src/HOL/SetInterval.thy
 author nipkow Mon Mar 09 12:24:01 2009 +0100 (2009-03-09) changeset 30384 2f24531b2d3e parent 30372 96d508968153 child 31017 2c227493ea56 permissions -rw-r--r--
fixed typing of UN/INT syntax
     1 (*  Title:      HOL/SetInterval.thy

     2     Author:     Tobias Nipkow and Clemens Ballarin

     3                 Additions by Jeremy Avigad in March 2004

     4     Copyright   2000  TU Muenchen

     5

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

     7 *)

     8

     9 header {* Set intervals *}

    10

    11 theory SetInterval

    12 imports Int

    13 begin

    14

    15 context ord

    16 begin

    17 definition

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

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

    20

    21 definition

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

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

    24

    25 definition

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

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

    28

    29 definition

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

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

    32

    33 definition

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

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

    36

    37 definition

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

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

    40

    41 definition

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

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

    44

    45 definition

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

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

    48

    49 end

    50

    51

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

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

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

    55

    56 syntax

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

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

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

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

    61

    62 syntax (xsymbols)

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

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

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

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

    67

    68 syntax (latex output)

    69   "@UNION_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ \<le> _)/ _)" 10)

    70   "@UNION_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ < _)/ _)" 10)

    71   "@INTER_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ \<le> _)/ _)" 10)

    72   "@INTER_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ < _)/ _)" 10)

    73

    74 translations

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

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

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

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

    79

    80

    81 subsection {* Various equivalences *}

    82

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

    84 by (simp add: lessThan_def)

    85

    86 lemma Compl_lessThan [simp]:

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

    88 apply (auto simp add: lessThan_def atLeast_def)

    89 done

    90

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

    92 by auto

    93

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

    95 by (simp add: greaterThan_def)

    96

    97 lemma Compl_greaterThan [simp]:

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

    99   by (auto simp add: greaterThan_def atMost_def)

   100

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

   102 apply (subst Compl_greaterThan [symmetric])

   103 apply (rule double_complement)

   104 done

   105

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

   107 by (simp add: atLeast_def)

   108

   109 lemma Compl_atLeast [simp]:

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

   111   by (auto simp add: lessThan_def atLeast_def)

   112

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

   114 by (simp add: atMost_def)

   115

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

   117 by (blast intro: order_antisym)

   118

   119

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

   121

   122 lemma atLeast_subset_iff [iff]:

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

   124 by (blast intro: order_trans)

   125

   126 lemma atLeast_eq_iff [iff]:

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

   128 by (blast intro: order_antisym order_trans)

   129

   130 lemma greaterThan_subset_iff [iff]:

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

   132 apply (auto simp add: greaterThan_def)

   133  apply (subst linorder_not_less [symmetric], blast)

   134 done

   135

   136 lemma greaterThan_eq_iff [iff]:

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

   138 apply (rule iffI)

   139  apply (erule equalityE)

   140  apply simp_all

   141 done

   142

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

   144 by (blast intro: order_trans)

   145

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

   147 by (blast intro: order_antisym order_trans)

   148

   149 lemma lessThan_subset_iff [iff]:

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

   151 apply (auto simp add: lessThan_def)

   152  apply (subst linorder_not_less [symmetric], blast)

   153 done

   154

   155 lemma lessThan_eq_iff [iff]:

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

   157 apply (rule iffI)

   158  apply (erule equalityE)

   159  apply simp_all

   160 done

   161

   162

   163 subsection {*Two-sided intervals*}

   164

   165 context ord

   166 begin

   167

   168 lemma greaterThanLessThan_iff [simp,noatp]:

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

   170 by (simp add: greaterThanLessThan_def)

   171

   172 lemma atLeastLessThan_iff [simp,noatp]:

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

   174 by (simp add: atLeastLessThan_def)

   175

   176 lemma greaterThanAtMost_iff [simp,noatp]:

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

   178 by (simp add: greaterThanAtMost_def)

   179

   180 lemma atLeastAtMost_iff [simp,noatp]:

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

   182 by (simp add: atLeastAtMost_def)

   183

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

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

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

   187 end

   188

   189 subsubsection{* Emptyness and singletons *}

   190

   191 context order

   192 begin

   193

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

   195 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)

   196

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

   198 by (auto simp add: atLeastLessThan_def)

   199

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

   201 by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)

   202

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

   204 by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)

   205

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

   207 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)

   208

   209 end

   210

   211 subsection {* Intervals of natural numbers *}

   212

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

   214

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

   216 by (simp add: lessThan_def)

   217

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

   219 by (simp add: lessThan_def less_Suc_eq, blast)

   220

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

   222 by (simp add: lessThan_def atMost_def less_Suc_eq_le)

   223

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

   225 by blast

   226

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

   228

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

   230 apply (simp add: greaterThan_def)

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

   232 done

   233

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

   235 apply (simp add: greaterThan_def)

   236 apply (auto elim: linorder_neqE)

   237 done

   238

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

   240 by blast

   241

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

   243

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

   245 by (unfold atLeast_def UNIV_def, simp)

   246

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

   248 apply (simp add: atLeast_def)

   249 apply (simp add: Suc_le_eq)

   250 apply (simp add: order_le_less, blast)

   251 done

   252

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

   254   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)

   255

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

   257 by blast

   258

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

   260

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

   262 by (simp add: atMost_def)

   263

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

   265 apply (simp add: atMost_def)

   266 apply (simp add: less_Suc_eq order_le_less, blast)

   267 done

   268

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

   270 by blast

   271

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

   273

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

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

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

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

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

   279 specific concept to a more general one. *}

   280

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

   282 by(simp add:lessThan_def atLeastLessThan_def)

   283

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

   285 by(simp add:atMost_def atLeastAtMost_def)

   286

   287 declare atLeast0LessThan[symmetric, code unfold]

   288         atLeast0AtMost[symmetric, code unfold]

   289

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

   291 by (simp add: atLeastLessThan_def)

   292

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

   294

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

   296 lemma atLeastLessThanSuc:

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

   298 by (auto simp add: atLeastLessThan_def)

   299

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

   301 by (auto simp add: atLeastLessThan_def)

   302 (*

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

   304 by (induct k, simp_all add: atLeastLessThanSuc)

   305

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

   307 by (auto simp add: atLeastLessThan_def)

   308 *)

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

   310   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)

   311

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

   313   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def

   314     greaterThanAtMost_def)

   315

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

   317   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def

   318     greaterThanLessThan_def)

   319

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

   321 by (auto simp add: atLeastAtMost_def)

   322

   323 subsubsection {* Image *}

   324

   325 lemma image_add_atLeastAtMost:

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

   327 proof

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

   329 next

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

   331   proof

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

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

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

   335     ultimately show "n : ?A" by blast

   336   qed

   337 qed

   338

   339 lemma image_add_atLeastLessThan:

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

   341 proof

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

   343 next

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

   345   proof

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

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

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

   349     ultimately show "n : ?A" by blast

   350   qed

   351 qed

   352

   353 corollary image_Suc_atLeastAtMost[simp]:

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

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

   356

   357 corollary image_Suc_atLeastLessThan[simp]:

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

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

   360

   361 lemma image_add_int_atLeastLessThan:

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

   363   apply (auto simp add: image_def)

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

   365   apply auto

   366   done

   367

   368

   369 subsubsection {* Finiteness *}

   370

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

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

   373

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

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

   376

   377 lemma finite_greaterThanLessThan [iff]:

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

   379 by (simp add: greaterThanLessThan_def)

   380

   381 lemma finite_atLeastLessThan [iff]:

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

   383 by (simp add: atLeastLessThan_def)

   384

   385 lemma finite_greaterThanAtMost [iff]:

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

   387 by (simp add: greaterThanAtMost_def)

   388

   389 lemma finite_atLeastAtMost [iff]:

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

   391 by (simp add: atLeastAtMost_def)

   392

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

   394 lemma bounded_nat_set_is_finite:

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

   396 apply (rule finite_subset)

   397  apply (rule_tac [2] finite_lessThan, auto)

   398 done

   399

   400 lemma finite_less_ub:

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

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

   403

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

   405 subset is exactly that interval. *}

   406

   407 lemma subset_card_intvl_is_intvl:

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

   409 proof cases

   410   assume "finite A"

   411   thus "PROP ?P"

   412   proof(induct A rule:finite_linorder_induct)

   413     case empty thus ?case by auto

   414   next

   415     case (insert A b)

   416     moreover hence "b ~: A" by auto

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

   418       using b ~: A insert by fastsimp+

   419     ultimately show ?case by auto

   420   qed

   421 next

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

   423 qed

   424

   425

   426 subsubsection {* Cardinality *}

   427

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

   429   by (induct u, simp_all add: lessThan_Suc)

   430

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

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

   433

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

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

   436   apply (erule ssubst, rule card_lessThan)

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

   438   apply (erule subst)

   439   apply (rule card_image)

   440   apply (simp add: inj_on_def)

   441   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)

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

   443   apply arith

   444   done

   445

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

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

   448

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

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

   451

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

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

   454

   455

   456 lemma ex_bij_betw_nat_finite:

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

   458 apply(drule finite_imp_nat_seg_image_inj_on)

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

   460 done

   461

   462 lemma ex_bij_betw_finite_nat:

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

   464 by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)

   465

   466

   467 subsection {* Intervals of integers *}

   468

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

   470   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)

   471

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

   473   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)

   474

   475 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:

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

   477   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)

   478

   479 subsubsection {* Finiteness *}

   480

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

   482     {(0::int)..<u} = int  {..<nat u}"

   483   apply (unfold image_def lessThan_def)

   484   apply auto

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

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

   487   done

   488

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

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

   491   apply (subst image_atLeastZeroLessThan_int, assumption)

   492   apply (rule finite_imageI)

   493   apply auto

   494   done

   495

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

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

   498   apply (erule subst)

   499   apply (rule finite_imageI)

   500   apply (rule finite_atLeastZeroLessThan_int)

   501   apply (rule image_add_int_atLeastLessThan)

   502   done

   503

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

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

   506

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

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

   509

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

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

   512

   513

   514 subsubsection {* Cardinality *}

   515

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

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

   518   apply (subst image_atLeastZeroLessThan_int, assumption)

   519   apply (subst card_image)

   520   apply (auto simp add: inj_on_def)

   521   done

   522

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

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

   525   apply (erule ssubst, rule card_atLeastZeroLessThan_int)

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

   527   apply (erule subst)

   528   apply (rule card_image)

   529   apply (simp add: inj_on_def)

   530   apply (rule image_add_int_atLeastLessThan)

   531   done

   532

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

   534 apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])

   535 apply (auto simp add: algebra_simps)

   536 done

   537

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

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

   540

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

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

   543

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

   545 proof -

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

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

   548 qed

   549

   550 lemma card_less:

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

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

   553 proof -

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

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

   556 qed

   557

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

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

   560 apply simp

   561 apply fastsimp

   562 apply auto

   563 apply (rule inj_on_diff_nat)

   564 apply auto

   565 apply (case_tac x)

   566 apply auto

   567 apply (case_tac xa)

   568 apply auto

   569 apply (case_tac xa)

   570 apply auto

   571 done

   572

   573 lemma card_less_Suc:

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

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

   576 proof -

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

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

   579     by (auto simp only: insert_Diff)

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

   581   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}))"

   582     apply (subst card_insert)

   583     apply simp_all

   584     apply (subst b)

   585     apply (subst card_less_Suc2[symmetric])

   586     apply simp_all

   587     done

   588   with c show ?thesis by simp

   589 qed

   590

   591

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

   593

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

   595

   596 subsubsection {* Disjoint Unions *}

   597

   598 text {* Singletons and open intervals *}

   599

   600 lemma ivl_disj_un_singleton:

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

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

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

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

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

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

   607 by auto

   608

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

   610

   611 lemma ivl_disj_un_one:

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

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

   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<..u} Un {u<..} = {l<..}"

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

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

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

   620 by auto

   621

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

   623

   624 lemma ivl_disj_un_two:

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

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

   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 by auto

   634

   635 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two

   636

   637 subsubsection {* Disjoint Intersections *}

   638

   639 text {* Singletons and open intervals *}

   640

   641 lemma ivl_disj_int_singleton:

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

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

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

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

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

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

   648   by simp+

   649

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

   651

   652 lemma ivl_disj_int_one:

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

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

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

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

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

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

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

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

   661   by auto

   662

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

   664

   665 lemma ivl_disj_int_two:

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

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

   668   "{l..<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   by auto

   675

   676 lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two

   677

   678 subsubsection {* Some Differences *}

   679

   680 lemma ivl_diff[simp]:

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

   682 by(auto)

   683

   684

   685 subsubsection {* Some Subset Conditions *}

   686

   687 lemma ivl_subset [simp,noatp]:

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

   689 apply(auto simp:linorder_not_le)

   690 apply(rule ccontr)

   691 apply(insert linorder_le_less_linear[of i n])

   692 apply(clarsimp simp:linorder_not_le)

   693 apply(fastsimp)

   694 done

   695

   696

   697 subsection {* Summation indexed over intervals *}

   698

   699 syntax

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

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

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

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

   704 syntax (xsymbols)

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

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

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

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

   709 syntax (HTML output)

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

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

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

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

   714 syntax (latex_sum output)

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

   716  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

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

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

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

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

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

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

   723

   724 translations

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

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

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

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

   729

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

   731 summation over intervals:

   732 \begin{center}

   733 \begin{tabular}{lll}

   734 Old & New & \LaTeX\\

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

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

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

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

   739 \end{tabular}

   740 \end{center}

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

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

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

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

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

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

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

   748

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

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

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

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

   753

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

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

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

   757 the context. *}

   758

   759 lemma setsum_ivl_cong:

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

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

   762 by(rule setsum_cong, simp_all)

   763

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

   765 on intervals are not? *)

   766

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

   768 by (simp add:atMost_Suc add_ac)

   769

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

   771 by (simp add:lessThan_Suc add_ac)

   772

   773 lemma setsum_cl_ivl_Suc[simp]:

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

   775 by (auto simp:add_ac atLeastAtMostSuc_conv)

   776

   777 lemma setsum_op_ivl_Suc[simp]:

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

   779 by (auto simp:add_ac atLeastLessThanSuc)

   780 (*

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

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

   783 by (auto simp:add_ac atLeastAtMostSuc_conv)

   784 *)

   785

   786 lemma setsum_head:

   787   fixes n :: nat

   788   assumes mn: "m <= n"

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

   790 proof -

   791   from mn

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

   793     by (auto intro: ivl_disj_un_singleton)

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

   795     by (simp add: atLeast0LessThan)

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

   797   finally show ?thesis .

   798 qed

   799

   800 lemma setsum_head_Suc:

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

   802 by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)

   803

   804 lemma setsum_head_upt_Suc:

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

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

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

   808 done

   809

   810

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

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

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

   814

   815 lemma setsum_diff_nat_ivl:

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

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

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

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

   820 apply (simp add: add_ac)

   821 done

   822

   823

   824 subsection{* Shifting bounds *}

   825

   826 lemma setsum_shift_bounds_nat_ivl:

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

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

   829

   830 lemma setsum_shift_bounds_cl_nat_ivl:

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

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

   833 apply (simp add:image_add_atLeastAtMost o_def)

   834 done

   835

   836 corollary setsum_shift_bounds_cl_Suc_ivl:

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

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

   839

   840 corollary setsum_shift_bounds_Suc_ivl:

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

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

   843

   844 lemma setsum_shift_lb_Suc0_0:

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

   846 by(simp add:setsum_head_Suc)

   847

   848 lemma setsum_shift_lb_Suc0_0_upt:

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

   850 apply(cases k)apply simp

   851 apply(simp add:setsum_head_upt_Suc)

   852 done

   853

   854 subsection {* The formula for geometric sums *}

   855

   856 lemma geometric_sum:

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

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

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

   860

   861 subsection {* The formula for arithmetic sums *}

   862

   863 lemma gauss_sum:

   864   "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =

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

   866 proof (induct n)

   867   case 0

   868   show ?case by simp

   869 next

   870   case (Suc n)

   871   then show ?case by (simp add: algebra_simps)

   872 qed

   873

   874 theorem arith_series_general:

   875   "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

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

   877 proof cases

   878   assume ngt1: "n > 1"

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

   880   have

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

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

   883     by (rule setsum_addf)

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

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

   886     unfolding One_nat_def

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

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

   889     by (simp add: left_distrib right_distrib)

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

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

   892   also from ngt1

   893   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)"

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

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

   896   finally show ?thesis by (simp add: algebra_simps)

   897 next

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

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

   900   thus ?thesis by (auto simp: algebra_simps)

   901 qed

   902

   903 lemma arith_series_nat:

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

   905 proof -

   906   have

   907     "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =

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

   909     by (rule arith_series_general)

   910   thus ?thesis

   911     unfolding One_nat_def 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 subsection {* Products indexed over intervals *}

   952

   953 syntax

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

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

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

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

   958 syntax (xsymbols)

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

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

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

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

   963 syntax (HTML output)

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

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

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

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

   968 syntax (latex_prod output)

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

   970  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

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

   972  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

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

   974  ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

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

   976  ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

   977

   978 translations

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

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

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

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

   983

   984 end
`