src/HOL/SetInterval.thy
 author nipkow Thu Dec 11 08:52:50 2008 +0100 (2008-12-11) changeset 29106 25e28a4070f3 parent 28853 69eb69659bf3 child 29667 53103fc8ffa3 permissions -rw-r--r--
Testfile for Stefan's code generator
     1 (*  Title:      HOL/SetInterval.thy

     2     ID:         $Id$

     3     Author:     Tobias Nipkow and Clemens Ballarin

     4                 Additions by Jeremy Avigad in March 2004

     5     Copyright   2000  TU Muenchen

     6

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

     8 *)

     9

    10 header {* Set intervals *}

    11

    12 theory SetInterval

    13 imports Int

    14 begin

    15

    16 context ord

    17 begin

    18 definition

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

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

    21

    22 definition

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

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

    25

    26 definition

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

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

    29

    30 definition

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

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

    33

    34 definition

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

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

    37

    38 definition

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

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

    41

    42 definition

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

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

    45

    46 definition

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

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

    49

    50 end

    51

    52

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

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

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

    56

    57 syntax

    58   "@UNION_le"   :: "nat => nat => 'b set => 'b set"       ("(3UN _<=_./ _)" 10)

    59   "@UNION_less" :: "nat => nat => 'b set => 'b set"       ("(3UN _<_./ _)" 10)

    60   "@INTER_le"   :: "nat => nat => 'b set => 'b set"       ("(3INT _<=_./ _)" 10)

    61   "@INTER_less" :: "nat => nat => 'b set => 'b set"       ("(3INT _<_./ _)" 10)

    62

    63 syntax (input)

    64   "@UNION_le"   :: "nat => nat => 'b set => 'b set"       ("(3\<Union> _\<le>_./ _)" 10)

    65   "@UNION_less" :: "nat => nat => 'b set => 'b set"       ("(3\<Union> _<_./ _)" 10)

    66   "@INTER_le"   :: "nat => nat => 'b set => 'b set"       ("(3\<Inter> _\<le>_./ _)" 10)

    67   "@INTER_less" :: "nat => nat => 'b set => 'b set"       ("(3\<Inter> _<_./ _)" 10)

    68

    69 syntax (xsymbols)

    70   "@UNION_le"   :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Union>(00\<^bsub>_ \<le> _\<^esub>)/ _)" 10)

    71   "@UNION_less" :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Union>(00\<^bsub>_ < _\<^esub>)/ _)" 10)

    72   "@INTER_le"   :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Inter>(00\<^bsub>_ \<le> _\<^esub>)/ _)" 10)

    73   "@INTER_less" :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Inter>(00\<^bsub>_ < _\<^esub>)/ _)" 10)

    74

    75 translations

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

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

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

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

    80

    81

    82 subsection {* Various equivalences *}

    83

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

    85 by (simp add: lessThan_def)

    86

    87 lemma Compl_lessThan [simp]:

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

    89 apply (auto simp add: lessThan_def atLeast_def)

    90 done

    91

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

    93 by auto

    94

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

    96 by (simp add: greaterThan_def)

    97

    98 lemma Compl_greaterThan [simp]:

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

   100   by (auto simp add: greaterThan_def atMost_def)

   101

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

   103 apply (subst Compl_greaterThan [symmetric])

   104 apply (rule double_complement)

   105 done

   106

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

   108 by (simp add: atLeast_def)

   109

   110 lemma Compl_atLeast [simp]:

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

   112   by (auto simp add: lessThan_def atLeast_def)

   113

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

   115 by (simp add: atMost_def)

   116

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

   118 by (blast intro: order_antisym)

   119

   120

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

   122

   123 lemma atLeast_subset_iff [iff]:

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

   125 by (blast intro: order_trans)

   126

   127 lemma atLeast_eq_iff [iff]:

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

   129 by (blast intro: order_antisym order_trans)

   130

   131 lemma greaterThan_subset_iff [iff]:

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

   133 apply (auto simp add: greaterThan_def)

   134  apply (subst linorder_not_less [symmetric], blast)

   135 done

   136

   137 lemma greaterThan_eq_iff [iff]:

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

   139 apply (rule iffI)

   140  apply (erule equalityE)

   141  apply (simp_all add: greaterThan_subset_iff)

   142 done

   143

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

   145 by (blast intro: order_trans)

   146

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

   148 by (blast intro: order_antisym order_trans)

   149

   150 lemma lessThan_subset_iff [iff]:

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

   152 apply (auto simp add: lessThan_def)

   153  apply (subst linorder_not_less [symmetric], blast)

   154 done

   155

   156 lemma lessThan_eq_iff [iff]:

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

   158 apply (rule iffI)

   159  apply (erule equalityE)

   160  apply (simp_all add: lessThan_subset_iff)

   161 done

   162

   163

   164 subsection {*Two-sided intervals*}

   165

   166 context ord

   167 begin

   168

   169 lemma greaterThanLessThan_iff [simp,noatp]:

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

   171 by (simp add: greaterThanLessThan_def)

   172

   173 lemma atLeastLessThan_iff [simp,noatp]:

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

   175 by (simp add: atLeastLessThan_def)

   176

   177 lemma greaterThanAtMost_iff [simp,noatp]:

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

   179 by (simp add: greaterThanAtMost_def)

   180

   181 lemma atLeastAtMost_iff [simp,noatp]:

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

   183 by (simp add: atLeastAtMost_def)

   184

   185 text {* The above four lemmas could be declared as iffs.

   186   If we do so, a call to blast in Hyperreal/Star.ML, lemma @{text STAR_Int}

   187   seems to take forever (more than one hour). *}

   188 end

   189

   190 subsubsection{* Emptyness and singletons *}

   191

   192 context order

   193 begin

   194

   195 lemma atLeastAtMost_empty [simp]: "n < m ==> {m..n} = {}";

   196 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)

   197

   198 lemma atLeastLessThan_empty[simp]: "n \<le> m ==> {m..<n} = {}"

   199 by (auto simp add: atLeastLessThan_def)

   200

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

   202 by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)

   203

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

   205 by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)

   206

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

   208 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)

   209

   210 end

   211

   212 subsection {* Intervals of natural numbers *}

   213

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

   215

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

   217 by (simp add: lessThan_def)

   218

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

   220 by (simp add: lessThan_def less_Suc_eq, blast)

   221

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

   223 by (simp add: lessThan_def atMost_def less_Suc_eq_le)

   224

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

   226 by blast

   227

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

   229

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

   231 apply (simp add: greaterThan_def)

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

   233 done

   234

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

   236 apply (simp add: greaterThan_def)

   237 apply (auto elim: linorder_neqE)

   238 done

   239

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

   241 by blast

   242

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

   244

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

   246 by (unfold atLeast_def UNIV_def, simp)

   247

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

   249 apply (simp add: atLeast_def)

   250 apply (simp add: Suc_le_eq)

   251 apply (simp add: order_le_less, blast)

   252 done

   253

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

   255   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)

   256

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

   258 by blast

   259

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

   261

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

   263 by (simp add: atMost_def)

   264

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

   266 apply (simp add: atMost_def)

   267 apply (simp add: less_Suc_eq order_le_less, blast)

   268 done

   269

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

   271 by blast

   272

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

   274

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

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

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

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

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

   280 specific concept to a more general one. *}

   281

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

   283 by(simp add:lessThan_def atLeastLessThan_def)

   284

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

   286 by(simp add:atMost_def atLeastAtMost_def)

   287

   288 declare atLeast0LessThan[symmetric, code unfold]

   289         atLeast0AtMost[symmetric, code unfold]

   290

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

   292 by (simp add: atLeastLessThan_def)

   293

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

   295

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

   297 lemma atLeastLessThanSuc:

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

   299 by (auto simp add: atLeastLessThan_def)

   300

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

   302 by (auto simp add: atLeastLessThan_def)

   303 (*

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

   305 by (induct k, simp_all add: atLeastLessThanSuc)

   306

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

   308 by (auto simp add: atLeastLessThan_def)

   309 *)

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

   311   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)

   312

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

   314   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def

   315     greaterThanAtMost_def)

   316

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

   318   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def

   319     greaterThanLessThan_def)

   320

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

   322 by (auto simp add: atLeastAtMost_def)

   323

   324 subsubsection {* Image *}

   325

   326 lemma image_add_atLeastAtMost:

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

   328 proof

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

   330 next

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

   332   proof

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

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

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

   336     ultimately show "n : ?A" by blast

   337   qed

   338 qed

   339

   340 lemma image_add_atLeastLessThan:

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

   342 proof

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

   344 next

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

   346   proof

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

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

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

   350     ultimately show "n : ?A" by blast

   351   qed

   352 qed

   353

   354 corollary image_Suc_atLeastAtMost[simp]:

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

   356 using image_add_atLeastAtMost[where k=1] by simp

   357

   358 corollary image_Suc_atLeastLessThan[simp]:

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

   360 using image_add_atLeastLessThan[where k=1] by simp

   361

   362 lemma image_add_int_atLeastLessThan:

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

   364   apply (auto simp add: image_def)

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

   366   apply auto

   367   done

   368

   369

   370 subsubsection {* Finiteness *}

   371

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

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

   374

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

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

   377

   378 lemma finite_greaterThanLessThan [iff]:

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

   380 by (simp add: greaterThanLessThan_def)

   381

   382 lemma finite_atLeastLessThan [iff]:

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

   384 by (simp add: atLeastLessThan_def)

   385

   386 lemma finite_greaterThanAtMost [iff]:

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

   388 by (simp add: greaterThanAtMost_def)

   389

   390 lemma finite_atLeastAtMost [iff]:

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

   392 by (simp add: atLeastAtMost_def)

   393

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

   395 lemma bounded_nat_set_is_finite:

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

   397 apply (rule finite_subset)

   398  apply (rule_tac [2] finite_lessThan, auto)

   399 done

   400

   401 lemma finite_less_ub:

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

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

   404

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

   406 subset is exactly that interval. *}

   407

   408 lemma subset_card_intvl_is_intvl:

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

   410 proof cases

   411   assume "finite A"

   412   thus "PROP ?P"

   413   proof(induct A rule:finite_linorder_induct)

   414     case empty thus ?case by auto

   415   next

   416     case (insert A b)

   417     moreover hence "b ~: A" by auto

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

   419       using b ~: A insert by fastsimp+

   420     ultimately show ?case by auto

   421   qed

   422 next

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

   424 qed

   425

   426

   427 subsubsection {* Cardinality *}

   428

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

   430   by (induct u, simp_all add: lessThan_Suc)

   431

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

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

   434

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

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

   437   apply (erule ssubst, rule card_lessThan)

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

   439   apply (erule subst)

   440   apply (rule card_image)

   441   apply (simp add: inj_on_def)

   442   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)

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

   444   apply arith

   445   done

   446

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

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

   449

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

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

   452

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

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

   455

   456

   457 lemma ex_bij_betw_nat_finite:

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

   459 apply(drule finite_imp_nat_seg_image_inj_on)

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

   461 done

   462

   463 lemma ex_bij_betw_finite_nat:

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

   465 by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)

   466

   467

   468 subsection {* Intervals of integers *}

   469

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

   471   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)

   472

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

   474   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)

   475

   476 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:

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

   478   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)

   479

   480 subsubsection {* Finiteness *}

   481

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

   483     {(0::int)..<u} = int  {..<nat u}"

   484   apply (unfold image_def lessThan_def)

   485   apply auto

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

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

   488   done

   489

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

   491   apply (case_tac "0 \<le> u")

   492   apply (subst image_atLeastZeroLessThan_int, assumption)

   493   apply (rule finite_imageI)

   494   apply auto

   495   done

   496

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

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

   499   apply (erule subst)

   500   apply (rule finite_imageI)

   501   apply (rule finite_atLeastZeroLessThan_int)

   502   apply (rule image_add_int_atLeastLessThan)

   503   done

   504

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

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

   507

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

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

   510

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

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

   513

   514

   515 subsubsection {* Cardinality *}

   516

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

   518   apply (case_tac "0 \<le> u")

   519   apply (subst image_atLeastZeroLessThan_int, assumption)

   520   apply (subst card_image)

   521   apply (auto simp add: inj_on_def)

   522   done

   523

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

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

   526   apply (erule ssubst, rule card_atLeastZeroLessThan_int)

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

   528   apply (erule subst)

   529   apply (rule card_image)

   530   apply (simp add: inj_on_def)

   531   apply (rule image_add_int_atLeastLessThan)

   532   done

   533

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

   535   apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])

   536   apply (auto simp add: compare_rls)

   537   done

   538

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

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

   541

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

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

   544

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

   546 proof -

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

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

   549 qed

   550

   551 lemma card_less:

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

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

   554 proof -

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

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

   557 qed

   558

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

   560 apply (rule card_bij_eq [of "Suc" _ _ "\<lambda>x. x - 1"])

   561 apply simp

   562 apply fastsimp

   563 apply auto

   564 apply (rule inj_on_diff_nat)

   565 apply auto

   566 apply (case_tac x)

   567 apply auto

   568 apply (case_tac xa)

   569 apply auto

   570 apply (case_tac xa)

   571 apply auto

   572 apply (auto simp add: finite_M_bounded_by_nat)

   573 done

   574

   575 lemma card_less_Suc:

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

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

   578 proof -

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

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

   581     by (auto simp only: insert_Diff)

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

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

   584     apply (subst card_insert)

   585     apply simp_all

   586     apply (subst b)

   587     apply (subst card_less_Suc2[symmetric])

   588     apply simp_all

   589     done

   590   with c show ?thesis by simp

   591 qed

   592

   593

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

   595

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

   597

   598 subsubsection {* Disjoint Unions *}

   599

   600 text {* Singletons and open intervals *}

   601

   602 lemma ivl_disj_un_singleton:

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

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

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

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

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

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

   609 by auto

   610

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

   612

   613 lemma ivl_disj_un_one:

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

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

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

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

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

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

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

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

   622 by auto

   623

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

   625

   626 lemma ivl_disj_un_two:

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

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

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

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

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

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

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

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

   635 by auto

   636

   637 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two

   638

   639 subsubsection {* Disjoint Intersections *}

   640

   641 text {* Singletons and open intervals *}

   642

   643 lemma ivl_disj_int_singleton:

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

   645   "{..<u} Int {u} = {}"

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

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

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

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

   650   by simp+

   651

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

   653

   654 lemma ivl_disj_int_one:

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

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

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

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

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

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

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

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

   663   by auto

   664

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

   666

   667 lemma ivl_disj_int_two:

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

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

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

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

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

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

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

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

   676   by auto

   677

   678 lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two

   679

   680 subsubsection {* Some Differences *}

   681

   682 lemma ivl_diff[simp]:

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

   684 by(auto)

   685

   686

   687 subsubsection {* Some Subset Conditions *}

   688

   689 lemma ivl_subset [simp,noatp]:

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

   691 apply(auto simp:linorder_not_le)

   692 apply(rule ccontr)

   693 apply(insert linorder_le_less_linear[of i n])

   694 apply(clarsimp simp:linorder_not_le)

   695 apply(fastsimp)

   696 done

   697

   698

   699 subsection {* Summation indexed over intervals *}

   700

   701 syntax

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

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

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

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

   706 syntax (xsymbols)

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

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

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

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

   711 syntax (HTML output)

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

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

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

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

   716 syntax (latex_sum output)

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

   718  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

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

   720  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

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

   722  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

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

   724  ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

   725

   726 translations

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

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

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

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

   731

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

   733 summation over intervals:

   734 \begin{center}

   735 \begin{tabular}{lll}

   736 Old & New & \LaTeX\\

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

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

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

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

   741 \end{tabular}

   742 \end{center}

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

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

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

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

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

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

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

   750

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

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

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

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

   755

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

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

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

   759 the context. *}

   760

   761 lemma setsum_ivl_cong:

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

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

   764 by(rule setsum_cong, simp_all)

   765

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

   767 on intervals are not? *)

   768

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

   770 by (simp add:atMost_Suc add_ac)

   771

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

   773 by (simp add:lessThan_Suc add_ac)

   774

   775 lemma setsum_cl_ivl_Suc[simp]:

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

   777 by (auto simp:add_ac atLeastAtMostSuc_conv)

   778

   779 lemma setsum_op_ivl_Suc[simp]:

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

   781 by (auto simp:add_ac atLeastLessThanSuc)

   782 (*

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

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

   785 by (auto simp:add_ac atLeastAtMostSuc_conv)

   786 *)

   787

   788 lemma setsum_head:

   789   fixes n :: nat

   790   assumes mn: "m <= n"

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

   792 proof -

   793   from mn

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

   795     by (auto intro: ivl_disj_un_singleton)

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

   797     by (simp add: atLeast0LessThan)

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

   799   finally show ?thesis .

   800 qed

   801

   802 lemma setsum_head_Suc:

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

   804 by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)

   805

   806 lemma setsum_head_upt_Suc:

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

   808 apply(insert setsum_head_Suc[of m "n - 1" f])

   809 apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] ring_simps)

   810 done

   811

   812

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

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

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

   816

   817 lemma setsum_diff_nat_ivl:

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

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

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

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

   822 apply (simp add: add_ac)

   823 done

   824

   825

   826 subsection{* Shifting bounds *}

   827

   828 lemma setsum_shift_bounds_nat_ivl:

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

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

   831

   832 lemma setsum_shift_bounds_cl_nat_ivl:

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

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

   835 apply (simp add:image_add_atLeastAtMost o_def)

   836 done

   837

   838 corollary setsum_shift_bounds_cl_Suc_ivl:

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

   840 by (simp add:setsum_shift_bounds_cl_nat_ivl[where k=1,simplified])

   841

   842 corollary setsum_shift_bounds_Suc_ivl:

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

   844 by (simp add:setsum_shift_bounds_nat_ivl[where k=1,simplified])

   845

   846 lemma setsum_shift_lb_Suc0_0:

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

   848 by(simp add:setsum_head_Suc)

   849

   850 lemma setsum_shift_lb_Suc0_0_upt:

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

   852 apply(cases k)apply simp

   853 apply(simp add:setsum_head_upt_Suc)

   854 done

   855

   856 subsection {* The formula for geometric sums *}

   857

   858 lemma geometric_sum:

   859   "x ~= 1 ==> (\<Sum>i=0..<n. x ^ i) =

   860   (x ^ n - 1) / (x - 1::'a::{field, recpower})"

   861 by (induct "n") (simp_all add:field_simps power_Suc)

   862

   863 subsection {* The formula for arithmetic sums *}

   864

   865 lemma gauss_sum:

   866   "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =

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

   868 proof (induct n)

   869   case 0

   870   show ?case by simp

   871 next

   872   case (Suc n)

   873   then show ?case by (simp add: ring_simps)

   874 qed

   875

   876 theorem arith_series_general:

   877   "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

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

   879 proof cases

   880   assume ngt1: "n > 1"

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

   882   have

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

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

   885     by (rule setsum_addf)

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

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

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

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

   890     by (simp add: left_distrib right_distrib)

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

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

   893   also from ngt1

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

   895     by (simp only: mult_ac gauss_sum [of "n - 1"])

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

   897   finally show ?thesis by (simp add: mult_ac add_ac right_distrib)

   898 next

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

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

   901   thus ?thesis by (auto simp: mult_ac right_distrib)

   902 qed

   903

   904 lemma arith_series_nat:

   905   "Suc (Suc 0) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"

   906 proof -

   907   have

   908     "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =

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

   910     by (rule arith_series_general)

   911   thus ?thesis by (auto simp add: of_nat_id)

   912 qed

   913

   914 lemma arith_series_int:

   915   "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

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

   917 proof -

   918   have

   919     "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

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

   921     by (rule arith_series_general)

   922   thus ?thesis by simp

   923 qed

   924

   925 lemma sum_diff_distrib:

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

   927   shows

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

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

   930 proof (induct n)

   931   case 0 show ?case by simp

   932 next

   933   case (Suc n)

   934

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

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

   937

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

   939   moreover

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

   941   moreover

   942   from Suc have

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

   944     by (subst diff_diff_left[symmetric],

   945         subst diff_add_assoc2)

   946        (auto simp: diff_add_assoc2 intro: setsum_mono)

   947   ultimately

   948   show ?case by simp

   949 qed

   950

   951 end
`