(*  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

 imports IntArith

 begin

 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}"

 text{* A note of warning when using @{term"{..<n}"} on type @{typ

 nat}: it is equivalent to @{term"{0::nat..<n}"} but some lemmas involving

 @{term"{m..<n}"} may not exist in @{term"{..<n}"}-form as well. *}

 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_all add: greaterThan_subset_iff)

 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_all add: lessThan_subset_iff)

 done

 subsection {*Two-sided intervals*}

 lemma greaterThanLessThan_iff [simp,noatp]:

   "(i : {l<..<u}) = (l < i & i < u)"

 by (simp add: greaterThanLessThan_def)

 lemma atLeastLessThan_iff [simp,noatp]:

   "(i : {l..<u}) = (l <= i & i < u)"

 by (simp add: atLeastLessThan_def)

 lemma greaterThanAtMost_iff [simp,noatp]:

   "(i : {l<..u}) = (l < i & i <= u)"

 by (simp add: greaterThanAtMost_def)

 lemma atLeastAtMost_iff [simp,noatp]:

   "(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). *}

 subsubsection{* Emptyness and singletons *}

 lemma atLeastAtMost_empty [simp]: "n < m ==> {m::'a::order..n} = {}";

   by (auto simp add: atLeastAtMost_def atMost_def atLeast_def);

 lemma atLeastLessThan_empty[simp]: "n \<le> m ==> {m..<n::'a::order} = {}"

 by (auto simp add: atLeastLessThan_def)

 lemma greaterThanAtMost_empty[simp]:"l \<le> k ==> {k<..(l::'a::order)} = {}"

 by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)

 lemma greaterThanLessThan_empty[simp]:"l \<le> k ==> {k<..(l::'a::order)} = {}"

 by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)

 lemma atLeastAtMost_singleton [simp]: "{a::'a::order..a} = {a}";

 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def);

 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{*The orientation of the following rule is tricky. The lhs is

 defined in terms of the rhs.  Hence the chosen orientation makes sense

 in this theory --- the reverse orientation complicates proofs (eg

 nontermination). But outside, when the definition of the lhs is rarely

 used, the opposite orientation seems preferable because it reduces a

 specific concept to a more general one. *}

 lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"

 by(simp add:lessThan_def atLeastLessThan_def)

 declare atLeast0LessThan[symmetric, code unfold]

 lemma atLeastLessThan0: "{m..<0::nat} = {}"

 by (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)

 lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"

 by (auto simp add: atLeastAtMost_def)

 subsubsection {* Image *}

 lemma image_add_atLeastAtMost:

   "(%n::nat. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")

 proof

   show "?A \<subseteq> ?B" by auto

 next

   show "?B \<subseteq> ?A"

   proof

     fix n assume a: "n : ?B"

     hence "n - k : {i..j}" by auto

     moreover have "n = (n - k) + k" using a by auto

     ultimately show "n : ?A" by blast

   qed

 qed

 lemma image_add_atLeastLessThan:

   "(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")

 proof

   show "?A \<subseteq> ?B" by auto

 next

   show "?B \<subseteq> ?A"

   proof

     fix n assume a: "n : ?B"

     hence "n - k : {i..<j}" by auto

     moreover have "n = (n - k) + k" using a by auto

     ultimately show "n : ?A" by blast

   qed

 qed

 corollary image_Suc_atLeastAtMost[simp]:

   "Suc ` {i..j} = {Suc i..Suc j}"

 using image_add_atLeastAtMost[where k=1] by simp

 corollary image_Suc_atLeastLessThan[simp]:

   "Suc ` {i..<j} = {Suc i..<Suc j}"

 using image_add_atLeastLessThan[where k=1] by simp

 lemma image_add_int_atLeastLessThan:

     "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"

   apply (auto simp add: image_def)

   apply (rule_tac x = "x - l" in bexI)

   apply auto

   done

 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 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 (simp add: inj_on_def)

   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)

   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

   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_add_int_atLeastLessThan)

   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 (simp add: inj_on_def)

   apply (rule image_add_int_atLeastLessThan)

   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/UnivPoly2.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

 subsubsection {* Some Differences *}

 lemma ivl_diff[simp]:

  "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"

 by(auto)

 subsubsection {* Some Subset Conditions *}

 lemma ivl_subset [simp,noatp]:

  "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"

 apply(auto simp:linorder_not_le)

 apply(rule ccontr)

 apply(insert linorder_le_less_linear[of i n])

 apply(clarsimp simp:linorder_not_le)

 apply(fastsimp)

 done

 subsection {* Summation indexed over intervals *}

 syntax

   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)

   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)

   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)

   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [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)

   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)

   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)

   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [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)

   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)

   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)

   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)

 syntax (latex_sum output)

   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

 translations

   "\<Sum>x=a..b. t" == "setsum (%x. t) {a..b}"

   "\<Sum>x=a..<b. t" == "setsum (%x. t) {a..<b}"

   "\<Sum>i\<le>n. t" == "setsum (\<lambda>i. t) {..n}"

   "\<Sum>i<n. t" == "setsum (\<lambda>i. t) {..<n}"

 text{* The above introduces some pretty alternative syntaxes for

 summation over intervals:

 \begin{center}

 \begin{tabular}{lll}

 Old & New & \LaTeX\\

 @{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\

 @{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\

 @{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\

 @{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}

 \end{tabular}

 \end{center}

 The left column shows the term before introduction of the new syntax,

 the middle column shows the new (default) syntax, and the right column

 shows a special syntax. The latter is only meaningful for latex output

 and has to be activated explicitly by setting the print mode to

 @{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in

 antiquotations). It is not the default \LaTeX\ output because it only

 works well with italic-style formulae, not tt-style.

 Note that for uniformity on @{typ nat} it is better to use

 @{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may

 not provide all lemmas available for @{term"{m..<n}"} also in the

 special form for @{term"{..<n}"}. *}

 text{* This congruence rule should be used for sums over intervals as

 the standard theorem @{text[source]setsum_cong} does not work well

 with the simplifier who adds the unsimplified premise @{term"x:B"} to

 the context. *}

 lemma setsum_ivl_cong:

  "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>

  setsum f {a..<b} = setsum g {c..<d}"

 by(rule setsum_cong, simp_all)

 (* FIXME why are the following simp rules but the corresponding eqns

 on intervals are not? *)

 lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"

 by (simp add:atMost_Suc add_ac)

 lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"

 by (simp add:lessThan_Suc add_ac)

 lemma setsum_cl_ivl_Suc[simp]:

   "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"

 by (auto simp:add_ac atLeastAtMostSuc_conv)

 lemma setsum_op_ivl_Suc[simp]:

   "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"

 by (auto simp:add_ac atLeastLessThanSuc)

 (*

 lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>

     (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"

 by (auto simp:add_ac atLeastAtMostSuc_conv)

 *)

 lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>

   setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"

 by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)

 lemma setsum_diff_nat_ivl:

 fixes f :: "nat \<Rightarrow> 'a::ab_group_add"

 shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>

   setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"

 using setsum_add_nat_ivl [of m n p f,symmetric]

 apply (simp add: add_ac)

 done

 subsection{* Shifting bounds *}

 lemma setsum_shift_bounds_nat_ivl:

   "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"

 by (induct "n", auto simp:atLeastLessThanSuc)

 lemma setsum_shift_bounds_cl_nat_ivl:

   "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"

 apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])

 apply (simp add:image_add_atLeastAtMost o_def)

 done

 corollary setsum_shift_bounds_cl_Suc_ivl:

   "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"

 by (simp add:setsum_shift_bounds_cl_nat_ivl[where k=1,simplified])

 corollary setsum_shift_bounds_Suc_ivl:

   "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"

 by (simp add:setsum_shift_bounds_nat_ivl[where k=1,simplified])

 lemma setsum_head:

   fixes n :: nat

   assumes mn: "m <= n" 

   shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")

 proof -

   from mn

   have "{m..n} = {m} \<union> {m<..n}"

     by (auto intro: ivl_disj_un_singleton)

   hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"

     by (simp add: atLeast0LessThan)

   also have "\<dots> = ?rhs" by simp

   finally show ?thesis .

 qed

 lemma setsum_head_upt:

   fixes m::nat

   assumes m: "0 < m"

   shows "(\<Sum>x<m. P x) = P 0 + (\<Sum>x\<in>{1..<m}. P x)"

 proof -

   have "(\<Sum>x<m. P x) = (\<Sum>x\<in>{0..<m}. P x)" 

     by (simp add: atLeast0LessThan)

   also 

   from m 

   have "\<dots> = (\<Sum>x\<in>{0..m - 1}. P x)"

     by (cases m) (auto simp add: atLeastLessThanSuc_atLeastAtMost)

   also

   have "\<dots> = P 0 + (\<Sum>x\<in>{0<..m - 1}. P x)"

     by (simp add: setsum_head)

   also 

   from m 

   have "{0<..m - 1} = {1..<m}" 

     by (cases m) (auto simp add: atLeastLessThanSuc_atLeastAtMost)

   finally show ?thesis .

 qed

 subsection {* The formula for geometric sums *}

 lemma geometric_sum:

   "x ~= 1 ==> (\<Sum>i=0..<n. x ^ i) =

   (x ^ n - 1) / (x - 1::'a::{field, recpower})"

 by (induct "n") (simp_all add:field_simps power_Suc)

 subsection {* The formula for arithmetic sums *}

 lemma gauss_sum:

   "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =

    of_nat n*((of_nat n)+1)"

 proof (induct n)

   case 0

   show ?case by simp

 next

   case (Suc n)

   then show ?case by (simp add: ring_simps)

 qed

 theorem arith_series_general:

   "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

   of_nat n * (a + (a + of_nat(n - 1)*d))"

 proof cases

   assume ngt1: "n > 1"

   let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"

   have

     "(\<Sum>i\<in>{..<n}. a+?I i*d) =

      ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"

     by (rule setsum_addf)

   also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp

   also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"

     by (simp add: setsum_right_distrib setsum_head_upt mult_ac)

   also have "(1+1)*\<dots> = (1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..<n}. ?I i)"

     by (simp add: left_distrib right_distrib)

   also from ngt1 have "{1..<n} = {1..n - 1}"

     by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)    

   also from ngt1 

   have "(1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..n - 1}. ?I i) = ((1+1)*?n*a + d*?I (n - 1)*?I n)"

     by (simp only: mult_ac gauss_sum [of "n - 1"])

        (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])

   finally show ?thesis by (simp add: mult_ac add_ac right_distrib)

 next

   assume "\<not>(n > 1)"

   hence "n = 1 \<or> n = 0" by auto

   thus ?thesis by (auto simp: mult_ac right_distrib)

 qed

 lemma arith_series_nat:

   "Suc (Suc 0) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"

 proof -

   have

     "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =

     of_nat(n) * (a + (a + of_nat(n - 1)*d))"

     by (rule arith_series_general)

   thus ?thesis by (auto simp add: of_nat_id)

 qed

 lemma arith_series_int:

   "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

   of_nat n * (a + (a + of_nat(n - 1)*d))"

 proof -

   have

     "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

     of_nat(n) * (a + (a + of_nat(n - 1)*d))"

     by (rule arith_series_general)

   thus ?thesis by simp

 qed

 lemma sum_diff_distrib:

   fixes P::"nat\<Rightarrow>nat"

   shows

   "\<forall>x. Q x \<le> P x  \<Longrightarrow>

   (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"

 proof (induct n)

   case 0 show ?case by simp

 next

   case (Suc n)

   let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"

   let ?rhs = "\<Sum>x<n. P x - Q x"

   from Suc have "?lhs = ?rhs" by simp

   moreover

   from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp

   moreover

   from Suc have

     "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"

     by (subst diff_diff_left[symmetric],

         subst diff_add_assoc2)

        (auto simp: diff_add_assoc2 intro: setsum_mono)

   ultimately

   show ?case by simp

 qed

 ML

 {*

 val Compl_atLeast = thm "Compl_atLeast";

 val Compl_atMost = thm "Compl_atMost";

 val Compl_greaterThan = thm "Compl_greaterThan";

 val Compl_lessThan = thm "Compl_lessThan";

 val INT_greaterThan_UNIV = thm "INT_greaterThan_UNIV";

 val UN_atLeast_UNIV = thm "UN_atLeast_UNIV";

 val UN_atMost_UNIV = thm "UN_atMost_UNIV";

 val UN_lessThan_UNIV = thm "UN_lessThan_UNIV";

 val atLeastAtMost_def = thm "atLeastAtMost_def";

 val atLeastAtMost_iff = thm "atLeastAtMost_iff";

 val atLeastLessThan_def  = thm "atLeastLessThan_def";

 val atLeastLessThan_iff = thm "atLeastLessThan_iff";

 val atLeast_0 = thm "atLeast_0";

 val atLeast_Suc = thm "atLeast_Suc";

 val atLeast_def      = thm "atLeast_def";

 val atLeast_iff = thm "atLeast_iff";

 val atMost_0 = thm "atMost_0";

 val atMost_Int_atLeast = thm "atMost_Int_atLeast";

 val atMost_Suc = thm "atMost_Suc";

 val atMost_def       = thm "atMost_def";

 val atMost_iff = thm "atMost_iff";

 val greaterThanAtMost_def  = thm "greaterThanAtMost_def";

 val greaterThanAtMost_iff = thm "greaterThanAtMost_iff";

 val greaterThanLessThan_def  = thm "greaterThanLessThan_def";

 val greaterThanLessThan_iff = thm "greaterThanLessThan_iff";

 val greaterThan_0 = thm "greaterThan_0";

 val greaterThan_Suc = thm "greaterThan_Suc";

 val greaterThan_def  = thm "greaterThan_def";

 val greaterThan_iff = thm "greaterThan_iff";

 val ivl_disj_int = thms "ivl_disj_int";

 val ivl_disj_int_one = thms "ivl_disj_int_one";

 val ivl_disj_int_singleton = thms "ivl_disj_int_singleton";

 val ivl_disj_int_two = thms "ivl_disj_int_two";

 val ivl_disj_un = thms "ivl_disj_un";

 val ivl_disj_un_one = thms "ivl_disj_un_one";

 val ivl_disj_un_singleton = thms "ivl_disj_un_singleton";

 val ivl_disj_un_two = thms "ivl_disj_un_two";

 val lessThan_0 = thm "lessThan_0";

 val lessThan_Suc = thm "lessThan_Suc";

 val lessThan_Suc_atMost = thm "lessThan_Suc_atMost";

 val lessThan_def     = thm "lessThan_def";

 val lessThan_iff = thm "lessThan_iff";

 val single_Diff_lessThan = thm "single_Diff_lessThan";

 val bounded_nat_set_is_finite = thm "bounded_nat_set_is_finite";

 val finite_atMost = thm "finite_atMost";

 val finite_lessThan = thm "finite_lessThan";

 *}

 end