(* 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
text {* Explicit dictionaries for code generation *}
lemma min_ord_min [code, code_unfold, code_inline del]:
"min = ord.min (op \<le>)"
by (rule ext)+ (simp add: min_def ord.min_def)
declare ord.min_def [code]
lemma max_ord_max [code, code_unfold, code_inline del]:
"max = ord.max (op \<le>)"
by (rule ext)+ (simp add: max_def ord.max_def)
declare ord.max_def [code]
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 = (#prop (rep_thm thm) = t);
fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) =
let val prems = prems_of_ss 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 = prems_of_ss 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 (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 (fn ss => ss addSolver
mk_solver' name (fn ss => tac (Simplifier.the_context ss) (Simplifier.prems_of_ss 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:
"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:
"(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: "(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: "(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: "(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: "(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: "(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: "(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: "(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, 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_leastL: "(!!x. least <= x) ==> min least x = least"
by (simp add: min_def)
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
by (simp add: max_def)
lemma min_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
apply (simp add: min_def)
apply (blast intro: antisym)
done
lemma max_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> max x least = x"
apply (simp add: max_def)
apply (blast intro: antisym)
done
subsection {* Top and bottom elements *}
class bot = preorder +
fixes bot :: 'a
assumes bot_least [simp]: "bot \<le> x"
class top = preorder +
fixes top :: 'a
assumes top_greatest [simp]: "x \<le> top"
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 bool *}
instantiation bool :: "{order, bot, top}"
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]: "bot \<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: "bot \<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 functions *}
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 intro!: ext)
instance "fun" :: (type, order) order proof
qed (auto simp add: le_fun_def intro: antisym ext)
instantiation "fun" :: (type, bot) bot
begin
definition
"bot = (\<lambda>x. bot)"
lemma bot_apply:
"bot x = bot"
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:
"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 {* 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