(*  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}"

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 *}

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

text {* Intervals of nats with @{text 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 \<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 natural numbers *}

text{* Legacy, only used in HoareParallel and Isar-Examples. Really

needed? Probably better to replace it with a more generic operator

like ``SUM i = m..n. b''. *}

syntax

  "_Summation" :: "id => nat => 'a => nat"    ("\<Sum>_<_. _" [0, 51, 10] 10)

translations

  "\<Sum>i < n. b" == "setsum (\<lambda>i::nat. 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