src/HOL/SetInterval.thy
 author wenzelm Fri Mar 28 19:43:54 2008 +0100 (2008-03-28) changeset 26462 dac4e2bce00d parent 26105 ae06618225ec child 27656 d4f6e64ee7cc permissions -rw-r--r--
avoid rebinding of existing facts;
     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 Int

    14 begin

    15

    16 context ord

    17 begin

    18 definition

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

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

    21

    22 definition

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

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

    25

    26 definition

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

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

    29

    30 definition

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

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

    33

    34 definition

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

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

    37

    38 definition

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

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

    41

    42 definition

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

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

    45

    46 definition

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

    48   "{l..u} == {l..} Int {..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<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<i)"

   121 by (simp add: greaterThan_def)

   122

   123 lemma Compl_greaterThan [simp]:

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

   125   by (auto simp add: greaterThan_def atMost_def)

   126

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

   128 apply (subst Compl_greaterThan [symmetric])

   129 apply (rule double_complement)

   130 done

   131

   132 lemma (in ord) atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"

   133 by (simp add: atLeast_def)

   134

   135 lemma Compl_atLeast [simp]:

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

   137   by (auto simp add: lessThan_def atLeast_def)

   138

   139 lemma (in ord) atMost_iff [iff]: "(i: atMost k) = (i<=k)"

   140 by (simp add: atMost_def)

   141

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

   143 by (blast intro: order_antisym)

   144

   145

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

   147

   148 lemma atLeast_subset_iff [iff]:

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

   150 by (blast intro: order_trans)

   151

   152 lemma atLeast_eq_iff [iff]:

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

   154 by (blast intro: order_antisym order_trans)

   155

   156 lemma greaterThan_subset_iff [iff]:

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

   158 apply (auto simp add: greaterThan_def)

   159  apply (subst linorder_not_less [symmetric], blast)

   160 done

   161

   162 lemma greaterThan_eq_iff [iff]:

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

   164 apply (rule iffI)

   165  apply (erule equalityE)

   166  apply (simp_all add: greaterThan_subset_iff)

   167 done

   168

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

   170 by (blast intro: order_trans)

   171

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

   173 by (blast intro: order_antisym order_trans)

   174

   175 lemma lessThan_subset_iff [iff]:

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

   177 apply (auto simp add: lessThan_def)

   178  apply (subst linorder_not_less [symmetric], blast)

   179 done

   180

   181 lemma lessThan_eq_iff [iff]:

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

   183 apply (rule iffI)

   184  apply (erule equalityE)

   185  apply (simp_all add: lessThan_subset_iff)

   186 done

   187

   188

   189 subsection {*Two-sided intervals*}

   190

   191 context ord

   192 begin

   193

   194 lemma greaterThanLessThan_iff [simp,noatp]:

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

   196 by (simp add: greaterThanLessThan_def)

   197

   198 lemma atLeastLessThan_iff [simp,noatp]:

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

   200 by (simp add: atLeastLessThan_def)

   201

   202 lemma greaterThanAtMost_iff [simp,noatp]:

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

   204 by (simp add: greaterThanAtMost_def)

   205

   206 lemma atLeastAtMost_iff [simp,noatp]:

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

   208 by (simp add: atLeastAtMost_def)

   209

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

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

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

   213 end

   214

   215 subsubsection{* Emptyness and singletons *}

   216

   217 context order

   218 begin

   219

   220 lemma atLeastAtMost_empty [simp]: "n < m ==> {m..n} = {}";

   221 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)

   222

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

   224 by (auto simp add: atLeastLessThan_def)

   225

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

   227 by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)

   228

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

   230 by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)

   231

   232 lemma atLeastAtMost_singleton [simp]: "{a..a} = {a}"

   233 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)

   234

   235 end

   236

   237 subsection {* Intervals of natural numbers *}

   238

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

   240

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

   242 by (simp add: lessThan_def)

   243

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

   245 by (simp add: lessThan_def less_Suc_eq, blast)

   246

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

   248 by (simp add: lessThan_def atMost_def less_Suc_eq_le)

   249

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

   251 by blast

   252

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

   254

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

   256 apply (simp add: greaterThan_def)

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

   258 done

   259

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

   261 apply (simp add: greaterThan_def)

   262 apply (auto elim: linorder_neqE)

   263 done

   264

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

   266 by blast

   267

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

   269

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

   271 by (unfold atLeast_def UNIV_def, simp)

   272

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

   274 apply (simp add: atLeast_def)

   275 apply (simp add: Suc_le_eq)

   276 apply (simp add: order_le_less, blast)

   277 done

   278

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

   280   by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)

   281

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

   283 by blast

   284

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

   286

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

   288 by (simp add: atMost_def)

   289

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

   291 apply (simp add: atMost_def)

   292 apply (simp add: less_Suc_eq order_le_less, blast)

   293 done

   294

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

   296 by blast

   297

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

   299

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

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

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

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

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

   305 specific concept to a more general one. *}

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

   307 by(simp add:lessThan_def atLeastLessThan_def)

   308

   309 declare atLeast0LessThan[symmetric, code unfold]

   310

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

   312 by (simp add: atLeastLessThan_def)

   313

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

   315

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

   317 lemma atLeastLessThanSuc:

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

   319 by (auto simp add: atLeastLessThan_def)

   320

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

   322 by (auto simp add: atLeastLessThan_def)

   323 (*

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

   325 by (induct k, simp_all add: atLeastLessThanSuc)

   326

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

   328 by (auto simp add: atLeastLessThan_def)

   329 *)

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

   331   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)

   332

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

   334   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def

   335     greaterThanAtMost_def)

   336

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

   338   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def

   339     greaterThanLessThan_def)

   340

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

   342 by (auto simp add: atLeastAtMost_def)

   343

   344 subsubsection {* Image *}

   345

   346 lemma image_add_atLeastAtMost:

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

   348 proof

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

   350 next

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

   352   proof

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

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

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

   356     ultimately show "n : ?A" by blast

   357   qed

   358 qed

   359

   360 lemma image_add_atLeastLessThan:

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

   362 proof

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

   364 next

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

   366   proof

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

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

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

   370     ultimately show "n : ?A" by blast

   371   qed

   372 qed

   373

   374 corollary image_Suc_atLeastAtMost[simp]:

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

   376 using image_add_atLeastAtMost[where k=1] by simp

   377

   378 corollary image_Suc_atLeastLessThan[simp]:

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

   380 using image_add_atLeastLessThan[where k=1] by simp

   381

   382 lemma image_add_int_atLeastLessThan:

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

   384   apply (auto simp add: image_def)

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

   386   apply auto

   387   done

   388

   389

   390 subsubsection {* Finiteness *}

   391

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

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

   394

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

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

   397

   398 lemma finite_greaterThanLessThan [iff]:

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

   400 by (simp add: greaterThanLessThan_def)

   401

   402 lemma finite_atLeastLessThan [iff]:

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

   404 by (simp add: atLeastLessThan_def)

   405

   406 lemma finite_greaterThanAtMost [iff]:

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

   408 by (simp add: greaterThanAtMost_def)

   409

   410 lemma finite_atLeastAtMost [iff]:

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

   412 by (simp add: atLeastAtMost_def)

   413

   414 lemma bounded_nat_set_is_finite:

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

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

   417   apply (rule finite_subset)

   418    apply (rule_tac [2] finite_lessThan, auto)

   419   done

   420

   421 text{* Any subset of an interval of natural numbers the size of the

   422 subset is exactly that interval. *}

   423

   424 lemma subset_card_intvl_is_intvl:

   425   "A <= {k..<k+card A} \<Longrightarrow> A = {k..<k+card A}" (is "PROP ?P")

   426 proof cases

   427   assume "finite A"

   428   thus "PROP ?P"

   429   proof(induct A rule:finite_linorder_induct)

   430     case empty thus ?case by auto

   431   next

   432     case (insert A b)

   433     moreover hence "b ~: A" by auto

   434     moreover have "A <= {k..<k+card A}" and "b = k+card A"

   435       using b ~: A insert by fastsimp+

   436     ultimately show ?case by auto

   437   qed

   438 next

   439   assume "~finite A" thus "PROP ?P" by simp

   440 qed

   441

   442

   443 subsubsection {* Cardinality *}

   444

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

   446   by (induct u, simp_all add: lessThan_Suc)

   447

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

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

   450

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

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

   453   apply (erule ssubst, rule card_lessThan)

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

   455   apply (erule subst)

   456   apply (rule card_image)

   457   apply (simp add: inj_on_def)

   458   apply (auto simp add: image_def atLeastLessThan_def lessThan_def)

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

   460   apply arith

   461   done

   462

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

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

   465

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

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

   468

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

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

   471

   472

   473 lemma ex_bij_betw_nat_finite:

   474   "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"

   475 apply(drule finite_imp_nat_seg_image_inj_on)

   476 apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)

   477 done

   478

   479 lemma ex_bij_betw_finite_nat:

   480   "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"

   481 by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)

   482

   483

   484 subsection {* Intervals of integers *}

   485

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

   487   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)

   488

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

   490   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)

   491

   492 lemma atLeastPlusOneLessThan_greaterThanLessThan_int:

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

   494   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)

   495

   496 subsubsection {* Finiteness *}

   497

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

   499     {(0::int)..<u} = int  {..<nat u}"

   500   apply (unfold image_def lessThan_def)

   501   apply auto

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

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

   504   done

   505

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

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

   508   apply (subst image_atLeastZeroLessThan_int, assumption)

   509   apply (rule finite_imageI)

   510   apply auto

   511   done

   512

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

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

   515   apply (erule subst)

   516   apply (rule finite_imageI)

   517   apply (rule finite_atLeastZeroLessThan_int)

   518   apply (rule image_add_int_atLeastLessThan)

   519   done

   520

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

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

   523

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

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

   526

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

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

   529

   530

   531 subsubsection {* Cardinality *}

   532

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

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

   535   apply (subst image_atLeastZeroLessThan_int, assumption)

   536   apply (subst card_image)

   537   apply (auto simp add: inj_on_def)

   538   done

   539

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

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

   542   apply (erule ssubst, rule card_atLeastZeroLessThan_int)

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

   544   apply (erule subst)

   545   apply (rule card_image)

   546   apply (simp add: inj_on_def)

   547   apply (rule image_add_int_atLeastLessThan)

   548   done

   549

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

   551   apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])

   552   apply (auto simp add: compare_rls)

   553   done

   554

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

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

   557

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

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

   560

   561

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

   563

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

   565

   566 subsubsection {* Disjoint Unions *}

   567

   568 text {* Singletons and open intervals *}

   569

   570 lemma ivl_disj_un_singleton:

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

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

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

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

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

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

   577 by auto

   578

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

   580

   581 lemma ivl_disj_un_one:

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

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

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

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

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

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

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

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

   590 by auto

   591

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

   593

   594 lemma ivl_disj_un_two:

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

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

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

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

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

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

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

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

   603 by auto

   604

   605 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two

   606

   607 subsubsection {* Disjoint Intersections *}

   608

   609 text {* Singletons and open intervals *}

   610

   611 lemma ivl_disj_int_singleton:

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

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

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

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

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

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

   618   by simp+

   619

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

   621

   622 lemma ivl_disj_int_one:

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

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

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

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

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

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

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

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

   631   by auto

   632

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

   634

   635 lemma ivl_disj_int_two:

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

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

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

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

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

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

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

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

   644   by auto

   645

   646 lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two

   647

   648 subsubsection {* Some Differences *}

   649

   650 lemma ivl_diff[simp]:

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

   652 by(auto)

   653

   654

   655 subsubsection {* Some Subset Conditions *}

   656

   657 lemma ivl_subset [simp,noatp]:

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

   659 apply(auto simp:linorder_not_le)

   660 apply(rule ccontr)

   661 apply(insert linorder_le_less_linear[of i n])

   662 apply(clarsimp simp:linorder_not_le)

   663 apply(fastsimp)

   664 done

   665

   666

   667 subsection {* Summation indexed over intervals *}

   668

   669 syntax

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

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

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

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

   674 syntax (xsymbols)

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

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

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

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

   679 syntax (HTML output)

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

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

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

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

   684 syntax (latex_sum output)

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

   686  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)

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

   688  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)

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

   690  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)

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

   692  ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)

   693

   694 translations

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

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

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

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

   699

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

   701 summation over intervals:

   702 \begin{center}

   703 \begin{tabular}{lll}

   704 Old & New & \LaTeX\\

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

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

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

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

   709 \end{tabular}

   710 \end{center}

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

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

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

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

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

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

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

   718

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

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

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

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

   723

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

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

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

   727 the context. *}

   728

   729 lemma setsum_ivl_cong:

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

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

   732 by(rule setsum_cong, simp_all)

   733

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

   735 on intervals are not? *)

   736

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

   738 by (simp add:atMost_Suc add_ac)

   739

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

   741 by (simp add:lessThan_Suc add_ac)

   742

   743 lemma setsum_cl_ivl_Suc[simp]:

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

   745 by (auto simp:add_ac atLeastAtMostSuc_conv)

   746

   747 lemma setsum_op_ivl_Suc[simp]:

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

   749 by (auto simp:add_ac atLeastLessThanSuc)

   750 (*

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

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

   753 by (auto simp:add_ac atLeastAtMostSuc_conv)

   754 *)

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

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

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

   758

   759 lemma setsum_diff_nat_ivl:

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

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

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

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

   764 apply (simp add: add_ac)

   765 done

   766

   767 subsection{* Shifting bounds *}

   768

   769 lemma setsum_shift_bounds_nat_ivl:

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

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

   772

   773 lemma setsum_shift_bounds_cl_nat_ivl:

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

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

   776 apply (simp add:image_add_atLeastAtMost o_def)

   777 done

   778

   779 corollary setsum_shift_bounds_cl_Suc_ivl:

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

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

   782

   783 corollary setsum_shift_bounds_Suc_ivl:

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

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

   786

   787 lemma setsum_head:

   788   fixes n :: nat

   789   assumes mn: "m <= n"

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

   791 proof -

   792   from mn

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

   794     by (auto intro: ivl_disj_un_singleton)

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

   796     by (simp add: atLeast0LessThan)

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

   798   finally show ?thesis .

   799 qed

   800

   801 lemma setsum_head_upt:

   802   fixes m::nat

   803   assumes m: "0 < m"

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

   805 proof -

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

   807     by (simp add: atLeast0LessThan)

   808   also

   809   from m

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

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

   812   also

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

   814     by (simp add: setsum_head)

   815   also

   816   from m

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

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

   819   finally show ?thesis .

   820 qed

   821

   822 subsection {* The formula for geometric sums *}

   823

   824 lemma geometric_sum:

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

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

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

   828

   829 subsection {* The formula for arithmetic sums *}

   830

   831 lemma gauss_sum:

   832   "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =

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

   834 proof (induct n)

   835   case 0

   836   show ?case by simp

   837 next

   838   case (Suc n)

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

   840 qed

   841

   842 theorem arith_series_general:

   843   "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

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

   845 proof cases

   846   assume ngt1: "n > 1"

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

   848   have

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

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

   851     by (rule setsum_addf)

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

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

   854     by (simp add: setsum_right_distrib setsum_head_upt mult_ac)

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

   856     by (simp add: left_distrib right_distrib)

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

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

   859   also from ngt1

   860   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)"

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

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

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

   864 next

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

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

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

   868 qed

   869

   870 lemma arith_series_nat:

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

   872 proof -

   873   have

   874     "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =

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

   876     by (rule arith_series_general)

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

   878 qed

   879

   880 lemma arith_series_int:

   881   "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

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

   883 proof -

   884   have

   885     "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =

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

   887     by (rule arith_series_general)

   888   thus ?thesis by simp

   889 qed

   890

   891 lemma sum_diff_distrib:

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

   893   shows

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

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

   896 proof (induct n)

   897   case 0 show ?case by simp

   898 next

   899   case (Suc n)

   900

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

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

   903

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

   905   moreover

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

   907   moreover

   908   from Suc have

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

   910     by (subst diff_diff_left[symmetric],

   911         subst diff_add_assoc2)

   912        (auto simp: diff_add_assoc2 intro: setsum_mono)

   913   ultimately

   914   show ?case by simp

   915 qed

   916

   917 end
`