src/HOL/SetInterval.thy
 author nipkow Mon May 23 19:39:45 2005 +0200 (2005-05-23) changeset 16052 880b0e786c1b parent 16041 5a8736668ced child 16102 c5f6726d9bb1 permissions -rw-r--r--
tuned setsum rewrites
     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 {* Finiteness *}

   304

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

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

   307

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

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

   310

   311 lemma finite_greaterThanLessThan [iff]:

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

   313 by (simp add: greaterThanLessThan_def)

   314

   315 lemma finite_atLeastLessThan [iff]:

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

   317 by (simp add: atLeastLessThan_def)

   318

   319 lemma finite_greaterThanAtMost [iff]:

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

   321 by (simp add: greaterThanAtMost_def)

   322

   323 lemma finite_atLeastAtMost [iff]:

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

   325 by (simp add: atLeastAtMost_def)

   326

   327 lemma bounded_nat_set_is_finite:

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

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

   330   apply (rule finite_subset)

   331    apply (rule_tac [2] finite_lessThan, auto)

   332   done

   333

   334 subsubsection {* Cardinality *}

   335

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

   337   by (induct u, simp_all add: lessThan_Suc)

   338

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

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

   341

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

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

   344   apply (erule ssubst, rule card_lessThan)

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

   346   apply (erule subst)

   347   apply (rule card_image)

   348   apply (simp add: inj_on_def)

   349   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)

   350   apply arith

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

   352   apply arith

   353   done

   354

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

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

   357

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

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

   360

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

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

   363

   364 subsection {* Intervals of integers *}

   365

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

   367   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)

   368

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

   370   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)

   371

   372 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:

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

   374   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)

   375

   376 subsubsection {* Finiteness *}

   377

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

   379     {(0::int)..<u} = int  {..<nat u}"

   380   apply (unfold image_def lessThan_def)

   381   apply auto

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

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

   384   done

   385

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

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

   388   apply (subst image_atLeastZeroLessThan_int, assumption)

   389   apply (rule finite_imageI)

   390   apply auto

   391   done

   392

   393 lemma image_atLeastLessThan_int_shift:

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

   395   apply (auto simp add: image_def)

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

   397   apply auto

   398   done

   399

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

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

   402   apply (erule subst)

   403   apply (rule finite_imageI)

   404   apply (rule finite_atLeastZeroLessThan_int)

   405   apply (rule image_atLeastLessThan_int_shift)

   406   done

   407

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

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

   410

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

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

   413

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

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

   416

   417 subsubsection {* Cardinality *}

   418

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

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

   421   apply (subst image_atLeastZeroLessThan_int, assumption)

   422   apply (subst card_image)

   423   apply (auto simp add: inj_on_def)

   424   done

   425

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

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

   428   apply (erule ssubst, rule card_atLeastZeroLessThan_int)

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

   430   apply (erule subst)

   431   apply (rule card_image)

   432   apply (simp add: inj_on_def)

   433   apply (rule image_atLeastLessThan_int_shift)

   434   done

   435

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

   437   apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])

   438   apply (auto simp add: compare_rls)

   439   done

   440

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

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

   443

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

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

   446

   447

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

   449

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

   451

   452 subsubsection {* Disjoint Unions *}

   453

   454 text {* Singletons and open intervals *}

   455

   456 lemma ivl_disj_un_singleton:

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

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

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

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

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

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

   463 by auto

   464

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

   466

   467 lemma ivl_disj_un_one:

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

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

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

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

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

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

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

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

   476 by auto

   477

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

   479

   480 lemma ivl_disj_un_two:

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

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

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

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

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

   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 by auto

   490

   491 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two

   492

   493 subsubsection {* Disjoint Intersections *}

   494

   495 text {* Singletons and open intervals *}

   496

   497 lemma ivl_disj_int_singleton:

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

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

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

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

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

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

   504   by simp+

   505

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

   507

   508 lemma ivl_disj_int_one:

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

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

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

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

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

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

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

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

   517   by auto

   518

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

   520

   521 lemma ivl_disj_int_two:

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

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

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

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

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

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

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

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

   530   by auto

   531

   532 lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two

   533

   534 subsubsection {* Some Differences *}

   535

   536 lemma ivl_diff[simp]:

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

   538 by(auto)

   539

   540

   541 subsubsection {* Some Subset Conditions *}

   542

   543 lemma ivl_subset[simp]:

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

   545 apply(auto simp:linorder_not_le)

   546 apply(rule ccontr)

   547 apply(insert linorder_le_less_linear[of i n])

   548 apply(clarsimp simp:linorder_not_le)

   549 apply(fastsimp)

   550 done

   551

   552

   553 subsection {* Summation indexed over intervals *}

   554

   555 syntax

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

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

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

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

   560 syntax (xsymbols)

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

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

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

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

   565 syntax (HTML output)

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

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

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

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

   570 syntax (latex_sum output)

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

   572  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

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

   574  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

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

   576  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

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

   578  ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

   579

   580 translations

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

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

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

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

   585

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

   587 summation over intervals:

   588 \begin{center}

   589 \begin{tabular}{lll}

   590 Old & New & \LaTeX\\

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

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

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

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

   595 \end{tabular}

   596 \end{center}

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

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

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

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

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

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

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

   604

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

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

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

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

   609

   610 (* FIXME change the simplifier's treatment of congruence rules?? *)

   611

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

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

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

   615 the context. *}

   616

   617 lemma setsum_ivl_cong:

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

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

   620 by(rule setsum_cong, simp_all)

   621

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

   623 on intervals are not? *)

   624

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

   626 by (simp add:atMost_Suc add_ac)

   627

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

   629 by (simp add:lessThan_Suc add_ac)

   630

   631 lemma setsum_cl_ivl_Suc[simp]:

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

   633 by (auto simp:add_ac atLeastAtMostSuc_conv)

   634

   635 lemma setsum_op_ivl_Suc[simp]:

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

   637 by (auto simp:add_ac atLeastLessThanSuc)

   638 (*

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

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

   641 by (auto simp:add_ac atLeastAtMostSuc_conv)

   642 *)

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

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

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

   646

   647 lemma setsum_diff_nat_ivl:

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

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

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

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

   652 apply (simp add: add_ac)

   653 done

   654

   655 lemma setsum_shift_bounds_nat_ivl:

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

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

   658

   659

   660 ML

   661 {*

   662 val Compl_atLeast = thm "Compl_atLeast";

   663 val Compl_atMost = thm "Compl_atMost";

   664 val Compl_greaterThan = thm "Compl_greaterThan";

   665 val Compl_lessThan = thm "Compl_lessThan";

   666 val INT_greaterThan_UNIV = thm "INT_greaterThan_UNIV";

   667 val UN_atLeast_UNIV = thm "UN_atLeast_UNIV";

   668 val UN_atMost_UNIV = thm "UN_atMost_UNIV";

   669 val UN_lessThan_UNIV = thm "UN_lessThan_UNIV";

   670 val atLeastAtMost_def = thm "atLeastAtMost_def";

   671 val atLeastAtMost_iff = thm "atLeastAtMost_iff";

   672 val atLeastLessThan_def  = thm "atLeastLessThan_def";

   673 val atLeastLessThan_iff = thm "atLeastLessThan_iff";

   674 val atLeast_0 = thm "atLeast_0";

   675 val atLeast_Suc = thm "atLeast_Suc";

   676 val atLeast_def      = thm "atLeast_def";

   677 val atLeast_iff = thm "atLeast_iff";

   678 val atMost_0 = thm "atMost_0";

   679 val atMost_Int_atLeast = thm "atMost_Int_atLeast";

   680 val atMost_Suc = thm "atMost_Suc";

   681 val atMost_def       = thm "atMost_def";

   682 val atMost_iff = thm "atMost_iff";

   683 val greaterThanAtMost_def  = thm "greaterThanAtMost_def";

   684 val greaterThanAtMost_iff = thm "greaterThanAtMost_iff";

   685 val greaterThanLessThan_def  = thm "greaterThanLessThan_def";

   686 val greaterThanLessThan_iff = thm "greaterThanLessThan_iff";

   687 val greaterThan_0 = thm "greaterThan_0";

   688 val greaterThan_Suc = thm "greaterThan_Suc";

   689 val greaterThan_def  = thm "greaterThan_def";

   690 val greaterThan_iff = thm "greaterThan_iff";

   691 val ivl_disj_int = thms "ivl_disj_int";

   692 val ivl_disj_int_one = thms "ivl_disj_int_one";

   693 val ivl_disj_int_singleton = thms "ivl_disj_int_singleton";

   694 val ivl_disj_int_two = thms "ivl_disj_int_two";

   695 val ivl_disj_un = thms "ivl_disj_un";

   696 val ivl_disj_un_one = thms "ivl_disj_un_one";

   697 val ivl_disj_un_singleton = thms "ivl_disj_un_singleton";

   698 val ivl_disj_un_two = thms "ivl_disj_un_two";

   699 val lessThan_0 = thm "lessThan_0";

   700 val lessThan_Suc = thm "lessThan_Suc";

   701 val lessThan_Suc_atMost = thm "lessThan_Suc_atMost";

   702 val lessThan_def     = thm "lessThan_def";

   703 val lessThan_iff = thm "lessThan_iff";

   704 val single_Diff_lessThan = thm "single_Diff_lessThan";

   705

   706 val bounded_nat_set_is_finite = thm "bounded_nat_set_is_finite";

   707 val finite_atMost = thm "finite_atMost";

   708 val finite_lessThan = thm "finite_lessThan";

   709 *}

   710

   711 end
`