src/HOL/SetInterval.thy
 author paulson Thu Sep 17 14:59:58 2009 +0100 (2009-09-17) changeset 32596 bd68c04dace1 parent 32456 341c83339aeb child 32960 69916a850301 permissions -rw-r--r--
New theorems for proving equalities and inclusions involving unions
     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. Unfortunately this

   185 breaks many proofs. Since it only helps blast, it is better to leave well

   186 alone *}

   187

   188 end

   189

   190 subsubsection{* Emptyness, singletons, subset *}

   191

   192 context order

   193 begin

   194

   195 lemma atLeastatMost_empty[simp]:

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

   197 by(auto simp: atLeastAtMost_def atLeast_def atMost_def)

   198

   199 lemma atLeastatMost_empty_iff[simp]:

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

   201 by auto (blast intro: order_trans)

   202

   203 lemma atLeastatMost_empty_iff2[simp]:

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

   205 by auto (blast intro: order_trans)

   206

   207 lemma atLeastLessThan_empty[simp]:

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

   209 by(auto simp: atLeastLessThan_def)

   210

   211 lemma atLeastLessThan_empty_iff[simp]:

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

   213 by auto (blast intro: le_less_trans)

   214

   215 lemma atLeastLessThan_empty_iff2[simp]:

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

   217 by auto (blast intro: le_less_trans)

   218

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

   220 by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)

   221

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

   223 by auto (blast intro: less_le_trans)

   224

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

   226 by auto (blast intro: less_le_trans)

   227

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

   229 by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)

   230

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

   232 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)

   233

   234 lemma atLeastatMost_subset_iff[simp]:

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

   236 unfolding atLeastAtMost_def atLeast_def atMost_def

   237 by (blast intro: order_trans)

   238

   239 lemma atLeastatMost_psubset_iff:

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

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

   242 by(simp add: psubset_eq expand_set_eq less_le_not_le)(blast intro: order_trans)

   243

   244 end

   245

   246 lemma (in linorder) atLeastLessThan_subset_iff:

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

   248 apply (auto simp:subset_eq Ball_def)

   249 apply(frule_tac x=a in spec)

   250 apply(erule_tac x=d in allE)

   251 apply (simp add: less_imp_le)

   252 done

   253

   254 subsubsection {* Intersection *}

   255

   256 context linorder

   257 begin

   258

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

   260 by auto

   261

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

   263 by auto

   264

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

   266 by auto

   267

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

   269 by auto

   270

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

   272 by auto

   273

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

   275 by auto

   276

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

   278 by auto

   279

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

   281 by auto

   282

   283 end

   284

   285

   286 subsection {* Intervals of natural numbers *}

   287

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

   289

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

   291 by (simp add: lessThan_def)

   292

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

   294 by (simp add: lessThan_def less_Suc_eq, blast)

   295

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

   297 by (simp add: lessThan_def atMost_def less_Suc_eq_le)

   298

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

   300 by blast

   301

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

   303

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

   305 apply (simp add: greaterThan_def)

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

   307 done

   308

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

   310 apply (simp add: greaterThan_def)

   311 apply (auto elim: linorder_neqE)

   312 done

   313

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

   315 by blast

   316

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

   318

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

   320 by (unfold atLeast_def UNIV_def, simp)

   321

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

   323 apply (simp add: atLeast_def)

   324 apply (simp add: Suc_le_eq)

   325 apply (simp add: order_le_less, blast)

   326 done

   327

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

   329   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)

   330

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

   332 by blast

   333

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

   335

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

   337 by (simp add: atMost_def)

   338

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

   340 apply (simp add: atMost_def)

   341 apply (simp add: less_Suc_eq order_le_less, blast)

   342 done

   343

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

   345 by blast

   346

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

   348

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

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

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

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

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

   354 specific concept to a more general one. *}

   355

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

   357 by(simp add:lessThan_def atLeastLessThan_def)

   358

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

   360 by(simp add:atMost_def atLeastAtMost_def)

   361

   362 declare atLeast0LessThan[symmetric, code_unfold]

   363         atLeast0AtMost[symmetric, code_unfold]

   364

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

   366 by (simp add: atLeastLessThan_def)

   367

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

   369

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

   371 lemma atLeastLessThanSuc:

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

   373 by (auto simp add: atLeastLessThan_def)

   374

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

   376 by (auto simp add: atLeastLessThan_def)

   377 (*

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

   379 by (induct k, simp_all add: atLeastLessThanSuc)

   380

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

   382 by (auto simp add: atLeastLessThan_def)

   383 *)

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

   385   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)

   386

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

   388   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def

   389     greaterThanAtMost_def)

   390

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

   392   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def

   393     greaterThanLessThan_def)

   394

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

   396 by (auto simp add: atLeastAtMost_def)

   397

   398 subsubsection {* Image *}

   399

   400 lemma image_add_atLeastAtMost:

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

   402 proof

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

   404 next

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

   406   proof

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

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

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

   410     ultimately show "n : ?A" by blast

   411   qed

   412 qed

   413

   414 lemma image_add_atLeastLessThan:

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

   416 proof

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

   418 next

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

   420   proof

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

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

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

   424     ultimately show "n : ?A" by blast

   425   qed

   426 qed

   427

   428 corollary image_Suc_atLeastAtMost[simp]:

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

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

   431

   432 corollary image_Suc_atLeastLessThan[simp]:

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

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

   435

   436 lemma image_add_int_atLeastLessThan:

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

   438   apply (auto simp add: image_def)

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

   440   apply auto

   441   done

   442

   443

   444 subsubsection {* Finiteness *}

   445

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

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

   448

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

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

   451

   452 lemma finite_greaterThanLessThan [iff]:

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

   454 by (simp add: greaterThanLessThan_def)

   455

   456 lemma finite_atLeastLessThan [iff]:

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

   458 by (simp add: atLeastLessThan_def)

   459

   460 lemma finite_greaterThanAtMost [iff]:

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

   462 by (simp add: greaterThanAtMost_def)

   463

   464 lemma finite_atLeastAtMost [iff]:

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

   466 by (simp add: atLeastAtMost_def)

   467

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

   469 lemma bounded_nat_set_is_finite:

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

   471 apply (rule finite_subset)

   472  apply (rule_tac [2] finite_lessThan, auto)

   473 done

   474

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

   476 lemma finite_nat_set_iff_bounded:

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

   478 proof

   479   assume f:?F  show ?B

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

   481 next

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

   483 qed

   484

   485 lemma finite_nat_set_iff_bounded_le:

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

   487 apply(simp add:finite_nat_set_iff_bounded)

   488 apply(blast dest:less_imp_le_nat le_imp_less_Suc)

   489 done

   490

   491 lemma finite_less_ub:

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

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

   494

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

   496 subset is exactly that interval. *}

   497

   498 lemma subset_card_intvl_is_intvl:

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

   500 proof cases

   501   assume "finite A"

   502   thus "PROP ?P"

   503   proof(induct A rule:finite_linorder_max_induct)

   504     case empty thus ?case by auto

   505   next

   506     case (insert A b)

   507     moreover hence "b ~: A" by auto

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

   509       using b ~: A insert by fastsimp+

   510     ultimately show ?case by auto

   511   qed

   512 next

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

   514 qed

   515

   516

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

   518

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

   520   by (auto simp add: atLeast0LessThan)

   521

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

   523   by (subst UN_UN_finite_eq [symmetric]) blast

   524

   525 lemma UN_finite2_subset:

   526   assumes sb: "!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> (\<Union>i\<in>{0..<n}. B i)"

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

   528 proof (rule UN_finite_subset)

   529   fix n

   530   have "(\<Union>i\<in>{0..<n}. A i) \<subseteq> (\<Union>i\<in>{0..<n}. B i)" by (rule sb)

   531   also have "...  \<subseteq> (\<Union>n::nat. \<Union>i\<in>{0..<n}. B i)" by blast

   532   also have "... = (\<Union>n. B n)" by (simp add: UN_UN_finite_eq)

   533   finally show "(\<Union>i\<in>{0..<n}. A i) \<subseteq> (\<Union>n. B n)" .

   534 qed

   535

   536 lemma UN_finite2_eq:

   537   "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) = (\<Union>i\<in>{0..<n}. B i)) \<Longrightarrow> (\<Union>n. A n) = (\<Union>n. B n)"

   538   by (iprover intro: subset_antisym UN_finite2_subset elim: equalityE)

   539

   540

   541 subsubsection {* Cardinality *}

   542

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

   544   by (induct u, simp_all add: lessThan_Suc)

   545

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

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

   548

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

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

   551   apply (erule ssubst, rule card_lessThan)

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

   553   apply (erule subst)

   554   apply (rule card_image)

   555   apply (simp add: inj_on_def)

   556   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)

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

   558   apply arith

   559   done

   560

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

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

   563

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

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

   566

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

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

   569

   570 lemma ex_bij_betw_nat_finite:

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

   572 apply(drule finite_imp_nat_seg_image_inj_on)

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

   574 done

   575

   576 lemma ex_bij_betw_finite_nat:

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

   578 by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)

   579

   580 lemma finite_same_card_bij:

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

   582 apply(drule ex_bij_betw_finite_nat)

   583 apply(drule ex_bij_betw_nat_finite)

   584 apply(auto intro!:bij_betw_trans)

   585 done

   586

   587 lemma ex_bij_betw_nat_finite_1:

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

   589 by (rule finite_same_card_bij) auto

   590

   591

   592 subsection {* Intervals of integers *}

   593

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

   595   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)

   596

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

   598   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)

   599

   600 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:

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

   602   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)

   603

   604 subsubsection {* Finiteness *}

   605

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

   607     {(0::int)..<u} = int  {..<nat u}"

   608   apply (unfold image_def lessThan_def)

   609   apply auto

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

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

   612   done

   613

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

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

   616   apply (subst image_atLeastZeroLessThan_int, assumption)

   617   apply (rule finite_imageI)

   618   apply auto

   619   done

   620

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

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

   623   apply (erule subst)

   624   apply (rule finite_imageI)

   625   apply (rule finite_atLeastZeroLessThan_int)

   626   apply (rule image_add_int_atLeastLessThan)

   627   done

   628

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

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

   631

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

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

   634

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

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

   637

   638

   639 subsubsection {* Cardinality *}

   640

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

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

   643   apply (subst image_atLeastZeroLessThan_int, assumption)

   644   apply (subst card_image)

   645   apply (auto simp add: inj_on_def)

   646   done

   647

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

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

   650   apply (erule ssubst, rule card_atLeastZeroLessThan_int)

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

   652   apply (erule subst)

   653   apply (rule card_image)

   654   apply (simp add: inj_on_def)

   655   apply (rule image_add_int_atLeastLessThan)

   656   done

   657

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

   659 apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])

   660 apply (auto simp add: algebra_simps)

   661 done

   662

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

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

   665

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

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

   668

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

   670 proof -

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

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

   673 qed

   674

   675 lemma card_less:

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

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

   678 proof -

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

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

   681 qed

   682

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

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

   685 apply simp

   686 apply fastsimp

   687 apply auto

   688 apply (rule inj_on_diff_nat)

   689 apply auto

   690 apply (case_tac x)

   691 apply auto

   692 apply (case_tac xa)

   693 apply auto

   694 apply (case_tac xa)

   695 apply auto

   696 done

   697

   698 lemma card_less_Suc:

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

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

   701 proof -

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

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

   704     by (auto simp only: insert_Diff)

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

   706   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}))"

   707     apply (subst card_insert)

   708     apply simp_all

   709     apply (subst b)

   710     apply (subst card_less_Suc2[symmetric])

   711     apply simp_all

   712     done

   713   with c show ?thesis by simp

   714 qed

   715

   716

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

   718

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

   720

   721 subsubsection {* Disjoint Unions *}

   722

   723 text {* Singletons and open intervals *}

   724

   725 lemma ivl_disj_un_singleton:

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

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

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

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

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

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

   732 by auto

   733

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

   735

   736 lemma ivl_disj_un_one:

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

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

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

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

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

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

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

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

   745 by auto

   746

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

   748

   749 lemma ivl_disj_un_two:

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

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

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

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

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

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

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

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

   758 by auto

   759

   760 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two

   761

   762 subsubsection {* Disjoint Intersections *}

   763

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

   765

   766 lemma ivl_disj_int_one:

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

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

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

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

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

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

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

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

   775   by auto

   776

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

   778

   779 lemma ivl_disj_int_two:

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

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

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

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

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

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

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

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

   788   by auto

   789

   790 lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two

   791

   792 subsubsection {* Some Differences *}

   793

   794 lemma ivl_diff[simp]:

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

   796 by(auto)

   797

   798

   799 subsubsection {* Some Subset Conditions *}

   800

   801 lemma ivl_subset [simp,noatp]:

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

   803 apply(auto simp:linorder_not_le)

   804 apply(rule ccontr)

   805 apply(insert linorder_le_less_linear[of i n])

   806 apply(clarsimp simp:linorder_not_le)

   807 apply(fastsimp)

   808 done

   809

   810

   811 subsection {* Summation indexed over intervals *}

   812

   813 syntax

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

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

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

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

   818 syntax (xsymbols)

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

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

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

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

   823 syntax (HTML output)

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

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

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

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

   828 syntax (latex_sum output)

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

   830  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

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

   832  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

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

   834  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

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

   836  ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

   837

   838 translations

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

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

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

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

   843

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

   845 summation over intervals:

   846 \begin{center}

   847 \begin{tabular}{lll}

   848 Old & New & \LaTeX\\

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

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

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

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

   853 \end{tabular}

   854 \end{center}

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

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

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

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

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

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

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

   862

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

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

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

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

   867

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

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

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

   871 the context. *}

   872

   873 lemma setsum_ivl_cong:

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

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

   876 by(rule setsum_cong, simp_all)

   877

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

   879 on intervals are not? *)

   880

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

   882 by (simp add:atMost_Suc add_ac)

   883

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

   885 by (simp add:lessThan_Suc add_ac)

   886

   887 lemma setsum_cl_ivl_Suc[simp]:

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

   889 by (auto simp:add_ac atLeastAtMostSuc_conv)

   890

   891 lemma setsum_op_ivl_Suc[simp]:

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

   893 by (auto simp:add_ac atLeastLessThanSuc)

   894 (*

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

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

   897 by (auto simp:add_ac atLeastAtMostSuc_conv)

   898 *)

   899

   900 lemma setsum_head:

   901   fixes n :: nat

   902   assumes mn: "m <= n"

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

   904 proof -

   905   from mn

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

   907     by (auto intro: ivl_disj_un_singleton)

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

   909     by (simp add: atLeast0LessThan)

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

   911   finally show ?thesis .

   912 qed

   913

   914 lemma setsum_head_Suc:

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

   916 by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)

   917

   918 lemma setsum_head_upt_Suc:

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

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

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

   922 done

   923

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

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

   926 proof-

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

   928   thus ?thesis by (auto simp: ivl_disj_int setsum_Un_disjoint

   929     atLeastSucAtMost_greaterThanAtMost)

   930 qed

   931

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

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

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

   935

   936 lemma setsum_diff_nat_ivl:

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

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

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

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

   941 apply (simp add: add_ac)

   942 done

   943

   944 lemma setsum_natinterval_difff:

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

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

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

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

   949

   950 lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]

   951

   952 lemma setsum_setsum_restrict:

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

   954   by (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)

   955      (rule setsum_commute)

   956

   957 lemma setsum_image_gen: assumes fS: "finite S"

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

   959 proof-

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

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

   962     by simp

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

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

   965   finally show ?thesis .

   966 qed

   967

   968 lemma setsum_multicount_gen:

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

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

   971 proof-

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

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

   974     using assms(3) by auto

   975   finally show ?thesis .

   976 qed

   977

   978 lemma setsum_multicount:

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

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

   981 proof-

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

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

   984   finally show ?thesis by auto

   985 qed

   986

   987

   988 subsection{* Shifting bounds *}

   989

   990 lemma setsum_shift_bounds_nat_ivl:

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

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

   993

   994 lemma setsum_shift_bounds_cl_nat_ivl:

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

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

   997 apply (simp add:image_add_atLeastAtMost o_def)

   998 done

   999

  1000 corollary setsum_shift_bounds_cl_Suc_ivl:

  1001   "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"

  1002 by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])

  1003

  1004 corollary setsum_shift_bounds_Suc_ivl:

  1005   "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"

  1006 by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])

  1007

  1008 lemma setsum_shift_lb_Suc0_0:

  1009   "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"

  1010 by(simp add:setsum_head_Suc)

  1011

  1012 lemma setsum_shift_lb_Suc0_0_upt:

  1013   "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"

  1014 apply(cases k)apply simp

  1015 apply(simp add:setsum_head_upt_Suc)

  1016 done

  1017

  1018 subsection {* The formula for geometric sums *}

  1019

  1020 lemma geometric_sum:

  1021   "x ~= 1 ==> (\<Sum>i=0..<n. x ^ i) =

  1022   (x ^ n - 1) / (x - 1::'a::{field})"

  1023 by (induct "n") (simp_all add:field_simps power_Suc)

  1024

  1025 subsection {* The formula for arithmetic sums *}

  1026

  1027 lemma gauss_sum:

  1028   "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =

  1029    of_nat n*((of_nat n)+1)"

  1030 proof (induct n)

  1031   case 0

  1032   show ?case by simp

  1033 next

  1034   case (Suc n)

  1035   then show ?case by (simp add: algebra_simps)

  1036 qed

  1037

  1038 theorem arith_series_general:

  1039   "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

  1040   of_nat n * (a + (a + of_nat(n - 1)*d))"

  1041 proof cases

  1042   assume ngt1: "n > 1"

  1043   let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"

  1044   have

  1045     "(\<Sum>i\<in>{..<n}. a+?I i*d) =

  1046      ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"

  1047     by (rule setsum_addf)

  1048   also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp

  1049   also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"

  1050     unfolding One_nat_def

  1051     by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)

  1052   also have "(1+1)*\<dots> = (1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..<n}. ?I i)"

  1053     by (simp add: left_distrib right_distrib)

  1054   also from ngt1 have "{1..<n} = {1..n - 1}"

  1055     by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)

  1056   also from ngt1

  1057   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)"

  1058     by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)

  1059        (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])

  1060   finally show ?thesis by (simp add: algebra_simps)

  1061 next

  1062   assume "\<not>(n > 1)"

  1063   hence "n = 1 \<or> n = 0" by auto

  1064   thus ?thesis by (auto simp: algebra_simps)

  1065 qed

  1066

  1067 lemma arith_series_nat:

  1068   "Suc (Suc 0) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"

  1069 proof -

  1070   have

  1071     "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =

  1072     of_nat(n) * (a + (a + of_nat(n - 1)*d))"

  1073     by (rule arith_series_general)

  1074   thus ?thesis

  1075     unfolding One_nat_def by (auto simp add: of_nat_id)

  1076 qed

  1077

  1078 lemma arith_series_int:

  1079   "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

  1080   of_nat n * (a + (a + of_nat(n - 1)*d))"

  1081 proof -

  1082   have

  1083     "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

  1084     of_nat(n) * (a + (a + of_nat(n - 1)*d))"

  1085     by (rule arith_series_general)

  1086   thus ?thesis by simp

  1087 qed

  1088

  1089 lemma sum_diff_distrib:

  1090   fixes P::"nat\<Rightarrow>nat"

  1091   shows

  1092   "\<forall>x. Q x \<le> P x  \<Longrightarrow>

  1093   (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"

  1094 proof (induct n)

  1095   case 0 show ?case by simp

  1096 next

  1097   case (Suc n)

  1098

  1099   let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"

  1100   let ?rhs = "\<Sum>x<n. P x - Q x"

  1101

  1102   from Suc have "?lhs = ?rhs" by simp

  1103   moreover

  1104   from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp

  1105   moreover

  1106   from Suc have

  1107     "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"

  1108     by (subst diff_diff_left[symmetric],

  1109         subst diff_add_assoc2)

  1110        (auto simp: diff_add_assoc2 intro: setsum_mono)

  1111   ultimately

  1112   show ?case by simp

  1113 qed

  1114

  1115 subsection {* Products indexed over intervals *}

  1116

  1117 syntax

  1118   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)

  1119   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)

  1120   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)

  1121   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)

  1122 syntax (xsymbols)

  1123   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)

  1124   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)

  1125   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)

  1126   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)

  1127 syntax (HTML output)

  1128   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)

  1129   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)

  1130   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)

  1131   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)

  1132 syntax (latex_prod output)

  1133   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1134  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

  1135   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1136  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

  1137   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1138  ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

  1139   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1140  ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

  1141

  1142 translations

  1143   "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"

  1144   "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"

  1145   "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"

  1146   "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"

  1147

  1148 end
`