(* Title: HOL/Orderings.thy
ID: $Id$
Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson
*)
header {* Abstract orderings *}
theory Orderings
imports Code_Generator Lattice_Locales
begin
section {* Abstract orderings *}
subsection {* Order signatures *}
class ord = eq +
constrains eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (*FIXME: class_package should do the job*)
fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
const_syntax
less ("op <")
less ("(_/ < _)" [50, 51] 50)
less_eq ("op <=")
less_eq ("(_/ <= _)" [50, 51] 50)
const_syntax (xsymbols)
less_eq ("op \<le>")
less_eq ("(_/ \<le> _)" [50, 51] 50)
const_syntax (HTML output)
less_eq ("op \<le>")
less_eq ("(_/ \<le> _)" [50, 51] 50)
abbreviation (in ord)
"less_eq_syn \<equiv> less_eq"
"less_syn \<equiv> less"
const_syntax (in ord)
less_eq_syn ("op \<^loc><=")
less_eq_syn ("(_/ \<^loc><= _)" [50, 51] 50)
less_syn ("op \<^loc><")
less_syn ("(_/ \<^loc>< _)" [50, 51] 50)
const_syntax (in ord) (xsymbols)
less_eq_syn ("op \<^loc>\<le>")
less_eq_syn ("(_/ \<^loc>\<le> _)" [50, 51] 50)
const_syntax (in ord) (HTML output)
less_eq_syn ("op \<^loc>\<le>")
less_eq_syn ("(_/ \<^loc>\<le> _)" [50, 51] 50)
abbreviation (input)
greater (infixl ">" 50)
"x > y \<equiv> y < x"
greater_eq (infixl ">=" 50)
"x >= y \<equiv> y <= x"
const_syntax (xsymbols)
greater_eq (infixl "\<ge>" 50)
abbreviation (in ord) (input)
greater (infix "\<^loc>>" 50)
"x \<^loc>> y \<equiv> y \<^loc>< x"
greater_eq (infix "\<^loc>>=" 50)
"x \<^loc>>= y \<equiv> y \<^loc><= x"
const_syntax (in ord) (xsymbols)
greater_eq (infixl "\<^loc>\<ge>" 50)
subsection {* Partial orderings *}
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 \<le> :: 'a::order \<Rightarrow> 'a \<Rightarrow> bool"]
apply(rule partial_order.intro)
apply(rule order_refl, erule (1) order_trans, erule (1) order_antisym)
done
text {* Reflexivity. *}
lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y"
-- {* This form is useful with the classical reasoner. *}
apply (erule ssubst)
apply (rule order_refl)
done
lemma order_less_irrefl [iff]: "~ x < (x::'a::order)"
by (simp add: order_less_le)
lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
-- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
apply (simp add: order_less_le, blast)
done
lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
by (simp add: order_less_le)
text {* Asymmetry. *}
lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"
by (simp add: order_less_le order_antisym)
lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P"
apply (drule order_less_not_sym)
apply (erule contrapos_np, simp)
done
lemma order_eq_iff: "!!x::'a::order. (x = y) = (x \<le> y & y \<le> x)"
by (blast intro: order_antisym)
lemma order_antisym_conv: "(y::'a::order) <= x ==> (x <= y) = (x = y)"
by(blast intro:order_antisym)
lemma less_imp_neq: "[| (x::'a::order) < y |] ==> x ~= y"
by (erule contrapos_pn, erule subst, rule order_less_irrefl)
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
lemma eq_neq_eq_imp_neq: "[| x = a ; a ~= b; b = y |] ==> x ~= y"
by (erule subst, erule ssubst, assumption)
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 {* 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 {* Total orderings *}
axclass linorder < order
linorder_linear: "x <= y | y <= x"
lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x"
apply (simp add: order_less_le)
apply (insert linorder_linear, blast)
done
lemma linorder_le_less_linear: "!!x::'a::linorder. x\<le>y | y<x"
by (simp add: order_le_less linorder_less_linear)
lemma linorder_le_cases [case_names le ge]:
"((x::'a::linorder) \<le> y ==> P) ==> (y \<le> x ==> P) ==> P"
by (insert linorder_linear, blast)
lemma linorder_cases [case_names less equal greater]:
"((x::'a::linorder) < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P"
by (insert linorder_less_linear, blast)
lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)"
apply (simp add: order_less_le)
apply (insert linorder_linear)
apply (blast intro: order_antisym)
done
lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
apply (simp add: order_less_le)
apply (insert linorder_linear)
apply (blast intro: order_antisym)
done
lemma linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)"
by (cut_tac x = x and y = y in linorder_less_linear, auto)
lemma linorder_neqE: "x ~= (y::'a::linorder) ==> (x < y ==> R) ==> (y < x ==> R) ==> R"
by (simp add: linorder_neq_iff, blast)
lemma linorder_antisym_conv1: "~ (x::'a::linorder) < y ==> (x <= y) = (x = y)"
by(blast intro:order_antisym dest:linorder_not_less[THEN iffD1])
lemma linorder_antisym_conv2: "(x::'a::linorder) <= y ==> (~ x < y) = (x = y)"
by(blast intro:order_antisym dest:linorder_not_less[THEN iffD1])
lemma linorder_antisym_conv3: "~ (y::'a::linorder) < x ==> (~ x < y) = (x = y)"
by(blast intro:order_antisym dest:linorder_not_less[THEN iffD1])
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)
subsection {* Reasoning tools setup *}
setup {*
let
val order_antisym_conv = thm "order_antisym_conv"
val linorder_antisym_conv1 = thm "linorder_antisym_conv1"
val linorder_antisym_conv2 = thm "linorder_antisym_conv2"
val linorder_antisym_conv3 = thm "linorder_antisym_conv3"
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("Orderings.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 linorder_antisym_conv1))
end
| SOME thm => SOME(mk_meta_eq(thm RS order_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("Orderings.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 linorder_antisym_conv3))
end
| SOME thm => SOME(mk_meta_eq(thm RS linorder_antisym_conv2))
end
handle THM _ => NONE;
val antisym_le = Simplifier.simproc (the_context())
"antisym le" ["(x::'a::order) <= y"] prove_antisym_le;
val antisym_less = Simplifier.simproc (the_context())
"antisym less" ["~ (x::'a::linorder) < y"] prove_antisym_less;
in
(fn thy => (Simplifier.change_simpset_of thy
(fn ss => ss addsimprocs [antisym_le, antisym_less]); thy))
end
*}
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 thy (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 thy (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 ("Orderings.less_eq", _) $ t1 $ t2) =
if of_sort t1
then SOME (t1, "<=", t2)
else NONE
| dec (Const ("Orderings.less", _) $ 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 *)
change_simpset (fn ss => ss
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 {* Bounded quantifiers *}
syntax
"_lessAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<_./ _)" [0, 0, 10] 10)
"_lessEx" :: "[idt, 'a, bool] => bool" ("(3EX _<_./ _)" [0, 0, 10] 10)
"_leAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10)
"_leEx" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10)
"_gtAll" :: "[idt, 'a, bool] => bool" ("(3ALL _>_./ _)" [0, 0, 10] 10)
"_gtEx" :: "[idt, 'a, bool] => bool" ("(3EX _>_./ _)" [0, 0, 10] 10)
"_geAll" :: "[idt, 'a, bool] => bool" ("(3ALL _>=_./ _)" [0, 0, 10] 10)
"_geEx" :: "[idt, 'a, bool] => bool" ("(3EX _>=_./ _)" [0, 0, 10] 10)
syntax (xsymbols)
"_lessAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_<_./ _)" [0, 0, 10] 10)
"_lessEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_<_./ _)" [0, 0, 10] 10)
"_leAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
"_leEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
"_gtAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_>_./ _)" [0, 0, 10] 10)
"_gtEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_>_./ _)" [0, 0, 10] 10)
"_geAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
"_geEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
syntax (HOL)
"_lessAll" :: "[idt, 'a, bool] => bool" ("(3! _<_./ _)" [0, 0, 10] 10)
"_lessEx" :: "[idt, 'a, bool] => bool" ("(3? _<_./ _)" [0, 0, 10] 10)
"_leAll" :: "[idt, 'a, bool] => bool" ("(3! _<=_./ _)" [0, 0, 10] 10)
"_leEx" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10)
syntax (HTML output)
"_lessAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_<_./ _)" [0, 0, 10] 10)
"_lessEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_<_./ _)" [0, 0, 10] 10)
"_leAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
"_leEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
"_gtAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_>_./ _)" [0, 0, 10] 10)
"_gtEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_>_./ _)" [0, 0, 10] 10)
"_geAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
"_geEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
translations
"ALL x<y. P" => "ALL x. x < y \<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
fun mk v v' c n P =
if v = v' andalso not (member (op =) (map fst (Term.add_frees n [])) v)
then Syntax.const c $ Syntax.mark_bound v' $ n $ P else raise Match;
fun mk_all "\\<^const>Scratch.less" f =
f ("_lessAll", "_gtAll")
| mk_all "\\<^const>Scratch.less_eq" f =
f ("_leAll", "_geAll")
fun mk_ex "\\<^const>Scratch.less" f =
f ("_lessEx", "_gtEx")
| mk_ex "\\<^const>Scratch.less_eq" f =
f ("_leEx", "_geEx");
fun tr_all' [Const ("_bound", _) $ Free (v, _), Const("op -->", _)
$ (Const (c, _) $ (Const ("_bound", _) $ Free (v', _)) $ n) $ P] =
mk v v' (mk_all c fst) n P
| tr_all' [Const ("_bound", _) $ Free (v, _), Const("op -->", _)
$ (Const (c, _) $ n $ (Const ("_bound", _) $ Free (v', _))) $ P] =
mk v v' (mk_all c snd) n P;
fun tr_ex' [Const ("_bound", _) $ Free (v, _), Const("op &", _)
$ (Const (c, _) $ (Const ("_bound", _) $ Free (v', _)) $ n) $ P] =
mk v v' (mk_ex c fst) n P
| tr_ex' [Const ("_bound", _) $ Free (v, _), Const("op &", _)
$ (Const (c, _) $ n $ (Const ("_bound", _) $ Free (v', _))) $ P] =
mk v v' (mk_ex c snd) n P;
in
[("ALL ", tr_all'), ("EX ", tr_ex')]
end
*}
subsection {* Transitivity reasoning on decreasing inequalities *}
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 {* Least value operator, monotonicity and min/max *}
(*FIXME: derive more of the min/max laws generically via semilattices*)
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 \<le> y |] ==> Q x |]
==> Q (Least P)"
apply (unfold Least_def)
apply (rule theI2)
apply (blast intro: order_antisym)+
done
lemma Least_equality:
"[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
apply (simp add: Least_def)
apply (rule the_equality)
apply (auto intro!: order_antisym)
done
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_leastR: "(\<And>x\<Colon>'a\<Colon>order. least \<le> x) \<Longrightarrow> min x least = least"
apply (simp add: min_def)
apply (blast intro: order_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: order_antisym)
done
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_of_mono:
"(!!x y. (f x <= f y) = (x <= y)) ==> min (f m) (f n) = f (min m n)"
by (simp add: min_def)
lemma max_of_mono:
"(!!x y. (f x <= f y) = (x <= y)) ==> max (f m) (f n) = f (max m n)"
by (simp add: max_def)
text{* Instantiate locales: *}
interpretation min_max:
lower_semilattice["op \<le>" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
apply unfold_locales
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 \<le>" "max :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a"]
apply unfold_locales
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 \<le>" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max"]
by unfold_locales
interpretation min_max:
distrib_lattice["op \<le>" "min :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max"]
apply unfold_locales
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)
end