src/HOL/Set_Interval.thy
 author wenzelm Sun Nov 02 18:21:45 2014 +0100 (2014-11-02) changeset 58889 5b7a9633cfa8 parent 57514 bdc2c6b40bf2 child 58970 2f65dcd32a59 permissions -rw-r--r--
     1 (*  Title:      HOL/Set_Interval.thy

     2     Author:     Tobias Nipkow

     3     Author:     Clemens Ballarin

     4     Author:     Jeremy Avigad

     5

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

     7

     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 *)

    13

    14 section {* Set intervals *}

    15

    16 theory Set_Interval

    17 imports Lattices_Big Nat_Transfer

    18 begin

    19

    20 context ord

    21 begin

    22

    23 definition

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

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

    26

    27 definition

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

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

    30

    31 definition

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

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

    34

    35 definition

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

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

    38

    39 definition

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

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

    42

    43 definition

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

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

    46

    47 definition

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

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

    50

    51 definition

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

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

    54

    55 end

    56

    57

    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. *}

    61

    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)

    67

    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)

    73

    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)

    79

    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"

    85

    86

    87 subsection {* Various equivalences *}

    88

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

    90 by (simp add: lessThan_def)

    91

    92 lemma Compl_lessThan [simp]:

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

    94 apply (auto simp add: lessThan_def atLeast_def)

    95 done

    96

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

    98 by auto

    99

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

   101 by (simp add: greaterThan_def)

   102

   103 lemma Compl_greaterThan [simp]:

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

   105   by (auto simp add: greaterThan_def atMost_def)

   106

   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

   111

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

   113 by (simp add: atLeast_def)

   114

   115 lemma Compl_atLeast [simp]:

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

   117   by (auto simp add: lessThan_def atLeast_def)

   118

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

   120 by (simp add: atMost_def)

   121

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

   123 by (blast intro: order_antisym)

   124

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

   126   by auto

   127

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

   129   by auto

   130

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

   132

   133 lemma atLeast_subset_iff [iff]:

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

   135 by (blast intro: order_trans)

   136

   137 lemma atLeast_eq_iff [iff]:

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

   139 by (blast intro: order_antisym order_trans)

   140

   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

   146

   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

   153

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

   155 by (blast intro: order_trans)

   156

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

   158 by (blast intro: order_antisym order_trans)

   159

   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

   165

   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

   172

   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)

   177

   178 lemma (in linorder) Ici_subset_Ioi_iff: "{a ..} \<subseteq> {b <..} \<longleftrightarrow> b < a"

   179   by auto

   180

   181 lemma (in linorder) Iic_subset_Iio_iff: "{.. a} \<subseteq> {..< b} \<longleftrightarrow> a < b"

   182   by auto

   183

   184 subsection {*Two-sided intervals*}

   185

   186 context ord

   187 begin

   188

   189 lemma greaterThanLessThan_iff [simp]:

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

   191 by (simp add: greaterThanLessThan_def)

   192

   193 lemma atLeastLessThan_iff [simp]:

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

   195 by (simp add: atLeastLessThan_def)

   196

   197 lemma greaterThanAtMost_iff [simp]:

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

   199 by (simp add: greaterThanAtMost_def)

   200

   201 lemma atLeastAtMost_iff [simp]:

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

   203 by (simp add: atLeastAtMost_def)

   204

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

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

   207 alone. *}

   208

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

   210   by auto

   211

   212 end

   213

   214 subsubsection{* Emptyness, singletons, subset *}

   215

   216 context order

   217 begin

   218

   219 lemma atLeastatMost_empty[simp]:

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

   221 by(auto simp: atLeastAtMost_def atLeast_def atMost_def)

   222

   223 lemma atLeastatMost_empty_iff[simp]:

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

   225 by auto (blast intro: order_trans)

   226

   227 lemma atLeastatMost_empty_iff2[simp]:

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

   229 by auto (blast intro: order_trans)

   230

   231 lemma atLeastLessThan_empty[simp]:

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

   233 by(auto simp: atLeastLessThan_def)

   234

   235 lemma atLeastLessThan_empty_iff[simp]:

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

   237 by auto (blast intro: le_less_trans)

   238

   239 lemma atLeastLessThan_empty_iff2[simp]:

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

   241 by auto (blast intro: le_less_trans)

   242

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

   244 by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)

   245

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

   247 by auto (blast intro: less_le_trans)

   248

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

   250 by auto (blast intro: less_le_trans)

   251

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

   253 by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)

   254

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

   256 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)

   257

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

   259

   260 lemma atLeastatMost_subset_iff[simp]:

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

   262 unfolding atLeastAtMost_def atLeast_def atMost_def

   263 by (blast intro: order_trans)

   264

   265 lemma atLeastatMost_psubset_iff:

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

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

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

   269

   270 lemma Icc_eq_Icc[simp]:

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

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

   273

   274 lemma atLeastAtMost_singleton_iff[simp]:

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

   276 proof

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

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

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

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

   281 qed simp

   282

   283 lemma Icc_subset_Ici_iff[simp]:

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

   285 by(auto simp: subset_eq intro: order_trans)

   286

   287 lemma Icc_subset_Iic_iff[simp]:

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

   289 by(auto simp: subset_eq intro: order_trans)

   290

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

   292 by(auto simp: set_eq_iff)

   293

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

   295 by(auto simp: set_eq_iff)

   296

   297 lemmas not_empty_eq_Ici_eq_empty[simp] = not_Ici_eq_empty[symmetric]

   298 lemmas not_empty_eq_Iic_eq_empty[simp] = not_Iic_eq_empty[symmetric]

   299

   300 end

   301

   302 context no_top

   303 begin

   304

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

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

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

   308

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

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

   311

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

   313 using gt_ex[of h']

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

   315

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

   317 using gt_ex[of h']

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

   319

   320 end

   321

   322 context no_bot

   323 begin

   324

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

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

   327

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

   329 using lt_ex[of l']

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

   331

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

   333 using lt_ex[of l']

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

   335

   336 end

   337

   338

   339 context no_top

   340 begin

   341

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

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

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

   345

   346 lemmas not_Icc_eq_UNIV[simp] = not_UNIV_eq_Icc[symmetric]

   347

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

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

   350

   351 lemmas not_Iic_eq_UNIV[simp] = not_UNIV_eq_Iic[symmetric]

   352

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

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

   355

   356 lemmas not_Ici_eq_Icc[simp] = not_Icc_eq_Ici[symmetric]

   357

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

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

   360 using not_Ici_le_Iic[of l' h] by blast

   361

   362 lemmas not_Ici_eq_Iic[simp] = not_Iic_eq_Ici[symmetric]

   363

   364 end

   365

   366 context no_bot

   367 begin

   368

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

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

   371

   372 lemmas not_Ici_eq_UNIV[simp] = not_UNIV_eq_Ici[symmetric]

   373

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

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

   376

   377 lemmas not_Iic_eq_Icc[simp] = not_Icc_eq_Iic[symmetric]

   378

   379 end

   380

   381

   382 context dense_linorder

   383 begin

   384

   385 lemma greaterThanLessThan_empty_iff[simp]:

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

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

   388

   389 lemma greaterThanLessThan_empty_iff2[simp]:

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

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

   392

   393 lemma atLeastLessThan_subseteq_atLeastAtMost_iff:

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

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

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

   397

   398 lemma greaterThanAtMost_subseteq_atLeastAtMost_iff:

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

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

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

   402

   403 lemma greaterThanLessThan_subseteq_atLeastAtMost_iff:

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

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

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

   407

   408 lemma atLeastAtMost_subseteq_atLeastLessThan_iff:

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

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

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

   412

   413 lemma greaterThanAtMost_subseteq_atLeastLessThan_iff:

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

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

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

   417

   418 lemma greaterThanLessThan_subseteq_atLeastLessThan_iff:

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

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

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

   422

   423 lemma greaterThanLessThan_subseteq_greaterThanAtMost_iff:

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

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

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

   427

   428 end

   429

   430 context no_top

   431 begin

   432

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

   434   using gt_ex[of x] by auto

   435

   436 end

   437

   438 context no_bot

   439 begin

   440

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

   442   using lt_ex[of x] by auto

   443

   444 end

   445

   446 lemma (in linorder) atLeastLessThan_subset_iff:

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

   448 apply (auto simp:subset_eq Ball_def)

   449 apply(frule_tac x=a in spec)

   450 apply(erule_tac x=d in allE)

   451 apply (simp add: less_imp_le)

   452 done

   453

   454 lemma atLeastLessThan_inj:

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

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

   457   shows "a = c" "b = d"

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

   459

   460 lemma atLeastLessThan_eq_iff:

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

   462   assumes "a < b" "c < d"

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

   464   using atLeastLessThan_inj assms by auto

   465

   466 lemma (in linorder) Ioc_inj: "{a <.. b} = {c <.. d} \<longleftrightarrow> (b \<le> a \<and> d \<le> c) \<or> a = c \<and> b = d"

   467   by (metis eq_iff greaterThanAtMost_empty_iff2 greaterThanAtMost_iff le_cases not_le)

   468

   469 lemma (in order) Iio_Int_singleton: "{..<k} \<inter> {x} = (if x < k then {x} else {})"

   470   by auto

   471

   472 lemma (in linorder) Ioc_subset_iff: "{a<..b} \<subseteq> {c<..d} \<longleftrightarrow> (b \<le> a \<or> c \<le> a \<and> b \<le> d)"

   473   by (auto simp: subset_eq Ball_def) (metis less_le not_less)

   474

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

   476 by (auto simp: set_eq_iff intro: le_bot)

   477

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

   479 by (auto simp: set_eq_iff intro: top_le)

   480

   481 lemma (in bounded_lattice) atLeastAtMost_eq_UNIV_iff:

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

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

   484

   485 lemma Iio_eq_empty_iff: "{..< n::'a::{linorder, order_bot}} = {} \<longleftrightarrow> n = bot"

   486   by (auto simp: set_eq_iff not_less le_bot)

   487

   488 lemma Iio_eq_empty_iff_nat: "{..< n::nat} = {} \<longleftrightarrow> n = 0"

   489   by (simp add: Iio_eq_empty_iff bot_nat_def)

   490

   491

   492 subsection {* Infinite intervals *}

   493

   494 context dense_linorder

   495 begin

   496

   497 lemma infinite_Ioo:

   498   assumes "a < b"

   499   shows "\<not> finite {a<..<b}"

   500 proof

   501   assume fin: "finite {a<..<b}"

   502   moreover have ne: "{a<..<b} \<noteq> {}"

   503     using a < b by auto

   504   ultimately have "a < Max {a <..< b}" "Max {a <..< b} < b"

   505     using Max_in[of "{a <..< b}"] by auto

   506   then obtain x where "Max {a <..< b} < x" "x < b"

   507     using dense[of "Max {a<..<b}" b] by auto

   508   then have "x \<in> {a <..< b}"

   509     using a < Max {a <..< b} by auto

   510   then have "x \<le> Max {a <..< b}"

   511     using fin by auto

   512   with Max {a <..< b} < x show False by auto

   513 qed

   514

   515 lemma infinite_Icc: "a < b \<Longrightarrow> \<not> finite {a .. b}"

   516   using greaterThanLessThan_subseteq_atLeastAtMost_iff[of a b a b] infinite_Ioo[of a b]

   517   by (auto dest: finite_subset)

   518

   519 lemma infinite_Ico: "a < b \<Longrightarrow> \<not> finite {a ..< b}"

   520   using greaterThanLessThan_subseteq_atLeastLessThan_iff[of a b a b] infinite_Ioo[of a b]

   521   by (auto dest: finite_subset)

   522

   523 lemma infinite_Ioc: "a < b \<Longrightarrow> \<not> finite {a <.. b}"

   524   using greaterThanLessThan_subseteq_greaterThanAtMost_iff[of a b a b] infinite_Ioo[of a b]

   525   by (auto dest: finite_subset)

   526

   527 end

   528

   529 lemma infinite_Iio: "\<not> finite {..< a :: 'a :: {no_bot, linorder}}"

   530 proof

   531   assume "finite {..< a}"

   532   then have *: "\<And>x. x < a \<Longrightarrow> Min {..< a} \<le> x"

   533     by auto

   534   obtain x where "x < a"

   535     using lt_ex by auto

   536

   537   obtain y where "y < Min {..< a}"

   538     using lt_ex by auto

   539   also have "Min {..< a} \<le> x"

   540     using x < a by fact

   541   also note x < a

   542   finally have "Min {..< a} \<le> y"

   543     by fact

   544   with y < Min {..< a} show False by auto

   545 qed

   546

   547 lemma infinite_Iic: "\<not> finite {.. a :: 'a :: {no_bot, linorder}}"

   548   using infinite_Iio[of a] finite_subset[of "{..< a}" "{.. a}"]

   549   by (auto simp: subset_eq less_imp_le)

   550

   551 lemma infinite_Ioi: "\<not> finite {a :: 'a :: {no_top, linorder} <..}"

   552 proof

   553   assume "finite {a <..}"

   554   then have *: "\<And>x. a < x \<Longrightarrow> x \<le> Max {a <..}"

   555     by auto

   556

   557   obtain y where "Max {a <..} < y"

   558     using gt_ex by auto

   559

   560   obtain x where "a < x"

   561     using gt_ex by auto

   562   also then have "x \<le> Max {a <..}"

   563     by fact

   564   also note Max {a <..} < y

   565   finally have "y \<le> Max { a <..}"

   566     by fact

   567   with Max {a <..} < y show False by auto

   568 qed

   569

   570 lemma infinite_Ici: "\<not> finite {a :: 'a :: {no_top, linorder} ..}"

   571   using infinite_Ioi[of a] finite_subset[of "{a <..}" "{a ..}"]

   572   by (auto simp: subset_eq less_imp_le)

   573

   574 subsubsection {* Intersection *}

   575

   576 context linorder

   577 begin

   578

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

   580 by auto

   581

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

   583 by auto

   584

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

   586 by auto

   587

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

   589 by auto

   590

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

   592 by auto

   593

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

   595 by auto

   596

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

   598 by auto

   599

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

   601 by auto

   602

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

   604   by (auto simp: min_def)

   605

   606 lemma Ioc_disjoint: "{a<..b} \<inter> {c<..d} = {} \<longleftrightarrow> b \<le> a \<or> d \<le> c \<or> b \<le> c \<or> d \<le> a"

   607   using assms by auto

   608

   609 end

   610

   611 context complete_lattice

   612 begin

   613

   614 lemma

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

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

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

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

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

   620   by (auto intro!: Sup_eqI)

   621

   622 lemma

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

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

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

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

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

   628   by (auto intro!: Inf_eqI)

   629

   630 end

   631

   632 lemma

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

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

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

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

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

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

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

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

   641

   642 subsection {* Intervals of natural numbers *}

   643

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

   645

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

   647 by (simp add: lessThan_def)

   648

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

   650 by (simp add: lessThan_def less_Suc_eq, blast)

   651

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

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

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

   655

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

   657 proof safe

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

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

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

   661 qed

   662

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

   664 by (simp add: lessThan_def atMost_def less_Suc_eq_le)

   665

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

   667 by blast

   668

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

   670

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

   672 apply (simp add: greaterThan_def)

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

   674 done

   675

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

   677 apply (simp add: greaterThan_def)

   678 apply (auto elim: linorder_neqE)

   679 done

   680

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

   682 by blast

   683

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

   685

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

   687 by (unfold atLeast_def UNIV_def, simp)

   688

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

   690 apply (simp add: atLeast_def)

   691 apply (simp add: Suc_le_eq)

   692 apply (simp add: order_le_less, blast)

   693 done

   694

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

   696   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)

   697

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

   699 by blast

   700

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

   702

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

   704 by (simp add: atMost_def)

   705

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

   707 apply (simp add: atMost_def)

   708 apply (simp add: less_Suc_eq order_le_less, blast)

   709 done

   710

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

   712 by blast

   713

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

   715

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

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

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

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

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

   721 specific concept to a more general one. *}

   722

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

   724 by(simp add:lessThan_def atLeastLessThan_def)

   725

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

   727 by(simp add:atMost_def atLeastAtMost_def)

   728

   729 declare atLeast0LessThan[symmetric, code_unfold]

   730         atLeast0AtMost[symmetric, code_unfold]

   731

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

   733 by (simp add: atLeastLessThan_def)

   734

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

   736

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

   738 lemma atLeastLessThanSuc:

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

   740 by (auto simp add: atLeastLessThan_def)

   741

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

   743 by (auto simp add: atLeastLessThan_def)

   744 (*

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

   746 by (induct k, simp_all add: atLeastLessThanSuc)

   747

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

   749 by (auto simp add: atLeastLessThan_def)

   750 *)

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

   752   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)

   753

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

   755   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def

   756     greaterThanAtMost_def)

   757

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

   759   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def

   760     greaterThanLessThan_def)

   761

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

   763 by (auto simp add: atLeastAtMost_def)

   764

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

   766 by auto

   767

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

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

   770 lemma atLeastAtMostPlus1_int_conv:

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

   772   by (auto intro: set_eqI)

   773

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

   775   apply (induct k)

   776   apply (simp_all add: atLeastLessThanSuc)

   777   done

   778

   779 subsubsection {* Intervals and numerals *}

   780

   781 lemma lessThan_nat_numeral:  --{*Evaluation for specific numerals*}

   782   "lessThan (numeral k :: nat) = insert (pred_numeral k) (lessThan (pred_numeral k))"

   783   by (simp add: numeral_eq_Suc lessThan_Suc)

   784

   785 lemma atMost_nat_numeral:  --{*Evaluation for specific numerals*}

   786   "atMost (numeral k :: nat) = insert (numeral k) (atMost (pred_numeral k))"

   787   by (simp add: numeral_eq_Suc atMost_Suc)

   788

   789 lemma atLeastLessThan_nat_numeral:  --{*Evaluation for specific numerals*}

   790   "atLeastLessThan m (numeral k :: nat) =

   791      (if m \<le> (pred_numeral k) then insert (pred_numeral k) (atLeastLessThan m (pred_numeral k))

   792                  else {})"

   793   by (simp add: numeral_eq_Suc atLeastLessThanSuc)

   794

   795 subsubsection {* Image *}

   796

   797 lemma image_add_atLeastAtMost:

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

   799 proof

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

   801 next

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

   803   proof

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

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

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

   807     ultimately show "n : ?A" by blast

   808   qed

   809 qed

   810

   811 lemma image_add_atLeastLessThan:

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

   813 proof

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

   815 next

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

   817   proof

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

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

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

   821     ultimately show "n : ?A" by blast

   822   qed

   823 qed

   824

   825 corollary image_Suc_atLeastAtMost[simp]:

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

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

   828

   829 corollary image_Suc_atLeastLessThan[simp]:

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

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

   832

   833 lemma image_add_int_atLeastLessThan:

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

   835   apply (auto simp add: image_def)

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

   837   apply auto

   838   done

   839

   840 lemma image_minus_const_atLeastLessThan_nat:

   841   fixes c :: nat

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

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

   844     (is "_ = ?right")

   845 proof safe

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

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

   848   proof cases

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

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

   851   next

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

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

   854   qed

   855 qed auto

   856

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

   858   by (auto intro!: image_eqI [where x = "nat x" for x])

   859

   860 context ordered_ab_group_add

   861 begin

   862

   863 lemma

   864   fixes x :: 'a

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

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

   867 proof safe

   868   fix y assume "y < -x"

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

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

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

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

   873 next

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

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

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

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

   878 qed simp_all

   879

   880 lemma

   881   fixes x :: 'a

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

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

   884 proof -

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

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

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

   888     by (simp_all add: image_image

   889         del: image_uminus_greaterThan image_uminus_atLeast)

   890 qed

   891

   892 lemma

   893   fixes x :: 'a

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

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

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

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

   898   by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def

   899       greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)

   900 end

   901

   902 subsubsection {* Finiteness *}

   903

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

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

   906

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

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

   909

   910 lemma finite_greaterThanLessThan [iff]:

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

   912 by (simp add: greaterThanLessThan_def)

   913

   914 lemma finite_atLeastLessThan [iff]:

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

   916 by (simp add: atLeastLessThan_def)

   917

   918 lemma finite_greaterThanAtMost [iff]:

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

   920 by (simp add: greaterThanAtMost_def)

   921

   922 lemma finite_atLeastAtMost [iff]:

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

   924 by (simp add: atLeastAtMost_def)

   925

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

   927 lemma bounded_nat_set_is_finite:

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

   929 apply (rule finite_subset)

   930  apply (rule_tac [2] finite_lessThan, auto)

   931 done

   932

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

   934 lemma finite_nat_set_iff_bounded:

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

   936 proof

   937   assume f:?F  show ?B

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

   939 next

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

   941 qed

   942

   943 lemma finite_nat_set_iff_bounded_le:

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

   945 apply(simp add:finite_nat_set_iff_bounded)

   946 apply(blast dest:less_imp_le_nat le_imp_less_Suc)

   947 done

   948

   949 lemma finite_less_ub:

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

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

   952

   953

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

   955 subset is exactly that interval. *}

   956

   957 lemma subset_card_intvl_is_intvl:

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

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

   960 proof (cases "finite A")

   961   case True

   962   from this and assms show ?thesis

   963   proof (induct A rule: finite_linorder_max_induct)

   964     case empty thus ?case by auto

   965   next

   966     case (insert b A)

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

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

   969       by fastforce+

   970     with insert * show ?case by auto

   971   qed

   972 next

   973   case False

   974   with assms show ?thesis by simp

   975 qed

   976

   977

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

   979

   980 lemma UN_le_eq_Un0:

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

   982 proof

   983   show "?A <= ?B"

   984   proof

   985     fix x assume "x : ?A"

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

   987     show "x : ?B"

   988     proof(cases i)

   989       case 0 with i show ?thesis by simp

   990     next

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

   992     qed

   993   qed

   994 next

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

   996 qed

   997

   998 lemma UN_le_add_shift:

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

  1000 proof

  1001   show "?A <= ?B" by fastforce

  1002 next

  1003   show "?B <= ?A"

  1004   proof

  1005     fix x assume "x : ?B"

  1006     then obtain i where i: "i : {k..n+k}" "x : M(i)" by auto

  1007     hence "i-k\<le>n & x : M((i-k)+k)" by auto

  1008     thus "x : ?A" by blast

  1009   qed

  1010 qed

  1011

  1012 lemma UN_UN_finite_eq: "(\<Union>n::nat. \<Union>i\<in>{0..<n}. A i) = (\<Union>n. A n)"

  1013   by (auto simp add: atLeast0LessThan)

  1014

  1015 lemma UN_finite_subset: "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> C) \<Longrightarrow> (\<Union>n. A n) \<subseteq> C"

  1016   by (subst UN_UN_finite_eq [symmetric]) blast

  1017

  1018 lemma UN_finite2_subset:

  1019      "(!!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)"

  1020   apply (rule UN_finite_subset)

  1021   apply (subst UN_UN_finite_eq [symmetric, of B])

  1022   apply blast

  1023   done

  1024

  1025 lemma UN_finite2_eq:

  1026   "(!!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)"

  1027   apply (rule subset_antisym)

  1028    apply (rule UN_finite2_subset, blast)

  1029  apply (rule UN_finite2_subset [where k=k])

  1030  apply (force simp add: atLeastLessThan_add_Un [of 0])

  1031  done

  1032

  1033

  1034 subsubsection {* Cardinality *}

  1035

  1036 lemma card_lessThan [simp]: "card {..<u} = u"

  1037   by (induct u, simp_all add: lessThan_Suc)

  1038

  1039 lemma card_atMost [simp]: "card {..u} = Suc u"

  1040   by (simp add: lessThan_Suc_atMost [THEN sym])

  1041

  1042 lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"

  1043 proof -

  1044   have "{l..<u} = (%x. x + l)  {..<u-l}"

  1045     apply (auto simp add: image_def atLeastLessThan_def lessThan_def)

  1046     apply (rule_tac x = "x - l" in exI)

  1047     apply arith

  1048     done

  1049   then have "card {l..<u} = card {..<u-l}"

  1050     by (simp add: card_image inj_on_def)

  1051   then show ?thesis

  1052     by simp

  1053 qed

  1054

  1055 lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"

  1056   by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)

  1057

  1058 lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"

  1059   by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)

  1060

  1061 lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"

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

  1063

  1064 lemma ex_bij_betw_nat_finite:

  1065   "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"

  1066 apply(drule finite_imp_nat_seg_image_inj_on)

  1067 apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)

  1068 done

  1069

  1070 lemma ex_bij_betw_finite_nat:

  1071   "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"

  1072 by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)

  1073

  1074 lemma finite_same_card_bij:

  1075   "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> EX h. bij_betw h A B"

  1076 apply(drule ex_bij_betw_finite_nat)

  1077 apply(drule ex_bij_betw_nat_finite)

  1078 apply(auto intro!:bij_betw_trans)

  1079 done

  1080

  1081 lemma ex_bij_betw_nat_finite_1:

  1082   "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"

  1083 by (rule finite_same_card_bij) auto

  1084

  1085 lemma bij_betw_iff_card:

  1086   assumes FIN: "finite A" and FIN': "finite B"

  1087   shows BIJ: "(\<exists>f. bij_betw f A B) \<longleftrightarrow> (card A = card B)"

  1088 using assms

  1089 proof(auto simp add: bij_betw_same_card)

  1090   assume *: "card A = card B"

  1091   obtain f where "bij_betw f A {0 ..< card A}"

  1092   using FIN ex_bij_betw_finite_nat by blast

  1093   moreover obtain g where "bij_betw g {0 ..< card B} B"

  1094   using FIN' ex_bij_betw_nat_finite by blast

  1095   ultimately have "bij_betw (g o f) A B"

  1096   using * by (auto simp add: bij_betw_trans)

  1097   thus "(\<exists>f. bij_betw f A B)" by blast

  1098 qed

  1099

  1100 lemma inj_on_iff_card_le:

  1101   assumes FIN: "finite A" and FIN': "finite B"

  1102   shows "(\<exists>f. inj_on f A \<and> f  A \<le> B) = (card A \<le> card B)"

  1103 proof (safe intro!: card_inj_on_le)

  1104   assume *: "card A \<le> card B"

  1105   obtain f where 1: "inj_on f A" and 2: "f  A = {0 ..< card A}"

  1106   using FIN ex_bij_betw_finite_nat unfolding bij_betw_def by force

  1107   moreover obtain g where "inj_on g {0 ..< card B}" and 3: "g  {0 ..< card B} = B"

  1108   using FIN' ex_bij_betw_nat_finite unfolding bij_betw_def by force

  1109   ultimately have "inj_on g (f  A)" using subset_inj_on[of g _ "f  A"] * by force

  1110   hence "inj_on (g o f) A" using 1 comp_inj_on by blast

  1111   moreover

  1112   {have "{0 ..< card A} \<le> {0 ..< card B}" using * by force

  1113    with 2 have "f  A  \<le> {0 ..< card B}" by blast

  1114    hence "(g o f)  A \<le> B" unfolding comp_def using 3 by force

  1115   }

  1116   ultimately show "(\<exists>f. inj_on f A \<and> f  A \<le> B)" by blast

  1117 qed (insert assms, auto)

  1118

  1119 subsection {* Intervals of integers *}

  1120

  1121 lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"

  1122   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)

  1123

  1124 lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"

  1125   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)

  1126

  1127 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:

  1128     "{l+1..<u} = {l<..<u::int}"

  1129   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)

  1130

  1131 subsubsection {* Finiteness *}

  1132

  1133 lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>

  1134     {(0::int)..<u} = int  {..<nat u}"

  1135   apply (unfold image_def lessThan_def)

  1136   apply auto

  1137   apply (rule_tac x = "nat x" in exI)

  1138   apply (auto simp add: zless_nat_eq_int_zless [THEN sym])

  1139   done

  1140

  1141 lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"

  1142   apply (cases "0 \<le> u")

  1143   apply (subst image_atLeastZeroLessThan_int, assumption)

  1144   apply (rule finite_imageI)

  1145   apply auto

  1146   done

  1147

  1148 lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"

  1149   apply (subgoal_tac "(%x. x + l)  {0..<u-l} = {l..<u}")

  1150   apply (erule subst)

  1151   apply (rule finite_imageI)

  1152   apply (rule finite_atLeastZeroLessThan_int)

  1153   apply (rule image_add_int_atLeastLessThan)

  1154   done

  1155

  1156 lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"

  1157   by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)

  1158

  1159 lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"

  1160   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)

  1161

  1162 lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"

  1163   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)

  1164

  1165

  1166 subsubsection {* Cardinality *}

  1167

  1168 lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"

  1169   apply (cases "0 \<le> u")

  1170   apply (subst image_atLeastZeroLessThan_int, assumption)

  1171   apply (subst card_image)

  1172   apply (auto simp add: inj_on_def)

  1173   done

  1174

  1175 lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"

  1176   apply (subgoal_tac "card {l..<u} = card {0..<u-l}")

  1177   apply (erule ssubst, rule card_atLeastZeroLessThan_int)

  1178   apply (subgoal_tac "(%x. x + l)  {0..<u-l} = {l..<u}")

  1179   apply (erule subst)

  1180   apply (rule card_image)

  1181   apply (simp add: inj_on_def)

  1182   apply (rule image_add_int_atLeastLessThan)

  1183   done

  1184

  1185 lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"

  1186 apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])

  1187 apply (auto simp add: algebra_simps)

  1188 done

  1189

  1190 lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"

  1191 by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)

  1192

  1193 lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"

  1194 by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)

  1195

  1196 lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"

  1197 proof -

  1198   have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto

  1199   with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)

  1200 qed

  1201

  1202 lemma card_less:

  1203 assumes zero_in_M: "0 \<in> M"

  1204 shows "card {k \<in> M. k < Suc i} \<noteq> 0"

  1205 proof -

  1206   from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto

  1207   with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)

  1208 qed

  1209

  1210 lemma card_less_Suc2: "0 \<notin> M \<Longrightarrow> card {k. Suc k \<in> M \<and> k < i} = card {k \<in> M. k < Suc i}"

  1211 apply (rule card_bij_eq [of Suc _ _ "\<lambda>x. x - Suc 0"])

  1212 apply auto

  1213 apply (rule inj_on_diff_nat)

  1214 apply auto

  1215 apply (case_tac x)

  1216 apply auto

  1217 apply (case_tac xa)

  1218 apply auto

  1219 apply (case_tac xa)

  1220 apply auto

  1221 done

  1222

  1223 lemma card_less_Suc:

  1224   assumes zero_in_M: "0 \<in> M"

  1225     shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"

  1226 proof -

  1227   from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp

  1228   hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"

  1229     by (auto simp only: insert_Diff)

  1230   have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto

  1231   from finite_M_bounded_by_nat[of "\<lambda>x. x \<in> M" "Suc i"]

  1232   have "Suc (card {k. Suc k \<in> M \<and> k < i}) = card (insert 0 ({k \<in> M. k < Suc i} - {0}))"

  1233     apply (subst card_insert)

  1234     apply simp_all

  1235     apply (subst b)

  1236     apply (subst card_less_Suc2[symmetric])

  1237     apply simp_all

  1238     done

  1239   with c show ?thesis by simp

  1240 qed

  1241

  1242

  1243 subsection {*Lemmas useful with the summation operator setsum*}

  1244

  1245 text {* For examples, see Algebra/poly/UnivPoly2.thy *}

  1246

  1247 subsubsection {* Disjoint Unions *}

  1248

  1249 text {* Singletons and open intervals *}

  1250

  1251 lemma ivl_disj_un_singleton:

  1252   "{l::'a::linorder} Un {l<..} = {l..}"

  1253   "{..<u} Un {u::'a::linorder} = {..u}"

  1254   "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"

  1255   "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"

  1256   "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"

  1257   "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"

  1258 by auto

  1259

  1260 text {* One- and two-sided intervals *}

  1261

  1262 lemma ivl_disj_un_one:

  1263   "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"

  1264   "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"

  1265   "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"

  1266   "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"

  1267   "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"

  1268   "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"

  1269   "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"

  1270   "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"

  1271 by auto

  1272

  1273 text {* Two- and two-sided intervals *}

  1274

  1275 lemma ivl_disj_un_two:

  1276   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"

  1277   "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"

  1278   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"

  1279   "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"

  1280   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"

  1281   "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"

  1282   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"

  1283   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"

  1284 by auto

  1285

  1286 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two

  1287

  1288 subsubsection {* Disjoint Intersections *}

  1289

  1290 text {* One- and two-sided intervals *}

  1291

  1292 lemma ivl_disj_int_one:

  1293   "{..l::'a::order} Int {l<..<u} = {}"

  1294   "{..<l} Int {l..<u} = {}"

  1295   "{..l} Int {l<..u} = {}"

  1296   "{..<l} Int {l..u} = {}"

  1297   "{l<..u} Int {u<..} = {}"

  1298   "{l<..<u} Int {u..} = {}"

  1299   "{l..u} Int {u<..} = {}"

  1300   "{l..<u} Int {u..} = {}"

  1301   by auto

  1302

  1303 text {* Two- and two-sided intervals *}

  1304

  1305 lemma ivl_disj_int_two:

  1306   "{l::'a::order<..<m} Int {m..<u} = {}"

  1307   "{l<..m} Int {m<..<u} = {}"

  1308   "{l..<m} Int {m..<u} = {}"

  1309   "{l..m} Int {m<..<u} = {}"

  1310   "{l<..<m} Int {m..u} = {}"

  1311   "{l<..m} Int {m<..u} = {}"

  1312   "{l..<m} Int {m..u} = {}"

  1313   "{l..m} Int {m<..u} = {}"

  1314   by auto

  1315

  1316 lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two

  1317

  1318 subsubsection {* Some Differences *}

  1319

  1320 lemma ivl_diff[simp]:

  1321  "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"

  1322 by(auto)

  1323

  1324 lemma (in linorder) lessThan_minus_lessThan [simp]:

  1325   "{..< n} - {..< m} = {m ..< n}"

  1326   by auto

  1327

  1328

  1329 subsubsection {* Some Subset Conditions *}

  1330

  1331 lemma ivl_subset [simp]:

  1332  "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"

  1333 apply(auto simp:linorder_not_le)

  1334 apply(rule ccontr)

  1335 apply(insert linorder_le_less_linear[of i n])

  1336 apply(clarsimp simp:linorder_not_le)

  1337 apply(fastforce)

  1338 done

  1339

  1340

  1341 subsection {* Summation indexed over intervals *}

  1342

  1343 syntax

  1344   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)

  1345   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)

  1346   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)

  1347   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)

  1348 syntax (xsymbols)

  1349   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)

  1350   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)

  1351   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)

  1352   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)

  1353 syntax (HTML output)

  1354   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)

  1355   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)

  1356   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)

  1357   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)

  1358 syntax (latex_sum output)

  1359   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1360  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

  1361   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1362  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

  1363   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1364  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

  1365   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1366  ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

  1367

  1368 translations

  1369   "\<Sum>x=a..b. t" == "CONST setsum (%x. t) {a..b}"

  1370   "\<Sum>x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"

  1371   "\<Sum>i\<le>n. t" == "CONST setsum (\<lambda>i. t) {..n}"

  1372   "\<Sum>i<n. t" == "CONST setsum (\<lambda>i. t) {..<n}"

  1373

  1374 text{* The above introduces some pretty alternative syntaxes for

  1375 summation over intervals:

  1376 \begin{center}

  1377 \begin{tabular}{lll}

  1378 Old & New & \LaTeX\\

  1379 @{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\

  1380 @{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\

  1381 @{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\

  1382 @{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}

  1383 \end{tabular}

  1384 \end{center}

  1385 The left column shows the term before introduction of the new syntax,

  1386 the middle column shows the new (default) syntax, and the right column

  1387 shows a special syntax. The latter is only meaningful for latex output

  1388 and has to be activated explicitly by setting the print mode to

  1389 @{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in

  1390 antiquotations). It is not the default \LaTeX\ output because it only

  1391 works well with italic-style formulae, not tt-style.

  1392

  1393 Note that for uniformity on @{typ nat} it is better to use

  1394 @{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may

  1395 not provide all lemmas available for @{term"{m..<n}"} also in the

  1396 special form for @{term"{..<n}"}. *}

  1397

  1398 text{* This congruence rule should be used for sums over intervals as

  1399 the standard theorem @{text[source]setsum.cong} does not work well

  1400 with the simplifier who adds the unsimplified premise @{term"x:B"} to

  1401 the context. *}

  1402

  1403 lemma setsum_ivl_cong:

  1404  "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>

  1405  setsum f {a..<b} = setsum g {c..<d}"

  1406 by(rule setsum.cong, simp_all)

  1407

  1408 (* FIXME why are the following simp rules but the corresponding eqns

  1409 on intervals are not? *)

  1410

  1411 lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"

  1412 by (simp add:atMost_Suc ac_simps)

  1413

  1414 lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"

  1415 by (simp add:lessThan_Suc ac_simps)

  1416

  1417 lemma setsum_cl_ivl_Suc[simp]:

  1418   "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"

  1419 by (auto simp:ac_simps atLeastAtMostSuc_conv)

  1420

  1421 lemma setsum_op_ivl_Suc[simp]:

  1422   "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"

  1423 by (auto simp:ac_simps atLeastLessThanSuc)

  1424 (*

  1425 lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>

  1426     (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"

  1427 by (auto simp:ac_simps atLeastAtMostSuc_conv)

  1428 *)

  1429

  1430 lemma setsum_head:

  1431   fixes n :: nat

  1432   assumes mn: "m <= n"

  1433   shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")

  1434 proof -

  1435   from mn

  1436   have "{m..n} = {m} \<union> {m<..n}"

  1437     by (auto intro: ivl_disj_un_singleton)

  1438   hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"

  1439     by (simp add: atLeast0LessThan)

  1440   also have "\<dots> = ?rhs" by simp

  1441   finally show ?thesis .

  1442 qed

  1443

  1444 lemma setsum_head_Suc:

  1445   "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"

  1446 by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)

  1447

  1448 lemma setsum_head_upt_Suc:

  1449   "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"

  1450 apply(insert setsum_head_Suc[of m "n - Suc 0" f])

  1451 apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)

  1452 done

  1453

  1454 lemma setsum_ub_add_nat: assumes "(m::nat) \<le> n + 1"

  1455   shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"

  1456 proof-

  1457   have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using m \<le> n+1 by auto

  1458   thus ?thesis by (auto simp: ivl_disj_int setsum.union_disjoint

  1459     atLeastSucAtMost_greaterThanAtMost)

  1460 qed

  1461

  1462 lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>

  1463   setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"

  1464 by (simp add:setsum.union_disjoint[symmetric] ivl_disj_int ivl_disj_un)

  1465

  1466 lemma setsum_diff_nat_ivl:

  1467 fixes f :: "nat \<Rightarrow> 'a::ab_group_add"

  1468 shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>

  1469   setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"

  1470 using setsum_add_nat_ivl [of m n p f,symmetric]

  1471 apply (simp add: ac_simps)

  1472 done

  1473

  1474 lemma setsum_natinterval_difff:

  1475   fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"

  1476   shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =

  1477           (if m <= n then f m - f(n + 1) else 0)"

  1478 by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)

  1479

  1480 lemma setsum_nat_group: "(\<Sum>m<n::nat. setsum f {m * k ..< m*k + k}) = setsum f {..< n * k}"

  1481   apply (subgoal_tac "k = 0 | 0 < k", auto)

  1482   apply (induct "n")

  1483   apply (simp_all add: setsum_add_nat_ivl add.commute atLeast0LessThan[symmetric])

  1484   done

  1485

  1486 subsection{* Shifting bounds *}

  1487

  1488 lemma setsum_shift_bounds_nat_ivl:

  1489   "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"

  1490 by (induct "n", auto simp:atLeastLessThanSuc)

  1491

  1492 lemma setsum_shift_bounds_cl_nat_ivl:

  1493   "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"

  1494   by (rule setsum.reindex_bij_witness[where i="\<lambda>i. i + k" and j="\<lambda>i. i - k"]) auto

  1495

  1496 corollary setsum_shift_bounds_cl_Suc_ivl:

  1497   "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"

  1498 by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])

  1499

  1500 corollary setsum_shift_bounds_Suc_ivl:

  1501   "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"

  1502 by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])

  1503

  1504 lemma setsum_shift_lb_Suc0_0:

  1505   "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"

  1506 by(simp add:setsum_head_Suc)

  1507

  1508 lemma setsum_shift_lb_Suc0_0_upt:

  1509   "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"

  1510 apply(cases k)apply simp

  1511 apply(simp add:setsum_head_upt_Suc)

  1512 done

  1513

  1514 lemma setsum_atMost_Suc_shift:

  1515   fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"

  1516   shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"

  1517 proof (induct n)

  1518   case 0 show ?case by simp

  1519 next

  1520   case (Suc n) note IH = this

  1521   have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))"

  1522     by (rule setsum_atMost_Suc)

  1523   also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"

  1524     by (rule IH)

  1525   also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) =

  1526              f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"

  1527     by (rule add.assoc)

  1528   also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))"

  1529     by (rule setsum_atMost_Suc [symmetric])

  1530   finally show ?case .

  1531 qed

  1532

  1533 lemma setsum_last_plus: fixes n::nat shows "m <= n \<Longrightarrow> (\<Sum>i = m..n. f i) = f n + (\<Sum>i = m..<n. f i)"

  1534   by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost add.commute)

  1535

  1536 lemma setsum_Suc_diff:

  1537   fixes f :: "nat \<Rightarrow> 'a::ab_group_add"

  1538   assumes "m \<le> Suc n"

  1539   shows "(\<Sum>i = m..n. f(Suc i) - f i) = f (Suc n) - f m"

  1540 using assms by (induct n) (auto simp: le_Suc_eq)

  1541

  1542 lemma nested_setsum_swap:

  1543      "(\<Sum>i = 0..n. (\<Sum>j = 0..<i. a i j)) = (\<Sum>j = 0..<n. \<Sum>i = Suc j..n. a i j)"

  1544   by (induction n) (auto simp: setsum.distrib)

  1545

  1546 lemma nested_setsum_swap':

  1547      "(\<Sum>i\<le>n. (\<Sum>j<i. a i j)) = (\<Sum>j<n. \<Sum>i = Suc j..n. a i j)"

  1548   by (induction n) (auto simp: setsum.distrib)

  1549

  1550 lemma setsum_zero_power [simp]:

  1551   fixes c :: "nat \<Rightarrow> 'a::division_ring"

  1552   shows "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"

  1553 apply (cases "finite A")

  1554   by (induction A rule: finite_induct) auto

  1555

  1556 lemma setsum_zero_power' [simp]:

  1557   fixes c :: "nat \<Rightarrow> 'a::field"

  1558   shows "(\<Sum>i\<in>A. c i * 0^i / d i) = (if finite A \<and> 0 \<in> A then c 0 / d 0 else 0)"

  1559   using setsum_zero_power [of "\<lambda>i. c i / d i" A]

  1560   by auto

  1561

  1562

  1563 subsection {* The formula for geometric sums *}

  1564

  1565 lemma geometric_sum:

  1566   assumes "x \<noteq> 1"

  1567   shows "(\<Sum>i<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"

  1568 proof -

  1569   from assms obtain y where "y = x - 1" and "y \<noteq> 0" by simp_all

  1570   moreover have "(\<Sum>i<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"

  1571     by (induct n) (simp_all add: field_simps y \<noteq> 0)

  1572   ultimately show ?thesis by simp

  1573 qed

  1574

  1575

  1576 subsection {* The formula for arithmetic sums *}

  1577

  1578 lemma gauss_sum:

  1579   "(2::'a::comm_semiring_1)*(\<Sum>i\<in>{1..n}. of_nat i) = of_nat n*((of_nat n)+1)"

  1580 proof (induct n)

  1581   case 0

  1582   show ?case by simp

  1583 next

  1584   case (Suc n)

  1585   then show ?case

  1586     by (simp add: algebra_simps add: one_add_one [symmetric] del: one_add_one)

  1587       (* FIXME: make numeral cancellation simprocs work for semirings *)

  1588 qed

  1589

  1590 theorem arith_series_general:

  1591   "(2::'a::comm_semiring_1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

  1592   of_nat n * (a + (a + of_nat(n - 1)*d))"

  1593 proof cases

  1594   assume ngt1: "n > 1"

  1595   let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"

  1596   have

  1597     "(\<Sum>i\<in>{..<n}. a+?I i*d) =

  1598      ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"

  1599     by (rule setsum.distrib)

  1600   also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp

  1601   also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"

  1602     unfolding One_nat_def

  1603     by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt ac_simps)

  1604   also have "2*\<dots> = 2*?n*a + d*2*(\<Sum>i\<in>{1..<n}. ?I i)"

  1605     by (simp add: algebra_simps)

  1606   also from ngt1 have "{1..<n} = {1..n - 1}"

  1607     by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)

  1608   also from ngt1

  1609   have "2*?n*a + d*2*(\<Sum>i\<in>{1..n - 1}. ?I i) = (2*?n*a + d*?I (n - 1)*?I n)"

  1610     by (simp only: mult.assoc gauss_sum [of "n - 1"], unfold One_nat_def)

  1611       (simp add:  mult.commute trans [OF add.commute of_nat_Suc [symmetric]])

  1612   finally show ?thesis

  1613     unfolding mult_2 by (simp add: algebra_simps)

  1614 next

  1615   assume "\<not>(n > 1)"

  1616   hence "n = 1 \<or> n = 0" by auto

  1617   thus ?thesis by (auto simp: mult_2)

  1618 qed

  1619

  1620 lemma arith_series_nat:

  1621   "(2::nat) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"

  1622 proof -

  1623   have

  1624     "2 * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =

  1625     of_nat(n) * (a + (a + of_nat(n - 1)*d))"

  1626     by (rule arith_series_general)

  1627   thus ?thesis

  1628     unfolding One_nat_def by auto

  1629 qed

  1630

  1631 lemma arith_series_int:

  1632   "2 * (\<Sum>i\<in>{..<n}. a + int i * d) = int n * (a + (a + int(n - 1)*d))"

  1633   by (fact arith_series_general) (* FIXME: duplicate *)

  1634

  1635 lemma sum_diff_distrib:

  1636   fixes P::"nat\<Rightarrow>nat"

  1637   shows

  1638   "\<forall>x. Q x \<le> P x  \<Longrightarrow>

  1639   (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"

  1640 proof (induct n)

  1641   case 0 show ?case by simp

  1642 next

  1643   case (Suc n)

  1644

  1645   let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"

  1646   let ?rhs = "\<Sum>x<n. P x - Q x"

  1647

  1648   from Suc have "?lhs = ?rhs" by simp

  1649   moreover

  1650   from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp

  1651   moreover

  1652   from Suc have

  1653     "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"

  1654     by (subst diff_diff_left[symmetric],

  1655         subst diff_add_assoc2)

  1656        (auto simp: diff_add_assoc2 intro: setsum_mono)

  1657   ultimately

  1658   show ?case by simp

  1659 qed

  1660

  1661 lemma nat_diff_setsum_reindex: "(\<Sum>i<n. f (n - Suc i)) = (\<Sum>i<n. f i)"

  1662   by (rule setsum.reindex_bij_witness[where i="\<lambda>i. n - Suc i" and j="\<lambda>i. n - Suc i"]) auto

  1663

  1664 subsection {* Products indexed over intervals *}

  1665

  1666 syntax

  1667   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)

  1668   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)

  1669   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)

  1670   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)

  1671 syntax (xsymbols)

  1672   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)

  1673   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)

  1674   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)

  1675   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)

  1676 syntax (HTML output)

  1677   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)

  1678   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)

  1679   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)

  1680   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)

  1681 syntax (latex_prod output)

  1682   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1683  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

  1684   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1685  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

  1686   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1687  ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

  1688   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1689  ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

  1690

  1691 translations

  1692   "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"

  1693   "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"

  1694   "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"

  1695   "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"

  1696

  1697 subsection {* Transfer setup *}

  1698

  1699 lemma transfer_nat_int_set_functions:

  1700     "{..n} = nat  {0..int n}"

  1701     "{m..n} = nat  {int m..int n}"  (* need all variants of these! *)

  1702   apply (auto simp add: image_def)

  1703   apply (rule_tac x = "int x" in bexI)

  1704   apply auto

  1705   apply (rule_tac x = "int x" in bexI)

  1706   apply auto

  1707   done

  1708

  1709 lemma transfer_nat_int_set_function_closures:

  1710     "x >= 0 \<Longrightarrow> nat_set {x..y}"

  1711   by (simp add: nat_set_def)

  1712

  1713 declare transfer_morphism_nat_int[transfer add

  1714   return: transfer_nat_int_set_functions

  1715     transfer_nat_int_set_function_closures

  1716 ]

  1717

  1718 lemma transfer_int_nat_set_functions:

  1719     "is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int  {nat m..nat n}"

  1720   by (simp only: is_nat_def transfer_nat_int_set_functions

  1721     transfer_nat_int_set_function_closures

  1722     transfer_nat_int_set_return_embed nat_0_le

  1723     cong: transfer_nat_int_set_cong)

  1724

  1725 lemma transfer_int_nat_set_function_closures:

  1726     "is_nat x \<Longrightarrow> nat_set {x..y}"

  1727   by (simp only: transfer_nat_int_set_function_closures is_nat_def)

  1728

  1729 declare transfer_morphism_int_nat[transfer add

  1730   return: transfer_int_nat_set_functions

  1731     transfer_int_nat_set_function_closures

  1732 ]

  1733

  1734 lemma setprod_int_plus_eq: "setprod int {i..i+j} =  \<Prod>{int i..int (i+j)}"

  1735   by (induct j) (auto simp add: atLeastAtMostSuc_conv atLeastAtMostPlus1_int_conv)

  1736

  1737 lemma setprod_int_eq: "setprod int {i..j} =  \<Prod>{int i..int j}"

  1738 proof (cases "i \<le> j")

  1739   case True

  1740   then show ?thesis

  1741     by (metis Nat.le_iff_add setprod_int_plus_eq)

  1742 next

  1743   case False

  1744   then show ?thesis

  1745     by auto

  1746 qed

  1747

  1748 end
`