# Theory Orderings

(*  Title:      HOL/Orderings.thy
Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
*)

section Abstract orderings

theory Orderings
imports HOL
keywords "print_orders" :: diag
begin

ML_file ~~/src/Provers/order_procedure.ML
ML_file ~~/src/Provers/order_tac.ML

subsection Abstract ordering

locale partial_preordering =
fixes less_eq :: 'a  'a  bool (infix  50)
assumes refl: a  a ― ‹not iff›: makes problems due to multiple (dual) interpretations
and trans: a  b  b  c  a  c

locale preordering = partial_preordering +
fixes less :: 'a  'a  bool (infix < 50)
assumes strict_iff_not: a < b  a  b  ¬ b  a
begin

lemma strict_implies_order:
a < b  a  b

lemma irrefl: ― ‹not iff›: makes problems due to multiple (dual) interpretations
¬ a < a

lemma asym:
a < b  b < a  False

lemma strict_trans1:
a  b  b < c  a < c
by (auto simp add: strict_iff_not intro: trans)

lemma strict_trans2:
a < b  b  c  a < c
by (auto simp add: strict_iff_not intro: trans)

lemma strict_trans:
a < b  b < c  a < c
by (auto intro: strict_trans1 strict_implies_order)

end

lemma preordering_strictI: ― ‹Alternative introduction rule with bias towards strict order
fixes less_eq (infix  50)
and less (infix < 50)
assumes less_eq_less: a b. a  b  a < b  a = b
assumes asym: a b. a < b  ¬ b < a
assumes irrefl: a. ¬ a < a
assumes trans: a b c. a < b  b < c  a < c
shows preordering () (<)
proof
fix a b
show a < b  a  b  ¬ b  a
by (auto simp add: less_eq_less asym irrefl)
next
fix a
show a  a
next
fix a b c
assume a  b and b  c then show a  c
by (auto simp add: less_eq_less intro: trans)
qed

lemma preordering_dualI:
fixes less_eq (infix  50)
and less (infix < 50)
assumes preordering (λa b. b  a) (λa b. b < a)
shows preordering () (<)
proof -
from assms interpret preordering λa b. b  a λa b. b < a .
show ?thesis
by standard (auto simp: strict_iff_not refl intro: trans)
qed

locale ordering = partial_preordering +
fixes less :: 'a  'a  bool (infix < 50)
assumes strict_iff_order: a < b  a  b  a  b
assumes antisym: a  b  b  a  a = b
begin

sublocale preordering () (<)
proof
show a < b  a  b  ¬ b  a for a b
by (auto simp add: strict_iff_order intro: antisym)
qed

lemma strict_implies_not_eq:
a < b  a  b

lemma not_eq_order_implies_strict:
a  b  a  b  a < b

lemma order_iff_strict:
a  b  a < b  a = b
by (auto simp add: strict_iff_order refl)

lemma eq_iff: a = b  a  b  b  a
by (auto simp add: refl intro: antisym)

end

lemma ordering_strictI: ― ‹Alternative introduction rule with bias towards strict order
fixes less_eq (infix  50)
and less (infix < 50)
assumes less_eq_less: a b. a  b  a < b  a = b
assumes asym: a b. a < b  ¬ b < a
assumes irrefl: a. ¬ a < a
assumes trans: a b c. a < b  b < c  a < c
shows ordering () (<)
proof
fix a b
show a < b  a  b  a  b
by (auto simp add: less_eq_less asym irrefl)
next
fix a
show a  a
next
fix a b c
assume a  b and b  c then show a  c
by (auto simp add: less_eq_less intro: trans)
next
fix a b
assume a  b and b  a then show a = b
by (auto simp add: less_eq_less asym)
qed

lemma ordering_dualI:
fixes less_eq (infix  50)
and less (infix < 50)
assumes ordering (λa b. b  a) (λa b. b < a)
shows ordering () (<)
proof -
from assms interpret ordering λa b. b  a λa b. b < a .
show ?thesis
by standard (auto simp: strict_iff_order refl intro: antisym trans)
qed

locale ordering_top = ordering +
fixes top :: 'a  ()
assumes extremum [simp]: a
begin

lemma extremum_uniqueI:
a  a =
by (rule antisym) auto

lemma extremum_unique:
a  a =
by (auto intro: antisym)

lemma extremum_strict [simp]:
¬ ( < a)
using extremum [of a] by (auto simp add: order_iff_strict intro: asym irrefl)

lemma not_eq_extremum:
a    a <
by (auto simp add: order_iff_strict intro: not_eq_order_implies_strict extremum)

end

subsection Syntactic orders

class ord =
fixes less_eq :: "'a  'a  bool"
and less :: "'a  'a  bool"
begin

notation
less_eq  ("'(≤')") and
less_eq  ("(_/  _)"  [51, 51] 50) and
less  ("'(<')") and
less  ("(_/ < _)"  [51, 51] 50)

abbreviation (input)
greater_eq  (infix "" 50)
where "x  y  y  x"

abbreviation (input)
greater  (infix ">" 50)
where "x > y  y < x"

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

notation (input)
greater_eq  (infix ">=" 50)

end

subsection Quasi orders

class preorder = ord +
assumes less_le_not_le: "x < y  x  y  ¬ (y  x)"
and order_refl [iff]: "x  x"
and order_trans: "x  y  y  z  x  z"
begin

sublocale order: preordering less_eq less + dual_order: preordering greater_eq greater
proof -
interpret preordering less_eq less
by standard (auto intro: order_trans simp add: less_le_not_le)
show
by (fact preordering_axioms)
then show
by (rule preordering_dualI)
qed

text Reflexivity.

lemma eq_refl: "x = y  x  y"
― ‹This form is useful with the classical reasoner.
by (erule ssubst) (rule order_refl)

lemma less_irrefl [iff]: "¬ x < x"

lemma less_imp_le: "x < y  x  y"

text Asymmetry.

lemma less_not_sym: "x < y  ¬ (y < x)"

lemma less_asym: "x < y  (¬ P  y < x)  P"
by (drule less_not_sym, erule contrapos_np) simp

text Transitivity.

lemma less_trans: "x < y  y < z  x < z"
by (auto simp add: less_le_not_le intro: order_trans)

lemma le_less_trans: "x  y  y < z  x < z"
by (auto simp add: less_le_not_le intro: order_trans)

lemma less_le_trans: "x < y  y  z  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  (¬ y < x)  True"
by (blast elim: less_asym)

lemma less_imp_triv: "x < y  (y < x  P)  True"
by (blast elim: less_asym)

text Transitivity rules for calculational reasoning

lemma less_asym': "a < b  b < a  P"
by (rule less_asym)

text Dual order

lemma dual_preorder:

by standard (auto simp add: less_le_not_le intro: order_trans)

end

lemma preordering_preorderI:
class.preorder () (<) if preordering () (<)
for less_eq (infix  50) and less (infix < 50)
proof -
from that interpret preordering () (<) .
show ?thesis
by standard (auto simp add: strict_iff_not refl intro: trans)
qed

subsection Partial orders

class order = preorder +
assumes order_antisym: "x  y  y  x  x = y"
begin

lemma less_le: "x < y  x  y  x  y"
by (auto simp add: less_le_not_le intro: order_antisym)

sublocale order: ordering less_eq less + dual_order: ordering greater_eq greater
proof -
interpret ordering less_eq less
by standard (auto intro: order_antisym order_trans simp add: less_le)
show
by (fact ordering_axioms)
then show
by (rule ordering_dualI)
qed

print_theorems

text Reflexivity.

lemma le_less: "x  y  x < y  x = y"
― ‹NOT suitable for iff, since it can cause PROOF FAILED.
by (fact order.order_iff_strict)

lemma le_imp_less_or_eq: "x  y  x < y  x = y"

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

lemma less_imp_not_eq: "x < y  (x = y)  False"
by auto

lemma less_imp_not_eq2: "x < y  (y = x)  False"
by auto

text Transitivity rules for calculational reasoning

lemma neq_le_trans: "a  b  a  b  a < b"
by (fact order.not_eq_order_implies_strict)

lemma le_neq_trans: "a  b  a  b  a < b"
by (rule order.not_eq_order_implies_strict)

text Asymmetry.

lemma order_eq_iff: "x = y  x  y  y  x"
by (fact order.eq_iff)

lemma antisym_conv: "y  x  x  y  x = y"

lemma less_imp_neq: "x < y  x  y"
by (fact order.strict_implies_not_eq)

lemma antisym_conv1: "¬ x < y  x  y  x = y"

lemma antisym_conv2: "x  y  ¬ x < y  x = y"

lemma leD: "y  x  ¬ x < y"
by (auto simp: less_le order.antisym)

text Least value operator

definition (in ord)
Least :: "('a  bool)  'a" (binder "LEAST " 10) where
"Least P = (THE x. P x  (y. P y  x  y))"

lemma Least_equality:
assumes "P x"
and "y. P y  x  y"
shows "Least P = x"
unfolding Least_def by (rule the_equality)
(blast intro: assms order.antisym)+

lemma LeastI2_order:
assumes "P x"
and "y. P y  x  y"
and "x. P x  y. P y  x  y  Q x"
shows "Q (Least P)"
unfolding Least_def by (rule theI2)
(blast intro: assms order.antisym)+

lemma Least_ex1:
assumes   "∃!x. P x  (y. P y  x  y)"
shows     Least1I: "P (Least P)" and Least1_le: "P z  Least P  z"
using     theI'[OF assms]
unfolding Least_def
by        auto

text Greatest value operator

definition Greatest :: "('a  bool)  'a" (binder "GREATEST " 10) where
"Greatest P = (THE x. P x  (y. P y  x  y))"

lemma GreatestI2_order:
" P x;
y. P y  x  y;
x.  P x; y. P y  x  y   Q x
Q (Greatest P)"
unfolding Greatest_def
by (rule theI2) (blast intro: order.antisym)+

lemma Greatest_equality:
" P x;  y. P y  x  y   Greatest P = x"
unfolding Greatest_def
by (rule the_equality) (blast intro: order.antisym)+

end

lemma ordering_orderI:
fixes less_eq (infix "" 50)
and less (infix "<" 50)
assumes "ordering less_eq less"
shows "class.order less_eq less"
proof -
from assms interpret ordering less_eq less .
show ?thesis
by standard (auto intro: antisym trans simp add: refl strict_iff_order)
qed

lemma order_strictI:
fixes less (infix "<" 50)
and less_eq (infix "" 50)
assumes "a b. a  b  a < b  a = b"
assumes "a b. a < b  ¬ b < a"
assumes "a. ¬ a < a"
assumes "a b c. a < b  b < c  a < c"
shows "class.order less_eq less"
by (rule ordering_orderI) (rule ordering_strictI, (fact assms)+)

context order
begin

text Dual order

lemma dual_order:

using dual_order.ordering_axioms by (rule ordering_orderI)

end

subsection Linear (total) orders

class linorder = order +
assumes linear: "x  y  y  x"
begin

lemma less_linear: "x < y  x = y  y < x"
unfolding less_le using less_le linear by blast

lemma le_less_linear: "x  y  y < x"

lemma le_cases [case_names le ge]:
"(x  y  P)  (y  x  P)  P"
using linear by blast

lemma (in linorder) le_cases3:
"x  y; y  z  P; y  x; x  z  P; x  z; z  y  P;
z  y; y  x  P; y  z; z  x  P; z  x; x  y  P  P"
by (blast intro: le_cases)

lemma linorder_cases [case_names less equal greater]:
"(x < y  P)  (x = y  P)  (y < x  P)  P"
using less_linear by blast

lemma linorder_wlog[case_names le sym]:
"(a b. a  b  P a b)  (a b. P b a  P a b)  P a b"
by (cases rule: le_cases[of a b]) blast+

lemma not_less: "¬ x < y  y  x"
unfolding less_le
using linear by (blast intro: order.antisym)

lemma not_less_iff_gr_or_eq: "¬(x < y)  (x > y  x = y)"

lemma not_le: "¬ x  y  y < x"
unfolding less_le
using linear by (blast intro: order.antisym)

lemma neq_iff: "x  y  x < y  y < x"
by (cut_tac x = x and y = y in less_linear, auto)

lemma neqE: "x  y  (x < y  R)  (y < x  R)  R"

lemma antisym_conv3: "¬ y < x  ¬ x < y  x = y"
by (blast intro: order.antisym dest: not_less [THEN iffD1])

lemma leI: "¬ x < y  y  x"
unfolding not_less .

lemma not_le_imp_less: "¬ y  x  x < y"
unfolding not_le .

lemma linorder_less_wlog[case_names less refl sym]:
"a b. a < b  P a b;  a. P a a;  a b. P b a  P a b  P a b"
using antisym_conv3 by blast

text Dual order

lemma dual_linorder:

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

end

text Alternative introduction rule with bias towards strict order

lemma linorder_strictI:
fixes less_eq (infix "" 50)
and less (infix "<" 50)
assumes "class.order less_eq less"
assumes trichotomy: "a b. a < b  a = b  b < a"
shows "class.linorder less_eq less"
proof -
interpret order less_eq less
by (fact class.order less_eq less)
show ?thesis
proof
fix a b
show "a  b  b  a"
using trichotomy by (auto simp add: le_less)
qed
qed

subsection Reasoning tools setup

ML
structure Logic_Signature : LOGIC_SIGNATURE = struct
val mk_Trueprop = HOLogic.mk_Trueprop
val dest_Trueprop = HOLogic.dest_Trueprop
val Trueprop_conv = HOLogic.Trueprop_conv
val Not = HOLogic.Not
val conj = HOLogic.conj
val disj = HOLogic.disj

val notI = @{thm notI}
val ccontr = @{thm ccontr}
val conjI = @{thm conjI}
val conjE = @{thm conjE}
val disjE = @{thm disjE}

val not_not_conv = Conv.rewr_conv @{thm eq_reflection[OF not_not]}
val de_Morgan_conj_conv = Conv.rewr_conv @{thm eq_reflection[OF de_Morgan_conj]}
val de_Morgan_disj_conv = Conv.rewr_conv @{thm eq_reflection[OF de_Morgan_disj]}
val conj_disj_distribL_conv = Conv.rewr_conv @{thm eq_reflection[OF conj_disj_distribL]}
val conj_disj_distribR_conv = Conv.rewr_conv @{thm eq_reflection[OF conj_disj_distribR]}
end

structure HOL_Base_Order_Tac = Base_Order_Tac(
structure Logic_Sig = Logic_Signature;
(* Exclude types with specialised solvers. *)
val excluded_types = [HOLogic.natT, HOLogic.intT, HOLogic.realT]
)

structure HOL_Order_Tac = Order_Tac(structure Base_Tac = HOL_Base_Order_Tac)

fun print_orders ctxt0 =
let
val ctxt = Config.put show_sorts true ctxt0
val orders = HOL_Order_Tac.Data.get (Context.Proof ctxt)
fun pretty_term t = Pretty.block
[Pretty.quote (Syntax.pretty_term ctxt t), Pretty.brk 1,
Pretty.str "::", Pretty.brk 1,
Pretty.quote (Syntax.pretty_typ ctxt (type_of t)), Pretty.brk 1]
fun pretty_order ({kind = kind, ops = ops, ...}, _) =
Pretty.block ([Pretty.str (@{make_string} kind), Pretty.str ":", Pretty.brk 1]
@ map pretty_term ops)
in
Pretty.writeln (Pretty.big_list "order structures:" (map pretty_order orders))
end

val _ =
Outer_Syntax.command command_keywordprint_orders
"print order structures available to transitivity reasoner"
(Scan.succeed (Toplevel.keep (print_orders o Toplevel.context_of)))

method_setup order =
Scan.succeed (fn ctxt => SIMPLE_METHOD' (HOL_Order_Tac.tac [] ctxt))
"transitivity reasoner"

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

context order
begin

lemma nless_le: "(¬ a < b)  (¬ a  b)  a = b"
using local.dual_order.order_iff_strict by blast

local_setup
HOL_Order_Tac.declare_order {
ops = {eq = @{term (=) :: 'a  'a  bool}, le = @{term (≤)}, lt = @{term (<)}},
thms = {trans = @{thm order_trans}, refl = @{thm order_refl}, eqD1 = @{thm eq_refl},
eqD2 = @{thm eq_refl[OF sym]}, antisym = @{thm order_antisym}, contr = @{thm notE}},
conv_thms = {less_le = @{thm eq_reflection[OF less_le]},
nless_le = @{thm eq_reflection[OF nless_le]}}
}

end

context linorder
begin

lemma nle_le: "(¬ a  b)  b  a  b  a"
using not_le less_le by simp

local_setup
HOL_Order_Tac.declare_linorder {
ops = {eq = @{term (=) :: 'a  'a  bool}, le = @{term (≤)}, lt = @{term (<)}},
thms = {trans = @{thm order_trans}, refl = @{thm order_refl}, eqD1 = @{thm eq_refl},
eqD2 = @{thm eq_refl[OF sym]}, antisym = @{thm order_antisym}, contr = @{thm notE}},
conv_thms = {less_le = @{thm eq_reflection[OF less_le]},
nless_le = @{thm eq_reflection[OF not_less]},
nle_le = @{thm eq_reflection[OF nle_le]}}
}

end

setup
map_theory_simpset (fn ctxt0 => ctxt0 addSolver
mk_solver "Transitivity" (fn ctxt => HOL_Order_Tac.tac (Simplifier.prems_of ctxt) ctxt))

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

fun antisym_le_simproc ctxt ct =
(case Thm.term_of ct of
(le as Const (_, T)) \$ r \$ s =>
(let
val prems = Simplifier.prems_of ctxt;
val less = Const (const_nameless, T);
val t = HOLogic.mk_Trueprop(le \$ s \$ r);
in
(case find_first (prp t) prems of
NONE =>
let val t = HOLogic.mk_Trueprop(HOLogic.Not \$ (less \$ r \$ s)) in
(case find_first (prp t) prems of
NONE => NONE
| SOME thm => SOME(mk_meta_eq(thm RS @{thm antisym_conv1})))
end
| SOME thm => SOME (mk_meta_eq (thm RS @{thm order_class.antisym_conv})))
end handle THM _ => NONE)
| _ => NONE);

fun antisym_less_simproc ctxt ct =
(case Thm.term_of ct of
NotC \$ ((less as Const(_,T)) \$ r \$ s) =>
(let
val prems = Simplifier.prems_of ctxt;
val le = Const (const_nameless_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 )))
end
| SOME thm => SOME (mk_meta_eq (thm RS @{thm antisym_conv2})))
end handle THM _ => NONE)
| _ => NONE);

end;

simproc_setup antisym_le ("(x::'a::order)  y") = "K antisym_le_simproc"
simproc_setup antisym_less ("¬ (x::'a::linorder) < y") = "K antisym_less_simproc"

subsection Bounded quantifiers

syntax (ASCII)
"_All_less" :: "[idt, 'a, bool] => bool"    ("(3ALL _<_./ _)"  [0, 0, 10] 10)
"_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3EX _<_./ _)"  [0, 0, 10] 10)
"_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _<=_./ _)" [0, 0, 10] 10)
"_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _<=_./ _)" [0, 0, 10] 10)

"_All_greater" :: "[idt, 'a, bool] => bool"    ("(3ALL _>_./ _)"  [0, 0, 10] 10)
"_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3EX _>_./ _)"  [0, 0, 10] 10)
"_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3ALL _>=_./ _)" [0, 0, 10] 10)
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3EX _>=_./ _)" [0, 0, 10] 10)

"_All_neq" :: "[idt, 'a, bool] => bool"    ("(3ALL _~=_./ _)"  [0, 0, 10] 10)
"_Ex_neq" :: "[idt, 'a, bool] => bool"    ("(3EX _~=_./ _)"  [0, 0, 10] 10)

syntax
"_All_less" :: "[idt, 'a, bool] => bool"    ("(3_<_./ _)"  [0, 0, 10] 10)
"_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3_<_./ _)"  [0, 0, 10] 10)
"_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3__./ _)" [0, 0, 10] 10)
"_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3__./ _)" [0, 0, 10] 10)

"_All_greater" :: "[idt, 'a, bool] => bool"    ("(3_>_./ _)"  [0, 0, 10] 10)
"_Ex_greater" :: "[idt, 'a, bool] => bool"    ("(3_>_./ _)"  [0, 0, 10] 10)
"_All_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3__./ _)" [0, 0, 10] 10)
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool"    ("(3__./ _)" [0, 0, 10] 10)

"_All_neq" :: "[idt, 'a, bool] => bool"    ("(3__./ _)"  [0, 0, 10] 10)
"_Ex_neq" :: "[idt, 'a, bool] => bool"    ("(3__./ _)"  [0, 0, 10] 10)

syntax (input)
"_All_less" :: "[idt, 'a, bool] => bool"    ("(3! _<_./ _)"  [0, 0, 10] 10)
"_Ex_less" :: "[idt, 'a, bool] => bool"    ("(3? _<_./ _)"  [0, 0, 10] 10)
"_All_less_eq" :: "[idt, 'a, bool] => bool"    ("(3! _<=_./ _)" [0, 0, 10] 10)
"_Ex_less_eq" :: "[idt, 'a, bool] => bool"    ("(3? _<=_./ _)" [0, 0, 10] 10)
"_All_neq" :: "[idt, 'a, bool] => bool"    ("(3! _~=_./ _)"  [0, 0, 10] 10)
"_Ex_neq" :: "[idt, 'a, bool] => bool"    ("(3? _~=_./ _)"  [0, 0, 10] 10)

translations
"x<y. P"  "x. x < y  P"
"x<y. P"  "x. x < y  P"
"xy. P"  "x. x  y  P"
"xy. P"  "x. x  y  P"
"x>y. P"  "x. x > y  P"
"x>y. P"  "x. x > y  P"
"xy. P"  "x. x  y  P"
"xy. P"  "x. x  y  P"
"xy. P"  "x. x  y  P"
"xy. P"  "x. x  y  P"

print_translation
let
val All_binder = Mixfix.binder_name const_syntaxAll;
val Ex_binder = Mixfix.binder_name const_syntaxEx;
val impl = const_syntaxHOL.implies;
val conj = const_syntaxHOL.conj;
val less = const_syntaxless;
val less_eq = const_syntaxless_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 x c n P = Syntax.const c \$ Syntax_Trans.mark_bound_body x \$ n \$ P;

fun tr' q = (q, fn _ =>
(fn [Const (syntax_const‹_bound›, _) \$ Free (v, T),
Const (c, _) \$ (Const (d, _) \$ t \$ u) \$ P] =>
(case AList.lookup (=) trans (q, c, d) of
NONE => raise Match
| SOME (l, g) =>
if matches_bound v t andalso not (contains_var v u) then mk (v, T) l u P
else if matches_bound v u andalso not (contains_var v t) then mk (v, T) g t P
else raise Match)
| _ => raise Match));
in [tr' All_binder, tr' Ex_binder] end

subsection Transitivity reasoning

context ord
begin

lemma ord_le_eq_trans: "a  b  b = c  a  c"
by (rule subst)

lemma ord_eq_le_trans: "a = b  b  c  a  c"
by (rule ssubst)

lemma ord_less_eq_trans: "a < b  b = c  a < c"
by (rule subst)

lemma ord_eq_less_trans: "a = b  b < c  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] =
order.antisym

lemmas (in ord) [trans] =
ord_le_eq_trans
ord_eq_le_trans
ord_less_eq_trans
ord_eq_less_trans

lemmas [trans] =
trans

lemmas order_trans_rules =
order_less_subst2
order_less_subst1
order_le_less_subst2
order_le_less_subst1
order_less_le_subst2
order_less_le_subst1
order_subst2
order_subst1
ord_le_eq_subst
ord_eq_le_subst
ord_less_eq_subst
ord_eq_less_subst
forw_subst
back_subst
rev_mp
mp
neq_le_trans
le_neq_trans
less_trans
less_asym'
le_less_trans
less_le_trans
order_trans
order.antisym
ord_le_eq_trans
ord_eq_le_trans
ord_less_eq_trans
ord_eq_less_trans
trans

text 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

(*
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

context order
begin

definition mono :: "('a  'b::order)  bool" where
"mono f  (x y. x  y  f x  f y)"

lemma monoI [intro?]:
fixes f :: "'a  'b::order"
shows "(x y. x  y  f x  f y)  mono f"
unfolding mono_def by iprover

lemma monoD [dest?]:
fixes f :: "'a  'b::order"
shows "mono f  x  y  f x  f y"
unfolding mono_def by iprover

lemma monoE:
fixes f :: "'a  'b::order"
assumes "mono f"
assumes "x  y"
obtains "f x  f y"
proof
from assms show "f x  f y" by (simp add: mono_def)
qed

definition antimono :: "('a  'b::order)  bool" where
"antimono f  (x y. x  y  f x  f y)"

lemma antimonoI [intro?]:
fixes f :: "'a  'b::order"
shows "(x y. x  y  f x  f y)  antimono f"
unfolding antimono_def by iprover

lemma antimonoD [dest?]:
fixes f :: "'a  'b::order"
shows "antimono f  x  y  f x  f y"
unfolding antimono_def by iprover

lemma antimonoE:
fixes f :: "'a  'b::order"
assumes "antimono f"
assumes "x  y"
obtains "f x  f y"
proof
from assms show "f x  f y" by (simp add: antimono_def)
qed

definition strict_mono :: "('a  'b::order)  bool" where
"strict_mono f  (x y. x < y  f x < f y)"

lemma strict_monoI [intro?]:
assumes "x y. x < y  f x < f y"
shows "strict_mono f"
using assms unfolding strict_mono_def by auto

lemma strict_monoD [dest?]:
"strict_mono f  x < y  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  y"
show "f x  f y"
proof (cases "x = y")
case True then show ?thesis by simp
next
case False with x  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 mono_invE:
fixes f :: "'a  'b::order"
assumes "mono f"
assumes "f x < f y"
obtains "x  y"
proof
show "x  y"
proof (rule ccontr)
assume "¬ x  y"
then have "y  x" by simp
with mono f obtain "f y  f x" by (rule monoE)
with f x < f y show False by simp
qed
qed

lemma mono_strict_invE:
fixes f :: "'a  'b::order"
assumes "mono f"
assumes "f x < f y"
obtains "x < y"
proof
show "x < y"
proof (rule ccontr)
assume "¬ x < y"
then have "y  x" by simp
with mono f obtain "f y  f x" by (rule monoE)
with f x < f y show False by simp
qed
qed

lemma strict_mono_eq:
assumes "strict_mono f"
shows "f x = f y  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  f y  x  y"
proof
assume "x  y"
with assms strict_mono_mono monoD show "f x  f y" by auto
next
assume "f x  f y"
show "x  y" proof (rule ccontr)
assume "¬ x  y" then have "y < x" by simp
with assms strict_monoD have "f y < f x" by auto
with f x  f y show False by simp
qed
qed

lemma strict_mono_less:
assumes "strict_mono f"
shows "f x < f y  x < y"
using assms
by (auto simp add: less_le Orderings.less_le strict_mono_eq strict_mono_less_eq)

end

subsection min and max -- fundamental

definition (in ord) min :: "'a  'a  'a" where
"min a b = (if a  b then a else b)"

definition (in ord) max :: "'a  'a  'a" where
"max a b = (if a  b then b else a)"

lemma min_absorb1: "x  y  min x y = x"

lemma max_absorb2: "x  y  max x y = y"

lemma min_absorb2: "(y::'a::order)  x  min x y = y"

lemma max_absorb1: "(y::'a::order)  x  max x y = x"

lemma max_min_same [simp]:
fixes x y :: "'a :: linorder"
shows "max x (min x y) = x" "max (min x y) x = x" "max (min x y) y = y" "max y (min x y) = y"

subsection (Unique) top and bottom elements

class bot =
fixes bot :: 'a ("")

class order_bot = order + bot +
assumes bot_least: "  a"
begin

sublocale bot: ordering_top greater_eq greater bot
by standard (fact bot_least)

lemma le_bot:
"a    a = "
by (fact bot.extremum_uniqueI)

lemma bot_unique:
"a    a = "
by (fact bot.extremum_unique)

lemma not_less_bot:
"¬ a < "
by (fact bot.extremum_strict)

lemma bot_less:
"a     < a"
by (fact bot.not_eq_extremum)

lemma max_bot[simp]: "max bot x = x"

lemma max_bot2[simp]: "max x bot = x"

lemma min_bot[simp]: "min bot x = bot"

lemma min_bot2[simp]: "min x bot = bot"

end

class top =
fixes top :: 'a ("")

class order_top = order + top +
assumes top_greatest: "a  "
begin

sublocale top: ordering_top less_eq less top
by standard (fact top_greatest)

lemma top_le:
"  a  a = "
by (fact top.extremum_uniqueI)

lemma top_unique:
"  a  a = "
by (fact top.extremum_unique)

lemma not_top_less:
"¬  < a"
by (fact top.extremum_strict)

lemma less_top:
"a    a < "
by (fact top.not_eq_extremum)

lemma max_top[simp]: "max top x = top"

lemma max_top2[simp]: "max x top = top"

lemma min_top[simp]: "min top x = x"

lemma min_top2[simp]: "min x top = x"

end

subsection Dense orders

class dense_order = order +
assumes dense: "x < y  (z. x < z  z < y)"

class dense_linorder = linorder + dense_order
begin

lemma dense_le:
fixes y z :: 'a
assumes "x. x < y  x  z"
shows "y  z"
proof (rule ccontr)
assume "¬ ?thesis"
hence "z < y" by simp
from dense[OF this]
obtain x where "x < y" and "z < x" by safe
moreover have "x  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 *: "w.  x < w ; w < y   w  z"
shows "y  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  z"
proof (rule disjE)
assume "u  w"
from less_le_trans[OF x < u u  w] w < y
show "w  z" by (rule *)
next
assume "w  u"
from w  u *[OF x < u u < y]
show "w  z" by (rule order_trans)
qed
qed

lemma dense_ge:
fixes y z :: 'a
assumes "x. z < x  y  x"
shows "y  z"
proof (rule ccontr)
assume "¬ ?thesis"
hence "z < y" by simp
from dense[OF this]
obtain x where "x < y" and "z < x" by safe
moreover have "y  x" using assms[OF z < x] .
ultimately show False by auto
qed

lemma dense_ge_bounded:
fixes x y z :: 'a
assumes "z < x"
assumes *: "w.  z < w ; w < x   y  w"
shows "y  z"
proof (rule dense_ge)
fix w assume "z < w"
from dense[OF z < x] obtain u where "z < u" "u < x" by safe
from linear[of u w]
show "y  w"
proof (rule disjE)
assume "w  u"
from z < w le_less_trans[OF w  u u < x]
show "y  w" by (rule *)
next
assume "u  w"
from *[OF z < u u < x] u  w
show "y  w" by (rule order_trans)
qed
qed

end

class no_top = order +
assumes gt_ex: "y. x < y"

class no_bot = order +
assumes lt_ex: "y. y < x"

class unbounded_dense_linorder = dense_linorder + no_top + no_bot

subsection Wellorders

class wellorder = linorder +
assumes less_induct [case_names less]: "(x. (y. y < x  P y)  P x)  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)  k"
proof -
have "P (LEAST x. P x)  (LEAST x. P x)  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: "¬ (P (LEAST a. P a)  (LEAST a. P a)  x)"
have "y. P y  x  y"
proof (rule classical)
fix y
assume "P y" and "¬ x  y"
with less have "P (LEAST a. P a)" and "(LEAST a. P a)  y"
with assm have "x < (LEAST a. P a)" and "(LEAST a. P a)  y"
by auto
then show "x  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)  k" by auto
qed

― ‹The following 3 lemmas are due to Brian Huffman
lemma LeastI_ex: "x. P x  P (Least P)"
by (erule exE) (erule LeastI)

lemma LeastI2:
"P a  (x. P x  Q x)  Q (Least P)"
by (blast intro: LeastI)

lemma LeastI2_ex:
"a. P a  (x. P x  Q x)  Q (Least P)"
by (blast intro: LeastI_ex)

lemma LeastI2_wellorder:
assumes "P a"
and "a.  P a; b. P b  a  b   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  y" by (rule Least_le)
next
fix x assume "P x" "y. P y  x  y" thus "Q x" by (rule assms(2))
qed

lemma LeastI2_wellorder_ex:
assumes "x. P x"
and "a.  P a; b. P b  a  b   Q a"
shows "Q (Least P)"
using assms by clarify (blast intro!: LeastI2_wellorder)

lemma not_less_Least: "k < (LEAST x. P x)  ¬ P k"
apply (erule contrapos_nn)
apply (erule Least_le)
done

lemma exists_least_iff: "(n. P n)  (n. P n  (m < n. ¬ P m))" (is "?lhs  ?rhs")
proof
assume ?rhs thus ?lhs by blast
next
assume H: ?lhs then obtain n where n: "P n" by blast
let ?x = "Least P"
{ fix m assume m: "m < ?x"
from not_less_Least[OF m] have "¬ P m" . }
with LeastI_ex[OF H] show ?rhs by blast
qed

end

subsection Order on typbool

instantiation bool ::
begin

definition
le_bool_def [simp]: "P  Q  P  Q"

definition
[simp]: "(P::bool) < Q  ¬ P  Q"

definition
[simp]: "  False"

definition
[simp]: "  True"

instance proof
qed auto

end

lemma le_boolI: "(P  Q)  P  Q"
by simp

lemma le_boolI': "P  Q  P  Q"
by simp

lemma le_boolE: "P  Q  P  (Q  R)  R"
by simp

lemma le_boolD: "P  Q  P  Q"
by simp

lemma bot_boolE: "  P"
by simp

lemma top_boolI:
by simp

lemma [code]:
"False  b  True"
"True  b  b"
"False < b  b"
"True < b  False"
by simp_all

subsection Order on typ_  _

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

definition
le_fun_def: "f  g  (x. f x  g x)"

definition
"(f::'a  'b) < g  f  g  ¬ (g  f)"

instance ..

end

instance "fun" :: (type, preorder) preorder proof
qed (auto simp add: le_fun_def less_fun_def
intro: order_trans order.antisym)

instance "fun" :: (type, order) order proof
qed (auto simp add: le_fun_def intro: order.antisym)

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

definition
" = (λx. )"

instance ..

end

instantiation "fun" :: (type, order_bot) order_bot
begin

lemma bot_apply [simp, code]:
" x = "

instance proof

end

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

definition
[no_atp]: " = (λx. )"

instance ..

end

instantiation "fun" :: (type, order_top) order_top
begin

lemma top_apply [simp, code]:
" x = "

instance proof

end

lemma le_funI: "(x. f x  g x)  f  g"
unfolding le_fun_def by simp

lemma le_funE: "f  g  (f x  g x  P)  P"
unfolding le_fun_def by simp

lemma le_funD: "f  g  f x  g x"
by (rule le_funE)

lemma mono_compose: "mono Q  mono (λi x. Q i (f x))"
unfolding mono_def le_fun_def by auto

subsection Order on unary and binary predicates

lemma predicate1I:
assumes PQ: "x. P x  Q x"
shows "P  Q"
apply (rule le_funI)
apply (rule le_boolI)
apply (rule PQ)
apply assumption
done

lemma predicate1D:
"P  Q  P x  Q x"
apply (erule le_funE)
apply (erule le_boolE)
apply assumption+
done

lemma rev_predicate1D:
"P x  P  Q  Q x"
by (rule predicate1D)

lemma predicate2I:
assumes PQ: "x y. P x y  Q x y"
shows "P  Q"
apply (rule le_funI)+
apply (rule le_boolI)
apply (rule PQ)
apply assumption
done

lemma predicate2D:
"P  Q  P x y  Q x y"
apply (erule le_funE)+
apply (erule le_boolE)
apply assumption+
done

lemma rev_predicate2D:
"P x y  P  Q  Q x y"
by (rule predicate2D)

lemma bot1E [no_atp]: " x  P"

lemma bot2E: " x y  P"

lemma top1I: " x"

lemma top2I: " x y"

subsection Name duplicates

lemmas antisym = order.antisym
lemmas eq_iff = order.eq_iff

lemmas order_eq_refl = preorder_class.eq_refl
lemmas order_less_irrefl = preorder_class.less_irrefl
lemmas order_less_imp_le = preorder_class.less_imp_le
lemmas order_less_not_sym = preorder_class.less_not_sym
lemmas order_less_asym = preorder_class.less_asym
lemmas order_less_trans = preorder_class.less_trans
lemmas order_le_less_trans = preorder_class.le_less_trans
lemmas order_less_le_trans = preorder_class.less_le_trans
lemmas order_less_imp_not_less = preorder_class.less_imp_not_less
lemmas order_less_imp_triv = preorder_class.less_imp_triv
lemmas order_less_asym' = preorder_class.less_asym'

lemmas order_less_le = order_class.less_le
lemmas order_le_less = order_class.le_less
lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq
lemmas order_less_imp_not_eq = order_class.less_imp_not_eq
lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2
lemmas order_neq_le_trans = order_class.neq_le_trans
lemmas order_le_neq_trans = order_class.le_neq_trans
lemmas order_eq_iff = order_class.order.eq_iff
lemmas order_antisym_conv = order_class.antisym_conv

lemmas linorder_linear = linorder_class.linear
lemmas linorder_less_linear = linorder_class.less_linear
lemmas linorder_le_less_linear = linorder_class.le_less_linear
lemmas linorder_le_cases = linorder_class.le_cases
lemmas linorder_not_less = linorder_class.not_less
lemmas linorder_not_le = linorder_class.not_le
lemmas linorder_neq_iff = linorder_class.neq_iff
lemmas linorder_neqE = linorder_class.neqE

end