src/HOL/Set_Interval.thy
 author haftmann Fri Jun 19 07:53:35 2015 +0200 (2015-06-19) changeset 60517 f16e4fb20652 parent 60162 645058aa9d6f child 60586 1d31e3a50373 permissions -rw-r--r--
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
     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 lemma mono_image_least:

   492   assumes f_mono: "mono f" and f_img: "f  {m ..< n} = {m' ..< n'}" "m < n"

   493   shows "f m = m'"

   494 proof -

   495   from f_img have "{m' ..< n'} \<noteq> {}"

   496     by (metis atLeastLessThan_empty_iff image_is_empty)

   497   with f_img have "m' \<in> f  {m ..< n}" by auto

   498   then obtain k where "f k = m'" "m \<le> k" by auto

   499   moreover have "m' \<le> f m" using f_img by auto

   500   ultimately show "f m = m'"

   501     using f_mono by (auto elim: monoE[where x=m and y=k])

   502 qed

   503

   504

   505 subsection {* Infinite intervals *}

   506

   507 context dense_linorder

   508 begin

   509

   510 lemma infinite_Ioo:

   511   assumes "a < b"

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

   513 proof

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

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

   516     using a < b by auto

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

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

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

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

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

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

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

   524     using fin by auto

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

   526 qed

   527

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

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

   530   by (auto dest: finite_subset)

   531

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

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

   534   by (auto dest: finite_subset)

   535

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

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

   538   by (auto dest: finite_subset)

   539

   540 end

   541

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

   543 proof

   544   assume "finite {..< a}"

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

   546     by auto

   547   obtain x where "x < a"

   548     using lt_ex by auto

   549

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

   551     using lt_ex by auto

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

   553     using x < a by fact

   554   also note x < a

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

   556     by fact

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

   558 qed

   559

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

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

   562   by (auto simp: subset_eq less_imp_le)

   563

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

   565 proof

   566   assume "finite {a <..}"

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

   568     by auto

   569

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

   571     using gt_ex by auto

   572

   573   obtain x where "a < x"

   574     using gt_ex by auto

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

   576     by fact

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

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

   579     by fact

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

   581 qed

   582

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

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

   585   by (auto simp: subset_eq less_imp_le)

   586

   587 subsubsection {* Intersection *}

   588

   589 context linorder

   590 begin

   591

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

   593 by auto

   594

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

   596 by auto

   597

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

   599 by auto

   600

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

   602 by auto

   603

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

   605 by auto

   606

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

   608 by auto

   609

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

   611 by auto

   612

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

   614 by auto

   615

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

   617   by (auto simp: min_def)

   618

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

   620   using assms by auto

   621

   622 end

   623

   624 context complete_lattice

   625 begin

   626

   627 lemma

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

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

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

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

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

   633   by (auto intro!: Sup_eqI)

   634

   635 lemma

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

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

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

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

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

   641   by (auto intro!: Inf_eqI)

   642

   643 end

   644

   645 lemma

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

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

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

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

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

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

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

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

   654

   655 subsection {* Intervals of natural numbers *}

   656

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

   658

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

   660 by (simp add: lessThan_def)

   661

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

   663 by (simp add: lessThan_def less_Suc_eq, blast)

   664

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

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

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

   668

   669 lemma zero_notin_Suc_image: "0 \<notin> Suc  A"

   670   by auto

   671

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

   673   by (auto simp: image_iff less_Suc_eq_0_disj)

   674

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

   676 by (simp add: lessThan_def atMost_def less_Suc_eq_le)

   677

   678 lemma Iic_Suc_eq_insert_0: "{.. Suc n} = insert 0 (Suc  {.. n})"

   679   unfolding lessThan_Suc_atMost[symmetric] lessThan_Suc_eq_insert_0[of "Suc n"] ..

   680

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

   682 by blast

   683

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

   685

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

   687 apply (simp add: greaterThan_def)

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

   689 done

   690

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

   692 apply (simp add: greaterThan_def)

   693 apply (auto elim: linorder_neqE)

   694 done

   695

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

   697 by blast

   698

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

   700

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

   702 by (unfold atLeast_def UNIV_def, simp)

   703

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

   705 apply (simp add: atLeast_def)

   706 apply (simp add: Suc_le_eq)

   707 apply (simp add: order_le_less, blast)

   708 done

   709

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

   711   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)

   712

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

   714 by blast

   715

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

   717

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

   719 by (simp add: atMost_def)

   720

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

   722 apply (simp add: atMost_def)

   723 apply (simp add: less_Suc_eq order_le_less, blast)

   724 done

   725

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

   727 by blast

   728

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

   730

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

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

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

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

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

   736 specific concept to a more general one. *}

   737

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

   739 by(simp add:lessThan_def atLeastLessThan_def)

   740

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

   742 by(simp add:atMost_def atLeastAtMost_def)

   743

   744 declare atLeast0LessThan[symmetric, code_unfold]

   745         atLeast0AtMost[symmetric, code_unfold]

   746

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

   748 by (simp add: atLeastLessThan_def)

   749

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

   751

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

   753 lemma atLeastLessThanSuc:

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

   755 by (auto simp add: atLeastLessThan_def)

   756

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

   758 by (auto simp add: atLeastLessThan_def)

   759 (*

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

   761 by (induct k, simp_all add: atLeastLessThanSuc)

   762

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

   764 by (auto simp add: atLeastLessThan_def)

   765 *)

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

   767   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)

   768

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

   770   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def

   771     greaterThanAtMost_def)

   772

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

   774   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def

   775     greaterThanLessThan_def)

   776

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

   778 by (auto simp add: atLeastAtMost_def)

   779

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

   781 by auto

   782

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

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

   785 lemma atLeastAtMostPlus1_int_conv:

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

   787   by (auto intro: set_eqI)

   788

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

   790   apply (induct k)

   791   apply (simp_all add: atLeastLessThanSuc)

   792   done

   793

   794 subsubsection {* Intervals and numerals *}

   795

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

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

   798   by (simp add: numeral_eq_Suc lessThan_Suc)

   799

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

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

   802   by (simp add: numeral_eq_Suc atMost_Suc)

   803

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

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

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

   807                  else {})"

   808   by (simp add: numeral_eq_Suc atLeastLessThanSuc)

   809

   810 subsubsection {* Image *}

   811

   812 lemma image_add_atLeastAtMost:

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

   814 proof

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

   816 next

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

   818   proof

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

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

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

   822     ultimately show "n : ?A" by blast

   823   qed

   824 qed

   825

   826 lemma image_add_atLeastLessThan:

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

   828 proof

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

   830 next

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

   832   proof

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

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

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

   836     ultimately show "n : ?A" by blast

   837   qed

   838 qed

   839

   840 corollary image_Suc_atLeastAtMost[simp]:

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

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

   843

   844 corollary image_Suc_atLeastLessThan[simp]:

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

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

   847

   848 lemma image_add_int_atLeastLessThan:

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

   850   apply (auto simp add: image_def)

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

   852   apply auto

   853   done

   854

   855 lemma image_minus_const_atLeastLessThan_nat:

   856   fixes c :: nat

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

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

   859     (is "_ = ?right")

   860 proof safe

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

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

   863   proof cases

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

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

   866   next

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

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

   869   qed

   870 qed auto

   871

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

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

   874

   875 context ordered_ab_group_add

   876 begin

   877

   878 lemma

   879   fixes x :: 'a

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

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

   882 proof safe

   883   fix y assume "y < -x"

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

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

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

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

   888 next

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

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

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

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

   893 qed simp_all

   894

   895 lemma

   896   fixes x :: 'a

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

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

   899 proof -

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

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

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

   903     by (simp_all add: image_image

   904         del: image_uminus_greaterThan image_uminus_atLeast)

   905 qed

   906

   907 lemma

   908   fixes x :: 'a

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

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

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

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

   913   by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def

   914       greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)

   915 end

   916

   917 subsubsection {* Finiteness *}

   918

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

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

   921

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

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

   924

   925 lemma finite_greaterThanLessThan [iff]:

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

   927 by (simp add: greaterThanLessThan_def)

   928

   929 lemma finite_atLeastLessThan [iff]:

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

   931 by (simp add: atLeastLessThan_def)

   932

   933 lemma finite_greaterThanAtMost [iff]:

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

   935 by (simp add: greaterThanAtMost_def)

   936

   937 lemma finite_atLeastAtMost [iff]:

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

   939 by (simp add: atLeastAtMost_def)

   940

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

   942 lemma bounded_nat_set_is_finite:

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

   944 apply (rule finite_subset)

   945  apply (rule_tac [2] finite_lessThan, auto)

   946 done

   947

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

   949 lemma finite_nat_set_iff_bounded:

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

   951 proof

   952   assume f:?F  show ?B

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

   954 next

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

   956 qed

   957

   958 lemma finite_nat_set_iff_bounded_le:

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

   960 apply(simp add:finite_nat_set_iff_bounded)

   961 apply(blast dest:less_imp_le_nat le_imp_less_Suc)

   962 done

   963

   964 lemma finite_less_ub:

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

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

   967

   968

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

   970 subset is exactly that interval. *}

   971

   972 lemma subset_card_intvl_is_intvl:

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

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

   975 proof (cases "finite A")

   976   case True

   977   from this and assms show ?thesis

   978   proof (induct A rule: finite_linorder_max_induct)

   979     case empty thus ?case by auto

   980   next

   981     case (insert b A)

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

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

   984       by fastforce+

   985     with insert * show ?case by auto

   986   qed

   987 next

   988   case False

   989   with assms show ?thesis by simp

   990 qed

   991

   992

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

   994

   995 lemma UN_le_eq_Un0:

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

   997 proof

   998   show "?A <= ?B"

   999   proof

  1000     fix x assume "x : ?A"

  1001     then obtain i where i: "i\<le>n" "x : M i" by auto

  1002     show "x : ?B"

  1003     proof(cases i)

  1004       case 0 with i show ?thesis by simp

  1005     next

  1006       case (Suc j) with i show ?thesis by auto

  1007     qed

  1008   qed

  1009 next

  1010   show "?B <= ?A" by auto

  1011 qed

  1012

  1013 lemma UN_le_add_shift:

  1014   "(\<Union>i\<le>n::nat. M(i+k)) = (\<Union>i\<in>{k..n+k}. M i)" (is "?A = ?B")

  1015 proof

  1016   show "?A <= ?B" by fastforce

  1017 next

  1018   show "?B <= ?A"

  1019   proof

  1020     fix x assume "x : ?B"

  1021     then obtain i where i: "i : {k..n+k}" "x : M(i)" by auto

  1022     hence "i-k\<le>n & x : M((i-k)+k)" by auto

  1023     thus "x : ?A" by blast

  1024   qed

  1025 qed

  1026

  1027 lemma UN_UN_finite_eq: "(\<Union>n::nat. \<Union>i\<in>{0..<n}. A i) = (\<Union>n. A n)"

  1028   by (auto simp add: atLeast0LessThan)

  1029

  1030 lemma UN_finite_subset: "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> C) \<Longrightarrow> (\<Union>n. A n) \<subseteq> C"

  1031   by (subst UN_UN_finite_eq [symmetric]) blast

  1032

  1033 lemma UN_finite2_subset:

  1034      "(!!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)"

  1035   apply (rule UN_finite_subset)

  1036   apply (subst UN_UN_finite_eq [symmetric, of B])

  1037   apply blast

  1038   done

  1039

  1040 lemma UN_finite2_eq:

  1041   "(!!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)"

  1042   apply (rule subset_antisym)

  1043    apply (rule UN_finite2_subset, blast)

  1044  apply (rule UN_finite2_subset [where k=k])

  1045  apply (force simp add: atLeastLessThan_add_Un [of 0])

  1046  done

  1047

  1048

  1049 subsubsection {* Cardinality *}

  1050

  1051 lemma card_lessThan [simp]: "card {..<u} = u"

  1052   by (induct u, simp_all add: lessThan_Suc)

  1053

  1054 lemma card_atMost [simp]: "card {..u} = Suc u"

  1055   by (simp add: lessThan_Suc_atMost [THEN sym])

  1056

  1057 lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"

  1058 proof -

  1059   have "{l..<u} = (%x. x + l)  {..<u-l}"

  1060     apply (auto simp add: image_def atLeastLessThan_def lessThan_def)

  1061     apply (rule_tac x = "x - l" in exI)

  1062     apply arith

  1063     done

  1064   then have "card {l..<u} = card {..<u-l}"

  1065     by (simp add: card_image inj_on_def)

  1066   then show ?thesis

  1067     by simp

  1068 qed

  1069

  1070 lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"

  1071   by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)

  1072

  1073 lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"

  1074   by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)

  1075

  1076 lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"

  1077   by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)

  1078

  1079 lemma ex_bij_betw_nat_finite:

  1080   "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"

  1081 apply(drule finite_imp_nat_seg_image_inj_on)

  1082 apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)

  1083 done

  1084

  1085 lemma ex_bij_betw_finite_nat:

  1086   "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"

  1087 by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)

  1088

  1089 lemma finite_same_card_bij:

  1090   "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> EX h. bij_betw h A B"

  1091 apply(drule ex_bij_betw_finite_nat)

  1092 apply(drule ex_bij_betw_nat_finite)

  1093 apply(auto intro!:bij_betw_trans)

  1094 done

  1095

  1096 lemma ex_bij_betw_nat_finite_1:

  1097   "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"

  1098 by (rule finite_same_card_bij) auto

  1099

  1100 lemma bij_betw_iff_card:

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

  1102   shows BIJ: "(\<exists>f. bij_betw f A B) \<longleftrightarrow> (card A = card B)"

  1103 using assms

  1104 proof(auto simp add: bij_betw_same_card)

  1105   assume *: "card A = card B"

  1106   obtain f where "bij_betw f A {0 ..< card A}"

  1107   using FIN ex_bij_betw_finite_nat by blast

  1108   moreover obtain g where "bij_betw g {0 ..< card B} B"

  1109   using FIN' ex_bij_betw_nat_finite by blast

  1110   ultimately have "bij_betw (g o f) A B"

  1111   using * by (auto simp add: bij_betw_trans)

  1112   thus "(\<exists>f. bij_betw f A B)" by blast

  1113 qed

  1114

  1115 lemma inj_on_iff_card_le:

  1116   assumes FIN: "finite A" and FIN': "finite B"

  1117   shows "(\<exists>f. inj_on f A \<and> f  A \<le> B) = (card A \<le> card B)"

  1118 proof (safe intro!: card_inj_on_le)

  1119   assume *: "card A \<le> card B"

  1120   obtain f where 1: "inj_on f A" and 2: "f  A = {0 ..< card A}"

  1121   using FIN ex_bij_betw_finite_nat unfolding bij_betw_def by force

  1122   moreover obtain g where "inj_on g {0 ..< card B}" and 3: "g  {0 ..< card B} = B"

  1123   using FIN' ex_bij_betw_nat_finite unfolding bij_betw_def by force

  1124   ultimately have "inj_on g (f  A)" using subset_inj_on[of g _ "f  A"] * by force

  1125   hence "inj_on (g o f) A" using 1 comp_inj_on by blast

  1126   moreover

  1127   {have "{0 ..< card A} \<le> {0 ..< card B}" using * by force

  1128    with 2 have "f  A  \<le> {0 ..< card B}" by blast

  1129    hence "(g o f)  A \<le> B" unfolding comp_def using 3 by force

  1130   }

  1131   ultimately show "(\<exists>f. inj_on f A \<and> f  A \<le> B)" by blast

  1132 qed (insert assms, auto)

  1133

  1134 subsection {* Intervals of integers *}

  1135

  1136 lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"

  1137   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)

  1138

  1139 lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"

  1140   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)

  1141

  1142 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:

  1143     "{l+1..<u} = {l<..<u::int}"

  1144   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)

  1145

  1146 subsubsection {* Finiteness *}

  1147

  1148 lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>

  1149     {(0::int)..<u} = int  {..<nat u}"

  1150   apply (unfold image_def lessThan_def)

  1151   apply auto

  1152   apply (rule_tac x = "nat x" in exI)

  1153   apply (auto simp add: zless_nat_eq_int_zless [THEN sym])

  1154   done

  1155

  1156 lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"

  1157   apply (cases "0 \<le> u")

  1158   apply (subst image_atLeastZeroLessThan_int, assumption)

  1159   apply (rule finite_imageI)

  1160   apply auto

  1161   done

  1162

  1163 lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"

  1164   apply (subgoal_tac "(%x. x + l)  {0..<u-l} = {l..<u}")

  1165   apply (erule subst)

  1166   apply (rule finite_imageI)

  1167   apply (rule finite_atLeastZeroLessThan_int)

  1168   apply (rule image_add_int_atLeastLessThan)

  1169   done

  1170

  1171 lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"

  1172   by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)

  1173

  1174 lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"

  1175   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)

  1176

  1177 lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"

  1178   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)

  1179

  1180

  1181 subsubsection {* Cardinality *}

  1182

  1183 lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"

  1184   apply (cases "0 \<le> u")

  1185   apply (subst image_atLeastZeroLessThan_int, assumption)

  1186   apply (subst card_image)

  1187   apply (auto simp add: inj_on_def)

  1188   done

  1189

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

  1191   apply (subgoal_tac "card {l..<u} = card {0..<u-l}")

  1192   apply (erule ssubst, rule card_atLeastZeroLessThan_int)

  1193   apply (subgoal_tac "(%x. x + l)  {0..<u-l} = {l..<u}")

  1194   apply (erule subst)

  1195   apply (rule card_image)

  1196   apply (simp add: inj_on_def)

  1197   apply (rule image_add_int_atLeastLessThan)

  1198   done

  1199

  1200 lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"

  1201 apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])

  1202 apply (auto simp add: algebra_simps)

  1203 done

  1204

  1205 lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"

  1206 by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)

  1207

  1208 lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"

  1209 by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)

  1210

  1211 lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"

  1212 proof -

  1213   have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto

  1214   with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)

  1215 qed

  1216

  1217 lemma card_less:

  1218 assumes zero_in_M: "0 \<in> M"

  1219 shows "card {k \<in> M. k < Suc i} \<noteq> 0"

  1220 proof -

  1221   from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto

  1222   with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)

  1223 qed

  1224

  1225 lemma card_less_Suc2: "0 \<notin> M \<Longrightarrow> card {k. Suc k \<in> M \<and> k < i} = card {k \<in> M. k < Suc i}"

  1226 apply (rule card_bij_eq [of Suc _ _ "\<lambda>x. x - Suc 0"])

  1227 apply auto

  1228 apply (rule inj_on_diff_nat)

  1229 apply auto

  1230 apply (case_tac x)

  1231 apply auto

  1232 apply (case_tac xa)

  1233 apply auto

  1234 apply (case_tac xa)

  1235 apply auto

  1236 done

  1237

  1238 lemma card_less_Suc:

  1239   assumes zero_in_M: "0 \<in> M"

  1240     shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"

  1241 proof -

  1242   from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp

  1243   hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"

  1244     by (auto simp only: insert_Diff)

  1245   have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto

  1246   from finite_M_bounded_by_nat[of "\<lambda>x. x \<in> M" "Suc i"]

  1247   have "Suc (card {k. Suc k \<in> M \<and> k < i}) = card (insert 0 ({k \<in> M. k < Suc i} - {0}))"

  1248     apply (subst card_insert)

  1249     apply simp_all

  1250     apply (subst b)

  1251     apply (subst card_less_Suc2[symmetric])

  1252     apply simp_all

  1253     done

  1254   with c show ?thesis by simp

  1255 qed

  1256

  1257

  1258 subsection {*Lemmas useful with the summation operator setsum*}

  1259

  1260 text {* For examples, see Algebra/poly/UnivPoly2.thy *}

  1261

  1262 subsubsection {* Disjoint Unions *}

  1263

  1264 text {* Singletons and open intervals *}

  1265

  1266 lemma ivl_disj_un_singleton:

  1267   "{l::'a::linorder} Un {l<..} = {l..}"

  1268   "{..<u} Un {u::'a::linorder} = {..u}"

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

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

  1271   "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"

  1272   "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"

  1273 by auto

  1274

  1275 text {* One- and two-sided intervals *}

  1276

  1277 lemma ivl_disj_un_one:

  1278   "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"

  1279   "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"

  1280   "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"

  1281   "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"

  1282   "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"

  1283   "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"

  1284   "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"

  1285   "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"

  1286 by auto

  1287

  1288 text {* Two- and two-sided intervals *}

  1289

  1290 lemma ivl_disj_un_two:

  1291   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"

  1292   "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"

  1293   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"

  1294   "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"

  1295   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"

  1296   "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"

  1297   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"

  1298   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"

  1299 by auto

  1300

  1301 lemma ivl_disj_un_two_touch:

  1302   "[| (l::'a::linorder) < m; m < u |] ==> {l<..m} Un {m..<u} = {l<..<u}"

  1303   "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m..<u} = {l..<u}"

  1304   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..m} Un {m..u} = {l<..u}"

  1305   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m..u} = {l..u}"

  1306 by auto

  1307

  1308 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two ivl_disj_un_two_touch

  1309

  1310 subsubsection {* Disjoint Intersections *}

  1311

  1312 text {* One- and two-sided intervals *}

  1313

  1314 lemma ivl_disj_int_one:

  1315   "{..l::'a::order} Int {l<..<u} = {}"

  1316   "{..<l} Int {l..<u} = {}"

  1317   "{..l} Int {l<..u} = {}"

  1318   "{..<l} Int {l..u} = {}"

  1319   "{l<..u} Int {u<..} = {}"

  1320   "{l<..<u} Int {u..} = {}"

  1321   "{l..u} Int {u<..} = {}"

  1322   "{l..<u} Int {u..} = {}"

  1323   by auto

  1324

  1325 text {* Two- and two-sided intervals *}

  1326

  1327 lemma ivl_disj_int_two:

  1328   "{l::'a::order<..<m} Int {m..<u} = {}"

  1329   "{l<..m} Int {m<..<u} = {}"

  1330   "{l..<m} Int {m..<u} = {}"

  1331   "{l..m} Int {m<..<u} = {}"

  1332   "{l<..<m} Int {m..u} = {}"

  1333   "{l<..m} Int {m<..u} = {}"

  1334   "{l..<m} Int {m..u} = {}"

  1335   "{l..m} Int {m<..u} = {}"

  1336   by auto

  1337

  1338 lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two

  1339

  1340 subsubsection {* Some Differences *}

  1341

  1342 lemma ivl_diff[simp]:

  1343  "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"

  1344 by(auto)

  1345

  1346 lemma (in linorder) lessThan_minus_lessThan [simp]:

  1347   "{..< n} - {..< m} = {m ..< n}"

  1348   by auto

  1349

  1350

  1351 subsubsection {* Some Subset Conditions *}

  1352

  1353 lemma ivl_subset [simp]:

  1354  "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"

  1355 apply(auto simp:linorder_not_le)

  1356 apply(rule ccontr)

  1357 apply(insert linorder_le_less_linear[of i n])

  1358 apply(clarsimp simp:linorder_not_le)

  1359 apply(fastforce)

  1360 done

  1361

  1362

  1363 subsection {* Summation indexed over intervals *}

  1364

  1365 syntax

  1366   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)

  1367   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)

  1368   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)

  1369   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)

  1370 syntax (xsymbols)

  1371   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)

  1372   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)

  1373   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)

  1374   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)

  1375 syntax (HTML output)

  1376   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)

  1377   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)

  1378   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)

  1379   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)

  1380 syntax (latex_sum output)

  1381   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1382  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

  1383   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1384  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

  1385   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1386  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

  1387   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1388  ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

  1389

  1390 translations

  1391   "\<Sum>x=a..b. t" == "CONST setsum (%x. t) {a..b}"

  1392   "\<Sum>x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"

  1393   "\<Sum>i\<le>n. t" == "CONST setsum (\<lambda>i. t) {..n}"

  1394   "\<Sum>i<n. t" == "CONST setsum (\<lambda>i. t) {..<n}"

  1395

  1396 text{* The above introduces some pretty alternative syntaxes for

  1397 summation over intervals:

  1398 \begin{center}

  1399 \begin{tabular}{lll}

  1400 Old & New & \LaTeX\\

  1401 @{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\

  1402 @{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\

  1403 @{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\

  1404 @{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}

  1405 \end{tabular}

  1406 \end{center}

  1407 The left column shows the term before introduction of the new syntax,

  1408 the middle column shows the new (default) syntax, and the right column

  1409 shows a special syntax. The latter is only meaningful for latex output

  1410 and has to be activated explicitly by setting the print mode to

  1411 @{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in

  1412 antiquotations). It is not the default \LaTeX\ output because it only

  1413 works well with italic-style formulae, not tt-style.

  1414

  1415 Note that for uniformity on @{typ nat} it is better to use

  1416 @{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may

  1417 not provide all lemmas available for @{term"{m..<n}"} also in the

  1418 special form for @{term"{..<n}"}. *}

  1419

  1420 text{* This congruence rule should be used for sums over intervals as

  1421 the standard theorem @{text[source]setsum.cong} does not work well

  1422 with the simplifier who adds the unsimplified premise @{term"x:B"} to

  1423 the context. *}

  1424

  1425 lemma setsum_ivl_cong:

  1426  "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>

  1427  setsum f {a..<b} = setsum g {c..<d}"

  1428 by(rule setsum.cong, simp_all)

  1429

  1430 (* FIXME why are the following simp rules but the corresponding eqns

  1431 on intervals are not? *)

  1432

  1433 lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"

  1434 by (simp add:atMost_Suc ac_simps)

  1435

  1436 lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"

  1437 by (simp add:lessThan_Suc ac_simps)

  1438

  1439 lemma setsum_cl_ivl_Suc[simp]:

  1440   "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"

  1441 by (auto simp:ac_simps atLeastAtMostSuc_conv)

  1442

  1443 lemma setsum_op_ivl_Suc[simp]:

  1444   "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"

  1445 by (auto simp:ac_simps atLeastLessThanSuc)

  1446 (*

  1447 lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>

  1448     (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"

  1449 by (auto simp:ac_simps atLeastAtMostSuc_conv)

  1450 *)

  1451

  1452 lemma setsum_head:

  1453   fixes n :: nat

  1454   assumes mn: "m <= n"

  1455   shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")

  1456 proof -

  1457   from mn

  1458   have "{m..n} = {m} \<union> {m<..n}"

  1459     by (auto intro: ivl_disj_un_singleton)

  1460   hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"

  1461     by (simp add: atLeast0LessThan)

  1462   also have "\<dots> = ?rhs" by simp

  1463   finally show ?thesis .

  1464 qed

  1465

  1466 lemma setsum_head_Suc:

  1467   "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"

  1468 by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)

  1469

  1470 lemma setsum_head_upt_Suc:

  1471   "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"

  1472 apply(insert setsum_head_Suc[of m "n - Suc 0" f])

  1473 apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)

  1474 done

  1475

  1476 lemma setsum_ub_add_nat: assumes "(m::nat) \<le> n + 1"

  1477   shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"

  1478 proof-

  1479   have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using m \<le> n+1 by auto

  1480   thus ?thesis by (auto simp: ivl_disj_int setsum.union_disjoint

  1481     atLeastSucAtMost_greaterThanAtMost)

  1482 qed

  1483

  1484 lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>

  1485   setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"

  1486 by (simp add:setsum.union_disjoint[symmetric] ivl_disj_int ivl_disj_un)

  1487

  1488 lemma setsum_diff_nat_ivl:

  1489 fixes f :: "nat \<Rightarrow> 'a::ab_group_add"

  1490 shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>

  1491   setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"

  1492 using setsum_add_nat_ivl [of m n p f,symmetric]

  1493 apply (simp add: ac_simps)

  1494 done

  1495

  1496 lemma setsum_natinterval_difff:

  1497   fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"

  1498   shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =

  1499           (if m <= n then f m - f(n + 1) else 0)"

  1500 by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)

  1501

  1502 lemma setsum_nat_group: "(\<Sum>m<n::nat. setsum f {m * k ..< m*k + k}) = setsum f {..< n * k}"

  1503   apply (subgoal_tac "k = 0 | 0 < k", auto)

  1504   apply (induct "n")

  1505   apply (simp_all add: setsum_add_nat_ivl add.commute atLeast0LessThan[symmetric])

  1506   done

  1507

  1508 lemma setsum_triangle_reindex:

  1509   fixes n :: nat

  1510   shows "(\<Sum>(i,j)\<in>{(i,j). i+j < n}. f i j) = (\<Sum>k<n. \<Sum>i\<le>k. f i (k - i))"

  1511   apply (simp add: setsum.Sigma)

  1512   apply (rule setsum.reindex_bij_witness[where j="\<lambda>(i, j). (i+j, i)" and i="\<lambda>(k, i). (i, k - i)"])

  1513   apply auto

  1514   done

  1515

  1516 lemma setsum_triangle_reindex_eq:

  1517   fixes n :: nat

  1518   shows "(\<Sum>(i,j)\<in>{(i,j). i+j \<le> n}. f i j) = (\<Sum>k\<le>n. \<Sum>i\<le>k. f i (k - i))"

  1519 using setsum_triangle_reindex [of f "Suc n"]

  1520 by (simp only: Nat.less_Suc_eq_le lessThan_Suc_atMost)

  1521

  1522 lemma nat_diff_setsum_reindex: "(\<Sum>i<n. f (n - Suc i)) = (\<Sum>i<n. f i)"

  1523   by (rule setsum.reindex_bij_witness[where i="\<lambda>i. n - Suc i" and j="\<lambda>i. n - Suc i"]) auto

  1524

  1525 subsection{* Shifting bounds *}

  1526

  1527 lemma setsum_shift_bounds_nat_ivl:

  1528   "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"

  1529 by (induct "n", auto simp:atLeastLessThanSuc)

  1530

  1531 lemma setsum_shift_bounds_cl_nat_ivl:

  1532   "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"

  1533   by (rule setsum.reindex_bij_witness[where i="\<lambda>i. i + k" and j="\<lambda>i. i - k"]) auto

  1534

  1535 corollary setsum_shift_bounds_cl_Suc_ivl:

  1536   "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"

  1537 by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])

  1538

  1539 corollary setsum_shift_bounds_Suc_ivl:

  1540   "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"

  1541 by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])

  1542

  1543 lemma setsum_shift_lb_Suc0_0:

  1544   "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"

  1545 by(simp add:setsum_head_Suc)

  1546

  1547 lemma setsum_shift_lb_Suc0_0_upt:

  1548   "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"

  1549 apply(cases k)apply simp

  1550 apply(simp add:setsum_head_upt_Suc)

  1551 done

  1552

  1553 lemma setsum_atMost_Suc_shift:

  1554   fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"

  1555   shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"

  1556 proof (induct n)

  1557   case 0 show ?case by simp

  1558 next

  1559   case (Suc n) note IH = this

  1560   have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))"

  1561     by (rule setsum_atMost_Suc)

  1562   also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"

  1563     by (rule IH)

  1564   also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) =

  1565              f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"

  1566     by (rule add.assoc)

  1567   also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))"

  1568     by (rule setsum_atMost_Suc [symmetric])

  1569   finally show ?case .

  1570 qed

  1571

  1572 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)"

  1573   by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost add.commute)

  1574

  1575 lemma setsum_Suc_diff:

  1576   fixes f :: "nat \<Rightarrow> 'a::ab_group_add"

  1577   assumes "m \<le> Suc n"

  1578   shows "(\<Sum>i = m..n. f(Suc i) - f i) = f (Suc n) - f m"

  1579 using assms by (induct n) (auto simp: le_Suc_eq)

  1580

  1581 lemma nested_setsum_swap:

  1582      "(\<Sum>i = 0..n. (\<Sum>j = 0..<i. a i j)) = (\<Sum>j = 0..<n. \<Sum>i = Suc j..n. a i j)"

  1583   by (induction n) (auto simp: setsum.distrib)

  1584

  1585 lemma nested_setsum_swap':

  1586      "(\<Sum>i\<le>n. (\<Sum>j<i. a i j)) = (\<Sum>j<n. \<Sum>i = Suc j..n. a i j)"

  1587   by (induction n) (auto simp: setsum.distrib)

  1588

  1589 lemma setsum_zero_power' [simp]:

  1590   fixes c :: "nat \<Rightarrow> 'a::field"

  1591   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)"

  1592   using setsum_zero_power [of "\<lambda>i. c i / d i" A]

  1593   by auto

  1594

  1595

  1596 subsection {* The formula for geometric sums *}

  1597

  1598 lemma geometric_sum:

  1599   assumes "x \<noteq> 1"

  1600   shows "(\<Sum>i<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"

  1601 proof -

  1602   from assms obtain y where "y = x - 1" and "y \<noteq> 0" by simp_all

  1603   moreover have "(\<Sum>i<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"

  1604     by (induct n) (simp_all add: power_Suc field_simps y \<noteq> 0)

  1605   ultimately show ?thesis by simp

  1606 qed

  1607

  1608 lemma diff_power_eq_setsum:

  1609   fixes y :: "'a::{comm_ring,monoid_mult}"

  1610   shows

  1611     "x ^ (Suc n) - y ^ (Suc n) =

  1612       (x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))"

  1613 proof (induct n)

  1614   case (Suc n)

  1615   have "x ^ Suc (Suc n) - y ^ Suc (Suc n) = x * (x * x^n) - y * (y * y ^ n)"

  1616     by (simp add: power_Suc)

  1617   also have "... = y * (x ^ (Suc n) - y ^ (Suc n)) + (x - y) * (x * x^n)"

  1618     by (simp add: power_Suc algebra_simps)

  1619   also have "... = y * ((x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x^n)"

  1620     by (simp only: Suc)

  1621   also have "... = (x - y) * (y * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x^n)"

  1622     by (simp only: mult.left_commute)

  1623   also have "... = (x - y) * (\<Sum>p<Suc (Suc n). x ^ p * y ^ (Suc n - p))"

  1624     by (simp add: power_Suc field_simps Suc_diff_le setsum_left_distrib setsum_right_distrib)

  1625   finally show ?case .

  1626 qed simp

  1627

  1628 corollary power_diff_sumr2: --{* @{text COMPLEX_POLYFUN} in HOL Light *}

  1629   fixes x :: "'a::{comm_ring,monoid_mult}"

  1630   shows   "x^n - y^n = (x - y) * (\<Sum>i<n. y^(n - Suc i) * x^i)"

  1631 using diff_power_eq_setsum[of x "n - 1" y]

  1632 by (cases "n = 0") (simp_all add: field_simps)

  1633

  1634 lemma power_diff_1_eq:

  1635   fixes x :: "'a::{comm_ring,monoid_mult}"

  1636   shows "n \<noteq> 0 \<Longrightarrow> x^n - 1 = (x - 1) * (\<Sum>i<n. (x^i))"

  1637 using diff_power_eq_setsum [of x _ 1]

  1638   by (cases n) auto

  1639

  1640 lemma one_diff_power_eq':

  1641   fixes x :: "'a::{comm_ring,monoid_mult}"

  1642   shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i<n. x^(n - Suc i))"

  1643 using diff_power_eq_setsum [of 1 _ x]

  1644   by (cases n) auto

  1645

  1646 lemma one_diff_power_eq:

  1647   fixes x :: "'a::{comm_ring,monoid_mult}"

  1648   shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i<n. x^i)"

  1649 by (metis one_diff_power_eq' [of n x] nat_diff_setsum_reindex)

  1650

  1651

  1652 subsection {* The formula for arithmetic sums *}

  1653

  1654 lemma gauss_sum:

  1655   "(2::'a::comm_semiring_1)*(\<Sum>i\<in>{1..n}. of_nat i) = of_nat n*((of_nat n)+1)"

  1656 proof (induct n)

  1657   case 0

  1658   show ?case by simp

  1659 next

  1660   case (Suc n)

  1661   then show ?case

  1662     by (simp add: algebra_simps add: one_add_one [symmetric] del: one_add_one)

  1663       (* FIXME: make numeral cancellation simprocs work for semirings *)

  1664 qed

  1665

  1666 theorem arith_series_general:

  1667   "(2::'a::comm_semiring_1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

  1668   of_nat n * (a + (a + of_nat(n - 1)*d))"

  1669 proof cases

  1670   assume ngt1: "n > 1"

  1671   let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"

  1672   have

  1673     "(\<Sum>i\<in>{..<n}. a+?I i*d) =

  1674      ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"

  1675     by (rule setsum.distrib)

  1676   also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp

  1677   also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"

  1678     unfolding One_nat_def

  1679     by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt ac_simps)

  1680   also have "2*\<dots> = 2*?n*a + d*2*(\<Sum>i\<in>{1..<n}. ?I i)"

  1681     by (simp add: algebra_simps)

  1682   also from ngt1 have "{1..<n} = {1..n - 1}"

  1683     by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)

  1684   also from ngt1

  1685   have "2*?n*a + d*2*(\<Sum>i\<in>{1..n - 1}. ?I i) = (2*?n*a + d*?I (n - 1)*?I n)"

  1686     by (simp only: mult.assoc gauss_sum [of "n - 1"], unfold One_nat_def)

  1687       (simp add:  mult.commute trans [OF add.commute of_nat_Suc [symmetric]])

  1688   finally show ?thesis

  1689     unfolding mult_2 by (simp add: algebra_simps)

  1690 next

  1691   assume "\<not>(n > 1)"

  1692   hence "n = 1 \<or> n = 0" by auto

  1693   thus ?thesis by (auto simp: mult_2)

  1694 qed

  1695

  1696 lemma arith_series_nat:

  1697   "(2::nat) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"

  1698 proof -

  1699   have

  1700     "2 * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =

  1701     of_nat(n) * (a + (a + of_nat(n - 1)*d))"

  1702     by (rule arith_series_general)

  1703   thus ?thesis

  1704     unfolding One_nat_def by auto

  1705 qed

  1706

  1707 lemma arith_series_int:

  1708   "2 * (\<Sum>i\<in>{..<n}. a + int i * d) = int n * (a + (a + int(n - 1)*d))"

  1709   by (fact arith_series_general) (* FIXME: duplicate *)

  1710

  1711 lemma sum_diff_distrib: "\<forall>x. Q x \<le> P x  \<Longrightarrow> (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x :: nat)"

  1712   by (subst setsum_subtractf_nat) auto

  1713

  1714 subsection {* Products indexed over intervals *}

  1715

  1716 syntax

  1717   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)

  1718   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)

  1719   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)

  1720   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)

  1721 syntax (xsymbols)

  1722   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)

  1723   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)

  1724   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)

  1725   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)

  1726 syntax (HTML output)

  1727   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)

  1728   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)

  1729   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)

  1730   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)

  1731 syntax (latex_prod output)

  1732   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1733  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

  1734   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1735  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

  1736   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1737  ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

  1738   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1739  ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

  1740

  1741 translations

  1742   "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"

  1743   "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"

  1744   "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"

  1745   "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"

  1746

  1747 subsection {* Transfer setup *}

  1748

  1749 lemma transfer_nat_int_set_functions:

  1750     "{..n} = nat  {0..int n}"

  1751     "{m..n} = nat  {int m..int n}"  (* need all variants of these! *)

  1752   apply (auto simp add: image_def)

  1753   apply (rule_tac x = "int x" in bexI)

  1754   apply auto

  1755   apply (rule_tac x = "int x" in bexI)

  1756   apply auto

  1757   done

  1758

  1759 lemma transfer_nat_int_set_function_closures:

  1760     "x >= 0 \<Longrightarrow> nat_set {x..y}"

  1761   by (simp add: nat_set_def)

  1762

  1763 declare transfer_morphism_nat_int[transfer add

  1764   return: transfer_nat_int_set_functions

  1765     transfer_nat_int_set_function_closures

  1766 ]

  1767

  1768 lemma transfer_int_nat_set_functions:

  1769     "is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int  {nat m..nat n}"

  1770   by (simp only: is_nat_def transfer_nat_int_set_functions

  1771     transfer_nat_int_set_function_closures

  1772     transfer_nat_int_set_return_embed nat_0_le

  1773     cong: transfer_nat_int_set_cong)

  1774

  1775 lemma transfer_int_nat_set_function_closures:

  1776     "is_nat x \<Longrightarrow> nat_set {x..y}"

  1777   by (simp only: transfer_nat_int_set_function_closures is_nat_def)

  1778

  1779 declare transfer_morphism_int_nat[transfer add

  1780   return: transfer_int_nat_set_functions

  1781     transfer_int_nat_set_function_closures

  1782 ]

  1783

  1784 lemma setprod_int_plus_eq: "setprod int {i..i+j} =  \<Prod>{int i..int (i+j)}"

  1785   by (induct j) (auto simp add: atLeastAtMostSuc_conv atLeastAtMostPlus1_int_conv)

  1786

  1787 lemma setprod_int_eq: "setprod int {i..j} =  \<Prod>{int i..int j}"

  1788 proof (cases "i \<le> j")

  1789   case True

  1790   then show ?thesis

  1791     by (metis Nat.le_iff_add setprod_int_plus_eq)

  1792 next

  1793   case False

  1794   then show ?thesis

  1795     by auto

  1796 qed

  1797

  1798 end