src/HOL/SetInterval.thy
 author haftmann Mon Jan 30 08:20:56 2006 +0100 (2006-01-30) changeset 18851 9502ce541f01 parent 17719 2e75155c5ed5 child 19022 0e6ec4fd204c 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 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 greaterThanAtMost_empty[simp]:"l \<le> k ==> {k<..(l::'a::order)} = {}"

   200 by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)

   201

   202 lemma greaterThanLessThan_empty[simp]:"l \<le> k ==> {k<..(l::'a::order)} = {}"

   203 by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)

   204

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

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

   207

   208 subsection {* Intervals of natural numbers *}

   209

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

   211

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

   213 by (simp add: lessThan_def)

   214

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

   216 by (simp add: lessThan_def less_Suc_eq, blast)

   217

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

   219 by (simp add: lessThan_def atMost_def less_Suc_eq_le)

   220

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

   222 by blast

   223

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

   225

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

   227 apply (simp add: greaterThan_def)

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

   229 done

   230

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

   232 apply (simp add: greaterThan_def)

   233 apply (auto elim: linorder_neqE)

   234 done

   235

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

   237 by blast

   238

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

   240

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

   242 by (unfold atLeast_def UNIV_def, simp)

   243

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

   245 apply (simp add: atLeast_def)

   246 apply (simp add: Suc_le_eq)

   247 apply (simp add: order_le_less, blast)

   248 done

   249

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

   251   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)

   252

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

   254 by blast

   255

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

   257

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

   259 by (simp add: atMost_def)

   260

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

   262 apply (simp add: atMost_def)

   263 apply (simp add: less_Suc_eq order_le_less, blast)

   264 done

   265

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

   267 by blast

   268

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

   270

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

   272   of @{term atLeastLessThan}*}

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

   274 by(simp add:lessThan_def atLeastLessThan_def)

   275 (*

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

   277 by (simp add: atLeastLessThan_def)

   278 *)

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

   280

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

   282 lemma atLeastLessThanSuc:

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

   284 by (auto simp add: atLeastLessThan_def)

   285

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

   287 by (auto simp add: atLeastLessThan_def)

   288 (*

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

   290 by (induct k, simp_all add: atLeastLessThanSuc)

   291

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

   293 by (auto simp add: atLeastLessThan_def)

   294 *)

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

   296   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)

   297

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

   299   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def

   300     greaterThanAtMost_def)

   301

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

   303   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def

   304     greaterThanLessThan_def)

   305

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

   307 by (auto simp add: atLeastAtMost_def)

   308

   309 subsubsection {* Image *}

   310

   311 lemma image_add_atLeastAtMost:

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

   313 proof

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

   315 next

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

   317   proof

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

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

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

   321     ultimately show "n : ?A" by blast

   322   qed

   323 qed

   324

   325 lemma image_add_atLeastLessThan:

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

   327 proof

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

   329 next

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

   331   proof

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

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

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

   335     ultimately show "n : ?A" by blast

   336   qed

   337 qed

   338

   339 corollary image_Suc_atLeastAtMost[simp]:

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

   341 using image_add_atLeastAtMost[where k=1] by simp

   342

   343 corollary image_Suc_atLeastLessThan[simp]:

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

   345 using image_add_atLeastLessThan[where k=1] by simp

   346

   347 lemma image_add_int_atLeastLessThan:

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

   349   apply (auto simp add: image_def)

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

   351   apply auto

   352   done

   353

   354

   355 subsubsection {* Finiteness *}

   356

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

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

   359

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

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

   362

   363 lemma finite_greaterThanLessThan [iff]:

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

   365 by (simp add: greaterThanLessThan_def)

   366

   367 lemma finite_atLeastLessThan [iff]:

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

   369 by (simp add: atLeastLessThan_def)

   370

   371 lemma finite_greaterThanAtMost [iff]:

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

   373 by (simp add: greaterThanAtMost_def)

   374

   375 lemma finite_atLeastAtMost [iff]:

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

   377 by (simp add: atLeastAtMost_def)

   378

   379 lemma bounded_nat_set_is_finite:

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

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

   382   apply (rule finite_subset)

   383    apply (rule_tac [2] finite_lessThan, auto)

   384   done

   385

   386 subsubsection {* Cardinality *}

   387

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

   389   by (induct u, simp_all add: lessThan_Suc)

   390

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

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

   393

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

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

   396   apply (erule ssubst, rule card_lessThan)

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

   398   apply (erule subst)

   399   apply (rule card_image)

   400   apply (simp add: inj_on_def)

   401   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)

   402   apply arith

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

   404   apply arith

   405   done

   406

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

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

   409

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

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

   412

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

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

   415

   416 subsection {* Intervals of integers *}

   417

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

   419   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)

   420

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

   422   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)

   423

   424 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:

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

   426   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)

   427

   428 subsubsection {* Finiteness *}

   429

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

   431     {(0::int)..<u} = int  {..<nat u}"

   432   apply (unfold image_def lessThan_def)

   433   apply auto

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

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

   436   done

   437

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

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

   440   apply (subst image_atLeastZeroLessThan_int, assumption)

   441   apply (rule finite_imageI)

   442   apply auto

   443   done

   444

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

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

   447   apply (erule subst)

   448   apply (rule finite_imageI)

   449   apply (rule finite_atLeastZeroLessThan_int)

   450   apply (rule image_add_int_atLeastLessThan)

   451   done

   452

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

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

   455

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

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

   458

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

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

   461

   462 subsubsection {* Cardinality *}

   463

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

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

   466   apply (subst image_atLeastZeroLessThan_int, assumption)

   467   apply (subst card_image)

   468   apply (auto simp add: inj_on_def)

   469   done

   470

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

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

   473   apply (erule ssubst, rule card_atLeastZeroLessThan_int)

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

   475   apply (erule subst)

   476   apply (rule card_image)

   477   apply (simp add: inj_on_def)

   478   apply (rule image_add_int_atLeastLessThan)

   479   done

   480

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

   482   apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])

   483   apply (auto simp add: compare_rls)

   484   done

   485

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

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

   488

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

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

   491

   492

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

   494

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

   496

   497 subsubsection {* Disjoint Unions *}

   498

   499 text {* Singletons and open intervals *}

   500

   501 lemma ivl_disj_un_singleton:

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

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

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

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

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

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

   508 by auto

   509

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

   511

   512 lemma ivl_disj_un_one:

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

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

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

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

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

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

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

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

   521 by auto

   522

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

   524

   525 lemma ivl_disj_un_two:

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

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

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

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

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

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

   534 by auto

   535

   536 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two

   537

   538 subsubsection {* Disjoint Intersections *}

   539

   540 text {* Singletons and open intervals *}

   541

   542 lemma ivl_disj_int_singleton:

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

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

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

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

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

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

   549   by simp+

   550

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

   552

   553 lemma ivl_disj_int_one:

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

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

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

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

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

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

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

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

   562   by auto

   563

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

   565

   566 lemma ivl_disj_int_two:

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

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

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

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

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

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

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

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

   575   by auto

   576

   577 lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two

   578

   579 subsubsection {* Some Differences *}

   580

   581 lemma ivl_diff[simp]:

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

   583 by(auto)

   584

   585

   586 subsubsection {* Some Subset Conditions *}

   587

   588 lemma ivl_subset[simp]:

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

   590 apply(auto simp:linorder_not_le)

   591 apply(rule ccontr)

   592 apply(insert linorder_le_less_linear[of i n])

   593 apply(clarsimp simp:linorder_not_le)

   594 apply(fastsimp)

   595 done

   596

   597

   598 subsection {* Summation indexed over intervals *}

   599

   600 syntax

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

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

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

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

   605 syntax (xsymbols)

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

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

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

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

   610 syntax (HTML output)

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

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

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

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

   615 syntax (latex_sum output)

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

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

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

   619  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

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

   621  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

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

   623  ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

   624

   625 translations

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

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

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

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

   630

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

   632 summation over intervals:

   633 \begin{center}

   634 \begin{tabular}{lll}

   635 Old & New & \LaTeX\\

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

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

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

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

   640 \end{tabular}

   641 \end{center}

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

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

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

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

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

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

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

   649

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

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

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

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

   654

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

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

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

   658 the context. *}

   659

   660 lemma setsum_ivl_cong:

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

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

   663 by(rule setsum_cong, simp_all)

   664

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

   666 on intervals are not? *)

   667

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

   669 by (simp add:atMost_Suc add_ac)

   670

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

   672 by (simp add:lessThan_Suc add_ac)

   673

   674 lemma setsum_cl_ivl_Suc[simp]:

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

   676 by (auto simp:add_ac atLeastAtMostSuc_conv)

   677

   678 lemma setsum_op_ivl_Suc[simp]:

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

   680 by (auto simp:add_ac atLeastLessThanSuc)

   681 (*

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

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

   684 by (auto simp:add_ac atLeastAtMostSuc_conv)

   685 *)

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

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

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

   689

   690 lemma setsum_diff_nat_ivl:

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

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

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

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

   695 apply (simp add: add_ac)

   696 done

   697

   698 subsection{* Shifting bounds *}

   699

   700 lemma setsum_shift_bounds_nat_ivl:

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

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

   703

   704 lemma setsum_shift_bounds_cl_nat_ivl:

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

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

   707 apply (simp add:image_add_atLeastAtMost o_def)

   708 done

   709

   710 corollary setsum_shift_bounds_cl_Suc_ivl:

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

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

   713

   714 corollary setsum_shift_bounds_Suc_ivl:

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

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

   717

   718 subsection {* The formula for geometric sums *}

   719

   720 lemma geometric_sum:

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

   722   (x ^ n - 1) / (x - 1::'a::{field, recpower, division_by_zero})"

   723   apply (induct "n", auto)

   724   apply (rule_tac c = "x - 1" in field_mult_cancel_right_lemma)

   725   apply (auto simp add: mult_assoc left_distrib)

   726   apply (simp add: right_distrib diff_minus mult_commute power_Suc)

   727   done

   728

   729

   730

   731 ML

   732 {*

   733 val Compl_atLeast = thm "Compl_atLeast";

   734 val Compl_atMost = thm "Compl_atMost";

   735 val Compl_greaterThan = thm "Compl_greaterThan";

   736 val Compl_lessThan = thm "Compl_lessThan";

   737 val INT_greaterThan_UNIV = thm "INT_greaterThan_UNIV";

   738 val UN_atLeast_UNIV = thm "UN_atLeast_UNIV";

   739 val UN_atMost_UNIV = thm "UN_atMost_UNIV";

   740 val UN_lessThan_UNIV = thm "UN_lessThan_UNIV";

   741 val atLeastAtMost_def = thm "atLeastAtMost_def";

   742 val atLeastAtMost_iff = thm "atLeastAtMost_iff";

   743 val atLeastLessThan_def  = thm "atLeastLessThan_def";

   744 val atLeastLessThan_iff = thm "atLeastLessThan_iff";

   745 val atLeast_0 = thm "atLeast_0";

   746 val atLeast_Suc = thm "atLeast_Suc";

   747 val atLeast_def      = thm "atLeast_def";

   748 val atLeast_iff = thm "atLeast_iff";

   749 val atMost_0 = thm "atMost_0";

   750 val atMost_Int_atLeast = thm "atMost_Int_atLeast";

   751 val atMost_Suc = thm "atMost_Suc";

   752 val atMost_def       = thm "atMost_def";

   753 val atMost_iff = thm "atMost_iff";

   754 val greaterThanAtMost_def  = thm "greaterThanAtMost_def";

   755 val greaterThanAtMost_iff = thm "greaterThanAtMost_iff";

   756 val greaterThanLessThan_def  = thm "greaterThanLessThan_def";

   757 val greaterThanLessThan_iff = thm "greaterThanLessThan_iff";

   758 val greaterThan_0 = thm "greaterThan_0";

   759 val greaterThan_Suc = thm "greaterThan_Suc";

   760 val greaterThan_def  = thm "greaterThan_def";

   761 val greaterThan_iff = thm "greaterThan_iff";

   762 val ivl_disj_int = thms "ivl_disj_int";

   763 val ivl_disj_int_one = thms "ivl_disj_int_one";

   764 val ivl_disj_int_singleton = thms "ivl_disj_int_singleton";

   765 val ivl_disj_int_two = thms "ivl_disj_int_two";

   766 val ivl_disj_un = thms "ivl_disj_un";

   767 val ivl_disj_un_one = thms "ivl_disj_un_one";

   768 val ivl_disj_un_singleton = thms "ivl_disj_un_singleton";

   769 val ivl_disj_un_two = thms "ivl_disj_un_two";

   770 val lessThan_0 = thm "lessThan_0";

   771 val lessThan_Suc = thm "lessThan_Suc";

   772 val lessThan_Suc_atMost = thm "lessThan_Suc_atMost";

   773 val lessThan_def     = thm "lessThan_def";

   774 val lessThan_iff = thm "lessThan_iff";

   775 val single_Diff_lessThan = thm "single_Diff_lessThan";

   776

   777 val bounded_nat_set_is_finite = thm "bounded_nat_set_is_finite";

   778 val finite_atMost = thm "finite_atMost";

   779 val finite_lessThan = thm "finite_lessThan";

   780 *}

   781

   782 end
`