(* Title: HOL/SetInterval.thy
ID: $Id$
Author: Tobias Nipkow and Clemens Ballarin
Copyright 2000 TU Muenchen
lessThan, greaterThan, atLeast, atMost and two-sided intervals
*)
theory SetInterval = NatArith:
constdefs
lessThan :: "('a::ord) => 'a set" ("(1{.._'(})")
"{..u(} == {x. x<u}"
atMost :: "('a::ord) => 'a set" ("(1{.._})")
"{..u} == {x. x<=u}"
greaterThan :: "('a::ord) => 'a set" ("(1{')_..})")
"{)l..} == {x. l<x}"
atLeast :: "('a::ord) => 'a set" ("(1{_..})")
"{l..} == {x. l<=x}"
greaterThanLessThan :: "['a::ord, 'a] => 'a set" ("(1{')_.._'(})")
"{)l..u(} == {)l..} Int {..u(}"
atLeastLessThan :: "['a::ord, 'a] => 'a set" ("(1{_.._'(})")
"{l..u(} == {l..} Int {..u(}"
greaterThanAtMost :: "['a::ord, 'a] => 'a set" ("(1{')_.._})")
"{)l..u} == {)l..} Int {..u}"
atLeastAtMost :: "['a::ord, 'a] => 'a set" ("(1{_.._})")
"{l..u} == {l..} Int {..u}"
(* Setup of transitivity reasoner *)
ML {*
structure Trans_Tac = Trans_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_less" 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";
fun decomp (Trueprop $ t) =
let fun dec (Const ("Not", _) $ t) = (
case dec t of
None => None
| Some (t1, rel, t2) => Some (t1, "~" ^ rel, t2))
| dec (Const (rel, _) $ t1 $ t2) =
Some (t1, implode (drop (3, explode rel)), t2)
| dec _ = None
in dec t end
| decomp _ = None
end);
val trans_tac = Trans_Tac.trans_tac;
*}
method_setup trans =
{* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (trans_tac)) *}
{* simple transitivity reasoner *}
(*** lessThan ***)
lemma lessThan_iff: "(i: lessThan k) = (i<k)"
apply (unfold lessThan_def)
apply blast
done
declare lessThan_iff [iff]
lemma lessThan_0: "lessThan (0::nat) = {}"
apply (unfold lessThan_def)
apply (simp (no_asm))
done
declare lessThan_0 [simp]
lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
apply (unfold lessThan_def)
apply (simp (no_asm) add: less_Suc_eq)
apply blast
done
lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
apply (unfold lessThan_def atMost_def)
apply (simp (no_asm) add: less_Suc_eq_le)
done
lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
apply blast
done
lemma Compl_lessThan:
"!!k:: 'a::linorder. -lessThan k = atLeast k"
apply (unfold lessThan_def atLeast_def)
apply auto
apply (blast intro: linorder_not_less [THEN iffD1])
apply (blast dest: order_le_less_trans)
done
lemma single_Diff_lessThan: "!!k:: 'a::order. {k} - lessThan k = {k}"
apply auto
done
declare single_Diff_lessThan [simp]
(*** greaterThan ***)
lemma greaterThan_iff: "(i: greaterThan k) = (k<i)"
apply (unfold greaterThan_def)
apply blast
done
declare greaterThan_iff [iff]
lemma greaterThan_0: "greaterThan 0 = range Suc"
apply (unfold greaterThan_def)
apply (blast dest: gr0_conv_Suc [THEN iffD1])
done
declare greaterThan_0 [simp]
lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
apply (unfold greaterThan_def)
apply (auto elim: linorder_neqE)
done
lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
apply blast
done
lemma Compl_greaterThan:
"!!k:: 'a::linorder. -greaterThan k = atMost k"
apply (unfold greaterThan_def atMost_def le_def)
apply auto
apply (blast intro: linorder_not_less [THEN iffD1])
apply (blast dest: order_le_less_trans)
done
lemma Compl_atMost: "!!k:: 'a::linorder. -atMost k = greaterThan k"
apply (simp (no_asm) add: Compl_greaterThan [symmetric])
done
declare Compl_greaterThan [simp] Compl_atMost [simp]
(*** atLeast ***)
lemma atLeast_iff: "(i: atLeast k) = (k<=i)"
apply (unfold atLeast_def)
apply blast
done
declare atLeast_iff [iff]
lemma atLeast_0: "atLeast (0::nat) = UNIV"
apply (unfold atLeast_def UNIV_def)
apply (simp (no_asm))
done
declare atLeast_0 [simp]
lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
apply (unfold atLeast_def)
apply (simp (no_asm) add: Suc_le_eq)
apply (simp (no_asm) add: order_le_less)
apply blast
done
lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
apply blast
done
lemma Compl_atLeast:
"!!k:: 'a::linorder. -atLeast k = lessThan k"
apply (unfold lessThan_def atLeast_def le_def)
apply auto
apply (blast intro: linorder_not_less [THEN iffD1])
apply (blast dest: order_le_less_trans)
done
declare Compl_lessThan [simp] Compl_atLeast [simp]
(*** atMost ***)
lemma atMost_iff: "(i: atMost k) = (i<=k)"
apply (unfold atMost_def)
apply blast
done
declare atMost_iff [iff]
lemma atMost_0: "atMost (0::nat) = {0}"
apply (unfold atMost_def)
apply (simp (no_asm))
done
declare atMost_0 [simp]
lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
apply (unfold atMost_def)
apply (simp (no_asm) add: less_Suc_eq order_le_less)
apply blast
done
lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
apply blast
done
(*** Combined properties ***)
lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
apply (blast intro: order_antisym)
done
(*** Two-sided intervals ***)
(* greaterThanLessThan *)
lemma greaterThanLessThan_iff [simp]:
"(i : {)l..u(}) = (l < i & i < u)"
by (simp add: greaterThanLessThan_def)
(* atLeastLessThan *)
lemma atLeastLessThan_iff [simp]:
"(i : {l..u(}) = (l <= i & i < u)"
by (simp add: atLeastLessThan_def)
(* greaterThanAtMost *)
lemma greaterThanAtMost_iff [simp]:
"(i : {)l..u}) = (l < i & i <= u)"
by (simp add: greaterThanAtMost_def)
(* atLeastAtMost *)
lemma atLeastAtMost_iff [simp]:
"(i : {l..u}) = (l <= i & i <= u)"
by (simp add: atLeastAtMost_def)
(* The above four lemmas could be declared as iffs.
If we do so, a call to blast in Hyperreal/Star.ML, lemma STAR_Int
seems to take forever (more than one hour). *)
(*** The following lemmas are useful with the summation operator setsum ***)
(* For examples, see Algebra/poly/UnivPoly.thy *)
(** Disjoint Unions **)
(* Singletons and open intervals *)
lemma ivl_disj_un_singleton:
"{l::'a::linorder} Un {)l..} = {l..}"
"{..u(} Un {u::'a::linorder} = {..u}"
"(l::'a::linorder) < u ==> {l} Un {)l..u(} = {l..u(}"
"(l::'a::linorder) < u ==> {)l..u(} Un {u} = {)l..u}"
"(l::'a::linorder) <= u ==> {l} Un {)l..u} = {l..u}"
"(l::'a::linorder) <= u ==> {l..u(} Un {u} = {l..u}"
by auto (elim linorder_neqE | trans+)+
(* One- and two-sided intervals *)
lemma ivl_disj_un_one:
"(l::'a::linorder) < u ==> {..l} Un {)l..u(} = {..u(}"
"(l::'a::linorder) <= u ==> {..l(} Un {l..u(} = {..u(}"
"(l::'a::linorder) <= u ==> {..l} Un {)l..u} = {..u}"
"(l::'a::linorder) <= u ==> {..l(} Un {l..u} = {..u}"
"(l::'a::linorder) <= u ==> {)l..u} Un {)u..} = {)l..}"
"(l::'a::linorder) < u ==> {)l..u(} Un {u..} = {)l..}"
"(l::'a::linorder) <= u ==> {l..u} Un {)u..} = {l..}"
"(l::'a::linorder) <= u ==> {l..u(} Un {u..} = {l..}"
by auto trans+
(* Two- and two-sided intervals *)
lemma ivl_disj_un_two:
"[| (l::'a::linorder) < m; m <= u |] ==> {)l..m(} Un {m..u(} = {)l..u(}"
"[| (l::'a::linorder) <= m; m < u |] ==> {)l..m} Un {)m..u(} = {)l..u(}"
"[| (l::'a::linorder) <= m; m <= u |] ==> {l..m(} Un {m..u(} = {l..u(}"
"[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {)m..u(} = {l..u(}"
"[| (l::'a::linorder) < m; m <= u |] ==> {)l..m(} Un {m..u} = {)l..u}"
"[| (l::'a::linorder) <= m; m <= u |] ==> {)l..m} Un {)m..u} = {)l..u}"
"[| (l::'a::linorder) <= m; m <= u |] ==> {l..m(} Un {m..u} = {l..u}"
"[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {)m..u} = {l..u}"
by auto trans+
lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
(** Disjoint Intersections **)
(* Singletons and open intervals *)
lemma ivl_disj_int_singleton:
"{l::'a::order} Int {)l..} = {}"
"{..u(} Int {u} = {}"
"{l} Int {)l..u(} = {}"
"{)l..u(} Int {u} = {}"
"{l} Int {)l..u} = {}"
"{l..u(} Int {u} = {}"
by simp+
(* One- and two-sided intervals *)
lemma ivl_disj_int_one:
"{..l::'a::order} Int {)l..u(} = {}"
"{..l(} Int {l..u(} = {}"
"{..l} Int {)l..u} = {}"
"{..l(} Int {l..u} = {}"
"{)l..u} Int {)u..} = {}"
"{)l..u(} Int {u..} = {}"
"{l..u} Int {)u..} = {}"
"{l..u(} Int {u..} = {}"
by auto trans+
(* Two- and two-sided intervals *)
lemma ivl_disj_int_two:
"{)l::'a::order..m(} Int {m..u(} = {}"
"{)l..m} Int {)m..u(} = {}"
"{l..m(} Int {m..u(} = {}"
"{l..m} Int {)m..u(} = {}"
"{)l..m(} Int {m..u} = {}"
"{)l..m} Int {)m..u} = {}"
"{l..m(} Int {m..u} = {}"
"{l..m} Int {)m..u} = {}"
by auto trans+
lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two
end