(* 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}"
subsection {*lessThan*}
lemma lessThan_iff [iff]: "(i: lessThan k) = (i<k)"
by (simp add: lessThan_def)
lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
by (simp add: lessThan_def)
lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
by (simp add: lessThan_def less_Suc_eq, blast)
lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
by (simp add: lessThan_def atMost_def less_Suc_eq_le)
lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
by blast
lemma Compl_lessThan [simp]:
"!!k:: 'a::linorder. -lessThan k = atLeast k"
apply (auto simp add: lessThan_def atLeast_def)
done
lemma single_Diff_lessThan [simp]: "!!k:: 'a::order. {k} - lessThan k = {k}"
by auto
subsection {*greaterThan*}
lemma greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"
by (simp add: greaterThan_def)
lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
apply (simp add: greaterThan_def)
apply (blast dest: gr0_conv_Suc [THEN iffD1])
done
lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
apply (simp add: greaterThan_def)
apply (auto elim: linorder_neqE)
done
lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
by blast
lemma Compl_greaterThan [simp]:
"!!k:: 'a::linorder. -greaterThan k = atMost k"
apply (simp add: greaterThan_def atMost_def le_def, auto)
done
lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"
apply (subst Compl_greaterThan [symmetric])
apply (rule double_complement)
done
subsection {*atLeast*}
lemma atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"
by (simp add: atLeast_def)
lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
by (unfold atLeast_def UNIV_def, simp)
lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
apply (simp add: atLeast_def)
apply (simp add: Suc_le_eq)
apply (simp add: order_le_less, blast)
done
lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
by blast
lemma Compl_atLeast [simp]:
"!!k:: 'a::linorder. -atLeast k = lessThan k"
apply (simp add: lessThan_def atLeast_def le_def, auto)
done
subsection {*atMost*}
lemma atMost_iff [iff]: "(i: atMost k) = (i<=k)"
by (simp add: atMost_def)
lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
by (simp add: atMost_def)
lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
apply (simp add: atMost_def)
apply (simp add: less_Suc_eq order_le_less, blast)
done
lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
by blast
subsection {*Logical Equivalences for Set Inclusion and Equality*}
lemma atLeast_subset_iff [iff]:
"(atLeast x \<subseteq> atLeast y) = (y \<le> (x::'a::order))"
by (blast intro: order_trans)
lemma atLeast_eq_iff [iff]:
"(atLeast x = atLeast y) = (x = (y::'a::linorder))"
by (blast intro: order_antisym order_trans)
lemma greaterThan_subset_iff [iff]:
"(greaterThan x \<subseteq> greaterThan y) = (y \<le> (x::'a::linorder))"
apply (auto simp add: greaterThan_def)
apply (subst linorder_not_less [symmetric], blast)
done
lemma greaterThan_eq_iff [iff]:
"(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"
apply (rule iffI)
apply (erule equalityE)
apply (simp add: greaterThan_subset_iff order_antisym, simp)
done
lemma atMost_subset_iff [iff]: "(atMost x \<subseteq> atMost y) = (x \<le> (y::'a::order))"
by (blast intro: order_trans)
lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::linorder))"
by (blast intro: order_antisym order_trans)
lemma lessThan_subset_iff [iff]:
"(lessThan x \<subseteq> lessThan y) = (x \<le> (y::'a::linorder))"
apply (auto simp add: lessThan_def)
apply (subst linorder_not_less [symmetric], blast)
done
lemma lessThan_eq_iff [iff]:
"(lessThan x = lessThan y) = (x = (y::'a::linorder))"
apply (rule iffI)
apply (erule equalityE)
apply (simp add: lessThan_subset_iff order_antisym, simp)
done
subsection {*Combined properties*}
lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
by (blast intro: order_antisym)
subsection {*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). *)
subsection {*Lemmas 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
(* 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
(* 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
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
(* 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
lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two
end