src/HOL/SetInterval.thy
 author wenzelm Mon Mar 22 20:58:52 2010 +0100 (2010-03-22) changeset 35898 c890a3835d15 parent 35828 46cfc4b8112e child 36307 1732232f9b27 permissions -rw-r--r--
     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 _<=_./ _)" 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,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_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 lemma atLeastLessThan_add_Un: "i \<le> j \<Longrightarrow> {i..<j+k} = {i..<j} \<union> {j..<j+k::nat}"

   399   apply (induct k)

   400   apply (simp_all add: atLeastLessThanSuc)

   401   done

   402

   403 subsubsection {* Image *}

   404

   405 lemma image_add_atLeastAtMost:

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

   407 proof

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

   409 next

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

   411   proof

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

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

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

   415     ultimately show "n : ?A" by blast

   416   qed

   417 qed

   418

   419 lemma image_add_atLeastLessThan:

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

   421 proof

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

   423 next

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

   425   proof

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

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

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

   429     ultimately show "n : ?A" by blast

   430   qed

   431 qed

   432

   433 corollary image_Suc_atLeastAtMost[simp]:

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

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

   436

   437 corollary image_Suc_atLeastLessThan[simp]:

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

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

   440

   441 lemma image_add_int_atLeastLessThan:

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

   443   apply (auto simp add: image_def)

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

   445   apply auto

   446   done

   447

   448 context ordered_ab_group_add

   449 begin

   450

   451 lemma

   452   fixes x :: 'a

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

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

   455 proof safe

   456   fix y assume "y < -x"

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

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

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

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

   461 next

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

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

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

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

   466 qed simp_all

   467

   468 lemma

   469   fixes x :: 'a

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

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

   472 proof -

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

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

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

   476     by (simp_all add: image_image

   477         del: image_uminus_greaterThan image_uminus_atLeast)

   478 qed

   479

   480 lemma

   481   fixes x :: 'a

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

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

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

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

   486   by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def

   487       greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)

   488 end

   489

   490 subsubsection {* Finiteness *}

   491

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

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

   494

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

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

   497

   498 lemma finite_greaterThanLessThan [iff]:

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

   500 by (simp add: greaterThanLessThan_def)

   501

   502 lemma finite_atLeastLessThan [iff]:

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

   504 by (simp add: atLeastLessThan_def)

   505

   506 lemma finite_greaterThanAtMost [iff]:

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

   508 by (simp add: greaterThanAtMost_def)

   509

   510 lemma finite_atLeastAtMost [iff]:

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

   512 by (simp add: atLeastAtMost_def)

   513

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

   515 lemma bounded_nat_set_is_finite:

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

   517 apply (rule finite_subset)

   518  apply (rule_tac [2] finite_lessThan, auto)

   519 done

   520

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

   522 lemma finite_nat_set_iff_bounded:

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

   524 proof

   525   assume f:?F  show ?B

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

   527 next

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

   529 qed

   530

   531 lemma finite_nat_set_iff_bounded_le:

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

   533 apply(simp add:finite_nat_set_iff_bounded)

   534 apply(blast dest:less_imp_le_nat le_imp_less_Suc)

   535 done

   536

   537 lemma finite_less_ub:

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

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

   540

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

   542 subset is exactly that interval. *}

   543

   544 lemma subset_card_intvl_is_intvl:

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

   546 proof cases

   547   assume "finite A"

   548   thus "PROP ?P"

   549   proof(induct A rule:finite_linorder_max_induct)

   550     case empty thus ?case by auto

   551   next

   552     case (insert b A)

   553     moreover hence "b ~: A" by auto

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

   555       using b ~: A insert by fastsimp+

   556     ultimately show ?case by auto

   557   qed

   558 next

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

   560 qed

   561

   562

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

   564

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

   566   by (auto simp add: atLeast0LessThan)

   567

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

   569   by (subst UN_UN_finite_eq [symmetric]) blast

   570

   571 lemma UN_finite2_subset:

   572      "(!!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)"

   573   apply (rule UN_finite_subset)

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

   575   apply blast

   576   done

   577

   578 lemma UN_finite2_eq:

   579   "(!!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)"

   580   apply (rule subset_antisym)

   581    apply (rule UN_finite2_subset, blast)

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

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

   584  done

   585

   586

   587 subsubsection {* Cardinality *}

   588

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

   590   by (induct u, simp_all add: lessThan_Suc)

   591

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

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

   594

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

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

   597   apply (erule ssubst, rule card_lessThan)

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

   599   apply (erule subst)

   600   apply (rule card_image)

   601   apply (simp add: inj_on_def)

   602   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)

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

   604   apply arith

   605   done

   606

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

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

   609

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

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

   612

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

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

   615

   616 lemma ex_bij_betw_nat_finite:

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

   618 apply(drule finite_imp_nat_seg_image_inj_on)

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

   620 done

   621

   622 lemma ex_bij_betw_finite_nat:

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

   624 by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)

   625

   626 lemma finite_same_card_bij:

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

   628 apply(drule ex_bij_betw_finite_nat)

   629 apply(drule ex_bij_betw_nat_finite)

   630 apply(auto intro!:bij_betw_trans)

   631 done

   632

   633 lemma ex_bij_betw_nat_finite_1:

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

   635 by (rule finite_same_card_bij) auto

   636

   637

   638 subsection {* Intervals of integers *}

   639

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

   641   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)

   642

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

   644   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)

   645

   646 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:

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

   648   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)

   649

   650 subsubsection {* Finiteness *}

   651

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

   653     {(0::int)..<u} = int  {..<nat u}"

   654   apply (unfold image_def lessThan_def)

   655   apply auto

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

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

   658   done

   659

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

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

   662   apply (subst image_atLeastZeroLessThan_int, assumption)

   663   apply (rule finite_imageI)

   664   apply auto

   665   done

   666

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

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

   669   apply (erule subst)

   670   apply (rule finite_imageI)

   671   apply (rule finite_atLeastZeroLessThan_int)

   672   apply (rule image_add_int_atLeastLessThan)

   673   done

   674

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

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

   677

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

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

   680

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

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

   683

   684

   685 subsubsection {* Cardinality *}

   686

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

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

   689   apply (subst image_atLeastZeroLessThan_int, assumption)

   690   apply (subst card_image)

   691   apply (auto simp add: inj_on_def)

   692   done

   693

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

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

   696   apply (erule ssubst, rule card_atLeastZeroLessThan_int)

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

   698   apply (erule subst)

   699   apply (rule card_image)

   700   apply (simp add: inj_on_def)

   701   apply (rule image_add_int_atLeastLessThan)

   702   done

   703

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

   705 apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])

   706 apply (auto simp add: algebra_simps)

   707 done

   708

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

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

   711

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

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

   714

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

   716 proof -

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

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

   719 qed

   720

   721 lemma card_less:

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

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

   724 proof -

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

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

   727 qed

   728

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

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

   731 apply simp

   732 apply fastsimp

   733 apply auto

   734 apply (rule inj_on_diff_nat)

   735 apply auto

   736 apply (case_tac x)

   737 apply auto

   738 apply (case_tac xa)

   739 apply auto

   740 apply (case_tac xa)

   741 apply auto

   742 done

   743

   744 lemma card_less_Suc:

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

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

   747 proof -

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

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

   750     by (auto simp only: insert_Diff)

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

   752   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}))"

   753     apply (subst card_insert)

   754     apply simp_all

   755     apply (subst b)

   756     apply (subst card_less_Suc2[symmetric])

   757     apply simp_all

   758     done

   759   with c show ?thesis by simp

   760 qed

   761

   762

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

   764

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

   766

   767 subsubsection {* Disjoint Unions *}

   768

   769 text {* Singletons and open intervals *}

   770

   771 lemma ivl_disj_un_singleton:

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

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

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

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

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

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

   778 by auto

   779

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

   781

   782 lemma ivl_disj_un_one:

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

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

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

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

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

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

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

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

   791 by auto

   792

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

   794

   795 lemma ivl_disj_un_two:

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

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

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

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

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

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

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

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

   804 by auto

   805

   806 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two

   807

   808 subsubsection {* Disjoint Intersections *}

   809

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

   811

   812 lemma ivl_disj_int_one:

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

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

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

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

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

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

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

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

   821   by auto

   822

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

   824

   825 lemma ivl_disj_int_two:

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

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

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

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

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

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

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

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

   834   by auto

   835

   836 lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two

   837

   838 subsubsection {* Some Differences *}

   839

   840 lemma ivl_diff[simp]:

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

   842 by(auto)

   843

   844

   845 subsubsection {* Some Subset Conditions *}

   846

   847 lemma ivl_subset [simp,no_atp]:

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

   849 apply(auto simp:linorder_not_le)

   850 apply(rule ccontr)

   851 apply(insert linorder_le_less_linear[of i n])

   852 apply(clarsimp simp:linorder_not_le)

   853 apply(fastsimp)

   854 done

   855

   856

   857 subsection {* Summation indexed over intervals *}

   858

   859 syntax

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

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

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

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

   864 syntax (xsymbols)

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

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

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

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

   869 syntax (HTML output)

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

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

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

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

   874 syntax (latex_sum output)

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

   876  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

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

   878  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

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

   880  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

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

   882  ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

   883

   884 translations

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

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

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

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

   889

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

   891 summation over intervals:

   892 \begin{center}

   893 \begin{tabular}{lll}

   894 Old & New & \LaTeX\\

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

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

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

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

   899 \end{tabular}

   900 \end{center}

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

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

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

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

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

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

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

   908

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

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

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

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

   913

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

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

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

   917 the context. *}

   918

   919 lemma setsum_ivl_cong:

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

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

   922 by(rule setsum_cong, simp_all)

   923

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

   925 on intervals are not? *)

   926

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

   928 by (simp add:atMost_Suc add_ac)

   929

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

   931 by (simp add:lessThan_Suc add_ac)

   932

   933 lemma setsum_cl_ivl_Suc[simp]:

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

   935 by (auto simp:add_ac atLeastAtMostSuc_conv)

   936

   937 lemma setsum_op_ivl_Suc[simp]:

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

   939 by (auto simp:add_ac atLeastLessThanSuc)

   940 (*

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

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

   943 by (auto simp:add_ac atLeastAtMostSuc_conv)

   944 *)

   945

   946 lemma setsum_head:

   947   fixes n :: nat

   948   assumes mn: "m <= n"

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

   950 proof -

   951   from mn

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

   953     by (auto intro: ivl_disj_un_singleton)

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

   955     by (simp add: atLeast0LessThan)

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

   957   finally show ?thesis .

   958 qed

   959

   960 lemma setsum_head_Suc:

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

   962 by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)

   963

   964 lemma setsum_head_upt_Suc:

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

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

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

   968 done

   969

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

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

   972 proof-

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

   974   thus ?thesis by (auto simp: ivl_disj_int setsum_Un_disjoint

   975     atLeastSucAtMost_greaterThanAtMost)

   976 qed

   977

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

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

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

   981

   982 lemma setsum_diff_nat_ivl:

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

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

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

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

   987 apply (simp add: add_ac)

   988 done

   989

   990 lemma setsum_natinterval_difff:

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

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

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

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

   995

   996 lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]

   997

   998 lemma setsum_setsum_restrict:

   999   "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"

  1000   by (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)

  1001      (rule setsum_commute)

  1002

  1003 lemma setsum_image_gen: assumes fS: "finite S"

  1004   shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f  S)"

  1005 proof-

  1006   { fix x assume "x \<in> S" then have "{y. y\<in> fS \<and> f x = y} = {f x}" by auto }

  1007   hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> fS \<and> f x = y}) S"

  1008     by simp

  1009   also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f  S)"

  1010     by (rule setsum_setsum_restrict[OF fS finite_imageI[OF fS]])

  1011   finally show ?thesis .

  1012 qed

  1013

  1014 lemma setsum_le_included:

  1015   fixes f :: "'a \<Rightarrow> 'b::{comm_monoid_add,ordered_ab_semigroup_add_imp_le}"

  1016   assumes "finite s" "finite t"

  1017   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)"

  1018   shows "setsum f s \<le> setsum g t"

  1019 proof -

  1020   have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"

  1021   proof (rule setsum_mono)

  1022     fix y assume "y \<in> s"

  1023     with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto

  1024     with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")

  1025       using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]

  1026       by (auto intro!: setsum_mono2)

  1027   qed

  1028   also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i  t)"

  1029     using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)

  1030   also have "... \<le> setsum g t"

  1031     using assms by (auto simp: setsum_image_gen[symmetric])

  1032   finally show ?thesis .

  1033 qed

  1034

  1035 lemma setsum_multicount_gen:

  1036   assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"

  1037   shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")

  1038 proof-

  1039   have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto

  1040   also have "\<dots> = ?r" unfolding setsum_setsum_restrict[OF assms(1-2)]

  1041     using assms(3) by auto

  1042   finally show ?thesis .

  1043 qed

  1044

  1045 lemma setsum_multicount:

  1046   assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"

  1047   shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")

  1048 proof-

  1049   have "?l = setsum (\<lambda>i. k) T" by(rule setsum_multicount_gen)(auto simp:assms)

  1050   also have "\<dots> = ?r" by(simp add: mult_commute)

  1051   finally show ?thesis by auto

  1052 qed

  1053

  1054

  1055 subsection{* Shifting bounds *}

  1056

  1057 lemma setsum_shift_bounds_nat_ivl:

  1058   "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"

  1059 by (induct "n", auto simp:atLeastLessThanSuc)

  1060

  1061 lemma setsum_shift_bounds_cl_nat_ivl:

  1062   "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"

  1063 apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])

  1064 apply (simp add:image_add_atLeastAtMost o_def)

  1065 done

  1066

  1067 corollary setsum_shift_bounds_cl_Suc_ivl:

  1068   "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"

  1069 by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])

  1070

  1071 corollary setsum_shift_bounds_Suc_ivl:

  1072   "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"

  1073 by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])

  1074

  1075 lemma setsum_shift_lb_Suc0_0:

  1076   "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"

  1077 by(simp add:setsum_head_Suc)

  1078

  1079 lemma setsum_shift_lb_Suc0_0_upt:

  1080   "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"

  1081 apply(cases k)apply simp

  1082 apply(simp add:setsum_head_upt_Suc)

  1083 done

  1084

  1085 subsection {* The formula for geometric sums *}

  1086

  1087 lemma geometric_sum:

  1088   "x ~= 1 ==> (\<Sum>i=0..<n. x ^ i) =

  1089   (x ^ n - 1) / (x - 1::'a::{field})"

  1090 by (induct "n") (simp_all add: field_simps)

  1091

  1092 subsection {* The formula for arithmetic sums *}

  1093

  1094 lemma gauss_sum:

  1095   "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =

  1096    of_nat n*((of_nat n)+1)"

  1097 proof (induct n)

  1098   case 0

  1099   show ?case by simp

  1100 next

  1101   case (Suc n)

  1102   then show ?case by (simp add: algebra_simps)

  1103 qed

  1104

  1105 theorem arith_series_general:

  1106   "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

  1107   of_nat n * (a + (a + of_nat(n - 1)*d))"

  1108 proof cases

  1109   assume ngt1: "n > 1"

  1110   let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"

  1111   have

  1112     "(\<Sum>i\<in>{..<n}. a+?I i*d) =

  1113      ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"

  1114     by (rule setsum_addf)

  1115   also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp

  1116   also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"

  1117     unfolding One_nat_def

  1118     by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)

  1119   also have "(1+1)*\<dots> = (1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..<n}. ?I i)"

  1120     by (simp add: left_distrib right_distrib)

  1121   also from ngt1 have "{1..<n} = {1..n - 1}"

  1122     by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)

  1123   also from ngt1

  1124   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)"

  1125     by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)

  1126        (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])

  1127   finally show ?thesis by (simp add: algebra_simps)

  1128 next

  1129   assume "\<not>(n > 1)"

  1130   hence "n = 1 \<or> n = 0" by auto

  1131   thus ?thesis by (auto simp: algebra_simps)

  1132 qed

  1133

  1134 lemma arith_series_nat:

  1135   "Suc (Suc 0) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"

  1136 proof -

  1137   have

  1138     "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =

  1139     of_nat(n) * (a + (a + of_nat(n - 1)*d))"

  1140     by (rule arith_series_general)

  1141   thus ?thesis

  1142     unfolding One_nat_def by auto

  1143 qed

  1144

  1145 lemma arith_series_int:

  1146   "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

  1147   of_nat n * (a + (a + of_nat(n - 1)*d))"

  1148 proof -

  1149   have

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

  1151     of_nat(n) * (a + (a + of_nat(n - 1)*d))"

  1152     by (rule arith_series_general)

  1153   thus ?thesis by simp

  1154 qed

  1155

  1156 lemma sum_diff_distrib:

  1157   fixes P::"nat\<Rightarrow>nat"

  1158   shows

  1159   "\<forall>x. Q x \<le> P x  \<Longrightarrow>

  1160   (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"

  1161 proof (induct n)

  1162   case 0 show ?case by simp

  1163 next

  1164   case (Suc n)

  1165

  1166   let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"

  1167   let ?rhs = "\<Sum>x<n. P x - Q x"

  1168

  1169   from Suc have "?lhs = ?rhs" by simp

  1170   moreover

  1171   from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp

  1172   moreover

  1173   from Suc have

  1174     "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"

  1175     by (subst diff_diff_left[symmetric],

  1176         subst diff_add_assoc2)

  1177        (auto simp: diff_add_assoc2 intro: setsum_mono)

  1178   ultimately

  1179   show ?case by simp

  1180 qed

  1181

  1182 subsection {* Products indexed over intervals *}

  1183

  1184 syntax

  1185   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)

  1186   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)

  1187   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)

  1188   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)

  1189 syntax (xsymbols)

  1190   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)

  1191   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)

  1192   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)

  1193   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)

  1194 syntax (HTML output)

  1195   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)

  1196   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)

  1197   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)

  1198   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)

  1199 syntax (latex_prod output)

  1200   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1201  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

  1202   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1203  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

  1204   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1205  ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

  1206   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"

  1207  ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

  1208

  1209 translations

  1210   "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"

  1211   "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"

  1212   "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"

  1213   "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"

  1214

  1215 subsection {* Transfer setup *}

  1216

  1217 lemma transfer_nat_int_set_functions:

  1218     "{..n} = nat  {0..int n}"

  1219     "{m..n} = nat  {int m..int n}"  (* need all variants of these! *)

  1220   apply (auto simp add: image_def)

  1221   apply (rule_tac x = "int x" in bexI)

  1222   apply auto

  1223   apply (rule_tac x = "int x" in bexI)

  1224   apply auto

  1225   done

  1226

  1227 lemma transfer_nat_int_set_function_closures:

  1228     "x >= 0 \<Longrightarrow> nat_set {x..y}"

  1229   by (simp add: nat_set_def)

  1230

  1231 declare transfer_morphism_nat_int[transfer add

  1232   return: transfer_nat_int_set_functions

  1233     transfer_nat_int_set_function_closures

  1234 ]

  1235

  1236 lemma transfer_int_nat_set_functions:

  1237     "is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int  {nat m..nat n}"

  1238   by (simp only: is_nat_def transfer_nat_int_set_functions

  1239     transfer_nat_int_set_function_closures

  1240     transfer_nat_int_set_return_embed nat_0_le

  1241     cong: transfer_nat_int_set_cong)

  1242

  1243 lemma transfer_int_nat_set_function_closures:

  1244     "is_nat x \<Longrightarrow> nat_set {x..y}"

  1245   by (simp only: transfer_nat_int_set_function_closures is_nat_def)

  1246

  1247 declare transfer_morphism_int_nat[transfer add

  1248   return: transfer_int_nat_set_functions

  1249     transfer_int_nat_set_function_closures

  1250 ]

  1251

  1252 end
`