(* Title: HOL/SetInterval.thy
ID: $Id$
Author: Tobias Nipkow and Clemens Ballarin
Additions by Jeremy Avigad in March 2004
Copyright 2000 TU Muenchen
lessThan, greaterThan, atLeast, atMost and two-sided intervals
*)
header {* Set intervals *}
theory SetInterval = IntArith:
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}"
(* Old syntax, will disappear! *)
syntax
"_lessThan" :: "('a::ord) => 'a set" ("(1{.._'(})")
"_greaterThan" :: "('a::ord) => 'a set" ("(1{')_..})")
"_greaterThanLessThan" :: "['a::ord, 'a] => 'a set" ("(1{')_.._'(})")
"_atLeastLessThan" :: "['a::ord, 'a] => 'a set" ("(1{_.._'(})")
"_greaterThanAtMost" :: "['a::ord, 'a] => 'a set" ("(1{')_.._})")
translations
"{..m(}" => "{..<m}"
"{)m..}" => "{m<..}"
"{)m..n(}" => "{m<..<n}"
"{m..n(}" => "{m..<n}"
"{)m..n}" => "{m<..n}"
syntax
"@UNION_le" :: "nat => nat => 'b set => 'b set" ("(3UN _<=_./ _)" 10)
"@UNION_less" :: "nat => nat => 'b set => 'b set" ("(3UN _<_./ _)" 10)
"@INTER_le" :: "nat => nat => 'b set => 'b set" ("(3INT _<=_./ _)" 10)
"@INTER_less" :: "nat => nat => 'b set => 'b set" ("(3INT _<_./ _)" 10)
syntax (input)
"@UNION_le" :: "nat => nat => 'b set => 'b set" ("(3\<Union> _\<le>_./ _)" 10)
"@UNION_less" :: "nat => nat => 'b set => 'b set" ("(3\<Union> _<_./ _)" 10)
"@INTER_le" :: "nat => nat => 'b set => 'b set" ("(3\<Inter> _\<le>_./ _)" 10)
"@INTER_less" :: "nat => nat => 'b set => 'b set" ("(3\<Inter> _<_./ _)" 10)
syntax (xsymbols)
"@UNION_le" :: "nat \<Rightarrow> nat => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_ \<le> _\<^esub>)/ _)" 10)
"@UNION_less" :: "nat \<Rightarrow> nat => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_ < _\<^esub>)/ _)" 10)
"@INTER_le" :: "nat \<Rightarrow> nat => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_ \<le> _\<^esub>)/ _)" 10)
"@INTER_less" :: "nat \<Rightarrow> nat => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_ < _\<^esub>)/ _)" 10)
translations
"UN i<=n. A" == "UN i:{..n}. A"
"UN i<n. A" == "UN i:{..<n}. A"
"INT i<=n. A" == "INT i:{..n}. A"
"INT i<n. A" == "INT i:{..<n}. A"
subsection {* Various equivalences *}
lemma lessThan_iff [iff]: "(i: lessThan k) = (i<k)"
by (simp add: lessThan_def)
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
lemma greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"
by (simp add: greaterThan_def)
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
lemma atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"
by (simp add: atLeast_def)
lemma Compl_atLeast [simp]:
"!!k:: 'a::linorder. -atLeast k = lessThan k"
apply (simp add: lessThan_def atLeast_def le_def, auto)
done
lemma atMost_iff [iff]: "(i: atMost k) = (i<=k)"
by (simp add: atMost_def)
lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
by (blast intro: order_antisym)
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 {*Two-sided intervals*}
text {* @{text greaterThanLessThan} *}
lemma greaterThanLessThan_iff [simp]:
"(i : {l<..<u}) = (l < i & i < u)"
by (simp add: greaterThanLessThan_def)
text {* @{text atLeastLessThan} *}
lemma atLeastLessThan_iff [simp]:
"(i : {l..<u}) = (l <= i & i < u)"
by (simp add: atLeastLessThan_def)
text {* @{text greaterThanAtMost} *}
lemma greaterThanAtMost_iff [simp]:
"(i : {l<..u}) = (l < i & i <= u)"
by (simp add: greaterThanAtMost_def)
text {* @{text atLeastAtMost} *}
lemma atLeastAtMost_iff [simp]:
"(i : {l..u}) = (l <= i & i <= u)"
by (simp add: atLeastAtMost_def)
text {* The above four lemmas could be declared as iffs.
If we do so, a call to blast in Hyperreal/Star.ML, lemma @{text STAR_Int}
seems to take forever (more than one hour). *}
subsection {* Intervals of natural numbers *}
subsubsection {* The Constant @{term lessThan} *}
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
subsubsection {* The Constant @{term greaterThan} *}
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
subsubsection {* The Constant @{term atLeast} *}
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 atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
by blast
subsubsection {* The Constant @{term atMost} *}
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
subsubsection {* The Constant @{term atLeastLessThan} *}
text{*But not a simprule because some concepts are better left in terms
of @{term atLeastLessThan}*}
lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
by(simp add:lessThan_def atLeastLessThan_def)
lemma atLeastLessThan0 [simp]: "{m..<0::nat} = {}"
by (simp add: atLeastLessThan_def)
lemma atLeastLessThan_self [simp]: "{n::'a::order..<n} = {}"
by (auto simp add: atLeastLessThan_def)
lemma atLeastLessThan_empty: "n \<le> m ==> {m..<n::'a::order} = {}"
by (auto simp add: atLeastLessThan_def)
subsubsection {* Intervals of nats with @{term Suc} *}
text{*Not a simprule because the RHS is too messy.*}
lemma atLeastLessThanSuc:
"{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
by (auto simp add: atLeastLessThan_def)
lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
by (auto simp add: atLeastLessThan_def)
lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
by (induct k, simp_all add: atLeastLessThanSuc)
lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
by (auto simp add: atLeastLessThan_def)
lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
greaterThanAtMost_def)
lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
greaterThanLessThan_def)
subsubsection {* Finiteness *}
lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
by (induct k) (simp_all add: lessThan_Suc)
lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
by (induct k) (simp_all add: atMost_Suc)
lemma finite_greaterThanLessThan [iff]:
fixes l :: nat shows "finite {l<..<u}"
by (simp add: greaterThanLessThan_def)
lemma finite_atLeastLessThan [iff]:
fixes l :: nat shows "finite {l..<u}"
by (simp add: atLeastLessThan_def)
lemma finite_greaterThanAtMost [iff]:
fixes l :: nat shows "finite {l<..u}"
by (simp add: greaterThanAtMost_def)
lemma finite_atLeastAtMost [iff]:
fixes l :: nat shows "finite {l..u}"
by (simp add: atLeastAtMost_def)
lemma bounded_nat_set_is_finite:
"(ALL i:N. i < (n::nat)) ==> finite N"
-- {* A bounded set of natural numbers is finite. *}
apply (rule finite_subset)
apply (rule_tac [2] finite_lessThan, auto)
done
subsubsection {* Cardinality *}
lemma card_lessThan [simp]: "card {..<u} = u"
by (induct_tac u, simp_all add: lessThan_Suc)
lemma card_atMost [simp]: "card {..u} = Suc u"
by (simp add: lessThan_Suc_atMost [THEN sym])
lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
apply (subgoal_tac "card {l..<u} = card {..<u-l}")
apply (erule ssubst, rule card_lessThan)
apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
apply (erule subst)
apply (rule card_image)
apply (rule finite_lessThan)
apply (simp add: inj_on_def)
apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
apply arith
apply (rule_tac x = "x - l" in exI)
apply arith
done
lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
subsection {* Intervals of integers *}
lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
"{l+1..<u} = {l<..<u::int}"
by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
subsubsection {* Finiteness *}
lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
{(0::int)..<u} = int ` {..<nat u}"
apply (unfold image_def lessThan_def)
apply auto
apply (rule_tac x = "nat x" in exI)
apply (auto simp add: zless_nat_conj zless_nat_eq_int_zless [THEN sym])
done
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
apply (case_tac "0 \<le> u")
apply (subst image_atLeastZeroLessThan_int, assumption)
apply (rule finite_imageI)
apply auto
apply (subgoal_tac "{0..<u} = {}")
apply auto
done
lemma image_atLeastLessThan_int_shift:
"(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
apply (auto simp add: image_def atLeastLessThan_iff)
apply (rule_tac x = "x - l" in bexI)
apply auto
done
lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
apply (erule subst)
apply (rule finite_imageI)
apply (rule finite_atLeastZeroLessThan_int)
apply (rule image_atLeastLessThan_int_shift)
done
lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
subsubsection {* Cardinality *}
lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
apply (case_tac "0 \<le> u")
apply (subst image_atLeastZeroLessThan_int, assumption)
apply (subst card_image)
apply (auto simp add: inj_on_def)
done
lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
apply (erule ssubst, rule card_atLeastZeroLessThan_int)
apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
apply (erule subst)
apply (rule card_image)
apply (rule finite_atLeastZeroLessThan_int)
apply (simp add: inj_on_def)
apply (rule image_atLeastLessThan_int_shift)
done
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
apply (auto simp add: compare_rls)
done
lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
subsection {*Lemmas useful with the summation operator setsum*}
text {* For examples, see Algebra/poly/UnivPoly.thy *}
subsubsection {* Disjoint Unions *}
text {* 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
text {* 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
text {* 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
subsubsection {* Disjoint Intersections *}
text {* 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+
text {* 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
text {* 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
subsection {* Summation indexed over intervals *}
text{* We introduce the obvious syntax @{text"\<Sum>x=a..b. e"} for
@{term"\<Sum>x\<in>{a..b}. e"}. *}
syntax
"_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
syntax (xsymbols)
"_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
syntax (HTML output)
"_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
translations "\<Sum>x=a..b. t" == "setsum (%x. t) {a..b}"
subsection {* Summation up to *}
text{* Legacy, only used in HoareParallel and Isar-Examples. Really
needed? Probably better to replace it with above syntax. *}
syntax
"_Summation" :: "idt => 'a => 'b => 'b" ("\<Sum>_<_. _" [0, 51, 10] 10)
translations
"\<Sum>i < n. b" == "setsum (\<lambda>i. b) {..<n}"
lemma Summation_Suc[simp]: "(\<Sum>i < Suc n. b i) = b n + (\<Sum>i < n. b i)"
by (simp add:lessThan_Suc)
end