(*  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}"

 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 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 *}

 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)

   "_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)

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

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

 syntax (latex 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)

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

  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [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<n. t" == "setsum (\<lambda>i. t) {..<n}"

 text{* The above introduces some pretty alternative syntaxes for

 summation over intervals as shown on the left-hand side:

 \begin{center}

 \begin{tabular}{lll}

 Sets & Indexed & TeX\\

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

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

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

 \end{tabular}

 \end{center}

 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}"}. *}

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

 by (simp add:lessThan_Suc)

 end