src/HOL/Set_Interval.thy
 author haftmann Thu Jul 25 08:57:16 2013 +0200 (2013-07-25) changeset 52729 412c9e0381a1 parent 52380 3cc46b8cca5e child 53216 ad2e09c30aa8 permissions -rw-r--r--
factored syntactic type classes for bot and top (by Alessandro Coglio)
     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 header {* Set intervals *}

    15

    16 theory Set_Interval

    17 imports Int 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 subsection {*Two-sided intervals*}

   179

   180 context ord

   181 begin

   182

   183 lemma greaterThanLessThan_iff [simp,no_atp]:

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

   185 by (simp add: greaterThanLessThan_def)

   186

   187 lemma atLeastLessThan_iff [simp,no_atp]:

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

   189 by (simp add: atLeastLessThan_def)

   190

   191 lemma greaterThanAtMost_iff [simp,no_atp]:

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

   193 by (simp add: greaterThanAtMost_def)

   194

   195 lemma atLeastAtMost_iff [simp,no_atp]:

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

   197 by (simp add: atLeastAtMost_def)

   198

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

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

   201 alone. *}

   202

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

   204   by auto

   205

   206 end

   207

   208 subsubsection{* Emptyness, singletons, subset *}

   209

   210 context order

   211 begin

   212

   213 lemma atLeastatMost_empty[simp]:

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

   215 by(auto simp: atLeastAtMost_def atLeast_def atMost_def)

   216

   217 lemma atLeastatMost_empty_iff[simp]:

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

   219 by auto (blast intro: order_trans)

   220

   221 lemma atLeastatMost_empty_iff2[simp]:

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

   223 by auto (blast intro: order_trans)

   224

   225 lemma atLeastLessThan_empty[simp]:

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

   227 by(auto simp: atLeastLessThan_def)

   228

   229 lemma atLeastLessThan_empty_iff[simp]:

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

   231 by auto (blast intro: le_less_trans)

   232

   233 lemma atLeastLessThan_empty_iff2[simp]:

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

   235 by auto (blast intro: le_less_trans)

   236

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

   238 by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)

   239

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

   241 by auto (blast intro: less_le_trans)

   242

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

   244 by auto (blast intro: less_le_trans)

   245

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

   247 by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)

   248

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

   250 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)

   251

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

   253

   254 lemma atLeastatMost_subset_iff[simp]:

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

   256 unfolding atLeastAtMost_def atLeast_def atMost_def

   257 by (blast intro: order_trans)

   258

   259 lemma atLeastatMost_psubset_iff:

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

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

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

   263

   264 lemma Icc_eq_Icc[simp]:

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

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

   267

   268 lemma atLeastAtMost_singleton_iff[simp]:

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

   270 proof

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

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

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

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

   275 qed simp

   276

   277 lemma Icc_subset_Ici_iff[simp]:

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

   279 by(auto simp: subset_eq intro: order_trans)

   280

   281 lemma Icc_subset_Iic_iff[simp]:

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

   283 by(auto simp: subset_eq intro: order_trans)

   284

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

   286 by(auto simp: set_eq_iff)

   287

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

   289 by(auto simp: set_eq_iff)

   290

   291 lemmas not_empty_eq_Ici_eq_empty[simp] = not_Ici_eq_empty[symmetric]

   292 lemmas not_empty_eq_Iic_eq_empty[simp] = not_Iic_eq_empty[symmetric]

   293

   294 end

   295

   296 context no_top

   297 begin

   298

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

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

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

   302

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

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

   305

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

   307 using gt_ex[of h']

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

   309

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

   311 using gt_ex[of h']

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

   313

   314 end

   315

   316 context no_bot

   317 begin

   318

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

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

   321

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

   323 using lt_ex[of l']

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

   325

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

   327 using lt_ex[of l']

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

   329

   330 end

   331

   332

   333 context no_top

   334 begin

   335

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

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

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

   339

   340 lemmas not_Icc_eq_UNIV[simp] = not_UNIV_eq_Icc[symmetric]

   341

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

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

   344

   345 lemmas not_Iic_eq_UNIV[simp] = not_UNIV_eq_Iic[symmetric]

   346

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

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

   349

   350 lemmas not_Ici_eq_Icc[simp] = not_Icc_eq_Ici[symmetric]

   351

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

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

   354 using not_Ici_le_Iic[of l' h] by blast

   355

   356 lemmas not_Ici_eq_Iic[simp] = not_Iic_eq_Ici[symmetric]

   357

   358 end

   359

   360 context no_bot

   361 begin

   362

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

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

   365

   366 lemmas not_Ici_eq_UNIV[simp] = not_UNIV_eq_Ici[symmetric]

   367

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

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

   370

   371 lemmas not_Iic_eq_Icc[simp] = not_Icc_eq_Iic[symmetric]

   372

   373 end

   374

   375

   376 context inner_dense_linorder

   377 begin

   378

   379 lemma greaterThanLessThan_empty_iff[simp]:

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

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

   382

   383 lemma greaterThanLessThan_empty_iff2[simp]:

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

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

   386

   387 lemma atLeastLessThan_subseteq_atLeastAtMost_iff:

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

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

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

   391

   392 lemma greaterThanAtMost_subseteq_atLeastAtMost_iff:

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

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

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

   396

   397 lemma greaterThanLessThan_subseteq_atLeastAtMost_iff:

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

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

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

   401

   402 lemma atLeastAtMost_subseteq_atLeastLessThan_iff:

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

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

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

   406

   407 lemma greaterThanAtMost_subseteq_atLeastLessThan_iff:

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

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

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

   411

   412 lemma greaterThanLessThan_subseteq_atLeastLessThan_iff:

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

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

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

   416

   417 end

   418

   419 context no_top

   420 begin

   421

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

   423   using gt_ex[of x] by auto

   424

   425 end

   426

   427 context no_bot

   428 begin

   429

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

   431   using lt_ex[of x] by auto

   432

   433 end

   434

   435 lemma (in linorder) atLeastLessThan_subset_iff:

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

   437 apply (auto simp:subset_eq Ball_def)

   438 apply(frule_tac x=a in spec)

   439 apply(erule_tac x=d in allE)

   440 apply (simp add: less_imp_le)

   441 done

   442

   443 lemma atLeastLessThan_inj:

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

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

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

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

   448

   449 lemma atLeastLessThan_eq_iff:

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

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

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

   453   using atLeastLessThan_inj assms by auto

   454

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

   456 by (auto simp: set_eq_iff intro: le_bot)

   457

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

   459 by (auto simp: set_eq_iff intro: top_le)

   460

   461 lemma (in bounded_lattice) atLeastAtMost_eq_UNIV_iff:

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

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

   464

   465

   466 subsubsection {* Intersection *}

   467

   468 context linorder

   469 begin

   470

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

   472 by auto

   473

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

   475 by auto

   476

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

   478 by auto

   479

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

   481 by auto

   482

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

   484 by auto

   485

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

   487 by auto

   488

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

   490 by auto

   491

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

   493 by auto

   494

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

   496   by (auto simp: min_def)

   497

   498 end

   499

   500 context complete_lattice

   501 begin

   502

   503 lemma

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

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

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

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

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

   509   by (auto intro!: Sup_eqI)

   510

   511 lemma

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

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

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

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

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

   517   by (auto intro!: Inf_eqI)

   518

   519 end

   520

   521 lemma

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

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

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

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

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

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

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

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

   530

   531 subsection {* Intervals of natural numbers *}

   532

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

   534

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

   536 by (simp add: lessThan_def)

   537

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

   539 by (simp add: lessThan_def less_Suc_eq, blast)

   540

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

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

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

   544

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

   546 proof safe

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

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

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

   550 qed

   551

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

   553 by (simp add: lessThan_def atMost_def less_Suc_eq_le)

   554

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

   556 by blast

   557

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

   559

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

   561 apply (simp add: greaterThan_def)

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

   563 done

   564

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

   566 apply (simp add: greaterThan_def)

   567 apply (auto elim: linorder_neqE)

   568 done

   569

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

   571 by blast

   572

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

   574

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

   576 by (unfold atLeast_def UNIV_def, simp)

   577

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

   579 apply (simp add: atLeast_def)

   580 apply (simp add: Suc_le_eq)

   581 apply (simp add: order_le_less, blast)

   582 done

   583

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

   585   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)

   586

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

   588 by blast

   589

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

   591

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

   593 by (simp add: atMost_def)

   594

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

   596 apply (simp add: atMost_def)

   597 apply (simp add: less_Suc_eq order_le_less, blast)

   598 done

   599

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

   601 by blast

   602

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

   604

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

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

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

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

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

   610 specific concept to a more general one. *}

   611

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

   613 by(simp add:lessThan_def atLeastLessThan_def)

   614

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

   616 by(simp add:atMost_def atLeastAtMost_def)

   617

   618 declare atLeast0LessThan[symmetric, code_unfold]

   619         atLeast0AtMost[symmetric, code_unfold]

   620

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

   622 by (simp add: atLeastLessThan_def)

   623

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

   625

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

   627 lemma atLeastLessThanSuc:

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

   629 by (auto simp add: atLeastLessThan_def)

   630

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

   632 by (auto simp add: atLeastLessThan_def)

   633 (*

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

   635 by (induct k, simp_all add: atLeastLessThanSuc)

   636

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

   638 by (auto simp add: atLeastLessThan_def)

   639 *)

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

   641   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)

   642

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

   644   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def

   645     greaterThanAtMost_def)

   646

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

   648   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def

   649     greaterThanLessThan_def)

   650

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

   652 by (auto simp add: atLeastAtMost_def)

   653

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

   655 by auto

   656

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

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

   659 lemma atLeastAtMostPlus1_int_conv:

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

   661   by (auto intro: set_eqI)

   662

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

   664   apply (induct k)

   665   apply (simp_all add: atLeastLessThanSuc)

   666   done

   667

   668 subsubsection {* Image *}

   669

   670 lemma image_add_atLeastAtMost:

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

   672 proof

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

   674 next

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

   676   proof

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

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

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

   680     ultimately show "n : ?A" by blast

   681   qed

   682 qed

   683

   684 lemma image_add_atLeastLessThan:

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

   686 proof

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

   688 next

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

   690   proof

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

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

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

   694     ultimately show "n : ?A" by blast

   695   qed

   696 qed

   697

   698 corollary image_Suc_atLeastAtMost[simp]:

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

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

   701

   702 corollary image_Suc_atLeastLessThan[simp]:

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

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

   705

   706 lemma image_add_int_atLeastLessThan:

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

   708   apply (auto simp add: image_def)

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

   710   apply auto

   711   done

   712

   713 lemma image_minus_const_atLeastLessThan_nat:

   714   fixes c :: nat

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

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

   717     (is "_ = ?right")

   718 proof safe

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

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

   721   proof cases

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

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

   724   next

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

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

   727   qed

   728 qed auto

   729

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

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

   732

   733 context ordered_ab_group_add

   734 begin

   735

   736 lemma

   737   fixes x :: 'a

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

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

   740 proof safe

   741   fix y assume "y < -x"

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

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

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

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

   746 next

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

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

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

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

   751 qed simp_all

   752

   753 lemma

   754   fixes x :: 'a

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

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

   757 proof -

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

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

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

   761     by (simp_all add: image_image

   762         del: image_uminus_greaterThan image_uminus_atLeast)

   763 qed

   764

   765 lemma

   766   fixes x :: 'a

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

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

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

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

   771   by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def

   772       greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)

   773 end

   774

   775 subsubsection {* Finiteness *}

   776

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

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

   779

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

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

   782

   783 lemma finite_greaterThanLessThan [iff]:

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

   785 by (simp add: greaterThanLessThan_def)

   786

   787 lemma finite_atLeastLessThan [iff]:

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

   789 by (simp add: atLeastLessThan_def)

   790

   791 lemma finite_greaterThanAtMost [iff]:

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

   793 by (simp add: greaterThanAtMost_def)

   794

   795 lemma finite_atLeastAtMost [iff]:

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

   797 by (simp add: atLeastAtMost_def)

   798

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

   800 lemma bounded_nat_set_is_finite:

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

   802 apply (rule finite_subset)

   803  apply (rule_tac [2] finite_lessThan, auto)

   804 done

   805

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

   807 lemma finite_nat_set_iff_bounded:

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

   809 proof

   810   assume f:?F  show ?B

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

   812 next

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

   814 qed

   815

   816 lemma finite_nat_set_iff_bounded_le:

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

   818 apply(simp add:finite_nat_set_iff_bounded)

   819 apply(blast dest:less_imp_le_nat le_imp_less_Suc)

   820 done

   821

   822 lemma finite_less_ub:

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

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

   825

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

   827 subset is exactly that interval. *}

   828

   829 lemma subset_card_intvl_is_intvl:

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

   831 proof cases

   832   assume "finite A"

   833   thus "PROP ?P"

   834   proof(induct A rule:finite_linorder_max_induct)

   835     case empty thus ?case by auto

   836   next

   837     case (insert b A)

   838     moreover hence "b ~: A" by auto

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

   840       using b ~: A insert by fastforce+

   841     ultimately show ?case by auto

   842   qed

   843 next

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

   845 qed

   846

   847

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

   849

   850 lemma UN_le_eq_Un0:

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

   852 proof

   853   show "?A <= ?B"

   854   proof

   855     fix x assume "x : ?A"

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

   857     show "x : ?B"

   858     proof(cases i)

   859       case 0 with i show ?thesis by simp

   860     next

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

   862     qed

   863   qed

   864 next

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

   866 qed

   867

   868 lemma UN_le_add_shift:

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

   870 proof

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

   872 next

   873   show "?B <= ?A"

   874   proof

   875     fix x assume "x : ?B"

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

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

   878     thus "x : ?A" by blast

   879   qed

   880 qed

   881

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

   883   by (auto simp add: atLeast0LessThan)

   884

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

   886   by (subst UN_UN_finite_eq [symmetric]) blast

   887

   888 lemma UN_finite2_subset:

   889      "(!!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)"

   890   apply (rule UN_finite_subset)

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

   892   apply blast

   893   done

   894

   895 lemma UN_finite2_eq:

   896   "(!!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)"

   897   apply (rule subset_antisym)

   898    apply (rule UN_finite2_subset, blast)

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

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

   901  done

   902

   903

   904 subsubsection {* Cardinality *}

   905

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

   907   by (induct u, simp_all add: lessThan_Suc)

   908

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

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

   911

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

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

   914   apply (erule ssubst, rule card_lessThan)

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

   916   apply (erule subst)

   917   apply (rule card_image)

   918   apply (simp add: inj_on_def)

   919   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)

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

   921   apply arith

   922   done

   923

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

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

   926

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

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

   929

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

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

   932

   933 lemma ex_bij_betw_nat_finite:

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

   935 apply(drule finite_imp_nat_seg_image_inj_on)

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

   937 done

   938

   939 lemma ex_bij_betw_finite_nat:

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

   941 by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)

   942

   943 lemma finite_same_card_bij:

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

   945 apply(drule ex_bij_betw_finite_nat)

   946 apply(drule ex_bij_betw_nat_finite)

   947 apply(auto intro!:bij_betw_trans)

   948 done

   949

   950 lemma ex_bij_betw_nat_finite_1:

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

   952 by (rule finite_same_card_bij) auto

   953

   954 lemma bij_betw_iff_card:

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

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

   957 using assms

   958 proof(auto simp add: bij_betw_same_card)

   959   assume *: "card A = card B"

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

   961   using FIN ex_bij_betw_finite_nat by blast

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

   963   using FIN' ex_bij_betw_nat_finite by blast

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

   965   using * by (auto simp add: bij_betw_trans)

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

   967 qed

   968

   969 lemma inj_on_iff_card_le:

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

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

   972 proof (safe intro!: card_inj_on_le)

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

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

   975   using FIN ex_bij_betw_finite_nat unfolding bij_betw_def by force

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

   977   using FIN' ex_bij_betw_nat_finite unfolding bij_betw_def by force

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

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

   980   moreover

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

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

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

   984   }

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

   986 qed (insert assms, auto)

   987

   988 subsection {* Intervals of integers *}

   989

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

   991   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)

   992

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

   994   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)

   995

   996 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:

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

   998   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)

   999

  1000 subsubsection {* Finiteness *}

  1001

  1002 lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>

  1003     {(0::int)..<u} = int  {..<nat u}"

  1004   apply (unfold image_def lessThan_def)

  1005   apply auto

  1006   apply (rule_tac x = "nat x" in exI)

  1007   apply (auto simp add: zless_nat_eq_int_zless [THEN sym])

  1008   done

  1009

  1010 lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"

  1011   apply (cases "0 \<le> u")

  1012   apply (subst image_atLeastZeroLessThan_int, assumption)

  1013   apply (rule finite_imageI)

  1014   apply auto

  1015   done

  1016

  1017 lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"

  1018   apply (subgoal_tac "(%x. x + l)  {0..<u-l} = {l..<u}")

  1019   apply (erule subst)

  1020   apply (rule finite_imageI)

  1021   apply (rule finite_atLeastZeroLessThan_int)

  1022   apply (rule image_add_int_atLeastLessThan)

  1023   done

  1024

  1025 lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"

  1026   by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)

  1027

  1028 lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"

  1029   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)

  1030

  1031 lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"

  1032   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)

  1033

  1034

  1035 subsubsection {* Cardinality *}

  1036

  1037 lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"

  1038   apply (cases "0 \<le> u")

  1039   apply (subst image_atLeastZeroLessThan_int, assumption)

  1040   apply (subst card_image)

  1041   apply (auto simp add: inj_on_def)

  1042   done

  1043

  1044 lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"

  1045   apply (subgoal_tac "card {l..<u} = card {0..<u-l}")

  1046   apply (erule ssubst, rule card_atLeastZeroLessThan_int)

  1047   apply (subgoal_tac "(%x. x + l)  {0..<u-l} = {l..<u}")

  1048   apply (erule subst)

  1049   apply (rule card_image)

  1050   apply (simp add: inj_on_def)

  1051   apply (rule image_add_int_atLeastLessThan)

  1052   done

  1053

  1054 lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"

  1055 apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])

  1056 apply (auto simp add: algebra_simps)

  1057 done

  1058

  1059 lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"

  1060 by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)

  1061

  1062 lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"

  1063 by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)

  1064

  1065 lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"

  1066 proof -

  1067   have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto

  1068   with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)

  1069 qed

  1070

  1071 lemma card_less:

  1072 assumes zero_in_M: "0 \<in> M"

  1073 shows "card {k \<in> M. k < Suc i} \<noteq> 0"

  1074 proof -

  1075   from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto

  1076   with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)

  1077 qed

  1078

  1079 lemma card_less_Suc2: "0 \<notin> M \<Longrightarrow> card {k. Suc k \<in> M \<and> k < i} = card {k \<in> M. k < Suc i}"

  1080 apply (rule card_bij_eq [of Suc _ _ "\<lambda>x. x - Suc 0"])

  1081 apply simp

  1082 apply fastforce

  1083 apply auto

  1084 apply (rule inj_on_diff_nat)

  1085 apply auto

  1086 apply (case_tac x)

  1087 apply auto

  1088 apply (case_tac xa)

  1089 apply auto

  1090 apply (case_tac xa)

  1091 apply auto

  1092 done

  1093

  1094 lemma card_less_Suc:

  1095   assumes zero_in_M: "0 \<in> M"

  1096     shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"

  1097 proof -

  1098   from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp

  1099   hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"

  1100     by (auto simp only: insert_Diff)

  1101   have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto

  1102   from finite_M_bounded_by_nat[of "\<lambda>x. x \<in> M" "Suc i"] have "Suc (card {k. Suc k \<in> M \<and> k < i}) = card (insert 0 ({k \<in> M. k < Suc i} - {0}))"

  1103     apply (subst card_insert)

  1104     apply simp_all

  1105     apply (subst b)

  1106     apply (subst card_less_Suc2[symmetric])

  1107     apply simp_all

  1108     done

  1109   with c show ?thesis by simp

  1110 qed

  1111

  1112

  1113 subsection {*Lemmas useful with the summation operator setsum*}

  1114

  1115 text {* For examples, see Algebra/poly/UnivPoly2.thy *}

  1116

  1117 subsubsection {* Disjoint Unions *}

  1118

  1119 text {* Singletons and open intervals *}

  1120

  1121 lemma ivl_disj_un_singleton:

  1122   "{l::'a::linorder} Un {l<..} = {l..}"

  1123   "{..<u} Un {u::'a::linorder} = {..u}"

  1124   "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"

  1125   "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"

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

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

  1128 by auto

  1129

  1130 text {* One- and two-sided intervals *}

  1131

  1132 lemma ivl_disj_un_one:

  1133   "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"

  1134   "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"

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

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

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

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

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

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

  1141 by auto

  1142

  1143 text {* Two- and two-sided intervals *}

  1144

  1145 lemma ivl_disj_un_two:

  1146   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"

  1147   "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"

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

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

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

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

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

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

  1154 by auto

  1155

  1156 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two

  1157

  1158 subsubsection {* Disjoint Intersections *}

  1159

  1160 text {* One- and two-sided intervals *}

  1161

  1162 lemma ivl_disj_int_one:

  1163   "{..l::'a::order} Int {l<..<u} = {}"

  1164   "{..<l} Int {l..<u} = {}"

  1165   "{..l} Int {l<..u} = {}"

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

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

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

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

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

  1171   by auto

  1172

  1173 text {* Two- and two-sided intervals *}

  1174

  1175 lemma ivl_disj_int_two:

  1176   "{l::'a::order<..<m} Int {m..<u} = {}"

  1177   "{l<..m} Int {m<..<u} = {}"

  1178   "{l..<m} Int {m..<u} = {}"

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

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

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

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

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

  1184   by auto

  1185

  1186 lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two

  1187

  1188 subsubsection {* Some Differences *}

  1189

  1190 lemma ivl_diff[simp]:

  1191  "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"

  1192 by(auto)

  1193

  1194

  1195 subsubsection {* Some Subset Conditions *}

  1196

  1197 lemma ivl_subset [simp,no_atp]:

  1198  "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"

  1199 apply(auto simp:linorder_not_le)

  1200 apply(rule ccontr)

  1201 apply(insert linorder_le_less_linear[of i n])

  1202 apply(clarsimp simp:linorder_not_le)

  1203 apply(fastforce)

  1204 done

  1205

  1206

  1207 subsection {* Summation indexed over intervals *}

  1208

  1209 syntax

  1210   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)

  1211   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)

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

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

  1214 syntax (xsymbols)

  1215   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)

  1216   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)

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

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

  1219 syntax (HTML output)

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

  1221   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)

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

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

  1224 syntax (latex_sum output)

  1225   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1226  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

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

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

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

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

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

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

  1233

  1234 translations

  1235   "\<Sum>x=a..b. t" == "CONST setsum (%x. t) {a..b}"

  1236   "\<Sum>x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"

  1237   "\<Sum>i\<le>n. t" == "CONST setsum (\<lambda>i. t) {..n}"

  1238   "\<Sum>i<n. t" == "CONST setsum (\<lambda>i. t) {..<n}"

  1239

  1240 text{* The above introduces some pretty alternative syntaxes for

  1241 summation over intervals:

  1242 \begin{center}

  1243 \begin{tabular}{lll}

  1244 Old & New & \LaTeX\\

  1245 @{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\

  1246 @{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\

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

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

  1249 \end{tabular}

  1250 \end{center}

  1251 The left column shows the term before introduction of the new syntax,

  1252 the middle column shows the new (default) syntax, and the right column

  1253 shows a special syntax. The latter is only meaningful for latex output

  1254 and has to be activated explicitly by setting the print mode to

  1255 @{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in

  1256 antiquotations). It is not the default \LaTeX\ output because it only

  1257 works well with italic-style formulae, not tt-style.

  1258

  1259 Note that for uniformity on @{typ nat} it is better to use

  1260 @{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may

  1261 not provide all lemmas available for @{term"{m..<n}"} also in the

  1262 special form for @{term"{..<n}"}. *}

  1263

  1264 text{* This congruence rule should be used for sums over intervals as

  1265 the standard theorem @{text[source]setsum_cong} does not work well

  1266 with the simplifier who adds the unsimplified premise @{term"x:B"} to

  1267 the context. *}

  1268

  1269 lemma setsum_ivl_cong:

  1270  "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>

  1271  setsum f {a..<b} = setsum g {c..<d}"

  1272 by(rule setsum_cong, simp_all)

  1273

  1274 (* FIXME why are the following simp rules but the corresponding eqns

  1275 on intervals are not? *)

  1276

  1277 lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"

  1278 by (simp add:atMost_Suc add_ac)

  1279

  1280 lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"

  1281 by (simp add:lessThan_Suc add_ac)

  1282

  1283 lemma setsum_cl_ivl_Suc[simp]:

  1284   "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"

  1285 by (auto simp:add_ac atLeastAtMostSuc_conv)

  1286

  1287 lemma setsum_op_ivl_Suc[simp]:

  1288   "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"

  1289 by (auto simp:add_ac atLeastLessThanSuc)

  1290 (*

  1291 lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>

  1292     (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"

  1293 by (auto simp:add_ac atLeastAtMostSuc_conv)

  1294 *)

  1295

  1296 lemma setsum_head:

  1297   fixes n :: nat

  1298   assumes mn: "m <= n"

  1299   shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")

  1300 proof -

  1301   from mn

  1302   have "{m..n} = {m} \<union> {m<..n}"

  1303     by (auto intro: ivl_disj_un_singleton)

  1304   hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"

  1305     by (simp add: atLeast0LessThan)

  1306   also have "\<dots> = ?rhs" by simp

  1307   finally show ?thesis .

  1308 qed

  1309

  1310 lemma setsum_head_Suc:

  1311   "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"

  1312 by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)

  1313

  1314 lemma setsum_head_upt_Suc:

  1315   "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"

  1316 apply(insert setsum_head_Suc[of m "n - Suc 0" f])

  1317 apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)

  1318 done

  1319

  1320 lemma setsum_ub_add_nat: assumes "(m::nat) \<le> n + 1"

  1321   shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"

  1322 proof-

  1323   have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using m \<le> n+1 by auto

  1324   thus ?thesis by (auto simp: ivl_disj_int setsum_Un_disjoint

  1325     atLeastSucAtMost_greaterThanAtMost)

  1326 qed

  1327

  1328 lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>

  1329   setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"

  1330 by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)

  1331

  1332 lemma setsum_diff_nat_ivl:

  1333 fixes f :: "nat \<Rightarrow> 'a::ab_group_add"

  1334 shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>

  1335   setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"

  1336 using setsum_add_nat_ivl [of m n p f,symmetric]

  1337 apply (simp add: add_ac)

  1338 done

  1339

  1340 lemma setsum_natinterval_difff:

  1341   fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"

  1342   shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =

  1343           (if m <= n then f m - f(n + 1) else 0)"

  1344 by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)

  1345

  1346 lemma setsum_restrict_set':

  1347   "finite A \<Longrightarrow> setsum f {x \<in> A. x \<in> B} = (\<Sum>x\<in>A. if x \<in> B then f x else 0)"

  1348   by (simp add: setsum_restrict_set [symmetric] Int_def)

  1349

  1350 lemma setsum_restrict_set'':

  1351   "finite A \<Longrightarrow> setsum f {x \<in> A. P x} = (\<Sum>x\<in>A. if P x  then f x else 0)"

  1352   by (simp add: setsum_restrict_set' [of A f "{x. P x}", simplified mem_Collect_eq])

  1353

  1354 lemma setsum_setsum_restrict:

  1355   "finite S \<Longrightarrow> finite T \<Longrightarrow>

  1356     setsum (\<lambda>x. setsum (\<lambda>y. f x y) {y. y \<in> T \<and> R x y}) S = setsum (\<lambda>y. setsum (\<lambda>x. f x y) {x. x \<in> S \<and> R x y}) T"

  1357   by (simp add: setsum_restrict_set'') (rule setsum_commute)

  1358

  1359 lemma setsum_image_gen: assumes fS: "finite S"

  1360   shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f  S)"

  1361 proof-

  1362   { fix x assume "x \<in> S" then have "{y. y\<in> fS \<and> f x = y} = {f x}" by auto }

  1363   hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> fS \<and> f x = y}) S"

  1364     by simp

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

  1366     by (rule setsum_setsum_restrict[OF fS finite_imageI[OF fS]])

  1367   finally show ?thesis .

  1368 qed

  1369

  1370 lemma setsum_le_included:

  1371   fixes f :: "'a \<Rightarrow> 'b::ordered_comm_monoid_add"

  1372   assumes "finite s" "finite t"

  1373   and "\<forall>y\<in>t. 0 \<le> g y" "(\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> f x \<le> g y)"

  1374   shows "setsum f s \<le> setsum g t"

  1375 proof -

  1376   have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"

  1377   proof (rule setsum_mono)

  1378     fix y assume "y \<in> s"

  1379     with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto

  1380     with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")

  1381       using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]

  1382       by (auto intro!: setsum_mono2)

  1383   qed

  1384   also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i  t)"

  1385     using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)

  1386   also have "... \<le> setsum g t"

  1387     using assms by (auto simp: setsum_image_gen[symmetric])

  1388   finally show ?thesis .

  1389 qed

  1390

  1391 lemma setsum_multicount_gen:

  1392   assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"

  1393   shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")

  1394 proof-

  1395   have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto

  1396   also have "\<dots> = ?r" unfolding setsum_setsum_restrict[OF assms(1-2)]

  1397     using assms(3) by auto

  1398   finally show ?thesis .

  1399 qed

  1400

  1401 lemma setsum_multicount:

  1402   assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"

  1403   shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")

  1404 proof-

  1405   have "?l = setsum (\<lambda>i. k) T" by(rule setsum_multicount_gen)(auto simp:assms)

  1406   also have "\<dots> = ?r" by(simp add: mult_commute)

  1407   finally show ?thesis by auto

  1408 qed

  1409

  1410

  1411 subsection{* Shifting bounds *}

  1412

  1413 lemma setsum_shift_bounds_nat_ivl:

  1414   "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"

  1415 by (induct "n", auto simp:atLeastLessThanSuc)

  1416

  1417 lemma setsum_shift_bounds_cl_nat_ivl:

  1418   "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"

  1419 apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])

  1420 apply (simp add:image_add_atLeastAtMost o_def)

  1421 done

  1422

  1423 corollary setsum_shift_bounds_cl_Suc_ivl:

  1424   "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"

  1425 by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])

  1426

  1427 corollary setsum_shift_bounds_Suc_ivl:

  1428   "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"

  1429 by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])

  1430

  1431 lemma setsum_shift_lb_Suc0_0:

  1432   "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"

  1433 by(simp add:setsum_head_Suc)

  1434

  1435 lemma setsum_shift_lb_Suc0_0_upt:

  1436   "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"

  1437 apply(cases k)apply simp

  1438 apply(simp add:setsum_head_upt_Suc)

  1439 done

  1440

  1441 lemma setsum_atMost_Suc_shift:

  1442   fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"

  1443   shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"

  1444 proof (induct n)

  1445   case 0 show ?case by simp

  1446 next

  1447   case (Suc n) note IH = this

  1448   have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))"

  1449     by (rule setsum_atMost_Suc)

  1450   also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"

  1451     by (rule IH)

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

  1453              f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"

  1454     by (rule add_assoc)

  1455   also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))"

  1456     by (rule setsum_atMost_Suc [symmetric])

  1457   finally show ?case .

  1458 qed

  1459

  1460

  1461 subsection {* The formula for geometric sums *}

  1462

  1463 lemma geometric_sum:

  1464   assumes "x \<noteq> 1"

  1465   shows "(\<Sum>i=0..<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"

  1466 proof -

  1467   from assms obtain y where "y = x - 1" and "y \<noteq> 0" by simp_all

  1468   moreover have "(\<Sum>i=0..<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"

  1469   proof (induct n)

  1470     case 0 then show ?case by simp

  1471   next

  1472     case (Suc n)

  1473     moreover with y \<noteq> 0 have "(1 + y) ^ n = (y * inverse y) * (1 + y) ^ n" by simp

  1474     ultimately show ?case by (simp add: field_simps divide_inverse)

  1475   qed

  1476   ultimately show ?thesis by simp

  1477 qed

  1478

  1479

  1480 subsection {* The formula for arithmetic sums *}

  1481

  1482 lemma gauss_sum:

  1483   "(2::'a::comm_semiring_1)*(\<Sum>i\<in>{1..n}. of_nat i) =

  1484    of_nat n*((of_nat n)+1)"

  1485 proof (induct n)

  1486   case 0

  1487   show ?case by simp

  1488 next

  1489   case (Suc n)

  1490   then show ?case

  1491     by (simp add: algebra_simps add: one_add_one [symmetric] del: one_add_one)

  1492       (* FIXME: make numeral cancellation simprocs work for semirings *)

  1493 qed

  1494

  1495 theorem arith_series_general:

  1496   "(2::'a::comm_semiring_1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

  1497   of_nat n * (a + (a + of_nat(n - 1)*d))"

  1498 proof cases

  1499   assume ngt1: "n > 1"

  1500   let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"

  1501   have

  1502     "(\<Sum>i\<in>{..<n}. a+?I i*d) =

  1503      ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"

  1504     by (rule setsum_addf)

  1505   also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp

  1506   also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"

  1507     unfolding One_nat_def

  1508     by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)

  1509   also have "2*\<dots> = 2*?n*a + d*2*(\<Sum>i\<in>{1..<n}. ?I i)"

  1510     by (simp add: algebra_simps)

  1511   also from ngt1 have "{1..<n} = {1..n - 1}"

  1512     by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)

  1513   also from ngt1

  1514   have "2*?n*a + d*2*(\<Sum>i\<in>{1..n - 1}. ?I i) = (2*?n*a + d*?I (n - 1)*?I n)"

  1515     by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)

  1516        (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])

  1517   finally show ?thesis

  1518     unfolding mult_2 by (simp add: algebra_simps)

  1519 next

  1520   assume "\<not>(n > 1)"

  1521   hence "n = 1 \<or> n = 0" by auto

  1522   thus ?thesis by (auto simp: mult_2)

  1523 qed

  1524

  1525 lemma arith_series_nat:

  1526   "(2::nat) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"

  1527 proof -

  1528   have

  1529     "2 * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =

  1530     of_nat(n) * (a + (a + of_nat(n - 1)*d))"

  1531     by (rule arith_series_general)

  1532   thus ?thesis

  1533     unfolding One_nat_def by auto

  1534 qed

  1535

  1536 lemma arith_series_int:

  1537   "2 * (\<Sum>i\<in>{..<n}. a + int i * d) = int n * (a + (a + int(n - 1)*d))"

  1538   by (fact arith_series_general) (* FIXME: duplicate *)

  1539

  1540 lemma sum_diff_distrib:

  1541   fixes P::"nat\<Rightarrow>nat"

  1542   shows

  1543   "\<forall>x. Q x \<le> P x  \<Longrightarrow>

  1544   (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"

  1545 proof (induct n)

  1546   case 0 show ?case by simp

  1547 next

  1548   case (Suc n)

  1549

  1550   let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"

  1551   let ?rhs = "\<Sum>x<n. P x - Q x"

  1552

  1553   from Suc have "?lhs = ?rhs" by simp

  1554   moreover

  1555   from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp

  1556   moreover

  1557   from Suc have

  1558     "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"

  1559     by (subst diff_diff_left[symmetric],

  1560         subst diff_add_assoc2)

  1561        (auto simp: diff_add_assoc2 intro: setsum_mono)

  1562   ultimately

  1563   show ?case by simp

  1564 qed

  1565

  1566 subsection {* Products indexed over intervals *}

  1567

  1568 syntax

  1569   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)

  1570   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)

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

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

  1573 syntax (xsymbols)

  1574   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)

  1575   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)

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

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

  1578 syntax (HTML output)

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

  1580   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)

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

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

  1583 syntax (latex_prod output)

  1584   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1585  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

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

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

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

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

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

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

  1592

  1593 translations

  1594   "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"

  1595   "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"

  1596   "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"

  1597   "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"

  1598

  1599 subsection {* Transfer setup *}

  1600

  1601 lemma transfer_nat_int_set_functions:

  1602     "{..n} = nat  {0..int n}"

  1603     "{m..n} = nat  {int m..int n}"  (* need all variants of these! *)

  1604   apply (auto simp add: image_def)

  1605   apply (rule_tac x = "int x" in bexI)

  1606   apply auto

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

  1608   apply auto

  1609   done

  1610

  1611 lemma transfer_nat_int_set_function_closures:

  1612     "x >= 0 \<Longrightarrow> nat_set {x..y}"

  1613   by (simp add: nat_set_def)

  1614

  1615 declare transfer_morphism_nat_int[transfer add

  1616   return: transfer_nat_int_set_functions

  1617     transfer_nat_int_set_function_closures

  1618 ]

  1619

  1620 lemma transfer_int_nat_set_functions:

  1621     "is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int  {nat m..nat n}"

  1622   by (simp only: is_nat_def transfer_nat_int_set_functions

  1623     transfer_nat_int_set_function_closures

  1624     transfer_nat_int_set_return_embed nat_0_le

  1625     cong: transfer_nat_int_set_cong)

  1626

  1627 lemma transfer_int_nat_set_function_closures:

  1628     "is_nat x \<Longrightarrow> nat_set {x..y}"

  1629   by (simp only: transfer_nat_int_set_function_closures is_nat_def)

  1630

  1631 declare transfer_morphism_int_nat[transfer add

  1632   return: transfer_int_nat_set_functions

  1633     transfer_int_nat_set_function_closures

  1634 ]

  1635

  1636 end