(* 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 *) theory SetInterval = IntArith: constdefs lessThan :: "('a::ord) => 'a set" ("(1{.._'(})") "{..u(} == {x. x 'a set" ("(1{.._})") "{..u} == {x. x<=u}" greaterThan :: "('a::ord) => 'a set" ("(1{')_..})") "{)l..} == {x. l '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}" 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\ _\_./ _)" 10) "@UNION_less" :: "nat => nat => 'b set => 'b set" ("(3\ _<_./ _)" 10) "@INTER_le" :: "nat => nat => 'b set => 'b set" ("(3\ _\_./ _)" 10) "@INTER_less" :: "nat => nat => 'b set => 'b set" ("(3\ _<_./ _)" 10) syntax (xsymbols) "@UNION_le" :: "nat \ nat => 'b set => 'b set" ("(3\\<^bsub>_ \ _\<^esub>/ _)" 10) "@UNION_less" :: "nat \ nat => 'b set => 'b set" ("(3\\<^bsub>_ < _\<^esub>/ _)" 10) "@INTER_le" :: "nat \ nat => 'b set => 'b set" ("(3\\<^bsub>_ \ _\<^esub>/ _)" 10) "@INTER_less" :: "nat \ nat => 'b set => 'b set" ("(3\\<^bsub>_ < _\<^esub>/ _)" 10) translations "UN i<=n. A" == "UN i:{..n}. A" "UN i atLeast y) = (y \ (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 \ greaterThan y) = (y \ (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 \ atMost y) = (x \ (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 \ lessThan y) = (x \ (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*} (* 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 {* Intervals of natural numbers *} 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 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 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 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 (* Intervals of nats with Suc *) 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 \ 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 \ 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 \ 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*} (* 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