= b >= c ... in Isar proofs. *}

    lemma xt1: "a = b ==> b > c ==> a > c"
    by simp

    lemma xt2: "a > b ==> b = c ==> a > c"
    by simp

    lemma xt3: "a = b ==> b >= c ==> a >= c"
    by simp

    lemma xt4: "a >= b ==> b = c ==> a >= c"
    by simp

    lemma xt5: "(x::'a::order) >= y ==> y >= x ==> x = y"
    by simp

    lemma xt6: "(x::'a::order) >= y ==> y >= z ==> x >= z"
    by simp

    lemma xt7: "(x::'a::order) > y ==> y >= z ==> x > z"
    by simp

    lemma xt8: "(x::'a::order) >= y ==> y > z ==> x > z"
    by simp

    lemma xt9: "(a::'a::order) > b ==> b > a ==> ?P"
    by simp

    lemma xt10: "(x::'a::order) > y ==> y > z ==> x > z"
    by simp

    lemma xt11: "(a::'a::order) >= b ==> a ~= b ==> a > b"
    by simp

    lemma xt12: "(a::'a::order) ~= b ==> a >= b ==> a > b"
    by simp

    lemma xt13: "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==>
        a > f c"
    by simp

    lemma xt14: "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==>
        f a > c"
    by auto

    lemma xt15: "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==>
        a >= f c"
    by simp

    lemma xt16: "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==>
        f a >= c"
    by auto

    lemma xt17: "(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 xt18: "(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 xt19: "(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 xt20: "(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 xt21: "(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 xt22: "(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 xt23: "(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 xt24: "(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 xt10 xt11 xt12
        xt13 xt14 xt15 xt15 xt17 xt18 xt19 xt20 xt21 xt22 xt23 xt24

    (*
      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.
    *)

    end

### Order signatures and orders

### Monotonicity

lemmas monoI:

      (!!A B. A ≤ B ==> f A ≤ f B) ==> mono f

and monoD:

      [| mono f; A ≤ B |] ==> f A ≤ f B

lemmas monoI:

      (!!A B. A ≤ B ==> f A ≤ f B) ==> mono f

and monoD:

      [| mono f; A ≤ B |] ==> f A ≤ f B

lemma min_leastL:

      (!!x. least ≤ x) ==> min least x = least

lemma min_of_mono:

      ∀x y. (f x ≤ f y) = (x ≤ y) ==> min (f m) (f n) = f (min m n)

lemma max_leastL:

      (!!x. least ≤ x) ==> max least x = x

lemma max_of_mono:

      ∀x y. (f x ≤ f y) = (x ≤ y) ==> max (f m) (f n) = f (max m n)

### Orders

lemma order_eq_refl:

      x = y ==> x ≤ y

lemma order_less_irrefl:

      ¬ x < x

lemma order_le_less:

      (x ≤ y) = (x < y ∨ x = y)

lemmas order_le_imp_less_or_eq:

      x ≤ y ==> x < y ∨ x = y

lemmas order_le_imp_less_or_eq:

      x ≤ y ==> x < y ∨ x = y

lemma order_less_imp_le:

      x < y ==> x ≤ y

lemma order_less_not_sym:

      x < y ==> ¬ y < x

lemma order_less_asym:

      [| x < y; ¬ P ==> y < x |] ==> P

lemma order_eq_iff:

      (x = y) = (x ≤ y ∧ y ≤ x)

lemma order_antisym_conv:

      y ≤ x ==> (x ≤ y) = (x = y)

lemma order_less_trans:

      [| x < y; y < z |] ==> x < z

lemma order_le_less_trans:

      [| x ≤ y; y < z |] ==> x < z

lemma order_less_le_trans:

      [| x < y; y ≤ z |] ==> x < z

lemma order_less_imp_not_less:

      x < y ==> (¬ y < x) = True

lemma order_less_imp_triv:

      x < y ==> (y < x --> P) = True

lemma order_less_imp_not_eq:

      x < y ==> (x = y) = False

lemma order_less_imp_not_eq2:

      x < y ==> (y = x) = False

lemma min_leastR:

      (!!x. least ≤ x) ==> min x least = least

lemma max_leastR:

      (!!x. least ≤ x) ==> max x least = x

### Transitivity rules for calculational reasoning

lemma order_neq_le_trans:

      [| a ≠ b; a ≤ b |] ==> a < b

lemma order_le_neq_trans:

      [| a ≤ b; a ≠ b |] ==> a < b

lemma order_less_asym':

      [| a < b; b < a |] ==> P

### Least value operator

lemma LeastI2_order:

      [| P x; !!y. P y ==> x ≤ y; !!x. [| P x; ∀y. P y --> x ≤ y |] ==> Q x |]
      ==> Q (Least P)

lemma Least_equality:

      [| P k; !!x. P x ==> k ≤ x |] ==> (LEAST x. P x) = k

### Linear / total orders

lemma linorder_less_linear:

      x < y ∨ x = y ∨ y < x

lemma linorder_le_less_linear:

      x ≤ y ∨ y < x

lemma linorder_le_cases:

      [| x ≤ y ==> P; y ≤ x ==> P |] ==> P

lemma linorder_cases:

      [| x < y ==> P; x = y ==> P; y < x ==> P |] ==> P

lemma linorder_not_less:

      (¬ x < y) = (y ≤ x)

lemma linorder_not_le:

      (¬ x ≤ y) = (y < x)

lemma linorder_neq_iff:

      (x ≠ y) = (x < y ∨ y < x)

lemma linorder_neqE:

      [| x ≠ y; x < y ==> R; y < x ==> R |] ==> R

lemma linorder_antisym_conv1:

      ¬ x < y ==> (x ≤ y) = (x = y)

lemma linorder_antisym_conv2:

      x ≤ y ==> (¬ x < y) = (x = y)

lemma linorder_antisym_conv3:

      ¬ y < x ==> (¬ x < y) = (x = y)

lemma leI:

      ¬ x < y ==> y ≤ x

lemma leD:

      y ≤ x ==> ¬ x < y

lemma not_leE:

      ¬ y ≤ x ==> x < y

### Setup of transitivity reasoner as Solver

lemma less_imp_neq:

      x < y ==> x ≠ y

lemma eq_neq_eq_imp_neq:

      [| x = a; a ≠ b; b = y |] ==> x ≠ y

### Min and max on (linear) orders

lemma le_max_iff_disj:

      (z ≤ max x y) = (z ≤ x ∨ z ≤ y)

lemmas le_maxI1:

      x ≤ max x y

lemmas le_maxI1:

      x ≤ max x y

lemmas le_maxI2:

      y ≤ max x y

lemmas le_maxI2:

      y ≤ max x y

lemma less_max_iff_disj:

      (z < max x y) = (z < x ∨ z < y)

lemma max_less_iff_conj:

      (max x y < z) = (x < z ∧ y < z)

lemma min_less_iff_conj:

      (z < min x y) = (z < x ∧ z < y)

lemma min_le_iff_disj:

      (min x y ≤ z) = (x ≤ z ∨ y ≤ z)

lemma min_less_iff_disj:

      (min x y < z) = (x < z ∨ y < z)

lemmas max_ac:

      max (max x y) z = max x (max y z)

      max x y = max y x

      max x (max y z) = max y (max x z)

lemmas max_ac:

      max (max x y) z = max x (max y z)

      max x y = max y x

      max x (max y z) = max y (max x z)

lemmas min_ac:

      min (min x y) z = min x (min y z)

      min x y = min y x

      min x (min y z) = min y (min x z)

lemmas min_ac:

      min (min x y) z = min x (min y z)

      min x y = min y x

      min x (min y z) = min y (min x z)

lemma split_min:

      P (min i j) = ((i ≤ j --> P i) ∧ (¬ i ≤ j --> P j))

lemma split_max:

      P (max i j) = ((i ≤ j --> P j) ∧ (¬ i ≤ j --> P i))

### Bounded quantifiers

### Extra transitivity rules

lemma xt1:

      [| a = b; c < b |] ==> c < a

lemma xt2:

      [| b < a; b = c |] ==> c < a

lemma xt3:

      [| a = b; c ≤ b |] ==> c ≤ a

lemma xt4:

      [| b ≤ a; b = c |] ==> c ≤ a

lemma xt5:

      [| y ≤ x; x ≤ y |] ==> x = y

lemma xt6:

      [| y ≤ x; z ≤ y |] ==> z ≤ x

lemma xt7:

      [| y < x; z ≤ y |] ==> z < x

lemma xt8:

      [| y ≤ x; z < y |] ==> z < x

lemma xt9:

      [| b < a; a < b |] ==> True

lemma xt10:

      [| y < x; z < y |] ==> z < x

lemma xt11:

      [| b ≤ a; a ≠ b |] ==> b < a

lemma xt12:

      [| a ≠ b; b ≤ a |] ==> b < a

lemma xt13:

      [| a = f b; c < b; !!x y. y < x ==> f y < f x |] ==> f c < a

lemma xt14:

      [| b < a; f b = c; !!x y. y < x ==> f y < f x |] ==> c < f a

lemma xt15:

      [| a = f b; c ≤ b; !!x y. y ≤ x ==> f y ≤ f x |] ==> f c ≤ a

lemma xt16:

      [| b ≤ a; f b = c; !!x y. y ≤ x ==> f y ≤ f x |] ==> c ≤ f a

lemma xt17:

      [| f b ≤ a; c ≤ b; !!x y. y ≤ x ==> f y ≤ f x |] ==> f c ≤ a

lemma xt18:

      [| b ≤ a; c ≤ f b; !!x y. y ≤ x ==> f y ≤ f x |] ==> c ≤ f a

lemma xt19:

      [| f b < a; c ≤ b; !!x y. y ≤ x ==> f y ≤ f x |] ==> f c < a

lemma xt20:

      [| b < a; c ≤ f b; !!x y. y < x ==> f y < f x |] ==> c < f a

lemma xt21:

      [| f b ≤ a; c < b; !!x y. y < x ==> f y < f x |] ==> f c < a

lemma xt22:

      [| b ≤ a; c < f b; !!x y. y ≤ x ==> f y ≤ f x |] ==> c < f a

lemma xt23:

      [| f b < a; c < b; !!x y. y < x ==> f y < f x |] ==> f c < a

lemma xt24:

      [| b < a; c < f b; !!x y. y < x ==> f y < f x |] ==> c < f a

lemmas xtrans:

      [| a = b; c < b |] ==> c < a

      [| b < a; b = c |] ==> c < a

      [| a = b; c ≤ b |] ==> c ≤ a

      [| b ≤ a; b = c |] ==> c ≤ a

      [| y ≤ x; x ≤ y |] ==> x = y

      [| y ≤ x; z ≤ y |] ==> z ≤ x

      [| y < x; z ≤ y |] ==> z < x

      [| y ≤ x; z < y |] ==> z < x

      [| b < a; a < b |] ==> True

      [| y < x; z < y |] ==> z < x

      [| b ≤ a; a ≠ b |] ==> b < a

      [| a ≠ b; b ≤ a |] ==> b < a

      [| a = f b; c < b; !!x y. y < x ==> f y < f x |] ==> f c < a

      [| b < a; f b = c; !!x y. y < x ==> f y < f x |] ==> c < f a

      [| a = f b; c ≤ b; !!x y. y ≤ x ==> f y ≤ f x |] ==> f c ≤ a

      [| a = f b; c ≤ b; !!x y. y ≤ x ==> f y ≤ f x |] ==> f c ≤ a

      [| f b ≤ a; c ≤ b; !!x y. y ≤ x ==> f y ≤ f x |] ==> f c ≤ a

      [| b ≤ a; c ≤ f b; !!x y. y ≤ x ==> f y ≤ f x |] ==> c ≤ f a

      [| f b < a; c ≤ b; !!x y. y ≤ x ==> f y ≤ f x |] ==> f c < a

      [| b < a; c ≤ f b; !!x y. y < x ==> f y < f x |] ==> c < f a

      [| f b ≤ a; c < b; !!x y. y < x ==> f y < f x |] ==> f c < a

      [| b ≤ a; c < f b; !!x y. y ≤ x ==> f y ≤ f x |] ==> c < f a

      [| f b < a; c < b; !!x y. y < x ==> f y < f x |] ==> f c < a

      [| b < a; c < f b; !!x y. y < x ==> f y < f x |] ==> c < f a

lemmas xtrans:

      [| a = b; c < b |] ==> c < a

      [| b < a; b = c |] ==> c < a

      [| a = b; c ≤ b |] ==> c ≤ a

      [| b ≤ a; b = c |] ==> c ≤ a

      [| y ≤ x; x ≤ y |] ==> x = y

      [| y ≤ x; z ≤ y |] ==> z ≤ x

      [| y < x; z ≤ y |] ==> z < x

      [| y ≤ x; z < y |] ==> z < x

      [| b < a; a < b |] ==> True

      [| y < x; z < y |] ==> z < x

      [| b ≤ a; a ≠ b |] ==> b < a

      [| a ≠ b; b ≤ a |] ==> b < a

      [| a = f b; c < b; !!x y. y < x ==> f y < f x |] ==> f c < a

      [| b < a; f b = c; !!x y. y < x ==> f y < f x |] ==> c < f a

      [| a = f b; c ≤ b; !!x y. y ≤ x ==> f y ≤ f x |] ==> f c ≤ a

      [| a = f b; c ≤ b; !!x y. y ≤ x ==> f y ≤ f x |] ==> f c ≤ a

      [| f b ≤ a; c ≤ b; !!x y. y ≤ x ==> f y ≤ f x |] ==> f c ≤ a

      [| b ≤ a; c ≤ f b; !!x y. y ≤ x ==> f y ≤ f x |] ==> c ≤ f a

      [| f b < a; c ≤ b; !!x y. y ≤ x ==> f y ≤ f x |] ==> f c < a

      [| b < a; c ≤ f b; !!x y. y < x ==> f y < f x |] ==> c < f a

      [| f b ≤ a; c < b; !!x y. y < x ==> f y < f x |] ==> f c < a

      [| b ≤ a; c < f b; !!x y. y ≤ x ==> f y ≤ f x |] ==> c < f a

      [| f b < a; c < b; !!x y. y < x ==> f y < f x |] ==> f c < a

      [| b < a; c < f b; !!x y. y < x ==> f y < f x |] ==> c < f a