src/HOL/SetInterval.thy
 author nipkow Mon Aug 16 14:22:27 2004 +0200 (2004-08-16) changeset 15131 c69542757a4d parent 15056 b75073d90bff child 15140 322485b816ac permissions -rw-r--r--
     1 (*  Title:      HOL/SetInterval.thy

     2     ID:         $Id$

     3     Author:     Tobias Nipkow and Clemens Ballarin

     4                 Additions by Jeremy Avigad in March 2004

     5     Copyright   2000  TU Muenchen

     6

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

     8 *)

     9

    10 header {* Set intervals *}

    11

    12 theory SetInterval

    13 import IntArith

    14 begin

    15

    16 constdefs

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

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

    19

    20   atMost      :: "('a::ord) => 'a set"	("(1{.._})")

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

    22

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

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

    25

    26   atLeast     :: "('a::ord) => 'a set"	("(1{_..})")

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

    28

    29   greaterThanLessThan :: "['a::ord, 'a] => 'a set"  ("(1{_<..<_})")

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

    31

    32   atLeastLessThan :: "['a::ord, 'a] => 'a set"      ("(1{_..<_})")

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

    34

    35   greaterThanAtMost :: "['a::ord, 'a] => 'a set"    ("(1{_<.._})")

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

    37

    38   atLeastAtMost :: "['a::ord, 'a] => 'a set"        ("(1{_.._})")

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

    40

    41 (* Old syntax, will disappear! *)

    42 syntax

    43   "_lessThan"    :: "('a::ord) => 'a set"	("(1{.._'(})")

    44   "_greaterThan" :: "('a::ord) => 'a set"	("(1{')_..})")

    45   "_greaterThanLessThan" :: "['a::ord, 'a] => 'a set"  ("(1{')_.._'(})")

    46   "_atLeastLessThan" :: "['a::ord, 'a] => 'a set"      ("(1{_.._'(})")

    47   "_greaterThanAtMost" :: "['a::ord, 'a] => 'a set"    ("(1{')_.._})")

    48 translations

    49   "{..m(}" => "{..<m}"

    50   "{)m..}" => "{m<..}"

    51   "{)m..n(}" => "{m<..<n}"

    52   "{m..n(}" => "{m..<n}"

    53   "{)m..n}" => "{m<..n}"

    54

    55

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

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

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

    59

    60 syntax

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

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

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

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

    65

    66 syntax (input)

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

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

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

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

    71

    72 syntax (xsymbols)

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

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

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

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

    77

    78 translations

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

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

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

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

    83

    84

    85 subsection {* Various equivalences *}

    86

    87 lemma lessThan_iff [iff]: "(i: lessThan k) = (i<k)"

    88 by (simp add: lessThan_def)

    89

    90 lemma Compl_lessThan [simp]:

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

    92 apply (auto simp add: lessThan_def atLeast_def)

    93 done

    94

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

    96 by auto

    97

    98 lemma greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"

    99 by (simp add: greaterThan_def)

   100

   101 lemma Compl_greaterThan [simp]:

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

   103 apply (simp add: greaterThan_def atMost_def le_def, auto)

   104 done

   105

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

   107 apply (subst Compl_greaterThan [symmetric])

   108 apply (rule double_complement)

   109 done

   110

   111 lemma atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"

   112 by (simp add: atLeast_def)

   113

   114 lemma Compl_atLeast [simp]:

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

   116 apply (simp add: lessThan_def atLeast_def le_def, auto)

   117 done

   118

   119 lemma atMost_iff [iff]: "(i: atMost k) = (i<=k)"

   120 by (simp add: atMost_def)

   121

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

   123 by (blast intro: order_antisym)

   124

   125

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

   127

   128 lemma atLeast_subset_iff [iff]:

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

   130 by (blast intro: order_trans)

   131

   132 lemma atLeast_eq_iff [iff]:

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

   134 by (blast intro: order_antisym order_trans)

   135

   136 lemma greaterThan_subset_iff [iff]:

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

   138 apply (auto simp add: greaterThan_def)

   139  apply (subst linorder_not_less [symmetric], blast)

   140 done

   141

   142 lemma greaterThan_eq_iff [iff]:

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

   144 apply (rule iffI)

   145  apply (erule equalityE)

   146  apply (simp add: greaterThan_subset_iff order_antisym, simp)

   147 done

   148

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

   150 by (blast intro: order_trans)

   151

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

   153 by (blast intro: order_antisym order_trans)

   154

   155 lemma lessThan_subset_iff [iff]:

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

   157 apply (auto simp add: lessThan_def)

   158  apply (subst linorder_not_less [symmetric], blast)

   159 done

   160

   161 lemma lessThan_eq_iff [iff]:

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

   163 apply (rule iffI)

   164  apply (erule equalityE)

   165  apply (simp add: lessThan_subset_iff order_antisym, simp)

   166 done

   167

   168

   169 subsection {*Two-sided intervals*}

   170

   171 text {* @{text greaterThanLessThan} *}

   172

   173 lemma greaterThanLessThan_iff [simp]:

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

   175 by (simp add: greaterThanLessThan_def)

   176

   177 text {* @{text atLeastLessThan} *}

   178

   179 lemma atLeastLessThan_iff [simp]:

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

   181 by (simp add: atLeastLessThan_def)

   182

   183 text {* @{text greaterThanAtMost} *}

   184

   185 lemma greaterThanAtMost_iff [simp]:

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

   187 by (simp add: greaterThanAtMost_def)

   188

   189 text {* @{text atLeastAtMost} *}

   190

   191 lemma atLeastAtMost_iff [simp]:

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

   193 by (simp add: atLeastAtMost_def)

   194

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

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

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

   198

   199

   200 subsection {* Intervals of natural numbers *}

   201

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

   203

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

   205 by (simp add: lessThan_def)

   206

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

   208 by (simp add: lessThan_def less_Suc_eq, blast)

   209

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

   211 by (simp add: lessThan_def atMost_def less_Suc_eq_le)

   212

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

   214 by blast

   215

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

   217

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

   219 apply (simp add: greaterThan_def)

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

   221 done

   222

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

   224 apply (simp add: greaterThan_def)

   225 apply (auto elim: linorder_neqE)

   226 done

   227

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

   229 by blast

   230

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

   232

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

   234 by (unfold atLeast_def UNIV_def, simp)

   235

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

   237 apply (simp add: atLeast_def)

   238 apply (simp add: Suc_le_eq)

   239 apply (simp add: order_le_less, blast)

   240 done

   241

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

   243   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)

   244

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

   246 by blast

   247

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

   249

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

   251 by (simp add: atMost_def)

   252

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

   254 apply (simp add: atMost_def)

   255 apply (simp add: less_Suc_eq order_le_less, blast)

   256 done

   257

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

   259 by blast

   260

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

   262

   263 text{*But not a simprule because some concepts are better left in terms

   264   of @{term atLeastLessThan}*}

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

   266 by(simp add:lessThan_def atLeastLessThan_def)

   267

   268 lemma atLeastLessThan0 [simp]: "{m..<0::nat} = {}"

   269 by (simp add: atLeastLessThan_def)

   270

   271 lemma atLeastLessThan_self [simp]: "{n::'a::order..<n} = {}"

   272 by (auto simp add: atLeastLessThan_def)

   273

   274 lemma atLeastLessThan_empty: "n \<le> m ==> {m..<n::'a::order} = {}"

   275 by (auto simp add: atLeastLessThan_def)

   276

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

   278

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

   280 lemma atLeastLessThanSuc:

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

   282 by (auto simp add: atLeastLessThan_def)

   283

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

   285 by (auto simp add: atLeastLessThan_def)

   286

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

   288 by (induct k, simp_all add: atLeastLessThanSuc)

   289

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

   291 by (auto simp add: atLeastLessThan_def)

   292

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

   294   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)

   295

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

   297   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def

   298     greaterThanAtMost_def)

   299

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

   301   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def

   302     greaterThanLessThan_def)

   303

   304 subsubsection {* Finiteness *}

   305

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

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

   308

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

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

   311

   312 lemma finite_greaterThanLessThan [iff]:

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

   314 by (simp add: greaterThanLessThan_def)

   315

   316 lemma finite_atLeastLessThan [iff]:

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

   318 by (simp add: atLeastLessThan_def)

   319

   320 lemma finite_greaterThanAtMost [iff]:

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

   322 by (simp add: greaterThanAtMost_def)

   323

   324 lemma finite_atLeastAtMost [iff]:

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

   326 by (simp add: atLeastAtMost_def)

   327

   328 lemma bounded_nat_set_is_finite:

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

   330   -- {* A bounded set of natural numbers is finite. *}

   331   apply (rule finite_subset)

   332    apply (rule_tac [2] finite_lessThan, auto)

   333   done

   334

   335 subsubsection {* Cardinality *}

   336

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

   338   by (induct_tac u, simp_all add: lessThan_Suc)

   339

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

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

   342

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

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

   345   apply (erule ssubst, rule card_lessThan)

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

   347   apply (erule subst)

   348   apply (rule card_image)

   349   apply (rule finite_lessThan)

   350   apply (simp add: inj_on_def)

   351   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)

   352   apply arith

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

   354   apply arith

   355   done

   356

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

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

   359

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

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

   362

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

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

   365

   366 subsection {* Intervals of integers *}

   367

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

   369   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)

   370

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

   372   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)

   373

   374 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:

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

   376   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)

   377

   378 subsubsection {* Finiteness *}

   379

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

   381     {(0::int)..<u} = int  {..<nat u}"

   382   apply (unfold image_def lessThan_def)

   383   apply auto

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

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

   386   done

   387

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

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

   390   apply (subst image_atLeastZeroLessThan_int, assumption)

   391   apply (rule finite_imageI)

   392   apply auto

   393   apply (subgoal_tac "{0..<u} = {}")

   394   apply auto

   395   done

   396

   397 lemma image_atLeastLessThan_int_shift:

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

   399   apply (auto simp add: image_def atLeastLessThan_iff)

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

   401   apply auto

   402   done

   403

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

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

   406   apply (erule subst)

   407   apply (rule finite_imageI)

   408   apply (rule finite_atLeastZeroLessThan_int)

   409   apply (rule image_atLeastLessThan_int_shift)

   410   done

   411

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

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

   414

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

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

   417

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

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

   420

   421 subsubsection {* Cardinality *}

   422

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

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

   425   apply (subst image_atLeastZeroLessThan_int, assumption)

   426   apply (subst card_image)

   427   apply (auto simp add: inj_on_def)

   428   done

   429

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

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

   432   apply (erule ssubst, rule card_atLeastZeroLessThan_int)

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

   434   apply (erule subst)

   435   apply (rule card_image)

   436   apply (rule finite_atLeastZeroLessThan_int)

   437   apply (simp add: inj_on_def)

   438   apply (rule image_atLeastLessThan_int_shift)

   439   done

   440

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

   442   apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])

   443   apply (auto simp add: compare_rls)

   444   done

   445

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

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

   448

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

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

   451

   452

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

   454

   455 text {* For examples, see Algebra/poly/UnivPoly.thy *}

   456

   457 subsubsection {* Disjoint Unions *}

   458

   459 text {* Singletons and open intervals *}

   460

   461 lemma ivl_disj_un_singleton:

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

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

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

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

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

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

   468 by auto

   469

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

   471

   472 lemma ivl_disj_un_one:

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

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

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

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

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

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

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

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

   481 by auto

   482

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

   484

   485 lemma ivl_disj_un_two:

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

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

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

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

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

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

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

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

   494 by auto

   495

   496 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two

   497

   498 subsubsection {* Disjoint Intersections *}

   499

   500 text {* Singletons and open intervals *}

   501

   502 lemma ivl_disj_int_singleton:

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

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

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

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

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

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

   509   by simp+

   510

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

   512

   513 lemma ivl_disj_int_one:

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

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

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

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

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

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

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

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

   522   by auto

   523

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

   525

   526 lemma ivl_disj_int_two:

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

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

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

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

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

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

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

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

   535   by auto

   536

   537 lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two

   538

   539

   540 subsection {* Summation indexed over intervals *}

   541

   542 syntax

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

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

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

   546 syntax (xsymbols)

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

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

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

   550 syntax (HTML output)

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

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

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

   554 syntax (latex_sum output)

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

   556  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

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

   558  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

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

   560  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

   561

   562 translations

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

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

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

   566

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

   568 summation over intervals:

   569 \begin{center}

   570 \begin{tabular}{lll}

   571 Old & New & \LaTeX\\

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

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

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

   575 \end{tabular}

   576 \end{center}

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

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

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

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

   581 \texttt{latex\_sum} (e.g.\ via \texttt{mode=latex\_sum} in

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

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

   584

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

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

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

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

   589

   590

   591 lemma Summation_Suc[simp]: "(\<Sum>i < Suc n. b i) = b n + (\<Sum>i < n. b i)"

   592 by (simp add:lessThan_Suc)

   593

   594 end
`