src/HOL/SetInterval.thy
 author wenzelm Thu Oct 04 20:29:42 2007 +0200 (2007-10-04) changeset 24850 0cfd722ab579 parent 24748 ee0a0eb6b738 child 24853 aab5798e5a33 permissions -rw-r--r--
Name.uu, Name.aT;
     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 context ord

    17 begin

    18 definition

    19   lessThan    :: "'a => 'a set"	("(1\<^loc>{..<_})") where

    20   "\<^loc>{..<u} == {x. x \<^loc>< u}"

    21

    22 definition

    23   atMost      :: "'a => 'a set"	("(1\<^loc>{.._})") where

    24   "\<^loc>{..u} == {x. x \<^loc>\<le> u}"

    25

    26 definition

    27   greaterThan :: "'a => 'a set"	("(1\<^loc>{_<..})") where

    28   "\<^loc>{l<..} == {x. l\<^loc><x}"

    29

    30 definition

    31   atLeast     :: "'a => 'a set"	("(1\<^loc>{_..})") where

    32   "\<^loc>{l..} == {x. l\<^loc>\<le>x}"

    33

    34 definition

    35   greaterThanLessThan :: "'a => 'a => 'a set"  ("(1\<^loc>{_<..<_})") where

    36   "\<^loc>{l<..<u} == \<^loc>{l<..} Int \<^loc>{..<u}"

    37

    38 definition

    39   atLeastLessThan :: "'a => 'a => 'a set"      ("(1\<^loc>{_..<_})") where

    40   "\<^loc>{l..<u} == \<^loc>{l..} Int \<^loc>{..<u}"

    41

    42 definition

    43   greaterThanAtMost :: "'a => 'a => 'a set"    ("(1\<^loc>{_<.._})") where

    44   "\<^loc>{l<..u} == \<^loc>{l<..} Int \<^loc>{..u}"

    45

    46 definition

    47   atLeastAtMost :: "'a => 'a => 'a set"        ("(1\<^loc>{_.._})") where

    48   "\<^loc>{l..u} == \<^loc>{l..} Int \<^loc>{..u}"

    49

    50 end

    51 (*

    52 constdefs

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

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

    55

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

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

    58

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

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

    61

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

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

    64

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

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

    67

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

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

    70

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

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

    73

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

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

    76 *)

    77

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

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

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

    81

    82 syntax

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

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

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

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

    87

    88 syntax (input)

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

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

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

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

    93

    94 syntax (xsymbols)

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

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

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

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

    99

   100 translations

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

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

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

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

   105

   106

   107 subsection {* Various equivalences *}

   108

   109 lemma (in ord) lessThan_iff [iff]: "(i: lessThan k) = (i\<^loc><k)"

   110 by (simp add: lessThan_def)

   111

   112 lemma Compl_lessThan [simp]:

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

   114 apply (auto simp add: lessThan_def atLeast_def)

   115 done

   116

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

   118 by auto

   119

   120 lemma (in ord) greaterThan_iff [iff]: "(i: greaterThan k) = (k\<^loc><i)"

   121 by (simp add: greaterThan_def)

   122

   123 lemma Compl_greaterThan [simp]:

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

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

   126 done

   127

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

   129 apply (subst Compl_greaterThan [symmetric])

   130 apply (rule double_complement)

   131 done

   132

   133 lemma (in ord) atLeast_iff [iff]: "(i: atLeast k) = (k\<^loc><=i)"

   134 by (simp add: atLeast_def)

   135

   136 lemma Compl_atLeast [simp]:

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

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

   139 done

   140

   141 lemma (in ord) atMost_iff [iff]: "(i: atMost k) = (i\<^loc><=k)"

   142 by (simp add: atMost_def)

   143

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

   145 by (blast intro: order_antisym)

   146

   147

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

   149

   150 lemma atLeast_subset_iff [iff]:

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

   152 by (blast intro: order_trans)

   153

   154 lemma atLeast_eq_iff [iff]:

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

   156 by (blast intro: order_antisym order_trans)

   157

   158 lemma greaterThan_subset_iff [iff]:

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

   160 apply (auto simp add: greaterThan_def)

   161  apply (subst linorder_not_less [symmetric], blast)

   162 done

   163

   164 lemma greaterThan_eq_iff [iff]:

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

   166 apply (rule iffI)

   167  apply (erule equalityE)

   168  apply (simp_all add: greaterThan_subset_iff)

   169 done

   170

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

   172 by (blast intro: order_trans)

   173

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

   175 by (blast intro: order_antisym order_trans)

   176

   177 lemma lessThan_subset_iff [iff]:

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

   179 apply (auto simp add: lessThan_def)

   180  apply (subst linorder_not_less [symmetric], blast)

   181 done

   182

   183 lemma lessThan_eq_iff [iff]:

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

   185 apply (rule iffI)

   186  apply (erule equalityE)

   187  apply (simp_all add: lessThan_subset_iff)

   188 done

   189

   190

   191 subsection {*Two-sided intervals*}

   192

   193 context ord

   194 begin

   195

   196 lemma greaterThanLessThan_iff [simp,noatp]:

   197   "(i : \<^loc>{l<..<u}) = (l \<^loc>< i & i \<^loc>< u)"

   198 by (simp add: greaterThanLessThan_def)

   199

   200 lemma atLeastLessThan_iff [simp,noatp]:

   201   "(i : \<^loc>{l..<u}) = (l \<^loc><= i & i \<^loc>< u)"

   202 by (simp add: atLeastLessThan_def)

   203

   204 lemma greaterThanAtMost_iff [simp,noatp]:

   205   "(i : \<^loc>{l<..u}) = (l \<^loc>< i & i \<^loc><= u)"

   206 by (simp add: greaterThanAtMost_def)

   207

   208 lemma atLeastAtMost_iff [simp,noatp]:

   209   "(i : \<^loc>{l..u}) = (l \<^loc><= i & i \<^loc><= u)"

   210 by (simp add: atLeastAtMost_def)

   211

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

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

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

   215 end

   216

   217 subsubsection{* Emptyness and singletons *}

   218

   219 context order

   220 begin

   221

   222 lemma atLeastAtMost_empty [simp]: "n \<^loc>< m ==> \<^loc>{m..n} = {}";

   223 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)

   224

   225 lemma atLeastLessThan_empty[simp]: "n \<^loc>\<le> m ==> \<^loc>{m..<n} = {}"

   226 by (auto simp add: atLeastLessThan_def)

   227

   228 lemma greaterThanAtMost_empty[simp]:"l \<^loc>\<le> k ==> \<^loc>{k<..l} = {}"

   229 by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)

   230

   231 lemma greaterThanLessThan_empty[simp]:"l \<^loc>\<le> k ==> \<^loc>{k<..l} = {}"

   232 by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)

   233

   234 lemma atLeastAtMost_singleton [simp]: "\<^loc>{a..a} = {a}"

   235 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)

   236

   237 end

   238

   239 subsection {* Intervals of natural numbers *}

   240

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

   242

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

   244 by (simp add: lessThan_def)

   245

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

   247 by (simp add: lessThan_def less_Suc_eq, blast)

   248

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

   250 by (simp add: lessThan_def atMost_def less_Suc_eq_le)

   251

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

   253 by blast

   254

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

   256

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

   258 apply (simp add: greaterThan_def)

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

   260 done

   261

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

   263 apply (simp add: greaterThan_def)

   264 apply (auto elim: linorder_neqE)

   265 done

   266

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

   268 by blast

   269

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

   271

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

   273 by (unfold atLeast_def UNIV_def, simp)

   274

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

   276 apply (simp add: atLeast_def)

   277 apply (simp add: Suc_le_eq)

   278 apply (simp add: order_le_less, blast)

   279 done

   280

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

   282   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)

   283

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

   285 by blast

   286

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

   288

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

   290 by (simp add: atMost_def)

   291

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

   293 apply (simp add: atMost_def)

   294 apply (simp add: less_Suc_eq order_le_less, blast)

   295 done

   296

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

   298 by blast

   299

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

   301

   302 text{*The orientation of the following rule is tricky. The lhs is

   303 defined in terms of the rhs.  Hence the chosen orientation makes sense

   304 in this theory --- the reverse orientation complicates proofs (eg

   305 nontermination). But outside, when the definition of the lhs is rarely

   306 used, the opposite orientation seems preferable because it reduces a

   307 specific concept to a more general one. *}

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

   309 by(simp add:lessThan_def atLeastLessThan_def)

   310

   311 declare atLeast0LessThan[symmetric, code unfold]

   312

   313 lemma atLeastLessThan0: "{m..<0::nat} = {}"

   314 by (simp add: atLeastLessThan_def)

   315

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

   317

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

   319 lemma atLeastLessThanSuc:

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

   321 by (auto simp add: atLeastLessThan_def)

   322

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

   324 by (auto simp add: atLeastLessThan_def)

   325 (*

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

   327 by (induct k, simp_all add: atLeastLessThanSuc)

   328

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

   330 by (auto simp add: atLeastLessThan_def)

   331 *)

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

   333   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)

   334

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

   336   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def

   337     greaterThanAtMost_def)

   338

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

   340   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def

   341     greaterThanLessThan_def)

   342

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

   344 by (auto simp add: atLeastAtMost_def)

   345

   346 subsubsection {* Image *}

   347

   348 lemma image_add_atLeastAtMost:

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

   350 proof

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

   352 next

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

   354   proof

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

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

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

   358     ultimately show "n : ?A" by blast

   359   qed

   360 qed

   361

   362 lemma image_add_atLeastLessThan:

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

   364 proof

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

   366 next

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

   368   proof

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

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

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

   372     ultimately show "n : ?A" by blast

   373   qed

   374 qed

   375

   376 corollary image_Suc_atLeastAtMost[simp]:

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

   378 using image_add_atLeastAtMost[where k=1] by simp

   379

   380 corollary image_Suc_atLeastLessThan[simp]:

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

   382 using image_add_atLeastLessThan[where k=1] by simp

   383

   384 lemma image_add_int_atLeastLessThan:

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

   386   apply (auto simp add: image_def)

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

   388   apply auto

   389   done

   390

   391

   392 subsubsection {* Finiteness *}

   393

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

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

   396

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

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

   399

   400 lemma finite_greaterThanLessThan [iff]:

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

   402 by (simp add: greaterThanLessThan_def)

   403

   404 lemma finite_atLeastLessThan [iff]:

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

   406 by (simp add: atLeastLessThan_def)

   407

   408 lemma finite_greaterThanAtMost [iff]:

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

   410 by (simp add: greaterThanAtMost_def)

   411

   412 lemma finite_atLeastAtMost [iff]:

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

   414 by (simp add: atLeastAtMost_def)

   415

   416 lemma bounded_nat_set_is_finite:

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

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

   419   apply (rule finite_subset)

   420    apply (rule_tac [2] finite_lessThan, auto)

   421   done

   422

   423 subsubsection {* Cardinality *}

   424

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

   426   by (induct u, simp_all add: lessThan_Suc)

   427

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

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

   430

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

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

   433   apply (erule ssubst, rule card_lessThan)

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

   435   apply (erule subst)

   436   apply (rule card_image)

   437   apply (simp add: inj_on_def)

   438   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)

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

   440   apply arith

   441   done

   442

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

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

   445

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

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

   448

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

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

   451

   452 subsection {* Intervals of integers *}

   453

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

   455   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)

   456

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

   458   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)

   459

   460 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:

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

   462   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)

   463

   464 subsubsection {* Finiteness *}

   465

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

   467     {(0::int)..<u} = int  {..<nat u}"

   468   apply (unfold image_def lessThan_def)

   469   apply auto

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

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

   472   done

   473

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

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

   476   apply (subst image_atLeastZeroLessThan_int, assumption)

   477   apply (rule finite_imageI)

   478   apply auto

   479   done

   480

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

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

   483   apply (erule subst)

   484   apply (rule finite_imageI)

   485   apply (rule finite_atLeastZeroLessThan_int)

   486   apply (rule image_add_int_atLeastLessThan)

   487   done

   488

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

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

   491

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

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

   494

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

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

   497

   498 subsubsection {* Cardinality *}

   499

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

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

   502   apply (subst image_atLeastZeroLessThan_int, assumption)

   503   apply (subst card_image)

   504   apply (auto simp add: inj_on_def)

   505   done

   506

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

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

   509   apply (erule ssubst, rule card_atLeastZeroLessThan_int)

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

   511   apply (erule subst)

   512   apply (rule card_image)

   513   apply (simp add: inj_on_def)

   514   apply (rule image_add_int_atLeastLessThan)

   515   done

   516

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

   518   apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])

   519   apply (auto simp add: compare_rls)

   520   done

   521

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

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

   524

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

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

   527

   528

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

   530

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

   532

   533 subsubsection {* Disjoint Unions *}

   534

   535 text {* Singletons and open intervals *}

   536

   537 lemma ivl_disj_un_singleton:

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

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

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

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

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

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

   544 by auto

   545

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

   547

   548 lemma ivl_disj_un_one:

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

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

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

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

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

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

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

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

   557 by auto

   558

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

   560

   561 lemma ivl_disj_un_two:

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

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

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

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

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

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

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

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

   570 by auto

   571

   572 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two

   573

   574 subsubsection {* Disjoint Intersections *}

   575

   576 text {* Singletons and open intervals *}

   577

   578 lemma ivl_disj_int_singleton:

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

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

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

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

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

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

   585   by simp+

   586

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

   588

   589 lemma ivl_disj_int_one:

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

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

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

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

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

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

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

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

   598   by auto

   599

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

   601

   602 lemma ivl_disj_int_two:

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

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

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

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

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

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

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

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

   611   by auto

   612

   613 lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two

   614

   615 subsubsection {* Some Differences *}

   616

   617 lemma ivl_diff[simp]:

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

   619 by(auto)

   620

   621

   622 subsubsection {* Some Subset Conditions *}

   623

   624 lemma ivl_subset [simp,noatp]:

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

   626 apply(auto simp:linorder_not_le)

   627 apply(rule ccontr)

   628 apply(insert linorder_le_less_linear[of i n])

   629 apply(clarsimp simp:linorder_not_le)

   630 apply(fastsimp)

   631 done

   632

   633

   634 subsection {* Summation indexed over intervals *}

   635

   636 syntax

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

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

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

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

   641 syntax (xsymbols)

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

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

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

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

   646 syntax (HTML output)

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

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

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

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

   651 syntax (latex_sum output)

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

   653  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

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

   655  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

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

   657  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

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

   659  ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

   660

   661 translations

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

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

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

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

   666

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

   668 summation over intervals:

   669 \begin{center}

   670 \begin{tabular}{lll}

   671 Old & New & \LaTeX\\

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

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

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

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

   676 \end{tabular}

   677 \end{center}

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

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

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

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

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

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

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

   685

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

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

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

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

   690

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

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

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

   694 the context. *}

   695

   696 lemma setsum_ivl_cong:

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

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

   699 by(rule setsum_cong, simp_all)

   700

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

   702 on intervals are not? *)

   703

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

   705 by (simp add:atMost_Suc add_ac)

   706

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

   708 by (simp add:lessThan_Suc add_ac)

   709

   710 lemma setsum_cl_ivl_Suc[simp]:

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

   712 by (auto simp:add_ac atLeastAtMostSuc_conv)

   713

   714 lemma setsum_op_ivl_Suc[simp]:

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

   716 by (auto simp:add_ac atLeastLessThanSuc)

   717 (*

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

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

   720 by (auto simp:add_ac atLeastAtMostSuc_conv)

   721 *)

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

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

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

   725

   726 lemma setsum_diff_nat_ivl:

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

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

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

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

   731 apply (simp add: add_ac)

   732 done

   733

   734 subsection{* Shifting bounds *}

   735

   736 lemma setsum_shift_bounds_nat_ivl:

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

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

   739

   740 lemma setsum_shift_bounds_cl_nat_ivl:

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

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

   743 apply (simp add:image_add_atLeastAtMost o_def)

   744 done

   745

   746 corollary setsum_shift_bounds_cl_Suc_ivl:

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

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

   749

   750 corollary setsum_shift_bounds_Suc_ivl:

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

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

   753

   754 lemma setsum_head:

   755   fixes n :: nat

   756   assumes mn: "m <= n"

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

   758 proof -

   759   from mn

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

   761     by (auto intro: ivl_disj_un_singleton)

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

   763     by (simp add: atLeast0LessThan)

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

   765   finally show ?thesis .

   766 qed

   767

   768 lemma setsum_head_upt:

   769   fixes m::nat

   770   assumes m: "0 < m"

   771   shows "(\<Sum>x<m. P x) = P 0 + (\<Sum>x\<in>{1..<m}. P x)"

   772 proof -

   773   have "(\<Sum>x<m. P x) = (\<Sum>x\<in>{0..<m}. P x)"

   774     by (simp add: atLeast0LessThan)

   775   also

   776   from m

   777   have "\<dots> = (\<Sum>x\<in>{0..m - 1}. P x)"

   778     by (cases m) (auto simp add: atLeastLessThanSuc_atLeastAtMost)

   779   also

   780   have "\<dots> = P 0 + (\<Sum>x\<in>{0<..m - 1}. P x)"

   781     by (simp add: setsum_head)

   782   also

   783   from m

   784   have "{0<..m - 1} = {1..<m}"

   785     by (cases m) (auto simp add: atLeastLessThanSuc_atLeastAtMost)

   786   finally show ?thesis .

   787 qed

   788

   789 subsection {* The formula for geometric sums *}

   790

   791 lemma geometric_sum:

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

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

   794 by (induct "n") (simp_all add:field_simps power_Suc)

   795

   796 subsection {* The formula for arithmetic sums *}

   797

   798 lemma gauss_sum:

   799   "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =

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

   801 proof (induct n)

   802   case 0

   803   show ?case by simp

   804 next

   805   case (Suc n)

   806   then show ?case by (simp add: ring_simps)

   807 qed

   808

   809 theorem arith_series_general:

   810   "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

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

   812 proof cases

   813   assume ngt1: "n > 1"

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

   815   have

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

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

   818     by (rule setsum_addf)

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

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

   821     by (simp add: setsum_right_distrib setsum_head_upt mult_ac)

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

   823     by (simp add: left_distrib right_distrib)

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

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

   826   also from ngt1

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

   828     by (simp only: mult_ac gauss_sum [of "n - 1"])

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

   830   finally show ?thesis by (simp add: mult_ac add_ac right_distrib)

   831 next

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

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

   834   thus ?thesis by (auto simp: mult_ac right_distrib)

   835 qed

   836

   837 lemma arith_series_nat:

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

   839 proof -

   840   have

   841     "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =

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

   843     by (rule arith_series_general)

   844   thus ?thesis by (auto simp add: of_nat_id)

   845 qed

   846

   847 lemma arith_series_int:

   848   "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

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

   850 proof -

   851   have

   852     "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

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

   854     by (rule arith_series_general)

   855   thus ?thesis by simp

   856 qed

   857

   858 lemma sum_diff_distrib:

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

   860   shows

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

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

   863 proof (induct n)

   864   case 0 show ?case by simp

   865 next

   866   case (Suc n)

   867

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

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

   870

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

   872   moreover

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

   874   moreover

   875   from Suc have

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

   877     by (subst diff_diff_left[symmetric],

   878         subst diff_add_assoc2)

   879        (auto simp: diff_add_assoc2 intro: setsum_mono)

   880   ultimately

   881   show ?case by simp

   882 qed

   883

   884

   885 ML

   886 {*

   887 val Compl_atLeast = thm "Compl_atLeast";

   888 val Compl_atMost = thm "Compl_atMost";

   889 val Compl_greaterThan = thm "Compl_greaterThan";

   890 val Compl_lessThan = thm "Compl_lessThan";

   891 val INT_greaterThan_UNIV = thm "INT_greaterThan_UNIV";

   892 val UN_atLeast_UNIV = thm "UN_atLeast_UNIV";

   893 val UN_atMost_UNIV = thm "UN_atMost_UNIV";

   894 val UN_lessThan_UNIV = thm "UN_lessThan_UNIV";

   895 val atLeastAtMost_def = thm "atLeastAtMost_def";

   896 val atLeastAtMost_iff = thm "atLeastAtMost_iff";

   897 val atLeastLessThan_def  = thm "atLeastLessThan_def";

   898 val atLeastLessThan_iff = thm "atLeastLessThan_iff";

   899 val atLeast_0 = thm "atLeast_0";

   900 val atLeast_Suc = thm "atLeast_Suc";

   901 val atLeast_def      = thm "atLeast_def";

   902 val atLeast_iff = thm "atLeast_iff";

   903 val atMost_0 = thm "atMost_0";

   904 val atMost_Int_atLeast = thm "atMost_Int_atLeast";

   905 val atMost_Suc = thm "atMost_Suc";

   906 val atMost_def       = thm "atMost_def";

   907 val atMost_iff = thm "atMost_iff";

   908 val greaterThanAtMost_def  = thm "greaterThanAtMost_def";

   909 val greaterThanAtMost_iff = thm "greaterThanAtMost_iff";

   910 val greaterThanLessThan_def  = thm "greaterThanLessThan_def";

   911 val greaterThanLessThan_iff = thm "greaterThanLessThan_iff";

   912 val greaterThan_0 = thm "greaterThan_0";

   913 val greaterThan_Suc = thm "greaterThan_Suc";

   914 val greaterThan_def  = thm "greaterThan_def";

   915 val greaterThan_iff = thm "greaterThan_iff";

   916 val ivl_disj_int = thms "ivl_disj_int";

   917 val ivl_disj_int_one = thms "ivl_disj_int_one";

   918 val ivl_disj_int_singleton = thms "ivl_disj_int_singleton";

   919 val ivl_disj_int_two = thms "ivl_disj_int_two";

   920 val ivl_disj_un = thms "ivl_disj_un";

   921 val ivl_disj_un_one = thms "ivl_disj_un_one";

   922 val ivl_disj_un_singleton = thms "ivl_disj_un_singleton";

   923 val ivl_disj_un_two = thms "ivl_disj_un_two";

   924 val lessThan_0 = thm "lessThan_0";

   925 val lessThan_Suc = thm "lessThan_Suc";

   926 val lessThan_Suc_atMost = thm "lessThan_Suc_atMost";

   927 val lessThan_def     = thm "lessThan_def";

   928 val lessThan_iff = thm "lessThan_iff";

   929 val single_Diff_lessThan = thm "single_Diff_lessThan";

   930

   931 val bounded_nat_set_is_finite = thm "bounded_nat_set_is_finite";

   932 val finite_atMost = thm "finite_atMost";

   933 val finite_lessThan = thm "finite_lessThan";

   934 *}

   935

   936 end
`