wneuper/isa: src/HOL/Set_Interval.thy@0cb8a2defb06

     1 (*  Title:      HOL/Set_Interval.thy

     2     Author:     Tobias Nipkow

     3     Author:     Clemens Ballarin

     4     Author:     Jeremy Avigad

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

     8 Modern convention: Ixy stands for an interval where x and y

     9 describe the lower and upper bound and x,y : {c,o,i}

    10 where c = closed, o = open, i = infinite.

    11 Examples: Ico = {_ ..< _} and Ici = {_ ..}

    12 *)

    14 header {* Set intervals *}

    16 theory Set_Interval

    17 imports Nat_Transfer

    18 begin

    20 context ord

    21 begin

    23 definition

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

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

    27 definition

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

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

    31 definition

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

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

    35 definition

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

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

    39 definition

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

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

    43 definition

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

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

    47 definition

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

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

    51 definition

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

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

    55 end

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

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

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

    62 syntax

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

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

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

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

    68 syntax (xsymbols)

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

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

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

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

    74 syntax (latex output)

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

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

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

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

    80 translations

    81   "UN i<=n. A"  == "UN i:{..n}. A"

    82   "UN i<n. A"   == "UN i:{..<n}. A"

    83   "INT i<=n. A" == "INT i:{..n}. A"

    84   "INT i<n. A"  == "INT i:{..<n}. A"

    87 subsection {* Various equivalences *}

    89 lemma (in ord) lessThan_iff [iff]: "(i: lessThan k) = (i<k)"

    90 by (simp add: lessThan_def)

    92 lemma Compl_lessThan [simp]:

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

    94 apply (auto simp add: lessThan_def atLeast_def)

    95 done

    97 lemma single_Diff_lessThan [simp]: "!!k:: 'a::order. {k} - lessThan k = {k}"

    98 by auto

   100 lemma (in ord) greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"

   101 by (simp add: greaterThan_def)

   103 lemma Compl_greaterThan [simp]:

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

   105   by (auto simp add: greaterThan_def atMost_def)

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

   108 apply (subst Compl_greaterThan [symmetric])

   109 apply (rule double_complement)

   110 done

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

   113 by (simp add: atLeast_def)

   115 lemma Compl_atLeast [simp]:

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

   117   by (auto simp add: lessThan_def atLeast_def)

   119 lemma (in ord) atMost_iff [iff]: "(i: atMost k) = (i<=k)"

   120 by (simp add: atMost_def)

   122 lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"

   123 by (blast intro: order_antisym)

   125 lemma (in linorder) lessThan_Int_lessThan: "{ a <..} \<inter> { b <..} = { max a b <..}"

   126   by auto

   128 lemma (in linorder) greaterThan_Int_greaterThan: "{..< a} \<inter> {..< b} = {..< min a b}"

   129   by auto

   131 subsection {* Logical Equivalences for Set Inclusion and Equality *}

   133 lemma atLeast_subset_iff [iff]:

   134      "(atLeast x \<subseteq> atLeast y) = (y \<le> (x::'a::order))"

   135 by (blast intro: order_trans)

   137 lemma atLeast_eq_iff [iff]:

   138      "(atLeast x = atLeast y) = (x = (y::'a::linorder))"

   139 by (blast intro: order_antisym order_trans)

   141 lemma greaterThan_subset_iff [iff]:

   142      "(greaterThan x \<subseteq> greaterThan y) = (y \<le> (x::'a::linorder))"

   143 apply (auto simp add: greaterThan_def)

   144  apply (subst linorder_not_less [symmetric], blast)

   145 done

   147 lemma greaterThan_eq_iff [iff]:

   148      "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"

   149 apply (rule iffI)

   150  apply (erule equalityE)

   151  apply simp_all

   152 done

   154 lemma atMost_subset_iff [iff]: "(atMost x \<subseteq> atMost y) = (x \<le> (y::'a::order))"

   155 by (blast intro: order_trans)

   157 lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::linorder))"

   158 by (blast intro: order_antisym order_trans)

   160 lemma lessThan_subset_iff [iff]:

   161      "(lessThan x \<subseteq> lessThan y) = (x \<le> (y::'a::linorder))"

   162 apply (auto simp add: lessThan_def)

   163  apply (subst linorder_not_less [symmetric], blast)

   164 done

   166 lemma lessThan_eq_iff [iff]:

   167      "(lessThan x = lessThan y) = (x = (y::'a::linorder))"

   168 apply (rule iffI)

   169  apply (erule equalityE)

   170  apply simp_all

   171 done

   173 lemma lessThan_strict_subset_iff:

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

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

   176   by (metis leD lessThan_subset_iff linorder_linear not_less_iff_gr_or_eq psubset_eq)

   178 subsection {*Two-sided intervals*}

   180 context ord

   181 begin

   183 lemma greaterThanLessThan_iff [simp]:

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

   185 by (simp add: greaterThanLessThan_def)

   187 lemma atLeastLessThan_iff [simp]:

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

   189 by (simp add: atLeastLessThan_def)

   191 lemma greaterThanAtMost_iff [simp]:

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

   193 by (simp add: greaterThanAtMost_def)

   195 lemma atLeastAtMost_iff [simp]:

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

   197 by (simp add: atLeastAtMost_def)

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

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

   201 alone. *}

   203 lemma greaterThanLessThan_eq: "{ a <..< b} = { a <..} \<inter> {..< b }"

   204   by auto

   206 end

   208 subsubsection{* Emptyness, singletons, subset *}

   210 context order

   211 begin

   213 lemma atLeastatMost_empty[simp]:

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

   215 by(auto simp: atLeastAtMost_def atLeast_def atMost_def)

   217 lemma atLeastatMost_empty_iff[simp]:

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

   219 by auto (blast intro: order_trans)

   221 lemma atLeastatMost_empty_iff2[simp]:

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

   223 by auto (blast intro: order_trans)

   225 lemma atLeastLessThan_empty[simp]:

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

   227 by(auto simp: atLeastLessThan_def)

   229 lemma atLeastLessThan_empty_iff[simp]:

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

   231 by auto (blast intro: le_less_trans)

   233 lemma atLeastLessThan_empty_iff2[simp]:

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

   235 by auto (blast intro: le_less_trans)

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

   238 by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)

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

   241 by auto (blast intro: less_le_trans)

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

   244 by auto (blast intro: less_le_trans)

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

   247 by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)

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

   250 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)

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

   254 lemma atLeastatMost_subset_iff[simp]:

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

   256 unfolding atLeastAtMost_def atLeast_def atMost_def

   257 by (blast intro: order_trans)

   259 lemma atLeastatMost_psubset_iff:

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

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

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

   264 lemma Icc_eq_Icc[simp]:

   265   "{l..h} = {l'..h'} = (l=l' \<and> h=h' \<or> \<not> l\<le>h \<and> \<not> l'\<le>h')"

   266 by(simp add: order_class.eq_iff)(auto intro: order_trans)

   268 lemma atLeastAtMost_singleton_iff[simp]:

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

   270 proof

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

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

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

   274   with * show "a = b \<and> b = c" by auto

   275 qed simp

   277 lemma Icc_subset_Ici_iff[simp]:

   278   "{l..h} \<subseteq> {l'..} = (~ l\<le>h \<or> l\<ge>l')"

   279 by(auto simp: subset_eq intro: order_trans)

   281 lemma Icc_subset_Iic_iff[simp]:

   282   "{l..h} \<subseteq> {..h'} = (~ l\<le>h \<or> h\<le>h')"

   283 by(auto simp: subset_eq intro: order_trans)

   285 lemma not_Ici_eq_empty[simp]: "{l..} \<noteq> {}"

   286 by(auto simp: set_eq_iff)

   288 lemma not_Iic_eq_empty[simp]: "{..h} \<noteq> {}"

   289 by(auto simp: set_eq_iff)

   291 lemmas not_empty_eq_Ici_eq_empty[simp] = not_Ici_eq_empty[symmetric]

   292 lemmas not_empty_eq_Iic_eq_empty[simp] = not_Iic_eq_empty[symmetric]

   294 end

   296 context no_top

   297 begin

   299 (* also holds for no_bot but no_top should suffice *)

   300 lemma not_UNIV_le_Icc[simp]: "\<not> UNIV \<subseteq> {l..h}"

   301 using gt_ex[of h] by(auto simp: subset_eq less_le_not_le)

   303 lemma not_UNIV_le_Iic[simp]: "\<not> UNIV \<subseteq> {..h}"

   304 using gt_ex[of h] by(auto simp: subset_eq less_le_not_le)

   306 lemma not_Ici_le_Icc[simp]: "\<not> {l..} \<subseteq> {l'..h'}"

   307 using gt_ex[of h']

   308 by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)

   310 lemma not_Ici_le_Iic[simp]: "\<not> {l..} \<subseteq> {..h'}"

   311 using gt_ex[of h']

   312 by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)

   314 end

   316 context no_bot

   317 begin

   319 lemma not_UNIV_le_Ici[simp]: "\<not> UNIV \<subseteq> {l..}"

   320 using lt_ex[of l] by(auto simp: subset_eq less_le_not_le)

   322 lemma not_Iic_le_Icc[simp]: "\<not> {..h} \<subseteq> {l'..h'}"

   323 using lt_ex[of l']

   324 by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)

   326 lemma not_Iic_le_Ici[simp]: "\<not> {..h} \<subseteq> {l'..}"

   327 using lt_ex[of l']

   328 by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)

   330 end

   333 context no_top

   334 begin

   336 (* also holds for no_bot but no_top should suffice *)

   337 lemma not_UNIV_eq_Icc[simp]: "\<not> UNIV = {l'..h'}"

   338 using gt_ex[of h'] by(auto simp: set_eq_iff  less_le_not_le)

   340 lemmas not_Icc_eq_UNIV[simp] = not_UNIV_eq_Icc[symmetric]

   342 lemma not_UNIV_eq_Iic[simp]: "\<not> UNIV = {..h'}"

   343 using gt_ex[of h'] by(auto simp: set_eq_iff  less_le_not_le)

   345 lemmas not_Iic_eq_UNIV[simp] = not_UNIV_eq_Iic[symmetric]

   347 lemma not_Icc_eq_Ici[simp]: "\<not> {l..h} = {l'..}"

   348 unfolding atLeastAtMost_def using not_Ici_le_Iic[of l'] by blast

   350 lemmas not_Ici_eq_Icc[simp] = not_Icc_eq_Ici[symmetric]

   352 (* also holds for no_bot but no_top should suffice *)

   353 lemma not_Iic_eq_Ici[simp]: "\<not> {..h} = {l'..}"

   354 using not_Ici_le_Iic[of l' h] by blast

   356 lemmas not_Ici_eq_Iic[simp] = not_Iic_eq_Ici[symmetric]

   358 end

   360 context no_bot

   361 begin

   363 lemma not_UNIV_eq_Ici[simp]: "\<not> UNIV = {l'..}"

   364 using lt_ex[of l'] by(auto simp: set_eq_iff  less_le_not_le)

   366 lemmas not_Ici_eq_UNIV[simp] = not_UNIV_eq_Ici[symmetric]

   368 lemma not_Icc_eq_Iic[simp]: "\<not> {l..h} = {..h'}"

   369 unfolding atLeastAtMost_def using not_Iic_le_Ici[of h'] by blast

   371 lemmas not_Iic_eq_Icc[simp] = not_Icc_eq_Iic[symmetric]

   373 end

   376 context dense_linorder

   377 begin

   379 lemma greaterThanLessThan_empty_iff[simp]:

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

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

   383 lemma greaterThanLessThan_empty_iff2[simp]:

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

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

   387 lemma atLeastLessThan_subseteq_atLeastAtMost_iff:

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

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

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

   392 lemma greaterThanAtMost_subseteq_atLeastAtMost_iff:

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

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

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

   397 lemma greaterThanLessThan_subseteq_atLeastAtMost_iff:

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

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

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

   402 lemma atLeastAtMost_subseteq_atLeastLessThan_iff:

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

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

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

   407 lemma greaterThanAtMost_subseteq_atLeastLessThan_iff:

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

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

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

   412 lemma greaterThanLessThan_subseteq_atLeastLessThan_iff:

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

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

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

   417 end

   419 context no_top

   420 begin

   422 lemma greaterThan_non_empty[simp]: "{x <..} \<noteq> {}"

   423   using gt_ex[of x] by auto

   425 end

   427 context no_bot

   428 begin

   430 lemma lessThan_non_empty[simp]: "{..< x} \<noteq> {}"

   431   using lt_ex[of x] by auto

   433 end

   435 lemma (in linorder) atLeastLessThan_subset_iff:

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

   437 apply (auto simp:subset_eq Ball_def)

   438 apply(frule_tac x=a in spec)

   439 apply(erule_tac x=d in allE)

   440 apply (simp add: less_imp_le)

   441 done

   443 lemma atLeastLessThan_inj:

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

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

   446   shows "a = c" "b = d"

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

   449 lemma atLeastLessThan_eq_iff:

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

   451   assumes "a < b" "c < d"

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

   453   using atLeastLessThan_inj assms by auto

   455 lemma (in order_bot) atLeast_eq_UNIV_iff: "{x..} = UNIV \<longleftrightarrow> x = bot"

   456 by (auto simp: set_eq_iff intro: le_bot)

   458 lemma (in order_top) atMost_eq_UNIV_iff: "{..x} = UNIV \<longleftrightarrow> x = top"

   459 by (auto simp: set_eq_iff intro: top_le)

   461 lemma (in bounded_lattice) atLeastAtMost_eq_UNIV_iff:

   462   "{x..y} = UNIV \<longleftrightarrow> (x = bot \<and> y = top)"

   463 by (auto simp: set_eq_iff intro: top_le le_bot)

   466 subsubsection {* Intersection *}

   468 context linorder

   469 begin

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

   472 by auto

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

   475 by auto

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

   478 by auto

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

   481 by auto

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

   484 by auto

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

   487 by auto

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

   490 by auto

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

   493 by auto

   495 lemma Int_atMost[simp]: "{..a} \<inter> {..b} = {.. min a b}"

   496   by (auto simp: min_def)

   498 end

   500 context complete_lattice

   501 begin

   503 lemma

   504   shows Sup_atLeast[simp]: "Sup {x ..} = top"

   505     and Sup_greaterThanAtLeast[simp]: "x < top \<Longrightarrow> Sup {x <..} = top"

   506     and Sup_atMost[simp]: "Sup {.. y} = y"

   507     and Sup_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Sup { x .. y} = y"

   508     and Sup_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Sup { x <.. y} = y"

   509   by (auto intro!: Sup_eqI)

   511 lemma

   512   shows Inf_atMost[simp]: "Inf {.. x} = bot"

   513     and Inf_atMostLessThan[simp]: "top < x \<Longrightarrow> Inf {..< x} = bot"

   514     and Inf_atLeast[simp]: "Inf {x ..} = x"

   515     and Inf_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Inf { x .. y} = x"

   516     and Inf_atLeastLessThan[simp]: "x < y \<Longrightarrow> Inf { x ..< y} = x"

   517   by (auto intro!: Inf_eqI)

   519 end

   521 lemma

   522   fixes x y :: "'a :: {complete_lattice, dense_linorder}"

   523   shows Sup_lessThan[simp]: "Sup {..< y} = y"

   524     and Sup_atLeastLessThan[simp]: "x < y \<Longrightarrow> Sup { x ..< y} = y"

   525     and Sup_greaterThanLessThan[simp]: "x < y \<Longrightarrow> Sup { x <..< y} = y"

   526     and Inf_greaterThan[simp]: "Inf {x <..} = x"

   527     and Inf_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Inf { x <.. y} = x"

   528     and Inf_greaterThanLessThan[simp]: "x < y \<Longrightarrow> Inf { x <..< y} = x"

   529   by (auto intro!: Inf_eqI Sup_eqI intro: dense_le dense_le_bounded dense_ge dense_ge_bounded)

   531 subsection {* Intervals of natural numbers *}

   533 subsubsection {* The Constant @{term lessThan} *}

   535 lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"

   536 by (simp add: lessThan_def)

   538 lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"

   539 by (simp add: lessThan_def less_Suc_eq, blast)

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

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

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

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

   546 proof safe

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

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

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

   550 qed

   552 lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"

   553 by (simp add: lessThan_def atMost_def less_Suc_eq_le)

   555 lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"

   556 by blast

   558 subsubsection {* The Constant @{term greaterThan} *}

   560 lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"

   561 apply (simp add: greaterThan_def)

   562 apply (blast dest: gr0_conv_Suc [THEN iffD1])

   563 done

   565 lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"

   566 apply (simp add: greaterThan_def)

   567 apply (auto elim: linorder_neqE)

   568 done

   570 lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"

   571 by blast

   573 subsubsection {* The Constant @{term atLeast} *}

   575 lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"

   576 by (unfold atLeast_def UNIV_def, simp)

   578 lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"

   579 apply (simp add: atLeast_def)

   580 apply (simp add: Suc_le_eq)

   581 apply (simp add: order_le_less, blast)

   582 done

   584 lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"

   585   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)

   587 lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"

   588 by blast

   590 subsubsection {* The Constant @{term atMost} *}

   592 lemma atMost_0 [simp]: "atMost (0::nat) = {0}"

   593 by (simp add: atMost_def)

   595 lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"

   596 apply (simp add: atMost_def)

   597 apply (simp add: less_Suc_eq order_le_less, blast)

   598 done

   600 lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"

   601 by blast

   603 subsubsection {* The Constant @{term atLeastLessThan} *}

   605 text{*The orientation of the following 2 rules is tricky. The lhs is

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

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

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

   609 used, the opposite orientation seems preferable because it reduces a

   610 specific concept to a more general one. *}

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

   613 by(simp add:lessThan_def atLeastLessThan_def)

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

   616 by(simp add:atMost_def atLeastAtMost_def)

   618 declare atLeast0LessThan[symmetric, code_unfold]

   619         atLeast0AtMost[symmetric, code_unfold]

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

   622 by (simp add: atLeastLessThan_def)

   624 subsubsection {* Intervals of nats with @{term Suc} *}

   626 text{*Not a simprule because the RHS is too messy.*}

   627 lemma atLeastLessThanSuc:

   628     "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"

   629 by (auto simp add: atLeastLessThan_def)

   631 lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"

   632 by (auto simp add: atLeastLessThan_def)

   633 (*

   634 lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"

   635 by (induct k, simp_all add: atLeastLessThanSuc)

   637 lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"

   638 by (auto simp add: atLeastLessThan_def)

   639 *)

   640 lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"

   641   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)

   643 lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"

   644   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def

   645     greaterThanAtMost_def)

   647 lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"

   648   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def

   649     greaterThanLessThan_def)

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

   652 by (auto simp add: atLeastAtMost_def)

   654 lemma atLeastAtMost_insertL: "m \<le> n \<Longrightarrow> insert m {Suc m..n} = {m ..n}"

   655 by auto

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

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

   659 lemma atLeastAtMostPlus1_int_conv:

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

   661   by (auto intro: set_eqI)

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

   664   apply (induct k)

   665   apply (simp_all add: atLeastLessThanSuc)

   666   done

   668 subsubsection {* Image *}

   670 lemma image_add_atLeastAtMost:

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

   672 proof

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

   674 next

   675   show "?B \<subseteq> ?A"

   676   proof

   677     fix n assume a: "n : ?B"

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

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

   680     ultimately show "n : ?A" by blast

   681   qed

   682 qed

   684 lemma image_add_atLeastLessThan:

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

   686 proof

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

   688 next

   689   show "?B \<subseteq> ?A"

   690   proof

   691     fix n assume a: "n : ?B"

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

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

   694     ultimately show "n : ?A" by blast

   695   qed

   696 qed

   698 corollary image_Suc_atLeastAtMost[simp]:

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

   700 using image_add_atLeastAtMost[where k="Suc 0"] by simp

   702 corollary image_Suc_atLeastLessThan[simp]:

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

   704 using image_add_atLeastLessThan[where k="Suc 0"] by simp

   706 lemma image_add_int_atLeastLessThan:

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

   708   apply (auto simp add: image_def)

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

   710   apply auto

   711   done

   713 lemma image_minus_const_atLeastLessThan_nat:

   714   fixes c :: nat

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

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

   717     (is "_ = ?right")

   718 proof safe

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

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

   721   proof cases

   722     assume "c < y" with a show ?thesis

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

   724   next

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

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

   727   qed

   728 qed auto

   730 lemma image_int_atLeastLessThan: "int ` {a..<b} = {int a..<int b}"

   731 by(auto intro!: image_eqI[where x="nat x", standard])

   733 context ordered_ab_group_add

   734 begin

   736 lemma

   737   fixes x :: 'a

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

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

   740 proof safe

   741   fix y assume "y < -x"

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

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

   744     by (rule imageI) (simp add: *)

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

   746 next

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

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

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

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

   751 qed simp_all

   753 lemma

   754   fixes x :: 'a

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

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

   757 proof -

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

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

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

   761     by (simp_all add: image_image

   762         del: image_uminus_greaterThan image_uminus_atLeast)

   763 qed

   765 lemma

   766   fixes x :: 'a

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

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

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

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

   771   by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def

   772       greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)

   773 end

   775 subsubsection {* Finiteness *}

   777 lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"

   778   by (induct k) (simp_all add: lessThan_Suc)

   780 lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"

   781   by (induct k) (simp_all add: atMost_Suc)

   783 lemma finite_greaterThanLessThan [iff]:

   784   fixes l :: nat shows "finite {l<..<u}"

   785 by (simp add: greaterThanLessThan_def)

   787 lemma finite_atLeastLessThan [iff]:

   788   fixes l :: nat shows "finite {l..<u}"

   789 by (simp add: atLeastLessThan_def)

   791 lemma finite_greaterThanAtMost [iff]:

   792   fixes l :: nat shows "finite {l<..u}"

   793 by (simp add: greaterThanAtMost_def)

   795 lemma finite_atLeastAtMost [iff]:

   796   fixes l :: nat shows "finite {l..u}"

   797 by (simp add: atLeastAtMost_def)

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

   800 lemma bounded_nat_set_is_finite:

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

   802 apply (rule finite_subset)

   803  apply (rule_tac [2] finite_lessThan, auto)

   804 done

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

   807 lemma finite_nat_set_iff_bounded:

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

   809 proof

   810   assume f:?F  show ?B

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

   812 next

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

   814 qed

   816 lemma finite_nat_set_iff_bounded_le:

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

   818 apply(simp add:finite_nat_set_iff_bounded)

   819 apply(blast dest:less_imp_le_nat le_imp_less_Suc)

   820 done

   822 lemma finite_less_ub:

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

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

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

   827 subset is exactly that interval. *}

   829 lemma subset_card_intvl_is_intvl:

   830   assumes "A \<subseteq> {k..<k+card A}"

   831   shows "A = {k..<k+card A}"

   832 proof (cases "finite A")

   833   case True

   834   from this and assms show ?thesis

   835   proof (induct A rule: finite_linorder_max_induct)

   836     case empty thus ?case by auto

   837   next

   838     case (insert b A)

   839     hence *: "b \<notin> A" by auto

   840     with insert have "A <= {k..<k+card A}" and "b = k+card A"

   841       by fastforce+

   842     with insert * show ?case by auto

   843   qed

   844 next

   845   case False

   846   with assms show ?thesis by simp

   847 qed

   850 subsubsection {* Proving Inclusions and Equalities between Unions *}

   852 lemma UN_le_eq_Un0:

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

   854 proof

   855   show "?A <= ?B"

   856   proof

   857     fix x assume "x : ?A"

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

   859     show "x : ?B"

   860     proof(cases i)

   861       case 0 with i show ?thesis by simp

   862     next

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

   864     qed

   865   qed

   866 next

   867   show "?B <= ?A" by auto

   868 qed

   870 lemma UN_le_add_shift:

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

   872 proof

   873   show "?A <= ?B" by fastforce

   874 next

   875   show "?B <= ?A"

   876   proof

   877     fix x assume "x : ?B"

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

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

   880     thus "x : ?A" by blast

   881   qed

   882 qed

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

   885   by (auto simp add: atLeast0LessThan)

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

   888   by (subst UN_UN_finite_eq [symmetric]) blast

   890 lemma UN_finite2_subset:

   891      "(!!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)"

   892   apply (rule UN_finite_subset)

   893   apply (subst UN_UN_finite_eq [symmetric, of B])

   894   apply blast

   895   done

   897 lemma UN_finite2_eq:

   898   "(!!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)"

   899   apply (rule subset_antisym)

   900    apply (rule UN_finite2_subset, blast)

   901  apply (rule UN_finite2_subset [where k=k])

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

   903  done

   906 subsubsection {* Cardinality *}

   908 lemma card_lessThan [simp]: "card {..<u} = u"

   909   by (induct u, simp_all add: lessThan_Suc)

   911 lemma card_atMost [simp]: "card {..u} = Suc u"

   912   by (simp add: lessThan_Suc_atMost [THEN sym])

   914 lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"

   915   apply (subgoal_tac "card {l..<u} = card {..<u-l}")

   916   apply (erule ssubst, rule card_lessThan)

   917   apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")

   918   apply (erule subst)

   919   apply (rule card_image)

   920   apply (simp add: inj_on_def)

   921   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)

   922   apply (rule_tac x = "x - l" in exI)

   923   apply arith

   924   done

   926 lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"

   927   by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)

   929 lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"

   930   by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)

   932 lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"

   933   by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)

   935 lemma ex_bij_betw_nat_finite:

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

   937 apply(drule finite_imp_nat_seg_image_inj_on)

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

   939 done

   941 lemma ex_bij_betw_finite_nat:

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

   943 by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)

   945 lemma finite_same_card_bij:

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

   947 apply(drule ex_bij_betw_finite_nat)

   948 apply(drule ex_bij_betw_nat_finite)

   949 apply(auto intro!:bij_betw_trans)

   950 done

   952 lemma ex_bij_betw_nat_finite_1:

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

   954 by (rule finite_same_card_bij) auto

   956 lemma bij_betw_iff_card:

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

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

   959 using assms

   960 proof(auto simp add: bij_betw_same_card)

   961   assume *: "card A = card B"

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

   963   using FIN ex_bij_betw_finite_nat by blast

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

   965   using FIN' ex_bij_betw_nat_finite by blast

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

   967   using * by (auto simp add: bij_betw_trans)

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

   969 qed

   971 lemma inj_on_iff_card_le:

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

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

   974 proof (safe intro!: card_inj_on_le)

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

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

   977   using FIN ex_bij_betw_finite_nat unfolding bij_betw_def by force

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

   979   using FIN' ex_bij_betw_nat_finite unfolding bij_betw_def by force

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

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

   982   moreover

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

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

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

   986   }

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

   988 qed (insert assms, auto)

   990 subsection {* Intervals of integers *}

   992 lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"

   993   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)

   995 lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"

   996   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)

   998 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:

   999     "{l+1..<u} = {l<..<u::int}"

  1000   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)

  1002 subsubsection {* Finiteness *}

  1004 lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>

  1005     {(0::int)..<u} = int ` {..<nat u}"

  1006   apply (unfold image_def lessThan_def)

  1007   apply auto

  1008   apply (rule_tac x = "nat x" in exI)

  1009   apply (auto simp add: zless_nat_eq_int_zless [THEN sym])

  1010   done

  1012 lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"

  1013   apply (cases "0 \<le> u")

  1014   apply (subst image_atLeastZeroLessThan_int, assumption)

  1015   apply (rule finite_imageI)

  1016   apply auto

  1017   done

  1019 lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"

  1020   apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")

  1021   apply (erule subst)

  1022   apply (rule finite_imageI)

  1023   apply (rule finite_atLeastZeroLessThan_int)

  1024   apply (rule image_add_int_atLeastLessThan)

  1025   done

  1027 lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"

  1028   by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)

  1030 lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"

  1031   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)

  1033 lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"

  1034   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)

  1037 subsubsection {* Cardinality *}

  1039 lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"

  1040   apply (cases "0 \<le> u")

  1041   apply (subst image_atLeastZeroLessThan_int, assumption)

  1042   apply (subst card_image)

  1043   apply (auto simp add: inj_on_def)

  1044   done

  1046 lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"

  1047   apply (subgoal_tac "card {l..<u} = card {0..<u-l}")

  1048   apply (erule ssubst, rule card_atLeastZeroLessThan_int)

  1049   apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")

  1050   apply (erule subst)

  1051   apply (rule card_image)

  1052   apply (simp add: inj_on_def)

  1053   apply (rule image_add_int_atLeastLessThan)

  1054   done

  1056 lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"

  1057 apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])

  1058 apply (auto simp add: algebra_simps)

  1059 done

  1061 lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"

  1062 by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)

  1064 lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"

  1065 by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)

  1067 lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"

  1068 proof -

  1069   have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto

  1070   with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)

  1071 qed

  1073 lemma card_less:

  1074 assumes zero_in_M: "0 \<in> M"

  1075 shows "card {k \<in> M. k < Suc i} \<noteq> 0"

  1076 proof -

  1077   from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto

  1078   with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)

  1079 qed

  1081 lemma card_less_Suc2: "0 \<notin> M \<Longrightarrow> card {k. Suc k \<in> M \<and> k < i} = card {k \<in> M. k < Suc i}"

  1082 apply (rule card_bij_eq [of Suc _ _ "\<lambda>x. x - Suc 0"])

  1083 apply simp

  1084 apply fastforce

  1085 apply auto

  1086 apply (rule inj_on_diff_nat)

  1087 apply auto

  1088 apply (case_tac x)

  1089 apply auto

  1090 apply (case_tac xa)

  1091 apply auto

  1092 apply (case_tac xa)

  1093 apply auto

  1094 done

  1096 lemma card_less_Suc:

  1097   assumes zero_in_M: "0 \<in> M"

  1098     shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"

  1099 proof -

  1100   from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp

  1101   hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"

  1102     by (auto simp only: insert_Diff)

  1103   have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto

  1104   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}))"

  1105     apply (subst card_insert)

  1106     apply simp_all

  1107     apply (subst b)

  1108     apply (subst card_less_Suc2[symmetric])

  1109     apply simp_all

  1110     done

  1111   with c show ?thesis by simp

  1112 qed

  1115 subsection {*Lemmas useful with the summation operator setsum*}

  1117 text {* For examples, see Algebra/poly/UnivPoly2.thy *}

  1119 subsubsection {* Disjoint Unions *}

  1121 text {* Singletons and open intervals *}

  1123 lemma ivl_disj_un_singleton:

  1124   "{l::'a::linorder} Un {l<..} = {l..}"

  1125   "{..<u} Un {u::'a::linorder} = {..u}"

  1126   "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"

  1127   "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"

  1128   "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"

  1129   "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"

  1130 by auto

  1132 text {* One- and two-sided intervals *}

  1134 lemma ivl_disj_un_one:

  1135   "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"

  1136   "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"

  1137   "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"

  1138   "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"

  1139   "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"

  1140   "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"

  1141   "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"

  1142   "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"

  1143 by auto

  1145 text {* Two- and two-sided intervals *}

  1147 lemma ivl_disj_un_two:

  1148   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"

  1149   "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"

  1150   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"

  1151   "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"

  1152   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"

  1153   "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"

  1154   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"

  1155   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"

  1156 by auto

  1158 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two

  1160 subsubsection {* Disjoint Intersections *}

  1162 text {* One- and two-sided intervals *}

  1164 lemma ivl_disj_int_one:

  1165   "{..l::'a::order} Int {l<..<u} = {}"

  1166   "{..<l} Int {l..<u} = {}"

  1167   "{..l} Int {l<..u} = {}"

  1168   "{..<l} Int {l..u} = {}"

  1169   "{l<..u} Int {u<..} = {}"

  1170   "{l<..<u} Int {u..} = {}"

  1171   "{l..u} Int {u<..} = {}"

  1172   "{l..<u} Int {u..} = {}"

  1173   by auto

  1175 text {* Two- and two-sided intervals *}

  1177 lemma ivl_disj_int_two:

  1178   "{l::'a::order<..<m} Int {m..<u} = {}"

  1179   "{l<..m} Int {m<..<u} = {}"

  1180   "{l..<m} Int {m..<u} = {}"

  1181   "{l..m} Int {m<..<u} = {}"

  1182   "{l<..<m} Int {m..u} = {}"

  1183   "{l<..m} Int {m<..u} = {}"

  1184   "{l..<m} Int {m..u} = {}"

  1185   "{l..m} Int {m<..u} = {}"

  1186   by auto

  1188 lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two

  1190 subsubsection {* Some Differences *}

  1192 lemma ivl_diff[simp]:

  1193  "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"

  1194 by(auto)

  1197 subsubsection {* Some Subset Conditions *}

  1199 lemma ivl_subset [simp]:

  1200  "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"

  1201 apply(auto simp:linorder_not_le)

  1202 apply(rule ccontr)

  1203 apply(insert linorder_le_less_linear[of i n])

  1204 apply(clarsimp simp:linorder_not_le)

  1205 apply(fastforce)

  1206 done

  1209 subsection {* Summation indexed over intervals *}

  1211 syntax

  1212   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)

  1213   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)

  1214   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)

  1215   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)

  1216 syntax (xsymbols)

  1217   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)

  1218   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)

  1219   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)

  1220   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)

  1221 syntax (HTML output)

  1222   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)

  1223   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)

  1224   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)

  1225   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)

  1226 syntax (latex_sum output)

  1227   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1228  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

  1229   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1230  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

  1231   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1232  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

  1233   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1234  ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

  1236 translations

  1237   "\<Sum>x=a..b. t" == "CONST setsum (%x. t) {a..b}"

  1238   "\<Sum>x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"

  1239   "\<Sum>i\<le>n. t" == "CONST setsum (\<lambda>i. t) {..n}"

  1240   "\<Sum>i<n. t" == "CONST setsum (\<lambda>i. t) {..<n}"

  1242 text{* The above introduces some pretty alternative syntaxes for

  1243 summation over intervals:

  1244 \begin{center}

  1245 \begin{tabular}{lll}

  1246 Old & New & \LaTeX\\

  1247 @{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\

  1248 @{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\

  1249 @{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\

  1250 @{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}

  1251 \end{tabular}

  1252 \end{center}

  1253 The left column shows the term before introduction of the new syntax,

  1254 the middle column shows the new (default) syntax, and the right column

  1255 shows a special syntax. The latter is only meaningful for latex output

  1256 and has to be activated explicitly by setting the print mode to

  1257 @{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in

  1258 antiquotations). It is not the default \LaTeX\ output because it only

  1259 works well with italic-style formulae, not tt-style.

  1261 Note that for uniformity on @{typ nat} it is better to use

  1262 @{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may

  1263 not provide all lemmas available for @{term"{m..<n}"} also in the

  1264 special form for @{term"{..<n}"}. *}

  1266 text{* This congruence rule should be used for sums over intervals as

  1267 the standard theorem @{text[source]setsum_cong} does not work well

  1268 with the simplifier who adds the unsimplified premise @{term"x:B"} to

  1269 the context. *}

  1271 lemma setsum_ivl_cong:

  1272  "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>

  1273  setsum f {a..<b} = setsum g {c..<d}"

  1274 by(rule setsum_cong, simp_all)

  1276 (* FIXME why are the following simp rules but the corresponding eqns

  1277 on intervals are not? *)

  1279 lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"

  1280 by (simp add:atMost_Suc add_ac)

  1282 lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"

  1283 by (simp add:lessThan_Suc add_ac)

  1285 lemma setsum_cl_ivl_Suc[simp]:

  1286   "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"

  1287 by (auto simp:add_ac atLeastAtMostSuc_conv)

  1289 lemma setsum_op_ivl_Suc[simp]:

  1290   "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"

  1291 by (auto simp:add_ac atLeastLessThanSuc)

  1292 (*

  1293 lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>

  1294     (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"

  1295 by (auto simp:add_ac atLeastAtMostSuc_conv)

  1296 *)

  1298 lemma setsum_head:

  1299   fixes n :: nat

  1300   assumes mn: "m <= n"

  1301   shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")

  1302 proof -

  1303   from mn

  1304   have "{m..n} = {m} \<union> {m<..n}"

  1305     by (auto intro: ivl_disj_un_singleton)

  1306   hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"

  1307     by (simp add: atLeast0LessThan)

  1308   also have "\<dots> = ?rhs" by simp

  1309   finally show ?thesis .

  1310 qed

  1312 lemma setsum_head_Suc:

  1313   "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"

  1314 by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)

  1316 lemma setsum_head_upt_Suc:

  1317   "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"

  1318 apply(insert setsum_head_Suc[of m "n - Suc 0" f])

  1319 apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)

  1320 done

  1322 lemma setsum_ub_add_nat: assumes "(m::nat) \<le> n + 1"

  1323   shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"

  1324 proof-

  1325   have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using `m \<le> n+1` by auto

  1326   thus ?thesis by (auto simp: ivl_disj_int setsum_Un_disjoint

  1327     atLeastSucAtMost_greaterThanAtMost)

  1328 qed

  1330 lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>

  1331   setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"

  1332 by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)

  1334 lemma setsum_diff_nat_ivl:

  1335 fixes f :: "nat \<Rightarrow> 'a::ab_group_add"

  1336 shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>

  1337   setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"

  1338 using setsum_add_nat_ivl [of m n p f,symmetric]

  1339 apply (simp add: add_ac)

  1340 done

  1342 lemma setsum_natinterval_difff:

  1343   fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"

  1344   shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =

  1345           (if m <= n then f m - f(n + 1) else 0)"

  1346 by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)

  1348 lemma setsum_restrict_set':

  1349   "finite A \<Longrightarrow> setsum f {x \<in> A. x \<in> B} = (\<Sum>x\<in>A. if x \<in> B then f x else 0)"

  1350   by (simp add: setsum_restrict_set [symmetric] Int_def)

  1352 lemma setsum_restrict_set'':

  1353   "finite A \<Longrightarrow> setsum f {x \<in> A. P x} = (\<Sum>x\<in>A. if P x  then f x else 0)"

  1354   by (simp add: setsum_restrict_set' [of A f "{x. P x}", simplified mem_Collect_eq])

  1356 lemma setsum_setsum_restrict:

  1357   "finite S \<Longrightarrow> finite T \<Longrightarrow>

  1358     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"

  1359   by (simp add: setsum_restrict_set'') (rule setsum_commute)

  1361 lemma setsum_image_gen: assumes fS: "finite S"

  1362   shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"

  1363 proof-

  1364   { fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto }

  1365   hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"

  1366     by simp

  1367   also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"

  1368     by (rule setsum_setsum_restrict[OF fS finite_imageI[OF fS]])

  1369   finally show ?thesis .

  1370 qed

  1372 lemma setsum_le_included:

  1373   fixes f :: "'a \<Rightarrow> 'b::ordered_comm_monoid_add"

  1374   assumes "finite s" "finite t"

  1375   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)"

  1376   shows "setsum f s \<le> setsum g t"

  1377 proof -

  1378   have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"

  1379   proof (rule setsum_mono)

  1380     fix y assume "y \<in> s"

  1381     with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto

  1382     with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")

  1383       using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]

  1384       by (auto intro!: setsum_mono2)

  1385   qed

  1386   also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i ` t)"

  1387     using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)

  1388   also have "... \<le> setsum g t"

  1389     using assms by (auto simp: setsum_image_gen[symmetric])

  1390   finally show ?thesis .

  1391 qed

  1393 lemma setsum_multicount_gen:

  1394   assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"

  1395   shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")

  1396 proof-

  1397   have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto

  1398   also have "\<dots> = ?r" unfolding setsum_setsum_restrict[OF assms(1-2)]

  1399     using assms(3) by auto

  1400   finally show ?thesis .

  1401 qed

  1403 lemma setsum_multicount:

  1404   assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"

  1405   shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")

  1406 proof-

  1407   have "?l = setsum (\<lambda>i. k) T" by(rule setsum_multicount_gen)(auto simp:assms)

  1408   also have "\<dots> = ?r" by(simp add: mult_commute)

  1409   finally show ?thesis by auto

  1410 qed

  1413 subsection{* Shifting bounds *}

  1415 lemma setsum_shift_bounds_nat_ivl:

  1416   "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"

  1417 by (induct "n", auto simp:atLeastLessThanSuc)

  1419 lemma setsum_shift_bounds_cl_nat_ivl:

  1420   "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"

  1421 apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])

  1422 apply (simp add:image_add_atLeastAtMost o_def)

  1423 done

  1425 corollary setsum_shift_bounds_cl_Suc_ivl:

  1426   "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"

  1427 by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])

  1429 corollary setsum_shift_bounds_Suc_ivl:

  1430   "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"

  1431 by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])

  1433 lemma setsum_shift_lb_Suc0_0:

  1434   "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"

  1435 by(simp add:setsum_head_Suc)

  1437 lemma setsum_shift_lb_Suc0_0_upt:

  1438   "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"

  1439 apply(cases k)apply simp

  1440 apply(simp add:setsum_head_upt_Suc)

  1441 done

  1443 lemma setsum_atMost_Suc_shift:

  1444   fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"

  1445   shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"

  1446 proof (induct n)

  1447   case 0 show ?case by simp

  1448 next

  1449   case (Suc n) note IH = this

  1450   have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))"

  1451     by (rule setsum_atMost_Suc)

  1452   also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"

  1453     by (rule IH)

  1454   also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) =

  1455              f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"

  1456     by (rule add_assoc)

  1457   also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))"

  1458     by (rule setsum_atMost_Suc [symmetric])

  1459   finally show ?case .

  1460 qed

  1463 subsection {* The formula for geometric sums *}

  1465 lemma geometric_sum:

  1466   assumes "x \<noteq> 1"

  1467   shows "(\<Sum>i=0..<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"

  1468 proof -

  1469   from assms obtain y where "y = x - 1" and "y \<noteq> 0" by simp_all

  1470   moreover have "(\<Sum>i=0..<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"

  1471   proof (induct n)

  1472     case 0 then show ?case by simp

  1473   next

  1474     case (Suc n)

  1475     moreover from Suc `y \<noteq> 0` have "(1 + y) ^ n = (y * inverse y) * (1 + y) ^ n" by simp

  1476     ultimately show ?case by (simp add: field_simps divide_inverse)

  1477   qed

  1478   ultimately show ?thesis by simp

  1479 qed

  1482 subsection {* The formula for arithmetic sums *}

  1484 lemma gauss_sum:

  1485   "(2::'a::comm_semiring_1)*(\<Sum>i\<in>{1..n}. of_nat i) =

  1486    of_nat n*((of_nat n)+1)"

  1487 proof (induct n)

  1488   case 0

  1489   show ?case by simp

  1490 next

  1491   case (Suc n)

  1492   then show ?case

  1493     by (simp add: algebra_simps add: one_add_one [symmetric] del: one_add_one)

  1494       (* FIXME: make numeral cancellation simprocs work for semirings *)

  1495 qed

  1497 theorem arith_series_general:

  1498   "(2::'a::comm_semiring_1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

  1499   of_nat n * (a + (a + of_nat(n - 1)*d))"

  1500 proof cases

  1501   assume ngt1: "n > 1"

  1502   let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"

  1503   have

  1504     "(\<Sum>i\<in>{..<n}. a+?I i*d) =

  1505      ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"

  1506     by (rule setsum_addf)

  1507   also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp

  1508   also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"

  1509     unfolding One_nat_def

  1510     by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)

  1511   also have "2*\<dots> = 2*?n*a + d*2*(\<Sum>i\<in>{1..<n}. ?I i)"

  1512     by (simp add: algebra_simps)

  1513   also from ngt1 have "{1..<n} = {1..n - 1}"

  1514     by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)

  1515   also from ngt1

  1516   have "2*?n*a + d*2*(\<Sum>i\<in>{1..n - 1}. ?I i) = (2*?n*a + d*?I (n - 1)*?I n)"

  1517     by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)

  1518        (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])

  1519   finally show ?thesis

  1520     unfolding mult_2 by (simp add: algebra_simps)

  1521 next

  1522   assume "\<not>(n > 1)"

  1523   hence "n = 1 \<or> n = 0" by auto

  1524   thus ?thesis by (auto simp: mult_2)

  1525 qed

  1527 lemma arith_series_nat:

  1528   "(2::nat) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"

  1529 proof -

  1530   have

  1531     "2 * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =

  1532     of_nat(n) * (a + (a + of_nat(n - 1)*d))"

  1533     by (rule arith_series_general)

  1534   thus ?thesis

  1535     unfolding One_nat_def by auto

  1536 qed

  1538 lemma arith_series_int:

  1539   "2 * (\<Sum>i\<in>{..<n}. a + int i * d) = int n * (a + (a + int(n - 1)*d))"

  1540   by (fact arith_series_general) (* FIXME: duplicate *)

  1542 lemma sum_diff_distrib:

  1543   fixes P::"nat\<Rightarrow>nat"

  1544   shows

  1545   "\<forall>x. Q x \<le> P x  \<Longrightarrow>

  1546   (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"

  1547 proof (induct n)

  1548   case 0 show ?case by simp

  1549 next

  1550   case (Suc n)

  1552   let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"

  1553   let ?rhs = "\<Sum>x<n. P x - Q x"

  1555   from Suc have "?lhs = ?rhs" by simp

  1556   moreover

  1557   from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp

  1558   moreover

  1559   from Suc have

  1560     "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"

  1561     by (subst diff_diff_left[symmetric],

  1562         subst diff_add_assoc2)

  1563        (auto simp: diff_add_assoc2 intro: setsum_mono)

  1564   ultimately

  1565   show ?case by simp

  1566 qed

  1568 subsection {* Products indexed over intervals *}

  1570 syntax

  1571   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)

  1572   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)

  1573   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)

  1574   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)

  1575 syntax (xsymbols)

  1576   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)

  1577   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)

  1578   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)

  1579   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)

  1580 syntax (HTML output)

  1581   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)

  1582   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)

  1583   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)

  1584   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)

  1585 syntax (latex_prod output)

  1586   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1587  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

  1588   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1589  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

  1590   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1591  ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

  1592   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1593  ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

  1595 translations

  1596   "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"

  1597   "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"

  1598   "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"

  1599   "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"

  1601 subsection {* Transfer setup *}

  1603 lemma transfer_nat_int_set_functions:

  1604     "{..n} = nat ` {0..int n}"

  1605     "{m..n} = nat ` {int m..int n}"  (* need all variants of these! *)

  1606   apply (auto simp add: image_def)

  1607   apply (rule_tac x = "int x" in bexI)

  1608   apply auto

  1609   apply (rule_tac x = "int x" in bexI)

  1610   apply auto

  1611   done

  1613 lemma transfer_nat_int_set_function_closures:

  1614     "x >= 0 \<Longrightarrow> nat_set {x..y}"

  1615   by (simp add: nat_set_def)

  1617 declare transfer_morphism_nat_int[transfer add

  1618   return: transfer_nat_int_set_functions

  1619     transfer_nat_int_set_function_closures

  1620 ]

  1622 lemma transfer_int_nat_set_functions:

  1623     "is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int ` {nat m..nat n}"

  1624   by (simp only: is_nat_def transfer_nat_int_set_functions

  1625     transfer_nat_int_set_function_closures

  1626     transfer_nat_int_set_return_embed nat_0_le

  1627     cong: transfer_nat_int_set_cong)

  1629 lemma transfer_int_nat_set_function_closures:

  1630     "is_nat x \<Longrightarrow> nat_set {x..y}"

  1631   by (simp only: transfer_nat_int_set_function_closures is_nat_def)

  1633 declare transfer_morphism_int_nat[transfer add

  1634   return: transfer_int_nat_set_functions

  1635     transfer_int_nat_set_function_closures

  1636 ]

  1638 end

author	blanchet
	Thu, 28 Nov 2013 13:58:11 +0100
changeset 55979	0cb8a2defb06
parent 55599	97a8ff4e4ac9
child 56427	0e8e4dc55866
permissions	-rw-r--r--