src/HOL/SetInterval.thy
 author nipkow Thu Jul 07 12:39:17 2005 +0200 (2005-07-07) changeset 16733 236dfafbeb63 parent 16102 c5f6726d9bb1 child 17149 e2b19c92ef51 permissions -rw-r--r--
linear arithmetic now takes "&" in assumptions apart.
     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 imports 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_all add: greaterThan_subset_iff)

   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_all add: lessThan_subset_iff)

   166 done

   167

   168

   169 subsection {*Two-sided intervals*}

   170

   171 lemma greaterThanLessThan_iff [simp]:

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

   173 by (simp add: greaterThanLessThan_def)

   174

   175 lemma atLeastLessThan_iff [simp]:

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

   177 by (simp add: atLeastLessThan_def)

   178

   179 lemma greaterThanAtMost_iff [simp]:

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

   181 by (simp add: greaterThanAtMost_def)

   182

   183 lemma atLeastAtMost_iff [simp]:

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

   185 by (simp add: atLeastAtMost_def)

   186

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

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

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

   190

   191 subsubsection{* Emptyness and singletons *}

   192

   193 lemma atLeastAtMost_empty [simp]: "n < m ==> {m::'a::order..n} = {}";

   194   by (auto simp add: atLeastAtMost_def atMost_def atLeast_def);

   195

   196 lemma atLeastLessThan_empty[simp]: "n \<le> m ==> {m..<n::'a::order} = {}"

   197 by (auto simp add: atLeastLessThan_def)

   198

   199 lemma atLeastAtMost_singleton [simp]: "{a::'a::order..a} = {a}";

   200   by (auto simp add: atLeastAtMost_def atMost_def atLeast_def);

   201

   202 subsection {* Intervals of natural numbers *}

   203

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

   205

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

   207 by (simp add: lessThan_def)

   208

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

   210 by (simp add: lessThan_def less_Suc_eq, blast)

   211

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

   213 by (simp add: lessThan_def atMost_def less_Suc_eq_le)

   214

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

   216 by blast

   217

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

   219

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

   221 apply (simp add: greaterThan_def)

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

   223 done

   224

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

   226 apply (simp add: greaterThan_def)

   227 apply (auto elim: linorder_neqE)

   228 done

   229

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

   231 by blast

   232

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

   234

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

   236 by (unfold atLeast_def UNIV_def, simp)

   237

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

   239 apply (simp add: atLeast_def)

   240 apply (simp add: Suc_le_eq)

   241 apply (simp add: order_le_less, blast)

   242 done

   243

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

   245   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)

   246

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

   248 by blast

   249

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

   251

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

   253 by (simp add: atMost_def)

   254

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

   256 apply (simp add: atMost_def)

   257 apply (simp add: less_Suc_eq order_le_less, blast)

   258 done

   259

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

   261 by blast

   262

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

   264

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

   266   of @{term atLeastLessThan}*}

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

   268 by(simp add:lessThan_def atLeastLessThan_def)

   269 (*

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

   271 by (simp add: atLeastLessThan_def)

   272 *)

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

   274

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

   276 lemma atLeastLessThanSuc:

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

   278 by (auto simp add: atLeastLessThan_def)

   279

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

   281 by (auto simp add: atLeastLessThan_def)

   282 (*

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

   284 by (induct k, simp_all add: atLeastLessThanSuc)

   285

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

   287 by (auto simp add: atLeastLessThan_def)

   288 *)

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

   290   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)

   291

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

   293   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def

   294     greaterThanAtMost_def)

   295

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

   297   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def

   298     greaterThanLessThan_def)

   299

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

   301 by (auto simp add: atLeastAtMost_def)

   302

   303 subsubsection {* Image *}

   304

   305 lemma image_add_atLeastAtMost:

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

   307 proof

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

   309 next

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

   311   proof

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

   313     hence "n - k : {i..j}" by auto arith+

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

   315     ultimately show "n : ?A" by blast

   316   qed

   317 qed

   318

   319 lemma image_add_atLeastLessThan:

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

   321 proof

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

   323 next

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

   325   proof

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

   327     hence "n - k : {i..<j}" by auto arith+

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

   329     ultimately show "n : ?A" by blast

   330   qed

   331 qed

   332

   333 corollary image_Suc_atLeastAtMost[simp]:

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

   335 using image_add_atLeastAtMost[where k=1] by simp

   336

   337 corollary image_Suc_atLeastLessThan[simp]:

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

   339 using image_add_atLeastLessThan[where k=1] by simp

   340

   341 lemma image_add_int_atLeastLessThan:

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

   343   apply (auto simp add: image_def)

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

   345   apply auto

   346   done

   347

   348

   349 subsubsection {* Finiteness *}

   350

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

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

   353

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

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

   356

   357 lemma finite_greaterThanLessThan [iff]:

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

   359 by (simp add: greaterThanLessThan_def)

   360

   361 lemma finite_atLeastLessThan [iff]:

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

   363 by (simp add: atLeastLessThan_def)

   364

   365 lemma finite_greaterThanAtMost [iff]:

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

   367 by (simp add: greaterThanAtMost_def)

   368

   369 lemma finite_atLeastAtMost [iff]:

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

   371 by (simp add: atLeastAtMost_def)

   372

   373 lemma bounded_nat_set_is_finite:

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

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

   376   apply (rule finite_subset)

   377    apply (rule_tac [2] finite_lessThan, auto)

   378   done

   379

   380 subsubsection {* Cardinality *}

   381

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

   383   by (induct u, simp_all add: lessThan_Suc)

   384

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

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

   387

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

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

   390   apply (erule ssubst, rule card_lessThan)

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

   392   apply (erule subst)

   393   apply (rule card_image)

   394   apply (simp add: inj_on_def)

   395   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)

   396   apply arith

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

   398   apply arith

   399   done

   400

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

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

   403

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

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

   406

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

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

   409

   410 subsection {* Intervals of integers *}

   411

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

   413   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)

   414

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

   416   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)

   417

   418 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:

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

   420   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)

   421

   422 subsubsection {* Finiteness *}

   423

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

   425     {(0::int)..<u} = int  {..<nat u}"

   426   apply (unfold image_def lessThan_def)

   427   apply auto

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

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

   430   done

   431

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

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

   434   apply (subst image_atLeastZeroLessThan_int, assumption)

   435   apply (rule finite_imageI)

   436   apply auto

   437   done

   438

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

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

   441   apply (erule subst)

   442   apply (rule finite_imageI)

   443   apply (rule finite_atLeastZeroLessThan_int)

   444   apply (rule image_add_int_atLeastLessThan)

   445   done

   446

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

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

   449

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

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

   452

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

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

   455

   456 subsubsection {* Cardinality *}

   457

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

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

   460   apply (subst image_atLeastZeroLessThan_int, assumption)

   461   apply (subst card_image)

   462   apply (auto simp add: inj_on_def)

   463   done

   464

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

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

   467   apply (erule ssubst, rule card_atLeastZeroLessThan_int)

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

   469   apply (erule subst)

   470   apply (rule card_image)

   471   apply (simp add: inj_on_def)

   472   apply (rule image_add_int_atLeastLessThan)

   473   done

   474

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

   476   apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])

   477   apply (auto simp add: compare_rls)

   478   done

   479

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

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

   482

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

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

   485

   486

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

   488

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

   490

   491 subsubsection {* Disjoint Unions *}

   492

   493 text {* Singletons and open intervals *}

   494

   495 lemma ivl_disj_un_singleton:

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

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

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

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

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

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

   502 by auto

   503

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

   505

   506 lemma ivl_disj_un_one:

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

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

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

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

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

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

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

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

   515 by auto

   516

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

   518

   519 lemma ivl_disj_un_two:

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

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

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

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

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

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

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

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

   528 by auto

   529

   530 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two

   531

   532 subsubsection {* Disjoint Intersections *}

   533

   534 text {* Singletons and open intervals *}

   535

   536 lemma ivl_disj_int_singleton:

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

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

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

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

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

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

   543   by simp+

   544

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

   546

   547 lemma ivl_disj_int_one:

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

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

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

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

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

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

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

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

   556   by auto

   557

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

   559

   560 lemma ivl_disj_int_two:

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

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

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

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

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

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

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

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

   569   by auto

   570

   571 lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two

   572

   573 subsubsection {* Some Differences *}

   574

   575 lemma ivl_diff[simp]:

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

   577 by(auto)

   578

   579

   580 subsubsection {* Some Subset Conditions *}

   581

   582 lemma ivl_subset[simp]:

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

   584 apply(auto simp:linorder_not_le)

   585 apply(rule ccontr)

   586 apply(insert linorder_le_less_linear[of i n])

   587 apply(clarsimp simp:linorder_not_le)

   588 apply(fastsimp)

   589 done

   590

   591

   592 subsection {* Summation indexed over intervals *}

   593

   594 syntax

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

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

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

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

   599 syntax (xsymbols)

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

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

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

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

   604 syntax (HTML output)

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

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

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

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

   609 syntax (latex_sum output)

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

   611  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

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

   613  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

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

   615  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

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

   617  ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

   618

   619 translations

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

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

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

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

   624

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

   626 summation over intervals:

   627 \begin{center}

   628 \begin{tabular}{lll}

   629 Old & New & \LaTeX\\

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

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

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

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

   634 \end{tabular}

   635 \end{center}

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

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

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

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

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

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

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

   643

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

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

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

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

   648

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

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

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

   652 the context. *}

   653

   654 lemma setsum_ivl_cong:

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

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

   657 by(rule setsum_cong, simp_all)

   658

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

   660 on intervals are not? *)

   661

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

   663 by (simp add:atMost_Suc add_ac)

   664

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

   666 by (simp add:lessThan_Suc add_ac)

   667

   668 lemma setsum_cl_ivl_Suc[simp]:

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

   670 by (auto simp:add_ac atLeastAtMostSuc_conv)

   671

   672 lemma setsum_op_ivl_Suc[simp]:

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

   674 by (auto simp:add_ac atLeastLessThanSuc)

   675 (*

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

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

   678 by (auto simp:add_ac atLeastAtMostSuc_conv)

   679 *)

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

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

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

   683

   684 lemma setsum_diff_nat_ivl:

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

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

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

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

   689 apply (simp add: add_ac)

   690 done

   691

   692 subsection{* Shifting bounds *}

   693

   694 lemma setsum_shift_bounds_nat_ivl:

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

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

   697

   698 lemma setsum_shift_bounds_cl_nat_ivl:

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

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

   701 apply (simp add:image_add_atLeastAtMost o_def)

   702 done

   703

   704 corollary setsum_shift_bounds_cl_Suc_ivl:

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

   706 by (simp add:setsum_shift_bounds_cl_nat_ivl[where k=1,simplified])

   707

   708 corollary setsum_shift_bounds_Suc_ivl:

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

   710 by (simp add:setsum_shift_bounds_nat_ivl[where k=1,simplified])

   711

   712

   713 ML

   714 {*

   715 val Compl_atLeast = thm "Compl_atLeast";

   716 val Compl_atMost = thm "Compl_atMost";

   717 val Compl_greaterThan = thm "Compl_greaterThan";

   718 val Compl_lessThan = thm "Compl_lessThan";

   719 val INT_greaterThan_UNIV = thm "INT_greaterThan_UNIV";

   720 val UN_atLeast_UNIV = thm "UN_atLeast_UNIV";

   721 val UN_atMost_UNIV = thm "UN_atMost_UNIV";

   722 val UN_lessThan_UNIV = thm "UN_lessThan_UNIV";

   723 val atLeastAtMost_def = thm "atLeastAtMost_def";

   724 val atLeastAtMost_iff = thm "atLeastAtMost_iff";

   725 val atLeastLessThan_def  = thm "atLeastLessThan_def";

   726 val atLeastLessThan_iff = thm "atLeastLessThan_iff";

   727 val atLeast_0 = thm "atLeast_0";

   728 val atLeast_Suc = thm "atLeast_Suc";

   729 val atLeast_def      = thm "atLeast_def";

   730 val atLeast_iff = thm "atLeast_iff";

   731 val atMost_0 = thm "atMost_0";

   732 val atMost_Int_atLeast = thm "atMost_Int_atLeast";

   733 val atMost_Suc = thm "atMost_Suc";

   734 val atMost_def       = thm "atMost_def";

   735 val atMost_iff = thm "atMost_iff";

   736 val greaterThanAtMost_def  = thm "greaterThanAtMost_def";

   737 val greaterThanAtMost_iff = thm "greaterThanAtMost_iff";

   738 val greaterThanLessThan_def  = thm "greaterThanLessThan_def";

   739 val greaterThanLessThan_iff = thm "greaterThanLessThan_iff";

   740 val greaterThan_0 = thm "greaterThan_0";

   741 val greaterThan_Suc = thm "greaterThan_Suc";

   742 val greaterThan_def  = thm "greaterThan_def";

   743 val greaterThan_iff = thm "greaterThan_iff";

   744 val ivl_disj_int = thms "ivl_disj_int";

   745 val ivl_disj_int_one = thms "ivl_disj_int_one";

   746 val ivl_disj_int_singleton = thms "ivl_disj_int_singleton";

   747 val ivl_disj_int_two = thms "ivl_disj_int_two";

   748 val ivl_disj_un = thms "ivl_disj_un";

   749 val ivl_disj_un_one = thms "ivl_disj_un_one";

   750 val ivl_disj_un_singleton = thms "ivl_disj_un_singleton";

   751 val ivl_disj_un_two = thms "ivl_disj_un_two";

   752 val lessThan_0 = thm "lessThan_0";

   753 val lessThan_Suc = thm "lessThan_Suc";

   754 val lessThan_Suc_atMost = thm "lessThan_Suc_atMost";

   755 val lessThan_def     = thm "lessThan_def";

   756 val lessThan_iff = thm "lessThan_iff";

   757 val single_Diff_lessThan = thm "single_Diff_lessThan";

   758

   759 val bounded_nat_set_is_finite = thm "bounded_nat_set_is_finite";

   760 val finite_atMost = thm "finite_atMost";

   761 val finite_lessThan = thm "finite_lessThan";

   762 *}

   763

   764 end
`