src/HOL/Set_Interval.thy
 author haftmann Mon Oct 09 19:10:47 2017 +0200 (2017-10-09) changeset 66836 4eb431c3f974 parent 66490 cc66ab2373ce child 66936 cf8d8fc23891 permissions -rw-r--r--
tuned imports
     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 \<open>Set intervals\<close>

    15

    16 theory Set_Interval

    17 imports Divides

    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\<open>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.\<close>

    61

    62 syntax (ASCII)

    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 (latex output)

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

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

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

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

    73

    74 syntax

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

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

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

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

    79

    80 translations

    81   "\<Union>i\<le>n. A" \<rightleftharpoons> "\<Union>i\<in>{..n}. A"

    82   "\<Union>i<n. A" \<rightleftharpoons> "\<Union>i\<in>{..<n}. A"

    83   "\<Inter>i\<le>n. A" \<rightleftharpoons> "\<Inter>i\<in>{..n}. A"

    84   "\<Inter>i<n. A" \<rightleftharpoons> "\<Inter>i\<in>{..<n}. A"

    85

    86

    87 subsection \<open>Various equivalences\<close>

    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 \<open>Logical Equivalences for Set Inclusion and Equality\<close>

   132

   133 lemma atLeast_empty_triv [simp]: "{{}..} = UNIV"

   134   by auto

   135

   136 lemma atMost_UNIV_triv [simp]: "{..UNIV} = UNIV"

   137   by auto

   138

   139 lemma atLeast_subset_iff [iff]:

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

   141 by (blast intro: order_trans)

   142

   143 lemma atLeast_eq_iff [iff]:

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

   145 by (blast intro: order_antisym order_trans)

   146

   147 lemma greaterThan_subset_iff [iff]:

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

   149 apply (auto simp add: greaterThan_def)

   150  apply (subst linorder_not_less [symmetric], blast)

   151 done

   152

   153 lemma greaterThan_eq_iff [iff]:

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

   155 apply (rule iffI)

   156  apply (erule equalityE)

   157  apply simp_all

   158 done

   159

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

   161 by (blast intro: order_trans)

   162

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

   164 by (blast intro: order_antisym order_trans)

   165

   166 lemma lessThan_subset_iff [iff]:

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

   168 apply (auto simp add: lessThan_def)

   169  apply (subst linorder_not_less [symmetric], blast)

   170 done

   171

   172 lemma lessThan_eq_iff [iff]:

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

   174 apply (rule iffI)

   175  apply (erule equalityE)

   176  apply simp_all

   177 done

   178

   179 lemma lessThan_strict_subset_iff:

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

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

   182   by (metis leD lessThan_subset_iff linorder_linear not_less_iff_gr_or_eq psubset_eq)

   183

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

   185   by auto

   186

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

   188   by auto

   189

   190 lemma (in preorder) Ioi_le_Ico: "{a <..} \<subseteq> {a ..}"

   191   by (auto intro: less_imp_le)

   192

   193 subsection \<open>Two-sided intervals\<close>

   194

   195 context ord

   196 begin

   197

   198 lemma greaterThanLessThan_iff [simp]:

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

   200 by (simp add: greaterThanLessThan_def)

   201

   202 lemma atLeastLessThan_iff [simp]:

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

   204 by (simp add: atLeastLessThan_def)

   205

   206 lemma greaterThanAtMost_iff [simp]:

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

   208 by (simp add: greaterThanAtMost_def)

   209

   210 lemma atLeastAtMost_iff [simp]:

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

   212 by (simp add: atLeastAtMost_def)

   213

   214 text \<open>The above four lemmas could be declared as iffs. Unfortunately this

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

   216 alone.\<close>

   217

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

   219   by auto

   220

   221 end

   222

   223 subsubsection\<open>Emptyness, singletons, subset\<close>

   224

   225 context order

   226 begin

   227

   228 lemma atLeastatMost_empty[simp]:

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

   230 by(auto simp: atLeastAtMost_def atLeast_def atMost_def)

   231

   232 lemma atLeastatMost_empty_iff[simp]:

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

   234 by auto (blast intro: order_trans)

   235

   236 lemma atLeastatMost_empty_iff2[simp]:

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

   238 by auto (blast intro: order_trans)

   239

   240 lemma atLeastLessThan_empty[simp]:

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

   242 by(auto simp: atLeastLessThan_def)

   243

   244 lemma atLeastLessThan_empty_iff[simp]:

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

   246 by auto (blast intro: le_less_trans)

   247

   248 lemma atLeastLessThan_empty_iff2[simp]:

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

   250 by auto (blast intro: le_less_trans)

   251

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

   253 by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)

   254

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

   256 by auto (blast intro: less_le_trans)

   257

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

   259 by auto (blast intro: less_le_trans)

   260

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

   262 by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)

   263

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

   265 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)

   266

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

   268

   269 lemma atLeastatMost_subset_iff[simp]:

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

   271 unfolding atLeastAtMost_def atLeast_def atMost_def

   272 by (blast intro: order_trans)

   273

   274 lemma atLeastatMost_psubset_iff:

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

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

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

   278

   279 lemma Icc_eq_Icc[simp]:

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

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

   282

   283 lemma atLeastAtMost_singleton_iff[simp]:

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

   285 proof

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

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

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

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

   290 qed simp

   291

   292 lemma Icc_subset_Ici_iff[simp]:

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

   294 by(auto simp: subset_eq intro: order_trans)

   295

   296 lemma Icc_subset_Iic_iff[simp]:

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

   298 by(auto simp: subset_eq intro: order_trans)

   299

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

   301 by(auto simp: set_eq_iff)

   302

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

   304 by(auto simp: set_eq_iff)

   305

   306 lemmas not_empty_eq_Ici_eq_empty[simp] = not_Ici_eq_empty[symmetric]

   307 lemmas not_empty_eq_Iic_eq_empty[simp] = not_Iic_eq_empty[symmetric]

   308

   309 end

   310

   311 context no_top

   312 begin

   313

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

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

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

   317

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

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

   320

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

   322 using gt_ex[of h']

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

   324

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

   326 using gt_ex[of h']

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

   328

   329 end

   330

   331 context no_bot

   332 begin

   333

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

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

   336

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

   338 using lt_ex[of l']

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

   340

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

   342 using lt_ex[of l']

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

   344

   345 end

   346

   347

   348 context no_top

   349 begin

   350

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

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

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

   354

   355 lemmas not_Icc_eq_UNIV[simp] = not_UNIV_eq_Icc[symmetric]

   356

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

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

   359

   360 lemmas not_Iic_eq_UNIV[simp] = not_UNIV_eq_Iic[symmetric]

   361

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

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

   364

   365 lemmas not_Ici_eq_Icc[simp] = not_Icc_eq_Ici[symmetric]

   366

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

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

   369 using not_Ici_le_Iic[of l' h] by blast

   370

   371 lemmas not_Ici_eq_Iic[simp] = not_Iic_eq_Ici[symmetric]

   372

   373 end

   374

   375 context no_bot

   376 begin

   377

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

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

   380

   381 lemmas not_Ici_eq_UNIV[simp] = not_UNIV_eq_Ici[symmetric]

   382

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

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

   385

   386 lemmas not_Iic_eq_Icc[simp] = not_Icc_eq_Iic[symmetric]

   387

   388 end

   389

   390

   391 context dense_linorder

   392 begin

   393

   394 lemma greaterThanLessThan_empty_iff[simp]:

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

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

   397

   398 lemma greaterThanLessThan_empty_iff2[simp]:

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

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

   401

   402 lemma atLeastLessThan_subseteq_atLeastAtMost_iff:

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

   408   "{a <.. b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> 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_atLeastAtMost_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 lemma atLeastAtMost_subseteq_atLeastLessThan_iff:

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

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

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

   421

   422 lemma greaterThanLessThan_subseteq_greaterThanLessThan:

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

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

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

   426

   427 lemma greaterThanAtMost_subseteq_atLeastLessThan_iff:

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

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

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

   431

   432 lemma greaterThanLessThan_subseteq_atLeastLessThan_iff:

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

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

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

   436

   437 lemma greaterThanLessThan_subseteq_greaterThanAtMost_iff:

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

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

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

   441

   442 end

   443

   444 context no_top

   445 begin

   446

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

   448   using gt_ex[of x] by auto

   449

   450 end

   451

   452 context no_bot

   453 begin

   454

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

   456   using lt_ex[of x] by auto

   457

   458 end

   459

   460 lemma (in linorder) atLeastLessThan_subset_iff:

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

   462 apply (auto simp:subset_eq Ball_def)

   463 apply(frule_tac x=a in spec)

   464 apply(erule_tac x=d in allE)

   465 apply (simp add: less_imp_le)

   466 done

   467

   468 lemma atLeastLessThan_inj:

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

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

   471   shows "a = c" "b = d"

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

   473

   474 lemma atLeastLessThan_eq_iff:

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

   476   assumes "a < b" "c < d"

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

   478   using atLeastLessThan_inj assms by auto

   479

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

   481   by (metis eq_iff greaterThanAtMost_empty_iff2 greaterThanAtMost_iff le_cases not_le)

   482

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

   484   by auto

   485

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

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

   488

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

   490 by (auto simp: set_eq_iff intro: le_bot)

   491

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

   493 by (auto simp: set_eq_iff intro: top_le)

   494

   495 lemma (in bounded_lattice) atLeastAtMost_eq_UNIV_iff:

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

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

   498

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

   500   by (auto simp: set_eq_iff not_less le_bot)

   501

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

   503   by (simp add: Iio_eq_empty_iff bot_nat_def)

   504

   505 lemma mono_image_least:

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

   507   shows "f m = m'"

   508 proof -

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

   510     by (metis atLeastLessThan_empty_iff image_is_empty)

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

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

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

   514   ultimately show "f m = m'"

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

   516 qed

   517

   518

   519 subsection \<open>Infinite intervals\<close>

   520

   521 context dense_linorder

   522 begin

   523

   524 lemma infinite_Ioo:

   525   assumes "a < b"

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

   527 proof

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

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

   530     using \<open>a < b\<close> by auto

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

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

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

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

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

   536     using \<open>a < Max {a <..< b}\<close> by auto

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

   538     using fin by auto

   539   with \<open>Max {a <..< b} < x\<close> show False by auto

   540 qed

   541

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

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

   544   by (auto dest: finite_subset)

   545

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

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

   548   by (auto dest: finite_subset)

   549

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

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

   552   by (auto dest: finite_subset)

   553

   554 lemma infinite_Ioo_iff [simp]: "infinite {a<..<b} \<longleftrightarrow> a < b"

   555   using not_less_iff_gr_or_eq by (fastforce simp: infinite_Ioo)

   556

   557 lemma infinite_Icc_iff [simp]: "infinite {a .. b} \<longleftrightarrow> a < b"

   558   using not_less_iff_gr_or_eq by (fastforce simp: infinite_Icc)

   559

   560 lemma infinite_Ico_iff [simp]: "infinite {a..<b} \<longleftrightarrow> a < b"

   561   using not_less_iff_gr_or_eq by (fastforce simp: infinite_Ico)

   562

   563 lemma infinite_Ioc_iff [simp]: "infinite {a<..b} \<longleftrightarrow> a < b"

   564   using not_less_iff_gr_or_eq by (fastforce simp: infinite_Ioc)

   565

   566 end

   567

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

   569 proof

   570   assume "finite {..< a}"

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

   572     by auto

   573   obtain x where "x < a"

   574     using lt_ex by auto

   575

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

   577     using lt_ex by auto

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

   579     using \<open>x < a\<close> by fact

   580   also note \<open>x < a\<close>

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

   582     by fact

   583   with \<open>y < Min {..< a}\<close> show False by auto

   584 qed

   585

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

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

   588   by (auto simp: subset_eq less_imp_le)

   589

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

   591 proof

   592   assume "finite {a <..}"

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

   594     by auto

   595

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

   597     using gt_ex by auto

   598

   599   obtain x where x: "a < x"

   600     using gt_ex by auto

   601   also from x have "x \<le> Max {a <..}"

   602     by fact

   603   also note \<open>Max {a <..} < y\<close>

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

   605     by fact

   606   with \<open>Max {a <..} < y\<close> show False by auto

   607 qed

   608

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

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

   611   by (auto simp: subset_eq less_imp_le)

   612

   613 subsubsection \<open>Intersection\<close>

   614

   615 context linorder

   616 begin

   617

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

   619 by auto

   620

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

   622 by auto

   623

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

   625 by auto

   626

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

   628 by auto

   629

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

   631 by auto

   632

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

   634 by auto

   635

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

   637 by auto

   638

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

   640 by auto

   641

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

   643   by (auto simp: min_def)

   644

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

   646   by auto

   647

   648 end

   649

   650 context complete_lattice

   651 begin

   652

   653 lemma

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

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

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

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

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

   659   by (auto intro!: Sup_eqI)

   660

   661 lemma

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

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

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

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

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

   667   by (auto intro!: Inf_eqI)

   668

   669 end

   670

   671 lemma

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

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

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

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

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

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

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

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

   680

   681 subsection \<open>Intervals of natural numbers\<close>

   682

   683 subsubsection \<open>The Constant @{term lessThan}\<close>

   684

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

   686 by (simp add: lessThan_def)

   687

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

   689 by (simp add: lessThan_def less_Suc_eq, blast)

   690

   691 text \<open>The following proof is convenient in induction proofs where

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

   693 @{term "{..<Suc n}"} to @{term "0::nat"} and @{term "{..< n}"}.\<close>

   694

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

   696   by auto

   697

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

   699   by (auto simp: image_iff less_Suc_eq_0_disj)

   700

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

   702 by (simp add: lessThan_def atMost_def less_Suc_eq_le)

   703

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

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

   706

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

   708 by blast

   709

   710 subsubsection \<open>The Constant @{term greaterThan}\<close>

   711

   712 lemma greaterThan_0: "greaterThan 0 = range Suc"

   713 apply (simp add: greaterThan_def)

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

   715 done

   716

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

   718 apply (simp add: greaterThan_def)

   719 apply (auto elim: linorder_neqE)

   720 done

   721

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

   723 by blast

   724

   725 subsubsection \<open>The Constant @{term atLeast}\<close>

   726

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

   728 by (unfold atLeast_def UNIV_def, simp)

   729

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

   731 apply (simp add: atLeast_def)

   732 apply (simp add: Suc_le_eq)

   733 apply (simp add: order_le_less, blast)

   734 done

   735

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

   737   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)

   738

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

   740 by blast

   741

   742 subsubsection \<open>The Constant @{term atMost}\<close>

   743

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

   745 by (simp add: atMost_def)

   746

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

   748 apply (simp add: atMost_def)

   749 apply (simp add: less_Suc_eq order_le_less, blast)

   750 done

   751

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

   753 by blast

   754

   755 subsubsection \<open>The Constant @{term atLeastLessThan}\<close>

   756

   757 text\<open>The orientation of the following 2 rules is tricky. The lhs is

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

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

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

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

   762 specific concept to a more general one.\<close>

   763

   764 lemma atLeast0LessThan [code_abbrev]: "{0::nat..<n} = {..<n}"

   765 by(simp add:lessThan_def atLeastLessThan_def)

   766

   767 lemma atLeast0AtMost [code_abbrev]: "{0..n::nat} = {..n}"

   768 by(simp add:atMost_def atLeastAtMost_def)

   769

   770 lemma lessThan_atLeast0:

   771   "{..<n} = {0::nat..<n}"

   772   by (simp add: atLeast0LessThan)

   773

   774 lemma atMost_atLeast0:

   775   "{..n} = {0::nat..n}"

   776   by (simp add: atLeast0AtMost)

   777

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

   779 by (simp add: atLeastLessThan_def)

   780

   781 lemma atLeast0_lessThan_Suc:

   782   "{0..<Suc n} = insert n {0..<n}"

   783   by (simp add: atLeast0LessThan lessThan_Suc)

   784

   785 lemma atLeast0_lessThan_Suc_eq_insert_0:

   786   "{0..<Suc n} = insert 0 (Suc  {0..<n})"

   787   by (simp add: atLeast0LessThan lessThan_Suc_eq_insert_0)

   788

   789

   790 subsubsection \<open>The Constant @{term atLeastAtMost}\<close>

   791

   792 lemma atLeast0_atMost_Suc:

   793   "{0..Suc n} = insert (Suc n) {0..n}"

   794   by (simp add: atLeast0AtMost atMost_Suc)

   795

   796 lemma atLeast0_atMost_Suc_eq_insert_0:

   797   "{0..Suc n} = insert 0 (Suc  {0..n})"

   798   by (simp add: atLeast0AtMost Iic_Suc_eq_insert_0)

   799

   800

   801 subsubsection \<open>Intervals of nats with @{term Suc}\<close>

   802

   803 text\<open>Not a simprule because the RHS is too messy.\<close>

   804 lemma atLeastLessThanSuc:

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

   806 by (auto simp add: atLeastLessThan_def)

   807

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

   809 by (auto simp add: atLeastLessThan_def)

   810 (*

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

   812 by (induct k, simp_all add: atLeastLessThanSuc)

   813

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

   815 by (auto simp add: atLeastLessThan_def)

   816 *)

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

   818   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)

   819

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

   821   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def

   822     greaterThanAtMost_def)

   823

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

   825   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def

   826     greaterThanLessThan_def)

   827

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

   829 by (auto simp add: atLeastAtMost_def)

   830

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

   832 by auto

   833

   834 text \<open>The analogous result is useful on @{typ int}:\<close>

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

   836 lemma atLeastAtMostPlus1_int_conv:

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

   838   by (auto intro: set_eqI)

   839

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

   841   apply (induct k)

   842   apply (simp_all add: atLeastLessThanSuc)

   843   done

   844

   845 subsubsection \<open>Intervals and numerals\<close>

   846

   847 lemma lessThan_nat_numeral:  \<comment>\<open>Evaluation for specific numerals\<close>

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

   849   by (simp add: numeral_eq_Suc lessThan_Suc)

   850

   851 lemma atMost_nat_numeral:  \<comment>\<open>Evaluation for specific numerals\<close>

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

   853   by (simp add: numeral_eq_Suc atMost_Suc)

   854

   855 lemma atLeastLessThan_nat_numeral:  \<comment>\<open>Evaluation for specific numerals\<close>

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

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

   858                  else {})"

   859   by (simp add: numeral_eq_Suc atLeastLessThanSuc)

   860

   861 subsubsection \<open>Image\<close>

   862

   863 lemma image_add_atLeastAtMost [simp]:

   864   fixes k ::"'a::linordered_semidom"

   865   shows "(\<lambda>n. n+k)  {i..j} = {i+k..j+k}" (is "?A = ?B")

   866 proof

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

   868 next

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

   870   proof

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

   872     hence "n - k : {i..j}"

   873       by (auto simp: add_le_imp_le_diff add_le_add_imp_diff_le)

   874     moreover have "n = (n - k) + k" using a

   875     proof -

   876       have "k + i \<le> n"

   877         by (metis a add.commute atLeastAtMost_iff)

   878       hence "k + (n - k) = n"

   879         by (metis (no_types) ab_semigroup_add_class.add_ac(1) add_diff_cancel_left' le_add_diff_inverse)

   880       thus ?thesis

   881         by (simp add: add.commute)

   882     qed

   883     ultimately show "n : ?A" by blast

   884   qed

   885 qed

   886

   887 lemma image_diff_atLeastAtMost [simp]:

   888   fixes d::"'a::linordered_idom" shows "(op - d  {a..b}) = {d-b..d-a}"

   889   apply auto

   890   apply (rule_tac x="d-x" in rev_image_eqI, auto)

   891   done

   892

   893 lemma image_mult_atLeastAtMost [simp]:

   894   fixes d::"'a::linordered_field"

   895   assumes "d>0" shows "(op * d  {a..b}) = {d*a..d*b}"

   896   using assms

   897   by (auto simp: field_simps mult_le_cancel_right intro: rev_image_eqI [where x="x/d" for x])

   898

   899 lemma image_affinity_atLeastAtMost:

   900   fixes c :: "'a::linordered_field"

   901   shows "((\<lambda>x. m*x + c)  {a..b}) = (if {a..b}={} then {}

   902             else if 0 \<le> m then {m*a + c .. m *b + c}

   903             else {m*b + c .. m*a + c})"

   904   apply (case_tac "m=0", auto simp: mult_le_cancel_left)

   905   apply (rule_tac x="inverse m*(x-c)" in rev_image_eqI, auto simp: field_simps)

   906   apply (rule_tac x="inverse m*(x-c)" in rev_image_eqI, auto simp: field_simps)

   907   done

   908

   909 lemma image_affinity_atLeastAtMost_diff:

   910   fixes c :: "'a::linordered_field"

   911   shows "((\<lambda>x. m*x - c)  {a..b}) = (if {a..b}={} then {}

   912             else if 0 \<le> m then {m*a - c .. m*b - c}

   913             else {m*b - c .. m*a - c})"

   914   using image_affinity_atLeastAtMost [of m "-c" a b]

   915   by simp

   916

   917 lemma image_affinity_atLeastAtMost_div:

   918   fixes c :: "'a::linordered_field"

   919   shows "((\<lambda>x. x/m + c)  {a..b}) = (if {a..b}={} then {}

   920             else if 0 \<le> m then {a/m + c .. b/m + c}

   921             else {b/m + c .. a/m + c})"

   922   using image_affinity_atLeastAtMost [of "inverse m" c a b]

   923   by (simp add: field_class.field_divide_inverse algebra_simps)

   924

   925 lemma image_affinity_atLeastAtMost_div_diff:

   926   fixes c :: "'a::linordered_field"

   927   shows "((\<lambda>x. x/m - c)  {a..b}) = (if {a..b}={} then {}

   928             else if 0 \<le> m then {a/m - c .. b/m - c}

   929             else {b/m - c .. a/m - c})"

   930   using image_affinity_atLeastAtMost_diff [of "inverse m" c a b]

   931   by (simp add: field_class.field_divide_inverse algebra_simps)

   932

   933 lemma image_add_atLeastLessThan:

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

   935 proof

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

   937 next

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

   939   proof

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

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

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

   943     ultimately show "n : ?A" by blast

   944   qed

   945 qed

   946

   947 corollary image_Suc_lessThan:

   948   "Suc  {..<n} = {1..n}"

   949   using image_add_atLeastLessThan [of 1 0 n]

   950   by (auto simp add: lessThan_Suc_atMost atLeast0LessThan)

   951

   952 corollary image_Suc_atMost:

   953   "Suc  {..n} = {1..Suc n}"

   954   using image_add_atLeastLessThan [of 1 0 "Suc n"]

   955   by (auto simp add: lessThan_Suc_atMost atLeast0LessThan)

   956

   957 corollary image_Suc_atLeastAtMost[simp]:

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

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

   960

   961 corollary image_Suc_atLeastLessThan[simp]:

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

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

   964

   965 lemma atLeast1_lessThan_eq_remove0:

   966   "{Suc 0..<n} = {..<n} - {0}"

   967   by auto

   968

   969 lemma atLeast1_atMost_eq_remove0:

   970   "{Suc 0..n} = {..n} - {0}"

   971   by auto

   972

   973 lemma image_add_int_atLeastLessThan:

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

   975   apply (auto simp add: image_def)

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

   977   apply auto

   978   done

   979

   980 lemma image_minus_const_atLeastLessThan_nat:

   981   fixes c :: nat

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

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

   984     (is "_ = ?right")

   985 proof safe

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

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

   988   proof cases

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

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

   991   next

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

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

   994   qed

   995 qed auto

   996

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

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

   999

  1000 context ordered_ab_group_add

  1001 begin

  1002

  1003 lemma

  1004   fixes x :: 'a

  1005   shows image_uminus_greaterThan[simp]: "uminus  {x<..} = {..<-x}"

  1006   and image_uminus_atLeast[simp]: "uminus  {x..} = {..-x}"

  1007 proof safe

  1008   fix y assume "y < -x"

  1009   hence *:  "x < -y" using neg_less_iff_less[of "-y" x] by simp

  1010   have "- (-y) \<in> uminus  {x<..}"

  1011     by (rule imageI) (simp add: *)

  1012   thus "y \<in> uminus  {x<..}" by simp

  1013 next

  1014   fix y assume "y \<le> -x"

  1015   have "- (-y) \<in> uminus  {x..}"

  1016     by (rule imageI) (insert \<open>y \<le> -x\<close>[THEN le_imp_neg_le], simp)

  1017   thus "y \<in> uminus  {x..}" by simp

  1018 qed simp_all

  1019

  1020 lemma

  1021   fixes x :: 'a

  1022   shows image_uminus_lessThan[simp]: "uminus  {..<x} = {-x<..}"

  1023   and image_uminus_atMost[simp]: "uminus  {..x} = {-x..}"

  1024 proof -

  1025   have "uminus  {..<x} = uminus  uminus  {-x<..}"

  1026     and "uminus  {..x} = uminus  uminus  {-x..}" by simp_all

  1027   thus "uminus  {..<x} = {-x<..}" and "uminus  {..x} = {-x..}"

  1028     by (simp_all add: image_image

  1029         del: image_uminus_greaterThan image_uminus_atLeast)

  1030 qed

  1031

  1032 lemma

  1033   fixes x :: 'a

  1034   shows image_uminus_atLeastAtMost[simp]: "uminus  {x..y} = {-y..-x}"

  1035   and image_uminus_greaterThanAtMost[simp]: "uminus  {x<..y} = {-y..<-x}"

  1036   and image_uminus_atLeastLessThan[simp]: "uminus  {x..<y} = {-y<..-x}"

  1037   and image_uminus_greaterThanLessThan[simp]: "uminus  {x<..<y} = {-y<..<-x}"

  1038   by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def

  1039       greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)

  1040 end

  1041

  1042 subsubsection \<open>Finiteness\<close>

  1043

  1044 lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"

  1045   by (induct k) (simp_all add: lessThan_Suc)

  1046

  1047 lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"

  1048   by (induct k) (simp_all add: atMost_Suc)

  1049

  1050 lemma finite_greaterThanLessThan [iff]:

  1051   fixes l :: nat shows "finite {l<..<u}"

  1052 by (simp add: greaterThanLessThan_def)

  1053

  1054 lemma finite_atLeastLessThan [iff]:

  1055   fixes l :: nat shows "finite {l..<u}"

  1056 by (simp add: atLeastLessThan_def)

  1057

  1058 lemma finite_greaterThanAtMost [iff]:

  1059   fixes l :: nat shows "finite {l<..u}"

  1060 by (simp add: greaterThanAtMost_def)

  1061

  1062 lemma finite_atLeastAtMost [iff]:

  1063   fixes l :: nat shows "finite {l..u}"

  1064 by (simp add: atLeastAtMost_def)

  1065

  1066 text \<open>A bounded set of natural numbers is finite.\<close>

  1067 lemma bounded_nat_set_is_finite:

  1068   "(ALL i:N. i < (n::nat)) ==> finite N"

  1069 apply (rule finite_subset)

  1070  apply (rule_tac [2] finite_lessThan, auto)

  1071 done

  1072

  1073 text \<open>A set of natural numbers is finite iff it is bounded.\<close>

  1074 lemma finite_nat_set_iff_bounded:

  1075   "finite(N::nat set) = (EX m. ALL n:N. n<m)" (is "?F = ?B")

  1076 proof

  1077   assume f:?F  show ?B

  1078     using Max_ge[OF \<open>?F\<close>, simplified less_Suc_eq_le[symmetric]] by blast

  1079 next

  1080   assume ?B show ?F using \<open>?B\<close> by(blast intro:bounded_nat_set_is_finite)

  1081 qed

  1082

  1083 lemma finite_nat_set_iff_bounded_le:

  1084   "finite(N::nat set) = (EX m. ALL n:N. n<=m)"

  1085 apply(simp add:finite_nat_set_iff_bounded)

  1086 apply(blast dest:less_imp_le_nat le_imp_less_Suc)

  1087 done

  1088

  1089 lemma finite_less_ub:

  1090      "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"

  1091 by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)

  1092

  1093 lemma bounded_Max_nat:

  1094   fixes P :: "nat \<Rightarrow> bool"

  1095   assumes x: "P x" and M: "\<And>x. P x \<Longrightarrow> x \<le> M"

  1096   obtains m where "P m" "\<And>x. P x \<Longrightarrow> x \<le> m"

  1097 proof -

  1098   have "finite {x. P x}"

  1099     using M finite_nat_set_iff_bounded_le by auto

  1100   then have "Max {x. P x} \<in> {x. P x}"

  1101     using Max_in x by auto

  1102   then show ?thesis

  1103     by (simp add: \<open>finite {x. P x}\<close> that)

  1104 qed

  1105

  1106

  1107 text\<open>Any subset of an interval of natural numbers the size of the

  1108 subset is exactly that interval.\<close>

  1109

  1110 lemma subset_card_intvl_is_intvl:

  1111   assumes "A \<subseteq> {k..<k + card A}"

  1112   shows "A = {k..<k + card A}"

  1113 proof (cases "finite A")

  1114   case True

  1115   from this and assms show ?thesis

  1116   proof (induct A rule: finite_linorder_max_induct)

  1117     case empty thus ?case by auto

  1118   next

  1119     case (insert b A)

  1120     hence *: "b \<notin> A" by auto

  1121     with insert have "A <= {k..<k + card A}" and "b = k + card A"

  1122       by fastforce+

  1123     with insert * show ?case by auto

  1124   qed

  1125 next

  1126   case False

  1127   with assms show ?thesis by simp

  1128 qed

  1129

  1130

  1131 subsubsection \<open>Proving Inclusions and Equalities between Unions\<close>

  1132

  1133 lemma UN_le_eq_Un0:

  1134   "(\<Union>i\<le>n::nat. M i) = (\<Union>i\<in>{1..n}. M i) \<union> M 0" (is "?A = ?B")

  1135 proof

  1136   show "?A <= ?B"

  1137   proof

  1138     fix x assume "x : ?A"

  1139     then obtain i where i: "i\<le>n" "x : M i" by auto

  1140     show "x : ?B"

  1141     proof(cases i)

  1142       case 0 with i show ?thesis by simp

  1143     next

  1144       case (Suc j) with i show ?thesis by auto

  1145     qed

  1146   qed

  1147 next

  1148   show "?B <= ?A" by fastforce

  1149 qed

  1150

  1151 lemma UN_le_add_shift:

  1152   "(\<Union>i\<le>n::nat. M(i+k)) = (\<Union>i\<in>{k..n+k}. M i)" (is "?A = ?B")

  1153 proof

  1154   show "?A <= ?B" by fastforce

  1155 next

  1156   show "?B <= ?A"

  1157   proof

  1158     fix x assume "x : ?B"

  1159     then obtain i where i: "i : {k..n+k}" "x : M(i)" by auto

  1160     hence "i-k\<le>n & x : M((i-k)+k)" by auto

  1161     thus "x : ?A" by blast

  1162   qed

  1163 qed

  1164

  1165 lemma UN_UN_finite_eq: "(\<Union>n::nat. \<Union>i\<in>{0..<n}. A i) = (\<Union>n. A n)"

  1166   by (auto simp add: atLeast0LessThan)

  1167

  1168 lemma UN_finite_subset:

  1169   "(\<And>n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> C) \<Longrightarrow> (\<Union>n. A n) \<subseteq> C"

  1170   by (subst UN_UN_finite_eq [symmetric]) blast

  1171

  1172 lemma UN_finite2_subset:

  1173   assumes "\<And>n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> (\<Union>i\<in>{0..<n + k}. B i)"

  1174   shows "(\<Union>n. A n) \<subseteq> (\<Union>n. B n)"

  1175 proof (rule UN_finite_subset, rule)

  1176   fix n and a

  1177   from assms have "(\<Union>i\<in>{0..<n}. A i) \<subseteq> (\<Union>i\<in>{0..<n + k}. B i)" .

  1178   moreover assume "a \<in> (\<Union>i\<in>{0..<n}. A i)"

  1179   ultimately have "a \<in> (\<Union>i\<in>{0..<n + k}. B i)" by blast

  1180   then show "a \<in> (\<Union>i. B i)" by (auto simp add: UN_UN_finite_eq)

  1181 qed

  1182

  1183 lemma UN_finite2_eq:

  1184   "(\<And>n::nat. (\<Union>i\<in>{0..<n}. A i) = (\<Union>i\<in>{0..<n + k}. B i)) \<Longrightarrow>

  1185     (\<Union>n. A n) = (\<Union>n. B n)"

  1186   apply (rule subset_antisym)

  1187    apply (rule UN_finite2_subset, blast)

  1188   apply (rule UN_finite2_subset [where k=k])

  1189   apply (force simp add: atLeastLessThan_add_Un [of 0])

  1190   done

  1191

  1192

  1193 subsubsection \<open>Cardinality\<close>

  1194

  1195 lemma card_lessThan [simp]: "card {..<u} = u"

  1196   by (induct u, simp_all add: lessThan_Suc)

  1197

  1198 lemma card_atMost [simp]: "card {..u} = Suc u"

  1199   by (simp add: lessThan_Suc_atMost [THEN sym])

  1200

  1201 lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"

  1202 proof -

  1203   have "{l..<u} = (%x. x + l)  {..<u-l}"

  1204     apply (auto simp add: image_def atLeastLessThan_def lessThan_def)

  1205     apply (rule_tac x = "x - l" in exI)

  1206     apply arith

  1207     done

  1208   then have "card {l..<u} = card {..<u-l}"

  1209     by (simp add: card_image inj_on_def)

  1210   then show ?thesis

  1211     by simp

  1212 qed

  1213

  1214 lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"

  1215   by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)

  1216

  1217 lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"

  1218   by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)

  1219

  1220 lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"

  1221   by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)

  1222

  1223 lemma subset_eq_atLeast0_lessThan_finite:

  1224   fixes n :: nat

  1225   assumes "N \<subseteq> {0..<n}"

  1226   shows "finite N"

  1227   using assms finite_atLeastLessThan by (rule finite_subset)

  1228

  1229 lemma subset_eq_atLeast0_atMost_finite:

  1230   fixes n :: nat

  1231   assumes "N \<subseteq> {0..n}"

  1232   shows "finite N"

  1233   using assms finite_atLeastAtMost by (rule finite_subset)

  1234

  1235 lemma ex_bij_betw_nat_finite:

  1236   "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"

  1237 apply(drule finite_imp_nat_seg_image_inj_on)

  1238 apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)

  1239 done

  1240

  1241 lemma ex_bij_betw_finite_nat:

  1242   "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"

  1243 by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)

  1244

  1245 lemma finite_same_card_bij:

  1246   "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> EX h. bij_betw h A B"

  1247 apply(drule ex_bij_betw_finite_nat)

  1248 apply(drule ex_bij_betw_nat_finite)

  1249 apply(auto intro!:bij_betw_trans)

  1250 done

  1251

  1252 lemma ex_bij_betw_nat_finite_1:

  1253   "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"

  1254 by (rule finite_same_card_bij) auto

  1255

  1256 lemma bij_betw_iff_card:

  1257   assumes "finite A" "finite B"

  1258   shows "(\<exists>f. bij_betw f A B) \<longleftrightarrow> (card A = card B)"

  1259 proof

  1260   assume "card A = card B"

  1261   moreover obtain f where "bij_betw f A {0 ..< card A}"

  1262     using assms ex_bij_betw_finite_nat by blast

  1263   moreover obtain g where "bij_betw g {0 ..< card B} B"

  1264     using assms ex_bij_betw_nat_finite by blast

  1265   ultimately have "bij_betw (g o f) A B"

  1266     by (auto simp: bij_betw_trans)

  1267   thus "(\<exists>f. bij_betw f A B)" by blast

  1268 qed (auto simp: bij_betw_same_card)

  1269

  1270 lemma inj_on_iff_card_le:

  1271   assumes FIN: "finite A" and FIN': "finite B"

  1272   shows "(\<exists>f. inj_on f A \<and> f  A \<le> B) = (card A \<le> card B)"

  1273 proof (safe intro!: card_inj_on_le)

  1274   assume *: "card A \<le> card B"

  1275   obtain f where 1: "inj_on f A" and 2: "f  A = {0 ..< card A}"

  1276   using FIN ex_bij_betw_finite_nat unfolding bij_betw_def by force

  1277   moreover obtain g where "inj_on g {0 ..< card B}" and 3: "g  {0 ..< card B} = B"

  1278   using FIN' ex_bij_betw_nat_finite unfolding bij_betw_def by force

  1279   ultimately have "inj_on g (f  A)" using subset_inj_on[of g _ "f  A"] * by force

  1280   hence "inj_on (g o f) A" using 1 comp_inj_on by blast

  1281   moreover

  1282   {have "{0 ..< card A} \<le> {0 ..< card B}" using * by force

  1283    with 2 have "f  A  \<le> {0 ..< card B}" by blast

  1284    hence "(g o f)  A \<le> B" unfolding comp_def using 3 by force

  1285   }

  1286   ultimately show "(\<exists>f. inj_on f A \<and> f  A \<le> B)" by blast

  1287 qed (insert assms, auto)

  1288

  1289 lemma subset_eq_atLeast0_lessThan_card:

  1290   fixes n :: nat

  1291   assumes "N \<subseteq> {0..<n}"

  1292   shows "card N \<le> n"

  1293 proof -

  1294   from assms finite_lessThan have "card N \<le> card {0..<n}"

  1295     using card_mono by blast

  1296   then show ?thesis by simp

  1297 qed

  1298

  1299

  1300 subsection \<open>Intervals of integers\<close>

  1301

  1302 lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"

  1303   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)

  1304

  1305 lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"

  1306   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)

  1307

  1308 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:

  1309     "{l+1..<u} = {l<..<u::int}"

  1310   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)

  1311

  1312 subsubsection \<open>Finiteness\<close>

  1313

  1314 lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>

  1315     {(0::int)..<u} = int  {..<nat u}"

  1316   apply (unfold image_def lessThan_def)

  1317   apply auto

  1318   apply (rule_tac x = "nat x" in exI)

  1319   apply (auto simp add: zless_nat_eq_int_zless [THEN sym])

  1320   done

  1321

  1322 lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"

  1323   apply (cases "0 \<le> u")

  1324   apply (subst image_atLeastZeroLessThan_int, assumption)

  1325   apply (rule finite_imageI)

  1326   apply auto

  1327   done

  1328

  1329 lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"

  1330   apply (subgoal_tac "(%x. x + l)  {0..<u-l} = {l..<u}")

  1331   apply (erule subst)

  1332   apply (rule finite_imageI)

  1333   apply (rule finite_atLeastZeroLessThan_int)

  1334   apply (rule image_add_int_atLeastLessThan)

  1335   done

  1336

  1337 lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"

  1338   by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)

  1339

  1340 lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"

  1341   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)

  1342

  1343 lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"

  1344   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)

  1345

  1346

  1347 subsubsection \<open>Cardinality\<close>

  1348

  1349 lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"

  1350   apply (cases "0 \<le> u")

  1351   apply (subst image_atLeastZeroLessThan_int, assumption)

  1352   apply (subst card_image)

  1353   apply (auto simp add: inj_on_def)

  1354   done

  1355

  1356 lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"

  1357   apply (subgoal_tac "card {l..<u} = card {0..<u-l}")

  1358   apply (erule ssubst, rule card_atLeastZeroLessThan_int)

  1359   apply (subgoal_tac "(%x. x + l)  {0..<u-l} = {l..<u}")

  1360   apply (erule subst)

  1361   apply (rule card_image)

  1362   apply (simp add: inj_on_def)

  1363   apply (rule image_add_int_atLeastLessThan)

  1364   done

  1365

  1366 lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"

  1367 apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])

  1368 apply (auto simp add: algebra_simps)

  1369 done

  1370

  1371 lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"

  1372 by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)

  1373

  1374 lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"

  1375 by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)

  1376

  1377 lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"

  1378 proof -

  1379   have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto

  1380   with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)

  1381 qed

  1382

  1383 lemma card_less:

  1384 assumes zero_in_M: "0 \<in> M"

  1385 shows "card {k \<in> M. k < Suc i} \<noteq> 0"

  1386 proof -

  1387   from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto

  1388   with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)

  1389 qed

  1390

  1391 lemma card_less_Suc2: "0 \<notin> M \<Longrightarrow> card {k. Suc k \<in> M \<and> k < i} = card {k \<in> M. k < Suc i}"

  1392 apply (rule card_bij_eq [of Suc _ _ "\<lambda>x. x - Suc 0"])

  1393 apply auto

  1394 apply (rule inj_on_diff_nat)

  1395 apply auto

  1396 apply (case_tac x)

  1397 apply auto

  1398 apply (case_tac xa)

  1399 apply auto

  1400 apply (case_tac xa)

  1401 apply auto

  1402 done

  1403

  1404 lemma card_less_Suc:

  1405   assumes zero_in_M: "0 \<in> M"

  1406     shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"

  1407 proof -

  1408   from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp

  1409   hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"

  1410     by (auto simp only: insert_Diff)

  1411   have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto

  1412   from finite_M_bounded_by_nat[of "\<lambda>x. x \<in> M" "Suc i"]

  1413   have "Suc (card {k. Suc k \<in> M \<and> k < i}) = card (insert 0 ({k \<in> M. k < Suc i} - {0}))"

  1414     apply (subst card_insert)

  1415     apply simp_all

  1416     apply (subst b)

  1417     apply (subst card_less_Suc2[symmetric])

  1418     apply simp_all

  1419     done

  1420   with c show ?thesis by simp

  1421 qed

  1422

  1423

  1424 subsection \<open>Lemmas useful with the summation operator sum\<close>

  1425

  1426 text \<open>For examples, see Algebra/poly/UnivPoly2.thy\<close>

  1427

  1428 subsubsection \<open>Disjoint Unions\<close>

  1429

  1430 text \<open>Singletons and open intervals\<close>

  1431

  1432 lemma ivl_disj_un_singleton:

  1433   "{l::'a::linorder} Un {l<..} = {l..}"

  1434   "{..<u} Un {u::'a::linorder} = {..u}"

  1435   "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"

  1436   "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"

  1437   "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"

  1438   "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"

  1439 by auto

  1440

  1441 text \<open>One- and two-sided intervals\<close>

  1442

  1443 lemma ivl_disj_un_one:

  1444   "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"

  1445   "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"

  1446   "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"

  1447   "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"

  1448   "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"

  1449   "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"

  1450   "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"

  1451   "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"

  1452 by auto

  1453

  1454 text \<open>Two- and two-sided intervals\<close>

  1455

  1456 lemma ivl_disj_un_two:

  1457   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"

  1458   "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"

  1459   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"

  1460   "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"

  1461   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"

  1462   "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"

  1463   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"

  1464   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"

  1465 by auto

  1466

  1467 lemma ivl_disj_un_two_touch:

  1468   "[| (l::'a::linorder) < m; m < u |] ==> {l<..m} Un {m..<u} = {l<..<u}"

  1469   "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m..<u} = {l..<u}"

  1470   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..m} Un {m..u} = {l<..u}"

  1471   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m..u} = {l..u}"

  1472 by auto

  1473

  1474 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two ivl_disj_un_two_touch

  1475

  1476 subsubsection \<open>Disjoint Intersections\<close>

  1477

  1478 text \<open>One- and two-sided intervals\<close>

  1479

  1480 lemma ivl_disj_int_one:

  1481   "{..l::'a::order} Int {l<..<u} = {}"

  1482   "{..<l} Int {l..<u} = {}"

  1483   "{..l} Int {l<..u} = {}"

  1484   "{..<l} Int {l..u} = {}"

  1485   "{l<..u} Int {u<..} = {}"

  1486   "{l<..<u} Int {u..} = {}"

  1487   "{l..u} Int {u<..} = {}"

  1488   "{l..<u} Int {u..} = {}"

  1489   by auto

  1490

  1491 text \<open>Two- and two-sided intervals\<close>

  1492

  1493 lemma ivl_disj_int_two:

  1494   "{l::'a::order<..<m} Int {m..<u} = {}"

  1495   "{l<..m} Int {m<..<u} = {}"

  1496   "{l..<m} Int {m..<u} = {}"

  1497   "{l..m} Int {m<..<u} = {}"

  1498   "{l<..<m} Int {m..u} = {}"

  1499   "{l<..m} Int {m<..u} = {}"

  1500   "{l..<m} Int {m..u} = {}"

  1501   "{l..m} Int {m<..u} = {}"

  1502   by auto

  1503

  1504 lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two

  1505

  1506 subsubsection \<open>Some Differences\<close>

  1507

  1508 lemma ivl_diff[simp]:

  1509  "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"

  1510 by(auto)

  1511

  1512 lemma (in linorder) lessThan_minus_lessThan [simp]:

  1513   "{..< n} - {..< m} = {m ..< n}"

  1514   by auto

  1515

  1516 lemma (in linorder) atLeastAtMost_diff_ends:

  1517   "{a..b} - {a, b} = {a<..<b}"

  1518   by auto

  1519

  1520

  1521 subsubsection \<open>Some Subset Conditions\<close>

  1522

  1523 lemma ivl_subset [simp]:

  1524  "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"

  1525 apply(auto simp:linorder_not_le)

  1526 apply(rule ccontr)

  1527 apply(insert linorder_le_less_linear[of i n])

  1528 apply(clarsimp simp:linorder_not_le)

  1529 apply(fastforce)

  1530 done

  1531

  1532

  1533 subsection \<open>Generic big monoid operation over intervals\<close>

  1534

  1535 lemma inj_on_add_nat' [simp]:

  1536   "inj_on (plus k) N" for k :: nat

  1537   by rule simp

  1538

  1539 context comm_monoid_set

  1540 begin

  1541

  1542 lemma atLeast_lessThan_shift_bounds:

  1543   fixes m n k :: nat

  1544   shows "F g {m + k..<n + k} = F (g \<circ> plus k) {m..<n}"

  1545 proof -

  1546   have "{m + k..<n + k} = plus k  {m..<n}"

  1547     by (auto simp add: image_add_atLeastLessThan [symmetric])

  1548   also have "F g (plus k  {m..<n}) = F (g \<circ> plus k) {m..<n}"

  1549     by (rule reindex) simp

  1550   finally show ?thesis .

  1551 qed

  1552

  1553 lemma atLeast_atMost_shift_bounds:

  1554   fixes m n k :: nat

  1555   shows "F g {m + k..n + k} = F (g \<circ> plus k) {m..n}"

  1556 proof -

  1557   have "{m + k..n + k} = plus k  {m..n}"

  1558     by (auto simp del: image_add_atLeastAtMost simp add: image_add_atLeastAtMost [symmetric])

  1559   also have "F g (plus k  {m..n}) = F (g \<circ> plus k) {m..n}"

  1560     by (rule reindex) simp

  1561   finally show ?thesis .

  1562 qed

  1563

  1564 lemma atLeast_Suc_lessThan_Suc_shift:

  1565   "F g {Suc m..<Suc n} = F (g \<circ> Suc) {m..<n}"

  1566   using atLeast_lessThan_shift_bounds [of _ _ 1] by simp

  1567

  1568 lemma atLeast_Suc_atMost_Suc_shift:

  1569   "F g {Suc m..Suc n} = F (g \<circ> Suc) {m..n}"

  1570   using atLeast_atMost_shift_bounds [of _ _ 1] by simp

  1571

  1572 lemma atLeast0_lessThan_Suc:

  1573   "F g {0..<Suc n} = F g {0..<n} \<^bold>* g n"

  1574   by (simp add: atLeast0_lessThan_Suc ac_simps)

  1575

  1576 lemma atLeast0_atMost_Suc:

  1577   "F g {0..Suc n} = F g {0..n} \<^bold>* g (Suc n)"

  1578   by (simp add: atLeast0_atMost_Suc ac_simps)

  1579

  1580 lemma atLeast0_lessThan_Suc_shift:

  1581   "F g {0..<Suc n} = g 0 \<^bold>* F (g \<circ> Suc) {0..<n}"

  1582   by (simp add: atLeast0_lessThan_Suc_eq_insert_0 atLeast_Suc_lessThan_Suc_shift)

  1583

  1584 lemma atLeast0_atMost_Suc_shift:

  1585   "F g {0..Suc n} = g 0 \<^bold>* F (g \<circ> Suc) {0..n}"

  1586   by (simp add: atLeast0_atMost_Suc_eq_insert_0 atLeast_Suc_atMost_Suc_shift)

  1587

  1588 lemma ivl_cong:

  1589   "a = c \<Longrightarrow> b = d \<Longrightarrow> (\<And>x. c \<le> x \<Longrightarrow> x < d \<Longrightarrow> g x = h x)

  1590     \<Longrightarrow> F g {a..<b} = F h {c..<d}"

  1591   by (rule cong) simp_all

  1592

  1593 lemma atLeast_lessThan_shift_0:

  1594   fixes m n p :: nat

  1595   shows "F g {m..<n} = F (g \<circ> plus m) {0..<n - m}"

  1596   using atLeast_lessThan_shift_bounds [of g 0 m "n - m"]

  1597   by (cases "m \<le> n") simp_all

  1598

  1599 lemma atLeast_atMost_shift_0:

  1600   fixes m n p :: nat

  1601   assumes "m \<le> n"

  1602   shows "F g {m..n} = F (g \<circ> plus m) {0..n - m}"

  1603   using assms atLeast_atMost_shift_bounds [of g 0 m "n - m"] by simp

  1604

  1605 lemma atLeast_lessThan_concat:

  1606   fixes m n p :: nat

  1607   shows "m \<le> n \<Longrightarrow> n \<le> p \<Longrightarrow> F g {m..<n} \<^bold>* F g {n..<p} = F g {m..<p}"

  1608   by (simp add: union_disjoint [symmetric] ivl_disj_un)

  1609

  1610 lemma atLeast_lessThan_rev:

  1611   "F g {n..<m} = F (\<lambda>i. g (m + n - Suc i)) {n..<m}"

  1612   by (rule reindex_bij_witness [where i="\<lambda>i. m + n - Suc i" and j="\<lambda>i. m + n - Suc i"], auto)

  1613

  1614 lemma atLeast_atMost_rev:

  1615   fixes n m :: nat

  1616   shows "F g {n..m} = F (\<lambda>i. g (m + n - i)) {n..m}"

  1617   by (rule reindex_bij_witness [where i="\<lambda>i. m + n - i" and j="\<lambda>i. m + n - i"]) auto

  1618

  1619 lemma atLeast_lessThan_rev_at_least_Suc_atMost:

  1620   "F g {n..<m} = F (\<lambda>i. g (m + n - i)) {Suc n..m}"

  1621   unfolding atLeast_lessThan_rev [of g n m]

  1622   by (cases m) (simp_all add: atLeast_Suc_atMost_Suc_shift atLeastLessThanSuc_atLeastAtMost)

  1623

  1624 end

  1625

  1626

  1627 subsection \<open>Summation indexed over intervals\<close>

  1628

  1629 syntax (ASCII)

  1630   "_from_to_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(SUM _ = _.._./ _)" [0,0,0,10] 10)

  1631   "_from_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)

  1632   "_upt_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(SUM _<_./ _)" [0,0,10] 10)

  1633   "_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(SUM _<=_./ _)" [0,0,10] 10)

  1634

  1635 syntax (latex_sum output)

  1636   "_from_to_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1637  ("(3\<^latex>\<open>$\\sum_{\<close>_ = _\<^latex>\<open>}^{\<close>_\<^latex>\<open>}$\<close> _)" [0,0,0,10] 10)

  1638   "_from_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1639  ("(3\<^latex>\<open>$\\sum_{\<close>_ = _\<^latex>\<open>}^{<\<close>_\<^latex>\<open>}$\<close> _)" [0,0,0,10] 10)

  1640   "_upt_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1641  ("(3\<^latex>\<open>$\\sum_{\<close>_ < _\<^latex>\<open>}$\<close> _)" [0,0,10] 10)

  1642   "_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1643  ("(3\<^latex>\<open>$\\sum_{\<close>_ \<le> _\<^latex>\<open>}$\<close> _)" [0,0,10] 10)

  1644

  1645 syntax

  1646   "_from_to_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)

  1647   "_from_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)

  1648   "_upt_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sum>_<_./ _)" [0,0,10] 10)

  1649   "_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)

  1650

  1651 translations

  1652   "\<Sum>x=a..b. t" == "CONST sum (\<lambda>x. t) {a..b}"

  1653   "\<Sum>x=a..<b. t" == "CONST sum (\<lambda>x. t) {a..<b}"

  1654   "\<Sum>i\<le>n. t" == "CONST sum (\<lambda>i. t) {..n}"

  1655   "\<Sum>i<n. t" == "CONST sum (\<lambda>i. t) {..<n}"

  1656

  1657 text\<open>The above introduces some pretty alternative syntaxes for

  1658 summation over intervals:

  1659 \begin{center}

  1660 \begin{tabular}{lll}

  1661 Old & New & \LaTeX\\

  1662 @{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\

  1663 @{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\

  1664 @{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\

  1665 @{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}

  1666 \end{tabular}

  1667 \end{center}

  1668 The left column shows the term before introduction of the new syntax,

  1669 the middle column shows the new (default) syntax, and the right column

  1670 shows a special syntax. The latter is only meaningful for latex output

  1671 and has to be activated explicitly by setting the print mode to

  1672 \<open>latex_sum\<close> (e.g.\ via \<open>mode = latex_sum\<close> in

  1673 antiquotations). It is not the default \LaTeX\ output because it only

  1674 works well with italic-style formulae, not tt-style.

  1675

  1676 Note that for uniformity on @{typ nat} it is better to use

  1677 @{term"\<Sum>x::nat=0..<n. e"} rather than \<open>\<Sum>x<n. e\<close>: \<open>sum\<close> may

  1678 not provide all lemmas available for @{term"{m..<n}"} also in the

  1679 special form for @{term"{..<n}"}.\<close>

  1680

  1681 text\<open>This congruence rule should be used for sums over intervals as

  1682 the standard theorem @{text[source]sum.cong} does not work well

  1683 with the simplifier who adds the unsimplified premise @{term"x:B"} to

  1684 the context.\<close>

  1685

  1686 lemmas sum_ivl_cong = sum.ivl_cong

  1687

  1688 (* FIXME why are the following simp rules but the corresponding eqns

  1689 on intervals are not? *)

  1690

  1691 lemma sum_atMost_Suc [simp]:

  1692   "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f (Suc n)"

  1693   by (simp add: atMost_Suc ac_simps)

  1694

  1695 lemma sum_lessThan_Suc [simp]:

  1696   "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"

  1697   by (simp add: lessThan_Suc ac_simps)

  1698

  1699 lemma sum_cl_ivl_Suc [simp]:

  1700   "sum f {m..Suc n} = (if Suc n < m then 0 else sum f {m..n} + f(Suc n))"

  1701   by (auto simp: ac_simps atLeastAtMostSuc_conv)

  1702

  1703 lemma sum_op_ivl_Suc [simp]:

  1704   "sum f {m..<Suc n} = (if n < m then 0 else sum f {m..<n} + f(n))"

  1705   by (auto simp: ac_simps atLeastLessThanSuc)

  1706 (*

  1707 lemma sum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>

  1708     (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"

  1709 by (auto simp:ac_simps atLeastAtMostSuc_conv)

  1710 *)

  1711

  1712 lemma sum_head:

  1713   fixes n :: nat

  1714   assumes mn: "m <= n"

  1715   shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")

  1716 proof -

  1717   from mn

  1718   have "{m..n} = {m} \<union> {m<..n}"

  1719     by (auto intro: ivl_disj_un_singleton)

  1720   hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"

  1721     by (simp add: atLeast0LessThan)

  1722   also have "\<dots> = ?rhs" by simp

  1723   finally show ?thesis .

  1724 qed

  1725

  1726 lemma sum_head_Suc:

  1727   "m \<le> n \<Longrightarrow> sum f {m..n} = f m + sum f {Suc m..n}"

  1728 by (simp add: sum_head atLeastSucAtMost_greaterThanAtMost)

  1729

  1730 lemma sum_head_upt_Suc:

  1731   "m < n \<Longrightarrow> sum f {m..<n} = f m + sum f {Suc m..<n}"

  1732 apply(insert sum_head_Suc[of m "n - Suc 0" f])

  1733 apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)

  1734 done

  1735

  1736 lemma sum_ub_add_nat: assumes "(m::nat) \<le> n + 1"

  1737   shows "sum f {m..n + p} = sum f {m..n} + sum f {n + 1..n + p}"

  1738 proof-

  1739   have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using \<open>m \<le> n+1\<close> by auto

  1740   thus ?thesis by (auto simp: ivl_disj_int sum.union_disjoint

  1741     atLeastSucAtMost_greaterThanAtMost)

  1742 qed

  1743

  1744 lemmas sum_add_nat_ivl = sum.atLeast_lessThan_concat

  1745

  1746 lemma sum_diff_nat_ivl:

  1747 fixes f :: "nat \<Rightarrow> 'a::ab_group_add"

  1748 shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>

  1749   sum f {m..<p} - sum f {m..<n} = sum f {n..<p}"

  1750 using sum_add_nat_ivl [of m n p f,symmetric]

  1751 apply (simp add: ac_simps)

  1752 done

  1753

  1754 lemma sum_natinterval_difff:

  1755   fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"

  1756   shows  "sum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =

  1757           (if m <= n then f m - f(n + 1) else 0)"

  1758 by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)

  1759

  1760 lemma sum_nat_group: "(\<Sum>m<n::nat. sum f {m * k ..< m*k + k}) = sum f {..< n * k}"

  1761   apply (subgoal_tac "k = 0 | 0 < k", auto)

  1762   apply (induct "n")

  1763   apply (simp_all add: sum_add_nat_ivl add.commute atLeast0LessThan[symmetric])

  1764   done

  1765

  1766 lemma sum_triangle_reindex:

  1767   fixes n :: nat

  1768   shows "(\<Sum>(i,j)\<in>{(i,j). i+j < n}. f i j) = (\<Sum>k<n. \<Sum>i\<le>k. f i (k - i))"

  1769   apply (simp add: sum.Sigma)

  1770   apply (rule sum.reindex_bij_witness[where j="\<lambda>(i, j). (i+j, i)" and i="\<lambda>(k, i). (i, k - i)"])

  1771   apply auto

  1772   done

  1773

  1774 lemma sum_triangle_reindex_eq:

  1775   fixes n :: nat

  1776   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))"

  1777 using sum_triangle_reindex [of f "Suc n"]

  1778 by (simp only: Nat.less_Suc_eq_le lessThan_Suc_atMost)

  1779

  1780 lemma nat_diff_sum_reindex: "(\<Sum>i<n. f (n - Suc i)) = (\<Sum>i<n. f i)"

  1781   by (rule sum.reindex_bij_witness[where i="\<lambda>i. n - Suc i" and j="\<lambda>i. n - Suc i"]) auto

  1782

  1783

  1784 subsubsection \<open>Shifting bounds\<close>

  1785

  1786 lemma sum_shift_bounds_nat_ivl:

  1787   "sum f {m+k..<n+k} = sum (%i. f(i + k)){m..<n::nat}"

  1788 by (induct "n", auto simp:atLeastLessThanSuc)

  1789

  1790 lemma sum_shift_bounds_cl_nat_ivl:

  1791   "sum f {m+k..n+k} = sum (%i. f(i + k)){m..n::nat}"

  1792   by (rule sum.reindex_bij_witness[where i="\<lambda>i. i + k" and j="\<lambda>i. i - k"]) auto

  1793

  1794 corollary sum_shift_bounds_cl_Suc_ivl:

  1795   "sum f {Suc m..Suc n} = sum (%i. f(Suc i)){m..n}"

  1796 by (simp add:sum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])

  1797

  1798 corollary sum_shift_bounds_Suc_ivl:

  1799   "sum f {Suc m..<Suc n} = sum (%i. f(Suc i)){m..<n}"

  1800 by (simp add:sum_shift_bounds_nat_ivl[where k="Suc 0", simplified])

  1801

  1802 lemma sum_shift_lb_Suc0_0:

  1803   "f(0::nat) = (0::nat) \<Longrightarrow> sum f {Suc 0..k} = sum f {0..k}"

  1804 by(simp add:sum_head_Suc)

  1805

  1806 lemma sum_shift_lb_Suc0_0_upt:

  1807   "f(0::nat) = 0 \<Longrightarrow> sum f {Suc 0..<k} = sum f {0..<k}"

  1808 apply(cases k)apply simp

  1809 apply(simp add:sum_head_upt_Suc)

  1810 done

  1811

  1812 lemma sum_atMost_Suc_shift:

  1813   fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"

  1814   shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"

  1815 proof (induct n)

  1816   case 0 show ?case by simp

  1817 next

  1818   case (Suc n) note IH = this

  1819   have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))"

  1820     by (rule sum_atMost_Suc)

  1821   also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"

  1822     by (rule IH)

  1823   also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) =

  1824              f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"

  1825     by (rule add.assoc)

  1826   also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))"

  1827     by (rule sum_atMost_Suc [symmetric])

  1828   finally show ?case .

  1829 qed

  1830

  1831 lemma sum_lessThan_Suc_shift:

  1832   "(\<Sum>i<Suc n. f i) = f 0 + (\<Sum>i<n. f (Suc i))"

  1833   by (induction n) (simp_all add: add_ac)

  1834

  1835 lemma sum_atMost_shift:

  1836   fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"

  1837   shows "(\<Sum>i\<le>n. f i) = f 0 + (\<Sum>i<n. f (Suc i))"

  1838 by (metis atLeast0AtMost atLeast0LessThan atLeastLessThanSuc_atLeastAtMost atLeastSucAtMost_greaterThanAtMost le0 sum_head sum_shift_bounds_Suc_ivl)

  1839

  1840 lemma sum_last_plus: fixes n::nat shows "m <= n \<Longrightarrow> (\<Sum>i = m..n. f i) = f n + (\<Sum>i = m..<n. f i)"

  1841   by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost add.commute)

  1842

  1843 lemma sum_Suc_diff:

  1844   fixes f :: "nat \<Rightarrow> 'a::ab_group_add"

  1845   assumes "m \<le> Suc n"

  1846   shows "(\<Sum>i = m..n. f(Suc i) - f i) = f (Suc n) - f m"

  1847 using assms by (induct n) (auto simp: le_Suc_eq)

  1848

  1849 lemma sum_Suc_diff':

  1850   fixes f :: "nat \<Rightarrow> 'a::ab_group_add"

  1851   assumes "m \<le> n"

  1852   shows "(\<Sum>i = m..<n. f (Suc i) - f i) = f n - f m"

  1853 using assms by (induct n) (auto simp: le_Suc_eq)

  1854

  1855 lemma nested_sum_swap:

  1856      "(\<Sum>i = 0..n. (\<Sum>j = 0..<i. a i j)) = (\<Sum>j = 0..<n. \<Sum>i = Suc j..n. a i j)"

  1857   by (induction n) (auto simp: sum.distrib)

  1858

  1859 lemma nested_sum_swap':

  1860      "(\<Sum>i\<le>n. (\<Sum>j<i. a i j)) = (\<Sum>j<n. \<Sum>i = Suc j..n. a i j)"

  1861   by (induction n) (auto simp: sum.distrib)

  1862

  1863 lemma sum_atLeast1_atMost_eq:

  1864   "sum f {Suc 0..n} = (\<Sum>k<n. f (Suc k))"

  1865 proof -

  1866   have "sum f {Suc 0..n} = sum f (Suc  {..<n})"

  1867     by (simp add: image_Suc_lessThan)

  1868   also have "\<dots> = (\<Sum>k<n. f (Suc k))"

  1869     by (simp add: sum.reindex)

  1870   finally show ?thesis .

  1871 qed

  1872

  1873

  1874 subsubsection \<open>Telescoping\<close>

  1875

  1876 lemma sum_telescope:

  1877   fixes f::"nat \<Rightarrow> 'a::ab_group_add"

  1878   shows "sum (\<lambda>i. f i - f (Suc i)) {.. i} = f 0 - f (Suc i)"

  1879   by (induct i) simp_all

  1880

  1881 lemma sum_telescope'':

  1882   assumes "m \<le> n"

  1883   shows   "(\<Sum>k\<in>{Suc m..n}. f k - f (k - 1)) = f n - (f m :: 'a :: ab_group_add)"

  1884   by (rule dec_induct[OF assms]) (simp_all add: algebra_simps)

  1885

  1886 lemma sum_lessThan_telescope:

  1887   "(\<Sum>n<m. f (Suc n) - f n :: 'a :: ab_group_add) = f m - f 0"

  1888   by (induction m) (simp_all add: algebra_simps)

  1889

  1890 lemma sum_lessThan_telescope':

  1891   "(\<Sum>n<m. f n - f (Suc n) :: 'a :: ab_group_add) = f 0 - f m"

  1892   by (induction m) (simp_all add: algebra_simps)

  1893

  1894

  1895 subsection \<open>The formula for geometric sums\<close>

  1896

  1897 lemma sum_power2: "(\<Sum>i=0..<k. (2::nat)^i) = 2^k-1"

  1898 by (induction k) (auto simp: mult_2)

  1899

  1900 lemma geometric_sum:

  1901   assumes "x \<noteq> 1"

  1902   shows "(\<Sum>i<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"

  1903 proof -

  1904   from assms obtain y where "y = x - 1" and "y \<noteq> 0" by simp_all

  1905   moreover have "(\<Sum>i<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"

  1906     by (induct n) (simp_all add: field_simps \<open>y \<noteq> 0\<close>)

  1907   ultimately show ?thesis by simp

  1908 qed

  1909

  1910 lemma diff_power_eq_sum:

  1911   fixes y :: "'a::{comm_ring,monoid_mult}"

  1912   shows

  1913     "x ^ (Suc n) - y ^ (Suc n) =

  1914       (x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))"

  1915 proof (induct n)

  1916   case (Suc n)

  1917   have "x ^ Suc (Suc n) - y ^ Suc (Suc n) = x * (x * x^n) - y * (y * y ^ n)"

  1918     by simp

  1919   also have "... = y * (x ^ (Suc n) - y ^ (Suc n)) + (x - y) * (x * x^n)"

  1920     by (simp add: algebra_simps)

  1921   also have "... = y * ((x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x^n)"

  1922     by (simp only: Suc)

  1923   also have "... = (x - y) * (y * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x^n)"

  1924     by (simp only: mult.left_commute)

  1925   also have "... = (x - y) * (\<Sum>p<Suc (Suc n). x ^ p * y ^ (Suc n - p))"

  1926     by (simp add: field_simps Suc_diff_le sum_distrib_right sum_distrib_left)

  1927   finally show ?case .

  1928 qed simp

  1929

  1930 corollary power_diff_sumr2: \<comment>\<open>\<open>COMPLEX_POLYFUN\<close> in HOL Light\<close>

  1931   fixes x :: "'a::{comm_ring,monoid_mult}"

  1932   shows   "x^n - y^n = (x - y) * (\<Sum>i<n. y^(n - Suc i) * x^i)"

  1933 using diff_power_eq_sum[of x "n - 1" y]

  1934 by (cases "n = 0") (simp_all add: field_simps)

  1935

  1936 lemma power_diff_1_eq:

  1937   fixes x :: "'a::{comm_ring,monoid_mult}"

  1938   shows "n \<noteq> 0 \<Longrightarrow> x^n - 1 = (x - 1) * (\<Sum>i<n. (x^i))"

  1939 using diff_power_eq_sum [of x _ 1]

  1940   by (cases n) auto

  1941

  1942 lemma one_diff_power_eq':

  1943   fixes x :: "'a::{comm_ring,monoid_mult}"

  1944   shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i<n. x^(n - Suc i))"

  1945 using diff_power_eq_sum [of 1 _ x]

  1946   by (cases n) auto

  1947

  1948 lemma one_diff_power_eq:

  1949   fixes x :: "'a::{comm_ring,monoid_mult}"

  1950   shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i<n. x^i)"

  1951 by (metis one_diff_power_eq' [of n x] nat_diff_sum_reindex)

  1952

  1953 lemma sum_gp_basic:

  1954   fixes x :: "'a::{comm_ring,monoid_mult}"

  1955   shows "(1 - x) * (\<Sum>i\<le>n. x^i) = 1 - x^Suc n"

  1956   by (simp only: one_diff_power_eq [of "Suc n" x] lessThan_Suc_atMost)

  1957

  1958 lemma sum_power_shift:

  1959   fixes x :: "'a::{comm_ring,monoid_mult}"

  1960   assumes "m \<le> n"

  1961   shows "(\<Sum>i=m..n. x^i) = x^m * (\<Sum>i\<le>n-m. x^i)"

  1962 proof -

  1963   have "(\<Sum>i=m..n. x^i) = x^m * (\<Sum>i=m..n. x^(i-m))"

  1964     by (simp add: sum_distrib_left power_add [symmetric])

  1965   also have "(\<Sum>i=m..n. x^(i-m)) = (\<Sum>i\<le>n-m. x^i)"

  1966     using \<open>m \<le> n\<close> by (intro sum.reindex_bij_witness[where j="\<lambda>i. i - m" and i="\<lambda>i. i + m"]) auto

  1967   finally show ?thesis .

  1968 qed

  1969

  1970 lemma sum_gp_multiplied:

  1971   fixes x :: "'a::{comm_ring,monoid_mult}"

  1972   assumes "m \<le> n"

  1973   shows "(1 - x) * (\<Sum>i=m..n. x^i) = x^m - x^Suc n"

  1974 proof -

  1975   have  "(1 - x) * (\<Sum>i=m..n. x^i) = x^m * (1 - x) * (\<Sum>i\<le>n-m. x^i)"

  1976     by (metis mult.assoc mult.commute assms sum_power_shift)

  1977   also have "... =x^m * (1 - x^Suc(n-m))"

  1978     by (metis mult.assoc sum_gp_basic)

  1979   also have "... = x^m - x^Suc n"

  1980     using assms

  1981     by (simp add: algebra_simps) (metis le_add_diff_inverse power_add)

  1982   finally show ?thesis .

  1983 qed

  1984

  1985 lemma sum_gp:

  1986   fixes x :: "'a::{comm_ring,division_ring}"

  1987   shows   "(\<Sum>i=m..n. x^i) =

  1988                (if n < m then 0

  1989                 else if x = 1 then of_nat((n + 1) - m)

  1990                 else (x^m - x^Suc n) / (1 - x))"

  1991 using sum_gp_multiplied [of m n x] apply auto

  1992 by (metis eq_iff_diff_eq_0 mult.commute nonzero_divide_eq_eq)

  1993

  1994 subsection\<open>Geometric progressions\<close>

  1995

  1996 lemma sum_gp0:

  1997   fixes x :: "'a::{comm_ring,division_ring}"

  1998   shows   "(\<Sum>i\<le>n. x^i) = (if x = 1 then of_nat(n + 1) else (1 - x^Suc n) / (1 - x))"

  1999   using sum_gp_basic[of x n]

  2000   by (simp add: mult.commute divide_simps)

  2001

  2002 lemma sum_power_add:

  2003   fixes x :: "'a::{comm_ring,monoid_mult}"

  2004   shows "(\<Sum>i\<in>I. x^(m+i)) = x^m * (\<Sum>i\<in>I. x^i)"

  2005   by (simp add: sum_distrib_left power_add)

  2006

  2007 lemma sum_gp_offset:

  2008   fixes x :: "'a::{comm_ring,division_ring}"

  2009   shows   "(\<Sum>i=m..m+n. x^i) =

  2010        (if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"

  2011   using sum_gp [of x m "m+n"]

  2012   by (auto simp: power_add algebra_simps)

  2013

  2014 lemma sum_gp_strict:

  2015   fixes x :: "'a::{comm_ring,division_ring}"

  2016   shows "(\<Sum>i<n. x^i) = (if x = 1 then of_nat n else (1 - x^n) / (1 - x))"

  2017   by (induct n) (auto simp: algebra_simps divide_simps)

  2018

  2019 subsubsection \<open>The formula for arithmetic sums\<close>

  2020

  2021 lemma gauss_sum:

  2022   "(2::'a::comm_semiring_1)*(\<Sum>i\<in>{1..n}. of_nat i) = of_nat n*((of_nat n)+1)"

  2023 proof (induct n)

  2024   case 0

  2025   show ?case by simp

  2026 next

  2027   case (Suc n)

  2028   then show ?case

  2029     by (simp add: algebra_simps add: one_add_one [symmetric] del: one_add_one)

  2030       (* FIXME: make numeral cancellation simprocs work for semirings *)

  2031 qed

  2032

  2033 theorem arith_series_general:

  2034   "(2::'a::comm_semiring_1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

  2035   of_nat n * (a + (a + of_nat(n - 1)*d))"

  2036 proof cases

  2037   assume ngt1: "n > 1"

  2038   let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"

  2039   have

  2040     "(\<Sum>i\<in>{..<n}. a+?I i*d) =

  2041      ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"

  2042     by (rule sum.distrib)

  2043   also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp

  2044   also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"

  2045     unfolding One_nat_def

  2046     by (simp add: sum_distrib_left atLeast0LessThan[symmetric] sum_shift_lb_Suc0_0_upt ac_simps)

  2047   also have "2*\<dots> = 2*?n*a + d*2*(\<Sum>i\<in>{1..<n}. ?I i)"

  2048     by (simp add: algebra_simps)

  2049   also from ngt1 have "{1..<n} = {1..n - 1}"

  2050     by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)

  2051   also from ngt1

  2052   have "2*?n*a + d*2*(\<Sum>i\<in>{1..n - 1}. ?I i) = (2*?n*a + d*?I (n - 1)*?I n)"

  2053     by (simp only: mult.assoc gauss_sum [of "n - 1"], unfold One_nat_def)

  2054       (simp add:  mult.commute trans [OF add.commute of_nat_Suc [symmetric]])

  2055   finally show ?thesis

  2056     unfolding mult_2 by (simp add: algebra_simps)

  2057 next

  2058   assume "\<not>(n > 1)"

  2059   hence "n = 1 \<or> n = 0" by auto

  2060   thus ?thesis by (auto simp: mult_2)

  2061 qed

  2062

  2063 lemma arith_series_nat:

  2064   "(2::nat) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"

  2065 proof -

  2066   have

  2067     "2 * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =

  2068     of_nat(n) * (a + (a + of_nat(n - 1)*d))"

  2069     by (rule arith_series_general)

  2070   thus ?thesis

  2071     unfolding One_nat_def by auto

  2072 qed

  2073

  2074 lemma arith_series_int:

  2075   "2 * (\<Sum>i\<in>{..<n}. a + int i * d) = int n * (a + (a + int(n - 1)*d))"

  2076   by (fact arith_series_general) (* FIXME: duplicate *)

  2077

  2078 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)"

  2079   by (subst sum_subtractf_nat) auto

  2080

  2081

  2082 subsubsection \<open>Division remainder\<close>

  2083

  2084 lemma range_mod:

  2085   fixes n :: nat

  2086   assumes "n > 0"

  2087   shows "range (\<lambda>m. m mod n) = {0..<n}" (is "?A = ?B")

  2088 proof (rule set_eqI)

  2089   fix m

  2090   show "m \<in> ?A \<longleftrightarrow> m \<in> ?B"

  2091   proof

  2092     assume "m \<in> ?A"

  2093     with assms show "m \<in> ?B"

  2094       by auto

  2095   next

  2096     assume "m \<in> ?B"

  2097     moreover have "m mod n \<in> ?A"

  2098       by (rule rangeI)

  2099     ultimately show "m \<in> ?A"

  2100       by simp

  2101   qed

  2102 qed

  2103

  2104

  2105 subsection \<open>Products indexed over intervals\<close>

  2106

  2107 syntax (ASCII)

  2108   "_from_to_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(PROD _ = _.._./ _)" [0,0,0,10] 10)

  2109   "_from_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)

  2110   "_upt_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(PROD _<_./ _)" [0,0,10] 10)

  2111   "_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(PROD _<=_./ _)" [0,0,10] 10)

  2112

  2113 syntax (latex_prod output)

  2114   "_from_to_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  2115  ("(3\<^latex>\<open>$\\prod_{\<close>_ = _\<^latex>\<open>}^{\<close>_\<^latex>\<open>}$\<close> _)" [0,0,0,10] 10)

  2116   "_from_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  2117  ("(3\<^latex>\<open>$\\prod_{\<close>_ = _\<^latex>\<open>}^{<\<close>_\<^latex>\<open>}$\<close> _)" [0,0,0,10] 10)

  2118   "_upt_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  2119  ("(3\<^latex>\<open>$\\prod_{\<close>_ < _\<^latex>\<open>}$\<close> _)" [0,0,10] 10)

  2120   "_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  2121  ("(3\<^latex>\<open>$\\prod_{\<close>_ \<le> _\<^latex>\<open>}$\<close> _)" [0,0,10] 10)

  2122

  2123 syntax

  2124   "_from_to_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)

  2125   "_from_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)

  2126   "_upt_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Prod>_<_./ _)" [0,0,10] 10)

  2127   "_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)

  2128

  2129 translations

  2130   "\<Prod>x=a..b. t" \<rightleftharpoons> "CONST prod (\<lambda>x. t) {a..b}"

  2131   "\<Prod>x=a..<b. t" \<rightleftharpoons> "CONST prod (\<lambda>x. t) {a..<b}"

  2132   "\<Prod>i\<le>n. t" \<rightleftharpoons> "CONST prod (\<lambda>i. t) {..n}"

  2133   "\<Prod>i<n. t" \<rightleftharpoons> "CONST prod (\<lambda>i. t) {..<n}"

  2134

  2135 lemma prod_int_plus_eq: "prod int {i..i+j} =  \<Prod>{int i..int (i+j)}"

  2136   by (induct j) (auto simp add: atLeastAtMostSuc_conv atLeastAtMostPlus1_int_conv)

  2137

  2138 lemma prod_int_eq: "prod int {i..j} =  \<Prod>{int i..int j}"

  2139 proof (cases "i \<le> j")

  2140   case True

  2141   then show ?thesis

  2142     by (metis le_iff_add prod_int_plus_eq)

  2143 next

  2144   case False

  2145   then show ?thesis

  2146     by auto

  2147 qed

  2148

  2149

  2150 subsubsection \<open>Shifting bounds\<close>

  2151

  2152 lemma prod_shift_bounds_nat_ivl:

  2153   "prod f {m+k..<n+k} = prod (%i. f(i + k)){m..<n::nat}"

  2154 by (induct "n", auto simp:atLeastLessThanSuc)

  2155

  2156 lemma prod_shift_bounds_cl_nat_ivl:

  2157   "prod f {m+k..n+k} = prod (%i. f(i + k)){m..n::nat}"

  2158   by (rule prod.reindex_bij_witness[where i="\<lambda>i. i + k" and j="\<lambda>i. i - k"]) auto

  2159

  2160 corollary prod_shift_bounds_cl_Suc_ivl:

  2161   "prod f {Suc m..Suc n} = prod (%i. f(Suc i)){m..n}"

  2162 by (simp add:prod_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])

  2163

  2164 corollary prod_shift_bounds_Suc_ivl:

  2165   "prod f {Suc m..<Suc n} = prod (%i. f(Suc i)){m..<n}"

  2166 by (simp add:prod_shift_bounds_nat_ivl[where k="Suc 0", simplified])

  2167

  2168 lemma prod_lessThan_Suc: "prod f {..<Suc n} = prod f {..<n} * f n"

  2169   by (simp add: lessThan_Suc mult.commute)

  2170

  2171 lemma prod_lessThan_Suc_shift:"(\<Prod>i<Suc n. f i) = f 0 * (\<Prod>i<n. f (Suc i))"

  2172   by (induction n) (simp_all add: lessThan_Suc mult_ac)

  2173

  2174 lemma prod_atLeastLessThan_Suc: "a \<le> b \<Longrightarrow> prod f {a..<Suc b} = prod f {a..<b} * f b"

  2175   by (simp add: atLeastLessThanSuc mult.commute)

  2176

  2177 lemma prod_nat_ivl_Suc':

  2178   assumes "m \<le> Suc n"

  2179   shows   "prod f {m..Suc n} = f (Suc n) * prod f {m..n}"

  2180 proof -

  2181   from assms have "{m..Suc n} = insert (Suc n) {m..n}" by auto

  2182   also have "prod f \<dots> = f (Suc n) * prod f {m..n}" by simp

  2183   finally show ?thesis .

  2184 qed

  2185

  2186

  2187 subsection \<open>Efficient folding over intervals\<close>

  2188

  2189 function fold_atLeastAtMost_nat where

  2190   [simp del]: "fold_atLeastAtMost_nat f a (b::nat) acc =

  2191                  (if a > b then acc else fold_atLeastAtMost_nat f (a+1) b (f a acc))"

  2192 by pat_completeness auto

  2193 termination by (relation "measure (\<lambda>(_,a,b,_). Suc b - a)") auto

  2194

  2195 lemma fold_atLeastAtMost_nat:

  2196   assumes "comp_fun_commute f"

  2197   shows   "fold_atLeastAtMost_nat f a b acc = Finite_Set.fold f acc {a..b}"

  2198 using assms

  2199 proof (induction f a b acc rule: fold_atLeastAtMost_nat.induct, goal_cases)

  2200   case (1 f a b acc)

  2201   interpret comp_fun_commute f by fact

  2202   show ?case

  2203   proof (cases "a > b")

  2204     case True

  2205     thus ?thesis by (subst fold_atLeastAtMost_nat.simps) auto

  2206   next

  2207     case False

  2208     with 1 show ?thesis

  2209       by (subst fold_atLeastAtMost_nat.simps)

  2210          (auto simp: atLeastAtMost_insertL[symmetric] fold_fun_left_comm)

  2211   qed

  2212 qed

  2213

  2214 lemma sum_atLeastAtMost_code:

  2215   "sum f {a..b} = fold_atLeastAtMost_nat (\<lambda>a acc. f a + acc) a b 0"

  2216 proof -

  2217   have "comp_fun_commute (\<lambda>a. op + (f a))"

  2218     by unfold_locales (auto simp: o_def add_ac)

  2219   thus ?thesis

  2220     by (simp add: sum.eq_fold fold_atLeastAtMost_nat o_def)

  2221 qed

  2222

  2223 lemma prod_atLeastAtMost_code:

  2224   "prod f {a..b} = fold_atLeastAtMost_nat (\<lambda>a acc. f a * acc) a b 1"

  2225 proof -

  2226   have "comp_fun_commute (\<lambda>a. op * (f a))"

  2227     by unfold_locales (auto simp: o_def mult_ac)

  2228   thus ?thesis

  2229     by (simp add: prod.eq_fold fold_atLeastAtMost_nat o_def)

  2230 qed

  2231

  2232 (* TODO: Add support for more kinds of intervals here *)

  2233

  2234 end