src/HOL/SetInterval.thy
 author haftmann Wed Jun 30 16:46:44 2010 +0200 (2010-06-30) changeset 37659 14cabf5fa710 parent 37388 793618618f78 child 37664 2946b8f057df permissions -rw-r--r--
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     1 (*  Title:      HOL/SetInterval.thy

     2     Author:     Tobias Nipkow

     3     Author:     Clemens Ballarin

     4     Author:     Jeremy Avigad

     5

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

     7 *)

     8

     9 header {* Set intervals *}

    10

    11 theory SetInterval

    12 imports Int Nat_Transfer

    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 _<=_./ _)" [0, 0, 10] 10)

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

    59   "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3INT _<=_./ _)" [0, 0, 10] 10)

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

    61

    62 syntax (xsymbols)

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

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

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

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

    67

    68 syntax (latex output)

    69   "_UNION_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ \<le> _)/ _)" [0, 0, 10] 10)

    70   "_UNION_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ < _)/ _)" [0, 0, 10] 10)

    71   "_INTER_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ \<le> _)/ _)" [0, 0, 10] 10)

    72   "_INTER_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ < _)/ _)" [0, 0, 10] 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,no_atp]:

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

   170 by (simp add: greaterThanLessThan_def)

   171

   172 lemma atLeastLessThan_iff [simp,no_atp]:

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

   174 by (simp add: atLeastLessThan_def)

   175

   176 lemma greaterThanAtMost_iff [simp,no_atp]:

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

   178 by (simp add: greaterThanAtMost_def)

   179

   180 lemma atLeastAtMost_iff [simp,no_atp]:

   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_singleton': "a = b \<Longrightarrow> {a .. b} = {a}" by simp

   235

   236 lemma atLeastatMost_subset_iff[simp]:

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

   238 unfolding atLeastAtMost_def atLeast_def atMost_def

   239 by (blast intro: order_trans)

   240

   241 lemma atLeastatMost_psubset_iff:

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

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

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

   245

   246 lemma atLeastAtMost_singleton_iff[simp]:

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

   248 proof

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

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

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

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

   253 qed simp

   254

   255 end

   256

   257 lemma (in linorder) atLeastLessThan_subset_iff:

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

   259 apply (auto simp:subset_eq Ball_def)

   260 apply(frule_tac x=a in spec)

   261 apply(erule_tac x=d in allE)

   262 apply (simp add: less_imp_le)

   263 done

   264

   265 subsubsection {* Intersection *}

   266

   267 context linorder

   268 begin

   269

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

   271 by auto

   272

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

   274 by auto

   275

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

   277 by auto

   278

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

   280 by auto

   281

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

   283 by auto

   284

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

   286 by auto

   287

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

   289 by auto

   290

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

   292 by auto

   293

   294 end

   295

   296

   297 subsection {* Intervals of natural numbers *}

   298

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

   300

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

   302 by (simp add: lessThan_def)

   303

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

   305 by (simp add: lessThan_def less_Suc_eq, blast)

   306

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

   308 by (simp add: lessThan_def atMost_def less_Suc_eq_le)

   309

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

   311 by blast

   312

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

   314

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

   316 apply (simp add: greaterThan_def)

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

   318 done

   319

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

   321 apply (simp add: greaterThan_def)

   322 apply (auto elim: linorder_neqE)

   323 done

   324

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

   326 by blast

   327

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

   329

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

   331 by (unfold atLeast_def UNIV_def, simp)

   332

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

   334 apply (simp add: atLeast_def)

   335 apply (simp add: Suc_le_eq)

   336 apply (simp add: order_le_less, blast)

   337 done

   338

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

   340   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)

   341

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

   343 by blast

   344

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

   346

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

   348 by (simp add: atMost_def)

   349

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

   351 apply (simp add: atMost_def)

   352 apply (simp add: less_Suc_eq order_le_less, blast)

   353 done

   354

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

   356 by blast

   357

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

   359

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

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

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

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

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

   365 specific concept to a more general one. *}

   366

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

   368 by(simp add:lessThan_def atLeastLessThan_def)

   369

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

   371 by(simp add:atMost_def atLeastAtMost_def)

   372

   373 declare atLeast0LessThan[symmetric, code_unfold]

   374         atLeast0AtMost[symmetric, code_unfold]

   375

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

   377 by (simp add: atLeastLessThan_def)

   378

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

   380

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

   382 lemma atLeastLessThanSuc:

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

   384 by (auto simp add: atLeastLessThan_def)

   385

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

   387 by (auto simp add: atLeastLessThan_def)

   388 (*

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

   390 by (induct k, simp_all add: atLeastLessThanSuc)

   391

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

   393 by (auto simp add: atLeastLessThan_def)

   394 *)

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

   396   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)

   397

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

   399   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def

   400     greaterThanAtMost_def)

   401

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

   403   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def

   404     greaterThanLessThan_def)

   405

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

   407 by (auto simp add: atLeastAtMost_def)

   408

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

   410   apply (induct k)

   411   apply (simp_all add: atLeastLessThanSuc)

   412   done

   413

   414 subsubsection {* Image *}

   415

   416 lemma image_add_atLeastAtMost:

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

   418 proof

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

   420 next

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

   422   proof

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

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

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

   426     ultimately show "n : ?A" by blast

   427   qed

   428 qed

   429

   430 lemma image_add_atLeastLessThan:

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

   432 proof

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

   434 next

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

   436   proof

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

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

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

   440     ultimately show "n : ?A" by blast

   441   qed

   442 qed

   443

   444 corollary image_Suc_atLeastAtMost[simp]:

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

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

   447

   448 corollary image_Suc_atLeastLessThan[simp]:

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

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

   451

   452 lemma image_add_int_atLeastLessThan:

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

   454   apply (auto simp add: image_def)

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

   456   apply auto

   457   done

   458

   459 context ordered_ab_group_add

   460 begin

   461

   462 lemma

   463   fixes x :: 'a

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

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

   466 proof safe

   467   fix y assume "y < -x"

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

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

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

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

   472 next

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

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

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

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

   477 qed simp_all

   478

   479 lemma

   480   fixes x :: 'a

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

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

   483 proof -

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

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

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

   487     by (simp_all add: image_image

   488         del: image_uminus_greaterThan image_uminus_atLeast)

   489 qed

   490

   491 lemma

   492   fixes x :: 'a

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

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

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

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

   497   by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def

   498       greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)

   499 end

   500

   501 subsubsection {* Finiteness *}

   502

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

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

   505

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

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

   508

   509 lemma finite_greaterThanLessThan [iff]:

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

   511 by (simp add: greaterThanLessThan_def)

   512

   513 lemma finite_atLeastLessThan [iff]:

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

   515 by (simp add: atLeastLessThan_def)

   516

   517 lemma finite_greaterThanAtMost [iff]:

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

   519 by (simp add: greaterThanAtMost_def)

   520

   521 lemma finite_atLeastAtMost [iff]:

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

   523 by (simp add: atLeastAtMost_def)

   524

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

   526 lemma bounded_nat_set_is_finite:

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

   528 apply (rule finite_subset)

   529  apply (rule_tac [2] finite_lessThan, auto)

   530 done

   531

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

   533 lemma finite_nat_set_iff_bounded:

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

   535 proof

   536   assume f:?F  show ?B

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

   538 next

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

   540 qed

   541

   542 lemma finite_nat_set_iff_bounded_le:

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

   544 apply(simp add:finite_nat_set_iff_bounded)

   545 apply(blast dest:less_imp_le_nat le_imp_less_Suc)

   546 done

   547

   548 lemma finite_less_ub:

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

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

   551

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

   553 subset is exactly that interval. *}

   554

   555 lemma subset_card_intvl_is_intvl:

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

   557 proof cases

   558   assume "finite A"

   559   thus "PROP ?P"

   560   proof(induct A rule:finite_linorder_max_induct)

   561     case empty thus ?case by auto

   562   next

   563     case (insert b A)

   564     moreover hence "b ~: A" by auto

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

   566       using b ~: A insert by fastsimp+

   567     ultimately show ?case by auto

   568   qed

   569 next

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

   571 qed

   572

   573

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

   575

   576 lemma UN_le_eq_Un0:

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

   578 proof

   579   show "?A <= ?B"

   580   proof

   581     fix x assume "x : ?A"

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

   583     show "x : ?B"

   584     proof(cases i)

   585       case 0 with i show ?thesis by simp

   586     next

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

   588     qed

   589   qed

   590 next

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

   592 qed

   593

   594 lemma UN_le_add_shift:

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

   596 proof

   597   show "?A <= ?B" by fastsimp

   598 next

   599   show "?B <= ?A"

   600   proof

   601     fix x assume "x : ?B"

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

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

   604     thus "x : ?A" by blast

   605   qed

   606 qed

   607

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

   609   by (auto simp add: atLeast0LessThan)

   610

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

   612   by (subst UN_UN_finite_eq [symmetric]) blast

   613

   614 lemma UN_finite2_subset:

   615      "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> (\<Union>i\<in>{0..<n+k}. B i)) \<Longrightarrow> (\<Union>n. A n) \<subseteq> (\<Union>n. B n)"

   616   apply (rule UN_finite_subset)

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

   618   apply blast

   619   done

   620

   621 lemma UN_finite2_eq:

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

   623   apply (rule subset_antisym)

   624    apply (rule UN_finite2_subset, blast)

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

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

   627  done

   628

   629

   630 subsubsection {* Cardinality *}

   631

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

   633   by (induct u, simp_all add: lessThan_Suc)

   634

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

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

   637

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

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

   640   apply (erule ssubst, rule card_lessThan)

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

   642   apply (erule subst)

   643   apply (rule card_image)

   644   apply (simp add: inj_on_def)

   645   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)

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

   647   apply arith

   648   done

   649

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

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

   652

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

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

   655

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

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

   658

   659 lemma ex_bij_betw_nat_finite:

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

   661 apply(drule finite_imp_nat_seg_image_inj_on)

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

   663 done

   664

   665 lemma ex_bij_betw_finite_nat:

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

   667 by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)

   668

   669 lemma finite_same_card_bij:

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

   671 apply(drule ex_bij_betw_finite_nat)

   672 apply(drule ex_bij_betw_nat_finite)

   673 apply(auto intro!:bij_betw_trans)

   674 done

   675

   676 lemma ex_bij_betw_nat_finite_1:

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

   678 by (rule finite_same_card_bij) auto

   679

   680

   681 subsection {* Intervals of integers *}

   682

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

   684   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)

   685

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

   687   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)

   688

   689 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:

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

   691   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)

   692

   693 subsubsection {* Finiteness *}

   694

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

   696     {(0::int)..<u} = int  {..<nat u}"

   697   apply (unfold image_def lessThan_def)

   698   apply auto

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

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

   701   done

   702

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

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

   705   apply (subst image_atLeastZeroLessThan_int, assumption)

   706   apply (rule finite_imageI)

   707   apply auto

   708   done

   709

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

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

   712   apply (erule subst)

   713   apply (rule finite_imageI)

   714   apply (rule finite_atLeastZeroLessThan_int)

   715   apply (rule image_add_int_atLeastLessThan)

   716   done

   717

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

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

   720

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

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

   723

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

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

   726

   727

   728 subsubsection {* Cardinality *}

   729

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

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

   732   apply (subst image_atLeastZeroLessThan_int, assumption)

   733   apply (subst card_image)

   734   apply (auto simp add: inj_on_def)

   735   done

   736

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

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

   739   apply (erule ssubst, rule card_atLeastZeroLessThan_int)

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

   741   apply (erule subst)

   742   apply (rule card_image)

   743   apply (simp add: inj_on_def)

   744   apply (rule image_add_int_atLeastLessThan)

   745   done

   746

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

   748 apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])

   749 apply (auto simp add: algebra_simps)

   750 done

   751

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

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

   754

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

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

   757

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

   759 proof -

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

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

   762 qed

   763

   764 lemma card_less:

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

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

   767 proof -

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

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

   770 qed

   771

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

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

   774 apply simp

   775 apply fastsimp

   776 apply auto

   777 apply (rule inj_on_diff_nat)

   778 apply auto

   779 apply (case_tac x)

   780 apply auto

   781 apply (case_tac xa)

   782 apply auto

   783 apply (case_tac xa)

   784 apply auto

   785 done

   786

   787 lemma card_less_Suc:

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

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

   790 proof -

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

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

   793     by (auto simp only: insert_Diff)

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

   795   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}))"

   796     apply (subst card_insert)

   797     apply simp_all

   798     apply (subst b)

   799     apply (subst card_less_Suc2[symmetric])

   800     apply simp_all

   801     done

   802   with c show ?thesis by simp

   803 qed

   804

   805

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

   807

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

   809

   810 subsubsection {* Disjoint Unions *}

   811

   812 text {* Singletons and open intervals *}

   813

   814 lemma ivl_disj_un_singleton:

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

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

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

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

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

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

   821 by auto

   822

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

   824

   825 lemma ivl_disj_un_one:

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

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

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

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

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

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

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

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

   834 by auto

   835

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

   837

   838 lemma ivl_disj_un_two:

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

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

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

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

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

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

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

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

   847 by auto

   848

   849 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two

   850

   851 subsubsection {* Disjoint Intersections *}

   852

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

   854

   855 lemma ivl_disj_int_one:

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

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

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

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

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

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

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

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

   864   by auto

   865

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

   867

   868 lemma ivl_disj_int_two:

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

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

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

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

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

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

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

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

   877   by auto

   878

   879 lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two

   880

   881 subsubsection {* Some Differences *}

   882

   883 lemma ivl_diff[simp]:

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

   885 by(auto)

   886

   887

   888 subsubsection {* Some Subset Conditions *}

   889

   890 lemma ivl_subset [simp,no_atp]:

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

   892 apply(auto simp:linorder_not_le)

   893 apply(rule ccontr)

   894 apply(insert linorder_le_less_linear[of i n])

   895 apply(clarsimp simp:linorder_not_le)

   896 apply(fastsimp)

   897 done

   898

   899

   900 subsection {* Summation indexed over intervals *}

   901

   902 syntax

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

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

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

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

   907 syntax (xsymbols)

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

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

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

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

   912 syntax (HTML output)

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

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

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

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

   917 syntax (latex_sum output)

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

   919  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

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

   921  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

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

   923  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

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

   925  ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

   926

   927 translations

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

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

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

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

   932

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

   934 summation over intervals:

   935 \begin{center}

   936 \begin{tabular}{lll}

   937 Old & New & \LaTeX\\

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

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

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

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

   942 \end{tabular}

   943 \end{center}

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

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

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

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

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

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

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

   951

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

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

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

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

   956

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

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

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

   960 the context. *}

   961

   962 lemma setsum_ivl_cong:

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

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

   965 by(rule setsum_cong, simp_all)

   966

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

   968 on intervals are not? *)

   969

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

   971 by (simp add:atMost_Suc add_ac)

   972

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

   974 by (simp add:lessThan_Suc add_ac)

   975

   976 lemma setsum_cl_ivl_Suc[simp]:

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

   978 by (auto simp:add_ac atLeastAtMostSuc_conv)

   979

   980 lemma setsum_op_ivl_Suc[simp]:

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

   982 by (auto simp:add_ac atLeastLessThanSuc)

   983 (*

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

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

   986 by (auto simp:add_ac atLeastAtMostSuc_conv)

   987 *)

   988

   989 lemma setsum_head:

   990   fixes n :: nat

   991   assumes mn: "m <= n"

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

   993 proof -

   994   from mn

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

   996     by (auto intro: ivl_disj_un_singleton)

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

   998     by (simp add: atLeast0LessThan)

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

  1000   finally show ?thesis .

  1001 qed

  1002

  1003 lemma setsum_head_Suc:

  1004   "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"

  1005 by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)

  1006

  1007 lemma setsum_head_upt_Suc:

  1008   "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"

  1009 apply(insert setsum_head_Suc[of m "n - Suc 0" f])

  1010 apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)

  1011 done

  1012

  1013 lemma setsum_ub_add_nat: assumes "(m::nat) \<le> n + 1"

  1014   shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"

  1015 proof-

  1016   have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using m \<le> n+1 by auto

  1017   thus ?thesis by (auto simp: ivl_disj_int setsum_Un_disjoint

  1018     atLeastSucAtMost_greaterThanAtMost)

  1019 qed

  1020

  1021 lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>

  1022   setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"

  1023 by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)

  1024

  1025 lemma setsum_diff_nat_ivl:

  1026 fixes f :: "nat \<Rightarrow> 'a::ab_group_add"

  1027 shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>

  1028   setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"

  1029 using setsum_add_nat_ivl [of m n p f,symmetric]

  1030 apply (simp add: add_ac)

  1031 done

  1032

  1033 lemma setsum_natinterval_difff:

  1034   fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"

  1035   shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =

  1036           (if m <= n then f m - f(n + 1) else 0)"

  1037 by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)

  1038

  1039 lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]

  1040

  1041 lemma setsum_setsum_restrict:

  1042   "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"

  1043   by (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)

  1044      (rule setsum_commute)

  1045

  1046 lemma setsum_image_gen: assumes fS: "finite S"

  1047   shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f  S)"

  1048 proof-

  1049   { fix x assume "x \<in> S" then have "{y. y\<in> fS \<and> f x = y} = {f x}" by auto }

  1050   hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> fS \<and> f x = y}) S"

  1051     by simp

  1052   also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f  S)"

  1053     by (rule setsum_setsum_restrict[OF fS finite_imageI[OF fS]])

  1054   finally show ?thesis .

  1055 qed

  1056

  1057 lemma setsum_le_included:

  1058   fixes f :: "'a \<Rightarrow> 'b::ordered_comm_monoid_add"

  1059   assumes "finite s" "finite t"

  1060   and "\<forall>y\<in>t. 0 \<le> g y" "(\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> f x \<le> g y)"

  1061   shows "setsum f s \<le> setsum g t"

  1062 proof -

  1063   have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"

  1064   proof (rule setsum_mono)

  1065     fix y assume "y \<in> s"

  1066     with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto

  1067     with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")

  1068       using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]

  1069       by (auto intro!: setsum_mono2)

  1070   qed

  1071   also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i  t)"

  1072     using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)

  1073   also have "... \<le> setsum g t"

  1074     using assms by (auto simp: setsum_image_gen[symmetric])

  1075   finally show ?thesis .

  1076 qed

  1077

  1078 lemma setsum_multicount_gen:

  1079   assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"

  1080   shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")

  1081 proof-

  1082   have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto

  1083   also have "\<dots> = ?r" unfolding setsum_setsum_restrict[OF assms(1-2)]

  1084     using assms(3) by auto

  1085   finally show ?thesis .

  1086 qed

  1087

  1088 lemma setsum_multicount:

  1089   assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"

  1090   shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")

  1091 proof-

  1092   have "?l = setsum (\<lambda>i. k) T" by(rule setsum_multicount_gen)(auto simp:assms)

  1093   also have "\<dots> = ?r" by(simp add: mult_commute)

  1094   finally show ?thesis by auto

  1095 qed

  1096

  1097

  1098 subsection{* Shifting bounds *}

  1099

  1100 lemma setsum_shift_bounds_nat_ivl:

  1101   "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"

  1102 by (induct "n", auto simp:atLeastLessThanSuc)

  1103

  1104 lemma setsum_shift_bounds_cl_nat_ivl:

  1105   "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"

  1106 apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])

  1107 apply (simp add:image_add_atLeastAtMost o_def)

  1108 done

  1109

  1110 corollary setsum_shift_bounds_cl_Suc_ivl:

  1111   "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"

  1112 by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])

  1113

  1114 corollary setsum_shift_bounds_Suc_ivl:

  1115   "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"

  1116 by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])

  1117

  1118 lemma setsum_shift_lb_Suc0_0:

  1119   "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"

  1120 by(simp add:setsum_head_Suc)

  1121

  1122 lemma setsum_shift_lb_Suc0_0_upt:

  1123   "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"

  1124 apply(cases k)apply simp

  1125 apply(simp add:setsum_head_upt_Suc)

  1126 done

  1127

  1128 subsection {* The formula for geometric sums *}

  1129

  1130 lemma geometric_sum:

  1131   assumes "x \<noteq> 1"

  1132   shows "(\<Sum>i=0..<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"

  1133 proof -

  1134   from assms obtain y where "y = x - 1" and "y \<noteq> 0" by simp_all

  1135   moreover have "(\<Sum>i=0..<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"

  1136   proof (induct n)

  1137     case 0 then show ?case by simp

  1138   next

  1139     case (Suc n)

  1140     moreover with y \<noteq> 0 have "(1 + y) ^ n = (y * inverse y) * (1 + y) ^ n" by simp

  1141     ultimately show ?case by (simp add: field_simps divide_inverse)

  1142   qed

  1143   ultimately show ?thesis by simp

  1144 qed

  1145

  1146

  1147 subsection {* The formula for arithmetic sums *}

  1148

  1149 lemma gauss_sum:

  1150   "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =

  1151    of_nat n*((of_nat n)+1)"

  1152 proof (induct n)

  1153   case 0

  1154   show ?case by simp

  1155 next

  1156   case (Suc n)

  1157   then show ?case by (simp add: algebra_simps)

  1158 qed

  1159

  1160 theorem arith_series_general:

  1161   "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

  1162   of_nat n * (a + (a + of_nat(n - 1)*d))"

  1163 proof cases

  1164   assume ngt1: "n > 1"

  1165   let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"

  1166   have

  1167     "(\<Sum>i\<in>{..<n}. a+?I i*d) =

  1168      ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"

  1169     by (rule setsum_addf)

  1170   also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp

  1171   also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"

  1172     unfolding One_nat_def

  1173     by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)

  1174   also have "(1+1)*\<dots> = (1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..<n}. ?I i)"

  1175     by (simp add: left_distrib right_distrib)

  1176   also from ngt1 have "{1..<n} = {1..n - 1}"

  1177     by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)

  1178   also from ngt1

  1179   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)"

  1180     by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)

  1181        (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])

  1182   finally show ?thesis by (simp add: algebra_simps)

  1183 next

  1184   assume "\<not>(n > 1)"

  1185   hence "n = 1 \<or> n = 0" by auto

  1186   thus ?thesis by (auto simp: algebra_simps)

  1187 qed

  1188

  1189 lemma arith_series_nat:

  1190   "Suc (Suc 0) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"

  1191 proof -

  1192   have

  1193     "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =

  1194     of_nat(n) * (a + (a + of_nat(n - 1)*d))"

  1195     by (rule arith_series_general)

  1196   thus ?thesis

  1197     unfolding One_nat_def by auto

  1198 qed

  1199

  1200 lemma arith_series_int:

  1201   "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

  1202   of_nat n * (a + (a + of_nat(n - 1)*d))"

  1203 proof -

  1204   have

  1205     "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

  1206     of_nat(n) * (a + (a + of_nat(n - 1)*d))"

  1207     by (rule arith_series_general)

  1208   thus ?thesis by simp

  1209 qed

  1210

  1211 lemma sum_diff_distrib:

  1212   fixes P::"nat\<Rightarrow>nat"

  1213   shows

  1214   "\<forall>x. Q x \<le> P x  \<Longrightarrow>

  1215   (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"

  1216 proof (induct n)

  1217   case 0 show ?case by simp

  1218 next

  1219   case (Suc n)

  1220

  1221   let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"

  1222   let ?rhs = "\<Sum>x<n. P x - Q x"

  1223

  1224   from Suc have "?lhs = ?rhs" by simp

  1225   moreover

  1226   from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp

  1227   moreover

  1228   from Suc have

  1229     "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"

  1230     by (subst diff_diff_left[symmetric],

  1231         subst diff_add_assoc2)

  1232        (auto simp: diff_add_assoc2 intro: setsum_mono)

  1233   ultimately

  1234   show ?case by simp

  1235 qed

  1236

  1237 subsection {* Products indexed over intervals *}

  1238

  1239 syntax

  1240   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)

  1241   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)

  1242   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)

  1243   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)

  1244 syntax (xsymbols)

  1245   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)

  1246   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)

  1247   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)

  1248   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)

  1249 syntax (HTML output)

  1250   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)

  1251   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)

  1252   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)

  1253   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)

  1254 syntax (latex_prod output)

  1255   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1256  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

  1257   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1258  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

  1259   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1260  ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

  1261   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1262  ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

  1263

  1264 translations

  1265   "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"

  1266   "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"

  1267   "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"

  1268   "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"

  1269

  1270 subsection {* Transfer setup *}

  1271

  1272 lemma transfer_nat_int_set_functions:

  1273     "{..n} = nat  {0..int n}"

  1274     "{m..n} = nat  {int m..int n}"  (* need all variants of these! *)

  1275   apply (auto simp add: image_def)

  1276   apply (rule_tac x = "int x" in bexI)

  1277   apply auto

  1278   apply (rule_tac x = "int x" in bexI)

  1279   apply auto

  1280   done

  1281

  1282 lemma transfer_nat_int_set_function_closures:

  1283     "x >= 0 \<Longrightarrow> nat_set {x..y}"

  1284   by (simp add: nat_set_def)

  1285

  1286 declare transfer_morphism_nat_int[transfer add

  1287   return: transfer_nat_int_set_functions

  1288     transfer_nat_int_set_function_closures

  1289 ]

  1290

  1291 lemma transfer_int_nat_set_functions:

  1292     "is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int  {nat m..nat n}"

  1293   by (simp only: is_nat_def transfer_nat_int_set_functions

  1294     transfer_nat_int_set_function_closures

  1295     transfer_nat_int_set_return_embed nat_0_le

  1296     cong: transfer_nat_int_set_cong)

  1297

  1298 lemma transfer_int_nat_set_function_closures:

  1299     "is_nat x \<Longrightarrow> nat_set {x..y}"

  1300   by (simp only: transfer_nat_int_set_function_closures is_nat_def)

  1301

  1302 declare transfer_morphism_int_nat[transfer add

  1303   return: transfer_int_nat_set_functions

  1304     transfer_int_nat_set_function_closures

  1305 ]

  1306

  1307 end
`