(*  Title:      HOL/SetInterval.thy

     Author:     Tobias Nipkow

     Author:     Clemens Ballarin

     Author:     Jeremy Avigad

 lessThan, greaterThan, atLeast, atMost and two-sided intervals

 *)

 header {* Set intervals *}

 theory SetInterval

 imports Int Nat_Transfer

 begin

 context ord

 begin

 definition

   lessThan    :: "'a => 'a set" ("(1{..<_})") where

   "{..<u} == {x. x < u}"

 definition

   atMost      :: "'a => 'a set" ("(1{.._})") where

   "{..u} == {x. x \<le> u}"

 definition

   greaterThan :: "'a => 'a set" ("(1{_<..})") where

   "{l<..} == {x. l<x}"

 definition

   atLeast     :: "'a => 'a set" ("(1{_..})") where

   "{l..} == {x. l\<le>x}"

 definition

   greaterThanLessThan :: "'a => 'a => 'a set"  ("(1{_<..<_})") where

   "{l<..<u} == {l<..} Int {..<u}"

 definition

   atLeastLessThan :: "'a => 'a => 'a set"      ("(1{_..<_})") where

   "{l..<u} == {l..} Int {..<u}"

 definition

   greaterThanAtMost :: "'a => 'a => 'a set"    ("(1{_<.._})") where

   "{l<..u} == {l<..} Int {..u}"

 definition

   atLeastAtMost :: "'a => 'a => 'a set"        ("(1{_.._})") where

   "{l..u} == {l..} Int {..u}"

 end

 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"   :: "'a => 'a => 'b set => 'b set"       ("(3UN _<=_./ _)" [0, 0, 10] 10)

   "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3UN _<_./ _)" [0, 0, 10] 10)

   "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3INT _<=_./ _)" [0, 0, 10] 10)

   "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3INT _<_./ _)" [0, 0, 10] 10)

 syntax (xsymbols)

   "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _\<le>_./ _)" [0, 0, 10] 10)

   "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _<_./ _)" [0, 0, 10] 10)

   "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _\<le>_./ _)" [0, 0, 10] 10)

   "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _<_./ _)" [0, 0, 10] 10)

 syntax (latex output)

   "_UNION_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ \<le> _)/ _)" [0, 0, 10] 10)

   "_UNION_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ < _)/ _)" [0, 0, 10] 10)

   "_INTER_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ \<le> _)/ _)" [0, 0, 10] 10)

   "_INTER_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ < _)/ _)" [0, 0, 10] 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 (in ord) 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 (in ord) greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"

 by (simp add: greaterThan_def)

 lemma Compl_greaterThan [simp]:

     "!!k:: 'a::linorder. -greaterThan k = atMost k"

   by (auto simp add: greaterThan_def atMost_def)

 lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"

 apply (subst Compl_greaterThan [symmetric])

 apply (rule double_complement)

 done

 lemma (in ord) atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"

 by (simp add: atLeast_def)

 lemma Compl_atLeast [simp]:

     "!!k:: 'a::linorder. -atLeast k = lessThan k"

   by (auto simp add: lessThan_def atLeast_def)

 lemma (in ord) 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

 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

 done

 lemma lessThan_strict_subset_iff:

   fixes m n :: "'a::linorder"

   shows "{..<m} < {..<n} \<longleftrightarrow> m < n"

   by (metis leD lessThan_subset_iff linorder_linear not_less_iff_gr_or_eq psubset_eq)

 subsection {*Two-sided intervals*}

 context ord

 begin

 lemma greaterThanLessThan_iff [simp,no_atp]:

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

 by (simp add: greaterThanLessThan_def)

 lemma atLeastLessThan_iff [simp,no_atp]:

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

 by (simp add: atLeastLessThan_def)

 lemma greaterThanAtMost_iff [simp,no_atp]:

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

 by (simp add: greaterThanAtMost_def)

 lemma atLeastAtMost_iff [simp,no_atp]:

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

 by (simp add: atLeastAtMost_def)

 text {* The above four lemmas could be declared as iffs. Unfortunately this

 breaks many proofs. Since it only helps blast, it is better to leave well

 alone *}

 end

 subsubsection{* Emptyness, singletons, subset *}

 context order

 begin

 lemma atLeastatMost_empty[simp]:

   "b < a \<Longrightarrow> {a..b} = {}"

 by(auto simp: atLeastAtMost_def atLeast_def atMost_def)

 lemma atLeastatMost_empty_iff[simp]:

   "{a..b} = {} \<longleftrightarrow> (~ a <= b)"

 by auto (blast intro: order_trans)

 lemma atLeastatMost_empty_iff2[simp]:

   "{} = {a..b} \<longleftrightarrow> (~ a <= b)"

 by auto (blast intro: order_trans)

 lemma atLeastLessThan_empty[simp]:

   "b <= a \<Longrightarrow> {a..<b} = {}"

 by(auto simp: atLeastLessThan_def)

 lemma atLeastLessThan_empty_iff[simp]:

   "{a..<b} = {} \<longleftrightarrow> (~ a < b)"

 by auto (blast intro: le_less_trans)

 lemma atLeastLessThan_empty_iff2[simp]:

   "{} = {a..<b} \<longleftrightarrow> (~ a < b)"

 by auto (blast intro: le_less_trans)

 lemma greaterThanAtMost_empty[simp]: "l \<le> k ==> {k<..l} = {}"

 by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)

 lemma greaterThanAtMost_empty_iff[simp]: "{k<..l} = {} \<longleftrightarrow> ~ k < l"

 by auto (blast intro: less_le_trans)

 lemma greaterThanAtMost_empty_iff2[simp]: "{} = {k<..l} \<longleftrightarrow> ~ k < l"

 by auto (blast intro: less_le_trans)

 lemma greaterThanLessThan_empty[simp]:"l \<le> k ==> {k<..<l} = {}"

 by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)

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

 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)

 lemma atLeastAtMost_singleton': "a = b \<Longrightarrow> {a .. b} = {a}" by simp

 lemma atLeastatMost_subset_iff[simp]:

   "{a..b} <= {c..d} \<longleftrightarrow> (~ a <= b) | c <= a & b <= d"

 unfolding atLeastAtMost_def atLeast_def atMost_def

 by (blast intro: order_trans)

 lemma atLeastatMost_psubset_iff:

   "{a..b} < {c..d} \<longleftrightarrow>

    ((~ a <= b) | c <= a & b <= d & (c < a | b < d))  &  c <= d"

 by(simp add: psubset_eq set_eq_iff less_le_not_le)(blast intro: order_trans)

 lemma atLeastAtMost_singleton_iff[simp]:

   "{a .. b} = {c} \<longleftrightarrow> a = b \<and> b = c"

 proof

   assume "{a..b} = {c}"

   hence "\<not> (\<not> a \<le> b)" unfolding atLeastatMost_empty_iff[symmetric] by simp

   moreover with `{a..b} = {c}` have "c \<le> a \<and> b \<le> c" by auto

   ultimately show "a = b \<and> b = c" by auto

 qed simp

 end

 context dense_linorder

 begin

 lemma greaterThanLessThan_empty_iff[simp]:

   "{ a <..< b } = {} \<longleftrightarrow> b \<le> a"

   using dense[of a b] by (cases "a < b") auto

 lemma greaterThanLessThan_empty_iff2[simp]:

   "{} = { a <..< b } \<longleftrightarrow> b \<le> a"

   using dense[of a b] by (cases "a < b") auto

 lemma atLeastLessThan_subseteq_atLeastAtMost_iff:

   "{a ..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"

   using dense[of "max a d" "b"]

   by (force simp: subset_eq Ball_def not_less[symmetric])

 lemma greaterThanAtMost_subseteq_atLeastAtMost_iff:

   "{a <.. b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"

   using dense[of "a" "min c b"]

   by (force simp: subset_eq Ball_def not_less[symmetric])

 lemma greaterThanLessThan_subseteq_atLeastAtMost_iff:

   "{a <..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"

   using dense[of "a" "min c b"] dense[of "max a d" "b"]

   by (force simp: subset_eq Ball_def not_less[symmetric])

 lemma atLeastAtMost_subseteq_atLeastLessThan_iff:

   "{a .. b} \<subseteq> { c ..< d } \<longleftrightarrow> (a \<le> b \<longrightarrow> c \<le> a \<and> b < d)"

   using dense[of "max a d" "b"]

   by (force simp: subset_eq Ball_def not_less[symmetric])

 lemma greaterThanAtMost_subseteq_atLeastLessThan_iff:

   "{a <.. b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b < d)"

   using dense[of "a" "min c b"]

   by (force simp: subset_eq Ball_def not_less[symmetric])

 lemma greaterThanLessThan_subseteq_atLeastLessThan_iff:

   "{a <..< b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"

   using dense[of "a" "min c b"] dense[of "max a d" "b"]

   by (force simp: subset_eq Ball_def not_less[symmetric])

 end

 lemma (in linorder) atLeastLessThan_subset_iff:

   "{a..<b} <= {c..<d} \<Longrightarrow> b <= a | c<=a & b<=d"

 apply (auto simp:subset_eq Ball_def)

 apply(frule_tac x=a in spec)

 apply(erule_tac x=d in allE)

 apply (simp add: less_imp_le)

 done

 lemma atLeastLessThan_inj:

   fixes a b c d :: "'a::linorder"

   assumes eq: "{a ..< b} = {c ..< d}" and "a < b" "c < d"

   shows "a = c" "b = d"

 using assms by (metis atLeastLessThan_subset_iff eq less_le_not_le linorder_antisym_conv2 subset_refl)+

 lemma atLeastLessThan_eq_iff:

   fixes a b c d :: "'a::linorder"

   assumes "a < b" "c < d"

   shows "{a ..< b} = {c ..< d} \<longleftrightarrow> a = c \<and> b = d"

   using atLeastLessThan_inj assms by auto

 subsubsection {* Intersection *}

 context linorder

 begin

 lemma Int_atLeastAtMost[simp]: "{a..b} Int {c..d} = {max a c .. min b d}"

 by auto

 lemma Int_atLeastAtMostR1[simp]: "{..b} Int {c..d} = {c .. min b d}"

 by auto

 lemma Int_atLeastAtMostR2[simp]: "{a..} Int {c..d} = {max a c .. d}"

 by auto

 lemma Int_atLeastAtMostL1[simp]: "{a..b} Int {..d} = {a .. min b d}"

 by auto

 lemma Int_atLeastAtMostL2[simp]: "{a..b} Int {c..} = {max a c .. b}"

 by auto

 lemma Int_atLeastLessThan[simp]: "{a..<b} Int {c..<d} = {max a c ..< min b d}"

 by auto

 lemma Int_greaterThanAtMost[simp]: "{a<..b} Int {c<..d} = {max a c <.. min b d}"

 by auto

 lemma Int_greaterThanLessThan[simp]: "{a<..<b} Int {c<..<d} = {max a c <..< min b d}"

 by auto

 end

 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)

 text {* The following proof is convenient in induction proofs where

 new elements get indices at the beginning. So it is used to transform

 @{term "{..<Suc n}"} to @{term "0::nat"} and @{term "{..< n}"}. *}

 lemma lessThan_Suc_eq_insert_0: "{..<Suc n} = insert 0 (Suc ` {..<n})"

 proof safe

   fix x assume "x < Suc n" "x \<notin> Suc ` {..<n}"

   then have "x \<noteq> Suc (x - 1)" by auto

   with `x < Suc n` show "x = 0" by auto

 qed

 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 2 rules 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)

 lemma atLeast0AtMost: "{0..n::nat} = {..n}"

 by(simp add:atMost_def atLeastAtMost_def)

 declare atLeast0LessThan[symmetric, code_unfold]

         atLeast0AtMost[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)

 text {* The analogous result is useful on @{typ int}: *}

 (* here, because we don't have an own int section *)

 lemma atLeastAtMostPlus1_int_conv:

   "m <= 1+n \<Longrightarrow> {m..1+n} = insert (1+n) {m..n::int}"

   by (auto intro: set_eqI)

 lemma atLeastLessThan_add_Un: "i \<le> j \<Longrightarrow> {i..<j+k} = {i..<j} \<union> {j..<j+k::nat}"

   apply (induct k) 

   apply (simp_all add: atLeastLessThanSuc)   

   done

 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="Suc 0"] by simp

 corollary image_Suc_atLeastLessThan[simp]:

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

 using image_add_atLeastLessThan[where k="Suc 0"] 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

 lemma image_minus_const_atLeastLessThan_nat:

   fixes c :: nat

   shows "(\<lambda>i. i - c) ` {x ..< y} =

       (if c < y then {x - c ..< y - c} else if x < y then {0} else {})"

     (is "_ = ?right")

 proof safe

   fix a assume a: "a \<in> ?right"

   show "a \<in> (\<lambda>i. i - c) ` {x ..< y}"

   proof cases

     assume "c < y" with a show ?thesis

       by (auto intro!: image_eqI[of _ _ "a + c"])

   next

     assume "\<not> c < y" with a show ?thesis

       by (auto intro!: image_eqI[of _ _ x] split: split_if_asm)

   qed

 qed auto

 context ordered_ab_group_add

 begin

 lemma

   fixes x :: 'a

   shows image_uminus_greaterThan[simp]: "uminus ` {x<..} = {..<-x}"

   and image_uminus_atLeast[simp]: "uminus ` {x..} = {..-x}"

 proof safe

   fix y assume "y < -x"

   hence *:  "x < -y" using neg_less_iff_less[of "-y" x] by simp

   have "- (-y) \<in> uminus ` {x<..}"

     by (rule imageI) (simp add: *)

   thus "y \<in> uminus ` {x<..}" by simp

 next

   fix y assume "y \<le> -x"

   have "- (-y) \<in> uminus ` {x..}"

     by (rule imageI) (insert `y \<le> -x`[THEN le_imp_neg_le], simp)

   thus "y \<in> uminus ` {x..}" by simp

 qed simp_all

 lemma

   fixes x :: 'a

   shows image_uminus_lessThan[simp]: "uminus ` {..<x} = {-x<..}"

   and image_uminus_atMost[simp]: "uminus ` {..x} = {-x..}"

 proof -

   have "uminus ` {..<x} = uminus ` uminus ` {-x<..}"

     and "uminus ` {..x} = uminus ` uminus ` {-x..}" by simp_all

   thus "uminus ` {..<x} = {-x<..}" and "uminus ` {..x} = {-x..}"

     by (simp_all add: image_image

         del: image_uminus_greaterThan image_uminus_atLeast)

 qed

 lemma

   fixes x :: 'a

   shows image_uminus_atLeastAtMost[simp]: "uminus ` {x..y} = {-y..-x}"

   and image_uminus_greaterThanAtMost[simp]: "uminus ` {x<..y} = {-y..<-x}"

   and image_uminus_atLeastLessThan[simp]: "uminus ` {x..<y} = {-y<..-x}"

   and image_uminus_greaterThanLessThan[simp]: "uminus ` {x<..<y} = {-y<..<-x}"

   by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def

       greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)

 end

 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)

 text {* A bounded set of natural numbers is finite. *}

 lemma bounded_nat_set_is_finite:

   "(ALL i:N. i < (n::nat)) ==> finite N"

 apply (rule finite_subset)

  apply (rule_tac [2] finite_lessThan, auto)

 done

 text {* A set of natural numbers is finite iff it is bounded. *}

 lemma finite_nat_set_iff_bounded:

   "finite(N::nat set) = (EX m. ALL n:N. n<m)" (is "?F = ?B")

 proof

   assume f:?F  show ?B

     using Max_ge[OF `?F`, simplified less_Suc_eq_le[symmetric]] by blast

 next

   assume ?B show ?F using `?B` by(blast intro:bounded_nat_set_is_finite)

 qed

 lemma finite_nat_set_iff_bounded_le:

   "finite(N::nat set) = (EX m. ALL n:N. n<=m)"

 apply(simp add:finite_nat_set_iff_bounded)

 apply(blast dest:less_imp_le_nat le_imp_less_Suc)

 done

 lemma finite_less_ub:

      "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"

 by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)

 text{* Any subset of an interval of natural numbers the size of the

 subset is exactly that interval. *}

 lemma subset_card_intvl_is_intvl:

   "A <= {k..<k+card A} \<Longrightarrow> A = {k..<k+card A}" (is "PROP ?P")

 proof cases

   assume "finite A"

   thus "PROP ?P"

   proof(induct A rule:finite_linorder_max_induct)

     case empty thus ?case by auto

   next

     case (insert b A)

     moreover hence "b ~: A" by auto

     moreover have "A <= {k..<k+card A}" and "b = k+card A"

       using `b ~: A` insert by fastsimp+

     ultimately show ?case by auto

   qed

 next

   assume "~finite A" thus "PROP ?P" by simp

 qed

 subsubsection {* Proving Inclusions and Equalities between Unions *}

 lemma UN_le_eq_Un0:

   "(\<Union>i\<le>n::nat. M i) = (\<Union>i\<in>{1..n}. M i) \<union> M 0" (is "?A = ?B")

 proof

   show "?A <= ?B"

   proof

     fix x assume "x : ?A"

     then obtain i where i: "i\<le>n" "x : M i" by auto

     show "x : ?B"

     proof(cases i)

       case 0 with i show ?thesis by simp

     next

       case (Suc j) with i show ?thesis by auto

     qed

   qed

 next

   show "?B <= ?A" by auto

 qed

 lemma UN_le_add_shift:

   "(\<Union>i\<le>n::nat. M(i+k)) = (\<Union>i\<in>{k..n+k}. M i)" (is "?A = ?B")

 proof

   show "?A <= ?B" by fastsimp

 next

   show "?B <= ?A"

   proof

     fix x assume "x : ?B"

     then obtain i where i: "i : {k..n+k}" "x : M(i)" by auto

     hence "i-k\<le>n & x : M((i-k)+k)" by auto

     thus "x : ?A" by blast

   qed

 qed

 lemma UN_UN_finite_eq: "(\<Union>n::nat. \<Union>i\<in>{0..<n}. A i) = (\<Union>n. A n)"

   by (auto simp add: atLeast0LessThan) 

 lemma UN_finite_subset: "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> C) \<Longrightarrow> (\<Union>n. A n) \<subseteq> C"

   by (subst UN_UN_finite_eq [symmetric]) blast

 lemma UN_finite2_subset: 

      "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> (\<Union>i\<in>{0..<n+k}. B i)) \<Longrightarrow> (\<Union>n. A n) \<subseteq> (\<Union>n. B n)"

   apply (rule UN_finite_subset)

   apply (subst UN_UN_finite_eq [symmetric, of B]) 

   apply blast

   done

 lemma UN_finite2_eq:

   "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) = (\<Union>i\<in>{0..<n+k}. B i)) \<Longrightarrow> (\<Union>n. A n) = (\<Union>n. B n)"

   apply (rule subset_antisym)

    apply (rule UN_finite2_subset, blast)

  apply (rule UN_finite2_subset [where k=k])

  apply (force simp add: atLeastLessThan_add_Un [of 0])

  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)

 lemma ex_bij_betw_nat_finite:

   "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"

 apply(drule finite_imp_nat_seg_image_inj_on)

 apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)

 done

 lemma ex_bij_betw_finite_nat:

   "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"

 by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)

 lemma finite_same_card_bij:

   "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> EX h. bij_betw h A B"

 apply(drule ex_bij_betw_finite_nat)

 apply(drule ex_bij_betw_nat_finite)

 apply(auto intro!:bij_betw_trans)

 done

 lemma ex_bij_betw_nat_finite_1:

   "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"

 by (rule finite_same_card_bij) auto

 lemma bij_betw_iff_card:

   assumes FIN: "finite A" and FIN': "finite B"

   shows BIJ: "(\<exists>f. bij_betw f A B) \<longleftrightarrow> (card A = card B)"

 using assms

 proof(auto simp add: bij_betw_same_card)

   assume *: "card A = card B"

   obtain f where "bij_betw f A {0 ..< card A}"

   using FIN ex_bij_betw_finite_nat by blast

   moreover obtain g where "bij_betw g {0 ..< card B} B"

   using FIN' ex_bij_betw_nat_finite by blast

   ultimately have "bij_betw (g o f) A B"

   using * by (auto simp add: bij_betw_trans)

   thus "(\<exists>f. bij_betw f A B)" by blast

 qed

 lemma inj_on_iff_card_le:

   assumes FIN: "finite A" and FIN': "finite B"

   shows "(\<exists>f. inj_on f A \<and> f ` A \<le> B) = (card A \<le> card B)"

 proof (safe intro!: card_inj_on_le)

   assume *: "card A \<le> card B"

   obtain f where 1: "inj_on f A" and 2: "f ` A = {0 ..< card A}"

   using FIN ex_bij_betw_finite_nat unfolding bij_betw_def by force

   moreover obtain g where "inj_on g {0 ..< card B}" and 3: "g ` {0 ..< card B} = B"

   using FIN' ex_bij_betw_nat_finite unfolding bij_betw_def by force

   ultimately have "inj_on g (f ` A)" using subset_inj_on[of g _ "f ` A"] * by force

   hence "inj_on (g o f) A" using 1 comp_inj_on by blast

   moreover

   {have "{0 ..< card A} \<le> {0 ..< card B}" using * by force

    with 2 have "f ` A  \<le> {0 ..< card B}" by blast

    hence "(g o f) ` A \<le> B" unfolding comp_def using 3 by force

   }

   ultimately show "(\<exists>f. inj_on f A \<and> f ` A \<le> B)" by blast

 qed (insert assms, auto)

 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_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: algebra_simps)

 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)

 lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"

 proof -

   have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto

   with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)

 qed

 lemma card_less:

 assumes zero_in_M: "0 \<in> M"

 shows "card {k \<in> M. k < Suc i} \<noteq> 0"

 proof -

   from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto

   with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)

 qed

 lemma card_less_Suc2: "0 \<notin> M \<Longrightarrow> card {k. Suc k \<in> M \<and> k < i} = card {k \<in> M. k < Suc i}"

 apply (rule card_bij_eq [of Suc _ _ "\<lambda>x. x - Suc 0"])

 apply simp

 apply fastsimp

 apply auto

 apply (rule inj_on_diff_nat)

 apply auto

 apply (case_tac x)

 apply auto

 apply (case_tac xa)

 apply auto

 apply (case_tac xa)

 apply auto

 done

 lemma card_less_Suc:

   assumes zero_in_M: "0 \<in> M"

     shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"

 proof -

   from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp

   hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"

     by (auto simp only: insert_Diff)

   have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto

   from finite_M_bounded_by_nat[of "\<lambda>x. x \<in> M" "Suc i"] have "Suc (card {k. Suc k \<in> M \<and> k < i}) = card (insert 0 ({k \<in> M. k < Suc i} - {0}))"

     apply (subst card_insert)

     apply simp_all

     apply (subst b)

     apply (subst card_less_Suc2[symmetric])

     apply simp_all

     done

   with c show ?thesis by simp

 qed

 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 {* 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_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,no_atp]:

  "({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" == "CONST setsum (%x. t) {a..b}"

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

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

   "\<Sum>i<n. t" == "CONST 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_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_Suc:

   "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"

 by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)

 lemma setsum_head_upt_Suc:

   "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"

 apply(insert setsum_head_Suc[of m "n - Suc 0" f])

 apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)

 done

 lemma setsum_ub_add_nat: assumes "(m::nat) \<le> n + 1"

   shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"

 proof-

   have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using `m \<le> n+1` by auto

   thus ?thesis by (auto simp: ivl_disj_int setsum_Un_disjoint

     atLeastSucAtMost_greaterThanAtMost)

 qed

 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

 lemma setsum_natinterval_difff:

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

   shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =

           (if m <= n then f m - f(n + 1) else 0)"

 by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)

 lemma setsum_restrict_set':

   "finite A \<Longrightarrow> setsum f {x \<in> A. x \<in> B} = (\<Sum>x\<in>A. if x \<in> B then f x else 0)"

   by (simp add: setsum_restrict_set [symmetric] Int_def)

 lemma setsum_restrict_set'':

   "finite A \<Longrightarrow> setsum f {x \<in> A. P x} = (\<Sum>x\<in>A. if P x  then f x else 0)"

   by (simp add: setsum_restrict_set' [of A f "{x. P x}", simplified mem_Collect_eq])

 lemma setsum_setsum_restrict:

   "finite S \<Longrightarrow> finite T \<Longrightarrow>

     setsum (\<lambda>x. setsum (\<lambda>y. f x y) {y. y \<in> T \<and> R x y}) S = setsum (\<lambda>y. setsum (\<lambda>x. f x y) {x. x \<in> S \<and> R x y}) T"

   by (simp add: setsum_restrict_set'') (rule setsum_commute)

 lemma setsum_image_gen: assumes fS: "finite S"

   shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"

 proof-

   { fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto }

   hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"

     by simp

   also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"

     by (rule setsum_setsum_restrict[OF fS finite_imageI[OF fS]])

   finally show ?thesis .

 qed

 lemma setsum_le_included:

   fixes f :: "'a \<Rightarrow> 'b::ordered_comm_monoid_add"

   assumes "finite s" "finite t"

   and "\<forall>y\<in>t. 0 \<le> g y" "(\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> f x \<le> g y)"

   shows "setsum f s \<le> setsum g t"

 proof -

   have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"

   proof (rule setsum_mono)

     fix y assume "y \<in> s"

     with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto

     with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")

       using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]

       by (auto intro!: setsum_mono2)

   qed

   also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i ` t)"

     using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)

   also have "... \<le> setsum g t"

     using assms by (auto simp: setsum_image_gen[symmetric])

   finally show ?thesis .

 qed

 lemma setsum_multicount_gen:

   assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"

   shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")

 proof-

   have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto

   also have "\<dots> = ?r" unfolding setsum_setsum_restrict[OF assms(1-2)]

     using assms(3) by auto

   finally show ?thesis .

 qed

 lemma setsum_multicount:

   assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"

   shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")

 proof-

   have "?l = setsum (\<lambda>i. k) T" by(rule setsum_multicount_gen)(auto simp:assms)

   also have "\<dots> = ?r" by(simp add: mult_commute)

   finally show ?thesis by auto

 qed

 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="Suc 0", 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="Suc 0", simplified])

 lemma setsum_shift_lb_Suc0_0:

   "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"

 by(simp add:setsum_head_Suc)

 lemma setsum_shift_lb_Suc0_0_upt:

   "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"

 apply(cases k)apply simp

 apply(simp add:setsum_head_upt_Suc)

 done

 subsection {* The formula for geometric sums *}

 lemma geometric_sum:

   assumes "x \<noteq> 1"

   shows "(\<Sum>i=0..<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"

 proof -

   from assms obtain y where "y = x - 1" and "y \<noteq> 0" by simp_all

   moreover have "(\<Sum>i=0..<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"

   proof (induct n)

     case 0 then show ?case by simp

   next

     case (Suc n)

     moreover with `y \<noteq> 0` have "(1 + y) ^ n = (y * inverse y) * (1 + y) ^ n" by simp 

     ultimately show ?case by (simp add: field_simps divide_inverse)

   qed

   ultimately show ?thesis by simp

 qed

 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: algebra_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))"

     unfolding One_nat_def

     by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_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"], unfold One_nat_def)

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

   finally show ?thesis by (simp add: algebra_simps)

 next

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

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

   thus ?thesis by (auto simp: algebra_simps)

 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

     unfolding One_nat_def by auto

 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

 subsection {* Products indexed over intervals *}

 syntax

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

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

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

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

 syntax (xsymbols)

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

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

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

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

 syntax (HTML output)

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

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

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

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

 syntax (latex_prod output)

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

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

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

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

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

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

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

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

 translations

   "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"

   "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"

   "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"

   "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"

 subsection {* Transfer setup *}

 lemma transfer_nat_int_set_functions:

     "{..n} = nat ` {0..int n}"

     "{m..n} = nat ` {int m..int n}"  (* need all variants of these! *)

   apply (auto simp add: image_def)

   apply (rule_tac x = "int x" in bexI)

   apply auto

   apply (rule_tac x = "int x" in bexI)

   apply auto

   done

 lemma transfer_nat_int_set_function_closures:

     "x >= 0 \<Longrightarrow> nat_set {x..y}"

   by (simp add: nat_set_def)

 declare transfer_morphism_nat_int[transfer add

   return: transfer_nat_int_set_functions

     transfer_nat_int_set_function_closures

 ]

 lemma transfer_int_nat_set_functions:

     "is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int ` {nat m..nat n}"

   by (simp only: is_nat_def transfer_nat_int_set_functions

     transfer_nat_int_set_function_closures

     transfer_nat_int_set_return_embed nat_0_le

     cong: transfer_nat_int_set_cong)

 lemma transfer_int_nat_set_function_closures:

     "is_nat x \<Longrightarrow> nat_set {x..y}"

   by (simp only: transfer_nat_int_set_function_closures is_nat_def)

 declare transfer_morphism_int_nat[transfer add

   return: transfer_int_nat_set_functions

     transfer_int_nat_set_function_closures

 ]

 end